Three-loop banana integrals with four unequal masses (2507.23061v1)

Abstract: We present a system of canonical differential equations satisfied by the three-loop banana integrals with four distinct non-zero masses in $D = 2-2\eps$ dimensions. Together with the initial condition in the small-mass limit, this provides all the ingredients to find analytic results for three-loop banana integrals in terms of iterated integrals to any desired order in the dimensional regulator. To obtain this result, we rely on recent advances in understanding the K3 geometry underlying these integrals and in how to construct rotations to an $\eps$-factorized basis. This rotation typically involves the introduction of objects defined as integrals of (derivatives of) K3 periods and rational functions. We apply and extend a method based on results from twisted cohomology to identify relations among these functions, which allows us to reduce their number considerably. We expect that the methods that we have applied here will prove useful to compute further multiloop multiscale Feynman integrals attached to non-trivial geometries.

• The paper presents the first construction of canonical differential equations for the complete set of master integrals of the three-loop banana integral with four unequal masses.
• It introduces a canonical basis via transformations aligned with the K3 geometry and employs twisted cohomology to reduce and constrain the transcendental functions.
• The analytic representation using iterated integrals enables both symbolic manipulation and high-precision numerical evaluation, streamlining computations in multiloop Feynman integral analysis.

Canonical Differential Equations for Three-Loop Banana Integrals with Four Unequal Masses

Introduction and Motivation

The computation of multiloop Feynman integrals is a central challenge in perturbative quantum field theory, underpinning precision predictions for collider and gravitational wave experiments. While the class of Feynman integrals expressible in terms of multiple polylogarithms is well understood, more complex geometries—such as elliptic curves and Calabi-Yau varieties—arise at higher loop orders and with more general mass configurations. The three-loop banana (or sunset) integral with four distinct non-zero masses is the simplest known example associated with a family of K3 surfaces, a class of Calabi-Yau twofolds. Analytic results for such integrals have been elusive due to the complexity of the underlying geometry and the associated transcendental functions.

This work presents, for the first time, a construction of canonical (i.e., $\epsilon$-factorized) differential equations for the full set of master integrals of the three-loop banana family with four unequal masses in $D=2-2\epsilon$ dimensions. The approach leverages recent advances in the understanding of the K3 geometry underlying these integrals, the construction of canonical bases, and the application of twisted cohomology to constrain the transcendental functions appearing in the differential equations.

Structure of the Three-Loop Banana Integral Family

The three-loop banana integral with four unequal masses is defined as a family of nine-propagator integrals, with the propagators corresponding to the four massive lines and five irreducible scalar products (ISPs). Integration-by-parts (IBP) reduction identifies 15 master integrals, which are organized to respect the natural filtration by the number of positive propagator powers. The top sector, corresponding to all four propagators present, is associated with the nontrivial K3 geometry.

The master integrals are functions of the dimensionless ratios $x_i = m_i^2/p^2$, where $p$ is the external momentum. The system of master integrals satisfies a first-order linear system of differential equations in these variables, with the differential equation matrix depending rationally on the kinematics and the dimensional regulator $\epsilon$.

Canonical Differential Equations and the Role of K3 Geometry

A central goal is to bring the system of differential equations into canonical ($\epsilon$-factorized) form:

$dJ(x, \epsilon) = \epsilon\, A(x)\, J(x, \epsilon)$

where $A(x)$ is independent of $\epsilon$ and constructed from a set of linearly independent differential one-forms. In the polylogarithmic case, $A(x)$ is a $d\log$-form matrix; for the banana integral, $A(x)$ involves more general transcendental functions associated with the periods of the underlying K3 surface.

The K3 geometry manifests through the Baikov representation of the maximal cut of the integral, which defines a family of K3 surfaces parameterized by the $x_i$. The periods of the holomorphic $(2,0)$-form on the K3 surface, and their derivatives, play a central role in the analytic structure of the integrals. The Picard-Fuchs system governing these periods is used to construct the canonical basis and to express the differential equation matrix.

Construction of the Canonical Basis

The canonical basis is constructed via a sequence of transformations:

1. Alignment with the Mixed Hodge Structure (MHS): The initial basis is chosen to align with the MHS of the K3 surface, ensuring that the maximal cuts of the master integrals correspond to the periods and their derivatives.
2. Semi-simple and Unipotent Decomposition: The period matrix is decomposed into semi-simple and unipotent parts, with the semi-simple part capturing the pure Hodge structure.
3. $\epsilon$-Scaling and Triangularization: Further transformations adjust the $\epsilon$-scaling and bring the system into lower-triangular form, after which a final transformation achieves full $\epsilon$-factorization.

The resulting canonical differential equation matrix $A(x)$ involves 23 functions (the so-called $\epsilon$-functions), defined as iterated integrals over rational functions and (derivatives of) K3 periods.

Reduction of Transcendental Functions via Twisted Cohomology

A key technical advance is the use of twisted cohomology and the constancy of the intersection matrix in a canonical basis to derive relations among the $\epsilon$-functions. The intersection matrix, computed in the initial basis and rotated to the canonical basis, is shown to be constant and independent of the kinematics. This imposes linear and nonlinear constraints on the $\epsilon$-functions.

By decomposing the transformation matrix into $\Delta$-symmetric and $\Delta$-orthogonal parts, it is shown that 10 of the 23 $\epsilon$-functions can be expressed in terms of rational functions and K3 periods and their derivatives. The remaining 13 functions are further reduced, using permutation symmetries, to just two genuinely new functions (and their images under permutations), which cannot be expressed in terms of periods and rational functions alone.

Analytic Representation and Iterated Integrals

The analytic solution for the master integrals is given in terms of iterated integrals over a set of integration kernels involving rational functions, K3 periods, and the two new transcendental functions. The explicit expressions for the $\epsilon$-functions are provided as (possibly nested) integrals, and their series expansions around the small-mass (MUM) point are given for practical evaluation.

The canonical differential equation matrix is shown to have only logarithmic singularities at the MUM point, enabling the use of tangential base-point regularization for the initial conditions. This ensures that the analytic solution is well-defined and can be systematically expanded in $\epsilon$.

Numerical and Symbolic Implementation

The full analytic expressions for the canonical differential equation matrix, the intersection matrix, and the initial conditions are provided in computer-readable form (Mathematica files). This enables both symbolic manipulation and high-precision numerical evaluation of the master integrals to any desired order in $\epsilon$.

The computational requirements are dominated by the evaluation of the K3 periods and the two new transcendental functions, which can be handled via series expansions or numerical integration of the Picard-Fuchs system. The reduction in the number of genuinely new functions significantly simplifies both analytic and numerical work compared to a naive approach.

Implications and Future Directions

This work provides the first complete analytic framework for the three-loop banana integral with four unequal masses, a prototypical example of a Feynman integral associated with a nontrivial Calabi-Yau geometry. The techniques developed—particularly the use of twisted cohomology to constrain transcendental functions—are expected to be broadly applicable to other multiloop, multiscale Feynman integrals associated with higher-dimensional Calabi-Yau varieties, higher-genus curves, and beyond.

The explicit reduction of the number of new transcendental functions required for the analytic solution has important implications for the classification of special functions arising in quantum field theory and for the practical computation of physical observables. Further mathematical developments, including a rigorous classification of the new functions and their properties, as well as the extension to other integral families, are anticipated.

Conclusion

The construction of canonical differential equations for the three-loop banana integral with four unequal masses represents a significant advance in the analytic computation of Feynman integrals associated with nontrivial algebraic geometry. By combining geometric insights, canonical basis construction, and twisted cohomology, the authors have provided a framework that both clarifies the transcendental structure of these integrals and enables their practical evaluation. The methods and results are expected to inform future developments in the analytic computation of multiloop Feynman integrals and the paper of special functions in quantum field theory.