The chirally rotated Schrödinger functional: theoretical expectations and perturbative tests

Abstract

The chirally rotated Schrödinger functional (χSF) with massless Wilson-type fermions provides an alternative lattice regularization of the Schrödinger functional (SF), with different lattice symmetries and a common continuum limit expected from universality. The explicit breaking of flavour and parity symmetries needs to be repaired by tuning the bare fermion mass and the coefficient of a dimension 3 boundary counterterm. Once this is achieved one expects the mechanism of automatic O(a) improvement to be operational in the χSF, in contrast to the standard formulation of the SF. This is expected to significantly improve the attainable precision for step-scaling functions of some composite operators. Furthermore, the χSF offers new strategies to determine finite renormalization constants which are traditionally obtained from chiral Ward identities. In this paper we consider a complete set of fermion bilinear operators, define corresponding correlation functions and explain the relation to their standard SF counterparts. We discuss renormalization and O(a) improvement and then use this set-up to formulate the theoretical expectations which follow from universality. Expanding the correlation functions to one-loop order of perturbation theory we then perform a number of non-trivial checks. In the process we obtain the action counterterm coefficients to one-loop order and reproduce some known perturbative results for renormalization constants of fermion bilinears. By confirming the theoretical expectations, this perturbative study lends further support to the soundness of the χSF framework and prepares the ground for non-perturbative applications.

Non-perturbative renormalization of tensor currents: strategy and results for $$N_f=0$$ N f = 0 and $$N_

“…(2.2), with the anomalous dimensions of the operators γ in the place of τ . Starting from the RGE for correlation functions, we can write the renormalization group equation for the insertion of a multiplicatively renormaliz- 1 During the development of this work, Dalla Brida, Sint and Vilaseca have performed a related perturbative study as part of the setup of the chirally rotated Schrödinger Functional [35]. Their results for the oneloop matching factor required to compute the NLO tensor anomalous dimensions in SF schemes coincide with ours (cf.…”

Non-perturbative quark mass renormalisation and running in $$N_{\scriptstyle \mathrm{f}}=3$$ N f = 3 QCD

“…Since the values of the axial current normalisation Z A are available from a separate computation [80][81][82] Fig. 4 Non-perturbative running of light quark masses as a function of the energy scale, down to our hadronic matching scale.…”

Nonperturbative renormalization of the axial current inNf=3lattice QCD with Wilson fermions and a tree-level improved gauge action

“…An alternative renormalization condition, using the Schrödinger functional with chirally rotated boundary conditions, has been recently proposed and tested in perturbation theory [19] and non-perturbatively in the quenched and in the N f = 2 cases [20,21]. The main advantage of the approach being that it entails automatic O(a) improvement, it seems very promising and, in two-flavor QCD, turned out to yield more precise results than with standard Schrödinger functional boundary conditions [21].…”