# Supersymmetric QCD on the Lattice: An Exploratory Study

## TL;DR

This paper presents a perturbative lattice study of ${\cal N}=1$ supersymmetric QCD, calculating renormalization factors and mass corrections for various fields, advancing the understanding of supersymmetric theories on the lattice.

## Contribution

It provides the first one-loop perturbative renormalization calculations for supersymmetric QCD on the lattice, including mixing matrices for squark fields.

## Key findings

- Analytic expressions for renormalization factors of fields and coupling.
- Critical mass values for gluino, quark, and squark determined.
- Mixing matrix for squark fields computed.

## 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 na\"ive 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.