Abstract
We determine non-perturbatively the normalisation factor r_{mathrm{m}}equiv Z_{mathrm{S}}/Z_{mathrm{S}}^{0}, where Z_{mathrm{S}} and Z_{mathrm{S}}^{0} 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, N_{mathrm{f}}= 3 mass-degenerate {mathrm{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 equiv Z_{mathrm{P}}/(Z_{mathrm{S}}Z_{mathrm{A}}) in order to obtain r_{mathrm{m}}. 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 N_{mathrm{f}}= 2+1 QCD simulations in large volumes with the same lattice action are typically performed.
Highlights
IntroductionP (where all valence flavours are chiral and the sea quark masses are small). These slopes are combined with
In the final step of our analysis we combine the values of Zrm obtained in a unitary setup, Z in a non-unitary setup, and Z from a chiral Ward identity, in order to arrive at different estimates for rm
With the non-perturbative computation of the ratio of the renormalisation constants of non-singlet and singlet scalar densities, rm ≡ ZS/ZS0, presented in this paper we have addressed a quantity, which enters the renormalisation pattern of quark masses in lattice QCD with Wilson fermions, and constitutes an important ingredient in calculations of renormalised nucleon matrix elements of singlet scalar densities, known as sigma terms
Summary
P (where all valence flavours are chiral and the sea quark masses are small). These slopes are combined with. Example, the renormalisation parameters of the non-singlet scalar and pseudoscalar densities (denoted as ZS and ZP, respectively) have a finite ratio which is a polynomial of the bare gauge coupling g0. This ratio can be determined by chiral Ward identities; see Refs. Since ZP and ZS 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 ZS/ZP.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. Non-perturbative estimates of this quantity have been reported in Ref. [13] at two values of
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