The multiloop sunset to all orders

TL;DR

This paper presents exact, convergent formulas for multiloop sunset Feynman integrals in two dimensions, expressed as sums of symmetric polynomials, facilitating both analysis and numerical evaluation, and enabling reconstruction of four-dimensional integrals.

Contribution

It introduces a novel representation of multiloop sunset integrals in two dimensions, including a dimension-raising relation and a method for reconstructing four-dimensional integrals from two-dimensional results.

Findings

Exact formulas for multiloop sunset integrals in 2D

Dimension-raising relation for equal-mass case

Method for reconstructing 4D integrals from 2D boundary conditions

Abstract

We derive exact, convergent representations of multiloop sunset Feynman integrals in two dimensions for arbitrary mass configurations and all loop orders valid for large Euclidean momentum. The integrals are expressed as sums of symmetric polynomials in logarithmic mass ratios, normalized by the external momentum squared, with coefficients determined by analytic series expansions. For the equal-mass case, we establish a dimension-raising relation expressing the $L$ loop sunset integrals in $D+2$ as the one in $D$ dimensions acted on a differential operator of order $L-1$. These representations are free of complicated transcendental functions, making them well-suited to both formal analysis and high-precision numerical evaluation. The two-dimensional results serve as boundary conditions for dimension-shifting relations, enabling systematic reconstruction of four-dimensional sunset integrals via analytic continuation to $D = 4 - 2\epsilon$.