Solution of Canonical Differential Equations for Integrals on Arbitrary Geometries

Technical Summary: Solution of Canonical Differential Equations for Integrals on Arbitrary Geometries

Problem Statement
The evaluation of multi-loop scattering amplitudes in quantum field theory often relies on reducing Feynman integrals to a set of master integrals (MIs) via integration-by-parts identities. A powerful method for determining these MIs involves solving systems of first-order linear differential equations with respect to kinematic invariants and masses. The introduction of $\epsilon$-factorized canonical forms has streamlined this process for polylogarithmic cases, where the connection matrix consists of $d\log$ forms with rational coefficients.

However, for problems involving massive particles and multiple scales, the underlying geometry often extends beyond simple polylogarithms to include elliptic curves, Calabi-Yau manifolds, and higher-genus structures. In these cases, achieving a canonical form requires a transformation involving transcendental functions (periods of the geometry). While such forms exist, a practical bottleneck arises: the numerical evaluation of these systems is hindered by the lack of closed-form expressions for the required special functions and the complexity of their analytic continuation. Existing numerical approaches either rely on series expansions (limited in multiscale regimes) or require the explicit implementation of complex special functions, which is not universally feasible.

Methodology
The authors propose a strategy to bypass the need for explicit evaluation of transcendental functions. The core observation is that the special functions appearing in the transformation to a canonical basis are not arbitrary; they originate from the original system of master integrals, which satisfy linear differential equations with rational coefficients.

Consequently, the authors demonstrate that these transcendental building blocks (e.g., elliptic periods, square roots) themselves satisfy systems of differential equations with rational coefficients. The proposed method involves:

1. Augmentation: Constructing an enlarged system of differential equations that couples the canonical differential equation for the master integrals with the auxiliary differential equations governing the special functions.
2. Rationalization: This enlarged system contains only rational coefficients. The derivatives of the special functions are computed directly from their defining rational differential equations rather than from closed-form expressions.
3. Numerical Integration: The entire system is solved numerically along a complexified path in the kinematic space to avoid singularities. The analytic continuation of the special functions is handled automatically by the integration of their differential equations, eliminating the need for ad-hoc analytic continuation algorithms.
4. Boundary Conditions: The method requires boundary values for the special functions at a non-singular point. These values can be chosen arbitrarily (e.g., set to 1) provided they are consistent with any algebraic relations used to simplify the connection matrix. The canonical master integrals are then rotated to the canonical basis using these boundary values.

Key Contributions

• Theoretical Insight: The paper establishes that the obstruction to numerically solving canonical systems with transcendental entries is only apparent. By treating the special functions as dynamical variables satisfying their own rational differential equations, the problem reduces to solving a larger, purely rational system.
• Software Implementation: The authors provide a standalone C++ package (diffeqs) designed for high-performance numerical evaluation. The software:
  • Is written in C++ and relies only on the standard Boost library, avoiding dependencies on commercial software (e.g., Mathematica) at runtime.
  • Supports arbitrary floating-point precision via the QD library and templated types.
  • Implements a modular interface allowing users to define the connection matrix, the vector field for special functions, and the integration path.
  • Includes features for complex path deformation to ensure numerical stability and analytic continuation.
• Realistic Application: The method is applied to a non-trivial physical case: the two-loop master integrals for di-photon, di-jet, and $\gamma$+jet hadro-production via a heavy-quark loop. This system involves 220 master integrals, 16 square roots, and transcendental functions related to elliptic periods.

Results
The authors tested the implementation on three specific phase-space points.

• Accuracy: In double precision, the method achieved absolute errors of $O(10^{-6})$ and $O(10^{-9})$ for the tested points. In quadruple precision, errors were reduced to $O(10^{-8})$ and $O(10^{-19})$ respectively, validating the correctness of the analytic continuation and the numerical stability of the approach.
• Performance: The evaluation time for the connection matrix was found to be efficient (sub-millisecond to a few milliseconds depending on precision and hardware). The authors note that the quadruple-precision version is approximately 10 times slower than the double-precision version regarding connection matrix evaluation.
• Comparison: The results were cross-checked against values obtained using the AMFlow package, confirming consistency. The performance was observed to be superior on Apple Silicon chips compared to AMD processors.

Significance and Claims
The paper claims that this strategy makes canonical differential equations a universally applicable tool for arbitrary Feynman integrals, including those involving elliptic and Calabi-Yau geometries. The significance lies in the ability to perform numerical phenomenology without requiring closed-form implementations of complex special functions. By propagating geometric building blocks via their own rational differential equations, the method avoids the "black box" implementation of special functions and the associated difficulties in analytic continuation.

The authors position this work as a practical step toward automating higher-order perturbative calculations for complex scattering processes. They emphasize that while the current implementation is functional without heavy optimization, further gains can be achieved through expression optimization and partial-fraction decomposition strategies. The method is presented as ready for large-scale applications in collider phenomenology, particularly for processes involving multiple scales and massive particles where traditional series expansion methods struggle.