The three-loop beta-fuction of QCD with the clover action

TL;DR

The paper computes the three-loop bare lattice beta function $\beta_L(g_0)$ for QCD with clover fermions, yielding the lattice coefficient $b_2^{L}$ as a function of $N$, $N_f$, and $c_{\rm SW}$, and derives the two-loop relation between the bare coupling and the MS-bar coupling as well as the three-loop relation between the lattice $\Lambda$-parameter and $g_0$, with a pronounced correction factor $q$ for typical $c_{\rm SW}$. The methodology combines standard lattice perturbation theory with the background-field formalism to relate $Z_g$ to the renormalization constants, and cross-checks results against Schrödinger-functional computations. A comprehensive perturbative analysis of $\nu^{(1)}(p)$ and $\nu^{(2)}(p)$, including 18 fermionic two-loop diagrams and various counterterms, enables precise determination of $l_1$, $l_{11}$, and $l_{12}$, from which $b_2^{L}$ and the $d_1,d_2$ coefficients are obtained. These results provide essential inputs for assessing asymptotic scaling and for tuning $c_{\rm SW}$ to minimize lattice artifacts in simulations. All notable formulas are expressed with $...$ delimiters for clarity in mathematical notation.

Abstract

We calculate, to 3 loops in perturbation theory, the bare $β$-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 $N_f$, as well as 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 $Λ$-parameter and $g_0$, which is important in checks of asymptotic scaling. For typical values of $c_{SW}$, this correction is found to be very pronounced.