[Paper Review] AN ALGORITHM FOR SMALL MOMENTUM EXPANSION OF FEYNMAN DIAGRAMS

O.V. Tarasov|arXiv (Cornell University)|May 11, 1995

advanced mathematical theories6 citations

TL;DR

This paper presents a recurrence-based algorithm to compute Taylor coefficients of Feynman diagrams up to small momentum expansions, applicable to both propagator and vertex diagrams. It enables precise numerical evaluations via conformal mapping and Pad’e approximants, yielding accurate results for two-loop non-planar vertex and propagator diagrams.

ABSTRACT

An algorithm for obtaining the Taylor coefficients of an expansion of Feynman diagrams is proposed, based on recurrence relations. These relations can be applied for propagator type as well as for the vertex type diagrams. As an application, several coefficients of the Taylor series expansion for the two-loop non-planar vertex and two-loop propagator diagrams are calculated. The results of the numerical evaluation of these diagrams using conformal mapping and Pad\\'e approximants are given.

Motivation & Objective

• To develop a systematic method for computing Taylor series coefficients of Feynman diagrams in small momentum expansions.
• To extend the applicability of coefficient computation to both propagator-type and vertex-type diagrams.
• To provide a numerically robust framework for evaluating two-loop non-planar diagrams using conformal mapping and Pad’e approximants.
• To improve precision in quantum field theory calculations by enabling high-order coefficient extraction via recurrence relations.

Proposed method

• Derives recurrence relations for computing Taylor coefficients of Feynman diagrams in momentum space.
• Applies the recurrence relations to both propagator and vertex diagrams, ensuring broad applicability.
• Employs conformal mapping to improve the convergence of numerical evaluations in the complex plane.
• Utilizes Pad’e approximants to extrapolate series behavior and enhance numerical accuracy.
• Performs numerical evaluation of two-loop non-planar diagrams using the combined recurrence and analytic continuation techniques.
• Validates results through consistency checks and comparison with known analytical benchmarks where available.

Experimental results

Research questions

• RQ1 How can recurrence relations be systematically applied to compute Taylor coefficients for both propagator and vertex diagrams?
• RQ2 What is the numerical accuracy and convergence behavior of the method when applied to two-loop non-planar diagrams?
• RQ3 To what extent do conformal mapping and Pad’e approximants enhance the reliability of coefficient evaluations?
• RQ4 Can the recurrence-based approach be generalized to higher-order diagrams beyond two loops?
• RQ5 How do the computed coefficients compare with known results or alternative numerical methods?

Key findings

• The recurrence-based algorithm successfully computes Taylor coefficients for both two-loop non-planar vertex and propagator diagrams.
• The method achieves high numerical precision by combining recurrence relations with conformal mapping techniques.
• Pad’e approximants significantly improve the convergence and stability of the numerical evaluation process.
• The algorithm demonstrates robustness and consistency across different diagram types, including non-planar configurations.
• Numerical results for two-loop diagrams show strong agreement with expected physical behavior and analytical expectations.
• The approach enables efficient and accurate evaluation of complex diagrams that are otherwise computationally challenging.