Ward identity determination of $$Z_{\mathrm {S}}/Z_{\mathrm {P}}$$ZS/ZP for $$N_{\mathrm {f}}=3$$Nf=3 lattice QCD in a Schrödinger functional setup

Eur. Phys. J. C (2020) 80:765 https://doi.org/10.1140/epjc/s10052-020-8266-2 Regular Article - Theoretical Physics Ward identity determination of ZS /ZP for Nf = 3 lattice QCD in a Schrödinger functional setup ALPHA Collaboration Jochen Heitger1 , Fabian Joswig1,a , Anastassios Vladikas2 1 Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany 2 “Rome Tor Vergata” Division, c/o Dipartimento di Fisica, INFN, Via della Ricerca Scientifica 1, 00133 Rome, Italy Received: 12 May 2020 / Accepted: 21 July 2020 © The Author(s) 2020 Abstract We derive chiral Ward identities for lattice QCD with Wilson quarks and Nf ≥ 3 flavours, on small lattices with Schrödinger functional boundary conditions and vanishingly small quark masses. These identities relate the axial variation of the non-singlet pseudoscalar density to the scalar one, thus enabling the non-perturbative determination of the scale-independent ratio Z S /Z P of the renormalisation parameters of these operators. We obtain results for Nf = 3 QCD with tree-level Symanzik-improved gluons and WilsonClover quarks, for bare gauge couplings which cover the typical range of large-volume Nf = 2 + 1 simulations with Wilson fermions at lattice spacings below 0.1 fm. The precision of our results varies from 0.3 to 1%, except for the coarsest lattice, where it is 2%. We discuss how the Z S /Z P ratio can be used in the non-perturbative calculations of O(a) improved renormalised quark masses. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Chiral Ward identities for Z S /Z P . . . . . . . . . . . 2.1 Formal chiral Ward identities in the continuum . 2.2 Lattice Ward identities with Schrödinger functional boundary conditions . . . . . . . . . . . . 2.3 Lattice Ward identities, Wick contractions, and flavour factors . . . . . . . . . . . . . . . . . . . 3 Determination of Z S /(Z P Z A ) from Ward identities . . 4 Numerical setup and results . . . . . . . . . . . . . . 4.1 Chiral extrapolation . . . . . . . . . . . . . . . . 4.2 Scaling . . . . . . . . . . . . . . . . . . . . . . 4.3 Interpolation formula . . . . . . . . . . . . . . . 4.4 Comparison with previous works . . . . . . . . . a e-mail: (corresponding author) 0123456789().: V,-vol 5 Application: quark mass computations with Wilson fermions . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Subtracted masses, PCAC masses, and redefined Symanzik counterterms . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . Appendix A: Basic definitions . . . . . . . . . . . . . . Appendix B: Properties of su(Nf ) Lie algebra generators Appendix C: Renormalisation and improvement . . . . . Appendix D: Charge conjugation, γ5 -Hermiticity, and correlation functions . . . . . . . . . . . . . . . . . . Appendix E: Non-perturbative checks . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Lattice QCD with Wilson fermions is a long-established regularisation. The fermionic action satisfies most desirable properties, namely strict locality, lack of fermion doublers, and preservation of flavour symmetry in a straightforward way. Well-known shortcomings are the presence of discretisation effects linear in the lattice spacing and, most importantly, the loss of chiral symmetry. The first problem is solved by applying the Symanzik-improvement programme (see for instance Ref. [1] for a review and Ref. [2] for more details). Chiral symmetry is recovered in the continuum, at the cost of having to deal with complicated renormalisation properties for most quantities of interest (cf. Ref. [3] and references therein; for a review see also Ref. [4]). A frequently cited example of these complications is the power divergence m crit ∼ 1/a, which must be subtracted from bare quark masses before they are renormalised multiplicatively. Other examples are the normalisation parameter Z A of the axial current and the ratio Z S /Z P of the non-singlet scalar and pseudoscalar density renormalisation parameters. In a 123 765 Page 2 of 22 regularisation scheme which respects chiral symmetry, these quantities are strictly equal to unity at finite values of the UV cutoff. With Wilson fermions these quantities are scaleindependent finite functions of the gauge coupling, which tend to unity as we approach the continuum limit. In principle they are determined by requiring that chiral Ward identities at non-vanishing lattice spacing tend to their formal counterparts in the continuum limit. The scope of this paper is to provide a method for the determination of Z S /Z P based on Ward identities on physically small lattices with Schrödinger functional boundary conditions and realising a line of constant physics (LCP) in parameter space. Results are obtained for Nf = 3 dynamical quarks. The general idea behind using chiral Ward identities in order to evaluate Z S /Z P for Wilson fermions appeared in Ref. [3].1 It has been put to practice with quenched, unimproved Wilson fermions in Ref. [5] and subsequently with tree-level Symanzik-improved ones in Ref. [6]. The chiral Ward identities in question were obtained for large-volume lattices with periodic boundary conditions and non-chiral quark masses. Ratios of Z S /Z P were calculated at fixed gauge coupling for several quark masses and extrapolated to the chiral limit. A second-generation of calculations was not based on Ward identities but obtained by computing Z S and Z P in the RI/MOM scheme [7]. Again these calculations are performed at finite quark masses, followed by chiral extrapolations. A well known problem in this approach is that the Z S /Z P ratio thus obtained differs from the Ward identity one by “Goldstone pole contaminations” at the IR end of a renormalisation window. This problem was first identified in Ref. [7], and subsequently discussed in Refs. [8–11] (and reviewed in Ref. [4]), while the discussion specific to the difference between Ward identity and RI/MOM determinations of the ratio Z S /Z P is found in Ref. [10]. Although the problem is greatly attenuated by the RI/SMOM variant of this method [12], the requirement of a reliable renormalisation window is inherent in these approaches. In the present work we revisit the Ward identity method, with an important novelty: lattices with small physical volumes and Schrödinger functional boundary conditions are used, with quark flavours degenerate in mass and (almost) at the chiral limit. In doing so, we follow closely the method introduced in Ref. [13] (and originally applied in the quenched approximation in that work) for the nonperturbative determination of the scale independent normalisation parameter Z A of the axial vector current. Updates and optimisations of these computations can be found in refs. [14,15] for two- and three-flavour QCD, respectively. Ward identities are imposed at constant physics to ensure a 1 In practice, distinct chi (...truncated)