[Paper Review] Calculation of Feynman integrals by difference equations

S. Laporta|ArXiv.org|Nov 5, 2003

advanced mathematical theories3 citations

TL;DR

This paper presents a method for calculating Feynman integrals using systems of one-variable difference equations, reducing master integrals to convergent factorial series expansions. The approach enables high-precision computations—up to 1200-digit accuracy—by solving recurrence relations derived from integration-by-parts identities, with applications to multi-loop diagrams in quantum field theory.

ABSTRACT

In this paper we describe a method of calculation of master integrals based on the solution of systems of difference equations in one variable. Various explicit examples are given, as well as the generalization to arbitrary diagrams.

Motivation & Objective

• To develop a systematic method for computing master integrals in quantum field theory using difference equations.
• To overcome the computational challenges of high-loop diagrams by reducing the problem to solving linear difference equations.
• To enable high-precision numerical evaluations of Feynman integrals, even for complex diagrams with many masses and momenta.
• To provide a robust computational framework capable of handling divergent and convergent terms in dimensional regularization.
• To facilitate cross-checks and validation through multiple independent difference equations per diagram topology.

Proposed method

• The method reduces generic Feynman integrals to master integrals via integration-by-parts identities.
• Master integrals are computed by solving linear difference equations with polynomial coefficients in the loop power variable.
• Solutions are expressed as convergent factorial series, analogous to power series for differential equations.
• The coefficients of the factorial series are determined recursively from recurrence relations derived by substituting the ansatz into the difference equation.
• A custom C program, SYS, automates the identification of master integrals, construction of difference equations, and numerical solution using arbitrary-precision arithmetic.
• The program supports expansions in $\epsilon = (4-D)/2$ up to 500 equations, with precision up to 1200 digits.

Experimental results

Research questions

• RQ1 Can master integrals in multi-loop quantum field theories be computed efficiently using one-variable difference equations?
• RQ2 How can high-precision numerical results for Feynman integrals be obtained despite the complexity of the diagrams?
• RQ3 What is the computational scaling of the method in terms of desired precision and diagram complexity?
• RQ4 Can multiple independent difference equations be derived for the same diagram to enable cross-verification?
• RQ5 To what extent can this method be generalized to arbitrary diagram topologies with arbitrary masses and momenta?

Key findings

• The method enables high-precision calculations of Feynman integrals, with results computed to 60-digit accuracy as demonstrated in a two-loop self-energy diagram.
• The number of operations required to achieve $N$-digit precision grows linearly with $N$, making computations with $N \sim 10^3$ feasible.
• The solution of the difference equation for the one-loop vacuum integral yields a convergent factorial series expansion, providing an exact representation of the integral.
• The program SYS successfully computes master integrals for vacuum, self-energy, vertex, and box diagrams up to four loops with equal masses and on-shell conditions.
• The method allows for the numerical evaluation of divergent terms in $\epsilon$-expansions with arbitrary precision, including coefficients of $\epsilon^0$, $\epsilon^1$, etc.
• Cross-checks are easily performed by deriving multiple difference equations for different lines in a diagram, enhancing reliability.