Nonholomorphic N = 2 terms in N = 4 SYM : 1 - Loop Calculation in N = 2 superspace

The effective action of N = 2 gauge multiplets in general includes higher-dimension UV finite nonholomorphic corrections integrated with the full N = 2 superspace measure. By adding a hypermultiplet in the adjoint representation we study the effective action of N = 4 SYM. The nonanomalous SU(4) R-symmetry of the classical N = 4 theory must be also present in the on-shell effective action, and therefore we expect to find similar non-holomorphic terms for each of the scalars in the hypermultiplet. The N = 2 path integral quantization formalism developed in projective superspace allows us to compute these hy-permultiplet nonholomorphic terms directly in N = 2 superspace. The corresponding gauge multiplet expression can be successfully compared with the result inferred from a N = 1 calculation in the abelian subsector.