Effective Action in N=2,4 Supersymmetric Yang-Mills Theories

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We review the approach to calculation of one-loop effective action in N=2,4 SYM theories. We compute the non-holomorphic corrections to low-energy effective action (higher derivative terms) in N=2, SU(2) SYM theory coupled to hypermultiplets on a non-abelian background for Rξ-gauge fixing conditions. A general procedure for calculating the gauge parameters depending contributions to one-loop superfield effective action is developed. The one-loop non-holomorphic effective potential is exactly found in terms of the Euler dilogarithm function for a specific choice of gauge parameters.We also discuss the calculations of hypermultiplet dependence of N=4 SYM effective action.