Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in $$N_\mathrm {f}=3$$ N f = 3 QCD with Wilson quarks

Eur. Phys. J. C (2021) 81:606 https://doi.org/10.1140/epjc/s10052-021-09387-z Regular Article - Theoretical Physics Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in Nf = 3 QCD with Wilson quarks ALPHA Collaboration Jochen Heitger1 , Fabian Joswig1 , Pia L. J. Petrak1,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: 18 February 2021 / Accepted: 27 June 2021 © The Author(s) 2021 Abstract We determine non-perturbatively the normalisation factor rm ≡ Z S /Z S0 , where Z S and Z S0 are the renormalisation parameters of the flavour non-singlet and singlet scalar densities, respectively. This quantity is required in the computation of quark masses with Wilson fermions and for instance the renormalisation of nucleon matrix elements of scalar densities. Our calculation involves simulations of finite-volume lattice QCD with the tree-level Symanzik-improved gauge action, Nf = 3 mass-degenerate O(a) improved Wilson fermions and Schrödinger functional boundary conditions. The slope of the current quark mass, as a function of the subtracted Wilson quark mass is extracted both in a unitary setup (where nearly chiral valence and sea quark masses are degenerate) and in a non-unitary setup (where all valence flavours are chiral and the sea quark masses are small). These slopes are then combined with Z ≡ Z P /(Z S Z A ) in order to obtain rm . A novel chiral Ward identity is employed for the calculation of the normalisation factor Z . Our results cover the range of gauge couplings corresponding to lattice spacings below 0.1 fm, for which Nf = 2 + 1 QCD simulations in large volumes with the same lattice action are typically performed. 1 Introduction Scalar and pseudoscalar flavour singlet and non-singlet dimension-3 bilinear operators have the same anomalous dimension, since they belong to the same chiral multiplet. The same is true for their renormalisation parameters, provided that the regularisation does not break chiral symmetry. Otherwise, the renormalisation parameters of the chiral multiplet components differ by finite terms. This is the case for the lattice regularisation with Wilson fermions. For a e-mail: (corresponding author) 0123456789().: V,-vol example, the renormalisation parameters of the non-singlet scalar and pseudoscalar densities (denoted as Z S and Z P , respectively) have a finite ratio which is a polynomial of the bare gauge coupling g0 . This ratio can be determined by chiral Ward identities1 ; see Refs. [1,2]. Since Z P and Z S are scale dependent, imposing a renormalisation scheme is necessary to fix one of them, and the other can be obtained using the scheme independent ratio Z S /Z P .2 In this way the renormalised scalar and pseudoscalar densities are defined consistently in the same scheme, with the same anomalous dimension and renormalisation group (RG) running, and chiral symmetry is restored in the continuum limit. The ratio Z S /Z P has been computed for several gauge and Wilson fermion actions (standard, improved etc.) in the quenched approximation [2,7–11], with two dynamical quarks (Nf = 2 QCD) [12], and with three dynamical quarks (Nf = 3 QCD) [13–16]. Far less progress has been made on the computation of the ratio of the renormalisation parameters of the non-singlet and singlet scalar densities, rm ≡ Z S /Z S0 . For chirally symmetric regularisations rm = 1 holds, while for Wilson fermions rm is a (finite) polynomial of the gauge coupling, arising from the sea fermion loops of the quark propagator. In the quenched approximation, rm = 1. As explained in Ref. [17], the lowest-order non-trivial perturbative contribution to this quantity is a two-loop effect; i.e., rm = 1 + O(g04 ). In Ref. [18] the O(g04 ) perturbative term has been calculated for several lattice actions. Non-perturbative estimates of this quantity have been reported in Ref. [13] at two values of 1 In practice, distinct chiral Ward identities are used for the computa- tion of the ratio Z S /(Z P Z A ) and Z A ; the two results are subsequently multiplied to give Z S /Z P . 2 Examples of renormalisation schemes are MS, RI/(S)MOM [3,4], the Schrödinger functional (SF) [5] and the chirally rotated Schrödinger functional (χSF) [6]. 123 606 Page 2 of 18 the gauge coupling for Nf = 2 + 1 QCD with the treelevel Symanzik-improved gauge action [19] and the nonperturbatively improved Wilson-clover fermion action [20]. This is the regularisation chosen by the CLS (Coordinated Lattice Simulations) initiative which carries out QCD simulations with Nf = 2 + 1 flavours, on large physical volumes, for a range of bare couplings corresponding to a hadronic regime [13,21–23]. These CLS ensembles are suitable for the computation of correlation functions, from which lowenergy hadronic quantities can be evaluated. In parallel, our group is performing Nf = 3 simulations in the same range of bare gauge couplings, but for small-volume lattices with Schrödinger functional boundary conditions and nearly-chiral quark masses. These ensembles are used for the numerical determination of the necessary renormalisation parameters and Symanzik improvement coefficients, see Refs. [14,15,24–27] that have various applications in lattice QCD when using this discretisation of Wilson fermions. The present work provides high-precision estimates of rm obtained in the same computational framework. As seen from Eq. (2.2) below, (rm − 1) contributes an O(g04 ) term to the renormalisation of the quark masses [17]. This is expected to be a small effect. Symanzik O(a) counterterms containing rm are often neglected in light quark mass determinations; cf. Ref. [28]. In practical computations, however, rm can be relevant at O(a), especially when dealing with heavy flavours, and should be taken into account in order to achieve full O(a) improvement; see, for example, Eq. (2.13) in Ref. [29]. Another application where rm plays a prominent rôle is the nucleon sigma-term, which is defined in terms of nucleon matrix elements of flavour singlet scalar densities; see Refs. [30,31] for example and [32–34] for more recent works. A direct determination of Z S0 is not as straightforward as that of Z S , the former also requiring the computation of two-boundary (“disconnected”) quark diagrams. This problem is circumvented by extracting Z S0 as the product of Z S and rm . Our computation of rm is based on the relation between the current (PCAC) mass m and the subtracted quark mass m q . Close to the chiral limit, m(m q ) is a linear function with a slope that depends on the details of the QCD model being simulated. In a unitary theory with degenerate sea and valence quark masses, the slope of m(m q ) is Zrm , where Z ≡ Z P /(Z S Z A ) and Z A is the non-singlet axial curre (...truncated)