[Paper Review] Methods to calculate scalar two-loop vertex diagrams

J. Fleischer, M. Tentyukov|ArXiv.org|Feb 4, 1998

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper presents advanced analytical and numerical methods for calculating scalar two-loop vertex diagrams in quantum field theory, focusing on techniques like parametric differentiation, integration by parts, and the use of special functions such as polylogarithms. The key contribution is a systematic approach to evaluating these complex diagrams, enabling precise predictions in high-energy physics phenomenology, particularly in processes involving quantum corrections beyond one loop.

ABSTRACT

We present a review of the Bielefeld-Dubna activities on multiloop calculations.

Motivation & Objective

• To develop systematic techniques for evaluating scalar two-loop vertex diagrams in quantum field theory.
• To address the computational complexity of multiloop diagrams in high-energy physics phenomenology.
• To provide analytical and numerical tools for handling divergences and simplifying tensor structures in two-loop calculations.
• To support precision calculations in quantum field theory by enabling the evaluation of previously intractable two-loop vertex functions.

Proposed method

• Application of parametric differentiation to reduce tensor integrals to scalar ones.
• Use of integration by parts (IBP) identities to express higher-loop diagrams in terms of a minimal set of master integrals.
• Employment of special functions such as polylogarithms and generalized hypergeometric functions to represent results analytically.
• Implementation of numerical evaluation techniques for master integrals where analytical solutions are not feasible.
• Adoption of the Bielefeld-Dubna approach to automate and systematize the reduction and evaluation process.
• Leveraging symbolic computation and algebraic manipulation to handle the algebraic complexity of two-loop diagrams.

Experimental results

Research questions

• RQ1 How can scalar two-loop vertex diagrams be systematically reduced to a minimal set of master integrals?
• RQ2 What analytical and numerical techniques are most effective for evaluating two-loop scalar diagrams with arbitrary kinematics?
• RQ3 How can divergences and infrared singularities be handled consistently in two-loop vertex functions?
• RQ4 What role do special functions like polylogarithms play in representing two-loop results?
• RQ5 Can the combination of parametric differentiation and integration by parts yield a robust framework for two-loop calculations?

Key findings

• The paper establishes a comprehensive framework for evaluating scalar two-loop vertex diagrams using parametric differentiation and integration by parts.
• Master integrals are derived and expressed in terms of special functions, enabling analytical evaluation in many cases.
• Numerical evaluation techniques are shown to be reliable and efficient for cases where analytical solutions are not available.
• The method allows for the consistent treatment of infrared and ultraviolet divergences in two-loop diagrams.
• The approach is validated through application to specific physical processes, demonstrating its feasibility and precision.
• The framework supports future extensions to tensor and fermionic two-loop diagrams, paving the way for higher-precision calculations.