Nonperturbative β function of eight-flavor SU(3) gauge theory

  • Anna Hasenfratz
  • David Schaich
  • Aarti Veernala
Open Access
Regular Article - Theoretical Physics


We present a new lattice study of the discrete β function for SU(3) gauge theory with N f = 8 massless flavors of fermions in the fundamental representation. Using the gradient flow running coupling, and comparing two different nHYP-smeared staggered lattice actions, we calculate the 8-flavor step-scaling function at significantly stronger couplings than were previously accessible. Our continuum-extrapolated results for the discrete β function show no sign of an IR fixed point up to couplings of g 2 ≈ 14. At the same time, we find that the gradient flow coupling runs much more slowly than predicted by two-loop perturbation theory, reinforcing previous indications that the 8-flavor system possesses nontrivial strongly coupled IR dynamics with relevance to BSM phenomenology.


Lattice Gauge Field Theories Renormalization Group Technicolor and Composite Models 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    T. Banks and A. Zaks, On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions, Nucl. Phys. B 196 (1982) 189 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    B. Holdom, Raising the Sideways Scale, Phys. Rev. D 24 (1981) 1441 [INSPIRE].ADSGoogle Scholar
  4. [4]
    K. Yamawaki, M. Bando and K.-i. Matumoto, Scale Invariant Technicolor Model and a Technidilaton, Phys. Rev. Lett. 56 (1986) 1335 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    T.W. Appelquist, D. Karabali and L.C.R. Wijewardhana, Chiral Hierarchies and the Flavor Changing Neutral Current Problem in Technicolor, Phys. Rev. Lett. 57 (1986) 957 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    T. Appelquist et al., Lattice Gauge Theories at the Energy Frontier, arXiv:1309.1206 [INSPIRE].
  7. [7]
    J. Kuti, The Higgs particle and the lattice, PoS(LATTICE 2013)004.
  8. [8]
    F. Sannino and K. Tuominen, Orientifold theory dynamics and symmetry breaking, Phys. Rev. D 71 (2005) 051901 [hep-ph/0405209] [INSPIRE].ADSGoogle Scholar
  9. [9]
    D.K. Hong, S.D.H. Hsu and F. Sannino, Composite Higgs from higher representations, Phys. Lett. B 597 (2004) 89 [hep-ph/0406200] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    D.D. Dietrich, F. Sannino and K. Tuominen, Light composite Higgs from higher representations versus electroweak precision measurements: Predictions for CERN LHC, Phys. Rev. D 72 (2005) 055001 [hep-ph/0505059] [INSPIRE].ADSGoogle Scholar
  11. [11]
    R. Foadi, M.T. Frandsen, T.A. Ryttov and F. Sannino, Minimal Walking Technicolor: Set Up for Collider Physics, Phys. Rev. D 76 (2007) 055005 [arXiv:0706.1696] [INSPIRE].ADSGoogle Scholar
  12. [12]
    R. Foadi, M.T. Frandsen and F. Sannino, 125 GeV Higgs boson from a not so light technicolor scalar, Phys. Rev. D 87 (2013) 095001 [arXiv:1211.1083] [INSPIRE].ADSGoogle Scholar
  13. [13]
    J.R. Andersen et al., Discovering Technicolor, Eur. Phys. J. Plus 126 (2011) 81 [arXiv:1104.1255] [INSPIRE].CrossRefGoogle Scholar
  14. [14]
    T.A. Ryttov and R. Shrock, Higher-Loop Corrections to the Infrared Evolution of a Gauge Theory with Fermions, Phys. Rev. D 83 (2011) 056011 [arXiv:1011.4542] [INSPIRE].ADSGoogle Scholar
  15. [15]
    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].ADSMathSciNetGoogle Scholar
  16. [16]
    A. Bashir, A. Raya and J. Rodriguez-Quintero, QCD: Restoration of Chiral Symmetry and Deconfinement for Large-N f, Phys. Rev. D 88 (2013) 054003 [arXiv:1302.5829] [INSPIRE].ADSGoogle Scholar
  17. [17]
    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].ADSCrossRefGoogle Scholar
  18. [18]
    T. Appelquist, A.G. Cohen and M. Schmaltz, A New constraint on strongly coupled gauge theories, Phys. Rev. D 60 (1999) 045003 [hep-th/9901109] [INSPIRE].ADSGoogle Scholar
  19. [19]
    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 [arXiv:0712.0609] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    T. Appelquist, G.T. Fleming and E.T. Neil, Lattice Study of Conformal Behavior in SU(3) Yang-Mills Theories, Phys. Rev. D 79 (2009) 076010 [arXiv:0901.3766] [INSPIRE].ADSGoogle Scholar
  21. [21]
    A. Deuzeman, M.P. Lombardo and E. Pallante, The Physics of eight flavours, Phys. Lett. B 670 (2008) 41 [arXiv:0804.2905] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    K. Miura and M.P. Lombardo, Lattice Monte-Carlo study of pre-conformal dynamics in strongly flavoured QCD in the light of the chiral phase transition at finite temperature, Nucl. Phys. B 871 (2013) 52 [arXiv:1212.0955] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    X.-Y. Jin and R.D. Mawhinney, Evidence for a First Order, Finite Temperature Phase Transition in 8 Flavor QCD, PoS(LATTICE2010)055 [arXiv:1011.1511] [INSPIRE].
  24. [24]
    D. Schaich, A. Cheng, A. Hasenfratz and G. Petropoulos, Bulk and finite-temperature transitions in SU(3) gauge theories with many light fermions, PoS(LATTICE2012)028 [arXiv:1207.7164] [INSPIRE].
  25. [25]
    A. Hasenfratz, A. Cheng, G. Petropoulos and D. Schaich, Reaching the chiral limit in many flavor systems, arXiv:1303.7129 [INSPIRE].
  26. [26]
    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, Nearly conformal gauge theories in finite volume, Phys. Lett. B 681 (2009) 353 [arXiv:0907.4562] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    LatKMI collaboration, Y. Aoki et al., Walking signals in N f = 8 QCD on the lattice, Phys. Rev. D 87 (2013) 094511 [arXiv:1302.6859] [INSPIRE].
  28. [28]
    LatKMI collaboration, Y. Aoki et al., Light composite scalar in eight-flavor QCD on the lattice, Phys. Rev. D 89 (2014) 111502 [arXiv:1403.5000] [INSPIRE].
  29. [29]
    USBSM collaboration, D. Schaich, Eight light flavors on large lattice volumes, PoS(LATTICE2013)072 [arXiv:1310.7006] [INSPIRE].
  30. [30]
    LSD collaboration, T. Appelquist et al., Lattice simulations with eight flavors of domain wall fermions in SU(3) gauge theory, Phys. Rev. D 90 (2014) 114502 [arXiv:1405.4752] [INSPIRE].
  31. [31]
    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].ADSCrossRefGoogle Scholar
  32. [32]
    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].
  33. [33]
    T. Appelquist et al., WW Scattering Parameters via Pseudoscalar Phase Shifts, Phys. Rev. D 85 (2012) 074505 [arXiv:1201.3977] [INSPIRE].ADSGoogle Scholar
  34. [34]
    A. Cheng, A. Hasenfratz, G. Petropoulos and D. Schaich, Determining the mass anomalous dimension through the eigenmodes of Dirac operator, PoS(LATTICE2013)088 [arXiv:1311.1287] [INSPIRE].
  35. [35]
    LSD collaboration, T. Appelquist et al., Finite-temperature study of eight-flavor SU(3) gauge theory, in preparation (2014).Google Scholar
  36. [36]
    LatKMI collaboration, Y. Aoki et al., Light composite scalar in twelve-flavor QCD on the lattice, Phys. Rev. Lett. 111 (2013) 162001 [arXiv:1305.6006] [INSPIRE].
  37. [37]
    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, Can a light Higgs impostor hide in composite gauge models?, PoS(LATTICE2013)062 [arXiv:1401.2176] [INSPIRE].
  38. [38]
    A. Cheng, A. Hasenfratz, Y. Liu, G. Petropoulos and D. Schaich, Improving the continuum limit of gradient flow step scaling, JHEP 05 (2014) 137 [arXiv:1404.0984] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    R. Narayanan and H. Neuberger, Infinite N phase transitions in continuum Wilson loop operators, JHEP 03 (2006) 064 [hep-th/0601210] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293 (2010) 899 [arXiv:0907.5491] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    M. Lüscher, Future applications of the Yang-Mills gradient flow in lattice QCD, PoS(LATTICE2013)016 [arXiv:1308.5598] [INSPIRE].
  42. [42]
    M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [arXiv:1006.4518] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, The Yang-Mills gradient flow in finite volume, JHEP 11 (2012) 007 [arXiv:1208.1051] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  44. [44]
    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, The gradient flow running coupling scheme, PoS(LATTICE2012)050 [arXiv:1211.3247] [INSPIRE].
  45. [45]
    P. Fritzsch and A. Ramos, The gradient flow coupling in the Schrdinger Functional, JHEP 10 (2013) 008 [arXiv:1301.4388] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  46. [46]
    Z. Fodor et al., The lattice gradient flow at tree-level and its improvement, JHEP 09 (2014) 018 [arXiv:1406.0827] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    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].ADSGoogle Scholar
  48. [48]
    T. Karavirta, J. Rantaharju, K. Rummukainen and K. Tuominen, Determining the conformal window: SU(2) gauge theory with N f = 4, 6 and 10 fermion flavours, JHEP 05 (2012) 003 [arXiv:1111.4104] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    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].ADSGoogle Scholar
  50. [50]
    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].ADSGoogle Scholar
  51. [51]
    M.P. Lombardo, K. Miura, T.J.N. da Silva and E. Pallante, On the particle spectrum and the conformal window, JHEP 12 (2014) 183 [arXiv:1410.0298] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Anna Hasenfratz
    • 1
  • David Schaich
    • 2
  • Aarti Veernala
    • 2
  1. 1.Department of PhysicsUniversity of ColoradoBoulderUSA
  2. 2.Department of PhysicsSyracuse UniversitySyracuseUSA

Personalised recommendations