On two-loop divergences of effective action in 6 D , 𝒩 = (1 , 1) SYM theory

We study the off-shell structure of the two-loop effective action in 6 D, 𝒩 = (1 , 1) supersymmetric gauge theories formulated in 𝒩 = (1 , 0) harmonic superspace. The off-shell effective action involving all fields of 6 D, 𝒩 = (1 , 1) supermultiplet is constructed by the harmonic superfield background field method, which ensures both manifest gauge covariance and manifest 𝒩 = (1 , 0) supersymmetry. We analyze the off-shell divergences dependent on both gauge and hypermultiplet superfields and argue that the gauge invariance of the divergences is consistent with the non-locality in harmonics. The two-loop contributions to the effective action are given by harmonic supergraphs with the background gauge and hypermultiplet superfields. The procedure is developed to operate with the harmonic-dependent superpropagators in the two-loop supergraphs within the superfield dimensional regularization. We explicitly calculate the gauge and the hypermultiplet-mixed divergences as the coefficients of 1/ε^2 and demonstrate that the corresponding expressions are non-local in harmonics.