Chiral effective potential in $4D$, $\mathcal{N}=1$ supersymmetric gauge theories

TL;DR

This paper computes the chiral effective superpotential in 4D $ ext{N}=1$ supersymmetric gauge theories, revealing finiteness at one- and two-loop levels and analyzing large N behavior in finite $ ext{N}=2$ models.

Contribution

It provides explicit calculations of the chiral effective superpotential at one- and two-loop orders, including finiteness properties and large N analysis in finite $ ext{N}=2$ supersymmetric theories.

Findings

One-loop chiral potential is finite and expressed via a triangle integral.

Two-loop contributions from purely chiral vertices are finite.

In finite $ ext{N}=2$ theories, the superpotential simplifies significantly.

Abstract

We calculate the chiral effective superpotential in $4D$ $\mathcal{N}=1$, $SU(N)$ super Yang-Mills theory coupled to chiral matter in one- and two-loop approximations. It is found that the one-loop contribution to the chiral effective potential is always finite and is expressed in terms of a specific triangle integral. The two-loop contributions generated by purely chiral vertices turned out to be finite as well. The chiral effective potential stipulated by supergraphs with gauge superfield subgraphs is finite for the supergraphs with no divergent subgraphs. In the case of the finite $\mathcal{N}=2$ SYM theory, the two-loop chiral contributions to the effective action are significanlty simplified. The leading large $N$ behavior of the chiral effective superpotential in finite $\mathcal{N}=2$ super-Yang-Mills models with $SU(N)$ gauge symmetry is studied and it is shown that the exact form in the coupling constant of the chiral effective superpotential can be found.