Two-loop computation of the Schrödinger functional in lattice QCD

Abstract

We compute the Schrödinger functional (SF) for the case of lattice QCD with Wilson fermions (with and without SW improvement) at two-loop order in lattice perturbation theory. This allows us to extract the three-loop β-function in the SF-scheme. These results are required to compute the running coupling, the Λ-parameter and quark masses by finite size techniques with negligible systematic errors. In addition our results enable the implementation of two-loop O(a) improvement in SF-simulations. This article is based on the revised version of ref. [11].

A non-perturbative exploration of the high energy regime in $$N_{\mathrm{f}}=3$$ N f = 3 QCD

“…For the observable we choose the traditional SF coupling [25,26] and a 1-parameter family of close relatives [27]. The most important reason for this choice is the existence of a 2-loop calculation in this case [14,15], which, in combination with [16,17] allows to infer the 3-loop β-function for these schemes. Furthermore, the values of the 3-loop β-function coefficients are reasonable and enable us to make contact with the asymptotic perturbative regime at energy scales in the range O(10-100) GeV.…”

“…The 2 boundary counterterm coefficients, c t (g 0 ) andc t (g 0 ) are set to their perturbative two-and one-loop expressions, respectively [15,25],…”

“…The 2-loop calculation in [15] has been carried out in the very same lattice regularized theory, and the two-loop lattice artefacts in the ν = 0 step-scaling functions, 21) are indeed available to this order. With the coefficients for N f = 3 from Table 2, one may thus define the improved step-scaling functions, 22) up to loop order i = 2.…”

“…[1] this fit parameter was denoted b eff 3 . 15) and determine the -parameter in units of L n and thus in units of L 0 = 2 n L n . Note that the expansion of the integral in the exponent…”

“…As part of the project to determine α s (m Z ) from low energy hadronic input in 3-flavour QCD [1,12,13], our collaboration has applied these techniques to a 1-parameter family of finite volume couplings in Schrödinger functional (SF) schemes, for which the 3-loop β-function is known [14][15][16][17]. We have measured these couplings in numerical simulations and for a range of lattice sizes with unprecedented precision.…”