Differential Equations for Scaling Relation in N=2 Supersymmetric SU(2) Yang-Mills Theory Coupled with Massive Hypermultiplet

TL;DR

This paper derives differential equations governing the scaling relations of the prepotential in N=2 supersymmetric SU(2) Yang-Mills theory with massive hypermultiplets, enabling explicit solutions and analysis in different coupling regimes.

Contribution

It introduces differential equations for the massive prepotential's scaling relation and demonstrates their solutions, advancing understanding of the theory's non-perturbative structure.

Findings

Derived differential equations for the prepotential's scaling relation.

Obtained closed-form solutions for the derivative of the prepotential.

Derived the massive prepotential in the strong coupling region.

Abstract

Differential equations for scaling relation of prepotential in N=2 supersymmetric SU(2) Yang-Mills theory coupled with massive matter hypermultiplet are proposed and are explicitly demonstrated in one flavour ($N_f =1$) theory. By applying Whitham dynamics, the first order derivative of the prepotential over the $T_0$ variable corresponding to the mass of the hypermultiplet, which has a line integral representation, is found to satisfy a differential equation. As the result, the closed form of this derivative can be obtained by solving this equation. In this way, the scaling relation of massive prepotential is established. Furthermore, as an application of another differential equation for the massive scaling relation, the massive prepotential in strong coupling region is derived.