Solving differential equations for Feynman integrals by expansions near singular points
Roman N. Lee, Alexander V. Smirnov, Vladimir A. Smirnov
TL;DR
The paper tackles the problem of evaluating Feynman integrals whose differential equations do not admit the canonical $\\epsilon$-form. It introduces a generalized series-expansion approach near regular singular points to construct the evolution operator $U(x)$ and uses a matching procedure to connect local solutions across multiple singularities. The authors derive finite-order recurrence relations for expansion coefficients and provide a public-style implementation for a four-loop generalized sunset with three equal masses, including boundary data from Mellin–Barnes representations to obtain the $\\epsilon$-expansion up to a given order. This work offers a practical route to compute challenging two-scale master integrals when analytic forms are unavailable, with potential generalization to elliptic sectors and broader two-scale problems.
Abstract
We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. nontrivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer implementation of our algorithm in a simple example of four-loop generalized sun-set integrals with three equal non-zero masses. Our code provides values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter $ε$.