O(a) improved QCD: The 3-loop beta-function, and the critical hopping parameter

TL;DR

This paper computes the three-loop beta-function for lattice QCD with clover fermions, providing precise relations between lattice and continuum parameters, and determines the critical hopping parameter with comparisons to non-perturbative results.

Contribution

It presents the first three-loop calculation of the lattice QCD beta-function with clover fermions and derives the relation between lattice and MS-bar schemes, including the critical hopping parameter.

Findings

Three-loop beta-function for lattice QCD with clover fermions calculated.

Two-loop relation between bare and renormalized coupling constants established.

Critical hopping parameter $$ computed to two loops and compared with Monte Carlo results.

Abstract

We calculate the 3-loop bare $\beta$-function of QCD, formulated on the lattice with the clover fermionic action. The dependence of our result on the number of colors $N$, the number of fermionic flavors $N_f$, and the clover parameter $c_{SW}$, is shown explicitly. A direct outcome of our calculation is the two-loop relation between the bare coupling constant $g_0$ and the one renormalized in the MS-bar scheme. Further, we can immediately derive the three-loop correction to the relation between the lattice $\Lambda$-parameter and $g_0$, which turns out to be very pronounced. We also calculate the critical value of the hopping parameter, $\kappa_c$, in the clover action, to two loops in perturbation theory. This quantity is an additive renormalization; as such, it exhibits a linear divergence in the lattice spacing. We compare our results to non perturbative evaluations of $\kappa_c$ coming from MC simulations.