The multiloop sunset to all orders

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-lowering 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 $D=4-2\hat{I}\mu$.

28 pages. Implementation code available at https://github.com/pierrevanhove/AllLoopSunset. v2: typographical errors corrected. Added some precisions on the choice of the branch cut of the logarithms. Version to appear in Annales Henri Poincaré

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