= Two-dimensional quartic scalar sunset mass shift
{title2=$|\Sigma_{\rm sun}(m^2)|=g_4^2\int_\Delta (xy+xz+yz-xyz)^{-1}/(96\pi^2m^2)$}
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,
$$
|\Sigma_{\rm sun}(s)|=\frac{g_4^2}{6(4\pi)^2}\int_{x,y,z\geq0}dx\,dy\,dz\,
\frac{\delta(1-x-y-z)}{m^2(xy+xz+yz)-sxyz}.
$$
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 $g_4\ne0$. Absence of new divergent <counterterms> does not imply absence of finite <mass counterterms> in an <on-shell renormalization scheme>.
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