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The multiloop sunset to all orders

Pierre Vanhove

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 $D = 4 - 2ε$.

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-raising relation expressing the loop sunset integrals in as the one in dimensions acted on a differential operator of order . 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 .
Paper Structure (19 sections, 4 theorems, 83 equations, 2 figures, 1 table)

This paper contains 19 sections, 4 theorems, 83 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mellin representation
  3. Mellin transform
  4. Inverse Mellin transform and residue computation
  5. Structure of the Picard--Fuchs differential equation
  6. The inhomogeneous Picard--Fuchs equation
  7. The holomorphic period
  8. Structure of the source term
  9. The one-loop case as a prototype
  10. The all-equal-mass case
  11. Frobenius basis near p2=infty
  12. The all-equal-mass sunset at all loop orders
  13. The two-loop sunset
  14. Higher-loop results and the Frobenius corrections
  15. The sunset integral in four dimensions
  16. ...and 4 more sections

Key Result

Theorem 2.1

The multiloop sunset integral has the exact expansion in equation e:IsunsetResult, where we capture the dependence on logarithms of $m_i^2/p^2$ via the logarithmic factors $\ell_i(r)$ defined in equation eq:elldef and the symmetric polynomials $P^k_{L+1}$, whose generating function is eq:polydef for $n\in\mathbb N$, $c_0(n)=0$ and for $r\geq1$ where and $Y_m(x_1,\dots,x_m)$ is the complete Bell p

Figures (2)

  • Figure 1: Multiloop sunset graph.
  • Figure 2: Contour integration strategy for extracting the series expansion. Left panel: In the $\zeta$-plane, we close the contour to the left (toward $\Re{e}(\zeta) \to -\infty$), capturing poles at $\zeta = -1-z_1-\cdots-z_L-r_{L+1}$ for all $r_{L+1} \in \mathbb{N}$. Each pole contributes a term in the final series. Right panel: In each $z_i$-plane, we close the contour to the right (toward $\Re{e}(z_i) \to +\infty$), capturing poles at $z_i = r_i \in \mathbb{N}$. The $(L+1)$-dimensional residue computation yields the coefficients of the expansion. The contours can be closed because the $\Gamma$-functions ensure exponential decay in the appropriate half-planes.

Theorems & Definitions (16)

  • Theorem 2.1
  • proof
  • Remark 2.1: Asymptotic of Feynman integrals
  • Proposition 3.1: Structure of the source term
  • proof
  • Remark 3.1
  • Proposition 4.1
  • Definition 4.1: Frobenius basis
  • Remark 4.1: Relation to the Frobenius basis of Bonisch:2021yfw
  • Theorem 4.1: All-equal-mass case
  • ...and 6 more