Nonminimal gradient flows in QCD-like theories

Abstract

The Yang-Mills gradient flow for QCD-like theories is generalized by including a fermionic matter term in the gauge field flow equation. We combine this with two different flow equations for the fermionic degrees of freedom. The solutions for the different gradient flow setups are used in the perturbative computations of the vacuum expectation value of the Yang-Mills Lagrangian density and the field renormalization factor of the evolved fermions up to next-to-leading order in the coupling. We find a one-parameter family of flow systems for which there exists a renormalization scheme in which the evolved fermion anomalous dimension vanishes to all orders in perturbation theory. The fermion number dependence of different flows is studied and applications to lattice studies are anticipated.

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References

  1. [1]

    M. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A 308 (1982) 523.

    ADS  MathSciNet  MATH  Google Scholar 

  2. [2]

    S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1.

    MathSciNet  Article  Google Scholar 

  3. [3]

    R. Narayanan and H. Neuberger, Infinite N phase transitions in continuum Wilson loop operators, JHEP 03 (2006) 064 [hep-th/0601210] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  4. [4]

    M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293 (2010) 899 [arXiv:0907.5491] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  5. [5]

    M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [Erratum ibid. 03 (2014) 092] [arXiv:1006.4518] [INSPIRE].

  6. [6]

    R. Lohmayer and H. Neuberger, Continuous smearing of Wilson Loops, PoS(LATTICE2011)249 (2011) [arXiv:1110.3522] [INSPIRE].

  7. [7]

    M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-Abelian gauge theories, JHEP 02 (2011) 051 [arXiv:1101.0963] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  8. [8]

    K. Hieda, H. Makino and H. Suzuki, Proof of the renormalizability of the gradient flow, Nucl. Phys. B 918 (2017) 23 [arXiv:1604.06200] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  9. [9]

    M. Lüscher, Chiral symmetry and the Yang-Mills gradient flow, JHEP 04 (2013) 123 [arXiv:1302.5246] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. [10]

    R. Sommer, Scale setting in lattice QCD, PoS(LATTICE2013)015 (2014) [arXiv:1401.3270] [INSPIRE].

  11. [11]

    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, Extended investigation of the twelve-flavor β-function, Phys. Lett. B 779 (2018) 230 [arXiv:1710.09262] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    A. Hasenfratz, C. Rebbi and O. Witzel, Testing Fermion universality at a conformal fixed point, EPJ Web Conf. 175 (2018) 03006 [arXiv:1708.03385] [INSPIRE].

    Article  Google Scholar 

  13. [13]

    A. Ramos, The Yang-Mills gradient flow and renormalization, PoS(LATTICE2014)017 (2015) [arXiv:1506.00118] [INSPIRE].

  14. [14]

    G. Parisi and Y.-S. Wu, Perturbation theory without gauge fixing, Sci. Sin. 24 (1981) 483.

    MathSciNet  Google Scholar 

  15. [15]

    P.H. Damgaard and H. Huffel, Stochastic quantization, Phys. Rept. 152 (1987) 227 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  16. [16]

    R. Tzani, Evaluation of the chiral anomaly by the stochastic quantization method, Phys. Rev. D 33 (1986) 1146 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    R.V. Harlander and T. Neumann, The perturbative QCD gradient flow to three loops, JHEP 06 (2016) 161 [arXiv:1606.03756] [INSPIRE].

    ADS  Article  Google Scholar 

  18. [18]

    K. Symanzik, Schrödinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  19. [19]

    W.E. Caswell, Asymptotic behavior of non-Abelian gauge theories to two loop order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].

    ADS  Article  Google Scholar 

  20. [20]

    J.C. Collins, Renormalization: an introduction to renormalization, the renormalization group, and the operator product expansion, Cambridge University Press, Cambridge, U.K. (1986) [INSPIRE].

    Google Scholar 

  21. [21]

    A. Hasenfratz and O. Witzel, Dislocations under gradient flow and their effect on the renormalized coupling, arXiv:2004.00758 [INSPIRE].

  22. [22]

    L.U. Ancarani and G. Gasaneo, Derivatives of any order of the Gaussian hypergeometric function 2F1 (a, b, c; a with respect to the parameters a, b and c, J. Phys. A 42 (2009) 395208 [INSPIRE].

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Correspondence to Marco Boers.

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ArXiv ePrint: 2011.05316

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Boers, M. Nonminimal gradient flows in QCD-like theories. J. High Energ. Phys. 2021, 204 (2021). https://doi.org/10.1007/JHEP01(2021)204

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Keywords

  • Lattice QCD
  • Perturbative QCD
  • Renormalization Group