Alleviating the window problem in large volume renormalization schemes

We propose a strategy for large volume non-perturbative renormalization which alleviates the window problem by reducing cut-off effects. We perform a proof-of-concept study using position space renormalization scheme and the CLS $N_f=2+1$ ensembles generated at 5 different lattice spacings. We show that in the advocated strategy results for the renormalization constants are to a large extend independent of the specific lattice direction used to define the renormalization condition. Hence, very short lattice distances become accessible even on coarse lattices and the contact with perturbation theory can be performed at much higher energy scales. Our results include non-perturbatively estimated renormalization constants for quark bilinear operators in the scalar, pseudoscalar and axial-vector channels using position space renormalization scheme which we subsequently translate to the $\overline{\mathrm{MS}}$ scheme perturbatively at $1.5$ GeV. Our proposal is applicable to other non-perturbative large volume renormalization schemes such as RI-MOM and its variants.