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, 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 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) 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.
Highlights
It has been shown that precise values of the parameter of the pure gauge theory can be used, via a non-perturbative matching between QCD and the pure gauge theory using heavy quarks [30], into a precise value for the strong coupling, a key quantity for phenomenology in high energy physics
Compared with the more customary finite volume renormalization schemes based on Schrödinger functional (SF) [31,32] or mixed open-SF boundary conditions [33], twisted boundary conditions [34,35] preserve the invariance under translations and are free of the linear O(a) cutoff effects present in other schemes
In this work we have investigated the running coupling using finite size scaling techniques in a scheme with twisted boundary conditions
Summary
The starting point for the determination of the parameter is the RG equation in a certain scheme (labeled s) βs(λs). Where s is an integration constant (the -parameter) As such, it is a renormalization group invariant (i.e. scale independent), but depends on the choice of renormalization scheme. Is called the step scaling function, and can be obtained via lattice simulations by first tuning the value of the bare coupling λ0 such that λs = u at several values of the lattice size L = l/a, and determining the value of the coupling in a lattice twice as large at the same values of the bare coupling λ0. After a reasonable number of steps (O(10)), large energy scales have been achieved, where perturbative corrections are expected to be small and one can explore the limit of Eq (2.8) by taking λs (μpt) = uk and μpt = 2k μhad μhad. We discuss the numerical determination of the step scaling function in the TGF set-up
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
