Renormalization of Supersymmetric QCD on the Lattice

TL;DR

This paper performs a perturbative one-loop renormalization study of ${\cal N}=1$ supersymmetric QCD on the lattice, deriving renormalization factors and addressing mass and field mixing issues for various particles.

Contribution

It provides the first analytic expressions for lattice renormalization factors in supersymmetric QCD with matter fields, including mass critical values and mixing matrices.

Findings

Derived analytic renormalization constants for all fields and coupling.

Calculated critical masses for gluino, quark, and squark fields.

Addressed squark mixing beyond tree level with a detailed mixing matrix.

Abstract

We perform a pilot study of the perturbative renormalization of a Supersymmetric gauge theory with matter fields on the lattice. As a specific example, we consider Supersymmetric ${\cal N}{=}1$ QCD (SQCD). We study the self-energies of all particles which appear in this theory, as well as the renormalization of the coupling constant. To this end we compute, perturbatively to one-loop, the relevant two-point and three-point Green's functions using both dimensional and lattice regularizations. Our lattice formulation involves the Wilson discretization for the gluino and quark fields; for gluons we employ the Wilson gauge action; for scalar fields (squarks) we use naive discretization. The gauge group that we consider is $SU(N_c)$, while the number of colors, $N_c$, the number of flavors, $N_f$, and the gauge parameter, $\alpha$, are left unspecified. We obtain analytic expressions for the renormalization factors of the coupling constant ($Z_g$) and of the quark ($Z_\psi$), gluon ($Z_u$), gluino ($Z_\lambda$), squark ($Z_{A_\pm}$), and ghost ($Z_c$) fields on the lattice. We also compute the critical values of the gluino, quark and squark masses. Finally, we address the mixing which occurs among squark degrees of freedom beyond tree level: we calculate the corresponding mixing matrix which is necessary in order to disentangle the components of the squark field via an additional finite renormalization.