Prepotential of N=2 SU(2) Yang-Mills gauge theory coupled with a massive matter multiplet

TL;DR

This work extends the Seiberg-Witten framework to $N=2$ $SU(2)$ Yang-Mills theory with one massive hypermultiplet by deriving a third-order Picard-Fuchs equation for the massive case and constructing the periods $a(u)$ and $a_D(u)$ from Frobenius-type solutions. It then obtains an all-orders prepotential $\mathcal{F}_1$ with instanton corrections, showing consistency with the massless limit and revealing mass-dependent contributions. A detailed double scaling analysis demonstrates how the theory flows to the pure $N_f=0$ limit, yielding a renormalized prepotential $\widehat{\mathcal{F}}_1$ that matches the $N_f=0$ instanton expansion (as per prior results). Overall, the paper strengthens the SW program for theories with massive matter, clarifies the behavior of instanton corrections with mass, and points to connections with integrable-system frameworks such as the Whitham hierarchy.

Abstract

We discuss the N=2 SU(2) Yang-Mills theory coupled with a massive matter in the weak coupling. In particular, we obtain the instanton expansion of its prepotential. Instanton contributions in the mass-less limit are completely reproduced. We study also the double scaling limit of this massive theory and find that the prepotential with instanton corrections in the double scaling limit coincides with that of N=2 SU(2) Yang-Mills theory without matter.