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The unequal-mass three-loop banana integral

Sebastian Pögel, Toni Teschke, Xing Wang, Stefan Weinzierl

TL;DR

We address the unequal-mass three-loop banana integral, a Feynman integral tied to a $K3$ Calabi–Yau geometry, and derive an $\varepsilon$-factorised differential equation using an algorithmic basis-rotation approach in the loop-by-loop Baikov framework. The authors construct two compatible bases, $J$ and $K$, by applying twisted-cohomology filtrations to separate geometric and combinatorial sectors, enabling a Laurent-polynomial and then an $\varepsilon$-factorised system $dK=\varepsilon\tilde{A}(y)K$. They provide explicit basis constructions, boundary data near the maximal unipotent monodromy point, and a numerical evaluation for the master integral $K_5$ that agrees with independent numerical methods, including a SM-mass configuration. This work demonstrates a fully algorithmic strategy for high-loop Feynman integrals with geometries beyond elliptic curves, offering a path to generic mass configurations and higher-loop generalisations.

Abstract

We compute the three-loop banana integral with four unequal masses in dimensional regularisation. This integral is associated to a family of K3 surfaces, thus representing an example for Feynman integrals with geometries beyond elliptic curves. We evaluate the integral by deriving an $\varepsilon$-factorised differential equation, for which we rely on the algorithm presented in a recent publication. Equipping the space of differential forms in Baikov representation by a set of filtrations inspired by Hodge theory, we first obtain a differential equation with entries as Laurent polynomials in $\varepsilon$. Via a sequence of basis rotations we then remove any non-$\varepsilon$-factorising terms. This procedure is algorithmic and at no point relies on prior knowledge of the underlying geometry.

The unequal-mass three-loop banana integral

TL;DR

We address the unequal-mass three-loop banana integral, a Feynman integral tied to a Calabi–Yau geometry, and derive an -factorised differential equation using an algorithmic basis-rotation approach in the loop-by-loop Baikov framework. The authors construct two compatible bases, and , by applying twisted-cohomology filtrations to separate geometric and combinatorial sectors, enabling a Laurent-polynomial and then an -factorised system . They provide explicit basis constructions, boundary data near the maximal unipotent monodromy point, and a numerical evaluation for the master integral that agrees with independent numerical methods, including a SM-mass configuration. This work demonstrates a fully algorithmic strategy for high-loop Feynman integrals with geometries beyond elliptic curves, offering a path to generic mass configurations and higher-loop generalisations.

Abstract

We compute the three-loop banana integral with four unequal masses in dimensional regularisation. This integral is associated to a family of K3 surfaces, thus representing an example for Feynman integrals with geometries beyond elliptic curves. We evaluate the integral by deriving an -factorised differential equation, for which we rely on the algorithm presented in a recent publication. Equipping the space of differential forms in Baikov representation by a set of filtrations inspired by Hodge theory, we first obtain a differential equation with entries as Laurent polynomials in . Via a sequence of basis rotations we then remove any non--factorising terms. This procedure is algorithmic and at no point relies on prior knowledge of the underlying geometry.

Paper Structure

This paper contains 19 sections, 2 theorems, 103 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and setup
  3. The construction of the basis $J$
  4. Weight four
  5. Weight three
  6. Weight two
  7. Pole order zero
  8. Pole order one
  9. Pole order two
  10. A basis for the twisted cohomology group
  11. The master integrals of the basis $J$
  12. The construction of the basis $K$
  13. The transformation $R_2^{(-2)}$
  14. The transformations $R_2^{(-1)}$ and $R_2^{0}$
  15. The neighbourhood of the point of maximal unipotent monodromy
  16. ...and 4 more sections

Key Result

Proposition 1

The holomorphic period $\psi_{0}$ and the single-logarithmic periods $\psi_{1,j}$ as defined by eq. (def_Frobenius) satisfy the system of differential equations from section sect:step_2 for $\psi$.

Figures (4)

  • Figure 1: The three-loop banana integral with unequal masses.
  • Figure 2: The sector of interest (sector $15$) together with the relevant super-sectors.
  • Figure 3: The (real) zero set of the polynomials $P_1, \dots, P_5$ in the chart $z_0=1$.
  • Figure 4: The real and the imaginary part of the master integral $K_5$ at order $\varepsilon^3$ (left plot) and order $\varepsilon^4$ (right plot) as a function of $\lambda$.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof