Ladders and rainbows in Minimal Subtraction

In dimensional regularization with $D=D_0-2\epsilon$, the minimal subtraction (MS) scheme is characterized by counterterms that only consist of singular terms in $\epsilon$. We develop a general method to compute the infinite sums of massless ladder or rainbow Feynman integrals in MS at $D_0$. Our method is based on relating the MS-solution to a kinematic solution at a coupling-dependent renormalization point. If the $\epsilon$-dependent Mellin transform of the kernel diagram of the insertions can be computed in closed form, we typically obtain a closed form expression for the all-order solution in MS. As examples, we consider Yukawa theory and ${\phi }^4$ theory in $D_0=4$, and ${\phi }^3$ theory in $D_0=6$.