Improving the continuum limit of gradient flow step scaling

  • Anqi Cheng
  • Anna Hasenfratz
  • Yuzhi Liu
  • Gregory Petropoulos
  • David Schaich
Open Access
Article

Abstract

We introduce a non-perturbative improvement for the renormalization group step scaling function based on the gradient flow running coupling, which may be applied to any lattice gauge theory of interest. Considering first SU(3) gauge theory with Nf = 4 massless staggered fermions, we demonstrate that this improvement can remove \( \mathcal{O}\left( {{a^2}} \right) \) lattice artifacts, and thereby increases our control over the continuum extrapolation. Turning to the 12-flavor system, we observe an infrared fixed point in the infinite-volume continuum limit. Applying our proposed improvement reinforces this conclusion by removing all observable \( \mathcal{O}\left( {{a^2}} \right) \) effects. For the finite-volume gradient flow renormalization scheme defined by \( c={{{\sqrt{8t }}} \left/ {L=0.2 } \right.} \), we find the continuum conformal fixed point to be located at \( g_{\star}^2=6.2(2) \).

Keywords

Renormalization Group Lattice Quantum Field Theory 

Notes

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.

References

  1. [1]
    W.E. Caswell, Asymptotic behavior of non-Abelian 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]
    Z. Fodor et al., Can the nearly conformal sextet gauge model hide the Higgs impostor?, Phys. Lett. B 718 (2012) 657 [arXiv:1209.0391] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    S. Matsuzaki and K. Yamawaki, Holographic techni-dilaton at 125 GeV, Phys. Rev. D 86 (2012) 115004 [arXiv:1209.2017] [INSPIRE].ADSGoogle Scholar
  5. [5]
    T. Appelquist et al., Lattice gauge theories at the energy frontier, arXiv:1309.1206 [INSPIRE].
  6. [6]
    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, Can a light Higgs impostor hide in composite gauge models?, PoS(LATTICE 2013)062 [arXiv:1401.2176] [INSPIRE].
  7. [7]
    LatKMI collaboration, Y. Aoki et al., Light composite scalar in eight-flavor QCD on the lattice, arXiv:1403.5000 [INSPIRE].
  8. [8]
    E.T. Neil, Exploring models for new physics on the lattice, PoS(LATTICE 2011)009 [arXiv:1205.4706] [INSPIRE].
  9. [9]
    J. Giedt, Lattice gauge theory and physics beyond the Standard Model, PoS(LATTICE 2012)006 [INSPIRE].
  10. [10]
    A. Hasenfratz, Conformal or walking? Monte Carlo renormalization group studies of SU(3) gauge models with fundamental fermions, Phys. Rev. D 82 (2010) 014506 [arXiv:1004.1004] [INSPIRE].ADSGoogle Scholar
  11. [11]
    C.-J.D. Lin, K. Ogawa, H. Ohki and E. Shintani, Lattice study of infrared behaviour in SU(3) gauge theory with twelve massless flavours, JHEP 08 (2012) 096 [arXiv:1205.6076] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    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].
  13. [13]
    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
  14. [14]
    A. Hasenfratz, Infrared fixed point of the 12-fermion SU(3) gauge model based on 2-lattice MCRG matching, Phys. Rev. Lett. 108 (2012) 061601 [arXiv:1106.5293] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    E. Itou, Properties of the twisted Polyakov loop coupling and the infrared fixed point in the SU(3) gauge theories, PTEP 2013 (2013) 083B01 [arXiv:1212.1353] [INSPIRE].Google Scholar
  16. [16]
    G. Petropoulos, A. Cheng, A. Hasenfratz and D. Schaich, Improved lattice renormalization group techniques, PoS(LATTICE 2013)079 [arXiv:1311.2679] [INSPIRE].
  17. [17]
    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].ADSGoogle Scholar
  18. [18]
    Z. Fodor et al., Twelve massless flavors and three colors below the conformal window, Phys. Lett. B 703 (2011) 348 [arXiv:1104.3124] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    T. Appelquist, G.T. Fleming, M.F. Lin, E.T. Neil and D.A. Schaich, Lattice simulations and infrared conformality, Phys. Rev. D 84 (2011) 054501 [arXiv:1106.2148] [INSPIRE].ADSGoogle Scholar
  20. [20]
    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].ADSGoogle Scholar
  21. [21]
    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
  22. [22]
    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
  23. [23]
    Z. Fodor et al., Confining force and running coupling with twelve fundamental and two sextet fermions, PoS(LATTICE 2012)025 [arXiv:1211.3548] [INSPIRE].
  24. [24]
    Z. Fodor et al., Conformal finite size scaling of twelve fermion flavors, PoS(LATTICE 2012)279 [arXiv:1211.4238] [INSPIRE].
  25. [25]
    Y. Aoki et al., Lattice study of conformality in twelve-flavor QCD, Phys. Rev. D 86 (2012) 054506 [arXiv:1207.3060] [INSPIRE].ADSGoogle Scholar
  26. [26]
    Y. Aoki et al., Light composite scalar in twelve-flavor QCD on the lattice, Phys. Rev. Lett. 111 (2013) 162001 [arXiv:1305.6006] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    X.-Y. Jin and R.D. Mawhinney, Lattice QCD with 12 degenerate quark flavors, PoS(LATTICE 2011)066 [arXiv:1203.5855] [INSPIRE].
  28. [28]
    D. Schaich, A. Cheng, A. Hasenfratz and G. Petropoulos, Bulk and finite-temperature transitions in SU(3) gauge theories with many light fermions, PoS(LATTICE 2012)028 [arXiv:1207.7164] [INSPIRE].
  29. [29]
    A. Hasenfratz, A. Cheng, G. Petropoulos and D. Schaich, Reaching the chiral limit in many flavor systems, published in KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective (SCGT 12), Nagoya Japan December 4-7 2012, pg. 44 [arXiv:1303.7129] [INSPIRE].
  30. [30]
    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, arXiv:1401.0195 [INSPIRE].
  31. [31]
    M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293 (2010) 899 [arXiv:0907.5491] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [arXiv:1006.4518] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    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
  34. [34]
    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, The gradient flow running coupling scheme, PoS(LATTICE 2012)050 [arXiv:1211.3247] [INSPIRE].
  35. [35]
    P. Fritzsch and A. Ramos, The gradient flow coupling in the Schrödinger functional, JHEP 10 (2013) 008 [arXiv:1301.4388] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  36. [36]
    ALPHA collaboration, F. Tekin, R. Sommer and U. Wolff, The running coupling of QCD with four flavors, Nucl. Phys. B 840 (2010) 114 [arXiv:1006.0672] [INSPIRE].ADSMATHGoogle Scholar
  37. [37]
    P. Perez-Rubio and S. Sint, Non-perturbative running of the coupling from four flavour lattice QCD with staggered quarks, PoS(LATTICE 2010)236 [arXiv:1011.6580] [INSPIRE].
  38. [38]
    J. Balog, F. Niedermayer and P. Weisz, Logarithmic corrections to O(a 2) lattice artifacts, Phys. Lett. B 676 (2009) 188 [arXiv:0901.4033] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    J. Balog, F. Niedermayer and P. Weisz, The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanziks solution, Nucl. Phys. B 824 (2010) 563 [arXiv:0905.1730] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    R. Sommer, Scale setting in lattice QCD, PoS(LATTICE 2013)015 [arXiv:1401.3270] [INSPIRE].
  41. [41]
    A. Hasenfratz and F. Knechtli, Flavor symmetry and the static potential with hypercubic blocking, Phys. Rev. D 64 (2001) 034504 [hep-lat/0103029] [INSPIRE].ADSGoogle Scholar
  42. [42]
    A. Hasenfratz, R. Hoffmann and S. Schaefer, Hypercubic smeared links for dynamical fermions, JHEP 05 (2007) 029 [hep-lat/0702028] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    A. Cheng, A. Hasenfratz, Y. Liu, G. Petropoulos and D. Schaich, Step scaling studies using the gradient flow running coupling, in preparation (2014).Google Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Anqi Cheng
    • 1
  • Anna Hasenfratz
    • 1
  • Yuzhi Liu
    • 1
  • Gregory Petropoulos
    • 1
  • David Schaich
    • 1
    • 2
  1. 1.Department of PhysicsUniversity of ColoradoBoulderUnited States
  2. 2.Department of PhysicsSyracuse UniversitySyracuseUnited States

Personalised recommendations