Abstract
We study the SU(3) gauge theory with twelve flavours of fermions in the fundamental representation as a prototype of non-Abelian gauge theories inside the conformal window. Guided by the pattern of underlying symmetries, chiral and conformal, we analyze the two-point functions theoretically and on the lattice, and determine the finite size scaling and the infinite volume fermion mass dependence of the would-be hadron masses. We show that the spectrum in the Coulomb phase of the system can be described in the context of a universal scaling analysis and we provide the nonperturbative determination of the fermion mass anomalous dimension γ∗ = 0.235(46) at the infrared fixed point. We comment on the agreement with the four-loop perturbative prediction for this quantity and we provide a unified description of all existing lattice results for the spectrum of this system, them being in the Coulomb phase or the asymptotically free phase. Our results corroborate the view that the fixed point we are studying is not associated to a physical singularity along the bare coupling line and estimates of physical observables can be attempted on either side of the fixed point. Finally, we observe the restoration of the U(1) axial symmetry in the two-point functions.
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T. Appelquist, A. Ratnaweera, J. Terning and L.C.R. Wijewardhana, The phase structure of an SU(N) gauge theory with N f flavors, Phys. Rev. D 58 (1998) 105017 [hep-ph/9806472] [INSPIRE].
V.A. Miransky and K. Yamawaki, Conformal phase transition in gauge theories, Phys. Rev. D 55 (1997) 5051 [Erratum ibid. D 56 (1997) 3768] [hep-th/9611142] [INSPIRE].
T. Banks and A. Zaks, On the phase structure of vector-like gauge theories with massless fermions, Nucl. Phys. B 196 (1982) 189 [INSPIRE].
A. Deuzeman, M.P. Lombardo and E. Pallante, Evidence for a conformal phase in SU(N) gauge theories, Phys. Rev. D 82 (2010) 074503 [arXiv:0904.4662] [INSPIRE].
C.G. Callan Jr., S.R. Coleman and R. Jackiw, A new improved energy-momentum tensor, Annals Phys. 59 (1970) 42 [INSPIRE].
A. Bzowski and K. Skenderis, Comments on scale and conformal invariance, JHEP 08 (2014) 027 [arXiv:1402.3208] [INSPIRE].
J.-F. Fortin, B. Grinstein and A. Stergiou, Limit cycles and conformal invariance, JHEP 01 (2013) 184 [arXiv:1208.3674] [INSPIRE].
M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the asymptotics of 4D quantum field theory, JHEP 01 (2013) 152 [arXiv:1204.5221] [INSPIRE].
A. Cheng, A. Hasenfratz, Y. Liu, G. Petropoulos and D. Schaich, Finite size scaling of conformal theories in the presence of a near-marginal operator, Phys. Rev. D 90 (2014) 014509 [arXiv:1401.0195] [INSPIRE].
A. Deuzeman, M.P. Lombardo, T. Nunes Da Silva and E. Pallante, The bulk transition of QCD with twelve flavors and the role of improvement, Phys. Lett. B 720 (2013) 358 [arXiv:1209.5720] [INSPIRE].
A. Cheng, A. Hasenfratz and D. Schaich, Novel phase in SU(3) lattice gauge theory with 12 light fermions, Phys. Rev. D 85 (2012) 094509 [arXiv:1111.2317] [INSPIRE].
T.N. da Silva and E. Pallante, The strong coupling regime of twelve flavors QCD, PoS(LATTICE 2012)052 [arXiv:1211.3656] [INSPIRE].
P. de Forcrand, S. Kim and W. Unger, Conformality in many-flavour lattice QCD at strong coupling, JHEP 02 (2013) 051 [arXiv:1208.2148] [INSPIRE].
L. Del Debbio and R. Zwicky, Hyperscaling relations in mass-deformed conformal gauge theories, Phys. Rev. D 82 (2010) 014502 [arXiv:1005.2371] [INSPIRE].
A. Deuzeman, E. Pallante and M.P. Lombardo, The bulk transition of many-flavour QCD and the search for a UVFP at strong coupling, PoS(LATTICE 2010)067 [arXiv:1012.5971] [INSPIRE].
K.-I. Ishikawa, Y. Iwasaki, Y. Nakayama and T. Yoshie, Global structure of conformal theories in the SU(3) gauge theory, Phys. Rev. D 89 (2014) 114503 [arXiv:1310.5049] [INSPIRE].
T. Appelquist, G.T. Fleming and E.T. Neil, Lattice study of the conformal window in QCD-like theories, Phys. Rev. Lett. 100 (2008) 171607 [Erratum ibid. 102 (2009) 149902] [arXiv:0712.0609] [INSPIRE].
G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].
S. Ferrara, R. Gatto and A.F. Grillo, Positivity restrictions on anomalous dimensions, Phys. Rev. D 9 (1974) 3564 [INSPIRE].
T. Appelquist, J. Terning and L.C.R. Wijewardhana, The zero temperature chiral phase transition in SU(N ) gauge theories, Phys. Rev. Lett. 77 (1996) 1214 [hep-ph/9602385] [INSPIRE].
A.G. Cohen and H. Georgi, Walking beyond the rainbow, Nucl. Phys. B 314 (1989) 7 [INSPIRE].
W.E. Caswell, Asymptotic behavior of nonabelian gauge theories to two loop order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].
T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The four loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].
J.A.M. Vermaseren, S.A. Larin and T. van Ritbergen, The four loop quark mass anomalous dimension and the invariant quark mass, Phys. Lett. B 405 (1997) 327 [hep-ph/9703284] [INSPIRE].
J.A. Gracey, The QCD β-function at O(1/N f ), Phys. Lett. B 373 (1996) 178 [hep-ph/9602214] [INSPIRE].
B. Holdom, Large-N flavor β-functions: a recap, Phys. Lett. B 694 (2010) 74 [arXiv:1006.2119] [INSPIRE].
R. Shrock, Generalized scheme transformations for the elimination of higher-loop terms in the β-function of a gauge theory, Phys. Rev. D 90 (2014) 045011 [arXiv:1405.6244] [INSPIRE].
T.A. Ryttov and R. Shrock, Scheme transformations in the vicinity of an infrared fixed point, Phys. Rev. D 86 (2012) 065032 [arXiv:1206.2366] [INSPIRE].
C. Pica and F. Sannino, UV and IR zeros of gauge theories at the four loop order and beyond, Phys. Rev. D 83 (2011) 035013 [arXiv:1011.5917] [INSPIRE].
M. Bochicchio, An asymptotic solution of large-N QCD and of large-N N = 1 SUSY YM, arXiv:1409.5149 [INSPIRE].
M. Bochicchio, Glueball and meson propagators of any spin in large-N QCD, Nucl. Phys. B 875 (2013) 621 [arXiv:1305.0273] [INSPIRE].
LSD collaboration, T. Appelquist et al., Parity doubling and the S parameter below the conformal window, Phys. Rev. Lett. 106 (2011) 231601 [arXiv:1009.5967] [INSPIRE].
T. DeGrand, Finite-size scaling tests for spectra in SU(3) lattice gauge theory coupled to 12 fundamental flavor fermions, Phys. Rev. D 84 (2011) 116901 [arXiv:1109.1237] [INSPIRE].
Y. Aoki et al., Lattice study of conformality in twelve-flavor QCD, Phys. Rev. D 86 (2012) 054506 [arXiv:1207.3060] [INSPIRE].
E. Itou, A novel scheme for the wave function renormalization of the composite operators, arXiv:1307.6645 [INSPIRE].
A. Cheng, A. Hasenfratz, G. Petropoulos and D. Schaich, Scale-dependent mass anomalous dimension from Dirac eigenmodes, JHEP 07 (2013) 061 [arXiv:1301.1355] [INSPIRE].
A. Deuzeman, M.P. Lombardo and E. Pallante, On the spectrum of QCD-like theories and the conformal window, PoS(LATTICE 2011)083 [arXiv:1201.1863] [INSPIRE].
K. Sasaki and S. Sasaki, Excited baryon spectroscopy from lattice QCD: finite size effect and hyperfine mass splitting, Phys. Rev. D 72 (2005) 034502 [hep-lat/0503026] [INSPIRE].
G. Boyd, S. Gupta, F. Karsch and E. Laermann, Spatial and temporal hadron correlators below and above the chiral phase transition, Z. Phys. C 64 (1994) 331 [hep-lat/9405006] [INSPIRE].
Z. Fodor et al., Twelve massless flavors and three colors below the conformal window, Phys. Lett. B 703 (2011) 348 [arXiv:1104.3124] [INSPIRE].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 2. Scattering states, Commun. Math. Phys. 105 (1986) 153 [INSPIRE].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 1. Stable particle states, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].
J. Gasser and H. Leutwyler, Spontaneously broken symmetries: effective Lagrangians at finite volume, Nucl. Phys. B 307 (1988) 763 [INSPIRE].
G. Colangelo, S. Dürr and R. Sommer, Finite size effects on M π in QCD from chiral perturbation theory, Nucl. Phys. Proc. Suppl. 119 (2003) 254 [hep-lat/0209110] [INSPIRE].
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Lombardo, M.P., Miura, K., Nunes da Silva, T.J. et al. On the particle spectrum and the conformal window. J. High Energ. Phys. 2014, 183 (2014). https://doi.org/10.1007/JHEP12(2014)183
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DOI: https://doi.org/10.1007/JHEP12(2014)183