Nonholomorphic terms inN=2supersymmetric Wilsonian actions and the renormalization group equation

In this paper we first investigate the ansatz of one of the present authors for $K(\Psi ,\mathrm{\Psi }\mathrm{¯}),$ the adimensional, modular-invariant, nonholomorphic correction to the Wilsonian effective Lagrangian of an $N=2\mathrm{}$ globally supersymmetric gauge theory. The renormalization group β function of the theory crucially allows us to express $K(\Psi ,\mathrm{\Psi }\mathrm{¯})$ in a form that easily generalizes to the case in which the theory is coupled to $N_F$ hypermultiplets. $K(\Psi ,\mathrm{\Psi }\mathrm{¯})$ satisfies an equation which should be viewed as a fully nonperturbative “nonchiral superconformal Ward identity.” We also determine its renormalization group equation. Furthermore, as a first step towards checking the validity of this ansatz, we compute the contribution to $K(\Psi ,\mathrm{\Psi }\mathrm{¯})$ from multi-instanton configurations of winding number $k=1\mathrm{}$ and $k=2.\mathrm{}$ As a by-product of our analysis we check a nonrenormalization theorem for $N_F=4.\mathrm{}$