Weak charge quantization on a superconducting island

We consider Coulomb blockade on a superconductive quantum dot strongly coupled to a lead through a tunneling barrier and/or normal diffusive metal. Andreev transport of the correlated pairs leads to quantum fluctuations of the charge on the dot. These fluctuations result in exponential renormalization of the effective charging energy. We employ two complimentary ways to approach the problem: the instanton and the functional renormalization group treatment of the nonlinear $\sigma$ model. We show that these two different methods produce identical results. We also derive the charging energy renormalization in terms of the arbitrary transmission matrix of the multichannel interface.