  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 beta-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 beta-deformed N = 4 SYM theory, we carry out explicit two-loop calculations
of the Kahler potential and F^4 term. Our considerations are restricted to the
case of beta real.
