# Super Heat Kernel of General Second Order Operators in $N=1$ Superspace and One-Loop Divergence of Dilaton-coupled SYM Theory

## TL;DR

This paper develops a method to compute super heat kernel coefficients for second order operators in $N=1$ superspace, enabling calculation of one-loop divergences in supersymmetric theories, including super Yang-Mills with a non-trivial gauge kinetic function.

## Contribution

It introduces a general technique for super heat kernel coefficients applicable to various superspaces and applies it to compute one-loop divergences in dilaton-coupled super Yang-Mills theory.

## Key findings

- Calculated the first three super heat kernel coefficients.
- Derived the one-loop logarithmic divergence for super Yang-Mills with a string dilaton.
- Extended the method to third order operators with spinor derivatives.

## Abstract

We shall develop a general technique to obtain the super heat kernel coefficients of an arbitrary second order operator in $N=1$ superspace. We focus on the space of conformal supergravity here but the method presented is equally applicable for other types of superspace. The first three coefficients which determine the one-loop divergence of the corresponding quantum theory will be calculated. As an application we shall present the one-loop logarithmic divergence of super Yang-Mills theory coupled to a string dilaton $S$. This is the first superfield calculation for SYM with a non-trivial gauge kinetic function, which generalize the previous result with a constant coupling strength. We also demonstrate that the method presented can be extended to the case of third order operators, with the restriction that its third order part is composed of only spinor derivatives.