Two-dimensional quartic scalar sunset mass shift ($|{\Sigma }_{\mathrm{sun}}(m^2)|=g_4^2{\int }_{\Delta }(xy+xz+yz-xyz{)}^{-1}/(96{\pi }^2{m}^2)$)

In a massive two-dimensional quartic scalar field theory, the two-loop sunset diagram is ultraviolet finite but changes the pole mass. Feynman parameters give, below the three-particle threshold,At $s=m^2$ the denominator is positive in the interior of the simplex, and its corners are integrable. Thus the result is finite and nonzero for $g_4\ne 0$. Absence of new divergent counterterms does not imply absence of finite mass counterterms in an on-shell renormalization scheme.