Strong Coupling Gauge Theories in LHC Era: Workshop in Honor of Toshihide Maskawa's 70th Birthday and 35th Anniversary of Dynamical Symmetry Breaking in SCGT - PDF Free Download (2024)

It is convenient to obtain these expressions from generating functions9 TrSk U =

I

I N N Y Y dt 1 dt U = , Tr 1 + teiφa A k k+1 iφ k+1 a 2πit 1 − te 2πit a=1 a=1

(8)

where the contour integral over t projects onto the term in a Taylor expansion of the integrand which contains k eigenvalues. The contour encircles the origin. It can be moved away from the origin if it does not cross singularities of the integrand. In the case of the anti-symmetric representation (8) when N is finite, the integrand is a polynomial and the contour can be moved anywhere. For the symmetric representation (8) it should remain within the unit circle. The covariant expressions for the free energies are gotten by taking logs of (8) I 1 1 1 ln (9) dt k+1 hexp [∓Tr ln(1 ∓ tU )]i , βΓSk /Ak = − N 2πi t (9) have center charge k modulo N . In the large N limit, the quantities in (9) can be computed using two saddle point approximations. The first integrates over unitary matrices in (6). At large N , the eigenvalues of U become classical variables and their distribution is found by minimizing Seff plus a Jacobian from the unitary integral measure. As long as k N 2 , the loop operators in (9) do not modify the eigenvalue distribution in the leading order at large N . It is given by a density ρ(φ). In the large N limit the

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expectation values in Eq. (9) are computed using the eigenvalue density,a I Z π 1 1 1 dφρ(φ) ln(1 ∓ teiφ ) − k ln t . (10) βΓSk /Ak = − ln dt exp ∓N N 2πi t −π The second use of a saddle-point approximation is to evaluate the integral over t in (10). Let tˆ satisfy the saddle-point equation Z π Z π tˆeiφ k k tˆeiφ ˆ ˆ dφρ(φ) , R ( t ) ≡ . (11) = = dφρ(φ) RSk (t) ≡ A k iφ iφ ˆ ˆ N N 1 − t e 1 + t e −π −π The functions RSk /Ak (t) in (11) are related to the resolvent of the matrix model and are holomorphic functions of t with cut singularities on the unit circle determined by the support of ρ(φ). Once the solution tˆ is determined, the free energy is given by Z π k dφρ(φ) ln(1 ∓ tˆeiφ ) + βΓSk /Ak = ± ln tˆ . (12) N −π Before we proceed further, let us consider a simple example, the confined phase. Center symmetry is an invariance under a simultaneous translation of all eigenvalues φa → φa + 2π/N . In the center-symmetric confined phase, the distribution is translation invariant, eigenvalues are uniformly distributed on the unit circle and 1 . We can integrate over φ in the saddle-point equations (11), ρconf = 2π RSk (tˆ) =

k 0 |tˆ| < 1 , RAk (tˆ) = = −1 |tˆ| > 1 N

k 0 |tˆ| < 1 . = 1 |tˆ| > 1 N

(13)

In the case of the symmetric representation, the saddle-point equation (13) has a k solution only when Nk = 0. We interpret the absence of a solution when N 6= 0 as meaning that the expectation value vanishes. Certainly, if there is no saddlepoint of a periodic function of a variable φ, the integration is not dominated by any particular value of φ and φ must be integrated over its entire range. This would average the expectation value of any operator with non-zero center charge to zero. It is in the other case, when there is a saddle point, where the large N limit forces one to evaluate the integrand at the saddle point and the expectation value is generically non-zero. Similarly, for the anti-symmetric representation, the saddlek point equation (13) has a solution only when either Nk = 0 or N = 1, the two cases where the antisymmetric representation is center neutral. This is also interpreted as confinement, the expectation value vanishes in all other cases. Note that it has an expected k → N − k duality, though it comes from interchanging two saddle points, one with |tˆ| < 1 and one with |tˆ| > 1. Neither of these saddle-points alone exhibit this duality. a We

will argue that using the leading order N 0 density ρ(φ) in (10) is sufficient to obtain βΓSk /Ak to leading order N 0 accuracy.

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5. Giant loop phase transition In the following, we will focus on the symmetric representation Sk . Consider, for 1 (1 + 2p cos φ) which solves the large example, the eigenvalue distribution ρ(φ) = 2π N limit of the matrix model (2) precisely at the critical point. It also describes the strong-coupling phase of large N 2-dimensional lattice Yang-Mills theory.8 The R 1 parameter p = N hTr U i = dφρ(φ)eiφ is the fundamental representation loop. Positivity of the density requires 0 ≤ p ≤ 21 . This distribution depends on φ and therefore is deconfined. k k k There is one solution of RSk (tˆ) = N in the region |tˆ| < 1 at tˆ = N /p. (If N and ˆ p are such that |t| > 1, both RSk and ΓSk should be extended there by analytic continuation.) The free energy is k/N k ln , (14) ΓSk = N ep k is increased, it where e = 2.718 . . .. ΓSk has the interesting feature that, as N changes sign from negative to positive. This results in a phase transition which k k k k occurs when N = N crit = ep. When N < N crit , ΓSk is negative and the k k > N loop expectation value, e−N Γ , is exponentially large. When N , ΓSk is crit positive and the loop vanishes for N → ∞. This phase transition implies that, even in the deconfined phase, where the fundamental representation loop is non-zero, sufficiently large symmetric representations are still confined. To see this behavior in another example, consider the semi-circle distribution √ which, for |φ| < 2 arcsin 2 − 2p, is r cos φ2 φ ρ(φ) = 2 − 2p − sin2 (15) π(2 − 2p) 2 √ and which vanishes in the gap 2 arcsin 2 − 2p ≤ |φ| ≤ π. We still use the fundamental loop, p, as a parameter and now 21 ≤ p ≤ 1. This is the distribution in the weak coupling phase of 2-dimensional lattice Yang-Mills theory.8 It is also an approximation to the deconfined distribution for weakly coupled N = 4 YangMills theory.3 For sufficiently weak coupling, it is accurate near the phase transition where p = 12 . The saddle point computation can be done explicitly near t = 0 and analytically continued. The free energy is p sinh θ + sinh2 θ + 2 − 2p ΓSk = (2θ cosh θ − sinh θ) 2 − 2p # " p 2 1 sinh θ + sinh θ + 2 − 2p − − ln , (16) 2 2 − 2p

where θ is defined by tˆ = e2θ and is determined by the saddle-point equation # " p k 1 sinh θ + sinh2 θ + 2 − 2p , (17) + = cosh θ N 2 2 − 2p

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Fig. 1.

The free energy ΓSk (θ) as a function of

k N

in the semi-circle distribution with p = 0.51.

which can be solved for sinh(θ). The free energy is plotted in Fig. 1. With p = 0.51, k ' 1.4. In the free energy becomes positive at θ ' 0.50 which corresponds to N crit. fact, we can see that the phase transition occurs over the whole allowed range of p and is surprisingly insensitive to the value of p. Now, we turn to the antisymmetric representation. For a large class of distribudR

(tˆ)

Ak tions gapped around φ = π and with > 0, which includes the semi-circle dtˆ distribution (15), it can be shown that that ΓAk is always negative and the phase transition that we are discussing does not occur. We refer the reader to Ref. [4] for the details. Our conclusion is that, in the weakly coupled Yang-Mills theory, the symmetric and antisymmetric representations have markedly different behavior. The symmetric one has a phase transition to a phase where the loop expectation value vanishes, even in the deconfined phase, whereas the antisymmetric one does not. This is qualitatively similar to what is seen in the string theory dual5 where the D5-brane k whereas dual to the antisymmetric representation exists for all allowed values of N the D3-brane dual to the antisymmetric representation is absent. The apparent k k at strong coupling implies tha N (T, λ) becomes absence of a critical value of N crit. coupling constant dependent and goes to zero faster than √4λ as λ → ∞.

References 1. E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998) [arXiv:hep-th/9803131]. 2. B. Sundborg, Nucl. Phys. B 573, 349 (2000) [arXiv:hep-th/9908001]. 3. O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, Adv. Theor. Math. Phys. 8, 603 (2004) [arXiv:hep-th/0310285]. 4. G. Grignani, J. L. Karczmarek and G. W. sem*noff, arXiv:0904.3750 [hep-th]. 5. S. A. Hartnoll and S. Prem Kumar, Phys. Rev. D 74, 026001 (2006) [arXiv:hepth/0603190]. 6. A. M. Polyakov, Phys. Lett. B 72, 477 (1978).

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7. 8. 9. 10. 11.

L. Susskind, Phys. Rev. D 20, 2610 (1979). D. J. Gross and E. Witten, Phys. Rev. D 21 (1980) 446. S. A. Hartnoll and S. P. Kumar, JHEP 0608, 026 (2006) [arXiv:hep-th/0605027]. J. M. Maldacena, Phys. Rev. Lett. 80, 4859 (1998) [arXiv:hep-th/9803002]. N. Drukker and B. Fiol, JHEP 0502, 010 (2005) [arXiv:hep-th/0501109]. S. Yamaguchi, Int. J. Mod. Phys. A 22, 1353 (2007) [arXiv:hep-th/0601089]; S. Yamaguchi, JHEP 0605, 037 (2006) [arXiv:hep-th/0603208]. J. Gomis and F. Passerini, JHEP 0608, 074 (2006) [arXiv:hep-th/0604007]; J. Gomis and F. Passerini, JHEP 0701, 097 (2007) [arXiv:hep-th/0612022]; K. Okuyama and G. W. sem*noff, JHEP 0606, 057 (2006) [arXiv:hep-th/0604209]; 12. J. K. Erickson, G. W. sem*noff and K. Zarembo, Nucl. Phys. B 582, 155 (2000) [arXiv:hep-th/0003055]; V. Pestun, arXiv:0712.2824 [hep-th].

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Effect of Vector–Axial-Vector Mixing to Dilepton Spectrum in Hot and/or Dense Matter∗ Masayasu Harada Department of Physics, Nagoya University, Nagoya, 464-8602, Japan E-mail: [emailprotected] Chihiro Sasaki Frankfurt Institute for Advanced Studies, J.W. Goethe University, D-60438 Frankfurt, Germany Physik-Department, Technische Universit¨ at M¨ unchen, D-85747 Garching, Germany E-mail: [emailprotected] In this write-up we summarize main results of our recent analyses on the mixing between transverse ρ and a1 mesons in hot and/or dense matter. We show that the axial-vector meson contributes significantly to the vector spectral function in hot matter through the mixing. In dense baryonic matter, we include a mixing through a set of ωρa1 -type interactions. We show that a clear enhancement of the vector spectral function appears √ below s = mρ for small three-momenta of the ρ meson, and thus the vector spectrum exhibits broadening.

1. Introduction In-medium modifications of hadrons have been extensively explored in the context of chiral dynamics of QCD.3,4 Due to an interaction with pions in the heat bath, the vector and axial-vector current correlators are mixed. At low temperatures or densities a low-energy theorem based on chiral symmetry describes this V-A mixing.5 The effects to the thermal vector spectral function have been studied through the theorem,6 or using chiral reduction formulas based on a virial expansion.7 In Ref. 1, it was shown that the effects of the V-A mixing, and how the axialvector mesons affect the spectral function near the chiral phase transition, within an effective field theory. The analysis was carried out assuming several possible patterns of chiral symmetry restoration: dropping or non-dropping ρ meson mass along with changing a1 meson mass, both considered to be options from a phenomenological point of view. In Ref. 2, we studied a novel effect of the V-A mixing through a set of ωρa1 -type interactions at finite baryon density, which was introduced by a Chern-Simons term in a holographic QCD model.8 We focused on the V-A mixing at tree level and its consequence on the in-medium spectral functions which are the main input to the ∗ This

talk is based on the work done in Refs. 1, 2.

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experimental observables. We showed that the mixing produces a clear enhance√ ment of the vector spectral function below s = mρ , and that the vector spectral function is broadened due to the mixing. We also discussed its relevance to dilepton measurements. In this write-up, we summarize main results of these papers especially focusing on the effect to the vector spectral function. 2. Effects of Vector–Axial-vector Mixing in Hot Matter We start with showing the vector spectral function at T /Tc = 0.6, without any dropping masses, in Fig. 1 (left). Two cases are compared; one includes the V-A mixing and the other does not. Both the spectral functions has a peak at Mρ . The √ effects of V-A mixing can be seen as a shoulder at s = Ma1 − mπ and a bump √ above s = Ma1 + mπ .

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Fig. 1. Left figure shows the vector spectral function at temperature T /Tc = 0.6. The solid (red) curve includes the effect of V-A mixing, while the dashed (green) curve does not. The middle figure shows the spectral function (option (A)) for mπ = 140 MeV at several temperatures T /Tc = 0.61.0. The right figure shows the vector spectral function (option (B)) for mπ = 140 MeV in type (I) at temperature T /Tc = 0.8.

In Ref. 1, two possible cases of chiral symmetry restoration are studied: (A) Dropping a1 meson mass but non-dropping ρ mass; (B) Dropping ρ and a1 meson masses. In both cases, the “flash temperature”9 Tf is introduced for controling how the mesons experience partial restoration of chiral symmetry. Then, the masses of vector and/or axial-vector mesons are assumed to have temperature dependences only above Tf . In option (A) at the chiral limit, the dropping a1 meson mass for T > Tf is taken as Ma21 (T ) − Mρ2 Tc2 − T 2 = , Ma21 (T = 0) − Mρ2 Tc2 − Tf2

(1)

where Tc is the critical temperature of the chiral symmetry restoration. The vector spectral function for this option (A), with the effect of pion mass included, is shown at several temperatures in Fig. 1 (middle). Below Tc one observes the previously mentioned threshold effects moving downward with increasing temperature. It is

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remarkable that at Tc the spectrum shows almost no traces of a1 -ρ-π threshold effects: The ρ to a1 mass ratio becomes almost 1 at T = Tc even though the effect of Mπ is included. Furthermore, one can show that a1 -ρ-π coupling constant becomes very tiny, ga1 ρπ ∼ 0.06 mπ . This indicates that at Tc the a1 meson mass nearly equals the ρ meson mass and the a1 -ρ-π coupling almost vanishes even in the presence of explicit chiral symmetry breaking. In option (B), two types of temperature dependences were used in Ref. 1: 2 Ma21 (T ) − Mρ2 (T ) Mρ2 (T ) Tc2 − T 2 Tc2 − T 2 = 2 = , , (2) (I) : Mρ2 (T = 0) Tc − Tf2 Ma21 (T = 0) − Mρ2 (T = 0) Tc2 − Tf2 2 Ma21 (T ) − Mρ2 (T ) Mρ2 (T ) Tc2 − T 2 T2 − T2 = = c2 , . (3) (II) : 2 2 2 2 2 Mρ (T = 0) Tc − Tf Ma1 (T = 0) − Mρ (T = 0) Tc − Tf2 Figure 1 (right) shows the vector spectrum using the type (I) parameterization at T = 0.8 Tc. The feature that the a1 meson suppresses the vector spectral function through the V-A mixing remains unchanged. Compared with the curve for T /Tc = 0.8 in Fig. 1 (middle), a bump through the V-A mixing and the ρ peak are shifted downward since both the ρ and a1 masses drop. The self-energy has a cusp at the √ threshold 2 Mρ and this appears as a dip at s ∼ 1.3 GeV. The influence of finite mπ turns out to be in threshold effects as before. We find that the nearly vanishing V-A mixing as seen for the non-dropping ρ mass, option (A). The result given here shows that the axial-vector meson contributes significantly to the vector spectral function; the presence of the a1 reduces the vector spectrum around Mρ and enhances it around Ma1 . A major change with both dropping ρ and a1 masses is a systematic downward shift of the vector spectrum. We observe that the a1 -ρ-π coupling almost vanishes at the critical temperature Tc and thus the V-A mixing becomes very tiny. 3. Effects of Vector–Axial-vector Mixing in Dense Matter In this section, we summarize main points shown in Ref. 2. At finite baryon density a system preserves parity but violates charge conjugation invariance. Chiral Lagrangians thus in general build in the term Lρa1 = 2C 0νλσ tr [∂ν Vλ · Aσ + ∂ν Aλ · Vσ ] ,

(4)

for the vector V µ and axial-vector Aµ mesons with the total anti-symmetric tensor 0123 = 1 and a parameter C. This mixing results in the dispersion relation8 1 2 (5) mρ + m2a1 ± (m2a1 − m2ρ )2 + 16C 2 p¯2 , p20 − p¯2 = 2 which describes the propagation of a mixture of the transverse ρ and a1 mesons with non-vanishing three-momentum | p| = p¯. The longitudinal polarizations, on the other hand, follow the standard dispersion relation, p20 − p¯2 = m2ρ,a1 . When the mixing vanishes as p¯ → 0, Eq. (5) with lower sign provides p0 = mρ and it

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with upper sign does p0 = ma1 . In the following, we call the mode following the dispersion relation with the lower sign in Eq. (5) “the ρ meson”, and that with the upper sign “the a1 meson”. The mixing strength C in Eq. (4) can be estimated assuming the ω-dominance in the following way: The gauged Wess-Zumino-Witten terms in an effective chiral Lagrangian include the ω-ρ-a1 term10 which leads to the following mixing term Lωρa1 = gωρa1 ω0 0νλσ tr [∂ν Vλ · Aσ + ∂ν Aλ · Vσ ] ,

(6)

where the ω field is replaced with its expectation value given by ω0 = gωN N · nB /m2ω . One finds with empirical numbers C = gωρa1 ω0 0.1 GeV at normal nuclear matter density. As we will show below, this is too small to have an importance in the correlation functions. In a holographic QCD approach, on the other hand, the effects from an infinite tower of the ω-type vector mesons are summed up to give C 1 GeV · (nB /n0 ) with normal nuclear matter density n0 = 0.16 fm−3 .8 In the following we assume an actual value of C in QCD in the range 0.1 < C < 1 GeV. Some importance of the higher Kaluza-Klein (KK) modes even in vacuum in the context of holographic QCD can be seen in the pion electromagnetic form factor at the photon on-shell: This is saturated by the lowest four vector mesons in a top-down holographic QCD model.11,12 In hot and dense environment those higher members get modified and the masses might be somewhat decreasing evidenced in an in-medium holographic model.13 This might provide a strong V-A mixing C > 0.1 GeV in three-color QCD and the dilepton measurements may give a good testing ground. longitudinal transv: C=0.5 GeV transv: C=1 GeV 2.5 3

p0 [GeV]

In Fig. 2, we show the dispersion relations (5) for the transverse modes together with those for the longitudinal modes with C = 1 and 0.5 GeV. This shows that, when C = 0.5 GeV, there are only small changes for both ρ and a1 mesons, while a substantial change for ρ meson when C = 1 GeV. For very large p¯ the longitudinal and transverse dispersions are in parallel with a finite gap, ±C.

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Fig. 2. The dispersion relations for the ρ (lower 3 curves) and a1 (upper 3 curves) mesons for C = 0.5, and 1 GeV.

In Fig. 3, we plot the integrated spectrum over three momentum, which is a main ingredient in dilepton production rates. Figure 3 (left) shows a clear enhancement of √ the spectrum below s = mρ due to the mixing. This enhancement becomes much suppressed when the ρ meson is moving with a large three-momentum as shown in Fig. 3 (right). The upper bump now emerges more remarkably and becomes a clear indication of the in-medium effect from the a1 via the mixing. As an application of the above in-medium spectrum, we calculate the production rate of a lepton pair emitted from dense matter through a decaying virtual photon.

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Fig. 3. The vector spectral function for C = 1 GeV. The curves of the left figure are calculated integrating over 0 < p¯ < 0.5 GeV, and those of the right figure over 0.5 < p¯ < 1 GeV. Here we use the values of masses given by mπ = 0.14 GeV, mρ = 0.77 GeV, ma1 = 1.26 GeV, and the widths given by the imaginary part of one-loop diagrams in a chiral Lagrangian approach as2,14 with the on-shell values of Γρ (s = m2ρ ) = 0.15 GeV and Γa1 (s = m2a1 ) = 0.33 GeV.

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Fig. 4. Left figure shows the dilepton production rate at T = 0.1 GeV for C = 1 GeV. Integration over 0 < p¯ < 0.5 GeV (dashed) and 0.5 < p¯ < 1 GeV (dashed-dotted) was carried out. The right figure shows the dilepton production rate at T = 0.1 GeV with various mixing strength C. Integration over 0 < p¯ < 1 GeV was done. We use the constant widths with values of Γω = 8.49 MeV, Γφ = 4.26 MeV, Γf1 (1285) = 24.3 MeV and Γf1 (1420) = 54.9 MeV.

Figure 4 (left) presents the integrated rate at T = 0.1 GeV for C = 1 GeV. One clearly observes a strong three-momentum dependence and an enhancement below √ s = mρ due to the Bose distribution function which result in a strong spectral broadening. The total rate is mostly governed by the spectrum with low momenta p¯ < 0.5 GeV due to the large mixing parameter C. When density is decreased, the √ mixing effect gets irrelevant and consequently in-medium effect in low s region is reduced in compared with that at higher density. The calculation performed in hadronic many-body theory in fact shows that the ρ spectral function with a low momentum carries details of medium modifications.15 One may have a chance to observe it in heavy-ion collisions with certain low-momentum binning at J-PARC, GSI/FAIR and RHIC low-energy running.

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It is straightforward to introduce other V-A mixing between ω-f1 (1285) and φ-f1 (1420). In Fig. 4 (right) we plot the integrated rate at T = 0.1 GeV with several mixing strength C which are phenomenological option. One observes that the enhancement below mρ is suppressed with decreasing mixing strength. This forms √ into a broad bump in low s region and its maximum moves toward mρ . Similarly, √ some contributions are seen just below mφ . This effect starts at threshold s = 2mK . Self-consistent calculations of the spectrum in dense medium will provide a smooth change and this eventually makes the φ meson peak somewhat broadened. Finally, we remark that the importance of the mixing effect studied here relies on the coupling strength C. Holographic QCD predicts an extremely strong mixing C ∼ 1 GeV at nB = n0 which leads to vector meson condensation at nB ∼ 1.1 n0 .8 This may be excluded by known properties of nuclear matter and therefore in reality the strength C will be smaller. We have discussed a possible range of C to be 0.1 < C < 1 GeV based on higher excitations and their in-medium modifications. The parameter C does carry an unknown density dependence. This will be determined in an elaborated treatment of hadronic matter along with the underlying QCD dynamics. If C ∼ 0.1 GeV at n0 were preferred as the lowest-omega dominance, the mixing effect is irrelevant there. However, it becomes more important at higher densities, e.g. C = 0.3 GeV at nB /n0 = 3 which leads to a distinct modification from the spectrum in free space.

4. Summary In this write-up, we summarized main results of our recent analyses on the mixing between ρ and a1 mesons in hot and/or dense matter. The analysis in Ref. 1 shows that the axial-vector meson contributes significantly to the vector spectral function in hot matter through the mixing: the presence of the a1 reduces the vector spectrum around Mρ and enhances it around Ma1 . The effect of dropping mass of a1 with or without dropping ρ mass associated with the chiral symmetry restoration is studied. It is shown that the a1 -ρ-π coupling almost vanishes at the critical temperature Tc and thus the V-A mixing becomes very tiny. In Ref. 2, we studied a novel effect of the V-A mixing through a set of ωρa1 -type interactions at finite baryon density. We showed that the mixing produces a clear √ enhancement of the vector spectral function below s = mρ , and that the vector spectral function is broadened due to the mixing. It is an interesting issue to address a change of the vector correlator with the V-A mixing at finite baryon density in section 3 toward chiral symmetry restoration. The mixing (4) is chirally symmetric and thus does not vanish at the chiral restoration in contrast to the vanishing V-A mixing near the critical temperature Tc without baryon density in section 2. A spontaneous breaking of Lorentz invariance via the omega condensation could increase the mixing strength C near chiral restoration.16 Furthermore, if meson masses drop due to partial restoration of chiral symmetry assuming a second- or weak first-order transition in high baryon density

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but low temperature region, the ground state near the critical point may favor vector condensation even for a moderate mixing strength. This will be reported elsewhere. Acknowledgment The work of M.H. is supported in part by the JSPS Grant-in-Aid for Scientific Research (c) 20540262, Grant-in-Aid for Scientific Research on Innovative Areas (No. 2104) “Quest on New Hadrons with Variety of Flavors” from MEXT and the Global COE Program of Nagoya University “Quest for Fundamental Principles in the Universe (QFPU)” from JSPS and MEXT of Japan. The work of C.S. is supported in part by the DFG cluster of excellence “Origin and Structure of the Universe”. References 1. C. Sasaki, M. Harada and W. Weise, Prog. Theor. Phys. Suppl. 174, 173 (2008); Phys. Rev. D 78, 114003 (2008); Nucl. Phys. A 827, 350C (2009). 2. M. Harada and C. Sasaki, Phys. Rev. C 80, 054912 (2009). 3. See, e.g., V. Bernard and U. G. Meissner, Nucl. Phys. A 489, 647 (1988); T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994); R. D. Pisarski, hep-ph/9503330; G. E. Brown and M. Rho, Phys. Rept. 269, 333 (1996); F. Klingl, N. Kaiser and W. Weise, Nucl. Phys. A 624, 527 (1997); F. Wilczek, hep-ph/0003183; G. E. Brown and M. Rho, Phys. Rept. 363, 85 (2002). 4. R. Rapp and J. Wambach, Adv. Nucl. Phys. 25, 1 (2000), R. S. Hayano and T. Hatsuda, arXiv:0812.1702 [nucl-ex], R. Rapp, J. Wambach and H. van Hees, arXiv:0901.3289 [hep-ph]. 5. M. Dey, V. L. Eletsky and B. L. Ioffe, Phys. Lett. B 252, 620 (1990), B. Krippa, Phys. Lett. B 427, 13 (1998). 6. E. Marco, R. Hofmann and W. Weise, Phys. Lett. B 530, 88 (2002), M. Urban, M. Buballa and J. Wambach, Phys. Rev. Lett. 88, 042002 (2002). 7. J. V. Steele, H. Yamagishi and I. Zahed, Phys. Lett. B 384, 255 (1996); Phys. Rev. D 56, 5605 (1997), K. Dusling, D. Teaney and I. Zahed, Phys. Rev. C 75, 024908 (2007), K. Dusling and I. Zahed, Nucl. Phys. A 825, 212 (2009). 8. S. K. Domokos and J. A. Harvey, Phys. Rev. Lett. 99, 141602 (2007). 9. G. E. Brown, C. H. Lee and M. Rho, Nucl. Phys. A 747, 530 (2005). 10. N. Kaiser and U. G. Meissner, Nucl. Phys. A 519, 671 (1990). 11. T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005); Prog. Theor. Phys. 114, 1083 (2005). 12. M. Harada, S. Matsuzaki and K. Yamawaki, in preparation. See also the contribution to the same proceedings. 13. K. Peeters, J. Sonnenschein and M. Zamaklar, Phys. Rev. D 74, 106008 (2006). 14. M. Harada and C. Sasaki, Phys. Rev. D 73, 036001 (2006). 15. F. Riek, R. Rapp, T. S. Lee and Y. Oh, Phys. Lett. B 677, 116 (2009). 16. K. Langfeld, H. Reinhardt and M. Rho, Nucl. Phys. A 622, 620 (1997), K. Langfeld and M. Rho, Nucl. Phys. A 660, 475 (1999).

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Infrared Behavior of Ghost and Gluon Propagators Compatible with Color Confinement in Yang-Mills Theory with the Gribov Horizon Kei-Ichi Kondo∗ Department of Physics, University of Tokyo, Tokyo 113-0033, Japan E-mail: [emailprotected] We discuss how the existence of the Gribov horizon affects the deep infrared behavior of ghost and gluon propagators of the D-dimensional SU (N ) Yang-Mills theory in the Landau gauge. If we use a horizon function in the Gribov-Zwanziger framework to restrict the functional integral to the first Gribov region for avoiding Gribov copies, we can show that the ghost propagator behaves like free, while the gluon propagator is non-vanishing in deep infrared region (the decoupling solution) in harmony with recent results obtained from numerical simulations on lattice, the Schwinger-Dyson equation and the functional renormalization group. This result should be compared with the Gribov prediction that such a restriction leads to the vanishing gluon propagator and enhanced ghost propagator (the scaling solution). We raise some questions on current understanding on the ghost propagator and its implications for quark confinement ` a la Wilson and color confinement a la Kugo-Ojima. ` Keywords: Ghost dressing function, Schwinger-Dyson equation, color confinement, KugoOjima, Gribov-Zwanziger, horizon function.

1. Introduction We consider the SU (N ) Yang-Mills theory in D-dimensional Euclidean space. For a quantum Yang-Mills theory to be well-defined, we must give a gauge fixing procedure which is able to select just one representative from each gauge orbit in the space of Yang-Mills fields. It is known that the usual Landau gauge fixing ∂A = 0 is not sufficient for this purpose, since there are many Yang-Mills field configurations (Gribov copies) connected by gauge transformations, which are obtained as intersection points of a gauge orbit with the gauge fixing hypersurface Γ := {A ; ∂A = 0}. Long ago, Gribov1 proposed to restrict the functional integral to the subspace Ω: ZGribov := [dA ]δ(∂A ) det(K[A ])) exp{−SY M [A ]}, (1) Ω

where SY M is the Yang-Mills action, K[A ] := −∂µ Dµ [A ] is the Faddeev-Popov operator, and Ω is the 1st Gribov region defined by Ω := {A ; ∂A = 0, −∂D[A ] > 0} ⊂ Γ.

∗ On

(2)

sabbatical leave of absence from Department of Physics, Chiba University, Chiba 263-8522, Japan.

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Ω is a bounded and convex region including the origin {A = 0}. In fact, −∂µ Dµ [A = 0] = −∂µ ∂µ > 0. The boundary of Ω is called the Gribov horizon: ∂Ω := {A ; ∂A = 0, −∂D[A ] = 0}.

(3)

In order to realize this restriction in a simpler manner, Zwanziger proposed a framework called the Gribov-Zwanziger theory2–4 described by µ D ZGZ := [dA ]δ(∂ Aµ ) det(K[A ]) exp −SY M [A ] − γ d xh(x) , (4) where h(x) = h[A ](x) is called the Zwanziger horizon function and the parameter γ called the Gribov parameter is determined by solving a gap equation, commonly called the horizon condition, h(x)GZ = (N 2 − 1)D.

(5)

The horizon function plays the role of restricting the integration region inside the Gribov horizon ∂Ω, since the horizon function is positive definite inside the 1st Gribov region and approaches infinity near the Gribov horizon: A → ∂Ω =⇒ K[A ] ↓ 0 =⇒

dD xh[A ](x) ↑ ∞ =⇒ exp{−γ

dD xh(x)} ↓ 0 (γ > 0).

The first proposal of the horizon function is2 h(x) = dD ygf ABC AµB (x)(K −1 )CE (x, y)gf AF E AµF (y).

The second proposal is the covariant derivative form:4 h(x) = dD yDµ [A ]AC (x)(K −1 )CE (x, y)Dµ [A ]AE (y).

(6)

(7)

(8)

2. A theoretical possibility of deriving the decoupling solution We use an exact relationship between the ghost dressing function G(k 2 ) and the Kugo-Ojima function u(k 2 ) with an additional function w(k 2 ): G−1 (k 2 ) = 1 + u(k 2 ) + w(k 2 ).

(9)

This identity was first derived in Ref. 7 and confirmed in Refs. 8,10. As a special case, we have a relationship between G(0) and the Kugo-Ojima parameter u(0): G(0) = [1 + u(0) + w(0)]−1 .

(10)

Therefore, the Kugo-Ojima criterion u(0) = −1 for color confinement is equivalent to the scaling solution G(0) = ∞, (Ref. 12) if and only if w(0) = 0. Here u and w are defined from the modified 1-particle irreducible (m1PI) part as kµ kν AB A B m1P I 2 2 ¯ λµν (k) := (gAµ × C ) (gAν × C ) k = δµν u(k ) + 2 w(k ) δ AB , (11) k

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where u(k 2 ) agrees with the Kugo-Ojima function usually defined by kµ kν A B ¯ (Dµ C ) (gAν × C ) k := δµν − 2 δ AB u(k 2 ). k

(12)

The m1PI part is defined from the two-point function of the composite operators AB (gAµ × C )A (gAν × C¯)B k = λAB µν (k) + ∆µν (k), λAB (k) := (gAµ × C )A (gAν × C¯)B m1PI , µν

k

A ¯C 1PI C ¯D D ¯ B 1PI ∆AB µν (k) := (gAµ × C ) C k C C k C (gAν × C ) k .

Φ1

Φ2

C

C

(13)

C

Fig. 1. Diagrammatic representation of (gAµ ×C )A (gAν × C¯)B k , (gAµ ×C )A (gAν × C¯)B conn , k and (gAµ × C )A (gAν × C¯)B m1PI . (gAµ × C )A (gAν × C¯)B 1PI k k

We propose a trick which enables one to incorporate the Gribov horizon directly into the self-consistent Schwinger-Dyson equation in the gauge-fixed Yang-Mills theory, G−1 (k 2 ) =

h(0) + u(k 2 ) + w(k 2 ). (dimG)D

(14)

We give a main result using the 1st horizon function.8,10,11 We consider G(0) and u(0) as functions of w(0). By applying the horizon condition (5) to the horizon function (7) and the identity (10), we have (15) G(0) = 1 − (D − 1)w(0)/2 + [1 − (D − 1)w(0)/2]2 − 1 + D > 0,

and u(0) = −1 − w(0) + G−1 (0) reads

1 (16) 2 − 3w(0) − 12 + [2 − 3w(0)]2 . u(0) = −1 − w(0) − 6 This implies that the horizon condition determines the boundary value G(0) as a function of w(0). Consequently, we have one-parameter family of solutions parameterized by w(0). Both G(0) and u(0) are monotonically decreasing functions in w(0); G(0), u(0) → +∞ as w(0) → −∞, while G(0) → 0 and u(0) → −5/3 as w(0) → +∞. Thus the scaling solution G(0) = +∞ is obtained only when w(0) = −∞. Otherwise w(0) > −∞, the decoupling solution 0 < G(0) < ∞ is obtained. Using a special value as an additional input w(0) = 0, we obtain √ √ (17) G(0) = 1 + D > 0, u(0) = (−D ± D)/(D − 1). In particular, for D = 4, G(0) = 3 > 0,

u(0) = −2/3 > −1 (D = 4).

(18)

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This leads to the decoupling solution, 0 < G(0) < ∞, (Ref. 13) which agrees with current data of numerical simulations. See Refs. 8–11 for related references. 3. Critical points We raise some questions on current understanding on the ghost propagator and its implications for quark confinement a la Wilson and color confinement a la KugoOjima. (1) Point.1 Both horizon functions (7) and (8) play the role of restricting the functional integral to the 1st Gribov regin in the sense of (6). However, the results depend on the choice of the horizon function. If one starts from the second horizon term (8), then one is led to the scaling solution (when w(0) = 0 up to Point.2). Whereas the first horizon function (7) leads to the decoupling solution. The difference between two horizon functions are just total derivatives. (2) Point.2 The conventional claim between the ghost propagator and the color confinement: the Kugo-Ojima criterion u(0) = −1 is equivalent to the scaling solution G(0) = ∞ follows from the relation G(0) = [1 + u(0)]−1 .

(19)

However, this is not necessarily precise, since the exact relationship between the ghost dressing function and the Kugo-Ojima parameter must be G(0) = [1 + u(0) + w(0)]−1 .

(20)

Therefore, the Kugo-Ojima criterion u(0) = −1 is equivalent to the scaling solution G(0) = ∞, if and only if w(0) = 0. In perturbation theory, indeed, w(0) = 0. However, no one knows whether w(0) = 0 or not exactly in the nonperturbative sense. (3) Point.3 The conventional multiplicative renormalization scheme is incompatible with the horizon condition. The inclusion of the horizon condition make the ghost propagator finite self-consistently. 4. Conclusion and discussion We have discussed how the existence of the Gribov horizon modifies the deep infrared behavior of the Landau gauge SU(N) Yang-Mills theory, using the GribovZwanziger framework with the horizon condition h(x) = (dimG)D. We have found that there exists one-parameter family of solutions parameterized by a parameter w(0) which was assumed to be zero implicitly. The family includes both the scaling and decoupling solutions, and specification of the parameter discriminates between them. For the first horizon function, (21) G(0) = 1 − 3w(0)/2 + [1 − 3w(0)/2]2 + 3 > 0,

1 (22) 2 − 3w(0) − 12 + [2 − 3w(0)]2 . u(0) = −1 − w(0) − 6

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In particular, G(0) = 3, G(0) = 2, G(0) = ∞,

u(0) = −2/3,

u(0) = −1,

for w(0) = 0.

(23)

for w(0) = 1/2.

(24)

u(0) = +∞ for w(0) = −∞(not permitted).

(25)

For the second horizon function: G(0) = w(0)−1 ,

(26)

u(0) = −1 irrespective of w(0).

(27)

Here the scaling solution is obtained as G(0) = ∞,

u(0) = −1 for

w(0) = 0,

(28)

w(0) = 0.

(29)

and the decoupling solution is obtained as G(0) = ∞,

u(0) = −1 for

In my opinion, one must fix the following issues: (1) The best choice of the horizon term: The horizon terms currently available are not necessarily agree with the definition of the Gribov horizon. Find the precise horizon term to specify the Gribov region in the Gribov-Zwanziger framework? (2) BRST symmetry and color confinement: The BRST symmetry is not maintained after being restricted to the Gribov region. (a) Find the BRST-invariant horizon term. Or, (b) find a modified (local) BRST transformation to make the horizon term invariant. It is interesting to consider how the existence of the horizon is relevant for color confinement. In the Gribov-Zwanziger theory (restricted to the 1st Gribov region), the BRST symmetry is broken by the existence of the horizon. δSGZ = δ S˜γ = 0 Nevertheless, there exists a “BRST” like symmetry (without nilpotency [Sorella,0905.1010[hep-th]] or with nilpotency [K.-I. K., 0905.1899[hep-th]]) which leaves the Gribov-Zwanziger action invariant. Then we could apply the Kugo-Ojima idea to the Gribov-Zwanziger theory, which opens the path to searching for the modified color confinement criterion a la Kugo and Ojima. (3) The value of the parameter w(0): In the perturbation theory, w(0) = 0, which is not guaranteed in the nonperturbative level. Determine the value w(0) which enables one to discriminate the scaling solution and decoupling one, once a horizon term is given. [However, a parameter w(0) might reflect the ambiguity originating from the Gribov copies inside the Gribov region. Then it may be undetermined.] (4) Gribov horizon and renormalization: The conventional multiplicative renormalization scheme is incompatible with the Gribov horizon condition. We observe that the inclusion of the horizon term cancels the ultraviolet divergence in the

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Schwinger-Dyson equation for the ghost propagator and the resulting (nonperturbative) self-consistent solution becomes ultraviolet finite. In other words, the horizon condition interpolates between the infrared behavior and the ultraviolet one. (5) UV divergence (convergence): UV renormalization might be changed if the horizon condition is taken into account. Numerical simulations on finer lattice are desirable, in addition to the larger size lattice. (6) Scaling or decoupling: It is really meaningful to discriminate between the scaling and the decoupling from physical point of view. Both solutions can give the same physical results. (7) Quark confinement from color confinement: It is claimed14 that both solutions (decoupling as well as scaling) lead to quark confinement by proving the vanishing of the order parameter of quark confinement, the Polyakov loop average. Therefore, one can consider both solutions are the same from the physical point of view. The difference comes just from the gauge artifact (insufficient implementation of the non-perturbative gauge fixing procedure). Acknowledgments This work is financially supported by Grant-in-Aid for Scientific Research (C) 21540256 from Japan Society for the Promotion of Science (JSPS). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

V.N. Gribov, Nucl. Phys. B139, 1–19 (1978). D. Zwanziger, Nucl. Phys. B323, 513–544 (1989). D. Zwanziger, Nucl. Phys. B378, 525–590 (1992). D. Zwanziger, Nucl. Phys. B399, 477–513 (1993). D. Zwanziger, Nucl.Phys.B412, 657-730 (1994). T. Kugo and I. Ojima, Suppl. Prog. Theor. Phys. 66, 1–130 (1979). T. Kugo, hep-th/9511033. K.-I. Kondo, arXiv:0904.4897 [hep-th], Phys.Lett.B. 678, 322-330 (2009). K.-I. Kondo, arXiv:0905.1899 [hep-th]. K.-I. Kondo, arXiv:0907.3249 [hep-th], Prog. Theor. Phys. 122, 1455-1475 (2009). K.-I. Kondo, arXiv:0909.4866 [hep-th]. R. Alkofer and L. von Smekal, hep-ph/0007355, Phys. Rept. 353, 281 (2001). Ph. Boucaud, J.P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene and J. RodriguezQuintero, hep-ph/0803.2161, JHEP 06, 099 (2008). Ph. Boucaud, J.P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene and J. RodriguezQuintero, arXiv:0801.2721[hep-ph], JHEP 06, 012 (2008). 14. J. Braun, H. Gies and J.M. Pawlowski, e-Print: arXiv:0708.2413 [hep-th]]

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Chiral Symmetry Breaking on the Lattice Hidenori f*ckaya [for JLQCD and TWQCD Collaborations] Department of Physics, Nagoya University, Nagoya 464-8602, Japan E-mail: [emailprotected] This talk presents a recent lattice study by JLQCD and TWQCD collaborations on spontaneous breaking of chiral symmetry. To maintain exact chiral symmetry in the zero quark mass limit, we employ the overlap Dirac operator for the dynamical quark action. The numerical cost for the overlap fermions is high but can be reduced by fixing the topological charge of the gauge fields along the Monte Carlo updates. By studying the low-lying eigenvalues of the Dirac operator, which is always UV finite, we confirm the presence of the chiral condensate. Correcting the finite size effects calculated within the chiral perturbation theory, we determine the value of the chiral condensate at a good accuracy. Keywords: Lattice QCD, chiral symmetry.

1. Introduction Spontaneous breaking of chiral symmetry1 plays a key role in low-energy limit of Quantum Chromo Dynamics (QCD). The chiral condensate is believed to be the source of the symmetry breaking and give a “constituent ” mass to the quark fields and thus to the hadron masses ∼ O(1) GeV. Moreover, the pion can be identified as the (pseudo) Nambu-Goldstone boson and described well by chiral perturbation theory (ChPT)2,3 of which interaction is highly restricted by the symmetries. Due to its universal properties, the chiral symmetry breaking has also been applied to “beyond ” QCD, or model buildings for the electro-weak symmetry breaking. The so-called Techni-Color models4–7 are candidates for the high energy theories which may be examined in LHC experiments.8 It is, however, still difficult to confirm the spontaneous chiral symmetry breaking from the 1st principle calculations of QCD nor more general strong coupling gauge theories (SCGT’s). The chiral condensate is 100 % non-perturbative effects : it never obtains a finite value in the perturbative QCD expansion. The numerical lattice QCD simulation has played a central role in nonperturbative analysis of QCD. Many kinds of masses, decay constants and matrix elements of hadrons are evaluated, which confirms, comparing with the experiments, QCD as the fundamental theory. But even with lattice QCD, it is a challenging problem to examine the chiral condensate, due to mainly three difficulties:

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(1) Nielsen-Ninomiya’s theorem9 : the chiral symmetry on the lattice must be broken to avoid appearance of the unphysical fermion modes (fermion doubling). (2) UV divergence : the chiral condensate at finite quark mass generally has power divergence which depends on the regularization scheme. (3) Finite V correction : the spontaneous symmetry breaking is never allowed at finite volume. In this talk, we present how these three problems are solved or circumvented in our recent works and show the numerical results for the chiral condensate. 2. Three problems and solutions 2.1. Chiral symmetry on the lattice Lattice QCD is a well-defined regularization of the gauge theory, which allows us to numerically perform the functional integrals. Disctetization of the space-time leads, however, to explicit breaking of a lot of symmetries the original continuum theory has.A typical example is violation of the translational invariance. It is well-known that the chiral symmetry is one of such symmetries incompatible with the lattice discretization. Nielsen and Ninomiya9 proved that the chiral symmetry must be broken otherwise unphysical fermion modes, known as doublers, appear, which is inconsistent with the continuum limit. In 1998, Neuberger solved this problem.10 He proposed a new type of subtraction operator, called overlap Dirac operator, m m + m0 − γ5 sgn[HW (−m0 )], (1) D(m) = m0 + 2 2

where m denotes the quark mass and HW ≡ γ5 DW (−m0 ) is the Hermitian WilsonDirac operator with a large negative mass −m0 (We take m0 = 1.6 in our numerical studies.). Here and in the following the parameters are given in the lattice units. In the chiral limit m → 0, the overlap-Dirac operator (1) satisfies the GinspargWilson relation:11 D(0)γ5 +γ5 D(0) = D(0)γ5 D(0)/m0 , with which the quark action has exact chiral symmetry under a modified chiral transformation.12 Moreover, it is known that the overlap-Dirac operator has an index which corresponds to the topological charge in the continuum limit.13 In spite of its promising features, the numerical study employing the overlap Dirac operator was far behind. It is simply due to its high numerical cost. For the approximation of the sign function in (1), a number of operations of the WilsonDirac operator are needed to keep a required precision. Moreover, the fermion determinant has discontinuity on the topology boundaries, configuration where the index of the overlap Dirac operator changes. This discontinuity of the determinant prevents smooth evolution of the molecular dynamics steps and requires a special treatment,14 which needs an additional numerical cost potentially proportional to the lattice volume squared.

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To avoid the problem of the large extra numerical cost, we introduce additional Wilson fermions and twisted-mass bosonic spinors to generate a weight det[HW (−m0 )2 ] , det[HW (−m0 )2 + µ2 ]

(2)

in the functional integrals.15–17 Both of fermions and ghosts are unphysical as their masses are of order of the lattice cutoff but prohibits the topology changes along the simulations. In our numerical studies, we set µ = 0.2. This topology fixing leads to a large reduction of the numerical cost.17 Since 2006, JLQCD and TWQCD collaborations have succeeded in performing Monte Carlo simulations with dynamical Nf =2 and 2+1 overlap quarks.18 We choose 5-6 different points for the up and down quark mass mud and 2 points for the strange quark mass ms . For the gauge part, we use the Iwasaki gauge action.19 The lattice volumes are V = L3 T = 163 × 32 (Nf = 2) and V = L3 T = 163 × 48 (Nf = 2 + 1). We also carry out a simulation on a V = L3 T = 243 × 48 lattice at one parameter choice, in order to check the finite volume effect. The lattice scales a−1 = 1.667 GeV (Nf = 2) and a−1 = 1.833 GeV (Nf = 2 + 1) are determined from the heavy quark potential, using the Sommer’s scale20 r0 = 0.49 fm as an input. The lattice size is then estimated as L ∼ 1.9 fm for Nf = 2, and L ∼ 1.7 fm for Nf = 2 + 1 runs. Note that we have reduced the lightest quark mass to almost the physical value mud ∼ 3 MeV. 2.2. Dirac spectrum and chiral symmetry breaking Even with an exact chiral symmetry on the lattice, the direct calculation of the chiral condensate is difficult, due to both of ultra-violet (UV) and infra-red (IR) problems. On the UV side, there is a power divergence at finite quark mass, of the form ∼ m/a2 where a denotes the lattice spacing. But in the limit of m → 0, an IR problem appears : the chiral condensate trivially vanishes since spontaneous breaking of symmetry is never allowed at finite volume. One can avoid both problems by analyzing Dirac operator spectrum rather than the chiral condensate itself. There is a non-perturbative relation called BanksCasher relation21 which relates the zero-mode density of the Dirac operator (here denoted by ρ) to the chiral condensate, ¯ q q = πρ(0). It is straightforward to extend this relation to finite value of λ: 1 ρ(λ) = Re¯ q q . π mv =iλ

(3)

(4)

Note here that the valence quark mass mv is analytically continued to a pure imaginary value iλ, while the sea quark masses remain real. The UV problem is avoided by taking the real part where the pure imaginary divergence is canceled. We expect

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that IR problem is also softened as the eigenvalue density (at finite λ) is non-trivial even at finite volume since the valence “mass ”, mv = iλ, is finite. The Banks Casher relation provides a picture of how chiral symmetry breaking occurs. In the free fermion case, the spectral density shows a cubic degeneracy: 3 3 λ . (5) 4π 2 Once the gauge coupling is turned on, however, this highly degenerate levels are split and the eigenvalues feel repulsive force from each other. As a result, the eigenvalues are accumulated around zero, which leads to the chiral condensate. We can further estimate how finite volume effect is seen in the spectrum. Finite volume never allows spontaneous breaking of symmetry, which means ρ(0) must be zero. Since only very long-wave length modes should reflect the difference, we expect the Dirac spectrum shows a steep drop around 1/V , above which the bulk mode q q. part still keeps a height around the chiral condensate Σ ≡ limm→0 limV →∞ ¯ Here we should note another advantage. The right-hand-side of (4) can be analyzed using (partially quenched) chiral perturbation theory, where one can estimate the effects from finite quark masses and fixed topological charge Q, too. ρ(λ) =

2.3. Chiral perturbation at finite volume Since there is a big mass gap between the pion mass and other hadron masses, the pions are most responsible for the finite size effects and chiral perturbation theory is useful to trace the volume scaling of QCD. In fact, a number of studies have been done on the eigenvalue distribution of the Dirac operator and a sharp drop behavior around zero, as we discussed in the previous subsection, was actually found.22–24 In the calculation a special type of expansion, the so-called -expansion of chiral Lagrangian was employed. The -expansion exactly treats the zero-mode and precisely reproduce the sharp drop around zero in the Dirac spectral density as a consequence of a non-perturbative vacuum fluctuation. The calculation was recently extended to the next-leading-order (NLO).25 The zero mode integral is exactly performed while the chiral logarithm, which comes from the conventional p-expansion, is also precisely treated. This means we calculate a hybrid system of zero-mode matrix model and perturbative modes. For the zeromode integrals we have used a technique developed in Refs.26. In this hybrid way of calculation, the spectral density in a fixed topological sector of Q at finite V and m is calculated, which is expressed by two contributions: ρQ (λ) = Σef f ρQ + ρp ,

(6)

where the first term has the same functional form as in the leading-order,22–24 except for Σef f having a shift from NLO contributions. The second term represents the purely NLO logarithmic curve, or “chiral logs” from the non-zero modes which is the p-expansion regime. Note that ρp is IR finite, and topology independent.

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β=2.35, Nf=2, Q=0, (m =0.002) 0.01

free theory 3 Σ=(240MeV)

πρ(λ)/V

0.008 0.006 0.004 0.002 0

0.05

0.1

0.15

0.2

0.25

λ Fig. 1.

Low-lying Dirac eigenvalue histogram in two-flavor lattice QCD.27

3. Numerical result for chiral condensate We are now ready to calculate the chiral condensate and judge whether QCD spontaneously breaks chiral symmetry or not. The histogram shown in Fig. 1 is the lattice data for the Dirac spectrum calculated by JLQCD collaboration.27 The result is very consistent with Banks-Casher’s scenario. A significant accumulation of the low-lying modes around zero is seen in contrast to that of the free theory shown in the solid curve. Moreover, the plateau suggests a reasonable value of the chiral condensate Σ1/3 ∼ 240 (MeV). A steep drop near zero is also seen, indicating the presence of finite volume effects. Recently we have extended the study to the 2+1 flavor QCD and compare the results with chiral perturbation theory to NLO at finite volume. On the top panel of Fig. 2, we plot the lattice data in grey histogram and one can see that the steep drop around zero and the height of the first peak are consistent with the NLO ChPT formula. Moreover, in the bulk region, the logarithmically negative dump is also reproduced by the NLO contribution in ChPT, which cannot be explained in the LO epsilon-expansion shown in the dashed line. In order to quantify the statistical error and quality of fitting, we use the modenumber29 or the integrated histogram shown in the bottom panel of Fig. 2. The agreement is good upto λ ∼ 0.03 which is around ms /2. We also find a good consistency with data at different masses, different volumes and different topological sectors. Details are shown in Ref.28. From the data, we 3 determine the chiral condensate as Σ = [242(4)(+19 −18 )MeV] (in the M S scheme at 2 GeV) at NLO accuracy. The pion decay constant F = 74(1)(8) and one of the NLO coefficient in ChPT: Lr6 (770MeV) = −0.0001(3)(1) are also extracted from the sub-leading contributions.

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Fig. 2. Low-lying Dirac eigenvalue histogram (top) and the mode-number (cumulative histogram) (bottom) in 2+1-flavor lattice QCD.28 ChPT result at LO (dashed) and NLO (solid) is compared.

4. Conclusion We have performed lattice QCD simulations with overlap Dirac operator which realizes an exact chiral symmetry. To confirm the presence of chiral condensate, we have studied Dirac operator spectrum which has no UV divergence. Chiral perturbation theory is useful to convert finite volume results to the one in the infinite volume. Our lattice data are consistent with the prediction of the Banks-Casher relation and chiral perturbation including the finite volume effects. All our data strongly suggest the breaking of chiral symmetry. The low-energy constants Σ, F and Lr6 are determined from the data. The author thanks P.H. Damgaard and members of JLQCD and TWQCD collaborations for discussions. Numerical simulations are performed on IBM System Blue Gene Solution at KEK under a support of its Large Scale Simulation Program (No. 06-13). This work was supported by the Global COE program of Nagoya Univ. Quest for Fundamental Principles in the Universe from JSPS and MEXT of Japan.

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References 1. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); Y. Nambu and G. JonaLasinio, Phys. Rev. 124, 246 (1961). 2. S. Weinberg, Phys. Rev. 166, 1568 (1968). 3. J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984); Nucl. Phys. B 250, 465 (1985). 4. S. Weinberg, Phys. Rev. D 13, 974 (1976); Phys. Rev. D 19, 1277 (1979). 5. L. Susskind, Phys. Rev. D 20, 2619 (1979). 6. B. Holdom, Phys. Lett. B 150, 301 (1985). 7. K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335 (1986). 8. S. Asai, in these proceedings. 9. H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 20 (1981) [Erratum-ibid. B 195, 541 (1982)]; Nucl. Phys. B 193, 173 (1981). 10. H. Neuberger, Phys. Lett. B 417, 141 (1998); Phys. Lett. B 427, 353 (1998). 11. P. H. Ginsparg and K. G. Wilson, Phys. Rev. D 25, 2649 (1982). 12. M. Luscher, Phys. Lett. B 428, 342 (1998) [arXiv:hep-lat/9802011]. 13. P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B 427, 125 (1998) [arXiv:hep-lat/9801021]. 14. Z. Fodor, S. D. Katz and K. K. Szabo, JHEP 0408, 003 (2004) [arXiv:heplat/0311010]; T. A. DeGrand and S. Schaefer, Phys. Rev. D 71, 034507 (2005) [arXiv:hep-lat/0412005]; T. A. DeGrand and S. Schaefer, Phys. Rev. D 72, 054503 (2005) [arXiv:hep-lat/0506021]; N. Cundy, S. Krieg, G. Arnold, A. Frommer, T. Lippert and K. Schilling, Comput. Phys. Commun. 180, 26 (2009) [arXiv:heplat/0502007]. 15. T. Izubuchi and C. Dawson [RBC Collaboration], Nucl. Phys. Proc. Suppl. 106, 748 (2002). 16. P. M. Vranas, Phys. Rev. D 74, 034512 (2006) [arXiv:hep-lat/0606014]. 17. H. f*ckaya et al. [JLQCD Collaboration], Phys. Rev. D 74, 094505 (2006) [arXiv:heplat/0607020]. 18. S. Aoki et al. [JLQCD Collaboration], Phys. Rev. D 78, 014508 (2008) [arXiv:0803.3197 [hep-lat]]. 19. Y. Iwasaki, Nucl. Phys. B 258 (1985) 141; Y. Iwasaki and T. Yoshie, Phys. Lett. B 143, 449 (1984). 20. R. Sommer, Nucl. Phys. B 411, 839 (1994) [arXiv:hep-lat/9310022]. 21. T. Banks and A. Casher, Nucl. Phys. B 169, 103 (1980). 22. J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70, 3852 (1993). 23. G. Akemann and P. H. Damgaard, Nucl. Phys. B 528, 411 (1998). 24. P. H. Damgaard and S. M. Nishigaki, Phys. Rev. D 63, 045012 (2001). 25. P. H. Damgaard and H. f*ckaya, JHEP 0901, 052 (2009) [arXiv:0812.2797 [hep-lat]]. 26. K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 90, 041601 (2003) [arXiv:cond-mat/0209594]; Y. V. Fyodorov and G. Akemann, JETP Lett. 77, 438 (2003) [Pisma Zh. Eksp. Teor. Fiz. 77, 513 (2003)] [arXiv:cond-mat/0210647]. K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 683, 467 (2004) [arXiv:hep-th/0310271]. 27. H. f*ckaya et al. [JLQCD Collaboration], Phys. Rev. Lett. 98, 172001 (2007) [arXiv:hep-lat/0702003]; H. f*ckaya et al., Phys. Rev. D 76, 054503 (2007) [arXiv:0705.3322 [hep-lat]]. 28. H. f*ckaya et al. [JLQCD collaboration], arXiv:0911.5555 [hep-lat]. 29. L. Giusti and M. Luscher, JHEP 0903, 013 (2009) [arXiv:0812.3638 [hep-lat]].

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Gauge-Higgs Unification: Stable Higgs Bosons as Cold Dark Matter Yutaka Hosotani Department of Physics, Osaka University Toyonaka, Osaka 560-0043, Japan In the gauge-Higgs unification the 4D Higgs field becomes a part of the extra-dimensional component of the gauge potentials. In the SO(5) × U (1) gauge-Higgs unification in the Randall-Sundrum warped spacetime the electroweak symmetry is dynamically broken through the Hosotani mechanism. The Higgs bosons become absolutely stable, and become the dark matter of the universe. The mass of the Higgs boson is determined from the WMAP data to be about 70 GeV. Keywords: Higgs boson, dark matter, gauge-Higgs unification, Hosotani mechanism.

1. Introduction Does the Higgs boson exist? What is it really like? What constitutes the dark matter in the universe? These are two of the most important problems in current physics. We would like to point out that these two mystery particles are really the same. In the gauge-Higgs unification scenario Higgs bosons become absolutely stable, and become the dark matter of the universe. In the gauge-Higgs unification the 4D Higgs boson is identified with a part of the extra-dimensional component of the gauge potentials. Its couplings with others particles are controlled by the gauge principle. The 4D Higgs field corresponds to 4D fluctuations of an AB phase in the extra dimensions.1–3 In the SO(5) × U (1) gauge-Higgs unification model in the Randall-Sundrum warped space the AB phase θH takes exactly the value 12 π in the vacuum as a consequence of quantum dynamics. At this particular value of θH the 4D Higgs boson becomes absolutely stable. The relic abundance of the Higgs bosons in the present universe is evaluated definitively with the mass of the Higgs boson as the only relevant variable parameter. Astonishingly the average mass density of the dark matter determined from the WMAP data is obtained with the Higgs mass around 70 GeV. It does not contradict with the LEP2 bound, because the ZZH coupling vanishes at θH = 12 π. The gaugeHiggs unification scenario gives a completely new viewpoint for the Higgs boson.4 Further it gives definitive predictions for electroweak gauge couplings of quarks and leptons, which can be tested experimentally.5

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2. SO(5) × U (1) gauge-Higgs unification in RS We consider an SO(5)×U (1) gauge theory in the five-dimensional Randall-Sundrum (RS) warped spacetime.4−14 Its metric is given by ds2 = e−2σ(y) ηµν dxµ dxν + dy 2 ,

(1)

where ηµν = diag(−1, 1, 1, 1), σ(y) = σ(y + 2L), and σ(y) = k|y| for |y| ≤ L. The fundamental region in the fifth dimension is given by 0 ≤ y ≤ L. The Planck brane and the TeV brane are located at y = 0 and y = L, respectively. The bulk region 0 < y < L is an anti-de Sitter spacetime with a cosmological constant Λ = −6k 2 . The RS spacetime has the same topology as the orbifold M 4 × (S 1 /Z2 ). Vector potentials AM (x, y) of the gauge group SO(5) and BM (x, y) of U (1) satisfy the orbifold boundary conditions Aµ Aµ (x, yj − y) = Pj (x, yj + y)Pj−1 , Ay −Ay Bµ Bµ (x, yj − y) = (x, yj + y) , By −By Pj = diag(−1, −1, −1, −1, +1) , (j = 0, 1),

(2)

where y0 = 0 and y1 = L. By the boundary conditions the SO(5)×U (1) symmetry is reduced to SO(4)×U (1) SU (2)L ×SU (2)R ×U (1). The symmetry SU (2)R ×U (1) is further spontaneously broken by a scalar field Φ(x) on the Planck brane to U (1)Y . Φ(x) belongs to (0, 12 ) representation of SU (2)L × SU (2)R . The pattern of the symmetry reduction11,13 is depicted in fig. 1. With the above boundary conditions the zero modes of 4D gauge fields reside in the 4-by-4 matrix part of Aµ , whereas the zero modes of Ay in the off-diagonal part of Ay . The latter is an SO(4) vector, or an SU (2)L doublet, corresponding to the Higgs doublet in the standard model. See fig. 2.

Fig. 1.

The symmetry reduction in the SO(5) × U (1) gauge-Higgs unification in RS.

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Fig. 2. of Ay .

W, Z, γ appear as zero modes of Aµ whereas the 4D Higgs field appears as a zero mode

Bulk fermions for quarks and leptons are introduced as multiplets in the vectorial representation of SO(5).5,14 In the quark sector two vector multiplets are introduced for each generation. In the lepton sector it suffices to introduce one multiplet for each generation to describe massless neutrinos, whereas it is necessary to have two multiplets to describe massive neutrinos. Each vector multiplet Ψa is decomposed to ( 12 , 12 )⊕(0, 0) in the SU (2)L ×SU (2)R . They satisfy the orbifold boundary condition Ψa (x, yj − y) = Pj Γ5 Ψa (x, yj + y) ,

(3)

which gives rise to chiral fermions as zero modes. The left-handed components of ( 12 , 12 ) and the right-handed components of (0, 0) have zero modes. In addition to the bulk fermions, right-handed brane fermions χˆα are introduced on the Planck brane. The brane fermions χ ˆα belong to the ( 12 , 0) representation of SU (2)L × SU (2)R . These brane fermions are necessary both to have the realistic quark-lepton spectrum at low energies and to have the cancellation of 4D chiral anomalies associated to the SO(4) × U (1) gauge fields. The matter content is summarized in fig. 3. The SO(4) × U (1) gauge invariance is maintained on the Planck ˆα , and the brane scalar brane as well. The bulk fermions Ψa , the brane fermions χ Φ form SO(4) × U (1) invariant interactions. When Φ develops a non-vanishing expectation value to spontaneously break SU (2)R × U (1) to U (1)Y , it simultaneously ˆα which, in turn, make all exotic fermions gives mass couplings between Ψa and χ heavy. 3. Higgs boson as an AB phase As shown in fig. 2, the 4D Higgs field appears as a zero mode of Ay . Without loss of generality one can suppose that the A45 y component develops a nonvanishing expectation value. We write 4k ˆ ˆ h0 (y) · T 4 + · · · , zL = ekL (4) Ay (x, y) = θH (x) · 2 zL − 1 2 1/2 2ky (0 ≤ y ≤ L). where the zero-mode wave function is h0 (y) = √ [2k/(zL − 1)] e ˆ ˆ 4 4 The generator T is given by (Tvec )ab = (i/ 2)(δa5 δb4 − δa4 δb5 ) in the vectorial

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Fig. 3.

Matter content in the model.

√ ˆ 4 representation and Tsp = (1/2 2)I2 ⊗ τ1 in the spinorial representation. θˆH (x) is decomposed as 2 mKK k H(x) 2 √ . (5) θˆH (x) = θH + , fH = 2 −1 ∼ fH gA zL kL πg −1 Here the Kaluza-Klein (KK) mass scale is mKK = πkzL , and the √ SO(5) gauge coupling gA is related to the 4D SU (2)L weak coupling g by g = gA / L. With H(x) = 0, in the spinorial representation, L i θH I2 ⊗ τ1 dyAy = exp (6) P exp igA 2 0

so that the constant part θH of θˆH (x) represents an Aharonov-Bohm phase in the extra-dimension. θˆH is a phase variable. There remains the residual gauge invariance AM → AM = ΩAM Ω−1 − (i/gA )Ω∂M Ω−1 which preserves the boundary conditions (2) and (3).7,15 In general, new gauge potentials satisfy new boundary conditions given by Pj = Ω(x, yj − y)Pj Ω(x, yj + y)−1 . The residual gauge invariance is defined with Ω(x, y) satisfying Pj = Pj . Consider a large gauge transformation y ˆ 2 − 1) h (y) · T 4 dy 4k/(zL . (7) Ωlarge (y; α) = exp − iα 0 0

(x) = θˆH (x) + α. With α = 2π, Pj = Pj and θˆH = θˆH + 2π. which shifts θˆH (x) to θˆH It implies that physics is invariant under

θH → θH + 2π .

(8)

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4. Effective theory Four-dimensional fluctuations of the AB phase θH correspond to the 4D neutral Higgs field H(x). In the effective theory of the low energy fields the Higgs field H(x) enters always in the combination of θˆH (x) in (5). The effective Lagrangian must be invariant under θˆH (x) → θˆH (x) + 2π. The effective Higgs interactions with the W , Z bosons, quarks and leptons at low energies are summarized as4,14,16,17 ˆ − Leff = −Veff (θˆH ) − m2W (θˆH )Wµ† W µ − 12 m2Z (θˆH )Zµ Z µ − mF ab (θH )ψ a ψb . (9) a,b

Veff (θˆH ) is the effective potential which arises at the one loop level. It is finite and independent of the cutoff.1 From the second derivative at the global minimum a finite Higgs mass mH is obtained. The finiteness of m2H gives a solution to the gauge-hierarchy problem.18 ˆ The mass functions mW (θˆH ), mZ (θˆH ) and mF ab (θH ) arise at the tree level. It is essential to include contributions coming from KK excited states in intermedium states. The effective Lagrangian (9) is obtained after integrating out all heavy KK modes. In the SO(5) × U (1) model under consideration these mass functions are found to be, to good accuracy in the warped space,10,11,14

gauge-Higgs SM 1 1 ˆ ˆ gfH sin θH , g(v + H) , mW (θH ) ∼ 2 2 1 1 ˆ ˆ gfH sin θH , g(v + H) , mZ (θH ) ∼ 2 cos θW 2 cos θW

F F ˆ yab fH sin θˆH , yab (v + H) . (10) mF ab (θH ) ∼

We have listed the formulas in the standard model in brackets on the right. It is seen that v + H in the standard model is replaced approximately by fH sin θˆH in the gauge-Higgs unification. In the standard model the mass functions are linear in the Higgs field H, whereas they become periodic, nonlinear functions of H in the gauge-Higgs unification. It is a consequence of the phase nature of θH . In other words the gauge invariance in the warped space naturally leads to the nonlinear behavior. The masses of the W , Z bosons and fermions are given by mW = 12 gfH sin θH , F mZ = mW / cos θW , and mF ab = yab fH sin θH . An immediate consequence is that the Higgs couplings to W , Z, and fermions deviate from those in the standard model. In particular,   WWH  ZZH  = SM × cos θH , Yukawa

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W W HH = SM × cos 2θH . ZZHH

(11)

As we shall see below, the effective potential is minimized at θH = 12 π so that the W W Z, ZZH and Yukawa couplings vanish. 5. Dynamical EW symmetry breaking The value of θH in the vacuum is determined by the location of the global minimum of the effective potential Veff (θH ), which is given at the one loop level by 1 d4 p 2 2 (12) ± ln p + m (θ ) Veff (θH ) = n H 2 (2π)4 n particles

where {mn (θH )} is a 4D mass spectrum in each KK tower with the AB phase θH . As originally shown in ref. 1 the θH -dependent part of Veff (θH ) is finite, independent of how the theory is regularized. Evaluation of Veff (θH ) in the RS warped space was initiated by Oda and Weiler.19 Since then a powerful method for evaluation has been developed by Falkowski.20 In the model under consideration the electroweak symmetry SU (2)L × U (1)Y remains unbroken for θH = 0, π. Otherwise the symmetry is broken to U (1)EM . If there were no fermions, the symmetry is unbroken. Among the fermions, multiplets in the bulk containing the top quark gives a dominant contribution to Veff (θH ). The fact that the top quark mass (172 GeV) is larger than mW is relevant. Contributions from other fermions, whose masses are much smaller than mW , are negligible in the warped space. Contributions fom charm quarks, for instance, are suppressed by a factor 10−5 . The existence of the top quark triggers the electroweak symmetry breaking. U (θH ) = (16π 4 /m4KK ) Veff (θH ) is plotted with the warp factor zL = 1010 in fig. 4. It is seen that the effective potential is minimized at θH = ± 12 π so that the electroweak symmetry is dynamically broken. It follows from (10) that mW = 12 gfH and fH ∼ 256 GeV. As the warp factor zL = ekL is decreased, there appear two kinds of phase transitions. For zL < zL1 ∼ 2200 the top quark mass mt ∼ 171 GeV cannot be achieved. It necessarily becomes smaller than the observed value. One can set the bulk mass parameter to be zero, and further decrease the value of zL . Then, for zL ∼ zL2 ∼ 1.67 there appears a weakly-first-order phase transition. For zL < zL2 the global minima move to θH = 0 and π so that the symmetry is unbroken.a The curvature of the Veff at the minimum is related to the Higgs mass mH by π 2 g 2 kL d2 Veff −1 m2H = , mKK = πkzL . (13) 2 4 m2KK dθH min a There

was a numerical error in the evaluation of Veff (θH ) in Ref. 13.

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Fig. 4.

Effective potential Veff (θH ) = m4KK /(16π 4 )U (θH ) with zL = 1010 .

For zL = 1010 , mKK = 1.2 TeV and mH = 108 GeV. At first sight one might think that this value for mH contradicts with the LEP2 bound mH > 114 GeV. However, the effective potential is minimized at θH = ± 21 π where the ZZH coupling vanishes. See (11). The LEP2 bound is evaded as the process e+ e− → Z ∗ → ZH does not take place in the present model. 6. The absolute stability of the Higgs bosons In the present gauge-Higgs unification model the AB phase is dynamically chosen to be θH = 12 π at the one loop level. It implies that all ZZH, W W H and Yukawa couplings of quarks and leptons vanish so that the Higgs boson cannot decay. Can the Higgs boson decay at higher orders? We show that the Higgs boson becomes absolutely stable in a class of the gauge-Higgs unification models including the present model.4 (i) Mirror reflection symmetry The theory is invariant under the mirror reflection in the fifth dimension; (xµ , y) → (xµ , y ) = (xµ , −y) ,

AM (x, y) → AM (x , y ) = (Aµ , −Ay )(x, y) , Ψa (x, y) → Ψa (x , y ) = ±γ 5 Ψa (x, y) .

(14)

It implies that physics is invariant under H(x) → θˆH (x ) = −θˆH (x) θˆH (x) = θH + fH while all other SM particles remain unchanged.

(15)

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(ii) Enhanced gauge invariance One may notice that the effective potential Veff (θH ) at the one loop level depicted in fig. 4 has periodicity π. Indeed, this remains true to all order. Gauge invariance is enhanced. Consider a lare gauge transformation (7) with α = π. With Ωlarge (y; π) the vec vec boundary conditions change to (P0 , P1 ) = (P0 vec , P1 vec ) in the vectorial represp sp sentation and (P0 , P1 ) = (P0 sp , −P1 sp ) in the spinorial representation. The AB phase changes to θH = θH + π. In the present model all bulk fermions are in the vector representation. The brane fermions and scalar field on the Planck brane are not affected by this transformation as Ωlarge (0; π) = 1. Hence the theory is invariant under θH → θH + π. All physical quantities become periodic in θH with a reduced period π. (iii) H-parity In a class of the SO(5) × U (1) gauge-Higgs unification models in the warped space which contains bulk fermions only in tensorial representations of SO(5) and brane fermions only on the Planck brane, the enhanced gauge symmetry with the mirror reflection symmetry leads to Veff (θˆH + π) = Veff (θˆH ) = Veff (−θˆH ) , m2W,Z (θˆH + π) = m2W,Z (θˆH ) = m2W,Z (−θˆH ) , F ˆ F ˆ ˆ mF ab (θH + π) = −mab (θH ) = mab (−θH ) .

(16)

mW,Z (0) = mF ab (0) = 0 as the EW symmetry is recovered at θH = 0. In the previous section we have seen that Veff (θH ) is minimized at θH = 12 π. ˆ It follows then from (16) that all of Veff (θˆH ), m2W,Z (θˆH ) and mF ab (θH ) satisfy a −1 −1 1 1 relation F ( 2 π + fH H) = F ( 2 π − fH H). They are even functions of H when expanded around θH = ± 21 π. All odd-power Higgs couplings H 2+1 , H 2+1 Wµ† W µ , − H 2+1 Zµ Z µ , and H 2+1 ψ a ψb , vanish. The effective interactions at low energies are invariant under H(x) → −H(x) with all other fields kept intact at θH = ± 21 π. There arises the H-parity invariance. Among low energy fields only the Higgs field is H-parity odd. The Higgs boson becomes absolutely stable, protected by the H-parity conservation.

7. Stable Higgs bosons as cold dark matter Abosolutely stable Higgs bosons become cold dark matter (CDM) in the present universe. They are copiously produced in the very early universe. As the annihilation rate of Higgs bosons falls below the expansion rate of the universe, the annihilation processes get effectively frozen and the remnant Higgs bosons become dark matter.4

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The annihilation rates can be estimated from the effective Lagrangian (9) with (10). At θH = 12 π one has H 1 H − Leff ∼ − m2W Wµ† W µ + m2Z Zµ Z µ cos2 − ma ψ a ψa cos . (17) 2 fH fH a

As mW ∼ 12 gfH , fH is determined to be ∼ 246 GeV. Although the Yukawa − − coupling ψ a ψa H vanishes, the ψ a ψa H 2 coupling is nonvanishing, given by (n) (n) 2 − )ψ a ψa H 2 . It is generated by two vertices ψa ψa H where ψa is the n-th (ma /2fH KK excited state of ψa . The Higgs mass mH is predicted in the present model in the range of 50 GeV to 130 GeV, depending on the value of the warp factor. If mH > mW , the dominant annihilation modes are HH → W W, ZZ. The rate is large so that the resultant relic abundance becomes very small. For mH < mW the relevant annihilation modes c, τ τ¯, and gg. Here W ∗ (Z ∗ ) indicates virtual W (Z) are HH → W ∗ W ∗ , Z ∗ Z ∗ , b¯b, c¯ which subsequently decays into a fermion pair. Annihilation into a gluon (g) pair takes place through a top quark loop. The relic abundance of Higgs bosons is evaluated as a function of mH . It is depicted in fig. 5. It is seen that ΩH h2 determined from the WMAP data is reproduced in the gauge-Higgs unification with mH ∼ 70 GeV. It is remarkable that the gauge-Higgs unification scenario gives ΩH h2 in the just right order of magnitude for 20 GeV < mH < 75 GeV. With mH = 70 GeV, the freeze-out temperature is Tf ∼ 3 GeV. The relative contributions of the HH → W ∗ W ∗ and b¯b modes are 61% and 34%, respectively.

Fig. 5. Thermal relic density of Higgs boson DM with fH = 246 GeV. The solid curve is obtained by the semi-analytic formulae. The horizontal band is the WMAP data ΩCDM h2 = 0.1131±0.0034. The value in the WMAP data is obtained with mH ∼ 70 GeV.

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If Higgs bosons constitute the cold dark matter of the universe, they can be detected by observing Higgs-nucleon elastic scattering process, HN → HN . The 2 ) f mf f¯f . With relevant part of the effective interaction (17) is Leff = (H 2 /2fH QCD corrections taken into account the effective interaction for the Higgs-nucleon coupling becomes 2 + 7fN mN 2 (18) LHN 2 H NN , 9 2fH where fN = q=u,d,s fqN and N |mq q¯q|N = mN fqN . With this coupling (18) the spin-independent (SI) Higgs-nucleon scattering cross section is found to be 2 1 2 + 7fN m4N , (19) σSI 4 4π 9 fH (mH + mN )2 in the non-relativistic limit. There is ambiguity in the value of fN . Our prediction is σSI (1.2 − 2.7) × 10−43 cm2 for fN = (0.1 − 0.3). The present experimental upper bounds for the spin-independent WIMP-nucleon cross sections come from CDMS II21 and XENON10.22 From the recent CDMS II data σSI 7 × 10−44 cm2 at 90 % CL with the WIMP mass 70 GeV. With many uncertainties and ambiguity in the analysis taken into account, this does not necessarily mean that the present gauge-Higgs unification model is excluded. The next generation experiments for the direct detection of WIMP-nucleon scattering are awaited to pin down the rate. 8. Stable Higgs bosons at Tevatron/LHC/ILC In all of the experiments performed so far, Higgs bosons are searched by trying to identify their decay products. If Higgs bosons are stable, however, this way of doing experiments becomes a vain effort. Higgs bosons are produced in pairs. Typical processes are Z ∗ → ZHH, W ∗ → W HH, W W → HH, ZZ → HH, and gg → HH. Higgs bosons are stable so that they appear as missing energies and momenta in collider experiments. The appearance of two particles of missing energies and momenta in the final state makes experiments hard, but not impossible. These events must be distinguished from those involving neutrinos. Since mH ∼ 70 GeV, Higgs bosons can be copiously produced at LHC. It is a challenging task to identify stable Higgs bosons at colliders. The effects of the vanishing W W H and ZZH couplings can be seen in W W , W Z, and ZZ elastic scattering, too. These scattering amplitudes become large as the energy is increased, much faster than in the standard model.23 9. Gauge couplings of quarks and leptons We have seen large deviations in the Higgs couplings from the standard model. There arise deviations in the gauge couplings of quarks and leptons as well, which have to be scrutinized with the electroweak precision measuremens.5,8,9 5D gauge couplings of fermions are universal. However, 4D gauge couplings of quarks and leptons are obtained by inserting their wave functions and W (Z) wave

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function into 5D Lagrangian and integrating over the fifth coordinate. Since the wave functions depend on quarks and leptons, slight variations appear in 4D gauge couplings, resulting in violation of the universality. The wave functions of W and Z bosons are determined in refs. 10 and 11, whereas those of quarks and leptons are determined in refs. 5 and 14. There arise deviations in the W boson couplings from the standard model. For the t and b quarks their interactions are given by 1 (W ) 1 (W ) g (Wµ¯bL γ µ tL + Wµ† t¯L γ µ bL ) + gtb,R (Wµ ¯bR γ µ tR + Wµ† t¯R γ µ bR ). (20) 2 tb,L 2 Not only left-handed components but also right-handed components couple to W in general. The W coupling defined experimentally is the coupling between left-handed (W ) (W ) (W ) e and νe , geν,L . In the standard model the couplings are universal; gf,L = geν,L and (W )

gf,R = 0. The violation of the universality for the left-handed quarks and leptons, (W )

(W )

gf,L /geν,L − 1, is summarized in Table 1. The W couplings to the right-handed quarks and leptons are summarized in Table 2. One can see that the deviation from the standard model is extremely tiny except for top-bottom, well below the current limits. For the left-handed top-bottom quarks the deviation is about 2%. Similarly the Z boson couplings deviate from the standard model. The couplings of t and b quarks take the form 1 (Z) ¯ µ (Z) ¯ (Z)¯ (Z)¯ µ µ µ Zµ gtL tL γ tL + gtR tR γ tR + gbL bL γ bL + gbR bR γ bR . (21) cos θW (Z)

(W )

The relevant couplings are gf,LR /geν,L . They are summarized in Table 3, in which the couplings in the standard model are also listed for comparison. It is seen from the table, the deviation from the standard model is rather small (0.1% to 1%) Table 1. The deviation of the W couplings for left-handed leptons and quarks from the standard model for the warp factor zL = 1015 . (W )

νµ , µ −1.022 ×10−8

ντ , τ −2.736 ×10−6

(W )

gf L /geν,L − 1

u, d −2.778 ×10−11

c, s −1.415 ×10−6

t, b −0.02329

Table 2. The W couplings for right-handed leptons and quarks for the warp factor zL = 1015 . They are all tiny. (W )

(W )

gf R /geν,L νe , e −4.761 ×10−21 u, d −1.392 ×10−11

νµ , µ −4.141 ×10−16 c, s −1.896 ×10−7

ντ , τ −1.133 ×10−13 t, b −0.001264

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except for the top quark coupling for zL = 1015 . The Z couplings of the left- and right-handed top quark deviate from those in the standard model by −7% and 18%, respectively. The deviations in the ZbL¯bL and ZbR¯bR couplings are 0.3% and 0.9%, respectively. Table 3. The deviation of the Z couplings for leptons and quarks from the standard model for the warp factor zL = 1015 . In the column SM the values in the standard model are quoted. (Z)

(Z)

(W )

g˜f,LR = gf,LR /geν,L f (Z) g˜f,L (Z) g˜f R

νe 0.500822

νµ 0.500822

ντ 0.500822

SM 0.5

−5.754 ×10−31

−5.392 ×10−29

−1.840 ×10−27

e −0.2692

µ −0.2692

τ −0.2692

−0.2688

u 0.3464

c 0.3464

t 0.3204

0.3459

−0.1556

−0.1556

−0.1823

−0.1541

d −0.4236

s −0.4236

b −0.4241

−0.4229

(Z) g˜f L (Z) g˜f R

(Z) g˜f L (Z) g˜f R

(Z) g˜f L (Z) g˜f R

0.2334

0.07779

0.2333

0.2333

0.07777

0.07774

0.2312

0.07707

There results an important prediction for the forward-backward asymmetry in the e+ e− collisions on the Z pole as pointed out by Uekusa.24 The asymmetry is given by (Z) 2 (Z) 2 gf L − gf R 3 e f f f AF B = ALR ALR , ALR = (Z) 2 (Z) 2 . (22) 4 g + g fL

fR

The predictions are summarized in Table 4. It is exciting that the gauge-Higgs unification model gives better fits to the observed data for the asymmetry in b¯b and c¯ c than the standard model. CP violation, anomalous magnetic moment, and electric dipole moment in gauge-Higgs unification have been also discussed.25

10. Summary The SO(5) × U (1) gauge-Higgs unification in the Randall-Sundrum warped spacetime is promising. The Higgs boson becomes a part of the gauge fields in higher dimensions, and is unified with gauge bosons. The resultant 4D gauge couplings of

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Table 4. The forward-backward asymmetry on the Z pole in e+ e− collisions. The numbers in the gauge-Higgs unification scenario with the warp factor zL = 1015 are quoted from Uekusa, ref. 24. AfF B f c s b e µ τ

Experiment 0.0707 ± 0.0035 0.0976 ± 0.0114 0.0992 ± 0.0016 0.0145 ± 0.0025 0.0169 ± 0.0013 0.0188 ± 0.0017

Gauge-Higgs 0.07073 0.09950 0.09952 0.01511 0.01513 0.01515

Standard Model 0.0738 ± 0.0006 0.1034 ± 0.0007 0.1033 ± 0.0007 0.01627 ± 0.00023

quarks and leptons are close to those in the standard model. The Higgs couplings, on the other hand, deviate significantly from those in the standard model. In a large class of the SO(5) × U (1) gauge-Higgs unification models the Higgs boson becomes absolutely stable to all order in perturbation theory. In the evolution of the universe Higgs bosons become cold dark matter. From the WMAP data the Higgs boson mass is determined to be around 70 GeV. In collider experiments Higgs bosons are produced in pairs. They appear as missing energies and momenta. The way of searching for Higgs bosons must be altered. Acknowledgment This work was supported in part by Scientific Grants from the Ministry of Education and Science, Grant No. 20244028, Grant No. 20025004, and Grant No. 50324744. References 1. Y. Hosotani, Phys. Lett. B 126, 309 (1983). 2. Y. Hosotani, Annals Phys. 190, 233 (1989). 3. A. T. Davies and A. McLachlan, Phys. Lett. B 200, 305 (1988); Nucl. Phys. B 317, 237 (1989). 4. Y. Hosotani, P. Ko and M. Tanaka, Phys. Lett. B 680, 179 (2009) [arXiv:0908.0212 [hep-ph]]. 5. Y. Hosotani, S. Noda, and N. Uekusa, arXiv:0912.1173 [hep-ph] (to appear in Prog. Theoret. Phys.). 6. R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B 671, 148 (2003) [arXiv:hepph/0306259]. 7. Y. Hosotani and M. Mabe, Phys. Lett. B 615, 257 (2005) [arXiv:hep-ph/0503020]. 8. K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B 719, 165 (2005) [arXiv:hepph/0412089]. 9. Y. Hosotani, S. Noda, Y. Sakamura and S. Shimasaki, Phys. Rev. D 73, 096006 (2006) [arXiv:hep-ph/0601241].

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10. Y. Sakamura and Y. Hosotani, Phys. Lett. B 645, 442 (2007) [arXiv:hep-ph/0607236]. 11. Y. Hosotani and Y. Sakamura, Prog. Theor. Phys. 118, 935 (2007) [arXiv:hepph/0703212]. 12. A. D. Medina, N. R. Shah and C. E. M. Wagner, Phys. Rev. D 76, 095010 (2007) [arXiv:0706.1281 [hep-ph]]. 13. Y. Hosotani, K. Oda, T. Ohnuma and Y. Sakamura, Phys. Rev. D 78, 096002 (2008) [Erratum-ibid. D 79, 079902 (2009)] [arXiv:0806.0480 [hep-ph]]. 14. Y. Hosotani and Y. Kobayashi, Phys. Lett. B 674, 192 (2009) [arXiv:0812.4782 [hepph]]. 15. N. Haba, M. Harada, Y. Hosotani and Y. Kawamura, Nucl. Phys. B 657, 169 (2003) [Erratum-ibid. B 669, 381 (2003)] [arXiv:hep-ph/0212035]. 16. Y. Sakamura, Phys. Rev. D 76, 065002 (2007) [arXiv:0705.1334 [hep-ph]]. 17. G. F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, JHEP 0706, 045 (2007) [arXiv:hep-ph/0703164]. 18. H. Hatanaka, T. Inami and C. S. Lim, Mod. Phys. Lett. A 13, 2601 (1998) [arXiv:hepth/9805067]. 19. K. Oda and A. Weiler, Phys. Lett. B 606, 408 (2005). 20. A. Falkowski, Phys. Rev. D 75, 025017(2007). 21. Z. Ahmed et al. [CDMS Collaboration], Phys. Rev. Lett. 102, 011301(2009) [arXiv:0802.3530 [astro-ph]]; arXiv:0912.3592 [astro-ph.CO]. 22. J. Angle et al. [XENON Collaboration], Phys. Rev. Lett. 100, 021303(2008) [arXiv:0706.0039 [astro-ph]]. 23. N. Haba, Y. Sakamura and T. Yamash*ta, arXiv:0908.1042 [hep-ph]. 24. N. Uekusa, arXiv:0912.1218 [hep-ph]. 25. Y. Adachi, C. S. Lim and N. Maru, Phys. Rev. D 76, 075009 (2007) [arXiv:0707.1735 [hep-ph]]; Phys. Rev. D 80, 055025 (2009) [arXiv:0905.1022 [hep-ph]].

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The Limits of Custodial Symmetry∗ R. Sekhar Chivukulaa , Roshan Foadib and Elizabeth H. Simmonsc Department of Physics and Astronomy Michigan State University, East Lansing, MI 48824, USA E-mails: a [emailprotected], b [emailprotected], c [emailprotected] Stefano Di Chiara CP3-Origins, Campusvej 55, DK-5230 Odense M, Denmark E-mail: [emailprotected] We introduce a toy model implementing the proposal of using a custodial symmetry to protect the ZbL¯bL coupling from large corrections. This “doublet-extended standard model” adds a weak doublet of fermions (including a heavy partner of the top quark) to the particle content of the standard model in order to implement an O(4) × U (1)X ∼ SU (2)L × SU (2)R × PLR × U (1)X symmetry in the top-quark mass generating sector. This symmetry is softly broken to the gauged SU (2)L × U (1)Y electroweak symmetry by a Dirac mass M for the new doublet; adjusting the value of M allows us to explore the range of possibilities between the O(4)-symmetric (M → 0) and standard-model-like (M → ∞) limits.

1. Introduction Agashe2 et al. have shown that the constraints on beyond the standard model physics related to the ZbL¯bL coupling can, in principle, be loosened if the global SU (2)L ×SU (2)R symmetry of the electroweak symmetry breaking sector is actually a subgroup of a larger global symmetry of both the symmetry breaking and top quark mass generating sectors of the theory. In particular, they propose that these interactions preserve an O(4) ∼ SU (2)L × SU (2)R × PLR symmetry, where PLR is a parity interchanging L ↔ R. The O(4) symmetry is then spontaneously broken to O(3) ∼ SU (2)V ×PLR , breaking the elecroweak interactions but protecting gLb from radiative corrections, so long as the left-handed bottom quark is a PLR eigenstate. In this talk we report on the construction of the simplest O(4)-symmetric extension of the SM. For reasons that will shortly become clear, we call this model the doublet-extended standard model or DESM. Because the DESM is minimal, it displays the essential ingredients protecting gLb without the burden of additional states, interactions, or symmetry patterns that might otherwise obscure the role ∗ Speaker at conference: R. Sekhar Chivukula. This report is a shortened version of previously published work.1

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played by custodial O(3). Because it is concrete, it also enables us to explore how the new symmetry impacts the model’s ability to conform with the constraints imposed by other precision electroweak data. In our model, all operators of dimension-4 in the Higgs potential and the sector generating the top quark mass respect a global O(4) × U (1)X symmetry; the U (1)X enables the SM-like fermions to obtain the appropriate electric charges and hypercharges. In addition to the particle content of the SM, we introduce a new weak doublet of Dirac fermions, Ψ = (Θ, T ), and combine ΨL with the left-handed top-bottom doublet (tL , bL ) to form a (2, 2∗ ) under the global SU (2)L × SU (2)R symmetry. The bL state is thereby endowed with identical charges under the two global SU (2) groups, TL3 = TR3 , making it a parity eigenstate, as desired. We also find that the T mixes with t to form a SM-like top quark and a heavy partner. The O(4) × U (1)X symmetric Yukawa interaction can, of course, be extended to the bottom quark and the remaining electroweak doublets, by adding further spectator fermions; here we focus exclusively on the partners of the top quark since they give the dominant contribution to gLb . To enable electroweak symmetry breaking and fermion mass generation to proceed, the global symmetry is explicitly broken to SU (2)L × U (1)Y by a dimensionthree Dirac mass M for Ψ. As M → ∞ the ordinary SM top Yukawa interaction is recovered; as M → 0 the model becomes exactly O(4)×U (1)X symmetric; adjusting the value of M allows us to interpolate between these extremes and to investigate the limits to which the custodial symmetry of the top-quark mass generating sector can be enhanced. When we calculate the dominant one-loop corrections to gLb in our model we find, consistent with previous results,2 that because bL is a PLR eigenstate, gLb is protected from radiative corrections in the M → 0 limit and these corrections return as M is switched on. However, when we study the behavior of oblique radiative corrections as M is varied, we find that in the small-M limit where gLb is closer to the experimental value,3 the oblique corrections become unacceptably large. In particular, in the M → 0 limit the enhanced custodial symmetry produces a potentially sizable negative contribution to αT . 2. Doublet-Extended Standard Model 2.1. Custodial Symmetry and Z Coupling The tree-level coupling of a SM fermion ψ to the Z boson is, e ¯ µψ , (T 3 − Q sin2 θW ) Z µ ψγ cw sw L

(1)

where TL3 and Q are, respectively, the weak isospin and electromagnetic charges of fermion ψ, e is the electromagnetic coupling; cw and sw are the cosine and sine of the weak mixing angle. Because the electromagnetic charge is conserved, ¯ coupling do not alter it; however, the weak symmetry loop corrections to the Z ψψ SU (2)L is broken at low energies, and radiative corrections to the TL3 coupling are present in the SM.

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Following Agashe2 et. al., we wish to construct a scenario in which the TL3 coupling is not subject to flavor-dependent radiative corrections. To start, we note that the accidental custodial symmetry of the SM implies that the vectorial charge TV3 ≡ TL3 + TR3 is conserved δTV3 = δTL3 + δTR3 = 0 .

(2)

This suggests a way to evade flavor-dependent corrections to TL3 itself, by adding a parity symmetry PLR that exchanges L ↔ R. If ψ is an eigenstate of this parity symmetry and the symmetry persists at the energies of interest, then δTL3 = δTR3 .

(3)

Now, we see that Eq. (2) is satisfied by having the two terms on the right hand side vanish separately, rather than remaining non-zero and canceling one another. ¯ coupling remains fixed even to higher-order In other words, δTL3 = 0 and the Z ψψ in this scenario. We will now show how to implement this idea for the b-quark in a toy model and examine the phenomenological consequences. 2.2. The Model Let us construct a simple extension of the SM that implements this parity idea for the third-generation quarks, in order to suppress radiative corrections to the Zb¯b vertex. We extend the global SU (2)L × SU (2)R symmetry of the Higgs sector of the SM to an O(4) × U (1)X ∼ SU (2)L × SU (2)R × PLR × U (1)X for both the symmetry breaking and top quark mass generating sectors of the theory. As usual, only the electroweak subgroup, SU (2)L × U (1)Y , of this global symmetry is gauged; our model does not include additional electroweak gauge bosons. The global O(4) spontaneously breaks to O(3) ∼ SU (2)V × PLR which will protect gLb from radiative corrections, as above, provided that the left-handed bottom quark is a parity eigenstate: PLR bL = ±bL. The additional global U (1)X group is included to ensure that the light t and b eigenstates, the ordinary top and bottom quarks, obtain the correct hypercharges. In light of the extended symmetry group, the relationships between electromagnetic charge Q, hypercharge Y , the left- and right-handed T 3 charges, and the new charge QX associated with U (1)X are as follows: Y = TR3 + QX , Q=

TL3

+Y =

TL3

(4) +

TR3

+ QX .

(5)

Since the bL state is supposed to correspond to the familiar bottom-quark, it has the familiar SM charges TL3 (bL ) = −1/2, and Q(bL ) = −1/3, and Y (bL ) = 1/6. Because bL must be an eigenstate under PLR , we deduce that TR3 (bL ) = TL3 (bL ) = −1/2. Then to be consistent with Eqs. (4) and (5), its charge under the new global U (1)X must be QX (bL ) = 2/3. Moreover, since the left-handed b quark is an SU (2)L partner of the left-handed t quark, the full left-handed top-bottom doublet must

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have the charges TR3 = −1/2 and QX = 2/3, just as the full doublet has hypercharge Y = 1/6. Finally, the non-zero T3R charge of the top-bottom doublet tells us that this doublet forms part of a larger multiplet under the SU (2)L × SU (2)R symmetry and it will be necessary to introduce some new fermions with TR3 = 1/2 to complete the multiplet. We therefore introduce a new doublet of fermions Ψ ≡ (Ω, T ). The left-handed component, ΨL joins with the top-bottom doublet qL ≡ (tL , bL ) to form an O(4) × U (1)X multiplet tL Ω L (6) QL = ≡ qL ΨL , bL T L which transforms as a (2, 2∗ )2/3 under SU (2)L × SU (2)R × U (1)X . The parity operation PLR , which exchanges the SU (2)L and SU (2)R transformation properties of the fields, acts on QL as: TL −ΩL T PLR QL = − [(iσ2 ) QL (iσ2 )] = (7) −bL tL

exchanging the diagonal components, while reversing the signs of the off-diagonal components. Thus tL and TL are constrained to share the same electromagnetic charge, in order to satisfy Eq. (5). In fact, we will later see that the t and T states mix to form mass eigenstates corresponding to the top quark (t) and a heavy partner (T ). The charges of the components of QL are listed in Table 1. Table 1. Charges of the fermions under the various symmetry groups in the model. Note that, as discussed in the text, 3 and Q other TR X assignments for the ΩR and TR states are possible. TL3 3 TR Q Y QX

tL

1 2 − 12 2 3 1 6 2 3

bL − 21 − 21 − 31 1 6 2 3

ΩL 1 2 1 2 5 3 7 6 2 3

TL − 12 1 2 2 3 7 6 2 3

tR 0 0 2 3 2 3 2 3

bR 0 −1 − 13 − 13 2 3

TR − 12 0

5 3 7 6 7 6

2 3 7 6 7 6

ΩR 1 2

We assign the minimal right-handed fermions charges that accord with the symmetry-breaking pattern we envision: the top and bottom quarks will receive mass via Yukawa terms that respect the full O(4) × U (1)X symmetry, while the exotic states will have a dimension-three mass term that explicitly breaks the large symmetry to SU (2)L × U (1). Moreover, to accord with experiment, the tR and bR must have TL3 = 0 and share the electric charges of their left-handed counterparts. The top and bottom quarks will receive mass through a Yukawa interaction with a SM-like Higgs multiplet that breaks the electroweak symmetry. The simplest choice is to assign the Higgs multiplet to be neutral under U (1)X ; in this case, both tR and bR share the QX = 2/3 charge of tL and bL . Therefore, from Eqs. (4) and (5),

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we find TR3 (tR ) = 0 (meaning that tR can be chosen to be an SU (2)R singlet) and TR3 (bR ) = −1 (so that bR is, minimally, part of an SU (2)R triplet if we extend the symmetry to the bottom quark mass generating sector). Turning now to the TR and ΩR states, we see that they must form an SU (2)L doublet with hypercharge 7/6 so that the Dirac mass term for Ψ preserves the electroweak symmetry as desired.a Finally, we choose TR3 (ΩR ) = TR3 (TR ) = 0, which implies QX = 7/6 for both states, as the minimal choice satisfying the constraint imposed by Eq. (4); other choices of TR3 charge would involve adding additional fermions to form complete SU (2)R multiplets. The charges of the fermions are listed in Table 1. Now, let us describe the symmetry-breaking pattern and fermion mass terms explicitly. Spontaneous electroweak symmetry breaking proceeds through a Higgs multiplet that transforms as a (2, 2∗ )0 under SU (2)L × SU (2)R × U (1)X : √ 1 v +√h + iφ0 i 2 φ+ Φ= √ . (8) i 2 φ− v + h − iφ0 2 Again, the parity operator PLR exchanges the diagonal fields and reverses the signs of the off-diagonal elements. When the Higgs acquires a vacuum expectation value, the longitudinal W and Z bosons acquire mass and a single Higgs boson remains in the low-energy spectrum. The Higgs multiplet has an O(4) × U (1)X symmetric Yukawa interaction with the top quark: (9) LYukawa = −λt Tr QL · Φ tR + h.c.

that contributes to generating a top quark mass. In principle, the same Higgs multiplet can also contribute to the bottom quark mass through a separate, and similarly O(4) × U (1)X symmetric, Yukawa interaction involving the SU (2)R triplet to which bR belongs. Since the phenomenological issues that concern us in this paper are affected far more strongly by mt than by the far-smaller mb , we will neglect this and any other Yukawa interaction. Next we break the full O(4) × U (1)X symmetry to its electroweak subgroup. We do so first by gauging SU (2)L × U (1)Y . In addition, we wish to preserve the O(4) symmetry of the top quark mass generating sector in all dimension-4 terms, but break it softly by introducing a dimension-3 Dirac mass term for Ψ, ¯ L · ΨR + h.c. Lmass = −M Ψ

(10)

that explicitly breaks the global symmetry to SU (2)L × U (1)Y . Note that we therefore expect that any flavor-dependent radiative corrections to the ZbL¯bL coupling will vanish in the limit M → 0, as the protective parity symmetry is restored; alternatively, as M → ∞, the larger symmetry is pushed off to such high energies that the resulting theory looks more and more like the SM. states do not fill out the SU (2) triplet to which b means that the ΩR and TR R R belongs – which is uncharged under SU (2)L and carries hypercharge 2/3; other exotic fermions must play that role if we wish to extend the symmetry to the bottom quark mass generating sector.

a This

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In addition to the fermions explicitly described above, a more complete version of this toy model must contain several other fermions to fill out the SU (2)R multiplet to which the bR belongs and also some spectator fermions that cancel U (1)Y anomalies. However, the toy model suffices for exploration of the issues related to the ZbL¯bL coupling that is the focus of this paper. 2.3. Mass Matrices and Eigenstates When the Higgs multiplet acquires a vacuum expectation value and breaks the electroweak symmetry, masses are generated for the top quark, its heavy partner T and the exotic fermion Ω through the mass matrix: m 0 tR ¯ L ΩR + h.c , (11) − MΩ Lmass = − tL TL TR mM

where

λt v m= √ . 2

(12)

Note that the Ω field is decoupled from the SM sector, and its mass is simply mΩ = M . The bottom quark remains massless because we have ignored its Yukawa coupling. Diagonalizing the top quark mass matrix yields mass eigenstates t (corresponding to the SM top quark) and T (a heavy partner quark), with corresponding eigenvalues 1 4m4 1 4m4 2 2 2 2 1− 1+ 1+ 1+ mT = M +m , M 2 +m2 . (13) mt = 2 M4 2 M4 The mass eigenstates are related to the original gauge eigenstates through the rotations: tR cos θR sin θR tR tL cos θL sin θL tL = , = , TR − sin θR cos θR TR TL − sin θL cos θL TL (14) whose mixing angles are given by 1 1 − 2m2 /M 2 1 1 1− , sin θL = √ 1− . (15) sin θR = √ 4 4 2 2 1 + 4m /M 1 + 4m4 /M 4

From these equations the decoupling limit M → ∞ is evident: mt approaches its SM value as in Eq. (12), the t − T mixing goes to zero, and T becomes degenerate with Ω. Conversely, in the limit M → 0, the full O(4) × U (1)X symmetry is restored and only the combination TL + tL couples to tR with mass m. For phenomenological discussion, it will be convenient to fix mt at its experimental value and express the other masses in terms of mt and the ratio µ ≡ M/m. Fig. 1 shows how m, M , and mT , vary with µ; the horizontal line represents mt which is being held fixed at 172 GeV. In the limit as µ becomes large, m → mt , mT ∼ M grows

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steadily, and the mixing angles decline toward zero; this is a physically-sensible limit that ultimately leads back to the SM. However we see that the opposite limit, where µ → 0 can only be achieved for m → ∞, which is not physically reasonable since it corresponds to taking λt → ∞. Hence, we will need to take care in talking about the case of small µ.

1200

Mass GeV

1000 mT

800 m

600 400

M 200 0

mt 0

1

2

3

4

Μ

Fig. 1. The curves show the behaviors of m (dotted), M (dashed), and mT (upper solid) as functions of µ ≡ M/m when mt is held fixed. The solid horizontal line corresponds to mt 172 GeV.

3. Phenomenology bL 3.1. Z Coupling to bL¯ We are now ready to study how the flavor-dependent corrections to the ZbL¯bL coupling behave in our toy model. Specifically, if we write the ZbL¯bL coupling as e 1 1 (16) − + δgLb + sin2 θL Zµ ¯bL γµ bL , cw sw 2 3 then all the flavor-dependence is captured by δgLb . At tree-level, the ZbL¯bL coupling in our model has its SM value, with δgLb = 0, because the bL has the same quantum numbers as in the SM. However, at one-loop, flavor-dependent vertex corrections arise and these give non-zero corrections to δgLb ; these corrections differ from those in the SM due to the presence of vertex corrections involving exchange of T , the heavy partner of the top quark. The calculation may be done conveniently in the “gaugeless” limit,4–7 in which the Z boson is treated as a non-propagating external field coupled to the current µ µ − jQ sin2 θL . Operationally, this involves replacing Zµ with ∂µ φ0 /mZ in the j3L gauge current interaction, where φ0 is the Goldstone boson eaten by the Z: e e µ µ µ µ Zµ (j3L − jQ sin2 θL ) → ∂µ φ0 (j3L − jQ sin2 θL ) cw sw cw sw m Z 2 µ µ = ∂µ φ0 (j3L − jQ sin2 θL ) (17) v

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The general vertex diagram shown in Fig. 2, will yield radiative corrections to the effective operator ∂µ φ0 ¯bL γ µ bL ; that is, the expression for this diagram will include a term of the form A ∂µ φ0 ¯bL γ µ bL .

(18)

Comparing the last three equations shows that the coefficient A is proportional to the quantity we are interested in: δgLb =

v A. 2

ti

(19)

b

πZ φ− t¯j

¯b

Fig. 2. One-loop vertex correction diagram for πZ → b¯b in our model. The ti,j may be either the top quark (t) or its heavy partner (T ).

We have calculated the several loop diagrams represented by Fig. 2 and obtain the following expression for δgLb :

m2T m2t 2 2 δgLb = cos 2θ + 2 sin2 θL cos2 θR − cos 2θL θ + sin θ cos L L R 2 2 16π v mt mT /mt sin 2θL sin 2θR (20) − 4 2 mT /m2t + 1 log(m2T /m2t ) mT /mt mT − sin 2θL sin 2θR − 2 sin 2θL , 2 4 mt m2T /m2t − 1 where the prefactor proportional to m2t is the SM result for this class of diagram. We expect to see δgLb vanish in the limit M → 0 as the parity symmetry is restored; this expectation is fulfilled, since mt → 0 in this limit. At the other extreme, for large M , we expect to find δgLb take on its SM value by having the factor within square brackets approach one. This may be readily verified if we take the equivalent limit as µ → ∞ for fixed mt :

m2t log(1/µ2 ) 4 + O(1/µ ) , (21) δgLb (µ → ∞) → 1+ 16π 2 v 2 2µ2

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since in this limit sin θL → 1/µ2 , sin θR → 1/µ and m2T /m2t → µ2 . In other words, we find that adjusting the value of M allows us to interpolate between the SM value for δgLb at large M and the absence of a radiative correction at small M . While the limit of small µ is less useful, as we mentioned earlier, for completeness we note that δgLb (µ → 0) →

3 log(2/µ) − 1 m2t 2 6 + log(2/µ) 4 + µ + O(µ ) . 16π 2 v 2 2 8

(22)

√ since in this limit sin θL → (1/ 2)(1 − µ2 /4), sin θR → (1 − µ2 /8), and m2T /m2t → 4/µ2 . This growth at small µ is visible in Fig. (3). We now use our results to compare the value of gLb in our model (as a function of µ for fixed mt ) with the values given by experiment and the SM, as illustrated ex = −0.4182 ± 0.0015 corresponds to the in Fig. (3). The experimental8 value gLb thick horizontal line; the thin (red) horizontal lines bordering the shaded band show the ±1σ deviations from the experimental value. We calculated the SM value SM = using ZFITTER9,10 with a reference Higgs mass mh = 115 GeV, and obtain gLb −0.42114. This is indicated by the dashed horizontal line, and may be seen to ex by 1.96σ. The (solid blue) curve shows how gLb varies with µ in deviate from gLb our model; we required gLb to match the SM value with mt = 172 GeV and v = 246 GeV as µ → ∞ and the shape of the curve reflects our results for δgLb in Eq. (21). We see that gLb in our model is slightly more negative than (i.e. slightly farther from the experimental value than) the SM value for µ > 1, agrees with the SM value for µ = 1, and comes within ±1σ of the experimental value only for µ < 1. Given the shortcomings of the small-µ limit, this is disappointing.

Fig. 3. The solid (blue) curve shows the DESM model’s prediction for gLb , Eq. (21). The thick ex = −0.4182, while the two horizontal upper and lower solid lines horizontal line corresponds to gLb bordering the shaded band correspond to the ±1σ deviations.8 The SM prediction is given by the dashed horizontal line. The leading-log contribution is shown by the dotted curve.

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3.2. Oblique Electroweak Parameters The flavor-universal corrections from new physics beyond the SM can be parametrized in a model independent way using the four oblique EW parameters αS, αT, αδ, ∆ρ; the first two are the oblique paramters11–13 for models without additional electroweak gauge bosons, while the other two incorporate the effects of an extended electroweak sector. In general, the oblique parameters are related as follows14,15 to the neutral-current −MN C = 4πα

QQ (T 3 − s2w Q)(T 3 − s2 Q )

+ 2 2 2 sw cw S P 2 + √1 P 1 − αT + − 4πα 16π 4 2G F

αδ 4s2w c2w

(23)

√ √ αδ + 2GF 2 2 T 3 T 3 + 4 2GF (∆ρ − αT ) (Q − T 3 )(Q − T 3 ), sw cw

and charged-current electroweak scattering amplitudes −MCC =

s2w 4πα

(T + T − + T − T + )/2

S 1 P 2 + 4√2G 1+ − 16π F

αδ 4s2w c2w

√ αδ (T + T − + T − T + )

+ 2GF 2 2 , sw cw 2

(24)

with P 2 a Euclidean momentum-squared. In the DESM we may set ∆ρ = αT , because the model contains no extra U (1) gauge group, and δ = 0, because there is no extra SU (2) gauge group. We therefore work purely in terms of αS and αT from here on. We take the origin of the αS, αT parameter space to correspond to the SM with mH = 115 GeV; this ensures that any non-zero prediction for the oblique parameters for a Higgs of this mass arises from physics beyond the SM. At the one-loop level, the only new contributions to αS and αT in the DESM come from heavy fermion loops in the vacuum polarization diagrams indicated in Figure 4. We therefore expect αS and αT to be of order a few percent.b In this section, we will first separately derive expressions for αT and αS in DESM and see how each compares to current constraints from.3 We then compare the DESM’s joint prediction for αS and αT as a function of µ to the region of the αS − αT plane that gives the best fit to existing data3 and thereby derive a 95% confidence level lower bound on µ. 3.2.1. Parameter αT The custodial-symmetry-breaking parameter αT is defined as11

ΠW W (0) ΠZZ (0) αT = − , 2 MW MZ2

(25)

b There are, in principle, additional oblique parameters such as αU that arise at higher order. These will be suppressed relative to αS or αT by a factor of order m2Z /m2T ; since we can see from Figure 1 that mT > 2mt , the suppression is by at least an order of magnitude and we shall neglect αU and its ilk from here on.

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Fig. 4. Vacuum polarization diagram contributing to the oblique electroweak parameters. The indices i, j = 1, 2, 3, Q, refer to weak (i = 1, 2, 3) or electromagnetic (Q) generators, while f, f run over the appropriate combinations of t, b, T and Ω.

where the contributions proportional to g µν in the vacuum polarization diagrams of Fig. (4) for the W and Z are labeled ΠW W and ΠZZ , respectively. Each contribution ¯ T Ω; ¯ while for Z, sums over various f f¯ pairs – for W we have f f¯ = t¯b, T ¯b, tΩ, ¯ ¯ ¯ ¯ ¯ ¯ ¯ we have f f = tt, T T , tT , ΩΩ, bb, bΩ. 2

ΑT103

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Fig. 5. The solid (blue) curve shows the DESM model’s prediction for αT th as a function of µ. The horizontal lines show the optimal fit value of αT = 0.16 × 10−3 (thick solid line) and the relative ±1σ deviations3 (solid lines bordering the shaded band). The line αT = 0 corresponds to the SM value (with mh = 115 GeV), by definition. The leading-log contribution to αT is shown by the dotted curve.

The analytical result for αT DESM cannot be written in compact form; the resultc in the limit µ >> 1 is: 3m2t 22 ln µ2 DESM αT = 1−4 2 + 2 . (26) 16π 2 v 2 µ 3µ One can see that, for µ → ∞, Eq. (26) reproduces the leading SM result αT SM (mt ) = 3m2t /(4πv)2 ,11 as expected. It interesting to note that the leading c This

is consistent with Eq. (33) in Pomarol21 et. al., when only the contributions from new fermions are included (cL = 0).

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log contribution arising from the heavy states reducesd the value of αT . This is to be expected, since the custodial symmetry is enhanced in the small-µ limit and αT measures the change in the amount of isospin violation relative to the standard model. Subtracting the SM contribution from the top-quark, the numerical value of αT th = αT DESM − αT SM (mt ),

(27)

as a function of µ is plotted as the solid blue curve in Fig. (5); the dotted curve shows just the leading-log term (second term of Eq. (26)). The thick solid horizontal line corresponds to the best-fit value of αT = 0.16 × 10−3 obtained by ref.3 when setting U = 0; the two horizontal solid lines bordering the shaded band show the relative ±1σ deviations from that central fit value. Unlike the case of δgLb , the experimental constraints on αT clearly favor large values of µ, closer to the SM limit. By way of comparison, it is interesting to note that in an extra-dimensional model19,20 where a SM-like weak-singlet top quark was in the same SO(5) multiplet as extra quarks forming a weak doublet, radiative corrections produced an experimentally-disfavored large negative contribution to αT at one loop. Given that the SO(5) multiplet in 4D includes an SO(4) = SU (2)L × SU (2)R bi-doublet, our results are consistent with theirs. 3.2.2. Parameter αS The parameter S is defined11 as

d d αS = 16πα Π33 (0) − Π3Q (0) , dq dq

(28)

where q is the gauge boson momentum. The complete expression for αS DESM cannot be written in compact form; the limiting case where µ >> 1 is given by: 1 8 mb + 2 (2 − ln µ) , (29) αS DESM = 3 + 2 ln 6π mt µ

where we reintroduce a non-zero mass for the b quark to cut off a divergence in the integral over the fermion loop momenta. One can check that Eq. (29) reproduces the SM result αS SM (mt , mb )11 for µ → ∞. Defining αS th = αS DESM (µ) − αS SM (mt , mb ) ,

(30) −3

we plot the result in Fig. (6), along with the value, αS = 0.31 × 10 , that provides an optimal fit to the data (for U = 0) and the ±1σ relative deviations.3 From Fig. (6) one can see that αS is within the ±1σ bounds unless µ < 3; as with αT , smaller values of µ are disfavored, though the constraint in this case is less severe. d This does not violate the theorem16,17 stating that ∆ρ ≥ 0 when mixing occurs only between particles of the same T 3 and hypercharge. In the DESM, there is significant mixing between the tL and TL which have different T 3 and hypercharge values. As a result, we also expect significant GIM violation in the third generation.

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ΑS103

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0.0

0.5

1.0

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Fig. 6. The solid curve shows the DESM model’s prediction for αS th as a function of µ. The horizontal lines show the optimal fit value of αS = −0.31 × 10−3 (thick solid line) and the relative ±1σ deviations (solid lines bordering the shaded band) from.3 The line αS = 0 corresponds to the SM value (with mh = 115 GeV), by definition. The leading-log contribution to αS is shown by the dotted curve.

3.3. The αS– αT Plane In Figure 7 we show the DESM predictions for [αS th (µ), αT th (µ)] from Eqs. (30, 26) using mh = 115 GeV, and illustrating the successive mass-ratio values µ = 3, 4, ..., 20, ∞; the point µ = ∞ corresponds to the SM limit of the DESM and therefore lies at the origin of the αS - αT plane. On the same plane we also plot the elliptical curves that define the 95% confidence level (CL) bounds on the αS αT plane, relative to the optimal values of αS and αT found in.3 Using the best-fit values3 and corresponding ±1σ deviations for mh = 115 GeV, 300 GeV, along with the correlation matrix, we obtained the approximate values appropriate to mh = 1 TeV by extrapolating based on the logarithmic dependence of αS and αT on mh . To calculate the 95% CL ellipses, we solved the equation ∆χ2 = χ2 − χ2min = 5.99, as appropriate to the χ2 probability distribution for two degrees of freedom. From this figure, we observe directly that the 95%CL lower limit on µ for mh = 115 GeV is about 20, while for any larger value of mh the DESM with µ ≤ 20 is excluded at 95%CL. In other words, the fact that a heavier mh tends to worsen the fit of even the SM (µ → ∞) to the electroweak data is exacerbated by the new physics contributions within the DESM. The bound µ ≥ 20 corresponding to a DESM with a 115 GeV Higgs boson also implies, at 95%CL, that mT ≥ µ mt ∼ = 3.4 TeV, so that the heavy partners of the top quark would likely be too heavy for detection at LHC. 4. Conclusions We have introduced the doublet-extended standard model (DESM) as a simple realization of the idea2 of using custodial symmetry to protect the ZbL¯bL coupling (gLb ) from receiving large radiative corrections. In this toy model, all terms of dimension-

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3

2

1

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ΑS103

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Μ3

Fig. 7. The dots represent the theoretical predictions of the DESM (with mh set to the reference value 115 GeV), showing how the values of αS and αT change as µ successively takes on the values 3, 4, 5, ..., 20, ∞. The three ellipses enclose the 95%CL regions of the αS - αT plane for the fit to the experimental data performed in;3 they correspond to Higgs boson mass values of mh = 115 GeV, 300 GeV, and 1 TeV. Comparing the theoretical curve with the ellipses shows that the minimum allowed value of µ is 20, for mh = 115 GeV.

4 in the top-quark mass generating sector obey a global O(4) × U (1)X symmetry, which includes a parity symmetry protecting gLb from radiative corrections. That global symmetry is softly broken to its SU (2)L × U (1)Y subgroup by a Dirac mass term for the extra fermion doublet that incorporates the heavy partner of the top quark. Varying the size of this Dirac mass M allows the model to interpolate between the O(4) × U (1)X -symmetric case (M = 0) in which δgLb = 0 and the SM-like case (M → ∞) in which the one-loop corrections to gLb are as in the SM, and enabled us to investigate the degree to which the custodial symmetry of the top-quark mass generating sector can be enhanced. By comparing the predictions of the DESM with experimental constraints on the oblique parameters αS and αT from,3 we found the DESM to be consistent with experiment only for µ > 20 at 95%CL, with a Higgs mass mh = 115 GeV. The bound on µ translates into the 95% CL lower bound of 3.4 TeV on the masses of the extra quarks – placing them out of reach of the LHC. This result demonstrates that electroweak data strongly limits the amount by which the custodial symmetry of the top-quark mass generating sector can be enhanced relative to the standard model. While we cannot discuss this here, the toy model reproduces the behavior of extra-dimensional models in which the extended custodial symmetry is invoked to control the size of additional contributions to αT and the ZbL¯bL coupling,19,20 while leaving the standard model contributions essentially unchanged. Finally, we also note that our results are also consistent with a general effective field-theory analysis,21 which confirms that the toy DEWSB model illustrates the electroweak physics operative in a broad class of models.

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Acknowledgments This work was supported in part by the US National Science Foundation under grant PHY-0354226. RSC and EHS acknowledge the support of the Aspen Center for Physics and the Institute for Advanced Study where part of this work was completed. References 1. R. Sekhar Chivukula, S. Di Chiara, R. Foadi and E. H. Simmons, Phys. Rev. D 80, 095001 (2009) [arXiv:0908.1079 [hep-ph]]. 2. K. Agashe, R. Contino, L. Da Rold and A. Pomarol, Phys. Lett. B 641, 62 (2006) [arXiv:hep-ph/0605341]. 3. C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008). 4. R. Barbieri, M. Beccaria, P. Ciafaloni, G. Curci and A. Vicere, Phys. Lett. B 288, 95 (1992) [Erratum-ibid. B 312, 511 (1993)] [arXiv:hep-ph/9205238]. 5. R. Barbieri, M. Beccaria, P. Ciafaloni, G. Curci and A. Vicere, Nucl. Phys. B 409, 105 (1993). 6. J. F. Oliver, J. Papavassiliou and A. Santamaria, Phys. Rev. D 67, 056002 (2003) [arXiv:hep-ph/0212391]. 7. T. Abe, R. S. Chivukula, N. D. Christensen, K. Hsieh, S. Matsuzaki, E. H. Simmons and M. Tanabashi, arXiv:0902.3910 [hep-ph]. 8. [ALEPH Collaboration and DELPHI Collaboration and L3 Collaboration and ], Phys. Rept. 427, 257 (2006) [arXiv:hep-ex/0509008]. 9. D. Y. Bardin, P. Christova, M. Jack, L. Kalinovskaya, A. Olchevski, S. Riemann and T. Riemann, Comput. Phys. Commun. 133, 229 (2001) [arXiv:hep-ph/9908433]. 10. A. B. Arbuzov et al., Comput. Phys. Commun. 174, 728 (2006) [arXiv:hepph/0507146]. 11. M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992). 12. G. Altarelli and R. Barbieri, Phys. Lett. B 253, 161 (1991). 13. G. Altarelli, R. Barbieri and S. Jadach, Nucl. Phys. B 369, 3 (1992) [Erratum-ibid. B 376, 444 (1992)]. 14. R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Lett. B 603, 210 (2004) [arXiv:hep-ph/0408262]. 15. R. Barbieri, A. Pomarol, R. Rattazzi and A. Strumia, Nucl. Phys. B 703, 127 (2004) [arXiv:hep-ph/0405040]. 16. M. B. Einhorn, D. R. T. Jones and M. J. G. Veltman, Nucl. Phys. B 191, 146 (1981). 17. A. G. Cohen, H. Georgi and B. Grinstein, Nucl. Phys. B 232, 61 (1984). 18. F. Braam, M. Flossdorf, R. S. Chivukula, S. Di Chiara and E. H. Simmons, Phys. Rev. D 77, 055005 (2008) [arXiv:0711.1127 [hep-ph]]. 19. M. S. Carena, E. Ponton, J. Santiago and C. E. M. Wagner, Nucl. Phys. B 759, 202 (2006) [hep-ph/0607106]. 20. M. S. Carena, E. Ponton, J. Santiago and C. E. M. Wagner, Phys. Rev. D 76, 035006 (2007) [arXiv:hep-ph/0701055]. 21. A. Pomarol and J. Serra, Phys. Rev. D 78, 074026 (2008) [arXiv:0806.3247 [hep-ph]].

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Higgs Searches at the Tevatron Kazuhiro Yamamoto [for the CDF and DØ Collaborations] Department of Physics, Osaka City University Osaka 558-8585, Japan We present the latest results on searches for the standard and beyond-the-standard model Higgs bosons in proton-antiproton collisions at √ = 1.96 TeV by the CDF and DØ experiments at the Fermilab Tevatron. No significant excess is observed above the expected background, and the cross section limits for the Higgs bosons are calculated. It is noticed that the standard model Higgs boson in the mass range 163 – 166 GeV/c2 is excluded at the 95% C.L.

1

Introduction

The Higgs mechanism plays a role of a cornerstone in the modern particle physics. It was originally proposed to explain the non-zero masses of the weak bosons while keeping the theory gauge-invariant in the standard model (SM). As a result of the spontaneous electroweak symmetry braking, the weak bosons acquire the masses and simultaneously a new fundamental scalar particle, the Higgs boson (H), appears. On the other hand in the minimal supersymmetric standard model (MSSM) which is one of the simplest extensions beyond the SM, two Higgs doublet fields result in five Higgs bosons denoted as h, H, A, H±. In spite of many experiments searching for these predicted particles, they have yet to be discovered. For the SM Higgs boson, the direct searches at the CERN LEP collider have set the lower limit on the Higgs boson mass (MH) to be 114.4 GeV/c2 at the 95% C.L. [1]. Also combining this limit with precision electroweak measurements provides the constraint on MH to be less than 186 GeV/c2 at the 95% C.L. [2]. The Tevatron experiments, CDF and DØ, are probing the Higgs boson in the most probable region, 100 < MH < 200 GeV/c2. 2

Experimental Apparatus

The Fermilab Tevatron collider is providing proton-antiproton collisions at √ = 1.96 TeV. The CDF and DØ detectors are general-purpose cylindrically symmetric detectors put into place at the Tevatron. Both detectors consist of a superconducting solenoid magnet, a precision tracking system, electromagnetic and hadronic calorimeters and muon spectrometers. The detailed description can be found elsewhere [3][4]. The accelerators and detectors keep running very well in stable condition. As of December 2009, the typical peak luminosity is 3 × 1032 cm-1s-1 with the week integration of 50 ~ 60 pb-1. The delivered luminosity to each detector is accumulated to be 7.4 fb-1, among of which 6.1 fb-1 is recorded on tape as the collision data.

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3

Search for the Standard Model Higgs Boson

There are three dominant processes for producing the SM Higgs boson at the Tevatron energy: gluon fusion (gg → H), associated production with a vector boson (qˉq(ʹ) → VH, V = W or Z), and vector boson fusion (qˉq (ʹ) →qˉq (ʹ)H). For the Higgs mass of 100 ~ 200 GeV/c2, the cross section of each process is predicted to be 0.2 ~ 1 pb, 0.01 ~ 0.3 pb, and 0.02 ~ 0.1 pb respectively [5]. As for the decay mode, H → bbˉ is dominant in the low mass case (MH < 140 GeV/c2), while H → WW becomes dominant for the high mass Higgs (MH > 140 GeV/c2). The Higgs searches are performed with the proper combinations of the above production and decay modes. Considering the huge backgrounds from QCD multijet production, it is appropriate to use VH production with H → bbˉ for the low mass Higgs because a high-PT lepton and/or missing ET coming from the vector boson decay allows us to trigger the events efficiently. On the other hand for the high mass Higgs, we take advantage of the large cross section of gg → H followed by H → WW. The analyses are optimized for every channel of the different topological signatures in terms of the decay mode and jet multiplicity. The event selections and analysis strategies of each Higgs search channel are described below. The data used here correspond to an integrated luminosity of 2.0 ~ 5.4 fb -1. The obtained results on the cross section upper limit are summarized in Table I. The further details can be found in Refs. [6] – [15]. We search for ZH → νˉνbbˉ by investigating the missing ET + bbˉ signature. Considering the charged leptons which escape from the detector, this signature is also sensitive to a ˉ The event selections used in the CDF and DØ portion of WH → νbbˉ and ZH → bb. analyses are similar. They are performed by vetoing events with isolated charged leptons (e or μ) and also by requiring large missing ET and 2 or 3 jets containing at least one btagged jet. For b-tagging, the secondary vertex tagger (SECVTX) and the jet probability tagger (JP) are utilized at CDF, while the neural network (NN) tagger is developed and used at DØ. The dominant backgrounds are W/Z + jets, tˉt , diboson (WW, WZ, ZZ), and QCD multi-jets. The multivariate discriminant technique is applied in order to effectively separate signal from background; the CDF uses a NN while the DØ uses a boosted decision tree (BDT). The event separation is performed in two steps; the first discriminant is trained to remove the multi-jet background, and the second discriminant separates the expected signal from the remaining SM background. The process WH → νb bˉ is also a very sensitive channel in the low mass region as ˉ The events are selected by requiring a single isolated high-pT well as ZH → νˉνb b. lepton, large missing ET and 2 or 3 energetic jets with at least one b-tagged jet. The dominant background in the selected events is Wb bˉ for the 2 jets sample and tˉt for the 3 jets sample. The other backgrounds include W/Z + jets, single top, diboson and non-W QCD jets. For the DØ analysis, a NN is employed to discriminate signal from background after the base selection. On the other hand for the CDF analysis, efforts to achieve better sensitivity are made by two analysis groups each of which employs a matrix element method (ME) and a NN respectively. The discriminant is optimized for each b-tag category. ˉ In the base Another important channel in the low mass region is ZH → b b. selection, events containing Z candidates reconstructed from e+e and μ+μ pairs are selected at first. Then, two or more energetic jets at least one of which is b-tagged are

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required. Backgrounds of this analysis are dominated by Z + jets, tˉt , WZ, ZZ, and QCD events where a hadron is misidentified as a lepton. In order to discriminate signal from background, a BDT is used in the DØ analysis, while a two dimensional NN is implemented in the CDF analysis, where Z + jets and tˉt backgrounds are separated out from signal simultaneously. The Higgs events of which final state includes a ττ pair also become significant by combining all relevant channels. For the DØ analysis, the following five channels: ZH(Z → ττ, H → qˉq), HZ(H → ττ, Z → qˉq), HW(H → ττ, W → qˉq ʹ), qˉq (ʹ) → Hq qˉ (ʹ)(H → ττ, vector boson fusion), and gg → H → ττ (gluon fusion) are considered. Identification of ττ decay is essential for this analysis, and events in which one τ decays leptonically and the other τ decays hadronically are selected in order to increase the signal to noise ratio. An isolated object consisting of one or three charged particles in a narrow cone is required as a signature of a hadronically decaying τ, and a NN is used for further τ identification. After selecting the τ rich sample, an analysis using a BDT is performed to separate the expected signal from the remaining backgrounds which are mainly tˉt , W/Z + jets, and QCD multi-jets. In the high mass region, H → WW* → ν νˉ is the most sensitive channel at the Tevatron. The most dominant process for producing the Higgs boson in this channel is gluon fusion (gg → H). The other contribution comes from associated production with a vector boson (WH, ZH) and vector boson fusion (q qˉ (ʹ)H). The event selection is performed by requiring high-PT opposite-sign dilepton with large missing ET. The main backgrounds are tˉt , Drell-Yan, Wγ, and W + jets. Considering the scalar nature of the Higgs boson and the parity violation in W decays, two charged leptons from the Higgs decay tend to go in the same direction, which is different from the SM backgrounds. The final discriminants are NN outputs which are obtained by feeding a number of kinematic variables as inputs. The opening angle between two leptons mentioned above is one of the effective input variables. The NN training is carefully performed for the separated subsamples; the CDF analysis separates the sample by the jet multiplicity (0 jet, 1 jet, 2 or more jets), while the DØ analysis makes classification by dilepton flavor (ee, μμ, eμ). Also the low dilepton mass region (M < 16 GeV/c2) and the same-sign dilepton + jets sample (WH → WWW*, ZH → ZWW*) are added to increase sensitivity. Due to the different background composition, the dedicated NN training is performed separately. All the available results on the Higgs boson searches are combined into a single limit on the Higgs boson production cross section at the 95% C.L. for each Higgs boson mass ranging from 100 GeV/c2 to 200 GeV/c2. The combination process is made in two steps; the results of various analysis channels are combined in each experiment of CDF and DØ separately at first [15][16], and then the both results are combined into the single Tevatron combination [17]. In calculating the combined result, systematic correlations among the analysis channels and also those between experiments are carefully taken into account. The obtained limit is presented as a ratio to the predicted cross section by the SM in Fig. 1. We obtain the observed (expected) values of 2.70 (1.78) at MH = 115 GeV/c2 and 0.94 (0.89) at MH = 165 GeV/c2. We exclude the SM Higgs boson in the mass range 163 GeV/c2 to 166 GeV/c2, while the corresponding expected excluded region is 159 GeV/c2 to 169 GeV/c2.

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TABLE I: Observed (Expected) limits at the 95% C.L. on the production cross section as a ratio to the SM Higgs cross section. The limits at MH = 115 GeV/c2 are presented for the first four processes, while 165 GeV/c2 for H → WW. Two analysis results are shown for the CDF WH → νbbˉ channel; one is a result by using a neural network (NN) and the other is obtained with a matrix element method (ME). Analysis channel

MH (GeV/c2)

VH →E/ T bbˉ

115

WH → νbbˉ

115

ZH → bbˉ

115 115 165

 

τ τ qˉq final state

H → WW

CDF

DØ

Observed (Expected) limit [σ/σSM]

L (fb-1)

Observed (Expected) limit [σ/σSM]

L (fb-1)

6.1 (4.2) [6] 5.3 (4.0), NN [8] 6.6 (4.1), ME [9] 5.9 (6.8) [11]

3.6

3.7 (4.6) [7]

5.2

4.3

6.9 (5.1) [10]

5.0

4.1

9.1 (8.0) [12] 27.0 (15.9) [13] 1.55 (1.36) [15]

4.2 4.9 5.4

Update in progress

1.23 (1.21) [14]

4.8

FIG. 1: Observed (solid line) and expected (dashed line) 95% C.L. upper limits on the ratio to the SM Higgs cross section as functions of the Higgs boson mass for the combined CDF and DØ analyses. The bands indicate the 68% and 95% probability regions where the limit is expected to fluctuate, in the absence of signal.

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4

Search for the Higgs Bosons Beyond the SM

The minimal supersymmetric standard model (MSSM) is one of the simplest extensions beyond the SM. The production cross section of the neutral Higgs bosons (ϕ = h, H, A) is enhanced by tan2β. At the Tevatron energy, the production cross section is 10 ~ 50 pb at tanβ = 40 [5], which is one order of magnitude larger than the SM. The main decay modes of ϕ are bbˉ (~90%) and ττ(~10%) in a wide mass range (100 ~ 200 GeV/c2). One of the most promising channels to search for the neutral MSSM Higgs bosons at the Tevatron is ϕ → ττ in combination with the Higgs production through gluon fusion (gg → ϕ) and bbˉ fusion (bbˉ → ϕ). Tau pairs are identified in decay modes of eμ, e + hadrons and μ + hadrons, where the hadronic τ decay is identified with the similar method as the SM Higgs search described in the previous section. The background is dominated by Z → ττ, diboson, tˉt and W + jets. Also certain portion of the background arises from Z → ee/μμ by misidentifying a lepton as a hadron. Figure 2 shows the combined result of the CDF and DØ data corresponding to an integrated luminosity of 1.0 ~ 2.2 fb -1. It presents the 95% C.L. upper limit on tanβ as a function of the A boson mass (MA) with an assumption of the typical MSSM scenario of the maximal stop mixing and μ = 200 GeV [19]. ˉ Three b-tagged jets are Another explorable channel of ϕ is gb → bϕ → bb b. required in the analysis, and a peak structure is searched for in the dijet mass spectrum above the large multi-jet background. Figure 3 shows the results released with 1.9 fb -1 at CDF [20] and 2.6 fb -1 at DØ [21]. The charged MSSM Higgs (H±) bosons can be investigated in top quark decays; a top quark decays into not only Wb but also H±b when H± is lighter than t. In this case, there would be some deviation from the SM prediction for the final states of tˉt decay. However, the results are consistent with the SM prediction at both CDF and DØ, and hence the upper limits are obtained as shown in Fig. 4 [22][23]. In some models beyond the SM, the neutral Higgs couplings to fermions are suppressed due to some specific mechanisms, thus the Higgs boson decay into vector bosons are significantly increased. This kind of “fermiophobic” Higgs boson mainly decays into γγ for the low mass region, while WW and ZZ become the main decay mode in the intermediate and high mass region. For simplicity, the benchmark scenario is A

FIG. 2: The 95% C.L. limits in the tanβ-MA plane for the typical MSSM scenario of the maximal stop mixing and μ = 200 GeV. The black line denotes the observed limit, the gray line denotes the expected limit, and the hatched regions indicate the ±1σ and ±2σ bands around the expectation.

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considered by assuming that the Higgs does not couple to any fermions but couples to electroweak bosons with the SM coupling strength. Figure 5 shows the results from CDF with 3.0 fb-1 and DØ with 4.2 fb-1 in γγ decay mode [24][25]. Both experiments exclude the fermiophobic Higgs boson with masses below ~110 GeV/c2.

(b)

(a)

FIG. 3: The 95% C.L. upper limits in the tanβ-MA plane for the MSSM scenario of maximal stop mixing and μ = 200 GeV, which is obtained from the analyses of gb → bϕ → bbbˉ channel. The left plot (a) is a result of the CDF analysis, and the right plot (b) is that of the DØ analysis.

(a)

(b)

FIG. 4: The left plot (a) shows the CDF result of the 95% C.L. upper limits on the branching fraction of t → H+b as a function of MH+ with an assumption that all H+’s decay into csˉ. The right plot (b) shows the DØ result of the excluded region in the tanβ-MH+ plane at the 95% C.L. for the no-mixing scenario.

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(b) (a)

FIG. 5: The 95% C.L. upper limits on the branching fraction for the fermiophobic Higgs boson decay to diphotons, as a function of the fermiophobic Higgs mass. The result on the left plot (a) is obtained at CDF and that on the right plot (b) is obtained at DØ.

5

Conclusions and Future Prospects

The Tevatron accelerators and the collider detectors, CDF and DØ, are performing very well. The integrated delivered luminosity as of December 2009 is 7.4 fb-1, while 6.1 fb-1 is acquired as the collision data on tape. The Higgs boson searches are in progress in the various production and decay channels. The preliminary results with up to 5.4 fb-1 of data are presented in this report. For the SM Higgs boson, the combined results of CDF and DØ exclude the mass range from 163 to 166 GeV/c2 at the 95% C.L., while the expected exclusion range is 159 to 168 GeV/c2. On the other hand for the Higgs bosons beyond the SM, no sign of discovery has been observed yet. However, the sensitivity is steadily increasing. By the end of FY2010, the luminosity is expected to be accumulated up to 9 fb-1, and the sensitivity for seeing the Higgs boson will significantly increase.

Acknowledgements I would like to thank the members of the CDF and DØ collaborations for their work and effort in achieving the results presented in this report. References 1. The ALEPH, DELPHI, L3 and OPAL Collaborations, and the LEP Working Group for Higgs Boson Searches, Phys. Lett. B 565, 61 (2003).

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2. The ALEPH, CDF, DØ, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group, the Tevatron Electroweak Working Group, and the SLD Electroweak and Heavy Flavor Groups, arXiv:0911.2604 [hep-ex] (2009). 3. T. Acosta et al. (CDF Collaboration), Phys. Rev. D 71, 032001 (2005). 4. V. M. Abazov et al. (DØ Collaboration), Nucl. Instr. Meth. A565, 463 (2006). 5. TeV4LHC Higgs Working Group, arXiv:hep-ph/0612172 (2007). 6. CDF Collaboration, CDF Note 9642 (2009). 7. DØ Collaboration, DØ Note 5872-CONF (2009). 8. CDF Collaboration, CDF Note 9997 (2009). 9. CDF Collaboration, CDF Note 9985 (2009). 10. DØ Collaboration, DØ Note 5972-CONF (2009). 11. CDF Collaboration, CDF Note 9889 (2009). 12. DØ Collaboration, DØ Note 5876-CONF (2009). 13. DØ Collaboration, DØ Note 5845-CONF (2009). 14. CDF Collaboration, CDF Note 9887 (2009). 15. DØ Collaboration, DØ Note 6006-CONF (2009). 16. CDF Collaboration, CDF Note 9999 (2009). 17. DØ Collaboration, DØ Note 6008-CONF (2009). 18. The TEVNPH Working Group, arXiv:0911.3930 [hep-ex] (2009). 19. The TEVNPH Working Group, FERMILAB-PUB-09-394-E, CDF Note 9888, DØ Note 5980-CONF (2009). 20. CDF Collaboration, CDF Note 9284 (2008). 21. DØ Collaboration, DØ Note 5726-CONF (2008). 22. T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 103, 101803 (2009). 23. DØ Collaboration, DØ Note 5715-CONF (2008). 24. T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 103, 061803 (2009). 25. DØ Collaboration, DØ Note 5880-CONF (2009).

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The Top Triangle Moose∗ R. S. Chivukula, N. D. Christensen, B. Coleppa and E. H. Simmons† Department of Physics and Astronomy, Michigan State University East Lansing, Michigan, 48824, USA † E-mail: [emailprotected] http://www.pa.msu.edu/hep/hept/index.php We introduce a deconstructed model that incorporates both Higgsless and top-color mechanisms. The model alleviates the typical tension in Higgsless models between obtaining the correct top quark mass and keeping ∆ρ small. It does so by singling out the top quark mass generation as arising from a Yukawa coupling to an effective top-Higgs which develops a small vacuum expectation value, while electroweak symmetry breaking results largely from a Higgsless mechanism. As a result, the heavy partners of the SM fermions can be light enough to be seen at the LHC. Keywords: Top quark, electroweak symmetry breaking, new strong dynamics.

1. Introduction Higgsless models1 have recently emerged as a novel way of understanding the mechanism of electroweak symmetry breaking (EWSB) without the presence of a scalar particle in the spectrum. In an extra dimensional context, these can be understood in terms of a SU (2) × SU (2) × U (1) gauge theory in the bulk of a finite AdS spacetime,2–5 with symmetry breaking encoded in the boundary conditions of the gauge fields. One can understand the low energy properties of such theories in a purely four dimensional picture by invoking the idea of deconstruction.6,7 The “bulk” of the extra dimension is replaced by a chain of gauge groups strung together by non linear sigma model fields. The spectrum typically includes extra sets of charged and neutral vector bosons and heavy fermions. A general analysis of Higgsless models8–13 suggests that to satisfy precision electroweak constraints, the standard model (SM) fermions must be ‘delocalized’ into the bulk. A useful realization of this idea, “ideal fermion delocalization’”,14 dictates that the light fermions be delocalized so as not to couple to the heavy charged gauge bosons. The simplest framework capturing these ideas is the “three site Higgsless model”,15 with just one gauge group in the bulk and correspondingly, only one set of heavy vector bosons. The twin constraints of getting the correct value of the top quark mass and having an admissible ρ parameter push the heavy fermion masses into the TeV regime15 in that model. ∗ Speaker

at SCGT09: E. H. Simmons.

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This presentation summarizes Ref. 16, in which we seek to decouple these constraints by separating the mechanisms that break the electroweak symmetry and generate the masses of the third family of fermions. In this way, one can obtain a massive top quark and heavy fermions in the sub TeV region, without altering tree level electroweak predictions. To present a minimal model with these features, we modify the three site model by adding a “top Higgs” field, Φ, that couples preferentially to the top quark. The resulting model is shown in Moose notation22 in Figure 1; we will refer to it as the “top triangle moose.” The idea of a top Higgs is motivated by top condensation models (see references in Ref. 16), and the specific framework shown here is most closely aligned with topcolor assisted technicolor theories first proposed in Ref. 17, in which EWSB occurs via technicolor18,19 interactions while the top mass has a dynamical component arising from topcolor20,21 interactions and a small component generated by an extended technicolor mechanism. The dynamical bound state arising from topcolor dynamics can be identified as a composite top Higgs field, and the low-energy spectrum includes a top Higgs boson. The extra link in our triangle moose that corresponds to the top Higgs field results in the presence of uneaten Goldstone bosons, the top pions, which couple preferentially to the third generation. The model can thus be thought of as the deconstructed version of a topcolor assisted technicolor model. 2. The Model We now introduce the essential features of the model, which are required in order to understand the LHC phenomenology. Full details are presented in Ref. 16. The electroweak gauge structure of our model is SU (2)0 × SU (2)1 × U (1)2 . This is shown using Moose notation22 in Figure 1, in which the SU (2) groups are associated with sites 0 and 1, and the U (1) group is associated with site 2. The SM fermions deriving their SU (2) charges mostly from site 0 (which is most closely associated with the SM SU (2)) and the bulk fermions mostly from site 1. The extended electroweak gauge structure of the theory is the same as that of the BESS models,23,24 motivated by models of hidden local symmetry.25–29 The non linear sigma field Σ01 breaks the SU (2)0 × SU (2)1 gauge symmetry down to SU (2), and field Σ12 breaks SU (2)1 × U (1)2 down to U (1). The left handed fermions are SU (2) doublets residing at sites 0 (ψL0 ) and 1 (ψL1 ), while the right handed fermions are a doublet under SU (2)1 (ψR1 ) and two SU (2)-singlet fermions at site 2 (uR2 and dR2 ). The fermions ψL0 , ψL1 , and ψR1 have SM-like U (1) charges (Y ): +1/6 for quarks and −1/2 for leptons. Similarly, the fermion uR2 (dR2 ) has an SM-like U (1) charge of +2/3 (−1/3); the right-handed leptons, likewise, have U (1) charges corresponding to their SM hypercharge values. The third component of isospin, T3 , takes values +1/2 for “up” type fermions and −1/2 for “down” type fermions, just like in the SM. The electric charges satisfy Q = T3 + Y .

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0 g Σ01 Φ

~ g

g’

1

Σ12

2 Fig. 1. The SU (2) × SU (2) × U (1) gauge structure of the model in Moose notation.22 The SU (2) coupling g and U (1) coupling g of sites 0 and 2 are approximately the SM SU (2) and hypercharge gauge couplings, while the SU (2) coupling g˜ represents the ’bulk’ gauge coupling.

We add a ‘top-Higgs’ link to separate top quark mass generation from EWSB. The top quark couple preferentially to the top Higgs link via the Largangian: Ltop = −λt ψ¯L0 Φ tR + h.c.

(1)

When the field Φ develops a non zero vacuum expectation value, Eqn.(1) generates a top quark mass term. Since we want most EWSB to come from the√ Higgsless side, we choose the vacuum expectation values of Σ01 and Σ02 to be F = 2 v cos ω and the one associated with the top Higgs sector to be f = Φ = v sin ω (where ω is small). The top Higgs sector also includes the uneaten Goldstone bosons, the top pions; we assume they are heavy enough not to affect electroweak phenomenology. The mass terms for the light fermions arise from Yukawa couplings of the fermionic fields with the non linear sigma fields uR 0 uR2 . (2) L = MD L ψ¯L0 Σ01 ψR1 + ψ¯R1 ψL1 + ψ¯L1 Σ12 dR2 0 dR We denote the Dirac mass setting the scale of the heavy fermion masses as MD . Here, L is a flavor-universal parameter describing delocalization of the left handed fermions. All the flavor violation for the light fermions is encoded in the last term; the delocalization parameters for the right handed fermions, f R , can be adjusted to realize the masses and mixings of the up and down type fermions. For our phenomenological study, we will, for the most part, assume that all the fermions, except the top, are massless and hence will set these f R parameters to zero. The tree level contributions to precision measurements in Higgsless models come from the coupling of standard model fermions to the heavy gauge bosons. Choosing the profile of a light fermion bilinear along the Moose to be proportional to the profile of the light W boson makes the fermion current’s coupling to the W vanish because the W and W fields are mutually orthogonal. This procedure (called ideal fermion delocalization14 ) keeps deviations from the SM values of all electroweak

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quantities at a phenomenologically acceptable level. We find that the ideal delocal2 2 /2MW ization condition in this model is 2L = MW , as in the three-site model. The top quark mass matrix may be read from Eqns. (1) and (2) and is given by: MD tL λt vsinω . (3) MD MD tR Diagonalizing the top quark mass matrix perturbatively in tL and tR , we find the mass of the top quark is: 2 + 2tR + a2 tL tR λt v sinω , (4) mt = λt v sin ω 1 + tL , a≡ 2 2(−1 + a ) MD Thus, we see that mt depends mainly on v and only slightly on tR , in contrast to the situation in the three-site model where mt ∝ MD L tR . Since the bL is the SU (2) partner of the tL , its delocalization is (to the extent that bR 0) also determined by tL . Thus, the tree level value of the ZbL¯bL coupling Zbb equals its tree-level SM value if the leftcan be used to constrain tL . We find gL handed top quark is delocalized exactly as the light fermions are: tL = L . Finally, the contribution of the heavy top-bottom doublet to ∆ρ is of the same 2 4 tR /16 π 2 v 2 . The key difference is that, form as in the three-site model:15 ∆ρ = MD since the top quark mass is dominated by the vev of the top Higgs instead of MD , tR can be as small as the R of any light fermion. There is no conflict between the twin goals of a large top quark mass and a small value of ∆ρ. Thus, the heavy fermions in the top triangle moose can be light enough to be seen at the LHC. 3. Heavy Quarks at the LHC We now summarize our analysis16 of the possible discovery modes of the heavy quarks at the LHC; this work employed the CalcHEP package.30 ¯ → W Zqq → lllνjj 3.1. Pair production: pp → QQ Pair production of heavy quarks occurs at LHC via gluon fusion and quark annihilation processes, with the former dominating for smaller MD . Each heavy quark decays to a vector boson and a light fermion. For MD < MW ,Z , the decay is purely to the standard model gauge bosons. We study the case where one heavy quark decays to Z + j and the other decays to W + j, with the gauge bosons subsequently decaying leptonically. Thus, the final state is lllνjj. To enhance the signal to background ratio, we have imposed a variety of cuts, as shown in Table 1. We note that the the two jets in the signal should have a high pT (∼ MD /2), since they each come from the 2-body decay of a heavy fermion. Thus, imposing strong pT cuts on the outgoing jets can eliminate much of the SM background without affecting the signal too much. We also expect the η distribution of the jets to be largely central, which suggests an η cut: |η| ≤ 2.5. We impose standard separation cuts between the two jets and between jets and leptons to ensure that they are observed as distinct final state particles. We also impose basic identification cuts on the leptons and missing transverse energy.

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Cut

pT j pT l Missing ET |ηj | |ηl | ∆Rjj ∆Rjl Mll

>100 GeV >15 GeV >15 GeV < 2.5 < 2.5 >0.4 >0.4 89 GeV< Mll < 93 GeV

Variable

Cut

pT j hard pT j soft pT l Missing ET |ηj hard | |ηj soft | |ηl | ∆Rjj ∆Rjl

>200 GeV >15 GeV >15 GeV >15 GeV < 2.5 2< |η| < 4 < 2.5 >0.4 >0.4

We identify the leptons that came from the Z by imposing the invariant mass cut (MZ − 2 GeV) < Mll < (MZ + 2 GeV). We then combine this lepton pair with a leading-pT light jet to reconstruct the heavy fermion mass. Because one cannot know which light jet came from the Q, we actually combine the lepton pair first with the light jet of largest pT and then, separately, with the light jet of next-largest pT , and include both reconstructed versions of each event in our analysis. This yields16 an invariant mass distribution with a narrow signal peak standing out cleanly at MD above a tiny “background” from the wrongly-reconstructed signal events. When generating the signal events, we included the four flavors of heavy quarks, U, D, C, S, that should have similar phenomenology. We estimate the size of the peak by counting the signal events in the invariant mass window: (MD −10 GeV) < Mjll < (MD + 10 GeV). To analyze the SM background, we fully calculated the irreducible pp → ZW jj process and subsequently decayed the W and Z leptonically. Imposing the full set of cuts on the final state lllνjj entirely eliminates the SM background. We find there are many signal events in the region of parameter space where Q → V q decays are allowed but Q → V q decays are kinematically forbidden. The precise number is controlled by the branching ratio of the heavy fermion into the SM vector bosons. Since the SM background is negligible, if we assume the signal events are Poisson-distributed, then we can take 10 events to represent a 5σ signal at 95% c.l. Figure 2 shows the results, recast in the form of the integrated luminosity required to achieve a 5σ discovery signal. We see that the pair-production process we have studied spans almost the entire parameter space. However, in the region where MD ≥ 900 GeV and MW ≤ MD there will not be enough signal events for the discovery of the heavy quark since the decay channel Q → W q becomes significant. To explore this region, we now investigate the single production channel where the heavy quark decays to a heavy gauge boson. 3.2. Single production: pp → Qq → W qq → W Zqq While the single production channel is electroweak, the smaller cross section is compensated by the fact that the u and d are valence quarks, and their parton

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MW' GeV

1000

1

10

50

100

300

800

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600

100 50 400 400

600

800

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MD GeV

Fig. 2. Luminosity required for a 5σ discovery of the heavy vector fermions at the LHC in the single (blue curves, nearly horizontal) and pair (red curves, nearly vertical) production channels. The shaded area is non perturbative and not included in the study. The two channels are complementary and allow almost the entire region to be covered in 300 f b−1 .

distribution functions fall less sharply than the gluon’s. Also, there is less phase space suppression in the single production channel than in the pair production case. We analyze the processes [u, u → u, U ], [d, d → d, D] and [u, d → u, D or U, d], which occur through a t channel exchange of a Z and Z . Since we want to look at the region of parameter space where MW is smaller than MD , we let the heavy quark decay to a W . The W decays 100% of the time to a W and Z, because its coupling to two SM fermions is zero in the limit of ideal fermion delocalization. We constrain both the Z and W to decay leptonically so the final state is lllνjj. Again, we expect the jet from the decay of the heavy quark to have a large pT , and we impose a strong pT cut on this “hard jet”. As before, this jet should be central, so we impose the same η cut on the hard jet. We expect the η distribution of the “soft jet” arising from the light quark in the production process to be in the forward region, 2 < |η| < 4. We impose the same ∆R jet separation and jet-lepton separation cuts as before, along with basic identification cuts on the leptons and missing transverse energy. The complete set of cuts is in the right side of Table 1. The leptonic W decay introduces the usual two fold ambiguity in determining the neutrino momentum and hence, we have performed a transverse mass analysis of the process, defining the transverse mass variable31 of interest as: 2 2 − → 2 2 2 MT = M (lllj) + pT (lllj) + |pT (missing)| − |− p→ T (lllj) + pT (missing)| (5)

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We expect the distribution to fall sharply at MD in the narrow width approximation, and indeed we find that there are typically few or no events beyond MD + 20 GeV in the distributions. Thus, we take the signal events to be those in the transverse mass window: (MD − 200 GeV) < MT < (MD + 20 GeV). The SM background for this process, pp → W Zjj → jjlνll, was calculated summing over the u, d, c, s and gluon jets and the first two families of leptons. Since we apply a strong pT cut on only one of the jets (unlike in the pair production case), there is a non zero SM background, as plotted in Ref. 16. The luminosity necessary √ for a 5σ discovery at 95% c.l. can be calculated by requiring (Nsignal / Nbkrnd ) ≥ 5, as per a Gaussian distribution. Figure 2) shows the results, again recast in the form of the integrated luminosity required to achieve a 5σ discovery signal. Almost the entire parameter space is covered, with the pair and single production channels nicely complementing each other. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

C. Csaki et al., Phys. Rev. D 69, 055006 (2004). K. Agashe et al., JHEP 0308, 050 (2003). C. Csaki et al., Phys. Rev. Lett. 92 (2004) 101802. G. Burdman and Y. Nomura, Phys. Rev. D 69, 115013 (2004). G. Cacciapaglia et al., Phys. Rev. D 70, (2004) 075014. N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Rev. Lett. 86 (2001) 4757-4761. C. T. Hill, S. Pokorski and J. Wang, Phys. Rev. D 64 (2001) 105005. R. S.Chivukula et al., Phys. Rev. D 71 (2005) 035007. G. Cacciapaglia et al., Phys. Rev. D 71, (2005) 035015. G. Cacciapaglia et al., Phys. Rev. D 72, (2005) 095018. R. Foadi, S. Gopalakrishna and C. Schmidt, Phys. Lett. B 606 (2005) 157. R. Casalbuoni et al., Phys. Rev. D 71, 075015 (2005). R. Foadi, and C. Schmidt, Phys. Rev. D 73 (2006) 075011. R. S. Chivukula et al., Phys. Rev. D 72 (2005) 015008. R. S. Chivukula et al., Phys. Rev. D 74 (2006) 075011. R. Sekhar Chivukula et al., Phys. Rev. D 80, 035011 (2009). C. T. Hill, B 345: 483-489 (1995). E. Eichten and K. D. Lane, Phys. Lett. B 90, 125 (1980). S. Dimopoulos and L. Susskind, Nucl. Phys. B 155, 237 (1979). C. T. Hill, Phys. Lett. B 266 , 419-424 (1991) C. T. Hill and S. Parke, Phys. Rev. D49, 4454 (1994). H. Georgi, Nucl. Phys. B266 (1986) 274. R. Casalbuoni, S. De Curtis, D. Dominici, and R. Gatto, Phys. Lett. B155 (1985) 95. R. Casalbuoni et. al., Phys. Rev. D53 (1996) 5201-5221. M. Bando et al., Phys. Rev. Lett. 54 (1985) 1215. M. Bando, T. Kugo, and K. Yamawaki, Nucl. Phys. B259 (1985) 493. M. Bando, T. Kugo, and K. Yamawaki, Phys. Rept. 164 (1988) 217-314. M. Bando, T. Fujiwara, and K. Yamawaki, Prog. Theor. Phys. 79 (1988) 1140. M. Harada and K. Yamawaki, Phys. Rept. 381 (2003) 1-233. A. Pukhov, arXiv: hep-ph/0412191. J. Bagger, et al., Phys. Rev. D 52, 3878 (1995).

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Conformal Phase Transition in QCD Like Theories and Beyond V. A. Miransky Department of Applied Mathematics, University of Western Ontario London, Ontario N6A 5B7, Canada E-mail: [emailprotected] The dynamics with an infrared stable fixed point in the conformal window in QCD like theories with a relatively large number of fermion flavors is reviewed. The emphasis is on the description of a clear signature for the conformal window, which in particular can be useful for lattice computer simulations of these gauge theories. Keywords: Conformal phase transition, infrared fixed point, scaling law for hadron masses, light glueballs.

1. Introduction The Landau, or σ-model-like, phase transition1 is characterized by the following basic feature. Around the critical point z = zc (where z is a generic notation for parameters of a theory, as the coupling constant α, number of particle flavors Nf , etc.), an order parameter X is X = Λf (z),

(1)

where Λ is an ultraviolet cutoff and the function f (z) has such a non-essential singularity at z = zc that lim f (z) = 0 as z goes to zc both in symmetric and nonsymmetric phases. The standard form for f (z) is f (z) ∼ (z − zc )ν , ν > 0, around z = zc [for convenience, we assume that z > zc (z < zc ) in the nonsymmetric (symmetric) phase].a The conformal phase transition (CPhT), whose conception was introduced in Ref. 3, is a very different continuous phase transition. It is defined as a phase transition in which an order parameter X is given by Eq. (1) where f (z) has such an essential singularity at z = zc that while lim f (z) = 0

z→zc

(2)

as z goes to zc from the side of the non-symmetric phase, lim f (z) 6= 0 as z → zc from the side of the symmetric phase (where X ≡ 0). Notice that since the relation (2) ensures that the order parameter X → 0 as z → zc , the phase transition is continuous. a Strictly speaking, Landau considered the mean-field phase transition. By the Landau phase transition, we understand a more general class, when fields may have anomalous dimensions. 2

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There are the following basic differences between the Landau phase transition (LPhT) and the CPhT one:3 (1) In the case of the LPhT, masses of light excitations are continuous functions of the parameters z around the critical point z = zc (though they are nonanalytic at z = zc ). In the case of the CPhT, the situation is different: there is an abrupt change of the spectrum of light excitations, as the critical point z = zc is crossed. This implies that the effective actions describing low energy dynamics in the phases with z < zc and z > zc are different in a system with CPhT. (2) Unlike the LPhT, the parameter z governing the CPhT is connected with a marginal operator [in the LPhT phase transition, such a parameter is connected with a relevant operator; it is usually a mass term]. (3) The fact that the parameter z is connected with a marginal operator in the CPhT implies that in the continuum limit, when z → zc + 0, the conformal symmetry is broken by a marginal operator in nonsymmetric phase, i.e., there is a conformal anomaly. (4) Unlike the LPhT, in the case of CPhT, the structures of renormalizations (i.e., the renormalization group at high momenta) are different in symmetric phase and nonsymmetric one. In relativistic field theory, the CPhT is realized in the two dimensional GrossNeveu (GN) model4 at the critical coupling constant gc = 0, reduced (or defect) QED,5,6 and quenched QED.7–10 It was suggested that the chiral phase transition with respect to the number of fermion flavors Nf in QCD is a CPhT one.3,11 In condensed matter physics, a CPhT like phase transition is realized in the BerezinskiiKosterlitz-Thouless (BKT) model12 and, possibly, graphene.13 Recently, the interest to the dynamics with the CPht phase transition has essentially increased. It is in particular connected with a progress in numerical lattice studies of gauge theories with a varied number of fermion flavors (for a recent review, see Ref. 14), the revival of the interest to the electroweak symmetry breaking based on the walking technicolor like dynamics15,16 (for a recent review, see Ref.17 ), and intensive studies of graphene, a single atomic layer of graphite (for a review, see Ref. 18). 2. Dynamics in the conformal window in QCD-like theories 2.1. General description In this section, we will consider the problem of the existence of a nontrivial conformal dynamics in 3+1 dimensional non-supersymmetric vector like gauge theories, with a relatively large number of fermion flavors Nf . We will discuss their phase diagram in the (α(0) , Nf ) plane, where α(0) is the bare coupling constant. We also discuss a clear signature for the conformal window in lattice computer simulations of these theories suggested quite time ago in Ref. 19.

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The roots of this problem go back to a work of Banks and Zaks20 who were first to discuss the consequences of the existence of an infrared-stable fixed point α = α ∗ for Nf > Nf∗ in vector-like gauge theories.21 The value Nf∗ depends on the gauge group: in the case of SU(3) gauge group, Nf∗ = 8 in the two-loop approximation. In Nineties, a new insight in this problem3,11 was, on the one hand, connected with using the results of the analysis of the Schwinger-Dyson (SD) equations describing chiral symmetry breaking in quenched QED7–10 and, on the other hand, with the discovery of the conformal window in N = 1 supersymmetric QCD.22 In particular, Appelquist, Terning, and Wijewardhana11 suggested that, in the case of the gauge group SU(Nc ), the critical value Nfcr ' 4Nc separates a phase with no confinement and chiral symmetry breaking (Nf > Nfcr ) and a phase with confinement and with chiral symmetry breaking (Nf < Nfcr ). The basic point for this suggestion was the observation that at Nf > Nfcr the value of the infrared c π , presumably needed to fixed point α∗ is smaller than a critical value αcr ' N2N 2 c −1 3 7–10 generate the chiral condensate. The authors of Ref. 11 considered only the case when the running coupling constant α(µ) is less than the fixed point α∗ . In this case the dynamics is asymptotically c free (at short distances) both at Nf < Nfcr and Nfcr < Nf < Nf∗∗ ≡ 11N 2 . Yamawaki 3 (0) and the author analyzed the dynamics in the whole (α , Nf ) plane and suggested the (α(0) , Nf )-phase diagram of the SU(Nc ) theory, where α(0) is the bare coupling constant (see Fig 1 below).b In particular, it was pointed out that one can get an interesting non-asymptotically free dynamics when the bare coupling constant α(0) is larger than α∗ , though not very large. The dynamics with α(0) > α∗ admits a continuum limit and is interesting in itself. Also, its better understanding can be important for establishing the conformal window in lattice computer simulations of the SU(Nc ) theory with such large values of Nf . In order to illustrate this, let us consider the following example. For Nc = 3 and Nf = 16, the value of the infrared fixed point α∗ calculated in the two-loop approximation is small: α∗ '0.04. To reach the asymptotically free phase, one needs to take the bare coupling α(0) less than this value of α∗ . However, because of large finite size effects, the lattice computer simulations of the SU(3) theory with such a small α(0) would be unreliable. Therefore, in this case, it is necessary to consider the dynamics with α(µ) > α∗ . In Ref. 19, this author suggested a clear signature of the existence of the infrared fixed point α∗ , which in particular can be useful for lattice computer simulations. The signature is based on two characteristic features of the the spectrum of low energy excitations in the presence of a bare fermion mass in the conformal window: a) a strong (and simple) dependence of the masses of all the colorless bound states (including glueballs) on the bare fermion mass, and b) unlike QCD with a small Nf (Nf = 2 or 3), glueballs are lighter than bound states composed of fermions, if the value of the infrared fixed point is not too large. b This

phase diagram is different from the original Banks-Zaks diagram.20

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2.2. Phase diagram The phase diagram in the (α(0) , Nf )-plane in the SU (Nc ) gauge theory is shown in Fig. 1. The left-hand portion of the curve in this figure coincides with the line of the infrared-stable fixed points α∗ (Nf ):21 b α(0) = α∗ = − , c

(3)

where 1 (11Nc − 2Nf ), 6π N2 − 1 1 (34Nc2 − 10Nc Nf − 3 c Nf ). c= 2 24π Nc

b=

(4) (5)

It separates two symmetric phases, S1 and S2 , with α(0) < α∗ and α(0) > α∗, c π respectively. Its lower end is Nf = Nfcr (with Nfcr ' 4Nc if αcr ' N2N ): at 2 c −1 3 ∗ cr Nf < Nf < Nf the infrared fixed point is washed out by generating a dynamical fermion mass (here Nf∗ is the value of Nf at which the coefficient c in Eq. (5) becomes positive and the fixed point disappears). The horizontal, Nf = Nfcr , line describes a phase transition between the symmetric phase S1 and the phase with confinement and chiral symmetry breaking. As it was suggested in Ref.11, based on a similarity of this phase transition with that in quenched QED4 ,8,9 there is the following scaling law for m2dyn :   C , (6) m2dyn ∼ Λ2cr exp − q ∗ α (Nf ) − 1 αcr

where the constant C is of order one and Λcr is a scale at which the running coupling is of order αcr . It is a CPhT phase transition with an essential singularity at Nf = Nfcr . At last, the right-hand portion of the curve on the diagram occurs because at large enough values of the bare coupling, spontaneous chiral symmetry breaking takes place for any number Nf of fermion flavors. This portion describes a phase transition called a bulk phase transition in the literature, and it is presumably a first order phase transition. c The vertical line ends above Nf =0 since in pure gluodynamics there is apparently no phase transition between weak-coupling and strong-coupling phases. 2.3. Signature for the conformal window Up to now we have considered the case of a chiral invariant action. But how will the dynamics change if a bare fermion mass term is added in the action? This question c The fact that spontaneous chiral symmetry breaking takes place for any number of fermion flavors, if α(0) is large enough, is valid at least for lattice theories with Kogut-Susskind fermions. Notice however that since the bulk phase transition is a lattice artifact, the form of this portion of the curve can depend on the type of fermions used in simulations.

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**

Nf

S2

cr

S1

Nf

A *

Nf

A

2

g(0) cr

g(0) Figure 3

Fig. 1. The phase diagram in an SU(Nc ) gauge model. The coupling constant g (0) = and S and A denote symmetric and asymmetric phases, respectively.

√ 4πα(0)

is in particular relevant for lattice computer simulations: for studying a chiral phase transition on a finite lattice, it is necessary to introduce a bare fermion mass. As was pointed out in Ref.,19 adding even an arbitrary small bare fermion mass results in a dramatic changing the dynamics both in the S1 and S2 phases. Recall that in the case of confinement SU(Nc ) theories, with a small, Nf < Nfcr , number of fermion flavors, the role of a bare fermion mass m(0) is minor if m(0) << ΛQCD (where ΛQCD is a confinement scale). The only relevant consequence is that massless Nambu-Goldstone pseudoscalars get a small mass (the PCAC dynamics). The reason for that is the fact that the scale ΛQCD , connected with a conformal anomaly, describes the breakdown of the conformal symmetry connected both with perturbative and nonperturbative dynamics: the running coupling and the formation of bound state. Certainly, a small bare mass m(0) << ΛQCD is irrelevant for the dynamics of those bound states. Now let us turn to the phases S1 and S2 , with Nf > Nfcr . There is still the conformal anomaly in these phases: because of the running of the effective coupling constant, the conformal symmetry is broken. It is restored only if α(0) is equal to the infrared fixed point α∗ . However, the essential difference with respect to confinement theories is that this conformal anomaly have nothing to do with the dynamics forming bound states: Since at Nf > Nfcr the effective coupling is relatively weak,

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it is impossible to form bound states from massless fermions and gluons (recall that the S1 and S2 phases are chiral invariant). Therefore the absence of a mass for fermions and gluons is a key point for not creating bound states in those phases. The situation changes dramatically if a bare fermion mass is introduced: indeed, even weak gauge, Coulomb-like, interactions can easily produce bound states composed of massive constituents, as it happens, for example, in QED, where electron-positron (positronium) bound states are present. To be concrete, let us consider the case when all fermions have the same bare mass m(0) . It leads to a mass function m(q 2 ) ≡ B(q 2 )/A(q 2 ) in the fermion propagator G(q) = (ˆ q A(q 2 ) − B(q 2 ))−1 . The current fermion mass m is given by the relation m(q 2 )|q2 =m2 = m.

(7)

For the clearest exposition, let us consider a particular theory with a finite cutoff Λ and the bare coupling constant α(0) = α(q)|q=Λ being not far away from the fixed point α∗ . Then, the mass function is changing in the “walking” regime16 with α(q 2 ) ' α∗ . It is γm M (8) m(q 2 ) ' m(0) q ¯ γm = 3 − d ¯ with where γm is the anomalous dimension of the operator ψψ: ψψ dψψ ¯ being the dynamical dimension of this operator. In the walking regime, γ m ' ∗ 1 − (1 − ααcr )1/2 (see Refs. 9,16). Eqs.(7) and (8) imply that m'Λ

m(0) Λ

1+γ1 m

.

(9)

Recall that the anomalous dimension γm ≥ 0, and γm . 2 in the “walking” regime. There are two main consequences of the presence of the bare mass: (a) bound states, composed of fermions, occur in the spectrum of the theory. The mass of a n-body bound state is M (n) ' nm. Therefore they satisfy the scaling 1+γ1 m . (10) M (n) ' nm ∼ n m(0) (b) At low momenta, q < m, fermions and their bound states decouple. There is a pure SU(Nc ) Yang-Mills theory with confinement. Its spectrum contains glueballs. To estimate glueball masses, notice that at momenta q < m, the running of the coupling is defined by the parameter ¯b of the Yang-Mills theory, ¯b = 11 Nc . 6π

(11)

Therefore the glueball masses Mgl are of order 1 ΛY M ' m exp(− ¯ ∗ ). bα

(12)

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For Nc = 3, we find from Eqs.(4), (5), and (11) that exp(− ¯bα1∗ ) is 6 × 10−7 , 2 × 10−2 , 10−1 , and 3 × 10−1 for Nf =16, 15, 14, and 13, respectively. Therefore at Nf =16, 15 and 14, the glueball masses are essentially lighter than the masses of the bound states composed of fermions. The situation is similar to that in confinement QCD with heavy (nonrelativistic) quarks, m >> ΛQCD . However, there is now a new important point. In the conformal window, any value of m(0) (and therefore m) is “heavy”: the fermion mass m sets a new scale in the theory, and the confinement scale ΛY M (12) is less, and rather often much less, than this scale m. One could say that the latter plays a role of a dynamical ultraviolet cutoff for the pure YM theory. This leads to a spectacular “experimental” signature of the conformal window in lattice computer simulations: the masses of all colorless bound states, including 1 glueballs, decrease as (m(0) ) 1+γm with the bare fermion mass m(0) for all values of m(0) less than cutoff Λ. Moreover, one should expect that glueball masses are lighter than the masses of the bound states composed of quarks. Few comments are in order: (1) The phases S1 and S2 have essentially the same long distance dynamics. They are distinguished only by their dynamics at short distances: while the dynamics of the phase S1 is asymptotically free, that of the phase S2 is not. Also, while around the infrared fixed point α∗ the sign of the beta function is negative in S1 , it is positive in S2 .3 When all fermions are massive (with the current mass m), the continuum limit Λ → ∞ of the S2 -theory is a non-asymptotically free confinement theory. Its spectrum includes colorless bound states composed of fermions and gluons. For q < m the running coupling α(q) is the same as in pure SU(Nc ) Yang-Mills theory, and for all q m α(q) is very close to α∗ (“walking”, actually, “standing” dynamics). For those values Nf for which α∗ is small (as Nf =16, 15 and 14 at Nc =3), glueballs are much lighter than the bound states composed of fermions. Notice that unlike the case with m = 0, corresponding to the unparticle dynamics,23 there exists a conventional S-matrix in this theory. (2) In order to get the clearest exposition, we assumed such estimates as Nfcr ' q ∗ 4Nc for Nfcr and γm = 1 − 1 − ααcr for the anomalous dimension γm . While the

latter should be reasonable for α∗ < αcr (and especially for α∗ << αcr ),9 the former c π which, though seems reasonable, is based on the assumption that αcr ' N2N 2 c −1 3 might be crude for some values of Nc . It is clear however that the dynamical picture presented above is essentially independent of those assumptions. 2.4. Lattice computer simulations During last two years, there has been an essential progress in the lattice computer simulations of gauge theories with a varied number of fermion flavors.d For a recent review, see Ref. 14 and the papers of Tom Appelquist, George Fleming, Kieran Holland, Julius Kuti, Maria Lombardo, and Donald Sinclair in this volume.

d For

pioneer papers in this area, see Refs. 24–26.

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This author is certainly not an expert in lattice computer simulations. Here I would like to discuss this topic only in the connection with the phase diagram and the signature of the conformal window considered in the Secs. 2.2 and 2.3 above. In Ref. 27, based on the fact that the sign of the beta function changes from negative to positive when the line between the S1 and S2 phases is crossed, the existence of the conformal window in QCD with Nf = 12 was studied by using the measurements of the chiral condensate and the mass spectrum. The analysis supports the existence of the conformal window in this theory. In Ref. 28, the scaling law (10) was rediscovered and applied to the study of the conformal window in the SU(3) lattice gauge theory with two flavors of color sextet fermions (the parameter ym in Ref. 28 is connected with the anomalous dimension γm as ym = 1+γm ). The main conclusion of that study was that ym ∼ 1.5 (γm ∼ 0.5). This value is smaller than γm ' 1 in walking technicolor and at this moment it is unclear whether this theory contains an infrared-stable fixed point. The authors of Ref. 29 studied the spectrum of mesons and glueballs in the SU(2) lattice gauge theory with adjoint fermions. They found that for light constituent fermions the lightest glueballs are lighter than the lightest mesons. It is tempting to speculate that in accordance with the signature for the conformal window discussed above, this fact indicates on the existence of a infrared fixed point in this theory. However, as the authors point out, a lot of issues should still be clarified in order to reach a solid conclusion. It is clear that lattice simulations of gauge theories with varied numbers of fermion flavors are crucial for further progress in our understanding of such dynamics. The important point is that CPhT is a long range interactions phenomenon, which is very sensitive to any screening and finite-size effects. The progress made in this area during last two years is certainly encouraging. Acknowledgments I am grateful to the organizers of SCGT09 Workshop, in particular Koichi Yamawaki, for their warm hospitality. This work was supported by the Natural Sciences and Engineering Research Council of Canada. References 1. L. D. Landau, Phys. Z. Sowjet. 11, 26 (1937). 2. K. G. Wilson, Phys. Rev. B4 (1971)3174; K. G. Wilson and J. B. Kogut, Phys. Rep. 12 (1974)75. 3. V. A. Miransky and K. Yamawaki, Phys. Rev. D 55, 5051 (1997). 4. D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974). 5. E. V. Gorbar, V. P. Gusynin and V. A. Miransky, Phys. Rev. D 64, 105028 (2001); S.-J. Rey, Prog. Theor. Phys. Suppl. No. 177, 128 (2009). 6. D. B. Kaplan, J. W. Lee, D. T. Son and M. A. Stephanov, Phys. Rev. D 80, 125005 (2009). 7. T. Maskawa and H. Nakajima, Prog. Theor. Phys. 52 (1974)1326; R. f*ckuda and T. Kugo, Nucl. Phys. B117 (1976)250.

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8. P. I. Fomin and V. A. Miransky, Phys. Lett. B 64, 166 (1976); P. I. Fomin, V. P. Gusynin and V. A. Miransky, Phys. Lett. B 78, 136 (1978); P. I. Fomin, V. P. Gusynin, V. A. Miransky and Yu. A. Sitenko, Riv. Nuovo Cim. 6N5, 1 (1983). 9. V. A. Miransky, Phys. Lett. B 91, 421 (1980); Nuovo Cim. A 90, 149 (1985). 10. W. A. Bardeen, C. N. Leung and S. T. Love, Phys. Rev. Lett. 56, 1230 (1986); C. N. Leung, S. T. Love and W. A. Bardeen, Nucl. Phys. B 273, 649 (1986). 11. T. Appelquist, J. Terning, and L. C. R. Wijewardhana, Phys. Rev. Lett. 75, 2081 (1995). 12. V. L. Berezinskii, Zh. Eksp. Teor. Fiz. 59, 907 (1970); J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973). 13. E. V. Gorbar, V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Phys. Rev. B 66, 045108 (2002); H. Leal and D. V. Khveshchenko, Nucl. Phys. B 687, 323 (2004). 14. E. Pallante, arXiv:0912.5188 [hep-lat]. 15. B. Holdom, Phys. Rev. D 24, 1441 (1981). 16. K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335 (1986); T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57, 957 (1986); T. Akiba and T. Yanagida, Phys. Lett. B 169, 432 (1986). 17. F. Sannino, arXiv:0911.0931 [hep-ph]. 18. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 19. V. A. Miransky, Phys. Rev. D 59, 105003 (1999). 20. T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982). 21. D. R. T. Jones, Nucl. Phys. B75, 531 (1974); W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974). 22. N. Seiberg, Phys. Rev. D49, 6857 (1994); K. Intriligator and N. Seiberg, Nucl. Phys. B (Proc. Suppl.) 45 B and C, 1 (1996). 23. H. Georgi, Phys. Rev. Lett. 98, 221601 (2007) 24. J. B. Kogut and D. R. Sinclair, Nucl. Phys. B295, 465 (1988); F. Brown et al., Phys. Rev. D 46, 5655 (1992). 25. Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie, Nucl Phys. B (Proc. Suppl.) 53, 449 (1997); D. Chen and R. D. Mawhinney, ibid. 53, 216 (1997). 26. P. H. Damgaard, U. M. Heller, A. Krasnitz and P. Olesen, Phys. Lett. B 400, 169 (1997). 27. A. Deuzeman, M. P. Lombardo and E. Pallante, arXiv:0904.4662 [hep-ph]. 28. T. DeGrand and A. Hasenfratz, Phys. Rev. D 80, 034506; T. DeGrand, arXiv:0910.3072. 29. L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Phys. Rev. D 80, 074507 (2009); B. Lucini, arXiv:0911.0020 [hep-ph].

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Gauge-Higgs Unification at LHC Nobuhito Maru∗ Department of Physics, Chuo University Tokyo, 112-8551, Japan E-mail: [emailprotected] Nobuchika Okada Department of Physics and Astronomy, University of Alabama Tuscaloosa, AL 35487, USA E-mail: [emailprotected] Higgs boson production by the gluon fusion and its decay into two photons at the LHC are investigated in the context of the gauge-Higgs unification scenario. The qualitative behaviors for these processes in the scenario are quite distinguishable from those of the Standard Model and the universal extra dimension scenario because of the overall sign difference for the effective couplings induced by one-loop corrections through KaluzaKlein (KK) modes. Keywords: Gauge-Higgs unification, Higgs production and decay at LHC.

1. Introduction Gauge-Higgs unification (GHU) is a fascinating scenario solving the hierarchy problem without invoking supersymmetry.1 In this scenario, Higgs scalar field in the Standard Model (SM) is identified with extra components of higher dimensional gauge field. The remarkable thing in this scenario is that the quantum correction to Higgs mass is finite due to the higher dimensional gauge symmetry regardless of the nonrenormalizability of the theory2 (See also3 ). Such a UV insensitivity in other physical observables has been also investigated for S and T parameters,4 g-2,5 the violation of gauge-Yukawa universality 6 and the gluon fusion (two photon decay) of Higgs boson.7 The last one is the issue discussed in this talk. The Large Hadron Collider (LHC) started its operation again and the collider signatures of various new physics models beyond the SM have been extensively studied. The GHU shares the similar structure with the universal extra dimension (UED) scenario,8 namely, Kaluza-Klein (KK) states of the SM particles appear. The collider phenomenology on the KK particles will be quite similar to the one in the UED scenario. A crucial difference should lie in the Higgs sector, because the Higgs ∗ Speaker

of the conference.

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doublet originates from the higher dimensional gauge field and its interactions are controlled by the higher dimensional gauge invariance. The discovery of Higgs boson is expected at the LHC, by which the origin of the electroweak symmetry breaking and the mechanism responsible for generating fermion masses will be revealed. Precise measurements of Higgs boson properties will provide us the information of a new physics relevant to the Higgs sector In this talk, we investigate the effect of GHU on Higgs boson phenomenology at the LHC, namely, the production and decay processes of Higgs boson9 (See also.10 ). At the LHC, the gluon fusion is the dominant Higgs boson production process and for light Higgs boson with mass mh < 150 GeV, and two photon decay mode of Higgs boson becomes the primary discovery mode11 nevertheless its branching ratio is O(10−3 ). The coupling between Higgs boson and these gauge bosons are induced through quantum corrections at one-loop level even in the SM. Therefore, we can expect a sizable effect from new particles if they contribute to the coupling at one-loop level. In a five dimensional GHU model, we calculate one-loop diagrams with KK fermions for the effective couplings between Higgs boson and the gauge bosons (gluons and photons). If the KK mass scale is small enough, we can see a sizable deviation from the SM couplings and as a result, the number of signal events from Higgs production at the LHC can be altered from the SM one. Interestingly, reflecting the special structure of Higgs sector in the GHU, there is a clear qualitative difference from the UED scenario, the signs of the effective couplings are opposite to those in the UED scenario. 2. Model Let us consider a toy model of five dimensional (5D) SU (3) GHU with an orbifold S 1 /Z2 compactification, in order to avoid unnecessary complications for our discussion. Although the predicted Weinberg angle in this toy model is unrealistic, sin2 θW = 43 , this does not affect our analysis. We introduce an SU (3) triplet fermion as a matter field, which is identified with top and bottom quarks and their KK excited states, although the top quark mass vanishes and the bottom quark mass mb = MW in this simple toy model. In our analysis, we will take into account a situation where a realistic top quark mass is realized and the bottom quark contributions are negligible comparing to the top quark ones. The SU (3) gauge symmetry is broken to SU (2) × U (1) by the orbifolding on 1 S /Z2 . The remaining gauge symmetry SU (2) × U (1) is supposed to be broken by the vacuum expectation value (VEV) of the zero-mode of A5 , the extra space component of the gauge field identified with the SM Higgs doublet. We do not address the origin of SU (2) × U (1) gauge symmetry breaking and the resultant Higgs boson mass in the one-loop effective Higgs potential, which is highly modeldependent and out of our scope of this work. The Lagrangian is simply given by 1 ¯ /Ψ. (1) L = − Tr(FM N F M N ) + iΨD 2

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The periodic boundary conditions are imposed along S 1 for all fields. The non-trivial Z2 parities are assigned for each field as follows, Aµ (yi − y) = PAµ (yi + y)P † ,

(2) †

Ay (yi − y) = −PAy (yi + y)P ,

(3)

5

Ψ(yi − y) = Pγ Ψ(yi + y)

(4)

where γ 5 ψL = ψL , P = diag(+, +, −) at fixed points yi = 0, πR. By this Z2 parity assignment, SU (3) is explicitly broken to SU (2) × U (1). Higgs scalar field is (0) identified with the off-diagonal block of zero mode Ay . After the electroweak gauge symmetry breaking, 4D effective Lagrangian among √ KK fermions, the SM gauge boson and Higgs boson (h) defined as h0 = (v + h)/ 2 ¯ /Ψ in Eq. (1). Integrating over the fifth can be derived from the term Lfermion = iΨD dimensional coordinate, we obtain a relevant 4D effective Lagrangian in the mass eigenstate of nonzero KK modes: (4D)

Lfermion = 

 × 

µ

∞ n X

(n) ¯(n) ¯(n) (ψ¯1 , ψ˜2 , ψ˜3 )

n=1

iγ ∂µ − mn 0

0 (n) µ iγ ∂µ − m+ + 0



 (n) ψ1  m   ˜(n)  0 v h   ψ2  (n) (n) ψ˜3 iγ µ ∂µ − m− − m v h

+ gauge interaction part + zero-mode part. where m =

gv 2 (=

√ g5 2πR n ± = R

MW ) is the bottom quark mass in this toy model, g = (n) m±

(5) is

m ≡ mn ± m the 4D gauge coupling. Note that the mass splitting occurs associated with a mixing between the SU (2) doublet component and singlet (n) component. Note that the mass eigenstate for m± has the Yukawa coupling ∓m/v, which is exactly the same as the one for the zero mode. In UED, however, the p KK mode mass spectrum and Yukawa couplings are given by Mn = m2n + m2t without mass splitting and −(mt /v) × (mt /Mn ), respectively.8 Together with the mass splitting of KK modes, this property is a general one realized in any GHU model and leads to a clear qualitative difference of the GHU from the UED scenario, as we will see. 3. Effective couplings between Higgs boson and gauge bosons Before calculating KK fermion contributions to one-loop effective couplings between Higgs boson and gauge bosons (gluons and photons), it is instructive to recall the SM result. We parameterize the effective coupling between Higgs boson and gluons or photons as Leff = CgSM h Gaµν Gaµν , Leff =

CγSM hF µν Fµν ,

(6) (7)

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where Gaµν (Fµν ) is a gluon (photon) field strength tensor. This coupling is generated by one-loop corrections (triangle diagram) on which quarks are running. The top quark loop diagram gives the dominant contribution and the coupling CgSM and CγSM is described in the following form: αs F1/2 (4m2t /m2h ) 1 mt αs × × ' , v 8πmt 2 12πv αem F1/2 (4m2t /m2h ) 4 m2W mt αem F1 (4m2W /m2h ) =− × × − × v 8πmt 3 v 8πm2W 47αem '− 72πv

CgSM = − CγSM

(8)

(9)

where the first (second) term in CγSM is KK top quark (KK W-boson) contributions, respectively, αs,em is the fine structure constant of QCD, electromagnetic coupling, the loop function F1/2,1 (τ ) given by √ 4 F1/2 (τ ) = −2τ 1 + (1 − τ ) [sin−1 (1/ τ )]2 → − for τ 1, 3 √ F1 (τ ) = 2 + 3τ + 3τ (2 − τ )[sin−1 (1/ τ )]2 → 7 for τ 1.

(10) (11)

It is well-known that in the top quark decoupling limit mt mh , F1/2,1 becomes a constant and the resultant effective coupling becomes independent of mt , mW and mh . Calculations of KK mode contributions are completely analogous to the top loop correction. The structure described in our toy model is common in any GHU model, (n) we will have KK modes of top quark with mass eigenvalue m± = mn ± mt and Yukawa couplings ∓mt /v, respectively. The KK mode contributions are found to be Leff = CgKK(GH) h Gaµν Gaµν , # " (n)2 ∞ X αs F1/2 (4m+ /m2h ) 1 mt (n) (n) KK(GH) × + (m+ ↔ m− ) × Cg =− (n) v 2 8πm+ n=1 # " ∞ ∞ X mt α s 1 αs X m2t 1 ' − ' − (n) 12πv n=1 m(n) 6πv n=1 m2n m− +

(12)

where we have taken the limit m2h , m2t m2n to simplify the results. Note that this result is finite due to the cancellation between two divergent corrections with opposite signs. Also, note that the KK mode contribution is subtractive against the top quark contribution in the SM. The results are depicted in Fig. 1 as a function of the mass of the lightest KK mode (diagonal) mass eigenvalue (m1 ). For the bulk fermion with the (half-)periodic boundary condition m1 = 1/R(1/(2R)). In this analysis, we take mh = 120 GeV. The result is not sensitive to the Higgs boson mass if mh < 2mt . For reference, the result in the UED scenario8 is also shown, for which only the periodic fermion has been considered. The KK fermion contribution is subtractive and the Higgs production cross section is reduced in the GHU, while

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Σ Hgg ® h; SML

Σ Hgg ® h; SM + KKL

2.0

1.5

1.0

0.5

0.0

600

800

m1 HGeVL 1000

1200

1400

Fig. 1. The ratio of the Higgs boson production cross sections in the GHU and in the SM as a function of the KK mode mass m1 . The solid (dashed) line corresponds to the result including the (half-)periodic fermion contributions, respectively. As a reference, the result in the UED scenario with top quark KK modes is also shown (dotted line). We have taken mh = 120 GeV.

it is increased in the UED scenario. This is a crucial point to distinguish the GHU from the UED scenario. The contribution of top quark KK modes to the effective coupling between Higgs boson and photons are calculated similarly. Leff = CγKK(GH) h F µν Fµν , " # (n)2 ∞ 2 X α F (4m /m ) m 4 em 1/2 t (n) (n) + h CγKK(GH) = − × + (m+ ↔ m− ) × (n) v 3 8πm+ n=1 # " ∞ ∞ 1 4αem X m2t 2mt αem X 1 − ' − ' . (13) (n) 9πv n=1 m(n) 9πv n=1 m2n m− + For the effective coupling with photons, in addition to the KK fermion contributions, we also have the KK W-boson loop corrections as in the SM. However, we neglect such contributions compared to those from the KK top quark ones by the following plausible reasons. • KK top (KK W-boson) contributions are decoupling effects, and are proportional to mass squared of top (W-boson), respectively. This indicates that KK top quark contributions are likely to be dominant. • In the GHU model on the flat space, a large dimensional representation in which the SM top quark is embedded must be introduced to reproduce a realistic top Yukawa coupling.13 Therefore, the effective 4D theory includes extra vector-like top-like quarks and its KK modes. Thus, KK top quark contributions are enhanced by a number of extra top-like quarks. • In some GHU models, bulk top-like quarks with the half-periodic boundary condition are often introduced to realize the correct electroweak symmetry breaking and a viable Higgs boson mass. The lowest KK mass of the half-periodic fermions is half of the lowest KK mass of periodic ones, so that their loop contributions can dominate over those by periodic KK mode fields.

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After these considerations, we see that the KK mode contributions to two photon decay is the same sign as those of the SM. 4. Effects on Higgs boson search at LHC As we have shown, the KK mode loop contribution to the effective coupling between Higgs boson and gluons (photons) is subtractive (slightly constructive) to the top quark loop contribution in the SM. This fact leads to remarkable effects on Higgs boson search at the LHC. Since the main production process of Higgs boson at the LHC is through gluon fusion, and the primary discovery mode of Higgs boson is its two photon decay channel if Higgs boson is light mh < 150 GeV. Therefore, the deviations of the effective coupling between Higgs boson and gluons or photons from the SM one give important effects on the Higgs boson production and the number of two photon events from Higgs boson decay. We show the ratio of the number of two photon events from Higgs decay produced through gluon fusion at the LHC. As a good approximation, this ratio is described as σ(gg → h; SM + KK) × BR(h → γγ; SM + KK) σ(gg → h; SM) × BR(h → γγ; SM) ! ! KK(GH) 2 KK(GH) 2 Cg Cγ ' 1+ 1+ CgSM CγSM

(14)

Σ Hgg ® h; SML ´ BR Hh ® ΓΓ; SML

Σ Hgg ® h; SM + KKL ´ BR Hh ® ΓΓ; SM + KKL

where σ is Higgs boson production cross section, BR denotes the branching ratio of two photon decay of Higgs boson. Fig. 2 shows the results for the periodic and halfperiodic KK modes as a function of m1 for the case of nt =1, 3 and 5 extra top-like quark fermions. Even for m1 = 1 TeV and nt = 1, the deviation is sizable '14%. When m1 is small and nt is large, the new physics contribution can dominate. 2.0

1.5

1.0

0.5

0.0

600

800

m1 HGeVL 1000

1200

1400

Fig. 2. The ratio of the number of two photon events in the GHU scenario to those in the SM as a function of the lowest KK mass m1 . The solid (dashed) lines represent the results including the nt (half-)periodic KK fermion contributions. nt = 1, 3, 5 are shown from the top to the bottom at m1 = 1500 GeV. Higgs mass is taken to be 120 GeV.

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5. Summary We have calculated the one-loop KK fermion contributions to the Higgs effective couplings between Higgs boson and gluons or photons and found them to be finite. This finiteness is achieved by a non-trivial cancellations between two KK mass eigenstates, although each contribution is divergent. The overall sign of the contributions is opposite compared to the SM result by top quark loop corrections and the similar result in the UED scenario. Therefore, this feature is a clue to distinguish the GHU from the UED scenario. Our analysis have shown that even with the KK mode mass is around 1 TeV, the KK mode loop corrections provide O(10%) deviations from the SM results in Higgs boson phenomenology at the LHC. In a realistic GHU model, some extra top-like quarks would be introduced to reproduce the top Yukawa coupling in the SM. In such a case, the KK mode contributions are enhanced and the signal events of Higgs boson production at the LHC are quite different from those in the SM. Acknowledgments The work of authors was supported in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture, No.18204024 and No.20025005. References 1. N. S. Manton, Nucl. Phys. B 158, 141 (1979); D. B. Fairlie, Phys. Lett. B 82, 97 (1979); J. Phys. G 5, L55 (1979); Y. Hosotani, Phys. Lett. B 126, 309 (1983); Phys. Lett. B 129, 193 (1983); Annals Phys. 190, 233 (1989). 2. H. Hatanaka, T. Inami and C. S. Lim, Mod. Phys. Lett. A 13, 2601 (1998). 3. I. Antoniadis, K. Benakli and M. Quiros, New J. Phys. 3, 20 (2001); G. von Gersdorff, N. Irges and M. Quiros, Nucl. Phys. B 635, 127 (2002); C. S. Lim, N. Maru and K. Hasegawa, J. Phys. Soc. Jap. 77, 074101 (2008); K. Hasegawa, C. S. Lim and N. Maru, Phys. Lett. B 604, 133 (2004); N. Maru and T. Yamash*ta, Nucl. Phys. B 754, 127 (2006); Y. Hosotani, N. Maru, K. Takenaga and T. Yamash*ta, Prog. Theor. Phys. 118, 1053 (2007). 4. C. S. Lim and N. Maru, Phys. Rev. D 75, 115011 (2007). 5. Y. Adachi, C. S. Lim and N. Maru, Phys. Rev. D 76, 075009 (2007); Phys. Rev. D 79, 075018 (2009). 6. C. S. Lim and N. Maru, arXiv:0904.0304 [hep-ph]. 7. N. Maru, Mod. Phys. Lett. A 23, 2737 (2008) [arXiv:0803.0380 [hep-ph]]. 8. T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev. D 64, 035002 (2001) . 9. N. Maru and N. Okada, Phys. Rev. D 77, 055010 (2008). 10. A. Falkowski, Phys. Rev. D 77, 055018 (2008); G. Cacciapaglia, A. Deandrea and J. Llodra-Perez, JHEP 0906, 054 (2009); I. Low, R. Rattazzi and A. Vichi, arXiv:0907.5413 [hep-ph]. 11. See, for example, A. Djouadi, arXiv:hep-ph/0503172, references therein. 12. F. J. Petriello, JHEP 0205, 003 (2002). 13. G. Cacciapaglia, C. Csaki and S. C. Park, JHEP 0603, 099 (2006).

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WL WL Scattering in Higgsless Models: Identifying Better Effective Theories Alexander S. Belyaeva , R. Sekhar Chivukulab , Neil D. Christensenb , Hong-Jian Hec , Masafumi Kurachid,∗ , Elizabeth H. Simmonsb and Masaharu Tanabashie a School

of Physics & Astronomy, University of Southampton Highfield, Southampton SO17 1BJ, UK Particle Physics Department, Rutherford Appleton Laboratory Chilton, Didcot, Oxon OX11 0QX, UK b Department of Physics and Astronomy, Michigan State University East Lansing, MI 48824, USA c Center for High Energy Physics, Tsinghua University, Beijing 100084, China d Department of Physics, Tohoku University, Sendai 980-8578, Japan e Department of Physics, Nagoya University, Nagoya 464-8602, Japan The three site Higgsless model has been offered as a benchmark for studying the collider phenomenology of Higgsless models. In this talk, we present how well the three site Higgsless model performs as a general representative of Higgsless models, in describing WL WL scattering, and which modifications can make it more representative. We employ general sum rules relating the masses and couplings of the Kaluza-Klein (KK) modes of the gauge fields in continuum and deconstructed Higgsless models as a way to compare the different theories. After comparing the three site Higgsless model to flat and warped continuum Higgsless models, we analyze an extensions of the three site Higgsless model, namely, the Hidden Local Symmetry (HLS) Higgsless model. We demonstrate that WL WL scattering in the HLS Higgsless model can very closely approximate scattering in the continuum models, provided that the parameter ‘a’ is chosen to mimic ρ-meson dominance of ππ scattering in QCD. Keywords: Higgsless model, WL WL scattering, sum rule, Hidden Local Symmetry.

1. Introduction Higgsless model1 is an attractive alternative to the Standard Model (SM) for describing the Electroweak symmetry breaking, in which the symmetry is broken by the boundary conditions of the five dimensional gauge theory. It turned out that dimensional deconstruction2 is quite useful to understand the important nature of the model, such as the delay of perturbative unitarity violation, boundary conditions, etc. (See Refs.3,4 ) It was also extensively used to study the constraints from the electroweak precision measurements.5 The three site Higgsless model6 was proposed as an extremely deconstructed version of five dimensional Higgsless models, in which only one copy of weak gauge ∗ Speaker,

Email: [emailprotected].

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Fig. 1. Gauge sector of the three site Higgsless model. The solid circles represent SU (2) gauge groups, with coupling strengths g0 and g1 , and the dashed circle is a U (1) gauge group with coupling g2 .

boson (W 0 , Z 0 ) is introduced as a new resonance. Such a model contains sufficient complexity to incorporate interesting physics issues related to fermion masses and electroweak observables, yet remains simple enough that it could be encoded in a Matrix Element Generator program for use with Monte Carlo simulations. Such program was already done by several groups.7 In this talk, we present how well the three site Higgsless model performs as a general representative of Higgsless models, in describing WL WL scattering, and which modifications can make it more representative. After briefly reviewing the three site Higgsless model, we compare the three site Higgsless model to flat and warped continuum Higgsless models. Then, we analyze a Hidden Local Symmetry (HLS) generalization of the three site Higgsless model.a

2. The three site Higgsless model In this section, we briefly review the three site Higgsless model.6 The gauge sector of the three site Higgsless model is illustrated in Fig. 1 using “Moose notation”.9 The model incorporates an SU (2) × SU (2) × U (1) gauge group (with couplings g 0 , g1 and g2 , respectively), and two nonlinear (SU (2)×SU (2))/SU (2) sigma models in which the global symmetry groups in adjacent sigma models are identified with the corresponding factors of the gauge group. The symmetry breaking between the middle SU (2) and the U (1) follows an SU (2)L × SU (2)R /SU (2)V symmetry breaking pattern with the U (1) embedded as the T3 -generator of SU (2)R . This extended electroweak gauge sector is in the same class as models of extended electroweak gauge symmetries,10 which are considered as an application of the hidden local symmetry11 to the electroweak sector.b The decay constants, f1 and f2 , of two nonlinear √ sigma models can be different in general, however, we take f1 = f2 = f (= 2v) for simplicity. Also, we work in the limit x ≡ g0 /g1 1, y ≡ g2 /g1 1 , in which case we expect a massless photon, light W and Z bosons, and a heavy set of bosons W 0 and Z 0 . Numerically, then, g0,2 are approximately equal to the standard model SU (2)W and U (1)Y couplings, and we therefore denote g0 ≡ g and g2 ≡ g 0 , and 0 sin θ s define an angle θ such that gg = cos ˜. θ ≡ c (≡ t). In addition, we denote g1 ≡ g a This b The

talk is based on the work done in Ref.8 new physics discussed in Ref.6 is related to the fermion sector.

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168 Table 1. Leading expressions of Z 0 W W coupling and ZW 0 W coupling in each models. gZ 0 W W gZW 0 W

Three Site MW 1 e −2 s M W0 MW 1 e − 2 sc M 0 W

5D Flat

√ MW − 4π 22 es M W0 √ MW e − 4π 22 sc MW 0

5D Warped MW −0.36 M W0 MW −0.36 1c M 0 W

3. Comparison between continuum and the three site Higgsless models The three site Higgsless model can be viewed as an extremely deconstructed version of 5-dimensional SU (2)×SU (2) Higgsless model in which SU (2)×SU (2) symmetry is broken down to its diagonal SU (2) by the boundary condition (BC) at one end of the extra dimension, while one of SU (2) is broken down to its U (1) subgroup by the BC at the other end. Thus, it is tempting to investigate how well the three site Higgsless model performs as a low energy effective theory of continuum Higgsless models. For this purpose, we consider SU (2) × SU (2) Higgsless model in the flat and the warped extra dimension as example models to compare. The electroweak gauge sector of SU (2)×SU (2) Higgsless models in both the flat and the warped extra dimension can be characterized (with certain simplifications) by four free parameters. In flat case, those are R (the size of the extra dimension), g5 (the bulk gauge coupling), g0 and gY (couplings for brane localized kinetic terms of SU (2) × U (1) gauge bosons). In warped case, those are R 0 (the seize of extra dimension), b (warp factor), g5 and gY . (See Ref.12 for detailed descriptions of the models.) Since the gauge sector of the three site Higgsless model is also characterized by four free parameter (see, Fig. 1), these three models can be compared by choosing four parameters so that they reproduce same values of four physical quantities. Three of four parameters should be chosen in a way that electroweak observables (e, MZ and MW , for example) are correctly reproduced. Then, there is still one free parameter to fix. Here, we use that parameter to fix the scale of KK mode (the mass of W 0 boson, for example). Now, there are no free parameter left in each model, and any physical quantities other than those four quantities can be calculated as predictions in each models, which should be compared among three models to see how much the three site Higgsless model approximates continuum Higgsless models. In the present analysis, we focus on the values of triple gauge boson couplings each model predicts. As for triple gauge boson couplings which involve only SM gauge bosons, it can be shown that those are of the same form as in the SM at the 2 2 /MW leading order in the expansion of MW 0 , and we require that the deviation from the SM value, which appears as next to leading order, should be within experimental bound (which put a lower bound on MW 0 ). In Table 1, we listed the leading expressions of Z 0 W W coupling and ZW 0 W √ 4 2e coupling in each models. Making the numerical approximations π2 s ' 0.36 and √ gZ 0 W W |warped−5d g 0 W |warped−5d 4 2 e ' ZW ' 1. In other words, the π 2 sc ' 0.41, we find, gZ 0 W W |flat−5d gZW 0 W |flat−5d values of gZ 0 W W and gZW 0 W in these continuum models are essentially independent

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of the geometry of the extra dimension to leading order. Then we can compare the couplings in continuum models to those in the three site Higgsless model, assuming 2 0 W |three−site π W |three−site √ a common value for MW /MW 0 : gZg 0 W ' gZW gZW 0 W |flat−5d ' 8 2 ' 0.87 . Z 0 W W |flat−5d The values of gZ 0 W W and gZW 0 W in the three site Higgsless model are about 13% smaller than those values in 5-dimensional SU (2) × SU (2) Higgsless models. Then, the question which naturally arises is — “Why do gZ 0 W W and gZW 0 W take similar values in different models?” There are two keywords to be addressed to answer to this question: sum rules and the lowest KK mode dominance. In any continuum five-dimensional gauge theory, the sum rules that guarantee the absence, respectively, of O(E 4 ) and O(E 2 ) growth in the amplitude for WL+ WL− → WL+ WL− elastic scattering have the following form,1,13 ∞ X i=1

3

∞ X i=1

2 2 2 gZ = gW W W W − gZW W − gγW W , iW W

2 2 2 2 gZ MZ2 i = 4gW W W W MW − 3gZW W MZ , iW W

(1)

(2)

where Zi represents the i-th KK mode of the neutral gauge boson. (Z1 is identified as Z 0 .) We focuses on the degree to which the first KK mode saturates the sum on the LHS of the identities (1) and (2). Suppose that we form the ratio of the n = 1 term in the sum on the LHS to the full combination of terms on the RHS, evaluated to leading order in (MW /MW 0 )2 . The ratios derived from Eqs. (1) and (2) are: gW W W W

2 gZ 0W W , 2 2 − gZW W − gγW W

2 2 3gZ 0 W W MZ 0 2 2 2 . 4gW W W W MW − 3gZW W MZ

(3) (4)

If the n = 1 KK mode saturates the identity, then the related ratio will be 1.0; ratio values less than 1.0 reflect contributions from higher KK modes. We see from Table 2 that each of these ratios is nearly 1.0 in both the SU (2) × SU (2) flat and warped Higgsless models, confirming that the first KK mode nearly saturates the sum rules in these continuum models. The similar behavior of the two extra dimensional models is consistent with our finding that the gZ 0 W W coupling is relatively independent of geometry. Because the first KK mode nearly saturates the identities (1) and (2) in these continuum models, the ratios (3) and (4) should be useful for drawing comparisons with the three site Higgsless model, which only possesses a single KK gauge mode. As shown in the 3rd column of Table 2, the first ratio has the value one in the three site Higgsless model, meaning that the identity (1) is still satisfied. The ratio related to identity (2), however, has the value 3/4 for the three site Higgsless model, meaning that the second identity is not satisfied; the longitudinal gauge boson scattering amplitude continues to grow as E 2 due to the underlying non-renormalizable interactions in the three site Higgsless model. Since the value of the denominator in

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170 Table 2. Ratios relevant to evaluating the degree of cancellation of growth in the WL WL scattering amplitude from the lowest lying KK resonance at order E 4 (top row, from Eq. (3)), and at order E 2 (second row, from Eq. (4)). A value close to one indicates a high degree of cancellation from the lowest lying resonance. Shown in successive columns for the SU (2) × SU (2) flat and warped continuum models discussed in the text, and the three site Higgsless model. 2 gZ 0W W

2 2 gW W W W −gZW −gγW W W 2 2 3gZ 0 W W MZ 0

2 −3g 2 2 4gW W W W MW MZ ZW W

5d 2 × 2 Flat 960 π6 96 π4

5d 2 × 2 Warped

Three-site

' 0.999

0.992

1

' 0.986

0.986

3/4

Eq. (4) has not changed appreciably, this indicates a difference between the values of the gZ 0 W W couplings in the continuum and three site Higgsless models. This is 2 the reason why gZ 0 W W in the three site Higgsless model is 13% (or 25% in gZ 0W W ) smaller than the value of gZ 0 W W in continuum models as we have shown in the previous section. 4. Hidden Local Symmetry generalization of the three site Higgsless models In this section, we consider an Hidden Local Symmetry (HLS) generalization of the three site Higgsless model, in which an extra parameter a is introduced and the Lagrangian takes the following form:11 i2 i2 h v2 h v2 LHLS = − Tr (Dµ Σ†1 )Σ1 − (Dµ Σ2 )Σ†2 − a Tr (Dµ Σ†1 )Σ1 + (Dµ Σ2 )Σ†2 , 4 4 v2 † µ (1 + a)Tr (Dµ Σ1 ) (D Σ1 ) + (Dµ Σ2 )† (Dµ Σ2 ) = 4 h i v2 + (1 − a)Tr (Dµ Σ1 )† Σ1 (Dµ Σ2 )Σ†2 , 2 av 2 Tr (Dµ Σ1 )† (Dµ Σ1 ) + (Dµ Σ2 )† (Dµ Σ2 ) = 2 v2 + (1 − a) Tr (Dµ (Σ1 Σ2 ))† Dµ (Σ1 Σ2 ) . (5) 4 Note that taking a = 1 reproduce the Lagrangian of the three site Higgsless model. Fermion sector of the model is also discussed in detail in Ref.,8 in which we showed that the model can accommodate the ideal fermion delocalization which is needed for the consistency with the precision EW experiments. It is straightforward to calculate triple gauge boson couplings of this model, from which we can evaluate the values in Eqs. (3) and (4): gW W W W

2 gZ 0W W = 1, 2 2 − gZW W − gγW W

2 2 3gZ 3 0 W W MZ 0 2 2 2 = 4 a. 4gW W W W MW − 3gZW M W Z

(6)

By comparing this result with the ones shown in Table 2, we see that the HLS Higgsless model can very closely approximate scattering in the continuum models if we take a = 43 .

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T00

0.6 cont.

0.4

a=43 0.2 0.0 a=2 -0.2

1000

2000

3000

s HGeVL

4000

5000

Fig. 2. Behavior of the partial wave amplitude T00 for NG boson scattering in the triangular moose model with various a. The values v = 250 GeV, M1 = 500 GeV are assumed. The curve labeled “cont.” shows T00 in the continuum flat SU (2) × SU (2) model for M1 = 500 GeV.

It is interesting to note that a = 34 is the choice in which the four-point NambuGoldstone (NG) boson coupling, gππππ = 1 − 34 a, vanishes in this model.14 Considering that, in the language of dimensional deconstruction,2 NG bosons are identified as a fifth-component of the gauge boson field (A5 ) in a discretized five-dimensional gauge theory, and also considering the fact that there is no four-point A5 coupling in five dimensional gauge theories, it is natural that the parameter choice in which the four-point NG boson coupling vanishes well approximates the continuum models. Fig. 2 shows the partial wave amplitude T00 for NG boson scattering in the global (which means we set g = g 0 = 0 for simplicity) continuum flat SU (2)×SU (2) model (with the mass of the first KK mode, M1 , taken to be M1 = 500 GeV) compared with T00 in the HLS Higgsless model for several values of the parameter a. The result in the globalthree site Higgsless model is shown by the curve labeled a = 1; the value a = 2 is motivated by the phenomenological KSRF relation.15 This plot ties our results together quite neatly: while the curves with three different values of a all give a reasonable description of T00 at very low energies, the best approximation to the continuum behavior of T00 over a wide range of energies is given by the HLS Higgsless model curve with a = 4/3. At low energies, the fact that the three-site and the HLS Higgsless models both prevent E 4 growth of the amplitude suffices; but at higher energies, the fact that the HLS Higgsless model with a = 4/3 has gππππ = 0 and enables it to cut off the E 2 growth of the amplitude as well, as consistent with the behavior in the continuum model. 5. Conclusions We have considered how well the three site Higgsless model performs as a general representative of Higgsless models, and have studied HLS generalization have the potential to improve upon its performance. Our comparisons have employed sum

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rules relating the masses and couplings of the gauge field KK modes. We find that the tendency of the sum rules to be saturated by contributions from the lowestlying KK resonances suggests a way to quantify the extent to which a highlydeconstructed theory like the three site Higgsless model can accurately describe the low-energy physics. We demonstrated that WL WL scattering in the HLS Higgsless model can very closely approximate scattering in the continuum models, provided that the HLS parameter a is chosen appropriately. This observation confirms that the collider phenomenology studied, such as in Ref.7 , are applicable not only to the three site Higgsless model, but also to extra-dimensional Higgsless models. References 1. C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Phys. Rev. D 69, 055006 (2004); C. Csaki, C. Grojean, L. Pilo and J. Terning, Phys. Rev. Lett. 92, 101802 (2004). 2. N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Rev. Lett. 86, 4757 (2001); C. T. Hill, S. Pokorski and J. Wang, Phys. Rev. D 64, 105005 (2001). 3. R. S. Chivukula, D. A. Dicus and H. J. He, Phys. Lett. B 525, 175 (2002); R. S. Chivukula and H. J. He, Phys. Lett. B 532, 121 (2002). 4. H. J. He, Int. J. Mod. Phys. A 20 (2005) 3362. 5. R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Rev. D 70, 075008 (2004); Phys. Lett. B 603, 210 (2004); Phys. Rev. D 71, 035007 (2005); Phys. Rev. D 71, 115001 (2005); Phys. Rev. D 72, 015008 (2005); Phys. Rev. D 72, 095013 (2005). 6. R. Sekhar Chivukula, B. Coleppa, S. Di Chiara, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Rev. D 74, 075011 (2006). 7. H. J. He et al., Phys. Rev. D 78, 031701 (2008); J. G. Bian et al., Nucl. Phys. B 819, 201 (2009). 8. A. S. Belyaev, R. Sekhar Chivukula, N. D. Christensen, H. J. He, M. Kurachi, E. H. Simmons and M. Tanabashi, Phys. Rev. D 80, 055022 (2009). 9. H. Georgi, Nucl. Phys. B 266, 274 (1986). 10. R. Casalbuoni, S. De Curtis, D. Dominici and R. Gatto, Phys. Lett. B 155, 95 (1985); R. Casalbuoni, A. Deandrea, S. De Curtis, D. Dominici, R. Gatto and M. Grazzini, Phys. Rev. D 53, 5201 (1996). 11. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54, 1215 (1985); M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B 259, 493 (1985); M. Bando, T. Fujiwara and K. Yamawaki, Prog. Theor. Phys. 79, 1140 (1988); M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988). 12. R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Rev. D 72, 075012 (2005). 13. R. S. Chivukula, H. J. He, M. Kurachi, E. H. Simmons and M. Tanabashi, Phys. Rev. D 78, 095003 (2008). 14. See section 3 in M. Harada and K. Yamawaki, Phys. Rept. 381, 1 (2003); The relation between a = 4/3 and ρ meson dominance was also pointed out in M. Harada, S. Matsuzaki and K. Yamawaki, Phys. Rev. D 74, 076004 (2006). 15. K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16, 255 (1966); Riazuddin and Fayyazuddin, Phys. Rev. 147, 1071 (1966).

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Holographic Estimate of Muon g − 2 Deog Ki Hong Department of Physics, Pusan National University Busan 609-735, Korea E-mail: [emailprotected] I present recent calculations of the hadronic contributions to muon anomalous magnetic moment in holographic QCD, based on gauge/gravity duality. The holographic estimates are compared well with the analysis based on recently revised BaBar measurements of e+ e− → π + π − cross-sections and also with other model calculations for the light-bylight scattering contributions. Keywords: Anomalous magnetic moment, muon, gauge/gravity duality.

1. Introduction The successful prediction of anomalous magnetic moments in early days of particle physics was one of the first triumphs of quantum field theory.1 Since then it has served as a critical test of the standard model of particle physics, which is based on quantum field theory, guided by gauge principle. Recent measurement of anomalous magnetic moment of muon (E821 at BNL)2 provides the most stringent test so far, at the level of sub parts per million, of the standard model, gµ − 2 = 11659208.0(5.4)(3.3) × 10−10 , 2 which exhibits currently 3.2σ deviation from the standard model estimate: = aexp µ

SM −11 ∆aµ = aexp . µ − aµ = 302(63)(61) × 10

(1)

(2)

If the discrepancy persists in more improved measurements, such as the E969 experiment, planned at BNL, or other experiments at the Fermi Lab, it will certainly indicate a hint of new physics. Therefore it is quite necessary to understand the standard model predictions more precisely to pin down the possible hint of new physics at the level of 5σ or more. The theoretical prediction of muon g − 2 in the standard model consists of three different contributions: QED + aweak + ahad ath µ = aµ µ µ .

Among them the QED contribution is most dominant one and has been calculated = 116584718.10(0.16) × 10−11 at 4.5 loops by Kinosh*ta et al,3 while the to be aQED µ

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weak interaction corrections are found to be aweak = 154(2) × 10−11 at the two-loop µ 4 level. As hadrons (or quarks) contribute to the anomalous magnetic moment of muon only through quantum fluctuations, the strong interaction contribution is suppressed, compared to that of QED. The current estimate of the hadronic contributions is = 116591778(2)(46)(40) × 10−11 . ahad µ

(3)

The most uncertainty in the SM estimate is however coming from the hadronic contributions, since we poorly understand strong dynamics, while the electroweak corrections can be calculated accurately in perturbation. Direct calculation of hadronic contributions from QCD requires lattice calculations, which are currently not accurate enough to give a meaningful result due to large systematic uncertainties.5 In this talk I present new estimates of the hadronic contributions, based on a holographic model of QCD. Holographic models have been proposed recently for QCD,6,7 inspired by the gauge/gravity duality, found in string theory,8 which states that certain strongly coupled gauge theories are equivalent to weakly coupled gravity in one-higher dimension, the extra dimension being the renormalization scale. The holographic models of QCD were found to be quite successful to account for the properties of hadrons at low energy and give relations to their couplings and also new sum rules,6,7,9,10 offering theoretical understanding of phenomenological rules for hadrons found in 60’s such as vector meson dominance, proposed by Sakurai.11 2. Holographic QCD Holographic QCD (hQCD) is a model for 5D gravity dual theory of QCD in the large Nc and large ’t Hooft coupling limit (λ ≡ gs2 Nc 1), describing QCD directly with hadrons. As a gravity dual of QCD with three light flavors we consider U (3)L ×U (3)R flavor gauge theory in a slice of AdS5 , √ 1 (4) S = d5 x g Tr |DX|2 + 3|X|2 − 2 (FL 2 + FR 2 ) + SY + SCS 4g5 where the metric is given as, taking the AdS radius R = 1, ds2 =

1 (dxµ dxµ − dz 2 ), z2

≤ z ≤ zm .

(5)

We take the ultraviolet (UV) regulator → 0 and introduce an infra-red (IR) brane at zm to implement the confinement of QCD, breaking the conformal symmetry.7 The bulk scalar (X) and the bulk gauge fields (A) are dual to q¯ q and q¯γ µ T a q, respectively. Following Katz and Schwartz,12 we have introduced a flavor-singlet bulk scalar (Y ) for the η meson, which is dual to F 2 (F F˜ ) and is described by 1 κ Nf 5 √ 2 d x g |DY | − (Y det(X) + h.c.) . (6) SY = 2 2

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µ

µ

had Fig. 1.

Leading hadronic-vacuum-polarization contribution to muon g − 2.

Finally we introduce a Chern-Simons term to reproduce the QCD flavor anomaly,13 Nc SCS = (7) [ω5 (AL ) − ω5 (AR )] , 24π 2 where d ω5 (A) = Tr F 3 . Our calculations can be easily applied to other holographic models like Sakai-Sugimoto model, which will give similar results. Gauge/gravity duality at large Nc and large λ implies that the generating functional of one-particle irreducible Green’s functions is given by the classical gravity action of the dual theory: W4D [φ0 (x)] = S5Deff [φ(x, )]

with φ(x, ) = φ0 (x) .

(8)

Using this equality we will be able to calculate the hadronic contributions at the leading order in 1/Nc and 1/λ expansion. 3. Holographic calculations of hadronic contributions The strong interaction contributions to the muon magnetic moment consist of three pieces; the hadronic vacuum polarization (HLO), the higher-order hadronic vacuumpolarization effect, and the hadronic light-by-light (LBL) scattering. The HLO contribution (see Fig. 1) is given as5 2 ∞ HLO 2 α 2 ¯ had aµ = 4π dQ2 f (Q2 ) Π (9) em (Q ) , π 0 2 had 2 had ¯ had where the vacuum polarization is defined as Π em (Q ) = Πem (Q ) − Πem (0) and the kernel is 2 2 3 2 Q2 − Q4 + 4m2µ Q2 m Q Z (1 − Q Z) µ f (Q2 ) = with Z = − . (10) 1 + m2µ Q2 Z 2 2m2µ Q2

From the AdS/CFT formula (8) the vector current correlator ΠV (q 2 ) can then be expressed in terms of the infinite set of the vector meson wave-functions ψVn (z), as

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had

=

=ρ,0 ω, ...

0(n)

V

Fig. 2. A diagram illustrating neutral vector meson exchanges giving dominant contributions to ΠV at the large Nc limit.

0.006 0.005

f Q2 emQ2

0.004 0.003 0.002 0.001 0.000 0.00

0.02

0.04 Q2 GeV2

0.06

0.08

¯ em (Q2 ) which has a peak around Q2 = m2µ : Fig. 3. Comparison of the integral kernel f (Q2 )Π The dashed curve corresponds to the result including full contributions from the infinite tower of vector mesons, while four bold curves are obtained by integrating out the infinite tower at the levels of n = 1, 2, 3, 4. The dashed curve is almost (within about 1% deviation) reproduced when n = 4. In the plot the number of flavors Nf is taken to be 2.

shown in Fig. 2, which is the holographic realization of the vector meson dominance proposed by Sakurai, ΠV (q 2 ) =

∞ 1 [ψ˙ Vn ()/]2 , g52 n=1 (q 2 − MV2n )MV2n

(11)

where the dot denotes the derivative with respect to z, ψ˙ ≡ ∂z ψ. Using the holographic renormalization to take care of the UV divergence and keeping only the first four low-lying states, we find for Euclidean momentum Q2 = −q 2 ≥ 014 ¯ V (Q2 ) Π

4

Q2 FV2n + O(Q2 /(MV25 )) 2 + M 2 )M 4 (Q Vn Vn n=1

(12)

where FVn is the decay constant of n-th excited vector mesons. Plugging the holographic vacuum polarization (12) into the formula (9), we obtain14 N =2

f aHLO |AdS/QCD = 470.5 × 10−10 , µ

(13)

which agrees, within 30% errors, with the currently updated value,15 estimated from new 2009 BaBar data16 aHLO [ππ]|BABAR = (514.1 ± 3.8) × 10−10 . µ

(14)

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k

q1 µ− Fig. 4.

Light-by-light corrections to muon g − 2. k

k

k

π 0 , η, η

π 0 , η, η

µ−

q3

q2

q1

Fig. 5.

q2

q3

µ−

q1

q2

π 0 , η, η

q3 µ−

q1

q2

q3

Light-by-light correction is dominated by the pseudo scalar mesons exchange.

For the hadronic light-by-light corrections, shown in Fig. 4, we need to calculate 4-point functions of flavor currents. Since there is no quartic term for AQem (Qem = 1/2 + I3 ), there is no 1PI 4-point function for the EM currents in hQCD, because higher order terms like F 4 or F 2 X 2 terms are suppressed. In hQCD the LBL diagram is therefore dominated by VVA or VVP diagrams, which come from the Chern-Simons term, Eq. 7: Fγ ∗ γ ∗ P (A) (q1 , q2 ) =

δ3 SCS δV (q1 )δV (q2 )δA(−q1 − q2 )

where the gauge fields satisfy in the axial gauge, V5 = 0 = A5 ,

1 q2 ∂z Vµaˆ (q, z) + Vµaˆ (q, z) = 0 , ∂z z z

⊥ 2 1 q aˆ g52 v 2 aˆ a ˆ ∂z Aµ + Aµ − 3 Aµ ∂z = 0, z z z ⊥

(15)

(16) (17)

where ⊥ denotes the projection onto the transversal components. For two flavors the longitudinal components, Aaµ = ∂µ φa , and the phase of bulk scalar X are related by EOM as

1 g2 v2 ∂z φa + 5 3 (π a − φa ) = 0 , (18) ∂z z z g2v2 −q 2 ∂z φa + 5 2 ∂z π a = 0 . (19) z

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Q2 FΠ ΓΓ 1

2

3

4

5

Q2

0.05

0.10

0.15

Fig. 6.

Fπγ ∗ γ (Q2 , 0) for lower part; Fπγ ∗ γ ∗ (Q2 , Q2 ) for upper part (Brodsky-Lepage). Table 1.

Muon g − 2 in unit of 10−10 from AdS/QCD.

Vector modes

aπ µ

aηµ

aηµ

aPS µ

4 6 8

7.5 7.1 6.9

2.1 2.5 2.7

1.0 0.9 1.1

10.6 10.5 10.7

The anomalous form factor is then given as,17 with ψ a (z) = φa − π a and Jq = V (iq, z) Nc Fπγ ∗ γ ∗ = ψ(zm )J(Q1 , zm )J(Q2 , zm ) − (20) ∂z ψJQ1 JQ2 , 12π 2 z where we take as boundary conditions ψ() = 1 and ∂z ψ(z)|zm = 0. To calculate the hadronic LBL contribution to the muon anomalous magnetic moment we expand the photon line in the anomalous form factor (20) as J(−iQ, z) = V (q, z) =

−g5 fρ ψρ (z) q 2 − m2ρ + i ρ

The result is shown in Table 1 for several choices of vector-mode truncations.18 Our results are comparable with recent results by A. Nyffeler,19 obtained in the LMD+V model; −10 aPS . µ = 9.9(1.6) × 10

(21)

4. Discussions In the era of electroweak precision, it becomes more important to understand precisely QCD corrections to the electroweak processes. As being non-perturbative strong dynamics, QCD corrections are often difficult to estimate and one resorts to lattice calculations, which are, however, not precise enough for certain measurements such as muon anomalous magnetic moment. Recent development in gauge/gravity

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duality shows that in the large Nc and large λ limit the estimate of QCD contributions can be made precisely in holographic QCD, thus may be useful in assessing the new physics effects. I present recent estimates14,18 of anomalous magnetic moment of muon in holographic QCD, which are found to be in consistent with other calculations. Acknowledgments D.K.H. thanks Doyoun Kim and Shinya Matsuzaki for the collaborations upon which this talk is based on. This work is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314- C00052). References 1. J. S. Schwinger, Phys. Rev. 82, 664 (1951). 2. G. W. Bennett et al. [Muon G-2 Collaboration], Phys. Rev. D 73, 072003 (2006). 3. For recent five-loop calculations, see T. Aoyama, M. Hayakawa, T. Kinosh*ta and M. Nio, Phys. Rev. D 78, 113006 (2008). 4. A. Czarnecki, B. Krause and W. J. Marciano, Phys. Rev. Lett. 76, 3267 (1996). 5. T. Blum, Phys. Rev. Lett. 91, 052001 (2003). 6. T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005). 7. J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 95, 261602 (2005); L. Da Rold and A. Pomarol, Nucl. Phys. B 721, 79 (2005). 8. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998); E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998). 9. D. K. Hong, T. Inami and H. U. Yee, Phys. Lett. B 646, 165 (2007). 10. D. K. Hong, M. Rho, H. U. Yee and P. Yi, Phys. Rev. D 76, 061901 (2007); JHEP 0709, 063 (2007); Phys. Rev. D 77, 014030 (2008). 11. J. J. Sakurai, Phys. Rev. Lett. 22, 981 (1969). 12. E. Katz and M. D. Schwartz, JHEP 0708, 077 (2007). 13. S. K. Domokos and J. A. Harvey, Phys. Rev. Lett. 99, 141602 (2007); C. T. Hill, Phys. Rev. D 73, 085001 (2006). 14. D. K. Hong, D. Kim and S. Matsuzaki, arXiv:0911.0560 [hep-ph]. 15. M. Davier, A. Hoecker, B. Malaescu, C. Z. Yuan and Z. Zhang, Eur. Phys. J. C 66, 1 (2010). 16. B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 103, 231801 (2009). 17. H. R. Grigoryan and A. V. Radyushkin, Phys. Rev. D 77, 115024 (2008). 18. D. K. Hong and D. Kim, Phys. Lett. B 680, 480 (2009). 19. A. Nyffeler, Phys. Rev. D 79, 073012 (2009).

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Gauge-Higgs Dark Matter T. Yamash*ta Department of Physics, Nagoya University Nagoya 464-8602, Japan E-mail: [emailprotected] We discuss a dark matter candidate peculiar in the gauge-Higgs unification scenario∗ .

1. Introduction The standard model (SM) of the particle physics successfully explains the experimental results with energies up to the electroweak scale. Nevertheless, we have several motivations to explore the physics beyond the SM. One of them is the hierarchy problem, and another one is the absence of dark matter (DM) candidate. In this talk, we discuss a possible solution to these two issues. Among the proposals for the former problem, we examine the so-called gaugeHiggs unification scenario, in which the Higgs field is identified with an extradimensional component of a higher-dimensional gauge field. This scenario, thus, works in higher-dimensional gauge theories, for example a five dimensional model with the S 1 /Z2 orbifold that we discuss here. To be more concrete, we write the bulk gauge symmetry as G, which is broken at the two endpoints down to H1 and H2 , respectively. This model can be viewed as four-dimensional theories with infinite copies of the hidden local symmetry G and one H1 and one H2 , written in a open moose diagram with H1 (H2 ) resides on a one (the other) boundary, via the deconstruction method [2] (latticization). In this view, the extra-dimensional component of the gauge field appears as the ”link” field which is the pseudo-NambuGoldstone (pNG) modes originated from the process of the ”deconstruction” [2]. The diagonalization of the mass matrix for these pNG modes leads to a massless modes on G/H1 ∩ G/H2 (at the tree-level). For instance, if we set G =SO(5) and H1 = H2 =SO(4) a , the remaining symmetry is SO(4)SU(2)L ×SU(2)R , and the massless pNG appears as a vector of SO(4), that is, a bi-doublet of SU(2)L ×SU(2)R which can be identified with our Higgs field. Since this pNG mode is for the collective breaking of the infinite symmetry groups, the Higgs mass is protected from ultraviolet divergences, solving the hierarchy problem. ∗ This

proceeding is based on our work Ref. [1], to appear in JHEP, in collaboration with Naoyuki Haba, Shigeki Matsumoto and Nobuchika Okada. a In our analysis, we add a U(1) symmetry to reproduce a correct Weinberg angle, in addition to the color SU(3) symmetry.

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So far, the latter issue has not been investigated in this scenario, except for a few literatures [3, 4], and so we examine a DM candidate peculiar in this scenario. We discuss first a new candidate of the DM in general higher-dimensional models, in Sec. 2. Then, we apply this to the gauge-Higgs unification scenario and show that a weakly interacting massive particle (WIMP) can be realized, in Sec. 3. 2. A New Candidate of the Dark Matter In this section, we claim that the lightest mode among the anti-periodic (AP) fields, if exist, can be stable and thus be a good DM candidate. In models with the toroidal compactification, no matter with or without further orbifolding, the Lagrangian should be invariant under a discrete coordinate shift along the compactified direction, for example as L(x, y + 2πR) = L(x, y),

(1)

where x (y) is the non-compact four (fifth) dimensional coordinate and R is radius of the toroidal compactification. This invariance, or periodicity, represents that the fundamental region of this direction is finite and thus the direction is actually compactified. A point is that we can impose the AP boundary condition under the shift to a field Φ: Φ(x, y + 2πR) = −Φ(x, y),

(2)

Then, they can not appear alone, but always in pairs, in the Lagrangian so as to make it periodic. This means that this Lagrangian has an accidental Z2 symmetry, under which the AP fields are odd while the periodic fields are even. This accidental symmetry forbids the lightest mode among the AP fields from decaying, and thus the lightest mode can be a good candidate of the DM if it is colorless and neutral. In this way, if we introduce AP fields in higher-dimensional models, a DM candidate can be accommodated. There are, however, in general no motivations to introduce AP fields, other than serving the DM candidate. It is, of course, not bad, but it is more desirable if there are. In the next section, we discuss a model with such a motivation. 3. Gauge-Higgs Dark Matter We examine the above new DM candidate in a model of the gauge-Higgs unification. It is known that the Kaluza-Klein (KK) scale, the top mass and the Higgs mass tend to be too small in this scenario with flat metric, while these can be made heavy enough easily in models with warped metric. In the latter models, as a general property of the warped models, the contributions to the electroweak parameters tend to be too large, and it is claimed in Ref. [5] that the Wilson line phase which corresponds to the Higgs vacuum expectation value (VEV) in this scenario should be small to suppress the contributions. In Ref. [6], we have shown that if we introduce

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WMAP

0.1

5 Te

V

10

0 Fig. 1.

0.2

c

0.4

0.6

WMAP

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10 TeV

AP

M

V Te

θW

CDMSII & XENON10 W

2

0.2

−42

10

0.1 TeV

0.5 TeV V 1 Te

0.3

1 0.1 0.5 Dark Matter Mass (TeV)

5

The relic abundance (left) and the direct detection (right).

AP fermions, a small VEV can be obtained. This is the motivation we mentioned aboveb . The lightest mode among these AP fermions introduced to suppress the contributions is stable and thus a good candidate of the DM, as discussed in Sec. 2. A bonus by applying this new way in the gauge-Higgs unification scenario is that the relevant interactions of the fermionic DM are the gauge interactions and those to the Higgs field, and both of them are controlled by the gauge symmetry since the Higgs field is a part of the gauge field. This leads to a peculiar predictions [1], and we call this candidate gauge-Higgs dark matter. We investigate this possibility in a model with the SO(5)×U(1) gauge symmetry which is broken down to SO(4)×U(1) by the boundary conditions, further down to SU(2)×U(1) by a boundary Higgs mechanism and finally to U(1) symmetry by the VEV of the Wilson line phase θW [1]. We do not specify the SM fermion sector, just by supposing it goes well, and treat the Higgs mass mh and θW , which should be determined by the loop induced effective potential which is calculable if the fermion sector is also fixed, as free parameters. We introduce a AP fermion with 50 quantum number of SO(5)×U(1), in addition to the SM fermion sector, as the DM candidate. For this fermion multiplet, we can introduce a parity odd bulk mass term [7], c, which is our third free parameter. Then, we derive the four-dimensional effective Lagrangian for light KK modes, as function of the three free parameters (mh , θW and c) [1]. Using the Lagrangian, we calculate the coannihilation cross sections, and then solve the Boltzmann equation to predict the thermal relic abundance of the DM candidate. In a similar way, we also evaluate the elastic scattering cross sections between the dark matter particle and nucleon to compare with the bounds from the direct detection experiments. The results are depicted in Figure 1, where the Higgs boson mass is set as mh = 120 GeV. The relic abundances consistent with the observations are drawn in the red b In

Ref. [3], with a similar purpose, a similar Z2 symmetry is imposed by hand.

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regions. There are two allowed regions: the very narrow region in upper-right and a band in upper-left of the left figure, where the contours of fixed dark matter masses are also shown. In the former region, the right relic abundance is achieved by the enhancement of the annihilation cross section through the s-channel Higgs boson resonance, so that the dark matter mass is mDM mh /2 = 60 GeV there. In the other one, the DM mass is around a few TeV, in accordance with the discussion of the WINP scenario. From the right figure, where the present (future expected) bound is shown in brown region (blue lines), we find that a part of the band has been excluded already and will be covered by the coming experiments. References 1. N. Haba S. Matsumoto, N. Okada and T. Yamash*ta, 2. deconstruction 3. M. Regis, M. Serone and P. Ullio, JHEP 0703 (2007) 084; G. Panico, E. Ponton, J. Santiago and M. Serone, Phys. Rev. D 77 (2008) 115012; M. Carena, A. D. Medina, N. R. Shah and C. E. M. Wagner, Phys. Rev. D 79 (2009) 096010. 4. Y. Hosotani, P. Ko and M. Tanaka, Phys. Lett. B 680 (2009) 179. 5. K. Agashe and R. Contino, Nucl. Phys. B 742 (2006) 59. 6. N. Haba, Y. Hosotani, Y. Kawamura and T. Yamash*ta, Phys. Rev. D 70 (2004) 015010. 7. T. Gherghetta and A. Pomarol, Nucl. Phys. B 586 (2000) 141.

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Topological and Curvature Effects in a Multi-Fermion Interaction Model T. Inagaki and M. Hayashi Department of Physics, Hiroshima University Higashi-Hiroshima, Hiroshima 739-8526, Japan A multi-fermion interaction model is investigated in a compact spacetime with nontrivial topology and in a weakly curved spacetime. Evaluating the effective potential in the leading order of the 1/N expansion, we show the phase boundary for a discrete chiral symmetry in an arbitrary dimensions, 2 ≤ D < 4. Keywords: Eight-fermion interaction, dynamical symmetry breaking.

1. Introduction It is believed that a fundamental theory with higher symmetry was realized at the early stage of our universe and the symmetry was broken down to the observed symmetry in the present universe. The mechanism of the symmetry breaking may be found in a dynamics of the fundamental theory. Thus the origin of symmetry breaking is quite important to find a consequence of the fundamental theory in cosmological and astrophysical phenomena. In QCD the chiral symmetry is dynamically broken according to a non-vanishing expectation value for a composite operator constructed by a quark and an antiquark field, q¯q. The chiral symmetry restoration is theoretically predicted in extreme conditions at the QCD scale. The symmetry restoration at high density may be found in phenomena of dense stars. The heavy ion collision at RIHC and LHC provide experimental data for the symmetry restoration at high temperature. We can apply the dynamical mechanism to symmetry breaking at GUT era. It is one of candidates to describe symmetry breaking of the fundamental symmetry in GUT. It is natural to expect that the broken symmetry is restored in extreme conditions at GUT scale. Thus we have launched a plan to study the symmetry restoration at high temperature, high density and strong curvature. A topological effect is also interesting before inflationally expansion of our spacetime. In this paper we focus on the topological and the curvature effects. A variety of works has been done in a simple four-fermion interaction model. The topological effect has been investigated in the spacetime with one compactified dimension, S 1 ,1 and the torus universe.2–6 It has been found that the finite size effect restore the broken symmetry if we adopt the anti-periodic boundary condition to

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the fermion fields. On the other hand, the fermion fields which possess the periodic boundary condition contribute to break the symmetry. The curvature effects has been studied in two,7,8 three,9 four10 and arbitrary dimensions.11 The broken symmetry is restored if the spacetime curvature is positive and strong enough. However, the symmetry is always broken in a negative curvature spacetime.12 A combined effect has been also discussed in a weakly curved spacetime,13 the maximally symmetric spacetime14,15 and Einstein space.16,17 For a review, see for example Ref. 18. In these works the four-fermion interaction model is considered to have something essential as a low energy effective model of a fundamental theory. To discuss the model dependence or independence of above results we have to extend the fourfermion interaction model. Here we consider a multi-fermion interaction model19–23 as a simple extension of the four-fermion interaction model and study the contribution from a higher dimensional operator. In Sec. 2 we introduce a multi-fermion interaction model which is considered in this paper. We show an explicit expression of the effective potential in an arbitrary dimensions, 2 ≤ D < 4. We consider a spacetime RD−1 ⊗S 1 in Sec. 3. Evaluating the effective potential, we study the topological effect. In Sec. 4 we assume that the spacetime curves slowly and investigate the curvature effect. In Sec. 5 we give some concluding remarks. 2. Multi-Fermion Interaction Model As in well-known, the chiral symmetry is dynamically broken by a simple scalar type four-fermion interaction model.24,25 It is a useful low energy effective theory of QCD to describe meson properties. The four-fermion interaction model is also useful as a simple toy model in the study of low energy phenomena of the strong coupling gauge theory at high energy scale in various environments. But there is no reason to neglect higher dimensional operators in extreme conditions at the early universe. In the present paper we extend the model to include scalar type multi-fermion interactions, N n N Gk ¯ D √ µ 2k ¯ ( , (1) ψl iγ (x)∇µ ψl + ψl ψl ) S = d x −g N 2k−1 l=1

k=1

l=1

where index l represents flavors of the fermion field ψ, N is the number of fermion flavors. We neglect the flavor index below. The multi-fermion interaction is unrenormalizable in four spacetime dimensions. The model depends on regularization methods. In this paper we adopt the dimensional regularization and regards the spacetime dimension, D, for the integration of internal fermion lines as one of parameters in the effective theory.26–30 In QCD it can be fixed to reproduce meson properties. Here we leave it as an arbitrary parameter to be fixed phenomenologically. ¯ → −ψψ, ¯ and the The action (1) possesses the discrete chiral symmetry, ψψ i a θa Ta ψ. The discrete chiral symmetry global SU (N ) flavor symmetry, ψ → e prohibits the fermion mass term. We can adopt the 1/N expansion as a non-

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perturbative approach to investigate the dynamical symmetry breaking under the global SU (N ) symmetry. For practical calculations it is more convenient to introduce the auxiliary fields31 and start from the action, n 2k N G σ N 2G k 1 D √ µ ¯ ¯ (x)∇µ ψ + ψψ . (2) Sy = d x −g ψiγ − s σ+ (2G1 )2k 2G1 N k=1

The multi-fermion interactions in the original action are replaced by the auxiliary fields σ and s. If the auxiliary field s develops a non-vanishing expectation value, the fermion field acquires a mass term and the chiral symmetry is eventually broken. In this paper we only consider the case n = 2 for simplicity and concentrate on the contribution from the eight-fermion interaction. To study the phase structure of the model we calculate the expectation value for the auxiliary field, s. It is obtained by observing the minimum of the effective potential. It should be noted ¯ is given by the value of that the expectation value for the composite operator ψψ the auxiliary field σ at the minimum of the effective potential. In the leading order of the 1/N expansion we can analytically integrate out the fermion field and get the effective potential, 1 2 N G2 4 N (σ − s)2 + s − σ + iTr ln x| [iγ µ (x)∇µ − s] |x . (3) V (s, σ) = − 4G1 4G1 16G41

The trace for the Dirac operator in Eq.(3) depends on the spacetime structure. The minimum of the effective potential satisfies the gap equation, ∂V ∂V = = 0. (4) ∂σ ∂s s

σ

If we consider the four-fermion interaction model, G2 = 0, in the Minkowski spacetime, RD , the gap equation allows a nontrivial solution only for a negative G1 . The solution is give by 1/(D−2) (4π)D/2 1 . (5) σ = s = m0 ≡ tr1Γ (1 − D/2) 2G1

Before investigating topological and curvature effects, we numerically calculate the effective potential in the Minkowski spacetime. We are interested in the model where the primordial symmetry is broken down at low energy scale. Thus we confine ourselves to a case of a negative G1 . In this case the effective potential (3) only depends on m0 and the rate of the coupling constants, G2 /G31 . We normalize all the mass scales by m0 and set g = G2 m20 /G31 . As is seen in Fig. 1, a positive g suppresses the symmetry breaking, while a negative g enhances it. A new local minimum appears for a negative g. 3. Phase Structure in a Spacetime with Non-Trivial Topology One of extreme conditions we have to consider at the early universe is the topological effect. It is expected that the boundary condition for matter fields restricts how to

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Fig. 1. Behavior of the effective potential in the tree dimensional Minkowski spacetime R3 . We set g = 0.1, 0 and −0.1 and draw the solid, dotted and dashed lines, respectively.

Fig. 2. Behavior of the effective potential on R2 ⊗ S 1 for δp,1 = 1, as L varies. We set g = 0.1, 0 and −0.1 and draw the solid, dotted and dashed lines, respectively.

compactify the spacetime. In this section we assume that one of space directions is compactified and investigate the multi-fermion interaction model on the cylindrical spacetime, RD−1 ⊗ S 1 . It is a flat spacetime with a non-trivial topology. On RD−1 ⊗ S 1 the effective potential, (3) is given by V (s, σ) (σ − s)2 tr1 D s2 g σ4 = − Γ 1 − − + 2 m20 m20 4 m40 N mD 2(4π)D/2 0 (D−1)/2 2 ∞ 1−D (2n + δp,1 )π 1 tr1 s2 + Γ + 2 , (6) 2 Lm0 n=−∞ Lm0 m0 2(4π)(D−1)/2

where we set δp,1 = 0 for the periodic and δp,1 = 1 for the anti-periodic boundary conditions. In Fig. 2 the effective potential is shown for δp,1 = 1, as L varies. It is clearly seen that the broken symmetry is restored through the second order phase transition as L is decreased. On the other hand only the broken phase is realized for the periodic boundary condition. To see the situation more precisely we perform a rigorous analysis on the critical length Lcr for δp,1 = 1 and find the boundary which divides the symmetric and the broken phases. For a non-negative g only the second order phase transition is realized. In this case we can find the explicit expression for the critical length,18

1/(D−2) 2Γ((3 − D)/2) 3−D (2 Lcr m0 = 2π √ − 1)ζ(3 − D) . (7) πΓ(1 − D/2) It is illustrated by the solid line in Fig. 3. The second local minimum appears for a negative g. The dashed line in Fig. 3 shows the length L where the effective potential satisfies V = 0 at the second local minimum for g = −0.1. We plot the effective potential on this dashed line in Fig. 4. The broken symmetry is restored below both the solid and the dashed lines on D − g plane for g = −0.1.

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Fig. 3. Critical length. The dashed line shows Fig. 4. Behavior of the effective potential on the length to satisfies V = 0 at the second local the dashed line in Fig. 3 for D = 2.25, 2.3, 2.35 minimum for g = −0.1. and 2.4.

4. Phase Structure in a Weakly Curved Spacetime At the early universe the symmetry breaking may be induced under the influence of the spacetime curvature. In this section we assume that the spacetime curved slowly and keep only terms independent of the curvature R and terms linear in R. We discuss the curvature induced phase transition by observing the minimum of the effective potential. Following the procedure developed in Ref. 11, we obtain the effective potential up to linear in R, (σ − s)2 tr1 D s2 g σ4 V (s, σ) = − Γ 1 − − + 2 m20 m20 4 m40 N mD 2(4π)D/2 0 D tr1 R tr1 D |s|D−2 D |s| − − Γ 1 − . (8) Γ 1 − 2 mD 2 mD−2 (4π)D/2 D (4π)D/2 24m20 0 0 In the four-fermion interaction model, g = 0, the phase transition takes place by varying the curvature R. The broken symmetry is restored for a large positive curvature R > Rcr ≥ 0. The phase transition is of the first order for 2 < D < 4. We evaluate the solution of the gap equation and find the analytic expression for the critical curvature,11 (4−D)/(D−2) (4 − D)D Rcr = 6(D − 2) . (9) m20 4 It is drawn as a function of the spacetime curvature in Fig. 5. The critical curvature Rcr is modified by the eight-fermion interaction. For a positive g we find a smaller Rcr except for two and four dimensions. The dashed line shows the curvature to satisfy V = 0 at the second minimum for g = −0.1. Thus the symmetric phase is realized above both the solid and the dashed lines for g = −0.1. In Fig. 6 we show the behavior of the effective potential near the point A

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Fig. 5. Critical curvature for g = −0.1, 0 and Fig. 6. Behavior of the first local minimum of 0.1. The dashed line shows the curvature to sat- the effective potential on the solid line for g = −0.1 near the point A in Fig.5. isfies V = 0 at the second local minimum.

on the solid line in Fig. 5. It is observed that the first local minimum disappears at the point A. Finally we noted that only the broken phase is realized in a spacetime with a negative curvature.12 5. Conclusion We have investigated the multi-fermion interaction model under the influence of the spacetime topology and curvature. In the practical calculation the four- and the eight-fermion interactions is studied. Evaluating the effective potential in the leading order of the 1/N expansion, we show the phase structure of the model in an arbitrary dimension, 2 ≤ D < 4. On RD−1 ⊗ S 1 the finite size effect restores the broken chiral symmetry for fermion fields with the anti-periodic boundary condition. In this case the theory is equivalent to the finite temperature field theory. The phase transition is of the second order for a non-positive g. We found that the phase boundary is not modified by the eight-fermion interaction with a positive g. For a negative g an additional local minimum contributes to the phase boundary. A new boundary appears in lower dimensions and the first order phase transition takes place. The spacetime curvature also contributes to the phase structure of the theory. For 2 ≤ D < 4 the broken symmetry is restored through the first order phase transition as increasing the curvature R. The eight-fermion interaction with a positive g suppresses the chiral symmetry breaking and changes the phase boundary. For a negative g we observe a contribution from an additional local minimum to enhance the broken symmetry in lower dimensions. In the present paper our study is restricted on the analysis of the phase structure of the theory. We are interested in applying our results to critical phenomena at the early universe. We will continue our work and hope to report on these problems.

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Acknowledgment The authors would like to thank H. Takata, D. Kimura, Y. Kitadono and Y. Mizutani for fruitful discussions. T. I. is supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), No. 18540276, 2009. References 1. A. S. Vshivtsev, K. G. Klimenko and B. V. Magnitsky, Phys. Atom. Nucl. 59, 529 (1996) [Yad. Fiz. 59, 557 (1996)]. 2. S. K. Kim, W. Namgung, K. S. Soh and J. H. Yee, Phys. Rev. D 36, 3172 (1987). 3. D. Y. Song and J. K. Kim, Phys. Rev. D 41, 3165 (1990). 4. D. K. Kim, Y. D. Han and I. G. Koh, Phys. Rev. D 49, 6943 (1994). 5. K. Ishikawa, T. Inagaki, K. Yamamoto and K. f*ckazawa, Prog. Theor. Phys. 99, 237 (1998). 6. L. M. Abreu, M. Gomes and A. J. da Silva, Phys. Lett. B 642, 551 (2006). 7. H. Itoyama, Prog. Theor. Phys. 64, 1886 (1980). 8. I. L. Buchbinder and E. N. Kirillova, Int. J. of Mod. Phys. A4, 143 (1989). 9. E. Elizalde, S. D. Odintsov and Yu. I. Shil’nov, Mod. Phys. Lett. A9, 913 (1994). 10. T. Inagaki, T. Muta and S. D. Odintsov, Mod. Phys. Lett. A8, 2117 (1993). 11. T. Inagaki, Int. J. of Mod. Phys. A11, 4561 (1996). 12. E. V. Gorbar, Phys. Rev. D 61, 024013 (2000). 13. E. Elizalde, S. Leseduarte and S. D. Odintsov, Phys. Rev. D49, 5551 (1994). 14. T. Inagaki, S. Mukaigawa, T. Muta, Phys. Rev. D52, 4267 (1995). 15. E. Elizalde, S. Leseduarte, S. D. Odintsov and Yu. I. Shilnov, Phys. Rev. D53, 1917 (1996). 16. K. Ishikawa, T. Inagaki, T. Muta, Mod. Phys. Lett. A11, 939 (1996). 17. D. Ebert, K. G. Klimenko, A. V. Tyukov and V. C. Zhukovsky, Eur. Phys. J. C 58, 57 (2008). 18. T. Inagaki, T. Muta and S. D. Odintsov, Prog. Theor. Phys. Suppl. 127, 93 (1997). 19. G. ’t Hooft, Phys. Rev. D14, 3432 (1976); D18, 2199 (E) (1978); Phys. Rep. 142, 357 (1986). 20. R. Alkofer and I. Zahed, Phys. Lett. B 238, 149 (1990). 21. J. Moreira, A. A. Osipov, B. Hiller, A. H. Blin, J. Providencia, Annals of Physics 322, 2021 (2007); A. A. Osipov, B. Hiller, A. H. Blin, J. da Providencia, Phys. Lett. B650, 262 (2007); A. A. Osipov, B. Hiller, J. Moreira, A. H. Blin, J. da Providencia, Phys. Lett. B646, 91 (2007). 22. M. Hayashi, T. Inagaki and H. Takata, to be published in Int. J. Mod. Phys. A, arXiv:0812.0900 (hep-ph). 23. T. Inagaki, The Problems of Modern Cosmology, ed. P. M. Lavrov, (Tomsk Pedagogical University Press, 2009), pp. 214-221. 24. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961). 25. D. J. Gross and A. Neveu, Phys. Rev. D10, 3235 (1974). 26. S. Krewald, K. Nakayama, Ann. Phys. 216, 201 (1992). 27. H.-J. He, Y.-P. Kuang, Q. Wang, Y.-P. Yi, Phys. Rev. D45, 4610 (1992). 28. T. Inagaki, T. Kouno and T. Muta, Int. J. Mod. Phys. A10, 2241 (1995). 29. R. G. Jafarov, V. E. Rochev, Russ. Phys. J. 49, 364 (2006). 30. T. Inagaki, D. Kimura and A. Kvinikhidze, Phys. Rev. D77, 116004 (2008); T. Fujihara, D. Kimura, T. Inagaki and A. Kvinikhidze, Phys. Rev. D79, 096008 (2009). 31. H. Reinhardt, R. Alkofer, Phys. Lett. B207, 482 (1988).

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A Model of Soft Mass Generation J. Hoˇsek Department of Theoretical Physics, Nuclear Physics Institute Academy of Sciences of the Czech Republic ˇ z (Prague), Czech Republic 25068 Reˇ E-mail: [emailprotected] We replace the Higgs sector of the electroweak gauge SU (2) L × U (1)Y model of three fermion families with its numerous free parameters by a horizontal gauge SU (3) F quantum flavor dynamics with eight flavor gluon fields Caµ and one coupling constant h. We suggest that the new strong low momentum dynamics generates spontaneously the masses of its eight flavor gluons, of leptons and quarks, and of the intermediate W and Z bosons. Absence of axial anomalies requires neutrino right-handed electroweak singlets and brings into the model a new non-Abelian global symmetry. Because the crucial test of the model, the reliable computation of the fermion mass spectrum is not available, we briefly discuss the experimental implications of the resulting neutrino sector. Keywords: Gauge principle, dynamical symmetry breaking, Majorana neutrinos.

1. Introduction Spontaneous (i.e. soft) fermion and intermediate boson mass generation in the electroweak gauge SU (2)L × U (1)Y interactions is a necessity: Hard fermion and W and Z masses ruin the unitary behavior of scattering amplitudes with longitudinally polarized intermediate vector bosons. Some interactions/concepts responsible for soft mass generation are necessary. With the advent of the LHC these will be harshly tested experimentally. We consider the glorious, weakly coupled Higgs sector responsible for the spontaneous mass generation in the Standard model beautiful but phenomenological by construction: (1) It contains too many theoretically arbitrary and numerically vastly different parameters. (2) The elementary scalar fields are unnatural. (3) It is a tree-level effect. (4) The fermion and the intermediate boson masses are not related. Phenomenological interpretation of the Higgs mechanism means that the massive spinless particle of the elementary real scalar field with properties given by the Standard model Lagrangian does not exist. Alternatives to the SM Higgs mechanism are numerous.1 If weakly coupled like SUSY extensions they parameterize the data with too many parameters. If strongly coupled like technicolor-like extensions they do not allow for quantitative computations of their properties and, frankly, have also many parameters. Fortunately, life

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with QCD taught us to be modest. New concepts like those referring to extra dimensions do not provide, to the best of our knowledge, ordinary operational framework. Our suggestion2 is conservative and nasty: We gauge the SU (3) flavor (family) index and argue that the resulting non-Abelian quantum SU (3)F flavor dynamics (QFD) is a viable3 candidate for spontaneous electroweak symmetry breaking i.e., for the dynamical generation of both the fermion (lepton and quark) and intermediate vector boson masses. 2. Basic reasoning (i) Because ’in QCD we trust’ the SU (3)F must not be vector-like. For if it were it would be confining. Instead, referring to4 we argue that all flavor gluons acquire masses by Schwinger mechanism5 i.e., by their self-interactions and by interactions with fermions. Operationally, in the transverse flavor gluon polarization tensor 2 µν Πµν − q µ q ν )Πab (q 2 ) ab (q) ≡ (q g

(1)

the scalars Πab develop dynamically the massless poles. Mediating the flavor changing electric charge conserving processes the flavor gluons have to be rather heavy. For definiteness we take Ma ∼ 106 GeV. (ii) The flavor gluon exchanges, strong at low momenta, build up the bridges (see Fig.1.) between the left- and right-handed fermion fields i.e., the fermion masses. Operationally, they are described by the chiral symmetry changing fermion proper ˆ in the inverse fermion propagators S(p)−1 = / ˆ 2 ) where Σ ˆ = self energies Σ p − Σ(p 1 + ΣPL + Σ PR and PL,R = 2 (1 ∓ γ5 ). Such matrices have an inverse + 2 + −1 2 + −1 S(p) = (p / + Σ )(p − ΣΣ ) PL + (p / + Σ)(p − Σ Σ) PR

(2)

Basically, the charged lepton and quark masses of three electroweakly identical fermion families differ due to a unique assignment of the the chiral fermion multiplets to triplet and antitriplet representations of QFD. The choice is in fact unique: Take qL as an SU (3)F triplet (i.e. both uL and dL are triplets). Then (uR , dR ) can be either (3, ¯ 3) or (¯ 3, 3), since for the choices (3, 3) and (¯ 3, ¯ 3) the mass matrices of the u- and d-type quarks would come out equal. Without lack of generality choose (uR , dR ) = (3, ¯ 3). Then lL (i.e. both νL and eL ) cannot be a triplet. For if it were the charged lepton mass matrix would be equal either to the u-type or the d-type quark matrix. Hence, lL (i.e. both νL and eL ) must be an antitriplet. Let eR be a triplet (case I). At this point we impose the the field theory restriction of the the absence of axial anomalies.6 It requires introduction of three neutrino right-handed flavor triplets, νN R , N = 1, 2, 3. For definiteness we conside this case here. (eR can also be an antitriplet (case II). Anomaly freedom requires introduction of five neutrino right-handed flavor triplets, νN R , N = 1, ..., 5. Knowledge of the charged lepton mass spectrum would select uniquely one of the two possibilities.) We will show that within the given electric charge the fermion masses differ due to the low-momentum effective sliding coupling depending upon the flavor gluon mass matrix.

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(iii) Dynamically generated fermion masses break spontaneously the electroweak SU (2)L × U (1)Y symmetry down to U (1)em . The resulting three multicomponent ’would-be’ composite Nambu-Goldstone bosons become the longitudinal polarization states of the intermediate bosons W and Z. Because of this realization of the general Schwinger mechanism5 the intermediate boson masses are expressed in terms of the fermion self-energies by sum rules. (iv) Neutrinos are peculiar in our model.7 Dynamical generation of both Dirac and Majorana mass terms results in 12 massive Majorana neutrinos and in 9 massless composite Majorons.8 3. Flavor gluon mass generation Here we follow the analysis of Ward identities with flavor gluons in accordance with:4 Divergences of the full vertices Γµνλ abc (p + q, p) (three-flavor-gluon vertex) and ;µ Γfij;c (p + q, p) (fermion-flavor-gluon vertex) at vanishing momenta are expressed in terms of the full inverse flavor gluon and fermion propagators i.e., in terms of the self energies Π and Σ. We assume that the ghost propagators do not play any dynamical role in the generically nonperturbative reasoning. This assumption is manifest in the ’pinch technique’.9 If the symmetry is unbroken the Ward identities are fulfilled trivially. If Πab and Σs develop the symmetry breaking parts the validity of the Ward identities requires the massless poles in the vertices themselves. They correspond to the ’would-be’ NG bosons composed by construction from both flavor gluons and from all fermion species in the world: νλ Γµνλ abc (p + q, p)|pole = Pbc;d (p + q, p)

i h(−iq µ )Λda (q 2 ) q2

(3)

;µ f Γfij;a (p + q, p)|pole = Pij;d (p + q, p)

i h(−iq µ )Λda (q 2 ) q2

(4)

where µ −iq µ Λda (q 2 ) ≡ [IC;da (q) +

X

Ifµ;da (q)]

(5)

f

Physical interpretation of this decomposition should be clear: (1) There are eight ’would-be’ NG bosons composed both of the flavor gluons and of all fermions in the νλ is the effective coupling of the NG boson with flavor gluons. (3) model. (2) Pbc;d f µ Pij;d is the effective coupling of the NG boson with the fermion f . (4) IC;da (q) and Ifµ;da (q) are the vectorial tadpole UV finite loop integrals. They convert in terms of the effective vertices P , the elementary vertices and the full flavor gluon and fermion propagators both the flavor gluon components and the fermion components of the ’would-be’ NG bosons to the flavor gluons. (5) The crucial effective bilinear derivative vertex between the flavor gluon octet and the ’would-be’ NG boson octet is given by (5).

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1/k 2 Σ(p2) = ψR

ψL

TR

ψR ψR

TL

ψL ψL

Σ[(p − k)2 − Σ† Σ]−1

Fig. 1. Structure of the SD equation for chirality changing Σ of a fermion (charged lepton or quark) ψ. TL,R are the triplet ( 12 λ) or antitriplet (− 12 λ∗ ) generators of a given chiral fermion ψL,R .

The vertex (5) gives rise to the massless pole in the longitudinal part of the flavor gluon polarization tensor (1). Although its transversality is saved by contributions which we cannot compute explicitly, it follows from it that the flavor gluon mass 2 matrix Mab (q 2 ) is given by the formula4 X 2 −q 2 Πab (q 2 ) ≡ Mab (q 2 ) = Λad (q 2 )Λbd (q 2 ) (6) d

2 Practical applications will demand diagonalization of the mass matrix Mab (0) and introduction of the flavor gluon mass eigenstates.

4. Fermion mass generation Structure of the SD equations for the chiral symmetry changing fermion proper self energies Σ(p2 ) (NJL type self-consistency condition10 ) of electrically charged fermions is easily read off the interaction Lagrangian of QFD using the general form of the massive fermion propagator. It is shown in Fig.1. The (bare) flavor gluon propagator is taken in the Feynman gauge as suggested by the pinch technique. 9 The integration in Fig.1. extends over all momenta and the SD equations must 3 be ¯ 2 (k 2 ) improved by taking into account the momentum-dependent sliding coupling h ab ¯ 2 (k 2 ) at low momenta vanishing at high momenta. We know no way of knowing h ab other than solving the theory.11 It is likely that it is dominated by the exchanges of νλ the composite ’would-be’ NG bosons with the effective vertices Pbc;d (p + q, p) and f Pij;d (p + q, p) to both gluons and fermions, respectively. To proceed we write 1 1 = 2 {[1 + Π(k 2 )]−1 + Π(k 2 )[1 + Π(k 2 )]−1 } 2 k k

(7)

and argue as follows: (1) At high momenta Π is given by perturbation theory. The first term in (7) when used in Fig.1. gives rise to flavor insensitive Σ due to massless flavor gluon ¯ 2 (k 2 ) = exchange with asymptotically free flavor insensitive interaction strength h h2 [1+Π(k 2 )]−1 . The corresponding SD equation giving the high-momentum asymptotics of Σ was studied in QCD in detail in.12 The second term in (7) corresponding to massless gluon exchange with the bare charge should be ignored.

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(2) At low momenta the non-perturbative Π is given by (6). The first term in (7) corresponds to the massive gluon exchanges with a bare charge ( k12 (1 + Π)−1 = (k 2 − M 2 )−1 ) and in Fig.1. it should be ignored. The second term in (7) when used in Fig.1. gives rise to Σs due to massless gluon exchange with the low-momentum ¯ 2 (k 2 ) running to a non-perturbative IR stable fixed point: h ab ¯ h2ab (k 2 ) = h2∗ [Π(k 2 )(1 + Π(k 2 ))−1 ]ab .

(8)

The corresponding matrix SD equations, still rather schematic, hopefully illustrate the main point: The symmetry-breaking form of Πab implies that at low momenta the fermion self energies Σ differ in different flavor channels by the low-momentum flavor sensitive interaction strengths (8), basically due to the low momentum symmetry breaking flavor gluon self energy. At this exploratory stage we are merely able to illustrate that the low momentum Ansatz (8) for the sliding coupling is bona fide responsible for strong suppression of the fermion mass with respect to the huge flavor gluon mass. We replace both 2 Mab (k 2 ) and the fermion self energies Σij (p2 ) =Σij (0) = mij by real numbers M 2 and m, respectively. The SD equation in Fig.1. turns into an algebraic equation Z ∞ m M2 h2 (9) dk 2 2 m = ∗2 16π 0 k + M 2 k 2 + m2

with solution m = M exp[−8π 2 /h∗2 ]. Indeed, for M = 106 GeV the ”neutrino” mass mν = 10−9 GeV is obtained with an effective interaction strength h2ν /4π = 2π/15 ln 10, and the ”top quark” mass mt = 102 GeV with h2t /4π = 2π/4 ln 10.

5. Intermediate boson mass generation Masses of the W and Z vector bosons are the necessary consequence of the dynamically generated fermion masses: The fermion proper self-energies Σ(p2 ) generated by strong QFD break spontaneously also the ’vertical’ SU (2)L × U (1)Y symmetry down to U (1)em . The properties of three composite ’would-be’ NG bosons can be extracted from the SU (2)L × U (1)Y Ward identities.13,14 For simplicity we consider the fermion proper self energies diagonal. g α Γα W (p + q, p) = √ {γ (1 − γ5 ) 2 2 − Γα Z (p + q, p) =

qα [(1 − γ5 )ΣU (p + q) − (1 + γ5 )ΣD (p)]}, q2

g {t3 γ α (1 − γ5 ) 2 cos θW qα − 2Qγ α sin2 θW − 2 t3 [Σ(p + q) + Σ(p)]γ5 }. q

When the electroweak gauge interactions are switched on as weak external perturbations the W and Z bosons dynamically acquire masses. Their squares are defined

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as the residues at single massless poles of the W and Z polarization tensors: 1 X 2 (mU IU ;D (0) + m2D ID;U (0)) (10) m2W = g 2 4 X 1 m2Z = (g 2 + g 02 ) (m2U IU ;U (0) + m2D ID;D (0)) (11) 4 In the formulas above U and D abbreviate upper and nether fermions in electroweak doublets, respectively. The neutrinos are considered as massive Dirac fermions for simplicity. The quantities I in (10, 11) defined in14 are the UV finite loop integrals depending upon Σs. The Weinberg relation m2W /m2Z cos2 θW = 1 is tolerably violated.14 6. Neutrino mass generation7 In the case I there is one SU (3)F antitriplet of left-handed neutrinos νf L , three SU (3)F triplets of the right-handed neutrinos νN f R , and the global sterility symmetry U (3). Building the neutrino left-right mass matrix bridge in accord with Fig.1. proceeds in steps: (1) It is convenient to collect all neutrino fields into one left-handed multiplet nL , nTL = (νf L , (νN f R )C ) and its corresponding right-handed charge conjugate, nR = (nL )C ((ψ)C = C ψ¯T , C + = C T = C −1 = −C). (2) Rewrite the known vertices of neutrinos with flavor gluons in terms of nL and nR resulting in the corresponding neutrino n vertices TL and TR . (3) The most general bilinear neutrino Lagrangian15 which defines the chirality changing neutrino proper self ˆ and the neutrino propagator S has in the momentum space the form energy Σ 0 + 1 1 C C p νL + νR / p ν R − ν L Σ+ L 0 = νL / D νR − νR ΣD νL − 2 νL ΣL (νL ) − 2 (νL ) ΣL νL − + 1 1 1 C −1 C ˆ n≡n ¯ (p ¯ S (p) n. Here n = nL + (nL )C is / − Σ) 2 νR ΣR (νR ) − 2 (νR ) ΣR νR = 2 n ˆ = ΣPL + Σ+ PR (PL,R = 1 (1 ∓ γ5 ))is the column of 12 Majorana fields and Σ in Σ 2 a general complex symmetric momentum dependent16 matrix. It can be cast into the real diagonal form with positive entries Σd by the transformation U ∗ ΣU + = Σd where U is a unitary matrix. Upon elimination of unphysical phases the lepton sector of the model has the fixed CP properties.7 (4) By construction the structure of the schematic SD equation for Σ is identical to that depicted in Fig.1. Identical is, unfortunately, also our inability to find reliable solutions. While computation of the charged fermion masses would ”merely” be a postdiction, a little is known at present experimentally about the neutrino sector. Because the model is uniquely defined, we conclude that the nontrivial non-degenerate solutions of the neutrino SD equations, if found, describe twelve massive Majorana neutrinos, cause complete spontaneous breakdown of the global U (3) sterility symmetry manifested by nine massless composite Majorons, and yield definite CP properties of the leptonic sector. 7. Conclusion Present model of soft generation of masses of the Standard model particles by a strong-coupling dynamics, having just one unknown parameter h, is either right or

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plainly wrong. Reliable computation of the fermion mass spectrum is, however, far away. Ultimately, masses should be related. (1) One elaborated example of mass relations is the sum rules for the intermediate boson masses mW (10) and mZ (11). The implication is interesting: There is no generic Fermi scale in the model. The intermediate boson masses are merely a manifestation of the large top quark mass. (2) Detailed analysis of the uniquely defined neutrino sector is a challenge. The very existence of almost sterile neutrinos introduced for anomaly freedom should have experimental consequences in neutrino oscillations and in astrophysics.17 Another low energy manifestation of the model are the elusive Majorons. (3) The fermion SD equations can fix also the fermion mixing parameters. (4) It is natural to expect that the unitarization of the scattering amplitudes with longitudinal polarization states of massive spin one particles proceeds in the present model via the massive composite ’cousins’ of the composite ’would-be’ NG bosons. Its practical implementation is obscured, however, by our ignorance of the detailed properties of the spectrum of strongly coupled SU (3)F . Acknowledgements I thank Petr Beneˇs and Adam Smetana for help with the manuscript and for collaboration. Financial support of this work by the Committee for the CR-CERN collaboration is gratefully acknowledged. References 1. E. Accomando et al., Workshop on CP studies and non-standard Higgs physics, arXiv:hep-ph/0608079. 2. J. Hoˇsek, arXiv:0909.0629. 3. H. Pagels, Phys. Rev. D21, 2336 (1980). 4. E.J.Eichten and F.L.Feinberg, Phys.Rev.D10,3254(1974). 5. J. Schwinger, Phys. Rev. 125, 397 (1962). 6. D. J. Gross and R. Jackiw, Phys. Rev. D6,477(1972). 7. P. Beneˇs, J. Hoˇsek and A. Smetana, Spontaneous breakdown of sterility, to be published. 8. Y. Chikashige, R. N. Mohapatra and R. D. Peccei, Phys. Lett. B98, 26 (1981). 9. D.Binosi and J. Papavassiliou, Phys.Reports 479:1-152(2009). 10. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961). 11. K. G. Wilson, Phys. Rev. D3, 1818(1970). 12. H. Pagels, Phys. Rev.D19,3080(1979). 13. J. Hoˇsek, preprint JINR E2-82-542 (1982); J. Hoˇsek, preprint CERN-TH.4104/85(1985). 14. J. Hoˇsek, Phys. Rev. D36,2093(1987). 15. S. M. Bilenky and S. T. Petcov, Rev. Mod. Phys.59,671(1987). 16. P. Beneˇs, arXiv:0904.0139. 17. A. Kusenko, New J. Phys.11: 105007, 2009.

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TeV Physics and Conformality Thomas Appelquist Department of Physics, Sloane Laboratory, Yale University New Haven, Connecticut, 06520, USA

1. Introduction Lattice gauge theory has been very successful in deepening our understanding of the strong nuclear interactions. During the past two years, stimulated to some extent by the start-up of the Large Hadron Collider, interest is growing in applying lattice methods to new, strongly interacting theories that could play a role in extending the standard model. I will focus here on the use of lattice methods to study strongly coupled gauge theories with some application to models of dynamical electroweak symmetry breaking. 2. The conformal window and walking 2.1. Perturbative RG flow in generalized Yang-Mills theories Consider a Yang-Mills theory with local gauge symmetry group SU (Nc ), coupled to Nf massless Dirac fermion flavors:

LY M = −

Nf Nc X 1 X a a,µν F F + ψ¯i (i6D)ψi , 4g 2 a=1 µν i=1

(1)

With the fermions in a representation R of the gauge group. The scale dependence of the renormalized coupling g = g(µ) is determined by the β-function, which we can expand perturbatively: β(α) ≡

∂α = −β0 α2 − β1 α3 − β2 α4 − ... ∂(log µ2 )

with α(µ) ≡ g(µ)2 /4π. The universal values for the first two coefficients are 1 11 4 β0 = Nc − T (R)Nf , 4π 3 3 1 34 2 20 β1 = N − 4C2 (R) + Nc T (R)Nf , (4π)2 3 c 3

(2)

(3) (4)

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dim(R)

T (R)

C2 (R)

F S2 A2 G

N

1 2 N +2 2 N −2 2

N 2 −1 2N (N +2)(N −1) N (N −2)(N +1) N

N (N +1) 2 N (N −1) 2 N2 − 1

N

N

where T (R) and C2 (R) are the trace normalization and quadratic Casimir invariant of the representation R, respectively. The Casimir invariants for a few commonly-used representations of SU (N ) are shown in Table 1. So long as Nf /Nc < 11/(4T (R)) so that β0 > 0, the theory is asymptotically free. One may continue on to higher order in this expansion, at the cost of specifying a renormalization scheme; in the commonly used MS scheme, the next two coefficients are known.1 For present purposes, a key step in studying the RG flow of the coupling constant is to identify any fixed points. Keeping the first two terms in the β-function, we see that in addition to the trivial ultraviolet fixed point α = 0, there is a second solution located at β0 (2L) (5) α? = − . β1 (2L)

Since β < 0 for all 0 < α < α? , this solution is an infrared-stable fixed point, describing the limit µ → 0. If the fixed-point coupling is sufficiently weak, the theory is perturbative at all scales. This condition will be satisfied if Nf is near the value 11Nc /4T (R) at which asymptotic freedom is lost.2,3 Since confinement of color charges and spontaneous breaking of chiral symmetry, are strong-coupling effects, they are absent in a theory that is perturbative at all scales. As Nf is decreased, the value of the fixed-point coupling increases, at some point reaching a critical value Nfc at which the infrared behavior changes from conformal to confining. Theories within the range Nfc < Nf <

11Nc 4T (R)

(6)

are said to lie in the conformal window, due to the approximate restoration of conformal symmetry in the infrared. Perturbation theory is a-priori unreliable to describe physics in the vicinity of the infrared fixed point as Nf approaches the transition point Nfc , so in order to determine the location of the transition, some non-perturbative estimate is required. 2.2. Infrared conformality and walking behavior Theories that lie inside the conformal window, although interesting in their own right, are not generally useful in describing electroweak symmetry breaking due to

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the lack of chiral symmetry breaking (although it is possible to force trigger symmetry breaking even in the conformal window by explicit construction.4 ) However, the basic picture outlined above suggests that a theory outside the window but very close to the transition could also have a novel structure while still breaking chiral symmetry spontaneously. The idea is as follows:5–7 suppose that there is some (scheme-dependent) critical coupling αc , which when exceeded will trigger the spontaneous breaking of chiral symmetry. Now consider a theory with a beta-function such that the coupling is approaching a somewhat supercritical fixed point α? > αc . When αc is exceeded, confinement and chiral symmetry breaking set in. The fermions which were responsible for the existence of the fixed point develop masses and are screened out of the theory, causing the coupling to run as in the Nf = 0 theory below the generated mass scale. This idea, known as walking, results in a separation of scale between the UV physics where the coupling runs perturbatively and the IR scale at which confinement sets in. This dynamical scale separation is what one needs to address the FCNC problem in extended technicolor. The conflict there was between trying to simultaneously match the standard model particle masses and suppress FCNCgenerating effects, both of which are tied to the same ETC scale Λi . With walking, the ultraviolet-sensitive condensate can pick up a large additional contribution from the scales between ΛT C and ΛET C , allowing recovery of standard model masses without violation of precision electroweak experimental bounds. While walking technicolor offers a solution to the difficulties with technicolor models, the existence of such a theory is speculative. The onset of walking is a strong-coupling effect, so that perturbative methods are unlikely to be useful in a walking theory. The search for walking is closely linked to more general questions about the location and nature of the conformal transition at Nf = Nfc . 3. Lattice studies of the conformal transition 3.1. Overview Lattice field theory provides an ideal way to study the conformal transition, and more generally the properties of Yang-Mills theories as we vary Nf . Lattice simulations are truly non-perturbative, although the continuum limit must be taken carefully to recover information about continuum physics (the ability to take this limit being made possible by the asymptotic freedom of the theories in which we are interested.) Lattice simulations allow broad investigation; a large number of different observables can be computed simultaneously on a single set of gauge configurations. 3.2. Running coupling A natural application of lattice simulation to investigate the conformal window is by the direct computation of a running coupling constant.

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The first step is to select a non-perturbative definition of a running coupling which we can measure on the lattice. There are a number of such choices possible, including the standard extraction of the static potential from Wilson loops, the Schr¨ odinger functional,8–12 the twisted Polyakov loop scheme,13 and constructions using ratios of Wilson loops.14,15 Regardless of the definition of the running coupling, the goal is to map out its evolution over a large range of distance scales R. If we work at a fixed lattice spacing a, then the range of available R at which we can measure the coupling is quite small, with the computational expense quickly becoming prohibitive even in QCD. The problem is exacerbated in a theory with large Nf , where the size of the β-function is small; to go from weak to strong coupling, a change in scale of many orders of magnitude is often required. To achieve our goal, then, we must find some way to match together lattice measurements of the running coupling taken at different lattice spacings and combine them into an overall measurement of continuum evolution. A technique known as step scaling 16,17 provides a systematic approach. Step scaling is a recursive procedure which describes the evolution of the coupling constant g(R) as the scale changes from R → sR, where s is a numerical scaling factor known as the step size. We define the relation between the coupling at these two scales in the continuum through the step-scaling function, σ(s, g 2 (R)) ≡ g 2 (sR).

(7)

The step-scaling function σ is nothing more than a discrete version of the usual continuum β-function, both of which describe the evolution of the coupling as a function of the coupling strength. In a lattice calculation, the step-scaling function which we extract also contains lattice artifacts in the form of a/R corrections. We denote the lattice version of the step-scaling function by Σ, and it is related to the continuum σ by extrapolation of the lattice spacing a to zero: σ(s, g 2 (R)) = lim Σ(s, g 2 (R), a/R). a→0

(8)

Generically, the implementation of step scaling begins with the choice of some initial value for g 2 (R). Several ensembles at different a/R are then generated, tuning the lattice bare coupling β = 2Nc /g02 so that on each ensemble we measure the chosen value of the renormalized coupling, g 2 (R). Then we generate a second ensemble at each β, but measure the coupling at a longer scale R → sR. The value of the coupling measured on the second lattice is exactly Σ(s, g 2 (R), a/R). One can then extrapolate a/R → 0 and recover the continuum value σ(s, g 2 (R)). Taking σ(s, g 2 (R)) to be the new starting value, we repeat the procedure, mapping g 2 (R) → g 2 (sR)... → g 2 (sn R) until we have sampled the coupling over a large range of R values. An efficient approach to step scaling is to measure g 2 (R) for a wide range of values in β and R/a, and then to generate an interpolating function. Step scaling may then be done analytically using the interpolated values. Such an interpolating

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function should reproduce the perturbative relation g 2 (R) = g02 + O(g04 ) at weak coupling, but otherwise its form is not strongly constrained. One possible choice which has worked quite well in some studies is an expansion of the inverse coupling 1/g 2 (β, R/a) as a set of polynomial series in the bare coupling g02 = 2Nc /β at each R/a: " i # n X 1 β 2Nc 1− . (9) = ci,R/a g 2 (β, R/a) 2Nc β i=1 The order n of the polynomial is arbitrary, and can be varied as a function of R/a to achieve the optimal fit to the available data. There is a natural caveat on the step-scaling procedure, especially in the context of studying theories with infrared fixed points. The procedure as outlined above depends crucially on the ability to take the limit a/R → 0. If we hold g 2 (R) fixed and take the limit a/R → 0, it is important that the bare coupling g02 (a/R), which depends on the short-distance behavior of the theory, does not become strong enough to trigger a bulk phase transition. This is satisfied automatically if the short-distance behavior is determined by asymptotic freedom, in which case g02 (a/R) vanishes as 1/ log(R/a). However, if we work in a theory with an infrared fixed point and measure values of g 2 (R) lying above the fixed point at g?2 , then g02 (a/R) will increase as a → 0, with no evidence that it remains bounded and therefore that a continuum limit exists. Even so, we can extrapolate to small enough values of a/R to render lattice artifact corrections negligible, providing that g02 (a/R) is kept small enough to avoid triggering a bulk transition into a strong-coupling phase. Our work has relied on one definition of a running coupling based on the Schr¨ odinger functional (SF). The SF running coupling is defined through the response of a system to variation in strength of a background chromoelectric field. It is a finite-volume method, with the coupling strength defined at the spatial box size L, so that we identify R = L and can discard finite-volume corrections. Formally, the Schr¨ odinger functional describes the quantum mechanical evolution of some system from a given state at time t = 0 to another given state at time t = T , in a spatial box of size L with periodic boundary conditions.8,9,18 The temporal extent T is fixed proportional to L, so that the Euclidean box size depends only on a single parameter. The initial and final states are described as Dirichlet boundary conditions which are imposed at t = 0 and t = T , and for measurement of the coupling constant are chosen such that the minimum-action configuration is a constant chromo-electric background field of strength O(1/L). This can be implemented both in the continuum8 and on the lattice.10 One can represent the Schr¨ odinger functional as the path integral 0

Z[W, ζ, ζ; W 0 , ζ 0 , ζ ] = Z 0 0 0 0 [DADψDψ]e−SG (W,W )−SF (W,W ,ζ,ζ,ζ ,ζ ) ,

(10)

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where A is the gauge field and ψ, ψ are the fermion fields. W and W 0 are the 0 boundary values of the gauge fields, and ζ, ζ, ζ 0 , ζ are the boundary values of the fermion fields at t = 0 and t = T , respectively. The fermionic boundary values are subject only to multiplicative renormalization,19 and as such are generally taken to be zero in order to simplify the calculation. The gauge boundary fields W, W 0 are chosen to given a constant chromo-electric field in the bulk, whose strength is of order 1/L and controlled by a dimensionless parameter η.20 The Schr¨ odinger functional (SF) running coupling is then defined by the response of the action to variation of η: ∂ k , (11) =− log Z 2 ∂η g (L, T ) η=0 where (with the standard choice of gauge boundary fields for SU (3)), the normalization factor k is 2 2 L 2πa2 πa k = 12 sin + sin . (12) a 3LT 3LT

The presence of k ensures that g 2 (L, T ) is equal to the bare coupling g02 at tree level in perturbation theory. In general, g 2 (L, T ) can be thought of as the response of the system to small variations in the background chromo-electric field. For most fermion discretizations, at this point we can take T = L in order to define the running coupling as a function of a single scale, g 2 (L). However, if staggered fermions are used (as they are often in order to offset the cost of simulating additional fermion flavors), then an additional complication arises which can be envisioned geometrically. The staggered approach to fermion discretization can be formulated as splitting the 16 spinor degrees of freedom available up over a 24 hypercubic sublattice. Clearly, such a framework requires an even number of lattice sites in all directions. If all boundaries are periodic or anti-periodic, then setting T = L can be done so long as L is even. However, with Dirichlet boundaries in the time direction, the site t = T is no longer identified with t = 0, so that a total of T /a + 1 lattice sites must exist. In order to accommodate staggered fermions, T /a must be odd. Thus when using staggered fermions, the closest we can come to our desired choice of T is T = L ± a. In the continuum limit, the desired relation T = L is recovered. However, at finite lattice spacing O(a) lattice artifacts are introduced into observables. This is undesirable, since staggered fermion simulations contain bulk artifacts only at O(a2 ) and above. Fortunately, there is a solution: simulating at both choices T = L±a and averaging over the results has been shown to eliminate the induced O(a) bulk artifact in the running coupling.21 We define g 2 (L) through the average: 1 1 1 1 = + . (13) 2 g 2 (L, L − a) g 2 (L, L + a) g 2 (L)

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I will now discuss simulation data and results from a Schr¨odinger functional running coupling study of the SU (3) fundamental, Nf = 8 and 12 theories, using staggered fermions.12 I begin with the Nf = 8 theory, for which data was gathered in the range 4.55 ≤ β ≤ 192 on lattice volumes given by L/a = 6, 8, 12, 16. The lower limit on β was determined to keep the lattice coupling too weak to trigger a bulk phase transition. A selection of the data, together with interpolating function fits of the form Eq. (9), are shown in Fig. 1. Note that at any fixed value of β, the coupling strength g 2 (L) increases with L/a, showing no evidence of the “backwards” running that we would expect to observe in a theory with an infrared fixed point. 20

L/a=6 L/a=8 L/a=12 L/a=16

15 g 2 !L"

10

5

4.6

4.8

5.0

5.2

5.4

5.6

5.8

6.0

Β Fig. 1. Measured values g 2 (L) versus β for Nf = 8. The interpolating curves shown represent the best fit to the data, using the functional form Eq. (9). The errors are statistical.

Although the study of Fig. 1 is indicative that the Nf = 8 theory lies outside the conformal window, it is possible for results at fixed β to be misleading; we must take the continuum limit in order to recover information about the continuum theory. We apply the step-scaling procedure detailed above in order to extract the continuum step-scaling function σ(2, g 2 (R)), by extrapolation of a/L → 0 with each doubling of the scale L. Our results for σ(2, g 2 (R)) will depend on the choice of continuum extrapolation, i.e. the model function for a/L dependence of Σ(2, g 2 (R), a/L). As this is a staggered fermion study, the leading bulk lattice artifacts are expected to be of O(a2 ), but there are additional boundary artifacts of O(a) which are only partially cancelled off by subtraction of their perturbative values. However, in this case the a/L dependence is weak, with the associated systematic error dominated

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by the statistical errors on the points, so that a constant extrapolation (i.e. weighted average of the two points) is used to extract σ(2, g 2 (R)) here. The resulting continuum running of g 2 (L) for Nf = 8 is shown in Fig. 2. L0 is an arbitrary length scale here defined by the condition g 2 (L) = 1.6, anchoring the step-scaling curve at a relatively weak value. The points shown correspond to repeated doubling of the scale L relative to L0 . Derivation of statistical errors uses a bootstrap technique.12 Perturbative running at two and three loops is also shown for comparison up through g 2 (L) ≈ 10, beyond which the accuracy of perturbation theory is expected to degrade. The coupling measured in this simulation follows the perturbative curve closely up through g 2 (L) ≈ 4, and then begins to increase more rapidly, reaching values that exceed typical estimates of the coupling strength needed to induce spontaneous chiral symmetry breaking. As there is no evidence for an infrared fixed point, or even for an inflection point in the running of g 2 (L), this study supports the assertion that the Nf = 8 theory lies outside the conformal window.

25

2-loop univ.

20

3-loop SF g $L%

15

2

10

5

2

4 Log!L"L0 #

6

8

Fig. 2. Continuum running for Nf = 8. Purple points are derived by step-scaling using the constant continuum-extrapolation. The error bars shown are purely statistical. Two-loop and threeloop perturbation theory curves are shown for comparison.

Data and interpolating fits for the Nf = 12 theory are shown in Fig. 3. Here, simulations were performed on lattice extents of L/a = 10, 20, in addition to the values of L/a = 6, 8, 12, 16 of the Nf = 8 case. The data and fits shown in Fig. 3 show striking qualitative differences with their counterparts at Nf = 8 (Fig. 1).

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12

L/a=6 L/a=8 L/a=10 L/a=12 L/a=16 L/a=20

10

g 2 !L"

8

6

4

2

4.2

4.4

4.6 Β

4.8

5.0

Fig. 3. Measured values g 2 (L) versus β, Nf = 12. The interpolating curves shown represent the best fit to the data, using the functional form of Eq. (9).

In particular, there are some hints of a “crossover” phenomenon taking place, in which the order of the curves in L/a from weak to strong coupling is inverted at small β. Such a crossover is indicative of a region in which the coupling decreases towards the infrared, and is thus a signature of an infrared fixed point. However, it should be emphasized again that it is important to go through the full stepscaling procedure in order to extract meaningful continuum physics, and any result indicated by working at fixed β should not be taken as definitive. As above, we choose a constant continuum extrapolation, i.e. weighted average of the three points. Results for continuum running, again from the starting value g 2 (L0 ) = 1.6, are shown in Fig. 4. Two-loop and three-loop perturbative curves are shown again for reference. The figure clearly shows the running coupling tracking towards an infrared fixed point, whose exact value lies within the statistical error band and which is consistent with the value predicted by three-loop perturbation theory. It should be noted that the error bars of Fig. 4 are highly correlated, with correlation approaching 100% near the fixed point, due to the use of an underlying interpolating function. This causes the error bars to approach a stable value asymptotically, even as we increase the number of steps towards infinity. The infrared fixed point here also governs the infrared behavior of the theory for values of g 2 (L) which lie above the fixed point. As discussed previously, we cannot naively apply the step-scaling procedure in this region, since we can no longer approach the ultraviolet fixed point at zero coupling strength in order to

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10

2-loop univ. 3-loop SF

8

g 2 $L%

6

4

2 0

10

20 Log!L"L0 #

30

40

Fig. 4. Continuum running for Nf = 12. Results shown for running from below the infrared fixed point (purple triangles) are based on g 2 (L0 ) ≡ 1.6. Also shown is continuum backwards running from above the fixed point (light blue squares), based on g 2 (L0 ) ≡ 9.0. Error bars are again purely statistical, although strongly correlated due to the underlying interpolating functions. Two-loop and three-loop perturbation theory curves are shown for comparison.

take the continuum limit. Instead, we can restrict our attention to finite but small value of a/L, small enough to keep lattice artifacts small and yet large enough so that g02 (a/L) does not trigger a bulk phase transition for g 2 (L) near (but above) the fixed point. With these caveats in mind, the step-scaling procedure can then be applied and leads to the running from above the fixed point shown in Fig. 4. The observation of this “backwards-running” region is crucial to distinguishing theories with true infrared fixed points from walking theories, in which the β-function may become vanishingly small before turning over and confining. Having shown evidence for the existence of an infrared fixed point in the Nf = 12 theory and demonstrated its absence up to strong coupling at Nf = 8, we have constrained the edge of the conformal window for the case of Nc = 3 with fermions in the fundamental rep, 8 < Nfc < 12. Similar measurements at other values of Nf can allow us to further constrain Nfc . Furthermore, if a walking theory exists just below the transition value, a lattice measurement of the scale dependence of the coupling could directly reveal the expected plateau behavior and resulting separation between infrared and ultraviolet scales. In addition, the non-perturbative β function can be used in conjunction with additional lattice measurements to extract the anomalous dimension γm of the mass operator.22

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3.3. Spectral and chiral properties I will next discuss the evolution with Nf of various observables on the broken side of the conformal transition. In order to meaningfully compare any quantity between theories with different Nf , we must first identify a physical scale to hold fixed. A natural choice is the Goldstone-boson decay constant F . However, the extraction of F from lattice simulations can be challenging. The rho meson mass mρ is much more easily determined, due to the lack of a chiral logarithm at next-to-leading order (NLO) in a χPT-derived fit.25 However, in the end we are more interested in the evolution of physics with respect to F than mρ . In QCD the two scales are connected, mρ ∼ 2πF , but it is not known a priori whether this connection will persist near the edge of the conformal window. The Sommer scale r0 ,26 associated with the scale of confinement, is another possible choice with similar advantages and drawbacks to mρ . In the present discussion we will assume that these scales do not evolve with respect to each other, so that holding any one constant with Nf is sufficient. This assumption is supported by available data, but the choice of scale is an open question going forward. As these lattice simulations are necessarily performed at finite mass, while we are interested in the behavior of theories in the chiral limit, extrapolation of results m → 0 is crucial. Chiral perturbation theory provides a consistent way to carry out this extrapolation. The familiar expressions for the Goldstone boson mass Mm , decay constant Fm and chiral condensate hψψim (with the subscript denoting evaluation at finite quark mass m) are easily generalized to theories with arbitrary Nf ≥ 2 by inclusion of known counting factors. The next-to-leading order (NLO) expressions for a theory with 3 colors are:27 1 2mhψψi 2 1 + zm α + Mm = log(zm) , (14) M F2 Nf Fm = F

hψψim

Nf 1 + zm αF − log(zm) , 2

(

"

#) Nf2 − 1 = hψψi 1 + zm αC − log(zm) , Nf

(15)

(16)

where z = 2hψψi/(4π)2 F 4 . These expressions have also been computed at next-tonext-to-leading order (NNLO) and for fermions in the adjoint representation and the pseudo-real representations of the 2-color theory.28 Each of the unknown coefficients αM , αF , αC also contain terms that grow linearly with Nf . αC also contains a unique, Nf -independent “contact term” which remains even in the absence of spontaneous chiral symmetry breaking. This contribution is linear in m, quadratically sensitive to the ultraviolet cutoff (here the lattice spacing a−1 .) This term dominates the chiral expansion of hψψim , making

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#$ log(zm) ,

#$ (zm) ,

(1)

(2)

#$ 1 log(zm) ,

(3)

inear in m and resent even withg. This “contact buse of notation) es αC , and in fact m values in our and #ψψ$m grow hiral-log term in ents αM , αF and shown explicitly,

ss Mρ,m can also d from χPT [13]. is no m log(m) iral expansion of

all fermions with

0.4

0.3

r-1 , M!,m 0,m

ice spacing fixed hoose a value for ale of several lato match the same ant enhancement atch qualitatively um cutoff. , then the extrapng the results for -to-leading-order

Nf=2 M!,m Nf=6 M!,m Nf=2 r-1 0,m Nf=6 r-1 0,m

0.2

0.1

0 0

0.01

m

0.02

0.03

−1 Fig. 5. From.29 Linear chiral extrapolations of Mρ,m and the Sommer scale r0,m , in lattice units, based on the (solid) points at mf = 0.01−0.02. Both show agreement ρ,mwithin error between Nf = 2 and Nf = 6−1 in the chiral limit.

FIG. 1: Linear chiral extrapolations of M and the Sommer scale r0,m , in lattice units, based on the (solid) points at mf = 0.01 − 0.02. Both show agreement within error between Nf = 2 numerically extrapolation of the condensate more difficult. Finally, the and Nf =accurate 6 in the chiral limit.

growth of the chiral log term in Fm with Nf forces the use of increasingly smaller fermion masses m as Nf is increased, in order to keep the NLO terms small enough relative to the leading order so that χPT is trustworthy. The goal here isSimulations to search for enhancement of the condensate relative masses to the scale lattice units. are performed for fermion 3 , and extrapolate directly F . One way to proceed is to construct the ratio hψψim /Fm m = 0.005 to 0.03. At finite lattice spacing, even with m →f 0; however, as noted above the presence of the contact term can make such an mf = 0,difficult the chiral is making not exact, with the vioextrapolation to carrysymmetry out precisely. By use of the additional quan2 2 2 the Gell-Mann-Oakes-Renner (GMOR) relation MmfF. mThe = 2mhψψi tity Mm and lation captured in a residual mass mres 'm totalm , incorporated into the NLO formulas shown above, we can construct other ratios at fermion mass m is then m ≡ mf 3+inm . finite m which will also extrapolate to hψψi/F theres chiral limit: the other two 1/2 , is an irrelevant 2 2 3/2 Although global possibilities are Mm /(2mFmtopological ), and (Mm /2m)charge, /hψψimQ . Due to the contact term 2 inquantity hψψim , Mm /(2mFmassless the mildest chiral extrapolation of theinthree with fermions and infinite volume, a m ) should have ratios. finite volume it becomes relevant [15]. On a discretized latA lattice study of the type outlined here, investigating the evolution from Nf = 2 not thebeen system between totice, Nf =Q 6 inisthe SUconserved, (3) fundamentalwith case, has carriedevolving out by the Lattice Strong 29 Dynamics (LSD) collaboration. I refer to this reference for details of the simulation sectors, an evolution crucial for the correct sampling of the and analysis, and here onlyvolume. selected details. Thevery physical scalesfermions, chosen to be path integral at give finite With light matched are the rho mass mρ and the Sommer scale r0−1 ; each observable was first the evolution slows dramatically using Monte measured in the Nf of = 6Q case, and then matched by tuning thecurrent bare lattice coupling

Carlo methods [16]. We find that Q evolves sufficiently for mf ≥ 0.01 for both Nf = 2 and 6. At mf = 0.005 it does not, leading to systematic shifts in #ψψ$m and Fm , which we will explore in a future paper. Here, we present results for mf = 0.005, but do not include them in our analysis. This also ensures that for each m, Mm L > 4, keeping the NGB Compton wavelength well inside the lattice. For Nf = 6, there are also results for mf = 0.025, 0.03, but

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2

2

1.5 Mm / 2 m

1.5

2

Rm

Nf=6 / Nf=2

1 0.5

1 0

0.005

0.01! m

0.015

0 0

0.02

2 /2mF 2 2 m ]6f /[Mm 2 Fig. 6. From.29 Rm ≡ [Mm ]2f , versus m ≡ (m(2f ) + m(6f ))/2, showing Rm ≡ [M /2mFm/2mF ] N m/[M m ≡ m /2mFm ]2f enhancement of hψψi/F 3 at Nfm = 6 relative to6f f = 2. The open symbol at m = 0.005 denotes the(m(2f presence )of + possible systematic m(6f ))/2 errors. "ψψ#/F 3

FIG. 1:

at

FIG. 2: The slope lattice units, as a is a joint fit to M

at Nf = 2. The resulting chiral extrapolation of these quantities is shown in Fig. 5, and shows good agreement, so that the lattice cutoffs are well matched between Nf = 2 and Nf = 6. Determination of thelittle presence condensate enhancement is done and Fm change in ortheabsence rangeof m f = 0.01 − 0.02, inthrough comparison of the quantity hψψi/F 3 between the Nf = 6 and Nf = 2 dicating along with the Nf M dependence in Eq. 3 that NLO 2 theories, by way of the equivalent ratio m /(2mFm ). We can directly construct a “ratio of ratios” χPT should provide a reliable fit. Our results for "ψψ#m 2 /2mF M m m show the expectedRdominance ofNfthe =6 large linear contact , (17) m ≡ 2 [Mm /2mFthe m ]Nf use =2 term (which does not preclude of χPT). We there-

mf = 0.01 − 0.0

A value of Rmout > 1 then of the condensate fore carry the implies Nf =enhancement 2 extrapolation to m as=Nf0increases. using The result is shown in Fig. 6, and indicates that Rm > 1.5 in the chiral limit, barring ∼ the combined NLO chiral expansions of Eqs. (1) – (3). The a downturn in Rm - an unlikely outcome, as the curvature of the NLO logarithm in Figs. 4, leads to extrapois five-parameter naturally upwards infit, the shown chiral expansion of R2m -itself. The magnitude of Rm 2 is significant and larger than expected; an MS perturbation theory estimate of the , lated values F = 0.0209(41) and "ψψ#/F = 0.99(17) enhancement from Nf = 2 to 6 by integrating the anomalous dimension of the mass giving the NLO χPT result operator γm leads to an expected increase on the order of 5-10%. Some care must be taken in comparing this value to our lattice result, since the condensate hψψi and by "ψψ# extension hψψi/F 3 depends on the renormalization scheme chosen. The conversion Nf = 2 : = 47.1(17.6). (4) MS factor Z between the lattice-cutoff scheme F 3 with domain wall fermions and MS

Fm

, versus , indicating enhancement of Nf = 6 relative to Nf = 2.

The other fit parameters are αM = 0.31(62), αF = 0.64(47) and αC = 83(29). The values of the fit parameters excluding αC indicate that we are near the limit of applicability of χPT. We note also that χ2 /d.o.f. = 6.50 with an uncorrelated fit and 4 degrees of freedom. We next compare our Nf = 2 results for "ψψ#/F 3 and Mρ /F to the QCD quantities "qq#/fπ3 and mρ /fπ , a reasonable comparison since the light-quark masses mq

0.05 0.04 0.03 0.02

N N

0.01 0 0

FIG. 3: The Gold as a function of 2 fit to Mm , Fm a 0.01 − 0.02, cons 0.04 0.001

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is known from Ref.30 From that reference, for this simulation the required factor to convert Rm is Z6MS /Z2MS = 1.449(29)/1.227(11) = 1.18(3). This increases the perturbative estimate of expected enhancement to the order of 20 − 30%, so the observed Rm > ∼ 1.5 is still significantly larger than anticipated. A direct computation of the S-parameter is also important to the study of general Yang-Mills theories with a focus on technicolor, and is well within the reach of existing lattice techniques. Some results have been reported,31,32 and more work is underway by the LSD collaboration. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, Phys. Lett. B400, 379 (1997). W. E. Caswell, Phys. Rev. Lett. 33, p. 244 (1974). T. Banks and A. Zaks, Nucl. Phys. B196, p. 189 (1982). M. A. Luty, JHEP 04, p. 050 (2009). B. Holdom, Phys. Lett. B150, p. 301 (1985). K. Yamawaki, M. Bando and K.-i. Matumoto, Phys. Rev. Lett. 56, p. 1335 (1986). T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57, p. 957 (1986). M. L¨ uscher, R. Narayanan, P. Weisz and U. Wolff, Nucl. Phys. B384, 168 (1992). S. Sint, Nucl. Phys. B421, 135 (1994). A. Bode, P. Weisz and U. Wolff, Nucl. Phys. B576, 517 (2000), Erratumibid.B608:481,2001. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100, p. 171607 (2008). T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D79, p. 076010 (2009). E. Bilgici et al. (2009). E. Bilgici et al., Phys. Rev. D80, p. 034507 (2009). Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, Phys. Lett. B681, 353 (2009). M. L¨ uscher, P. Weisz and U. Wolff, Nucl. Phys. B359, 221 (1991). S. Caracciolo, R. G. Edwards, S. J. Ferreira, A. Pelissetto and A. D. Sokal, Phys. Rev. Lett. 74, 2969 (1995). A. Bode et al., Phys. Lett. B515, 49 (2001). R. Sommer (2006). M. L¨ uscher, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B413, 481 (1994). U. M. Heller, Nucl. Phys. B504, 435 (1997). F. Bursa, L. Del Debbio, L. Keegan, C. Pica and T. Pickup, Phys. Rev. D81, p. 014505 (2010). X.-Y. Jin and R. D. Mawhinney, PoS LAT2009, p. 049 (2009). L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Phys. Rev. D80, p. 074507 (2009). D. B. Leinweber, A. W. Thomas, K. Tsushima and S. V. Wright, Phys. Rev. D64, p. 094502 (2001). R. Sommer, Nucl. Phys. B411, 839 (1994). J. Gasser and H. Leutwyler, Phys. Lett. B184, p. 83 (1987). J. Bijnens and J. Lu, JHEP 11, p. 116 (2009). T. Appelquist, A. Avakian, R. Babich, R. C. Brower, M. Cheng, M. A. Clark, S. D. Cohen, G. T. Fleming, J. Kiskis, E. T. Neil, J. C. Osborn, C. Rebbi, D. Schaich and P. Vranas, Phys. Rev. Lett. 104, p. 071601 (2010).

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30. S. Aoki, T. Izubuchi, Y. Kuramashi and Y. Taniguchi, Phys. Rev. D67, p. 094502 (2003). 31. E. Shintani et al., Phys. Rev. Lett. 101, p. 242001 (2008). 32. P. A. Boyle, L. Del Debbio, J. Wennekers and J. M. Zanotti, Phys. Rev. D81, p. 014504 (2010).

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Conformal Higgs, or Techni-Dilaton — Composite Higgs Near Conformality Koichi Yamawaki∗ Department of Physics, Nagoya University Nagoya, Japan E-mail: [emailprotected] In contrast to the folklore that Technicolor (TC) is a “Higgsless theory”, we shall discuss existence of a composite Higgs boson, Techni-Dilaton (TD), a pseudo-Nambu-Goldstone boson of the scale invariance in the Scale-invariant/Walking/Conformal TC (SWC TC) which generates a large anomalous dimension γm 1 in a wide region from the dynamical mass m = O (TeV) of the techni-fermion all the way up to the intrinsic scale ΛTC of the SWC TC (analogue of ΛQCD ), where ΛTC is taken typically as the scale of the Extended TC scale ΛETC : ΛTC ΛETC ∼ 103 TeV ( m). All the techni-hadrons have mass on the same order O(m), which in SWC TC is extremely smaller than the intrinsic scale ΛTC ΛETC , in sharp contrast to QCD where both are of the same order. The mass of TD arises from the non-perturbative scale anomaly associated with the techni-fermion mass generation and is typically 500-600 GeV, even smaller than other techni-hadrons of the same order of O(m), in another contrast to QCD which is believed to have no scalar q¯q bound state lighter than other hadrons. We discuss the TD mass in various methods, Gauged NJL model via ladder Schwinger-Dyson (SD) equation, straightforward calculations in the ladder SD/ Bethe-Salpeter equation, and the holographic approach including techni-gluon condensate. The TD may be discovered in LHC. Keywords: Walking technicolor, scale invariance, conformal symmetry, techni-dilaton, fixed point, composite Higgs, large anomalous dimension, holographic gauge theory.

1. Introduction Toshihide Maskawa is famous for 2008 Nobel prize-winning paper with Makoto Kobayashi on CP violation but did also fundamental contributions particularly to the SCGT, the topics of this workshop: Back in 1974 he found with Hideo Nakajima1 that spontaneous chiral symmetry breaking (SχSB) solution does exists for and only for the strong gauge coupling, with the critical coupling of order 1, based on the ladder Schwinger-Dyson (SD) equation with non-running (scale-invariant) coupling, namely the walking gauge dynamics what is called today. This turned out to be the origin of SCGT activities toward understanding the Origin of Mass. The present workshop SCGT 09 was held in honor of his 70th birthday on February 7, 2010 ∗ Present

Address: Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University.

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and the 35th anniversary of his crucial contributions to SCGT. I will later explain impact of Maskawa-Nakajima solution on the conformal gauge dynamics. The Origin of Mass is the most urgent issue of the particle physics today and is to be resolved at the LHC experiments. In the standard model (SM), all masses are attributed to a single parameter of the vacuum expectation value (VEV), H of the hypothetical elementary particle, the Higgs boson. The VEV simply picks up the mass scale of the input parameter M0 which is tuned to be tachyonic (M02 < 0) in such a way as to tune H 246 GeV (“naturalness problem”). As such SM does not explain the Origin of Mass. Technicolor (TC)2 is an attractive idea to account for the Origin of Mass without introducing ad hoc Higgs boson and tachyonic mass parameter: The mass arises dynamically from the condensate of the techni-fermion and the anti technifermion pair T¯T which is triggered by the attractive gauge forces between the pair analogously to the quark-antiquark condensate ¯ q q in QCD. For the TC with SU (NTC ) gauge symmetry and Nf flavors (Nf /2weak doublets) of technifermions, the techni-pion decay constant Fπ = H/ Nf /2 corresponds to the pion decay constant fπ 93 MeV in QCD, and hence the TC may be a scale-up of QCD by the factor Fπ /fπ 2650/ Nf /2. Then the mass scale of the condensate Λχ = (−T¯T /NTC )1/3 as the Origin of Mass may be estimated as √ 1/3 1/2 −¯ q q Nc /NTC Fπ / NTC √ Λχ · 450 GeV · , (1) Nc Nf /2 fπ / Nc where we have used a typical value (−¯ q q)1/3 250 MeV (Nc = 3). The dynamically generated mass scale of the condensate Λχ , or the dynamical mass of the techni-fermion, m (∼ Λχ ∼ Fπ ), in fact picks up the intrinsic mass scale ΛTC of the theory (analogue of ΛQCD in QCD) already generated by the scale anomaly through quantum effects (“dimensional transmutation”) in the gauge theory which is scale-invariant at classical level (for massless flavors): α(µ) α(Λ0 ) dα dα = Λ0 · exp − , (2) ΛTC = µ · exp − β(α) β(α) where the running of the coupling constant α(µ), with non-vanishing beta function β(α) ≡ µ dα(µ) dµ = 0, is a manifestation of the scale anomaly and Λ0 is a fundamental scale like Planck scale. Note that ΛTC is independent of the renormalization point µ, dΛdµTC = 0, and can largely be separated from Λ0 through logarithmic running (“naturalness”). Thus the Origin of Mass is eventually the quantum effect in this picture: In the simple scale-up of QCD we would have Naturalness (QCD scale up) :

m ∼ Λχ ∼ ΛTC Λ0 .

(3)

The original version of TC, just a simple scale-up of QCD, however, is plagued by the notorious problems: Excessive flavor-changing neutral currents (FCNCs), and excessive oblique corrections of O(1) to the Peskin-Takeuchi S parameter3 compared with the typical experimental bound about 0.1.

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The FCNC problem was resolved long time ago by the TC based on the near conformal gauge dynamics with γm 1,4,5 initially dubbed “scale-invariant TC” and then “walking TC”, with almost non-running (conformal) gauge coupling, based on the pioneering work by Maskawa and Nakajima1 who discovered nonzero critical coupling, αcr (= 0), for the SχSB to occur. We may call it “Scaleinvarinat/Walking/Conformal TC” (SWC TC) (For reviews see Ref.6 ). In addition to solving the FCNC problem, the theory made a definite prediction of “Techni-dilaton (TD)”,4 a pseudo Nambu-Goldstone (NG) boson of the spontaneous breaking of the (approximate) scale invariance of the theory. This will be the main topics of this talk in the light of modern version of SWC TC. The modern version7–9 of SWC TC is based on the Caswell-Banks-Zaks (CBZ) infrared (IR) fixed point 10 , α∗ = α∗ (Nf , NTC ), which appears at two-loop beta function for the number of massless flavors Nf (< 11NTC /2) larger than a certain number Nf∗ ( NTC ). See Fig. 1 and later discussions. Due to the IR fixed point the

«

¬

« «

¼

Á

ÁÁ

ÁÁÁ

Fig. 1.

The beta function and α(µ) for SWC-TC.

coupling is almost non-running (“walking”) all the way up to the intrinsic scale ΛTC which is generated by the the scale anomaly associated with the (two-loop) running of the coupling analogously to QCD scale-up in Eq. (2). For µ > ΛTC (Region I of Fig. 1) the coupling no longer walks and runs similarly to that of QCD. When we set α∗ slightly larger than αcr , we have a condensate or the dynamical mass of the techni-fermion m (∼ Λχ ), much smaller than the intrinsic scale of the theory m ΛTC . The CBZ-IR fixed point α∗ actually disappears (then becoming wouldbe IR fixed point) at the scale µ < ∼ m where the techni-fermions have acquired the mass m and get decoupled from the beta function for µ < m (Region III in Fig. 1). Nevertheless, the coupling is still walking due to the remnant of the CBZ-IR fixed point conformality in a wide region m < µ < ΛTC (Region II in Fig. 1). Thus the symmetry responsible for the natural hierarchy m ∼ Λχ ΛTC is the (approximate) conformal symmetry, while the naturalness for the hierarchy ΛTC Λ0 is the same as that of QCD scale-up in Eq.(3): Naturalness (SWC TC) :

m ∼ Λχ ΛTC

( Λ0 ) .

(4)

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The theory acts like the SWC-TC4,5 : It develops a large anomalous dimension γm 1 for the almost non-running coupling in the Region II .8,9 Here ΛTC plays a role of cutoff Λ identified with the ETC scale: ΛTC = Λ = ΛETC . Moreover, there also exists a possibility11,12 that the S parameter can be reduced in the case of SWC-TC. In this talk I will argue13 a that in contrast to the simple QCD scale-up which is widely believed to have no composite Higgs particle (“higgsless”), a salient feature of SWC TC is the conformality which manifests itself by the appearance of a composite Higgs boson (“conformal Higgs”) as the Techni-dilaton (TD)4 with mass relatively lighter than other techni-hadrons: MTD < Mρ , Ma1 · · · = O(Λχ ) ΛTC = ΛETC , where Mρ , Ma1 · · · denote the mass of techni-ρ, techni-a1 , etc. This is contrasted to the QCD dynamics where there are no scalar bound states lighter than others. Note that there is no idealized limit where the TD becomes exactly massless to be a true NG boson, in sharp contrast to the chiral symmetry breaking. Scale symmetry is always broken explicitly as well as spontaneously b . For the phenomenological purpose, I will argue through several different calculations13,15,16 that the techni-dilaton mass in the typical SWC TC models will be in the range (see the footnote below Eq.(30), however): mTD = 500 − 600 GeV,

(5)

which is definitely larger than the SM Higgs bound but still within the discovery region of the LHC experiments. 2. Scale-Invariant/Walking/Conformal Technicolor Let us briefly review the SWC TC. The FCNC problem is related with the mass generation of quarks/leptons mass. In order to communicate the techni-fermion condensate to the quarks/leptons masses mq/l , we would need interactions between the quarks/leptons and the technifermions which are typically introduced through Extended TC (ETC)17 c with much T¯T ΛETC , where T¯T ΛETC is the condenhigher scale ΛETC ( Λχ ): mq/l ∼ Λ−1 2 ETC sate measured at the scale of ΛETC . (We here do not refer to the origin of the mass scale ΛETC which should also be of dynamical origin such as the tumbling. ) Since the newly introduced ETC interactions characterized by the same scale ΛETC should induce extra FCNC’s, we should impose a constraint ΛETC > 106 GeV in a Preliminary

discussions on the revival of the techni-dilaton4 were given in several talks14 . straightforward calculations near the conformal edge indicated16 that there is no isolated massless spectrum: MTD /Fπ , MTD /Mρ , · · · → const. = 0 even in the limit of α∗ → αcr (Nf → Nfcrit ) where Fπ /ΛTC , MTD /ΛTC , Mρ /ΛTC , · · · → 0. In the case of holographic TD,13 this fact is realized in a different manner: Although there apparently exists an isolated massless spectrum, MTD /Fπ → 0 while Mρ /Fπ , Ma1 /Fπ → const. = 0, the decay constant of the TD diverges FTD /Fπ → ∞ in that limit and hence it gets decoupled. See later discussions. c The same can be done in a composite model where quarks/leptons and techni-fermions are composites on the same footing.18 b The

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order to avoid the excessive FCNC’s (typically involving s quark). If we assume a simple QCD scale up, T¯T ΛETC T¯T Λχ = −NTC · Λ3χ , we would have mq/l ∼

Λ3χ · NTC < 0.1 MeV · NTC Λ2ETC

Nc /NTC Nd

3/2

,

(6)

which implies that the typical mass (s-quark mass) would be roughly 10−3 smaller than the reality. We would desperately need 103 times enhancement. This was actually realized dynamically by the TC based on the near conformal gauge dynamics,4,5 based on the Maskawa-Nakajima solution1 of the (scaleinvariant) ladder Schwinger-Dyson (SD) equation for fermion full propagator SF (p) p − B(p2 ) with non-running (conformal, an ideal parameterized as iSF−1 (p) = A(p2 )/ limit of the “walking”) gauge coupling, α(Q) ≡ α = constant, with Q2 ≡ −p2 > 0. (See Fig. 2) Maskawa and Nakajima discovered that the SχSB can only take place for strong coupling α > -1 αcr = O(1), non-zero critical coui SF ( p ) p = pling.d The critical value reads:20 p q C2 (F )αcr = π/3, or full fermion propagator 2 αcr = (π/3) · 2NTC /(NTC − 1) (7)

Fig. 2. Graphical expression of the SD equation in the ladder approximation.

in the SU (NTC ) gauge theory, where C2 (F ) is the quadratic Casimir of the techni-fermion representation of the TC. The asymptotic form of the Maskawa-Nakajima SχSB solution of the fermion mass function Σ(Q) = B(p2 )/A(p2 ) in Landau gauge (A(p2 ) ≡ 1) reads,1,20 Σ(Q) ∼ 1/Q (Q Λχ ) .

(8)

We then proposed a “Scale-invariant TC” 4 , based on the observation that Eq.(8) implies a special value of the anomalous dimension γm = −Λ

∂ ln Zm = 1, ∂Λ

(9)

to be compared with the operator product expansion (OPE), Σ(Q) ∼ 1/Q2 · −1 · (Q/Λχ )γm . Accordingly, we had an enhanced condensate T¯T ΛETC = Zm 2 ¯ T T Λχ −NTC (ΛETC Λχ ), with the (inverse) mass renormalization constant being −1 = (ΛETC /Λχ )γm ΛETC /Λχ 103 , which in fact yields the desired enhanceZm ment. We actually obtained a different formula than Eq.(6):4 mq/l ∼ d Earlier

Λ2χ · NTC . ΛETC

(10)

works19 in the ladder SD equation with non-running coupling all confused explicit breaking solution with the SSB solution and thus implied αcr = 0

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The model4 was formulated in terms of the Renormalization Group Equation (RGE) a la Miransky 21 for the Maskawa-Nakajima solution Σ(m) = m which takes the form20,21 Λχ ∼ m ∼ 4Λ exp −π/ α/αcr − 1 , (11) where Λ is the cutoff of the SD equation. This has an essential singularity often called “Miransky scaling” and implies the non-perturbative beta function having a multiple zero e : β(α)NP = Λ

2 ∂α(Λ) =− ∂Λ 3C2 (F )

α −1 αcr

3/2

,

(12)

with the critical coupling αcr identified with a nontrivial ultraviolet (UV) stable fixed point α = α(Λ) → αcr as Λ/m → ∞. Subsequently, similar enhancement effects of the condensate were also studied5 within the same framework of the ladder SD equation, without use of the RGE concepts of anomalous dimension and fixed point, rather emphasizing the asymptotic freedom of the TC theories with slowly-running (walking) coupling which was implemented into the ladder SD equation (“improved ladder SD equation”). Today the Scale-invariant/Walking/Conformal TC (SWC TC) is simply characterized by near conformal property with γm 1 (For a review see Ref.6 ). Such a theory should have an almost non-running and strong gauge coupling (larger than a certain non-zero critical coupling for SχSB) to be realized either at UV fixed point or IR fixed point, or both (“fusion” of the IR and UV fixed points), as was characterized by “Conformal Phase Transition (CPT)”.9 The essential feature of the above is precisely what happens in the modern version 7–9 of the SWC TC based on the CBZ IR fixed point10 of the large Nf QCD, the QCD-like theory with many flavors Nf ( NTC ) of massless techni-fermions, f see Fig. 1. The two-loop beta function is given by d α(µ) = −bα2 (µ) − cα3 (µ), where b = (11NTC − 2Nf ) /(6π), c = β(α) = µ dµ

2 2 − 10Nf NTC − 3Nf (NTC − 1)/NTC /(24π 2 ) . When b > 0 and c < 0, i.e., 34NTC ∗ Nf∗ < Nf < 11 2 NTC (Nf 8.05 for NTC = 3), there exists an IR fixed point (CBZ IR fixed point) at α = α∗ , β(α∗ ) = 0, where α∗ = α∗ (NTC , Nf ) = −b/c.

(13)

Note that α∗ = α∗ (Nf , NTC ) → 0 as Nf → 11NTC /2 (b → 0) and hence there exists a certain range Nfcr < Nf < 11NTC /2 (“Conformal Window”) satisfying α∗ < αcr , where the gauge coupling α(µ) (< α∗ ) gets so weak that attractive forces are no longer strong enough to trigger the SχSB as was demonstrated by Maskawae Simple

zero of the beta function, β(α) ∼ (α − αcr )1 , never reproduces the essential singularity scaling, as is evident from Eq.(2). f For SWC TC based on higher representation/other gauge groups see, e.g., Ref.22

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Nakajima.1 Nfcr such that α∗ (NTC , Nfcr ) = αcr may be evaluated by using the value of αcr from the ladder SD equation Eq.(7): 8 Nfcr 4NTC (= 12 for NTC = 3) g . Here we are interested in the SχSB phase slightly off the conformal window, 0 < α∗ − αcr 1 (Nf Nfcr ). We may use the same equation as the ladder SD equation with α(µ) const. = α∗ , yielding the same form as Eq.(11) :8 m ∼ 4ΛTC exp −π/ α∗ /αcr − 1 ΛTC (α∗ αcr ) , (14) where the cutoff Λ was identified with ΛTC (= ΛETC ). We also have the same result as Eqs. (8),(9): Σ(Q) ∼ 1/Q ,

γm 1 .

(15)

Hence it acts like SWC TC. Incidentally, Eq.(14) implies a multiple zero at α∗ = αcr in a non-perturbative beta function for α∗ = α∗ (Λ) similar to Eq.(12), which would suggest “running” of the IR fixed point α∗ with its UV fixed point α∗ = αcr in the limit ΛTC /m → ∞. The actual running of the coupling largely based on two-loop perturbation is already depicted in Fig. 1. The critical coupling αcr can be regarded as the UV fixed point viewed from the IR part of the Region II (m < µ < µcr , with µcr such that α(µcr ) = αcr ), while it is regarded as the IR fixed point from the UV part of the Region II (ΛT C > µ > µcr ), with the Region II regarded as the fusion of the IR and UV fixed points in the idealized limit of non-running (perturbative) coupling in Region II (or ΛTC /m → ∞). Although the perturbative (two-loop) beta function has a simple zero, which never corresponds to the essential singularity scaling as we noted before, the coupling near αcr should be sensitive to the non-perturbative effects in such as way that the beta function looks like the multiple zero as in Eq. (12) from both sides, corresponding to the essential singularity scaling as in Eq. (11). This should be tested by the fully non-perturbative studies like lattice simulations. A possible phase diagram (Fig 3 of Ref.9 ) of the large Nf QCD on the lattice is also waiting for the test by simulations. 3. Conformal Phase Transition9,14 Such an essential singularity scaling law like Eq.(11),(14), or equivalently the multiple zero of the non-perturbative beta function, characterizes an unusual phase transition, what we called “Conformal Phase Transition (CPT)”, where the GinzburgLandau effective theory breaks down:9 Although it is a second order (continuous) phase transition where the order parameter m (α∗ > αcr ) is continuously changed to m = 0 in the symmetric phase (conformal window, α∗ < αcr ), the spectra do not, i.e., while there exist light composite particles whose mass vanishes at the critical g The value should not be taken seriously, since α = α ∗ cr is of O(1) and the perturbative estimate of α∗ is not so reliable there, although the chiral symmetry restoration in large Nf QCD has been supported by many other arguments, most notably the lattice QCD simulations,23,24 which however suggest diverse results as to Nfcr ; See e.g.,25 for recent results.

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point when approached from the side of the SSB phase, no isolated light particles do not exist in the conformal window, recently dubbed “unparticle”.26 This reflects the feature of the conformal symmetry in the conformal window. In fact explicit computations show no light (composite) spectra in the conformal window, in sharp contrast to the SχSB phase where light composite spectra do exist with mass of order O(m) which vanishes as we approach the conformal window Nf Nfcr .8,9,16 The essence of CPT was illustrated 9 by a simpler model, 2-dimensional GrossNeveu Model. This is the D → 2 limit of the D-dimensional Gross-Neveu model (2 < D < 4) which has the beta function and the anomalous dimension:27,28 β(g) = −2g(g − g∗ ),

γm = 2g ,

(16)

where g = g∗ (≡ D/2 − 1) = gcr and g = 0 are respectively the UV and IR fixed points of the dimensionless four-fermion coupling, g, properly normalized (as g∗ = 1 for the D = 4 NJL model). There exist light composites π, σ near the UV fixed point (phase boundary) g g∗ in both sides of symmetric (0 < g < g∗ ) and SSB (g > g∗ ) phases as in the NJL model. Now we consider D → 2 ( g∗ → 0) where we have a well-known effective potential: V (σ, π) ∼ (1/g−1)ρ2+ρ2 ln(ρ2 /Λ2 ), or ∂ 2 V /∂ρ2 |ρ=0 = −∞, where ρ2 = π 2 +σ 2 . This implies breakdown of the Ginzburg-Landau theory which distinguishes the SSB (< 0) and symmetric (> 0) phases by the signature of the finite ∂ 2 V /∂ρ2 at the critical point g = 0. Eq. (16 ) now reads: β(g) = −2g 2 , γm |g=0 = 0

(D = 2) ,

(17)

namely a fusion of the UV and IR fixed points at g = 0 as a result of multiple zero (not a simple zero) at g = 0. Now the symmetric phase is squeezed out to the region g < 0 (conformal phase) which corresponds to a repulsive four-fermion interaction and no composite states exist, while in the SSB phase (g > 0) there exists a composite state σ of mass Mσ = 2m where the dynamical mass of the fermion is given by m2 ∼ Λ2 exp(−1/g) → 0 (g → +0), which shows an essential singularity scaling, in accord with the beta function with multiple zero, β(g) = Λ∂g/∂Λ = −2g 2 . Note the would-be composite mass in the symmetric phase |M |2 ∼ Λ2 exp(−1/g) → ∞ (g → −0). Now look at the SWC TC as modeled by the large Nf QCD: When the walking coupling α(Q) α∗ is close to the critical coupling, α∗ αcr , we should include 2 ¯ ¯ 5 ψ 2 ], which becomes relthe induced four-fermion interaction, (G/2)[ ψψ + ψiγ evant operator due to the anomalous dimension γm = 1, and the system becomes “gauged Nambu-Jona-Lasinio” model29 whose solution in the full parameter space was obtained in Ref.30 Thus we may regard the SWC TC as the gauged Nambu-Jona-Lasinio (NJL) 30 model. It was found2 that SχSB solution exists for the parameter space g > g(+) = (1 + 1 − α∗ /αcr ) /4 (α∗ < αcr ) as well as the region α∗ > αcr , where the dimensionless four-fermion coupling g ≡ GΛ2 (NTC /4π 2 ) is normalized as g = 1 for α∗ = 0 (pure NJL model without gauge interaction). Based on the solution (including the

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running coupling case), the RGE flow in (α, g) space was found to be along the line of α = α∗ (α does not run), h on which the four-fermion coupling g runs, with the beta function and anomalous dimension given by 28,32,33 β(g) = −2(g − g(+) )(g − g(−) ), γm = 2g + α∗ /(2αcr ) (18) where g = g(±) ≡ (1 ± 1 − α∗ /αcr )2 /4 are regarded as the UV/IR fixed points (fixed ∗ ≤ αcr . The above anomalous dimension takes the values: γm = lines) for α31 1 + 1 − α∗ /αcr at the UV fixed line while γm = 1 − 1 − α∗ /αcr at the IR fixed line. Light composite spectra only exist near the UV fixed line (phase boundary) g g(+) in both SSB (g > g(+) ) and symmetric (g > g(+) ) phases as in NJL model. Thus it follows that as α∗ → αcr Eq. (18) takes the form β(g) = −2(g − g∗ )2 , γm |g=g∗ = 1 ,

(α∗ = αcr ) ,

(19)

with g(±) → 1/4 ≡ g∗ , and hence we again got a multiple zero and fusion of UV and IR fixed lines 28,32,33 which corresponds to the essential singularity scaling;30 m2 ∼ Λ2 exp(−1/(g − g∗ )). A similar observation was also made recently.34 In passing, it should be stressed that the anomalous dimension never changes discontinuously across the phase boundary as is seen from Eq.(16) and Eq.(18).27,28,32 The scale anomaly in this case is given by:9 2 β(g) G ¯ 2 ¯ · ψψ + ψiγ5 ψ g 2 −m4 · (4Nf NTC /π 4 ) = O(Λ4χ ) ,

∂ µ Dµ = θµµ = 4θ00 =

(20)

where the second line was from the explicit computation35 of the vacuum energy θ00 in the limit Λ/m → ∞ (g → g∗ ) at α ≡ αcr (The result coincides with the one for α → αcr with g ≡ 0, see Eq.(24).36 ). Again there is a composite state this time having mass15 √ Mσ → 2m , (21) as g → g∗ + 0, while there are no composites |M |2 ∼ Λ2 exp(−1/(g − g∗ )) → ∞ for g → g∗ − 0. Eq.(21) is compared with Mσ = 2m in the pure NJL case with α ≡ 0. This slightly lighter scalar may be identified with the techni-dilaton in the SWC TC. I will come back to this later. The absence of the composites in the symmetric phase g < g∗ may be understood as in the 2-dimensional Gross-Neveu model for g < 0, namely the repulsive fourfermion interactions: From the analysis of the RG flow, it was argued32 that the IR fixed line g = g(−) is due to the induced four-fermion interaction by the walking TC dynamics itself, while deviation from that line, g − g(−) , is due to the additional four-fermion interactions, repulsive (g < g(−) ) and attractive (g > g(−) ), from UV h The beta function in Eq.(12) may be regarded as an artificial one keeping g ≡ const. which is not along the renormalized trajectory in the extended parameter space (α, g).

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dynamics other than the TC (i.e., ETC). It is clear that no light composites exist for repulsive four-fermion interaction g < g(−) , which becomes g < g∗ at α∗ = αcr . 4. S Parameter Constraint Now we come to the next problem of TC, so-called S, T, U parameters3 measuring possible new physics in terms of the deviation of the LEP precision experiments from the SM. In particular, S parameter excludes the TC as a simple scale-up of QCD which yields S = (Nf /2) · Sˆ with SˆQCD = 0.32 ± 0.04. For a typical ETC model with one-family TC, Nf = 8,2 we would get S = O(1) which is much larger than the experiments S < 0.1. This is the reason why many people believe that the TC is dead. However, since the simple scale-up of QCD was already ruled out by the FCNC as was discussed before, the real problem is whether or not the walking/conformal TC which solved the FCNC problem is also consistent with the S parameter constraint above. There have been many arguments11,12 that the S parameter value could be reduced in the walking/conformal TC than in the simple scale-up of QCD. Recently such a reduction has also been argued37,38 in a version of the holographic QCD39 deformed to the walking/conformal TC by tuning a parameter to simulate the large anomalous dimension γm 1. Here we present the most straightforward computation of the S parameter for the large Nf QCD, based on the SD equation and (inhom*ogeneous) Bethe-Salpeter (BS) equation in the ladder approximation.12 The S parameter S = (Nf /2)Sˆ is defined by the slope of the the current correlators ΠJJ (Q2 ) at Q2 = 0:

d Sˆ = −4π 2 ΠV V (Q2 ) − ΠAA (Q2 )

, (22) dQ Q2 =0 where δ ab qµ qν /q 2 − gµν ΠJJ (q 2 ) = F .T . i0|T Jµa (x)Jνb (0)|0, (Jµa (x) = Vµa (x), Aaµ (x)) , with Fπ2 = ΠV V (0) − ΠAA (0) . The current correlators are ob(J)

tained by closing the fermion legs of the BS amplitudes χµ (p; q) ∼ F.T . ¯ 0|T ψ(r/2) ψ(−r/2) Jµ (x) |0, which is determined by the ladder BS equation (Fig.3). Solving the BS equation with the fermion propagator given as the solution p +q2

χ (p ; q ) p −q2

Fig. 3.

p +q2

p +q2 q

q

p −q2

+

k +q2

χ (k ; q ) p −q2

q

k −q2

Graphical expression of the BS equation in the ladder approximation.

of the ladder SD equation, we can evaluate the ΠV V (Q2 ) − ΠAA (Q2 ) numerically. ˆ From this result we may read its slope at Q2 = 0 to get S. ˆ The results show definitely smaller values of S than that in the ordinary QCD and moreover there is a tendency of the Sˆ getting reduced when approaching the

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conformal window α∗ αcr (Nf Nfcr ). However, due to technical limitation of the present computation getting very close to the conformal window, the reduction does not seem to be so dramatic as the walking TC being enough to be consistent with the experimental constraints. It is highly desirable to extend the computation further close to the conformal window. Another approach to this problem is the deformation of the holographic QCD by the anomalous dimension. The reduction of S parameter in the SWC TC has been argued in a version of the hard-wall type bottom up holographic QCD39 deformed to the SWCTC by tuning a parameter to simulate the large anomalous dimension γm 1.37 We examined38 such a possibility paying attention to the renormalization point dependence of the condensate. We explicitly calculated the S parameter in entire parameter space of the holographic SWC TC. We here take a set of Fπ /Mρ and γm . We find that S > 0 and it monotonically decreases to zero in accord with the previous results37 . However, our result turned out fairly independent of the value of the anomalous dimension γm , yielding no particular suppression solely by tuning the anomalous dimension large, Sˆ ∼ B(Fπ /Mρ )2 → 0 as Fπ /Mρ → 0, with B 27(32) for γm 1(0), in sharp contrast to the previous claim37 . Although B contains full contributions from the infinite tower of the vector/axial-vector KaluzaKlein modes (gauge bosons of hidden local symmetries)40 of the 5-dimensional gauge bosons, the resultant value of B turned out close to B 4πa 8π of the single ρ meson dominance, where a 2 is the parameter of the hidden local symmetry only for the ρ meson.40 This implies that as far as the pure TC dynamics (without ETC dynamics, etc.) is concerned, an obvious way to dynamically reduce S parameter is to tune Fπ /Mρ very small, namely techni-ρ mass very large to several TeV region. ( See, however, footnote below Eq.(30).) We would need more dynamical information other than the holographic recipe, since the parameter corresponding to Fπ /Mρ as well as the scale parameter is a pure input in all the holographic models, whether bottom up or top down approach, in contrast to the underlying gauge theory which has only a single parameter, a scale parameter like ΛQCD . Curiously enough, when we calculate Fπ /Mρ from the SD and the hom*ogeneous BS equations16 and S from the SD and the inhom*ogeneous BS equation12 both in the straightforward calculation in the ladder approximation, a set of the calculated values of (Fπ /Mρ , S) lies on the line of the holographic result.38 5. Techni-Dilaton Now we come to the discussions of Techni-dilaton (TD). Existence of two largely separated scales, Λχ ∼ m and ΛTC such that Λχ ΛTC , is the most important feature of SWC-TC, in sharp contrast to the ordinary QCD with small number of flavors (in the chiral limit) where all the mass parameters like dynamical mass of quarks are of order of the single scale parameter of the theory ΛQCD , m ∼ Λχ ∼ ΛQCD . See Fig. 1. The intrinsic scale ΛTC is related with the scale anomaly

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corresponding to the perturbative running effects of the coupling, with the ordinary two-loop beta function β(α) in the Region I, in the same sense as in QCD. ∂ µ Dµ = θµµ =

β(α) αG2µν = O(Λ4TC ), 4α2

(23)

which implies that all the techni-glue balls have mass of O(ΛTC ). On the other hand, the scale Λχ is related with totally different scale anomaly due to the dynamical generation of m (∼ Λχ ) which does exist even in the idealized case with non-running coupling α(µ) ≡ α(> αcr ) such as the Maskawa-Nakajima solution,1 as was discussed some time ago.36 Such an idealized case well simulates the dynamics of Region II of Fig. 1 ,8,9 with anomalous dimension γm 1 and m ΛTC in the numerical calculations,16 with the perturbative coupling constant in Region II being almost constant slightly larger than αcr , αcr < α(µ)(< α∗ ), for a wide infrared region. The coupling α ≡ α∗ in the “idealized Region II” actually runs non-perturbatively according to the essential-singularity scaling (Miransky scaling21 ) of mass generation, Eq.(11), with the non-perturbative beta function βNP (α), Eq.(12), having a multiple zero at α = αcr . Then the non-perturbative scale anomaly reads9,i ∂ µ Dµ NP = θµµ NP =

βNP (α) 4Nf NTC αG2µν NP = −m4 · = −O(Λ4χ ), 4α2 π4

(24)

where · · · NP is the quantity with the perturbative contributions subtracted:36 · · · NP ≡ · · · − · · · Perturbative . Eq.(24) coincides with Eq.(20) and ∂ µ Dµ NP /Λ4TC vanishes with ∂ µ Dµ NP /m4 → const. = 0, when we approach the conformal window from the broken phase α∗ αcr (m/ΛTC → 0). All the techni-fermion bound states have mass of order of m,41 while there are no light bound states in the symmetric phase (conformal window) α∗ < αcr , a characteristic feature of the conformal phase transition.9 The TD is associated with the latter scale anomaly and should have mass on order of m( ΛTC ). 5.1. Calculation from Gauged NJL Model in the Ladder SD Equation15 More concretely, the mass of TD or scalar bound state in the SWC-TC was estimated in various methods: The first method 15 was based on the the ladder SD equation for the gauged NJL model which well simulates8,9 the conformal phase transition in the large Nf QCD. The result was already given by Eq.(21): √ MTD 2m . (25) i In

terms of the gauged NJL model mentioned in Section 3 this is the expression of the scale anomaly for α → αcr with g = const. = g∗ , in contrast to Eq.(20) for g → g∗ with α = const. = α∗ = αcr . Both yield the same vacuum energy and hence the same scale anomaly.

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5.2. Straightforward Calculation from Ladder SD and BS Equations Also a straightforward calculation16 of mass of TD, the scalar bound state was made in the vicinity of the CBZ-IR fixed point in the large Nf QCD, based on the coupled use of the ladder SD equation and (hom*ogeneous) BS equation lacking the first term in Fig. 3: All the bound states masses are M = O(m) and M/ΛTC → 0, when approaching the conformal window α∗ → αcr (Nf → Nfcr ) such that m/ΛTC → 0. Near the conformal window (Nf Nfcr ) the calculated values are Mρ /Fπ 11, Ma1 /Fπ 12 (near degenerate !). On the other hand, the scalar mass sharply drops near the the conformal window, MTD /Fπ 4, or √ MTD 1.5m 2m (< Mρ , Ma1 ) . (26) Note that in this calculation MTD /Fπ → const. = 0 and hence there is no isolated massless scalar bound states even in the limit Nf → Nfcr . The result is consistent with Eq.(25) and is contrasted to the ordinary QCD where the scalar mass is larger than those of the vector mesons (“higgsless”) within the same framework of ladder SD/BS equation approach. The result would imply mTD 500 GeV

(27)

in the case of the one-family TC model with Fπ 125 GeV. 5.3. Holographic Techni-Dilaton13 Recently, we have calculated13 mass of TD in an extension of the previous paper38 on the hard-wall-type bottom-up holographic SWC-TC by including effects of (techni-) gluon condensation parameterized as 1/4 1 2 4 αG /F µν π (28) Γ ≡ 1 π 2 4 π αGµν /fπ QCD

through the bulk flavor/chiral-singlet scalar field ΦX , in addition to the conventional bulk scalar field Φ dual to the chiral condensate. The five-dimensional action is given by zm 1

√ 2 1 d4 x d z −g 2 ecg5 ΦX (z) − Tr LMN LMN + RMN RMN S5 = g5 4

1 (29) +Tr DM Φ† DM Φ − m2Φ Φ† Φ + ∂M ΦX ∂ M ΦX , 2 radius L of AdS where the anti-de-Sitter space (AdS5 ) with the curvature 5 is 2 described by the metric ds2 = gMN dxM dxN = (L/z) ηµν dxµ dxν − dz 2 with ηµν = diag[1, −1, −1, −1], g = det[gMN ] = −(L/z)10; g5 denotes the gauge coupling in five-dimension and c is the dimensionless coupling constant, and LM (RM ) =

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226 a LaM (RM )T a with the generators of SU (Nf ) are normalized by Tr[T a T b ] = δ ab ; L(R)MN = ∂M L(R)N − ∂N L(R)M − i[L(R)M , L(R)N ]. The covariant derivative acting on Φ is defined as DM Φ = ∂M Φ + iLM Φ − iΦRM . The TD, a flavor-singlet scalar bound state of techni-fermion and anti-technifermion, will be identified with the lowest KK mode coming from the bulk scalar field Φ, not ΦX . Thanks to the additional explicit bulk scalar field ΦX , we naturally improve the matching with the OPE of the underlying theory (QCD and SWCTC) for current correlators so as to reproduce gluonic 1/Q4 term, which is clearly distinguished from the same 1/Q4 terms from chiral condensate in the case of SWCTC with γm 1. Our model with γm = 0 and Nf = 3 well reproduces the real-life QCD. It is rather straightforward37,39 to compute masses of the techni-ρ meson (Mρ ), the techni-a1 meson (Ma1 ), while for that of the TD, flavor-singlet scalar meson (MTD ), we would need additional IR potential with quartic coupling λ to stabilize the SχSB vacuum.42 Such an IR potential might be regarded as generated by techni-fermion loop effects and we naturally expect λ ∼ NTC /(4π)2 . The S parameter was also calculated through the current correlators by the standard way. We found general tendency of the dependence of the meson masses relative to Fπ , (Mρ /Fπ , Ma1 /Fπ , MTD /Fπ ) on γm , S, and Γ. We find a characteristic feature of the techni-dilaton mass related to the conformality of SWC-TC: For fixed S and γm , absolute values of (Mρ /Fπ ) and (Ma1 /Fπ ) are not sensitive to Γ, although they get degenerate for large Γ. On the contrary, (MTD /Fπ ) substantially decreases as Γ increases. Actually, in the formal limit Γ → ∞ we would have (MTD /Fπ ) → 0 (This is contrast to the straightforward computation through ladder SD and BS equations mentioned before16 ). For fixed S and Γ, again (Mρ /Fπ ) and (Ma1 /Fπ ) are not sensitive to γm , while (MTD /Fπ ) substantially decreases as γm increases. Particularly for the case of γm = 1, we study the dependence of the S parameter on (Mρ /Fπ ) for typical values of Γ. It is shown that the techni-gluon contribution reduces the value of S about 10% in the region of Sˆ 0.1, although the general tendency is similar to the previous paper38 without techni-gluon condensation: Sˆ decreases to zero monotonically with respect to (Fπ /Mρ ). This implies (Mρ /Fπ ) necessarily increases when Sˆ is required to be smaller. To be more concrete, we consider a couple of typical models of SWC-TC with γm 1 and NTC = 2, 3, 4 based on the CBZ-IRFP in the large Nf QCD. Using the non-perturbative conformal anomaly Eq.(24) together with the non-perturbative beta function Eq.(12) and Eq.(11), we find a concrete relation between Γ and (ΛETC /Fπ ): In the case of NTC = 3 (Nf = 4NTC ) and S 0.1, we have Γ 7 for (ΛETC /Fπ ) = 104 –105 (required by the FCNC constraint). Thanks to the large anomalous dimension γm 1 and large techni-gluon condensation Γ 7, we obtain a relatively light techni-dilaton with mass

MTD 600 GeV

(30)

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compared with Mρ Ma1 3.8 TeV (almost degenerate). Eq. (30) is consistent with the perturbative unitarity of WL WL scattering even for large Mρ , Ma1 . Note that largeness of Mρ and Ma1 is essentially determined by the requirement of S = 0.1 fairly independently of techni-gluon condensation. j The essential reason for the large Γ is due to the existence of the wide conformal region Fπ (∼ m) < µ < ΛETC with ΛETC /Fπ = 104 –105 , which yields the smallness of the beta function (see Eq.(12) and Eq. (11) ) and hence amplifies the technigluon condensation compared with the ordinary QCD with Γ = 1. Actually, in the idealized (phenomenologically uninteresting) limit ΛETC /Fπ → ∞, we would have Γ → ∞, which in turn would imply MTD /Fπ → 0 as mentioned above. k To conclude, various methods predicted the mass of the techni-dilaton (“conformal Higgs”) in the range of MTD 500 − 600 GeV, which is within reach of LHC discovery. Since the SWC TC models are strong coupling theories and the ladder approximation/holographic calculations would be no more than a qualitative hint, more reliable calculations are certainly needed, including the lattice simulations, before drawing a definite conclusion about the physics predictions. Besides the phase diagram including the TC-induced/ETC-driven four-fermion couplings on the lattice, more reliable calculations such as the spectra as well as anomalous dimensions, non-perturbative beta functions, S parameter, etc. are highly desired. Acknowledgments We thank K. Haba, M. Harada, M. Hashimoto, M. Kurachi, and S. Matsuzaki for collaborations on the relevant topics and useful discussions. This work was supported by the JSPS Grant-in-Aid for Scientific Research(S) # 22224003 and by the Nagoya University Foundation. References 1. T. Maskawa and H. Nakajima, Prog. Theor. Phys. 52 (1974), 1326; 54 (1975), 860. 2. S. Weinberg, Phys. Rev. D 13 (1976), 974; D 19 (1979), 1277; L. Susskind, Phys. Rev. D 20 (1979), 2619: See for a review of earlier literature, E. Farhi and L. Susskind, Phys. Rep. 74 (1981), 277. j The calculated S parameter here was from the TC dynamics alone and could be drastically changed by incorporating contributions from the generation mechanism of the mass of SM fermions such as the strong ETC dynamics. For instance, the fermion delocalization43 in the Higgsless models as a possible analogue of certain ETC effects in fact can cancel large positive S arising from the 5-dimensional gauge sector which corresponds to the pure TC dynamics. If it is the case in the explicit ETC model, then the overall mass scale of techni-hadrons including TD would be much lower than the above estimate down to, say, 300 GeV. k This does not mean existence of true (exactly massless) NG boson of the broken scale symmetry, since such a would-be NG boson gets decoupled in our case: The decay constant of techni-dilaton FTD would diverge, FTD /Fπ → ∞, in that limit through the PCDC relation 2 = −4θ µ /M 2 4 2 4 2 FTD µ TD ∼ m /MTD ∼ Fπ /MTD . The scale symmetry is broken explicitly as well as spontaneously.

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3. M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990), 964; Phys. Rev. D 46 (1992), 381; B. Holdom and J. Terning, Phys. Lett. B 247 (1990), 88; M. Golden and L. Randall, Nucl. Phys. B 361 (1991), 3. 4. K. Yamawaki, M. Bando and K. Matumoto, Phys. Rev. Lett. 56 (1986), 1335; M. Bando, K. Matumoto and K. Yamawaki, Phys. Lett. B 178 (1986), 308. See also M. Bando, T. Morozumi, H. So and K. Yamawaki, Phys. Rev. Lett. 59 (1987), 389 . The solution of the FCNC problem by the large anomalous dimension was first considered by B. Holdom, based on a pure assumption of the existence of UV fixed point without explicit dynamics and hence without definite prediction of the value of the anomalous dimension. See B. Holdom, Phys. Rev. D 24 (1981), 1441. 5. T. Akiba and T. Yanagida, Phys. Lett. B 169 (1986), 432; T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57 (1986), 957; T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D 36 (1987), 568. For an earlier work on this line based on pure numerical analysis see B. Holdom, Phys. Lett. B 150 (1985), 301. 6. C. T. Hill and E. H. Simmons, Phys. Rept. 381 (2003), 235 [Erratum-ibid. 390 (2004), 553]; K. Yamawaki, Lecture at 14th Symposium on Theoretical Physics, Cheju, Korea, July 1995, arXiv:hep-ph/9603293. 7. K. D. Lane and M. V. Ramana, Phys. Rev. D 44, 2678 (1991). 8. T. Appelquist, J. Terning and L. C. Wijewardhana, Phys. Rev. Lett. 77 (1996), 1214; T. Appelquist, A. Ratnaweera, J. Terning and L. C. Wijewardhana, Phys. Rev. D 58 (1998), 105017. 9. V. A. Miransky and K. Yamawaki, Phys. Rev. D 55 (1997), 5051 [Errata; 56 (1997), 3768]. 10. W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974); T. Banks and A. Zaks, Nucl. Phys. B 196 (1982), 189. 11. R. Sundrum and S. D. H. Hsu, Nucl. Phys. B 391 (1993), 127; T. Appelquist and G. Triantaphyllou, Phys. Lett. B 278 (1992), 345; T. Appelquist and F. Sannino, Phys. Rev. D 59 (1999), 067702. 12. M. Harada, M. Kurachi and K. Yamawaki, Prog. Theor. Phys. 115 (2006), 765 ; M. Kurachi and R. Shrock, Phys. Rev. D 74, 056003 (2006); M. Kurachi, R. Shrock and K. Yamawaki, ibid D 76, 035003 (2007). 13. K. Haba, S. Matsuzaki and K. Yamawaki, arXiv:1003.2841 [hep-ph]; arXiv:1006.2526 [hep-ph]. 14. K. Yamawaki, Prog. Theor. Phys. Suppl. 180, 1 (2010) [arXiv:0907.5277 [hep-ph]]; See also Prog. Theor. Phys. Suppl. 167, 127 (2007). 15. S. Shuto, M. Tanabashi and K. Yamawaki, in Proc. 1989 Workshop on Dynamical Symmetry Breaking, Dec. 21-23, 1989, Nagoya, eds. T. Muta and K. Yamawaki (Nagoya Univ., Nagoya, 1990) 115-123; M. S. Carena and C. E. M. Wagner, Phys. Lett. B 285, 277 (1992); M. Hashimoto, Phys. Lett. B 441, 389 (1998). 16. M. Harada, M. Kurachi and K. Yamawaki, Phys. Rev. D 68 (2003), 076001; M. Kurachi and R. Shrock, JHEP 0612, 034 (2006). 17. S. Dimopoulos and L. Susskind, Nucl. Phys. B 155 (1979), 237; E. Eichten and K. D. Lane, Phys. Lett. B 90 (1980), 125. 18. K. Yamawaki and T. Yokota, Phys. Lett. B 113 (1982), 293; Nucl. Phys. B 223 (1983), 144. 19. K. Johnson, M. Baker and R. Willey, Phys. Rev. 136 (1964), B1111; M. Baker and K. Johnson, Phys. Rev. D3 (1971), 2516; R. Jackiw and K. Johnson Phys. Rev. D8 (1973), 2386; J.M. Cornwall and R.E. Norton, ibid, 3338. 20. R. f*ckuda and T. Kugo, Nucl. Phys. B 117 (1976), 250. 21. V. A. Miransky, Nuovo Cim. A 90 (1985), 149.

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22. F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901 (2005); D. K. Hong, S. D. H. Hsu and F. Sannino, Phys. Lett. B 597, 89 (2004); For a review F. Sannino, arXiv:0804.0182 [hep-ph]. 23. Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai and T. Yoshie, Nucl. Phys. (Proc. Suppl.) 53 (1997), 449; Prog. Theor. Phys. Suppl. No. 131 (1998), 415. 24. T. Appelquist, G. T. Fleming and E. T. Neil, Phys.Rev.Lett.100:171607,2008, Erratum-ibid.102:149902,2009; Phys. Rev. D 79, 076010 (2009); A. Deuzeman, M. P. Lombardo and E. Pallante, Phys. Lett. B 670, 41 (2008); Talks presented at SCGT 09, http://www.eken.phys.nagoya-u.ac.jp/scgt09/ . 25. Talks at SCGT 09: http://www.eken.phys.nagoya-u.ac.jp/scgt09/ . 26. H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). 27. Y. Kikukawa and K. Yamawaki, Phys. Lett. B 234, 497 (1990). 28. K. i. Kondo, M. Tanabashi and K. Yamawaki, Prog. Theor. Phys. 89, 1249 (1993). 29. W. A. Bardeen, C. N. Leung and S. T. Love, Phys. Rev. Lett. 56, 1230 (1986). 30. K. i. Kondo, H. Mino and K. Yamawaki, Phys. Rev. D 39, 2430 (1989); K. Yamawaki, in Proc. Johns Hopkins Workshop on Current Problems in Particle Theory 12, Baltimore, June 8-10, 1988, edited by G. Domokos and S. Kovesi-Domokos (World Scientific Pub. Co., Singapore 1988); T. Appelquist, M. Soldate, T. Takeuchi, and L. C. R. Wijewardhana, ibid. 31. V. A. Miransky and K. Yamawaki, Mod. Phys. Lett. A 4 (1989), 129. 32. K. i. Kondo, S. Shuto and K. Yamawaki, Mod. Phys. Lett. A 6, 3385 (1991). 33. K. I. Aoki, K. Morikawa, J. I. Sumi, H. Terao and M. Tomoyose, Prog. Theor. Phys. 102, 1151 (1999). 34. D. B. Kaplan, J. W. Lee, D. T. Son and M. A. Stephanov, Phys. Rev. D 80, 125005 (2009). 35. T. Nonoyama, T. B. Suzuki and K. Yamawaki, Prog. Theor. Phys. 81, 1238 (1989). 36. V. A. Miransky and V. P. Gusynin, Prog. Theor. Phys. 81 (1989) 426. 37. D. K. Hong and H. U. Yee, Phys. Rev. D 74 (2006), 015011; M. Piai, arXiv:hepph/0608241; K. Agashe, C. Csaki, C. Grojean and M. Reece, JHEP 0712, 003 (2007). 38. K. Haba, S. Matsuzaki and K. Yamawaki, Prog. Theor. Phys. 120, 691 (2008). 39. J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 95 (2005), 261602; L. Da Rold and A. Pomarol, Nucl. Phys. B 721, 79 (2005). 40. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54 (1985), 1215; M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B 259 (1985), 493; Phys. Rept. 164 (1988), 217; M. Harada and K. Yamawaki, Phys. Rept. 381 (2003), 1. 41. R. S. Chivukula, Phys. Rev. D 55 (1997), 5238. 42. L. Da Rold and A. Pomarol, JHEP 0601, 157 (2006). 43. G. Cacciapaglia, C. Csaki, C. Grojean and J. Terning, Phys. Rev. D 71, 035015 (2005); R. Foadi, S. Gopalakrishna and C. Schmidt, Phys. Lett. B 606, 157 (2005). R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Rev. D 72, 015008 (2005)

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Phase Diagram of Strongly Interacting Theories Francesco Sannino CP3 -Origins, University of Southern Denmark, Odense, 5230 M, Denmark E-mail: [emailprotected], [emailprotected] www.cp3-origins.dk We summarize the phase diagrams of SU, SO and Sp gauge theories as function of the number of flavors, colors, and matter representation as well as the ones of phenomenologically relevant chiral gauge theories such as the Bars-Yankielowicz and the generalized Georgi-Glashow models. We finally report on the intriguing possibility of the existence of gauge-duals for nonsupersymmetric gauge theories and the impact on their conformal window.

1. Phases of Gauge Theories Models of dynamical breaking of the electroweak symmetry are theoretically appealing and constitute one of the best motivated natural extensions of the standard model (SM). We have proposed several models1–8 possessing interesting dynamics relevant for collider phenomenology9–12 and cosmology.7,13–31 The structure of one of these models, known as Minimal Walking Technicolor, has led to the construction of a new supersymmetric extension of the SM featuring the maximal amount of supersymmetry in four dimension with a clear connection to string theory, i.e. Minimal Super Conformal Technicolor.32 These models are also being investigated via first principle lattice simulations33–48 a . An up-to-date review is Ref. 59 while an excellent review updated till 2003 is Ref. 60. These are also among the most challenging models to work with since they require deep knowledge of gauge dynamics in a regime where perturbation theory fails. In particular, it is of utmost importance to gain information on the nonperturbative dynamics of non-abelian four dimensional gauge theories. The phase diagram of SU (N ) gauge theories as functions of number of flavors, colors and matter representation has been investigated in.1,61–64 The analytical tools which will be used here for such an exploration are: i) The conjectured physical all orders beta function for nonsupersymmetric gauge theories with fermionic matter in arbitrary representations of the gauge group;63 ii) The truncated Schwinger-Dyson equation (SD)65–67 (referred also as the ladder approximation in the literature); The Appelquist-Cohen-Schmaltz (ACS) conjecture68 which makes use of the counting of the thermal degrees of freedom at high and low temperature. These are the methods which we have used in our investigations. However several very interesting and competing analytic approaches69–80 have been proa Earlier

interesting models49–51 have contributed triggering the lattice investigations for the conformal window with theories featuring fermions in the fundamental representation52–58

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posed in the literature. What is interesting is that despite the very different starting point the various methods agree qualitatively on the main features of the various conformal windows presented here. 1.1. Physical all orders Beta Function - Conjecture Recently we have conjectured an all orders beta function which allows for a bound of the conformal window63 of SU (N ) gauge theories for any matter representation. The predictions of the conformal window coming from the above beta function are nontrivially supported by all the recent lattice results.33,34,81–85 In86 we further assumed the form of the beta function to hold for SO(N ) and Sp(2N ) gauge groups and further extended in59 to chiral gauge theories. Consider a generic gauge group with Nf (ri ) Dirac flavors belonging to the representation ri , i = 1, . . . , k of the gauge group. The conjectured beta function reads: Pk 2 g 3 β0 − 32 i=1 T (ri ) Nf (ri ) γi (g ) β(g) = − , (1) 0 2β g2 (4π)2 1 − 8π 1 + β00 2 C2 (G) with

β0 =

k k X 4X 11 C2 (G) − T (ri )Nf (ri ) and β00 = C2 (G) − T (ri )Nf (ri ) . 3 3 i=1 i=1

(2)

The generators Tra , a = 1 . . . N2 − 1 of the gauge group in the representation r are normalized according to Tr Tra Trb = T (r)δ ab while the quadratic Casimir C2 (r) is given by Tra Tra = C2 (r)I. The trace normalization factor T (r) and the quadratic Casimir are connected via C2 (r)d(r) = T (r)d(G) where d(r) is the dimension of the representation r. The adjoint representation is denoted by G. The beta function is given in terms of the anomalous dimension of the fermion mass γ = −d ln m/d ln µ where m is the renormalized mass, similar to the supersymmetric case.87–89 The loss of asymptotic freedom is determined by the change of sign in the first coefficient β0 of the beta function. This occurs when k X 4 T (ri )Nf (ri ) = C2 (G) , 11 i=1

Loss of AF.

(3)

At the zero of the beta function we have

k X 2 T (ri )Nf (ri ) (2 + γi ) = C2 (G) , 11 i=1

(4)

Hence, specifying the value of the anomalous dimensions at the IRFP yields the last constraint needed to construct the conformal window. Having reached the zero of the beta function the theory is conformal in the infrared. For a theory to be conformal the dimension of the non-trivial spinless operators must be larger than one in order not to contain

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negative norm states.90–92 Since the dimension of the chiral condensate is 3 − γi we see that γi = 2, for all representations ri , yields the maximum possible bound k X 8 T (ri )Nf (ri ) = C2 (G) , 11 i=1

γi = 2 .

(5)

In the case of a single representation this constraint yields Nf (r)BF ≥

11 C2 (G) , 8 T (r)

γ =2.

(6)

The actual size of the conformal window can be smaller than the one determined by the bound above, Eq. (3) and (5). It may happen, in fact, that chiral symmetry breaking is triggered for a value of the anomalous dimension less than two. If this occurs the conformal window shrinks. Within the ladder approximation65,66 one finds that chiral symmetry breaking occurs when the anomalous dimension is close to one. Picking γi = 1 we find: k X 6 T (ri )Nf (ri ) = C2 (G) , 11 i=1

γ =1.

(7)

In the case of a single representation this constraint yields Nf (r)BF ≥

11 C2 (G) , 6 T (r)

γ =1.

(8)

When considering two distinct representations the conformal window becomes a three dimensional volume, i.e. the conformal house.62 Of course, we recover the results by Banks and Zaks93 valid in the perturbative regime of the conformal window. We note that the presence of a physical IRFP requires the vanishing of the beta function for a certain value of the coupling. The opposite however is not necessarily true; the vanishing of the beta function is not a sufficient condition to determine if the theory has a fixed point unless the beta function is physical. By physical we mean that the beta function allows to determine simultaneously other scheme-independent quantities at the fixed point such as the anomalous dimension of the mass of the fermions. This is exactly what our beta function does. In fact, in the case of a single representation, one finds that at the zero of the beta function one has: 11C2 (G) − 4T (r)Nf . (9) γ= 2T (r)Nf 1.2. Schwinger-Dyson in the Rainbow Approximation For nonsupersymmetric theories another way to get quantitative estimates is to use the rainbow approximation to the Schwinger-Dyson equation.94,95 After a series of approximations (see59 for a review) one deduces for an SU (N ) gauge theory with Nf Dirac fermions transforming according to the representation r the critical number of flavors above which chiral symmetry maybe unbroken: NfSD =

17C2 (G) + 66C2 (r) C2 (G) . 10C2 (G) + 30C2 (r) T (r)

(10)

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Comparing with the previous result obtained using the all orders beta function we see that it is the coefficient of C2 (G)/T (r) which is different. We note that in96 it has been advocated a coefficient similar to the one of the all-orders beta function. 1.3. The SU , SO and Sp phase diagrams We consider here gauge theories with fermions in any representation of the SU (N ) gauge group1,61–63,97 using the various analytic methods described above. Here we plot in Fig. 1 the conformal windows for various representations predicted with the physical all orders beta function and the SD approaches.

Fig. 1. Phase diagram for nonsupersymmetric theories with fermions in the: i) fundamental representation (black), ii) two-index antisymmetric representation (blue), iii) two-index symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. In between the upper and the lower solid curves the theories are expected to develop an infrared fixed point according to the all orders beta function. The area between the upper solid curve and the dashed curve corresponds to the conformal window obtained in the ladder approximation.

The ladder result provides a size of the window, for every fermion representation, smaller than the maximum bound found earlier. This is a consequence of the value of the anomalous dimension at the lower bound of the window. The unitarity constraint corresponds to γ = 2 while the ladder result is closer to γ ∼ 1. Indeed if we pick γ = 1 our conformal window approaches the ladder result. Incidentally, a value of γ larger than one, still allowed by unitarity, is a welcomed feature when using this window to construct walking technicolor theories. It may allow for the physical value of the mass of the top while avoiding a large violation of flavor changing neutral currents98 which were investigated in99 in the case of the ladder approximation for minimal walking models.

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1.3.1. The Sp(2N ) phase diagram Sp(2N ) is the subgroup of SU (2N ) which leaves the tensor J c1 c2 = (1N ×N ⊗ iσ2 )c1 c2 invariant. Irreducible tensors of Sp(2N ) must be traceless with respect to J c1 c2 . Here we consider Sp(2N ) gauge theories with fermions transforming according to a given irreducible representation. Since π 4 [Sp(2N )] = Z2 there is a Witten topological anomaly100 whenever the sum of the Dynkin indices of the various matter fields is odd. The adjoint of Sp(2N ) is the two-index symmetric tensor. In Figure 2 we summarize the relevant zero temperature and matter density phase diagram as function of the number of colors and Weyl flavors (NW f ) for Sp(2N ) gauge theories. For the vector representation NW f = 2Nf while for the two-index theories NW f = Nf . The shape of the various conformal windows are very similar to the ones for SU (N ) gauge theories1,61,63 with the difference that in this case the two-index symmetric representation is the adjoint representation and hence there is one less conformal window. 35

A.F.

B.F. & γ= 2

SD

30

NWf

25 20

SP(2N)

15 10 5 0 2

3

4 N

5

6

Fig. 2. Phase Diagram, from top to bottom, for Sp(2N ) Gauge Theories with NW f = 2Nf Weyl fermions in the vector representation (light blue), NW f = Nf in the two-index antisymmetric representation (light red) and finally in the two-index symmetric (adjoint) (light green). The arrows indicate that the conformal windows can be smaller and the associated solid curves correspond to the all orders beta function prediction for the maximum extension of the conformal windows.

1.3.2. The SO(N ) phase diagram We shall consider SO(N ) theories (for N > 5) since they do not suffer of a Witten anomaly100 and, besides, for N < 7 can always be reduced to either an SU or an Sp theory. In Figure 3 we summarize the relevant zero temperature and matter density phase diagram as function of the number of colors and Weyl flavors (Nf ) for SO(N ) gauge theories. The shape of the various conformal windows are very similar to the ones for SU (N ) and

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SD

B.F. & γ= 2

Nf

15

10

SO(N)

5

0 5

6

7

8

9

10

N

Fig. 3. Phase diagram of SO(N ) gauge theories with Nf Weyl fermions in the vector representation, in the two-index antisymmetric (adjoint) and finally in the two-index symmetric representation. The arrows indicate that the conformal windows can be smaller and the associated solid curves correspond to the all orders beta function prediction for the maximum extension of the conformal windows.

Sp(2N ) gauge with the difference that in this case the two-index antisymmetric representation is the adjoint representation. We have analyzed only the theories with N ≥ 6 since the remaining smaller N theories can be deduced from Sp and SU using the fact that SO(6) ∼ SU (4), SO(5) ∼ Sp(4), SO(4) ∼ SU (2) × SU (2), SO(3) ∼ SU (2), and SO(2) ∼ U (1). The phenomenological relevance of orthogonal gauge groups for models of dynamical electroweak symmetry breaking has been shown in.7

1.4. Phases of Chiral Gauge Theories Chiral gauge theories, in which at least part of the matter field content is in complex representations of the gauge group, play an important role in efforts to extend the SM. These include grand unified theories, dynamical breaking of symmetries, and theories of quark and lepton substructure. Chiral theories received much attention in the 1980’s.101,102 Here we confront the results obtained in Ref.103,104 using the thermal degree of count freedom with the generalization of the all orders beta function useful to constrain chiral gauge theories appeared in.59 The two important class of theories we are going to investigate are the Bars-Yankielowicz (BY)105 model involving fermions in the two-index symmetric tensor representation, and the other is a generalized Georgi-Glashow (GGG) model involving fermions in the two-index antisymmetric tensor representation. In each case, in addition to fermions in complex representations, a set of p anti fundamental-fundamental pairs are included and the allowed phases are considered as a function of p. An independent relevant study of the phase diagrams of chiral gauge theories appeared in.75 Here the authors also compare their results with the ones presented below.

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1.4.1. All-orders beta function for Chiral Gauge Theories A generic chiral gauge theory has always a set of matter fields for which one cannot provide a mass term, but it can also contain vector-like matter. We hence suggest the following minimal modification of the all orders beta function63 for any nonsupersymmetric chiral gauge theory: Pk g 3 β0 − 32 i=1 T (ri )p(ri )γi (g 2 ) , βχ (g) = − (11) (4π)2 1 − g22 C (G) 1 + 2βχ0 2 8π β0

where pi is the number of vector like pairs of fermions in the representation ri for which an anomalous dimension of the mass γi can be defined. β0 is the standard one loop coefficient of the beta function while βχ0 expression is readily obtained by imposing that when expanding βχ one recovers the two-loop coefficient correctly and its explicit expression is not relevant here. According to the new beta function gauge theories without vector-like matter but featuring several copies of purely chiral matter will be conformal when the number of copies is such that the first coefficient of the beta function vanishes identically. Using topological excitations an analysis of this case was performed in.74 1.4.2. The Bars Yankielowicz (BY) Model This model is based on the single gauge group SU (N ≥ 3) and includes fermions trans{ab} forming as a symmetric tensor representation, S = ψL , a, b = 1, · · · , N ; N + 4 + p c , where i = 1, · · · , N + 4 + p; and conjugate fundamental representations: F¯a,i = ψa,iL a,i a,i p fundamental representations, F = ψL , i = 1, · · · , p. The p = 0 theory is the basic chiral theory, free of gauge anomalies by virtue of cancellation between the antisymmetric tensor and the N + 4 conjugate fundamentals. The additional p pairs of fundamentals and conjugate fundamentals, in a real representation of the gauge group, lead to no gauge anomalies. The global symmetry group is Gf = SU (N + 4 + p) × SU (p) × U1 (1) × U2 (1) .

(12)

Two U (1)’s are the linear combination of the original U (1)’s generated by S → eiθS S , F¯ → eiθF¯ F¯ and F → eiθF F that are left invariant by instantons, namely that for which P j NRj T (Rj )QRj = 0, where QRj is the U (1) charge of Rj and NRj denotes the number of copies of Rj . Thus the fermionic content of the theory is where the first SU (N ) is the gauge group, indicated by the square brackets. Table 1.

The Bars Yankielowicz (BY) model.

Fields [SU (N )] SU (N + 4 + p) SU (p) S F¯ F

¯

1 ¯ 1

1 1

U1 (1)

U2 (1)

N +4 2p −(N + 2) −p N + 2 −(N − p)

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From the numerator of the chiral beta function and the knowledge of the one-loop coefficient of the BY perturbative beta function the predicted conformal window is: 3

3 (3N − 2) ≤ p ≤ (3N − 2) , 2 + γ∗ 2

(13)

with γ ∗ the largest possible value of the anomalous dimension of the mass. The maximum value of the number of p flavors is obtained by setting γ ∗ = 2: 3 3 (3N − 2) ≤ p ≤ (3N − 2) , γ∗ = 2 , (14) 4 2 while for γ ∗ = 1 one gets: 3 (3N − 2) ≤ p ≤ (3N − 2) , γ∗ = 1 . (15) 2 The chiral beta function predictions for the conformal window are compared with the thermal degree of freedom investigation as shown in the left panel of Fig. 4. In order to derive ϒ=1

ϒ=1

20

20

ACS

ϒ=2

ϒ=2 15

p

p

15

10

10

BY 5 2

4

GGG

5

ACS

6 N

8

10

2

4

6 N

8

10

Fig. 4. Left panel: Phase diagram of the BY generalized model. The upper solid (blue) line corresponds to the loss of asymptotic freedom; the dashed (blue) curve corresponds to the chiral beta function prediction for the breaking/restoring of chiral symmetry. The dashed black line corresponds to the ACS bound stating that the conformal region should start above this line. We have augmented the ACS method with the Appelquist-DuanSannino104 extra requirement that the phase with the lowest number of massless degrees of freedom wins among brk+sym all the possible phases in the infrared a chiral gauge theory can have. We hence used fIR and fU V to determine this curve. According to the all orders beta function (B.F.) the conformal window cannot extend below the solid (blue) line, as indicated by the arrows. This line corresponds to the anomalous dimension of the mass reaching the maximum value of 2. Right panel: The same plot for the GGG model.

a prediction from the ACS method we augmented it with the Appelquist-Duan-Sannino104 extra requirement that the phase with the lowest number of massless degrees of freedom wins among all the possible phases in the infrared a chiral gauge theory can have. The thermal critical number is: i p 1h pTherm = −16 + 3N + 208 − 196N + 69N 2 . (16) 4 1.4.3. The Generalized Georgi-Glashow (GGG) Model This model is similar to the BY model just considered. It is an SU (N ≥ 5) gauge theory, but with fermions in the anti-symmetric, rather than symmetric, tensor representation. The

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complete fermion content is A = ψL , a, b = 1, · · · , N ; an additional N − 4 + p fermions c in the conjugate fundamental representations: F¯a,i = ψa,iL , i = 1, · · · , N − 4 + p; and p a,i a,i fermions in the fundamental representations, F = ψL , i = 1, · · · , p. The global symmetry is Gf = SU (N − 4 + p) × SU (p) × U1 (1) × U2 (1) .

(17)

where the two U (1)’s are anomaly free. With respect to this symmetry, the fermion content is Table 2.

The Generalized Georgi-Glashow (GGG) model.

Fields [SU (N )] SU (N − 4 + p) SU (p) A F¯ F

¯

1 ¯

1 1

1

U1 (1)

U2 (1)

N −4 2p −(N − 2) −p N − 2 −(N − p)

Following the analysis for the BY model the chiral beta function predictions for the conformal window are compared with the thermal degree of freedom investigation and the result is shown in the right panel of Fig. 4. 1.5. Conformal Chiral Dynamics Our starting point is a nonsupersymmetric non-abelian gauge theory with sufficient massless fermionic matter to develop a nontrivial IRFP. The cartoon of the running of the coupling constant is represented in Fig. 5. In the plot ΛU is the dynamical scale below which the IRFP is essentially reached. It can be defined as the scale for which α is 2/3 of the fixed point value in a given renormalization scheme. If the theory possesses an IRFP the chiral condensate must vanish at large distances. Here we want to study the behavior of the condensate when a flavor singlet mass term is added to the underlying Lagrangian e + h.c. with m the fermion mass and ψ f as well as ψec left transforming ∆L = −m ψψ c f two component spinors, c and f represent color and flavor indices. The omitted color and � �*

�U

Energy

Fig. 5. Running of the coupling constant in an asymptotically free gauge theory developing an infrared fixed point for a value α = α∗ .

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flavor indices, in the Lagrangian term, are contracted. We consider the case of fermionic matter in the fundamental representation of the SU (N ) gauge group. The effect of such a term is to break the conformal symmetry together with some of the global symmetries of 0 f0 the underlying gauge theory. The composite operator Oψψ = ψef ψf has mass dimene f sion dψψ e = 3 − γ with γ the anomalous dimension of the mass term. At the fixed point γ is a positive number smaller than two.90 We assume m ΛU . Dimensional analysis demands ∆L → −m ΛγU Tr[Oψψ e ] + h.c. . The mass term is a relevant perturbation around the IRFP driving the theory away from the fixed point. It will induce a nonzero f0 vacuum expectation value for Oψψ e itself proportional to δf . It is convenient to define Tr[Oψψ e ] = Nf O with O a flavor singlet operator. The relevant low energy Lagrangian term is then −m ΛγU Nf O + h.c.. To determine the vacuum expectation value of O we follow.106,107 The induced physical mass gap is a natural infrared cutoff. We, hence, identify ΛIR with the physical value of the condensate. We find: hψecf ψfc i ∝ −mΛ2U ,

0<γ <1,

(18)

, γ →1,

(19)

hψecf ψfc i ∝ −m 1+γ ΛU1+γ , 1 < γ ≤ 2 .

(20)

hψecf ψfc i ∝ −mΛ2U log 3−γ

Λ2U |hOi|

4γ

e ∼ Λγ hOi to relate the expectation value of O to the one of the fermion We used hψψi U condensate. Via an allowed axial rotation m is now real and positive. The effects of the Instantons on the conformal dynamics has been investigated in.108 Here it was shown that the effects of the instantons can be sizable only for a very small number of flavors given that, otherwise, the instanton induced operators are highly irrelevant. 1.6. Gauge Duals and Conformal Window One of the most fascinating possibilities is that generic asymptotically free gauge theories have magnetic duals. In fact, in the late nineties, in a series of ground breaking papers Seiberg109,110 provided strong support for the existence of a consistent picture of such a duality within a supersymmetric framework. Arguably the existence of a possible dual of a generic nonsupersymmetric asymptotically free gauge theory able to reproduce its infrared dynamics must match the ’t Hooft anomaly conditions. We have exhibited several solutions of these conditions for QCD in111 and for certain gauge theories with higher dimensional representations in.112 An earlier exploration already appeared in the literature.113 The novelty with respect to these earlier results are: i) The request that the gauge singlet operators associated to the magnetic baryons should be interpreted as bound states of ordinary baryons;111 ii) The fact that the asymptotically free condition for the dual theory matches the lower bound on the conformal window obtained using the all orders beta function.63 These extra constraints help restricting further the number of possible gauge duals without diminishing the exactness of the associate solutions with respect to the ’t Hooft anomaly conditions.

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We will briefly summarize here the novel solutions to the ’t Hooft anomaly conditions for QCD. The resulting magnetic dual allows to predict the critical number of flavors above which the asymptotically free theory, in the electric variables, enters the conformal regime as predicted using the all orders conjectured beta function.63 1.6.1. QCD Duals The underlying gauge group is SU (3) while the quantum flavor group is SUL (Nf ) × SUR (Nf ) × UV (1) ,

(21)

and the classical UA (1) symmetry is destroyed at the quantum level by the Adler-BellJackiw anomaly. We indicate with Qiα;c the two component left spinor where α = 1, 2 is e α;c the spin index, c = 1, ..., 3 is the color index while i = 1, ..., Nf represents the flavor. Q i is the two component conjugated right spinor. The global anomalies are associated to the triangle diagrams featuring at the vertices three SU (Nf ) generators (either all right or all left), or two SU (Nf ) generators (all right or all left) and one UV (1) charge. We indicate these anomalies for short with: SUL/R (Nf )3 ,

SUL/R (Nf )2 UV (1) .

(22)

For a vector like theory there are no further global anomalies. The cubic anomaly factor, ˜ while the quadratic for fermions in fundamental representations, is 1 for Q and −1 for Q anomaly factor is 1 for both leading to SUL/R (Nf )3 ∝ ±3 ,

SUL/R (Nf )2 UV (1) ∝ ±3 .

(23)

If a magnetic dual of QCD does exist one expects it to be weakly coupled near the critical number of flavors below which one breaks large distance conformality in the electric variables. This idea is depicted in Fig 6. Determining a possible unique dual theory for QCD is, however, not simple given the few mathematical constraints at our disposal. The saturation of the global anomalies is an important tool but is not able to select out a unique solution. We shall see, however, that one of the solutions, when interpreted as the QCD dual, leads to a prediction of a critical number of flavors corresponding exactly to the one obtained via the conjectured all orders beta function. We seek solutions of the anomaly matching conditions for a gauge theory SU (X) with global symmetry group SUL (Nf ) × SUR (Nf ) × UV (1) featuring magnetic quarks q and qe together with SU (X) gauge singlet states identifiable as baryons built out of the electric quarks Q. Since mesons do not affect directly global anomaly matching conditions we could add them to the spectrum of the dual theory. We study the case in which X is a linear combination of number of flavors and colors of the type αNf + 3β with α and β integer numbers. We add to the magnetic quarks gauge singlet Weyl fermions which can be identified with the baryons of QCD but massless. Having defined the possible massless matter content of the gauge theory dual to QCD one computes the SUL (Nf )3 and SUL (Nf )2 UV (1) global anomalies in terms of the new

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Electric Weak Strong

Nf

Strong

Weak

Magnetic N Fig. 6. Schematic representation of the phase diagram as function of number of flavors and colors. For a given number of colors by increasing the number flavors within the conformal window we move from the lowest line (violet) to the upper (black) one. The upper black line corresponds to the one where one looses asymptotic freedom in the electric variables and the lower line where chiral symmetry breaks and long distance conformality is lost. In the magnetic variables the situation is reverted and the perturbative line, i.e. the one where one looses asymptotic freedom in the magnetic variables, correspond to the one where chiral symmetry breaks in the electric ones.

fields. We have found several solutions to the anomaly matching conditions presented above. Some were found previously in.113 Here we display a new solution in which the gauge group is SU (2Nf − 5N ) with the number of colors N equal to 3. It is, however, convenient to keep the dependence on N explicit. X must assume a value strictly larger Table 3. Massless spectrum of magnetic quarks and baryons and their transformation properties under the global symmetry group. The last column represents the multiplicity of each state and each state is a Weyl fermion. Fields SU (2Nf − 5N ) SUL (Nf ) SUR (Nf ) UV (1) # of copies q

1

qe

A

1

1 1

N (2Nf −5) 2Nf −5N N (2Nf −5) − 2N −5N f

1

3

2 −2

BA

1

3

DA

1

3

e A

1

−3

1

1

2 2

than one otherwise it is an abelian gauge theory. This provides the first nontrivial bound on the number of flavors: Nf >

5N + 1 , 2

(24)

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which for N = 3 requires Nf > 8. Asymptotic freedom of the newly found theory is dictated by the coefficient of the one-loop beta function : β0 =

11 2 (2Nf − 5N ) − Nf . 3 3

(25)

To this order in perturbation theory the gauge singlet states do not affect the magnetic quark sector and we can hence determine the number of flavors obtained by requiring the dual theory to be asymptotic free. i.e.: Nf ≥

11 N 4

Dual Asymptotic Freedom .

(26)

Quite remarkably this value coincides with the one predicted by means of the all orders conjectured beta function for the lowest bound of the conformal window, in the electric variables, when taking the anomalous dimension of the mass to be γ = 2. We recall that for any number of colors N the all orders beta function requires the critical number of flavors to be larger than: NfBF |γ=2 =

11 N. 4

(27)

For N=3 the two expressions yield 8.25 b . We consider this a nontrivial and interesting result lending further support to the all orders beta function conjecture and simultaneously suggesting that this theory might, indeed, be the QCD magnetic dual. The actual size of the conformal window matching this possible dual corresponds to setting γ = 2. We note that although for Nf = 9 and N = 3 the magnetic gauge group is SU (3) the theory is not trivially QCD given that it features new massless fermions and their interactions with massless mesonic type fields. Recent suggestions to analyze the conformal window of nonsupersymmetric gauge theories based on different model assumptions74 are in qualitative agreement with the precise results of the all orders beta function conjecture. It is worth noting that the combination 2Nf − 5N appears in the computation of the mass gap for gauge fluctuations presented in.74,114 It would be interesting to explore a possible link between these different approaches in the future. We have also find solutions for which the lower bound of the conformal window is saturated for γ = 1. The predictions from the gauge duals are, however, entirely and surprisingly consistent with the maximum extension of the conformal window obtained using the all orders beta function.63 Our main conclusion is that the ’t Hooft anomaly conditions alone do not exclude the possibility that the maximum extension of the QCD conformal window is the one obtained for a large anomalous dimension of the quark mass. By computing the same gauge singlet correlators in QCD and its suggested dual, one can directly validate or confute this proposal via lattice simulations. b Actually

given that X must be at least 2 we must have Nf ≥ 8.5 rather than 8.25

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1.7. Conclusions We investigated the conformal windows of chiral and non-chiral nonsupersymmetric gauge theories with fermions in any representation of the underlying gauge group using four independent analytic methods. For vector-like gauge theories one observes a universal value, i.e. independent of the representation, of the ratio of the area of the maximum extension of the conformal window, predicted using the all orders beta function, to the asymptotically free one, as defined in.62 It is easy to check from the results presented that this ratio is not only independent on the representation but also on the particular gauge group chosen. The four methods we used to unveil the conformal windows are the all orders beta function (BF), the SD truncated equation, the thermal degrees of freedom method and possible gauge - duality. They have vastly different starting points and there was no, a priori, reason to agree with each other, even at the qualitative level. Several questions remain open such as what happens on the right hand side of the infrared fixed point as we increase further the coupling. Does a generic strongly coupled theory develop a new UV fixed point as we increase the coupling beyond the first IR value115 ? If this were the case our beta function would still be a valid description of the running of the coupling of the constant in the region between the trivial UV fixed point and the neighborhood of the first IR fixed point. One might also consider extending our beta function to take into account of this possibility as done in.76 It is also possible that no non-trivial UV fixed point forms at higher values of the coupling constant for any value of the number of flavors within the conformal window. Gauge-duals seem to be in agreement with the simplest form of the beta function. The extension of the all orders beta function to take into account fermion masses has appeared in.116 Our analysis substantially increases the number of asymptotically free gauge theories which can be used to construct SM extensions making use of (near) conformal dynamics. Current Lattice simulations can test our predictions and lend further support or even disprove the emergence of a universal picture possibly relating the phase diagrams of gauge theories of fundamental interactions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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Resizing Conformal Windows O. Antipin Department of Physics, P.O.Box 35 FI-00014 University of Jyv¨ askyl¨ a, Finland and Helsinki Institute of Physics, P.O. Box 64 FI-00014, University of Helsinki, Finland E-mail: [emailprotected] K. Tuominen CP3-Origins, Campusvej 55 DK-5230, Odense N, Denmark and Helsinki Institute of Physics, P.O. Box 64 FI-00014 University of Helsinki, Finland E-mail: [emailprotected] Different mechanisms for the loss of conformality and analytic ans¨ atze for the betafunction of a generic gauge theory are reviewed and the implications on the conformal windows considered. Keywords: Conformal window, gauge theories.

1. Introduction Recent progress on the phase diagrams of strongly coupled gauge theory as a function of number of colors and flavors as well as matter representations1,2 shows that infrared conformal theories with simple particle content exist. The key feature in this analysis is the use of higher representations with respect to the new strong dynamics. Recently these models have been investigated also on the lattice.3,5,7–9 These theories are applied in beyond the Standard Model physics as phenomenologically viable candidates of walking Technicolor theories.10,11 If the β-function for a generic SU(N) gauge theory with matter was known beyond perturbation theory, the determination of the conformal window would be a simple matter. For N = 1 super-QCD (SQCD) the exact β-function is known,12 and for 3Nc /2 < Nf < 3Nc SQCD has an infrared stable fixed point.13,14 In15 a β-function ansatz for non-supersymmetric gauge theories was introduced. The parallel between SQCD and QCD was pursued further in16 where a proposal for the electromagnetic dual of QCD was constructed. In17 it was argued that there are three generic mechanisms how conformality is lost in quantum theories. Namely,

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1) the fixed point can go to zero coupling, 2) it can go to infinite coupling or 3) the Infrared fixed point (IRFP) may annihilate with an ultraviolet fixed point (UVFP) whence they both disappear into the complex plane. Mechanism 3) seems not to be allowed either with the β-function of SQCD or with the β-function ansatz of 15 for non-supersymmetric SU(Nc ) gauge theory. On the other hand, it was shown in17 that mechanism 3) is realized in a wide class of non-supersymmetric theories. Quantitative indications for this phenomena in QCD with many flavors were first presented in.18 In this talk we will discuss how the ideas of 15 and17 could be combined.19 We propose a concrete implementation of the mechanism 3) in an all order β-function for a generic Yang–Mills (YM) theory and show how this determines conformal windows; see also.19 2. An all order β-function ansatz We begin by a brief review of the basic results of 15 to define some notations. The rescaled ’t Hooft coupling is defined as a ≡ g 2 Nc /(4π)2 . In the perturbative βfunction for a generic non-supersymmetric YM theory with Nf Dirac fermions in a given representation R of the gauge group, the two lowest order coefficients are β 0 (x) = 11

C2 (G) − 4T (R)x Nc

(1)

β 1 (x) = 34

C22 (G) C2 (G) C2 (R) − 20 T (R)x − 12 T (R)x . 2 Nc Nc Nc

(2)

These two coefficients are universal, and the two-loop β-function is independent of the renormalization scheme, β(a) = − 23 β 0 (x)a2 − 23 β 1 (x)a3 . The conventions for the group theory factors for the representations we will consider can be found explicitly in.15 The proposal of 15 was to write all-order β-function as 2 β 0 (X) − X γ(a) , β(a) = − a2 3 1 − 2aC2 (G) 1 + 6β 0 (X) Nc

(3)

β 0 (X)

with β 0 (X) = 11 C2N(G) −2X, β 0 (X) = C2 (G)/Nc −X/2 and γ denotes the anomalous c dimension of the quark mass operator. For the group SU(Nc ), C2 (G) = Nc which simplifies the above equations. Perturbative two-loop β-function is obtained upon expansion to O(a3 ). Since only the two-loop β-function has universal coefficients, one assumes that there is a scheme in which the proposed β-function is complete. The ansatz in (3) determines the conformal window by assuming the value of γ at the lower boundary of the conformal window; by the unitarity bound γ ≤ 2. As discussed in the introduction, we will next modify Eq. (3) to be able to include conformality loss via mechanism 3).

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3. Fixed point merger in an all order β-function ansatz The simplest way to allow for the additional non-trivial UVFP in β-function of Eq. (3), and thus anticipate for the ‘fixed point merger’, is to include a term ∼ γ 2 in the numerator of the β-function suggesting a modified form 2 β (X) − X γ(a) + r(X)γ 2 (a) , β(a) = − a2 · 0 6β (X) 3 1 − 2aCN2c(G) 1 + β 0(X)

(4)

β 0 (X)

where β 0 (X) and are as defined earlier and we introduced unknown function r(X). We will determine this function first it in the limit Nc → ∞; possible O(1/Nc ) corrections will be discussed briefly in a later subsection. 3.1. Large Nc limit In the large Nc limit we will apply the holographic expectation17,20 that operator dimensions of the quark mass operators at the two fixed points satisfy ∆+ + ∆− = d = 4, which translates for the anomalous dimensions into equation γ1 + γ2 = 2.

(5)

Note, that this implies that the fixed point merger, a point where conformality is lost, will occur at γ1 = γ2 = 1. Setting the numerator of the β-function (4) to zero and solving the resulting quadratic equation, with the constraint Eq. (5) for the two roots, we find that r(X) = X/2. The numerator of Eq. (4) hence becomes X γ(γ − 2). (6) 2 Note the following features: First, the modified β-function, in the large Nc limit, predicts the lower end of the conformal window. For a given theory, we have to solve for X by setting the numerator of modified beta-function, Eq. (6), equal to zero under the condition γ = 1 on the basis of the constraint (5) from holography. Therefore, by construction, the maximum value of the anomalous dimension at the IRFP is γ ∗max = 1. This explicitly shows that the β-function of Eq. (4) is different from the one proposed in15 since scheme independent quantities, like the location of the conformal window and the values of the anomalous dimension, are different. Second, the upper end of the conformal window corresponds to γ = 0 and β0 = 0. Notice also that the second value of the anomalous dimension satisfying β0 = 0 in Eq. (6) is γ = 2 which coincides with the maximum value generally allowed in conformal field theory. We can then predict the size of the conformal window with the new β-function ansatz. Setting the expression in Eq. (6) to zero and using γ = 1 leads to the prediction that, in the large Nc limit, the lower end of the conformal window is at β 0 (X) +

Xmin =

22C2 (G) 22 . = 5Nc 5

(7)

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where we used G2 (G) = Nc . For the SU(Nc ) gauge theory with the fundamental quarks this predicts that Nfcr /Nc ≡ xmin = 22/5 in the large Nc limit. Note that for fundamental fermions this implies also that we have Nf → ∞, i.e. the Veneziano limit. The extrapolation of this result for the SU(3) theory with fundamental matter is shown in the left panel of Fig. 2. There, above x = 11/2 = 5.5 asymptotic freedom is lost (corresponding to the upper end of the conformal window), while below x = 22/5 = 4.4 the two zeros for γ become complex. Hence, for the SU(3), our extrapolation from large Nc result gives the critical number of flavors Nfcr = 13.2 below which the conformality is lost. This disagrees with the lattice result that the lower end of the conformal window occurs for 8 ≤ Nf ≤ 129 while agrees with e.g.21 Clearly more refined lattice studies are needed. As another example we consider calculating the lower end of the conformal window in terms of critical number of Dirac fermions Nfcr from (7) for fermions in the two-index symmetric (2S) representation of the SU(Nc ) gauge group. The results are presented in Fig.1 together with results corresponding to the β-function ansatz of 15 and with recent results22 obtained using deformation theory to establish the role of topological excitations in generating a mass gap in a gauge theory. Also the result from the ladder approximation is shown.

Nf

5

New Β

4

ladder D.T.

3

RSΓ1 ASF

2 2

RSΓ2

4

6

8

10

Nc

Fig. 1. Size of the conformal window determined using different methods discussed in the text with Nf fermions in 2-index symmetric representation of SU(Nc ). The topmost thick solid curve denotes the upper boundary of the conformal window, i.e. loss of asymptotic freedom. For the lower boundary the lower solid curve is the prediction from our β-function proposal, Eq.(7), the long dashed curve is the prediction from22 and the curve with shorter dashes correspond to the prediction from15 for γ = 2 (the dotted curve shows the corresponding result for γ = 1). The thin line between the solid and long-dashed curves corresponds to the ladder approximation.

3.2. Away from the large Nc limit However, in principle there could be important O(1/Nc ) corrections to (5) which, in turn, would modify the r(X) coefficient of the γ 2 term in (4). To get an idea of how such corrections would affect the results, we estimate these effects by means of

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a phenomenological modifying parameter as γ1 + γ2 = 2 + . Repeating then the exercise of Sec. 3.1, we arrive at the following coefficient of the γ 2 term X , 2+ and, thus, zero of the numerator of the new beta-function will occur at: γ − 1) = 0. β 0 (X) + X γ( 2+

(8)

r(X) =

X

X

5.4

5.4

5.2

5.2

5.0

5.0

4.8

4.8

4.6

4.6

4.4

4.4

4.2

4.2

0.0

(9)

0.5

1.0

1.5

2.0

Γ

0.0

Ε0.5

Ε0

Ε0.5

0.5

1.0

1.5

2.0

Γ

Fig. 2. Left: The line X(γ) along which β(a) = 0. Above X = 11/2 = 5.5 asymptotic freedom is lost while below X = 22/5 = 4.4 no real solutions exist. Right: The effects of 1/Nc corrections. Solid curve is the same as in the left panel. Dashed curve corresponds to = −0.5 and dotted to = 0.5

In the right panel of Fig. 2 we plot the solution X(γ) of this equation for the SU(3) fundamental with the illustrative values = ±0.5 and = 0. The latter coincides with the curve in the left panel of Fig. 2. The upper end of the conformal window at X=11/2=5.5 corresponding to γ1 = 0 remains unchanged, but the size of the conformal window increases (decreases) with positive (negative) . For positive the anomalous dimension at the UVFP will exceed the unitarity limit γ ≤ 2 before the IRFP merges with the free UVFP, i.e. before the other solution reaches γ = 0 value. In other words, as we decrease γ at the IRFP (by increasing with fixed X or vice versa) there is a point where UVFP disappears due to violation of the unitarity bound. Thus, past this point our picture becomes qualitatively similar to the one predicted by Ryttov-Sannino β-function. The value of γ at the IRFP below which the UVFP ceases to exist is given by γ = . Above, we have treated as a phenomenological parameter whose value is expected to be O(1). Considering positive and solving Eq. (9) at the point where the UV and IR fixed points merge (i.e. where γ1 = γ2 = 1 + /2) we obtain Xmin =

22 22C2 (G) = , Nc (5 + /2) 5 + /2

(10)

where the second equality again applies for SU(Nc ) gauge theory. For illustration we set = 1 which leads to Xmin = 4. Let us apply this result first for SU(3) with fundamental fermions. Using xmin = Xmin = 4 this translates to Nfmin = 12. Then let us consider various values of Nc

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252 Table 1. Lower end of the conformal window using procedure outlined in text for fundamental (F), 2-index (anti)symmetric (2AS) 2S and adjoint (A) representations. Nc 2 3 4 5 6 10 →∞

Nf,min Fund. 8 12 16 20 24 40 4Nc

Nf,min 2AS 12 8 6.67 6 5 4

Nf,min 2S 2 2.4 2.67 2.86 3 3.33 4

Nf,min A. 2 2 2 2 2 2 2

and also higher representations in addition to the fundamental one. We collect the results into the Table 1. Interestingly, our results are consistent with present lattice results. Remarkably, our results match exactly on the corresponding predictions from deformation theory both for fundamental23 and higher representations.22 Of course any relation between our β-function ansatz and some underlying microscopic dynamics is pure speculation; nevertheless this coincidence, although unexpected, is temptingly systematic. Our results also agree with the ones in.24 4. Conclusions In this paper we exploited recently proposed mechanism of IR and UV fixed point annihilation as a key difference between supersymmetric and non-supersymmetric YM gauge theories. We incorporated this mechanism of fixed point merging into a proposal for a nonperturbative β-function whose functional form is similar to the Ryttov-Sannino β-function. We also compared our results with results from Ryttov-Sannino β-function15 as well as with results of.22 As a speculative note, the agreements and disagreements between these three approaches suggest that there might be important differences in how conformality is lost in supersymmetric versus non-supersymmetric gauge theories. Combining the insights from the analytic approaches with the numerical studies, performed recently and currently in progress, is expected to lead to significant improvement in our understanding of non-perturbative dynamics of (quasi)conformal gauge theories. References 1. F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901 (2005) [arXiv:hep-ph/0405209]. 2. D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 72, 055001 (2005) [arXiv:hep-ph/0505059]; D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 73, 037701 (2006) [arXiv:hep-ph/0510217]. 3. A. J. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, JHEP 0905, 025 (2009) [arXiv:0812.1467 [hep-lat]]. 4. A. J. Hietanen, K. Rummukainen and K. Tuominen, Phys. Rev. D 80, 094504 (2009) [arXiv:0904.0864 [hep-lat]].

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5. S. Catterall and F. Sannino, Phys. Rev. D 76, 034504 (2007) [arXiv:0705.1664 [heplat]]; L. Del Debbio, A. Patella and C. Pica, arXiv:0805.2058 [hep-lat]; 6. S. Catterall, J. Giedt, F. Sannino and J. Schneible, JHEP 0811, 009 (2008) [arXiv:0807.0792 [hep-lat]]. 7. Y. Shamir, B. Svetit*ky and T. DeGrand, Phys. Rev. D 78, 031502 (2008) [arXiv:0803.1707 [hep-lat]]. 8. T. DeGrand, Y. Shamir and B. Svetit*ky, Phys. Rev. D 79, 034501 (2009) [arXiv:0812.1427 [hep-lat]]. 9. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100 (2008) 171607 [Erratum-ibid. 102 (2009) 149902] [arXiv:0712.0609 [hep-ph]]; T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D 79, 076010 (2009) [arXiv:0901.3766 [hep-ph]]. 10. O. Antipin, M. Heikinheimo and K. Tuominen, JHEP 0910, 018 (2009) [arXiv:0905.0622 [hep-ph]]. M. T. Frandsen, I. Masina and F. Sannino, arXiv:0905.1331 [hep-ph]; R. Foadi, M. T. Frandsen and F. Sannino, Phys. Rev. D 77, 097702 (2008) [arXiv:0712.1948 [hep-ph]]; O. Antipin and K. Tuominen, Phys. Rev. D 79, 075011 (2009) [arXiv:0901.4243 [hep-ph]]. 11. A. Belyaev, R. Foadi, M. T. Frandsen, M. Jarvinen, F. Sannino and A. Pukhov, Phys. Rev. D 79, 035006 (2009) [arXiv:0809.0793 [hep-ph]]. 12. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 229 (1983) 381; M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B 277 (1986) 456 [Sov. Phys. JETP 64 (1986 ZETFA,91,723-744.1986) 428]. 13. N. Seiberg, Nucl. Phys. B 435 (1995) 129 [arXiv:hep-th/9411149]. 14. K. A. Intriligator and N. Seiberg, Nucl. Phys. Proc. Suppl. 45BC, 1 (1996) [arXiv:hepth/9509066]. 15. T. A. Ryttov and F. Sannino, Phys. Rev. D 78 (2008) 065001 [arXiv:0711.3745 [hepth]]. 16. F. Sannino, Phys. Rev. D 80, 065011 (2009) [arXiv:0907.1364 [hep-th]]. 17. D. B. Kaplan, J. W. Lee, D. T. Son and M. A. Stephanov, Phys. Rev. D 80, 125005 (2009) [arXiv:0905.4752 [hep-th]]. 18. H. Gies and J. Jaeckel, Eur. Phys. J. C 46, 433 (2006) [arXiv:hep-ph/0507171]. 19. O. Antipin and K. Tuominen, arXiv:0909.4879 [hep-ph]. 20. I. R. Klebanov and E. Witten, Nucl. Phys. B 556, 89 (1999) [arXiv:hep-th/9905104]. 21. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0911.2463 [hep-lat]. 22. E. Poppitz and M. Unsal, JHEP 0909, 050 (2009) [arXiv:0906.5156 [hep-th]]. 23. E. Poppitz and M. Unsal, arXiv:0910.1245 [hep-th]. 24. A. Armoni, Nucl. Phys. B 826, 328 (2010) [arXiv:0907.4091 [hep-ph]].

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Nearly Conformal Gauge Theories on the Lattice Zolt´ an Fodor Department of Physics, University of Wuppertal, Gauss Strasse 20, D-42119, Germany and Institute for Theoretical Physics, Eotvos University Budapest, Pazmany P. 1, H-1117, Hungary E-mail: [emailprotected] Kieran Holland Department of Physics, University of the Pacific 3601 Pacific Ave, Stockton CA 95211, USA E-mail: [emailprotected] Julius Kuti∗ Department of Physics 0319, University of California, San Diego 9500 Gilman Drive, La Jolla CA 92093, USA E-mail: [emailprotected] D´ aniel N´ ogr´ adi Institute for Theoretical Physics, Eotvos University Budapest, Pazmany P. 1, H-1117, Hungary E-mail: [emailprotected] Chris Schroeder Department of Physics, University of Wuppertal, Gauss Strasse 20, D-42119, Germany E-mail: [emailprotected] We present selected new results on chiral symmetry breaking in nearly conformal gauge theories with fermions in the fundamental representation of the SU(3) color gauge group. We found chiral symmetry breaking (χSB) for all flavors between Nf = 4 and Nf = 12 with most of the results discussed here for Nf = 4, 8, 12 as we approach the conformal window. To identify χSB we apply several methods which include, within the framework of chiral perturbation theory, the analysis of the Goldstone spectrum in the p-regime and the spectrum of the fermion Dirac operator with eigenvalue distributions of random matrix theory in the -regime. Chiral condensate enhancement is observed with increasing Nf when the electroweak symmetry breaking scale F is held fixed in technicolor language. Important finite-volume consistency checks from the theoretical understanding of the SU (Nf ) rotator spectrum of the δ-regime are discussed. We also consider these gauge theories at Nf = 16 inside the conformal window. Our work on the running coupling is presented separately.1 Keywords: Beyond Standard Model, lattice, near-conformal, running coupling. ∗ Speaker

at the conference.

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1. Introduction It is an intriguing possibility that new physics beyond the Standard Model might take the form of some new strongly-interacting gauge theory building on the original technicolor idea.2–4 This approach has lately been revived by new explorations of the multi-dimensional theory space of nearly conformal gauge theories.5–8 Model building of a strongly interacting electroweak sector requires the knowledge of the phase diagram of nearly conformal gauge theories as the number of colors Nc , number of fermion flavors Nf , and the fermion representation R of the technicolor group are varied in theory space. For fixed Nc and R the theory is in the chirally broken phase for low Nf , and asymptotic freedom is maintained with a negative β function. On the other hand, if Nf is large enough, the β function is positive for all couplings, and the theory is trivial. There is some range of Nf for which the β function might have a non-trivial zero, an infrared fixed point, where the theory is in fact conformal.11,12 This method has been refined by estimating the critical value of Nf , above which spontaneous chiral symmetry breaking no longer occurs.13–15 Interesting models require the theory to be very close to, but below, the conformal window, with a running coupling which is almost constant over a large energy range .16–21 The nonperturbative knowledge of the critical Nfcrit separating the two phases is essential and this has generated much interest and many new lattice studies.22–55 To provide theoretical framework for the analysis of simulation results, we review first a series of tests expected to hold in the setting of χPT in finite volume and in the infinite volume limit. 2. Chiral symmetry breaking below the conformal window We will identify in lattice simulations the chirally broken phases with Nf = 4, 8, 12 flavors of staggered fermions in the fundamental SU(3) color representation using finite volume analysis. We deploy staggered fermions with exponential (stout) smearing56 in the lattice action to reduce well-known cutoff effects with taste breaking in the Goldstone spectrum.57 The presence of taste breaking requires careful analysis of staggered χPT following the important work of Lee, Sharpe, Aubin and Bernard.58–60 2.1. Finite volume analysis in the p-regime Three different regimes can be selected in simulations to identify the chirally broken phase from finite volume spectra and correlators. For a lattice size L3s × Lt in euclidean space and in the limit Lt Ls , the conditions Fπ Ls > 1 and Mπ Ls > 1 select the the p-regime, in analogy with low momentum counting.61,62 For arbitrary Nf , in the continuum and in infinite volume, the one-loop chiral corrections to Mπ and Fπ of the degenerate Goldstone pions are given by Λ3 M2 2 2 ln , (1) Mπ = M 1 − 2 8π Nf F 2 M

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Λ4 Nf M 2 ln Fπ = F 1 + , 16π 2 F 2 M

(2)

where M 2 = 2B · mq and F, B, Λ3 , Λ4 are four fundamental parameters of the chiral Lagrangian, and the small quark mass mq explicitly breaks the symmetry.63 The chiral parameters F, B appear in the leading part of the chiral Lagrangian, while Λ3 , Λ4 enter in next order. There is the well-known GMOR relation Σcond = BF 2 in the mq → 0 limit for the chiral condensate per unit flavor.64 It is important to note that the one-loop correction to the pion coupling constant Fπ is enhanced by a factor Nf2 compared to Mπ2 . The chiral expansion for large Nf will break down for Fπ much faster for a given Mπ /Fπ ratio. The NNLO terms have been recently calculated65 showing potentially dangerous Nf2 corrections to Eqs. (1,2). The finite volume corrections to Mπ and Fπ are given in the p-regime by M2 1 · ge1 (λ, η) , (3) Mπ (Ls , η) = Mπ 1 + 2Nf 16π 2 F 2 Nf M 2 Fπ (Ls , η) = Fπ 1 − · g e (λ, η) , 1 2 16π 2 F 2

(4)

where ge1 (λ, η) describes the finite volume corrections with λ = M · Ls and aspect ratio η = Lt /Ls . The form of ge1 (λ, η) is a complicated infinite sum which contains Bessel functions and requires numerical evaluation.62 Eqs. (1-4) provide the foundation of the p-regime fits in simulations. 1 Lt 1 Ls

4 F 2 Ls3

! chiral p-regime

!

3 2F 2 Ls3

rotator pion energy gap 1 F 2 Ls3

M = 2Bmq

1 Ls

Fig. 1. Schematic plot of the regions in which the three low energy chiral expansions are valid. The vertical axis shows the finite temperature scale (euclidean time in the path integral) which probes the rotator dynamics of the δ-regime and the -regime. The first two low lying rotator levels are also shown on the vertical axis for the simple case of Nf = 2. The fourfold degenerate lowest rotator excitation at mq = 0 will split into an isotriplet state (lowest energy level), which evolves into the p-regime pion as mq increases, and into an isosinglet state representing a multi-pion state in the p-regime. Higher rotator excitations have similar interpretations.

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2.2. δ-regime and -regime At fixed Ls and in cylindrical geometry Lt /Ls 1, a crossover occurs from the pregime to the δ-regime when mq → 0, as shown in Fig. 1. The dynamics is dominated by the rotator states of the chiral condensate in this limit66 which is characterized by the conditions F Ls > 1 and M Ls 1. The densely spaced rotator spectrum scales with gaps of the order ∼ 1/F 2 L3s , and at mq = 0 the chiral symmetry is apparently restored. However, the rotator spectrum, even at mq = 0 in the finite volume, will signal that the infinite system is in the chirally broken phase for the particular parameter set of the Lagrangian. This is often misunderstood in the interpretation of lattice simulations. Measuring finite energy levels with pion quantum numbers at fixed Ls in the mq → 0 limit is not a signal for chiral symmetry restoration of the infinite system.40 0.16

0.12 (a M)2

πi5

Nf = 4 4 Stout β = 3.80 163 x 32

π

0.08

0.04

0 0

0.01

0.02

0.03 a mq

0.04

0.05

Fig. 2. The crossover from the p-regime to the δ-regime is shown for the π and πi5 states at Nf = 4.

If Lt ∼ Ls under the conditions F Ls > 1 and M Ls 1, the system will be driven into the -regime which can be viewed as the high temperature limit of the δ-regime quantum rotator. Although the δ-regime and -regime have an overlapping region, there is an important difference in their dynamics. In the δ-regime of the quantum rotator, the mode of the pion field U (x) with zero spatial momentum dominates with time-dependent quantum dynamics. The -regime is dominated by the four-dimensional zero momentum mode of the chiral Lagrangian. We report simulation results of all three regimes in the chirally broken phase of the technicolor models we investigate. The analysis of the three regimes complement each other and provide cross-checks for the correct identification of the phases. First, we will probe Eqs. (1-4) in the p-regime, and follow with the study of Dirac spectra and RMT eigenvalue distributions in the -regime. The spectrum in the δ-regime is used as a signal to monitor p-regime spectra as mq decreases. Fig. 2 is an illustrative

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example of this crossover in our simulations. It is important to note that the energy levels in the chiral limit do not always match the rotator spectrum at the small F ·Ls values of the simulations. This squeezing of insufficiently large enough F · Ls for undistorted, finite volume chiral behavior in the p-regime, -regime, and δ-regime is a serious limitation of all recent simulations.

3. Goldstone spectrum and χSB at Nf = 12 We find the chiral symmetry breaking pattern for the controversial Nf = 12 case similar to the Nf = 8, 9 cases. The Goldstone spectrum remains separated from the technicolor scale of the ρ-meson. The true Goldstone pion and two additional split pseudo-Goldstone states are shown again in Fig. 3 with different slopes as a · mq increases. The trends and the underlying explanation are similar to the Nf = 8, 9 cases. The chiral fit to Mπ2 /mq shown at the top right side of Fig. 3 is based on Eq. (1) only since the Fπ data points are outside the convergence range of the chiral expansion. At β = 2.2 the fitted value of B is a · B = 2.7(2) in lattice units with a·F = 0.0120(1) and a·Λ3 = 0.50(3) also fitted. The fitted ρ-mass in the chiral limit is a · Mρ = 0.115(15) from a · mq = 0.025 − 0.045 with Mρ /F = 10(1). The fitted value of B/F = 223(17) is not very reliable but consistent with the enhancement of the chiral condensate found at Nf = 8, 9 without including renormalization scale effects. Again, at fixed lattice spacing, the small chiral condensate hψψi summed over all flavors is dominated by the linear term in mq from UV contributions. The linear fit gives hψψi = 0.0033(13) in the chiral limit which came out unexpectedly close the GMOR relation of hψψi = 12F 2 B with 12F 2 B = 0.0046(4) fitted. Issues and concerns in the systematics are similar to the Nf = 8, 9 cases but finite volume limitations and the convergence of the chiral expansion are more problematic. Currently we are investigating the important Nf = 12 model on larger lattices to probe the possible influence of unwanted squeezing effects on the spectra. This should also clarify the mass splitting pattern of the ρ and A1 states we are seeing in the chiral limit as Nf is varied. Our findings at Nf = 12 are not consistent with some earlier work31,32 where the runing coupling was studied. Lessons from the Dirac spectra and RMT to complement p-regime tests are discussed in the next section.

4. Epsilon regime, Dirac spectrum and RMT If the bare parameters of a gauge theory are tuned to the -regime in the chirally broken phase, the low-lying Dirac spectrum follows the predictions of random matrix theory. The corresponding random matrix model is only sensitive to the pattern of chiral symmetry breaking, the topological charge and the rescaled fermion mass once the eigenvalues are also rescaled by the same factor Σcond V . This idea has been confirmed in various settings both in quenched and fully dynamical simulations. The same method is applied here to nearly conformal gauge models.

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ρ

Nf = 12 2 Stout β = 2.00 243 x 32

0.3

Nf = 12 2 Stout β = 2.20 mq < 0.033: 324 mq > 0.033: 243x32

πi5 5

0.25

π

2

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0.4 0.35

0.2 0.15 0.1

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0.05 0

3.5

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Nf = 12 2 Stout β = 2.20 mq < 0.033: 324 mq > 0.033: 243x32

0.35 0.3 0.25 (a M)2

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πij 0.06 π

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0.02 0.03 a mq

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Nf = 12 2 Stout β = 2.20 mq < 0.033: 324 mq > 0.033: 243x32

0.08

ρ

πi5

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a Fπ

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0.1 0.05

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Nf = 12 2 Stout β = 2.40 243 x 32

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(a M)2

0.25 ρ πij πi5 π

0.2 0.15

a3 < ψ ψ >

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2 1.6 1.2

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0.8

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0.02 0.03 a mq

0.04

Nf = 12 2 Stout 203x32 - 324 β=1.0 β=1.4 β=1.8 β=2.0 β=2.2 β=2.4

0 0

0.01

0.02 0.03 a mq

0.04

0.01

0.02 a mq

0.03

0.04

0.05

Fig. 3. The pseudo-Goldstone spectrum and chiral fits are shown for Nf = 12 simulations with lattice size 243 ×32 and 324 . The left column shows the pseudo-Goldstone spectrum with decreasing taste breaking as the gauge coupling is varied from β = 2.0 to β = 2.4. Although the bottom figure on the left at β = 2.4 illustrates the continued restoration of taste symmetry, the volume is too small for the Goldstone spectrum. The middle value at β = 2.2 was chosen in the top right figure with fitting range a · mq = 0.015 − 0.035 of the NLO chiral fit to Mπ2 /mq which approaches 2B in the chiral limit. The middle figure on the right shows the Fπ data with no NLO fit far away from the chiral limit. The bottom right figure, with its additional features discussed in the text, is the linear fit to the chiral condensate with fitting range a·mq = 0.02−0.04. The physical fit parameters B, F, Λ3 are discussed in the text. Two stout steps were used in all Nf = 12 simulations.

The connection between the eigenvalues λ of the Dirac operator and chiral symmetry breaking is given in the Banks-Casher relation,70 Σcond = −hΨΨi = lim lim lim

λ→0 m→0 V →∞

πρ(λ) , V

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integrated distribution

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0.02 λn

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0.025

244

0.015

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integrated distribution

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0.008 0.012 0.016 quartet average, a units

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1

Nf = 4 β = 3.80

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λn

20

0.04 0.03

Nf = 4 β = 3.80 a mq = 0.001 244

0.8 0.6 0.4 0.2

1st quartet 2nd quartet

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4

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8

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12

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1

Nf = 4 β = 4.00 a mq = 0.001 2044 24

0.16

20

0.1 0.08

Nf = 4 β = 4.00 a mq = 0.001 204

0.8 0.6 0.4 0.2

1st quartet 2nd quartet

0.06 0

0.04 0

2

4

6

8

10 n

12

14

16

18

20

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 quartet average, a units

Fig. 4. From simulations at Nf = 4 the first column shows the approach to quartet degeneracy of the spectrum as β increases. The second column shows the integrated distribution of the two lowest quartets averaged. The solid line compares this procedure to RMT with Nf = 4.

where Σcond designates the quark condensate normalized to a single flavor. To generate a non-zero density ρ(0), the smallest eigenvalues must become densely packed as the volume increases, with an eigenvalue spacing ∆λ ≈ 1/ρ(0) = π/(Σcond V ). This allows a crude estimate of the quark condensate Σcond . One can do better by exploring the -regime: If chiral symmetry is spontaneously broken, tune the volume and quark mass such that F1π L M1π , so that the Goldstone pion is much lighter than the physical value, and finite volume effects are dominant as we discussed in Section 2. The chiral Lagrangian is dominated by the zero-momentum mode from the mass term and all kinetic terms are suppressed. In this limit, the distributions of the lowest eigenvalues are identical to those of random matrix theory, a theory of large matrices obeying certain symmetries.71–73 To connect with RMT,

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the eigenvalues and quark mass are rescaled as z = λΣcond V and µ = mq Σcond V , and the eigenvalue distributions also depend on the topological charge ν and the number of quark flavors Nf . RMT is a very useful tool to calculate analytically all of the eigenvalue distributions.74 The eigenvalue distributions in various topological sectors are measured via lattice simulations, and via comparison with RMT, the value of the condensate Σcond can be extracted. After we generate large thermalized ensembles, we calculate the lowest twenty eigenvalues of the Dirac operator using the PRIMME package.75 In the continuum limit, the staggered eigenvalues form degenerate quartets, with restored taste symmetry. The first column of Fig. 4 shows the change in the eigenvalue structure for Nf = 4 as the coupling constant is varied. At β = 3.6 grouping into quartets is not seen, the Goldstone pions are somewhat still split, and staggered perturbation theory is just beginning to kick in. At β = 3.8 doublet pairing appears and at β = 4.0 the quartets are nearly degenerate. The Dirac spectrum is collapsed as required by the Banks-Casher relation. In the second column we show the integrated distributions of the two lowest eigenvalue quartet averages, Z λ pk (λ0 )dλ0 , k = 1, 2 (5) 0

which is only justified close to quartet degeneracy. All low eigenvalues are selected with zero topology. To compare with RMT, we vary µ = mq Σcond V until we satisfy hz1 iRMT hλ1 isim = , m µ

(6)

where hλ1 isim is the lowest quartet average from simulations and the RMT average hziRMT depends implicitly on µ and Nf . With this optimal value of µ, we can predict the shapes of pk (λ) and their integrated distributions, and compare to the simulations. The agreement with the two lowest integrated RMT eigenvalue shapes is excellent for the larger β values. The main qualitative features of the RMT spectrum are very similar in our Nf = 8 simulations as shown in Fig. 5. One marked quantitative difference is a noticeable slowdown in response to change in the coupling constant. As β grows the recovery of the quartet degeneracy is considerably delayed in comparison with the onset of p-regime Goldstone dynamics. Overall, for the Nf = 4, 8 models we find consistency between the p-regime analysis and the RMT tests. Earlier, using Asqtad fermions at a particular β value, we found agreement with RMT even at Nf = 12 which indicated a chirally broken phase.23 Strong taste breaking with Asqtad fermions leaves the quartet averaging in question and the bulk pronounced crossover of the Asqtad action as β grows is also an issue. Currently we are investigating the RMT picture for Nf = 9, 10, 11, 12 with our much improved action with stout smearing. This action shows no artifact transitions and handles taste breaking much more effectively. Firm conclusions on the Nf = 12 model to support our findings of χSB in the p-regime will require continued investigations.

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integrated distribution

1 Nf = 8 β = 1.40 a mq = 0.001 244

0.8 0.6 0.4 0.2

1st quartet 2nd quartet 0 0

0.01

0.02 0.03 0.04 quartet average, a units

0.05

0.06

Fig. 5. The solid lines compare the integrated distribution of the two lowest quartet averages to RMT predictions with Nf = 8.

5. Inside the conformal window A distinguished feature of the Nf = 16 conformal model is how the renormalized coupling g 2 (L) runs with L, the linear size of the spatial volume in a Hamiltonian or Transfer Matrix description. On very small scales the running coupling g 2 (L) grows with L as in any other asymptotically free theory. However, g 2 (L) will not grow large, and in the L → ∞ limit it will converge to the fixed point g ∗2 which is rather weak,76 within the reach of perturbation theory. There is non-trivial, small-volume dynamics which is illustrated first in the pure gauge sector. At small g 2 , without fermions, the zero-momentum components of the gauge field are known to dominate the dynamics.77–79 With SU (3) gauge group, there are twenty-seven degenerate vacuum states, separated by energy barriers which are generated by the integrated effects of the non-zero momentum components of the gauge field in the Born-Oppenheimer approximation. The lowest-energy excitations of the gauge field Hamiltonian scale as ∼ g 2/3 (L)/L evolving into glueball states and becoming independent of the volume as the coupling constant grows with L. Non-trivial dynamics evolves through three stages as L grows. In the first regime, in very small boxes, tunneling is suppressed between vacua which remain isolated. In the second regime, for larger L, tunneling sets in and electric flux states will not be exponentially suppressed. Both regimes represent small worlds with zeromomentum spectra separated from higher momentum modes of the theory with energies on the scale of 2π/L. At large enough L the gauge dynamics overcomes the energy barrier, and wave functions spread over the vacuum valley. This third regime is the crossover to confinement where the electric fluxes collapse into thin string states wrapping around the box.

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It is likely that a conformal theory with a weak coupling fixed point at Nf = 16 will have only the first two regimes which are common with QCD. Now the calculations have to include fermion loops.80,81 The vacuum structure in small enough volumes, for which the wave functional is sufficiently localized around the vacuum configuration, remains calculable by adding in one-loop order the quantum effects of the fermion field fluctuations. The spatially constant abelian gauge fields parametrizing the vacuum valley are given by Ai (x) = T a Cia /L where Ta are the (N-1) generators for the Cartan subalgebra of SU (N ). For SU (3), T1 = λ3 /2 and T2 = λ8 /2. With Nf flavors of massless fermion fields the effective potential of the constant mode is given by X X (i) (j) (i) k Veff (Cb ) = V (Cb [µb − µb ]) − Nf V (Cb µb + πk), (7) i>j

i

with k = 0 for periodic, or k = (1, 1, 1), for antiperiodic boundary conditions on the fermion fields. The function V (C) is the one-loop effective potential for Nf = 0 and the weight vectors µ(i) are determined by the eigenvalues of the abelian genera√ tors. For SU(3) µ(1) = (1, 1, −2)/ 12 and µ(2) = 12 (1, −1, 0). The correct quantum vacuum is found at the minimum of this effective potential which is dramatically changed by the fermion loop contributions. The Polyakov loop observables remain center elements at the new vacuum configurations with complex values; for SU (N ) 1 X 1 (n) tr exp(iCjb Tb ) = exp(iµb Cjb ) = exp(2πilj /N ). (8) Pj = N N n (n)

k This implies µb Cb = 2πl/N (mod 2π), independent of n, and Veff = −Nf N V (2πl/N +πk). In the case of antiperiodic boundary conditions, k = (1, 1, 1), this is minimal only when l = 0 (mod 2π). The quantum vacuum in this case is the naive one, A = 0 (Pj = 1). In the case of periodic boundary conditions, k = 0, the vacua have l 6= 0, so that Pj correspond to non-trivial center elements. For

0.4

Imaginary

0.2 0

0.6

Nf = 16 3 Stout β = 30.0 m = 0.005 3 12 x 36 pbc

0.4 x 0.2 Imaginary

0.6

y -0.2 -0.4

Nf = 16 3 Stout β = 18.0 m = 0.001 3 12 x 36 apbc

x

-0.2 z

-0.6 -0.8 -0.6 -0.4 -0.2 0 Real

y

z

-0.4

0.2

0.4

0.6

-0.6 -0.8 -0.6 -0.4 -0.2 0 Real

0.2

0.4

0.6

Fig. 6. The time evolution of complex Polyakov loop distributions are shown from our Nf = 16 simulations with 123 × 36 lattice volume. Tree-level Symanzik-improved gauge action is used in the simulations and staggered fermions with three stout steps and very small fermion masses.

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SU(3), there are now 8 degenerate vacua characterized by eight different Polyakov loops, Pj = exp(±2πi/3). Since they are related by coordinate reflections, in a small volume parity (P) and charge conjugation (C) are spontaneously broken, although CP is still a good symmetry.80 Our simulations of the Nf = 16 model below the conformal fixed point g ∗2 confirm the theoretical vacuum structure. Fig. 6 shows the time evolution of Polyakov loop distributions monitored along the three separate spatial directions. On the left side, with periodic spatial boundary conditions, the time evolution is shown starting from randomized gauge configuration with the Polyakov loop at the origin. The system evolves into one of the eight degenerate vacua selected by the positive imaginary part of the complex Polyakov loop along the x and y direction and negative imaginary part along the z direction. On the right, with antiperiodic spatial boundary conditions, the vacuum is unique and trivial with real Polyakov loop in all three directions. The time evolution is particularly interesting in the z direction with a swing first from the randomized gauge configuration to a complex metastable minimum first, and eventually tunneling back to the trivial vacuum and staying there, as expected. The measured fermion-antifermion spectra and the spectrum of the Dirac operator further confirm this vacuum structure. Acknowledgments Julius Kuti is greatful to the organizers for a most enjoyable meeting. We thank Sandor Katz and Kalman Szabo for the Wuppertal RHMC code. For some calculations, we used the publicly available MILC code. We performed simulations on the Wuppertal GPU cluster, Fermilab clusters under the auspices of USQCD and SciDAC, and the Ranger cluster of the Teragrid organization. This research is supported by the NSF under grant 0704171, by the DOE under grants DOE-FG03-97ER40546, DOE-FG-02-97ER25308, by the DFG under grant FO 502/1 and by SFB-TR/55. References 1. K. Holland, Talk in this Proceedings; Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, PoS LATTICE2009, 058 (2009). 2. S. Weinberg, Phys. Rev. D 19, 1277 (1979). 3. L. Susskind, Phys. Rev. D 20, 2619 (1979). 4. E. Farhi and L. Susskind, Phys. Rept. 74, 277 (1981). 5. F. Sannino, arXiv:0902.3494 [hep-ph]. 6. T. A. Ryttov and F. Sannino, Phys. Rev. D 76, 105004 (2007). 7. D. D. Dietrich and F. Sannino, Phys. Rev. D 75, 085018 (2007). 8. D. K. Hong et al., Phys. Lett. B 597, 89 (2004). 9. H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). 10. M. A. Luty and T. Okui, JHEP 0609, 070 (2006). 11. W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974). 12. T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982). 13. T. Appelquist et al., Phys. Rev. Lett. 61, 1553 (1988). 14. A. G. Cohen and H. Georgi, Nucl. Phys. B 314, 7 (1989).

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Going Beyond QCD in Lattice Gauge Theory G. T. Fleming Sloane Physics Laboratory, Yale University New Haven, CT 06520-8120, USA E-mail: [emailprotected] Strongly coupled gauge theories (SCGT’s) have been studied theoretically for many decades using numerous techniques. The obvious motivation for these efforts stemmed from a desire to understand the source of the strong nuclear force: Quantum Chromodynamics (QCD). Guided by experimental results, theorists generally consider QCD to be a well-understood SCGT. Unfortunately, it is not clear how to extend the lessons learned from QCD to other SCGT’s. Particularly urgent motivators for new studies of other SCGT’s are the ongoing searches for physics beyond the standard model (BSM) at the Large Hadron Collider (LHC) and the Tevatron. Lattice gauge theory (LGT) is a technique for systematically-improvable calculations in many SCGT’s. It has become the standard for non-perturbative calculations in QCD and it is widely believed that it may be useful for study of other SCGT’s in the realm of BSM physics. We will discuss the prospects and potential pitfalls for these LGT studies, focusing primarily on the flavor dependence of SU(3) gauge theory. Keywords: Beyond the standard model, dynamical electroweak symmetry breaking, lattice gauge theory, technicolor.

1. Introduction The Tevatron is now scheduled to run through the end of 2011 with the expectation of providing three-sigma evidence against the existence of a standard model Higgs boson if, in fact, such a boson does not exist in nature. Given the current commissioning status, it is possible that the Large Hadron Collider (LHC) may be able to meaningfully contribute to the Higgs search in the same time frame. If the Higgs boson is ruled out, the case for new strong interactions to be discovered at higher energies in LHC detectors will be very much enhanced. If a strongly coupled gauge theory (SCGT) is responsible for the dynamical breaking of electroweak symmetry (DEWSB) in Nature, then the Tevatron and LHC may not observe a Standard Model Higgs boson. Such experimental nonobservation of the Higgs boson will accelerate already growing interest in DEWSB models. It is fortuitous that calculational capabilities of lattice gauge theory (LGT) are now able to make a significant impact just when new strong physics has the best chance of being discovered at the TeV scale. As a result, many lattice groups have now started calculations in theories which describe possible new TeV-scale strong interactions. See2,3 for reviews and additional references therein.

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95% CL Limit/SM

Tevatron Run II Preliminary,