Effective action of β-deformed [formula omitted] SYM theory: Farewell to two-loop BPS diagrams

Abstract

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 m 3 = m 1 + m 2 , 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 ( m 1 2 ) , J ( m 2 2 ) and J ( m 3 2 ) , 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 ( m 1 2 m 2 2 + m 1 2 m 3 2 + m 2 2 m 3 2 ) > m 1 4 + m 2 4 + m 3 4 . 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 F 4 term. Our considerations are restricted to the case of β real.