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Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph

Amedeo Primo, Lorenzo Tancredi

TL;DR

This work addresses the challenge of computing master integrals for the irreducible three-loop massive banana graph, whose master equations form a 3×3 coupled system with a third-order homogeneous equation in $d=2$ and four regular singular points. The authors show that all independent homogeneous solutions can be obtained from the maximal cut by integrating over three independent contours that avoid branch cuts, yielding three solutions that can be written as products of complete elliptic integrals with arguments tied to a carefully chosen parametrization. They reveal that the third-order equation is a symmetric square of a second-order equation, linking the banana graph solutions to Joyce’s Heun functions and ultimately to elliptic integrals, and they construct the inhomogeneous solution via variation of constants with boundary conditions fixed by regularity at pseudo-thresholds. The results provide explicit, real-valued integral representations in each kinematic region and a framework for extending maximal-cut methods to higher-order systems, offering analytic control beyond polylogarithmic functions for multiloop Feynman integrals.

Abstract

We consider the calculation of the master integrals of the three-loop massive banana graph. In the case of equal internal masses, the graph is reduced to three master integrals which satisfy an irreducible system of three coupled linear differential equations. The solution of the system requires finding a $3 \times 3$ matrix of homogeneous solutions. We show how the maximal cut can be used to determine all entries of this matrix in terms of products of elliptic integrals of first and second kind of suitable arguments. All independent solutions are found by performing the integration which defines the maximal cut on different contours. Once the homogeneous solution is known, the inhomogeneous solution can be obtained by use of Euler's variation of constants.

Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph

TL;DR

This work addresses the challenge of computing master integrals for the irreducible three-loop massive banana graph, whose master equations form a 3×3 coupled system with a third-order homogeneous equation in and four regular singular points. The authors show that all independent homogeneous solutions can be obtained from the maximal cut by integrating over three independent contours that avoid branch cuts, yielding three solutions that can be written as products of complete elliptic integrals with arguments tied to a carefully chosen parametrization. They reveal that the third-order equation is a symmetric square of a second-order equation, linking the banana graph solutions to Joyce’s Heun functions and ultimately to elliptic integrals, and they construct the inhomogeneous solution via variation of constants with boundary conditions fixed by regularity at pseudo-thresholds. The results provide explicit, real-valued integral representations in each kinematic region and a framework for extending maximal-cut methods to higher-order systems, offering analytic control beyond polylogarithmic functions for multiloop Feynman integrals.

Abstract

We consider the calculation of the master integrals of the three-loop massive banana graph. In the case of equal internal masses, the graph is reduced to three master integrals which satisfy an irreducible system of three coupled linear differential equations. The solution of the system requires finding a matrix of homogeneous solutions. We show how the maximal cut can be used to determine all entries of this matrix in terms of products of elliptic integrals of first and second kind of suitable arguments. All independent solutions are found by performing the integration which defines the maximal cut on different contours. Once the homogeneous solution is known, the inhomogeneous solution can be obtained by use of Euler's variation of constants.

Paper Structure

This paper contains 15 sections, 160 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Revisiting the two-loop massive sunrise graph
  3. The three-loop massive banana graph
  4. The homogeneous system
  5. The maximal cut and the homogeneous solution
  6. A basis of unit leading singularity
  7. The homogeneous solutions as product of elliptic integrals
  8. The third-order differential equation as a symmetric square
  9. The inhomogeneous solution
  10. Conclusions
  11. Analytic continuation
  12. Homogenous solutions
  13. Analytic continuation of the homogenous solution
  14. Analytic continuation of the inhomogeneous solution
  15. Proof of Eq.s \ref{['eq:rescut1']}-\ref{['eq:rescut2']}

Figures (3)

  • Figure 1: Left panel: The contours $\mathcal{C}_1$, $\mathcal{C}_3$ and $\mathcal{C}_\infty$. The branches of the integrand for the positive sign in the root in Eq. \ref{['eq:cutsun']} are drawn in red. Right panel: The contour $\mathcal{C}_2$. The branches of the integrand for the negative sign in the root in Eq. \ref{['eq:cutsun']} are drawn in red.
  • Figure 2: The lines represent the set of points where the argument of the square root in Eq. \ref{['eq:cutbanana']} changes sign.
  • Figure 3: Real \ref{['fig:PlotRe']} and imaginary \ref{['fig:PlotIm']} part of the finite term of the master integrals for the three-loop banana graph. The imaginary part is non-vanishing only in the range $0<x<1/4$, which corresponds to $s>16m^2$. The numerical evaluation of the result (solid curves) is compared to the values obtained with SecDec 3 (dots).