Effective action of β-deformed N=4 SYM theory: Farewell to two-loop BPS diagrams

Within the background field approach, all two-loop sunset vacuum diagrams, which occur in the Coulomb branch of N=2 superconformal theories (including N=4 SYM), obey the BPS condition m3=m1+m2, where the masses are generated by the scalars belonging to a background N=2 vector multiplet. These diagrams can be evaluated exactly, and prove to be homogeneous quadratic functions of the one-loop tadpoles J(m12), J(m22) and J(m32), with the coefficients being rational functions of the squared masses. We demonstrate that, if one switches on the β-deformation of the N=4 SYM theory, the BPS condition no longer holds, and then generic two-loop sunset vacuum diagrams with three non-vanishing masses prove to be characterized by the following property: 2(m12m22+m12m32+m22m32)>m14+m24+m34. In the literature, there exist several techniques to compute such diagrams. For the β-deformed N=4 SYM theory, we carry out explicit two-loop calculations of the Kähler potential and F4 term. Our considerations are restricted to the case of β real.