Memory efficient finite volume schemes with twisted boundary conditions

Abstract

In this paper we explore a finite volume renormalization scheme that combines three main ingredients: a coupling based on the gradient flow, the use of twisted boundary conditions and a particular asymmetric geometry, that for SU(N) gauge theories consists on a hypercubic box of size $$l^2 \times (Nl)^2$$ l 2 × ( N l ) 2 , a choice motivated by the study of volume independence in large N gauge theories. We argue that this scheme has several advantages that make it particularly suited for precision determinations of the strong coupling, among them translational invariance, an analytic expansion in the coupling and a reduced memory footprint with respect to standard simulations on symmetric lattices, allowing for a more efficient use of current GPU clusters. We test this scheme numerically with a determination of the $$\Lambda $$ Λ parameter in the SU(3) pure gauge theory. We show that the use of an asymmetric geometry has no significant impact in the size of scaling violations, obtaining a value $$\Lambda _{\overline{\mathrm{MS}}}\sqrt{8 t_0} =0.603(17)$$ Λ MS ¯ 8 t 0 = 0.603 ( 17 ) in good agreement with the existing literature. The role of topology freezing, that is relevant for the determination of the coupling in this particular scheme and for large N applications, is discussed in detail.

Results for $\alpha_s$ from the decoupling strategy

“…It is important to point out that the result of this work is largely independent from our previous determination [5]. Only the value of µ dec = 789 (15) MeV is shared between both determinations of the strong coupling (see Section 4.5.2). This amounts to 28% of the squared error.…”

“…MS /µ dec + O(1/z), see table 3. With µ dec = 789 (15) MeV obtained in N f = 3 QCD [35], we convert these ratios to the effective three flavor Λ-parameter, again equal to Λ…”

“…For the determination of massive simulation parameters detailed in section E we perform massless renormalization runs along the line of constant physics defined in eq. (2.12), i.e., g 2 GF (L dec ) = 3.949 and m = 0, corresponding to a scale µ dec = 789 (15) MeV or fixed physical volume with L dec = 0.250(5) fm [5,35]. We generate a large set of new ensembles around Lm PCAC = 0 for L/a = 12, 16, 20, 24, 32, 40 and 48, where we measure reweighting factors and the following observables, as well as the covariance between them:…”

“…A solution to this multiscale problem has been known for 30 years in the form of the recursive step-scaling method [13]. The method has since been applied to the running of the coupling in QCD with N f = 0 [2,3,14,15], N f = 2 [16,17], N f = 3 [5,18] and N f = 4 [19,20] quark flavours (for a review cf. [21]), and in various candidate models of physics beyond the Standard Model (for reviews, cf.…”

Scale setting for large-N SUSY Yang-Mills on the lattice

“…Let us finally mention that the twisted finite volume normalization factor, N (c(t), N ), has been computed and used before for a series of step scaling studies [25,65,66]; explicit formulas are given in appendix B, where we also give some more details on the perturbative expansion and the relevant references. This procedure also accounts for correcting lattice artefacts at leading order if the normalization factor is computed instead in lattice perturbation theory.…”