[Paper Review] New results for 5-point functions

J. Gluza, T. Riemann|DESY (CERN, DESY, Fermilab, IHEP, and SLAC)|Dec 18, 2007

Particle physics theoretical and experimental studies8 references3 citations

TL;DR

This paper investigates the use of Mellin-Barnes (MB) representations for evaluating one-loop 5-point functions in QED, particularly for radiative corrections in Bhabha scattering at NNLO. It demonstrates that while MB methods do not improve numerical evaluation of finite parts, they provide a systematic and efficient way to isolate and compute infrared (IR) singularities, showing that the IR structure of tensor 5-point functions is fully reducible to that of scalar functions via inverse binomial sums and polylogarithmic functions.

ABSTRACT

Bhabha scattering is one of the processes at the ILC where high precision data will be expected. The complete NNLO corrections include radiative loop corrections, with contributions from Feynman diagrams with five external legs. We take these diagrams as an example and discuss several features of the evaluation of pentagon diagrams. The tensor functions are usually reduced to simpler scalar functions. Here we study, as an alternative, the application of Mellin-Barnes representations to 5-point functions. There is no evidence for an improved numerical evaluation of their finite, physical parts. However, the approach gives interesting insights into the treatment of the IR-singularities.

Motivation & Objective

• To explore the applicability of Mellin-Barnes representations for computing one-loop 5-point functions in QED.
• To assess whether MB techniques improve numerical evaluation of the finite parts of scalar, vector, and tensor 5-point functions.
• To systematically treat infrared (IR) singularities in 5-point functions arising from radiative corrections in Bhabha scattering.
• To determine if the IR structure of tensor 5-point functions can be reduced to that of scalar functions.

Proposed method

• The paper employs Mellin-Barnes representations to express massive 5-point Feynman integrals by introducing seven successive MB integrals for each term in the F-form of the Feynman parameter integral.
• It uses the AMBRE.m package in Mathematica to derive the full MB representation and applies Barnes’ first lemma to reduce the integral to a five-dimensional form.
• The method involves analytical continuation in ǫ (dimensional regularization) using the MB.m package to obtain finite, multi-dimensional MB integrals.
• The residue theorem is applied to evaluate the integrals by closing contours to the left, extracting poles in ǫ that correspond to IR divergences.
• The IR-divergent parts are isolated and expressed in terms of inverse binomial sums and polylogarithmic functions via known summation identities.
• The tensor structure is handled by generalizing the numerator A(q) to include qμ, qμqν, etc., with subsequent x-integrations yielding vector and tensor integrals.

Experimental results

Research questions

• RQ1 Can Mellin-Barnes representations provide a numerically superior method for evaluating the finite parts of 5-point functions compared to standard reduction techniques?
• RQ2 How do the infrared singularities in 5-point functions emerge and can they be systematically isolated using MB techniques?
• RQ3 Is the IR structure of tensor 5-point functions fully reducible to that of scalar 5-point functions?
• RQ4 Can inverse binomial sums and polylogarithmic functions be used to express the IR-divergent parts of 5-point functions in a closed analytical form?

Key findings

• The Mellin-Barnes approach does not improve the numerical evaluation of the finite parts of 5-point functions, as no evidence for enhanced convergence or stability was found.
• The leading and non-leading infrared singularities in 5-point functions are systematically extracted via MB integrals and expressed as inverse binomial sums.
• The IR-divergent parts of scalar 5-point functions are given by I⁻¹(Vi)/ǫ + I⁰(Vi), with explicit expressions involving harmonic sums and polylogarithms.
• For vector and tensor 5-point functions, the IR-divergent parts are proportional to the chords Q₂ and Q₅ of massless internal lines and are fully reducible to the scalar case.
• The leading IR singularity I⁻¹(Vi)/ǫ is expressed as a series ∑ₙ₌₀^∞ tⁿ (2n choose n) / (2n+1), which can be summed in terms of logarithmic and polylogarithmic functions using known identities.
• The final IR structure of tensor functions matches that of scalar functions, confirming that the tensor structure is entirely determined by the scalar IR behavior.