[Paper Review] Tables of convolution integrals

A. B. Arbuzov|ArXiv.org|Apr 7, 2003

Mathematical functions and polynomials1 references3 citations

TL;DR

This paper presents an analytical method for computing convolution integrals of singular and non-singular functions arising in perturbative quantum field theory, particularly in renormalization group applications. Using $Δ$-regularization and the plus prescription, it derives explicit formulas for convolutions involving logarithmic and polylogarithmic functions, with results expressed in terms of dilogarithms, trilogarithms, and generalized polylogarithms, enabling precise analytical and numerical calculations in high-energy physics.

ABSTRACT

An analytical approach to convolution of functions, which appear in perturbative calculations, is discussed. An extended list of integrals is presented.

Motivation & Objective

• To develop a direct analytical approach for computing convolution integrals that commonly arise in perturbative quantum field theory.
• To provide a systematic table of convolution integrals involving singular functions with poles at $x=1$ or $x=z$, regularized via the plus prescription.
• To facilitate analytical and numerical computations in processes such as parton evolution and factorization theorems by avoiding Mellin transform techniques.
• To establish a consistent framework using $Δ$-regularization that ensures cancellation of divergences across contributions.
• To express results in terms of standard special functions like polylogarithms and generalized polylogarithms for broad applicability.

Proposed method

• The convolution of two functions $f(x)$ and $g(y)$ is defined via the integral $[f \otimes g](z) = \int_z^1 \frac{dx}{x} f(x) g(z/x)$, valid for $0 \leq z \leq 1$.
• Singular integrals with poles at $x=1$ or $x=z$ are regularized using a small parameter $\Delta \ll 1$, with the $Δ$-regularization scheme replacing divergent integrals by finite expressions.
• The plus prescription is implemented via the decomposition $[f]_+ = \lim_{\Delta \to 0} \left[ \delta\text{-part} + \Theta\text{-part} \right]$, where the $\delta$-part is the integral over $[0, 1-\Delta]$ and the $\Theta$-part is the function restricted to $x < 1$.
• Convolution of two singular functions is expressed as a sum of three terms: a finite integral over $[z/(1-\Delta), 1-\Delta]$, and two terms involving $f_\Delta g_\Theta(z)$ and $f_\Theta(z) g_\Delta$, ensuring cancellation of $\Delta$-dependence.
• The method relies on known definite integrals of logarithmic and rational functions, with results expressed in terms of polylogarithms $\mathrm{Li}_n(z)$ and generalized polylogarithms $\mathrm{S}_{m,n}(z)$.
• Relations between polylogarithms at $z$ and $1-z$ are used to express results in a standard form, enabling simplification and numerical evaluation.

Experimental results

Research questions

• RQ1 How can convolution integrals involving singular functions with poles at $x=1$ or $x=z$ be systematically evaluated in perturbative quantum field theory?
• RQ2 What is an effective analytical method for computing convolutions without relying on Mellin transforms, particularly for functions with logarithmic and polylogarithmic singularities?
• RQ3 How can divergent integrals be regularized such that the $\Delta$-dependence cancels across contributions, ensuring finite physical results?
• RQ4 What are the explicit analytical expressions for convolutions of functions like $\ln^n(1-x)/(1-x)$, $\ln^k x / (x-z)$, and $\mathrm{Li}_2(1-x)/(x-z)$?
• RQ5 How can results be expressed in terms of standard special functions like $\mathrm{Li}_n(z)$ and $\mathrm{S}_{m,n}(z)$ for practical use in high-energy physics calculations?

Key findings

• The paper derives explicit analytical expressions for the convolution of singular functions, such as $\int_{z/(1-\Delta)}^{1-\Delta} \frac{\ln^n(1-x)}{1-x} dx = \frac{1}{n+1} \left[ \ln^{n+1}(1-z) - \ln^{n+1}\Delta \right]$.
• For the divergent kernel $1/(x-z)$, the regularized integral yields $\int \frac{dx}{x-z} = -\ln\Delta + \ln(1-z) - \ln z$.
• Convolution of $\ln x / (x-z)$ results in $-\ln\Delta \ln z + \mathrm{Li}_2(1-z) + \ln(1-z)\ln z - \frac{1}{2}\ln^2 z$.
• Higher-order logarithmic kernels, such as $\ln^2 x / (x-z)$, produce results involving $\mathrm{S}_{1,2}(1-z)$, $\mathrm{Li}_2(1-z)\ln z$, and $\ln^3 z$ terms.
• The convolution of $\mathrm{Li}_2(1-x)/(x-z)$ is expressed as $\mathrm{Li}_2(1-z)(\ln(1-z) - \ln z - \ln\Delta) - \mathrm{S}_{1,2}(1-z) - \mathrm{Li}_3(1-z)$.
• The paper provides a complete set of relations between polylogarithms at $z$ and $1-z$, enabling transformation of results into a standard form for further use in physical calculations.