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Two-loop diagrams in non-relativistic QCD with elliptics

B. A. Kniehl, A. V. Kotikov, A. I Onishchenko, O. L. Veretin

TL;DR

This work analyzes two-loop NRQCD-relevant diagrams with elliptic structure arising from two masses $m$ and $M$ under threshold kinematics. It develops a master-parameter approach that reduces two-loop diagrams to integrals over one-loop building blocks with an $s$-dependent effective mass, and provides three distinct representations: a mass-ratio expansion in $x= m^2/M^2$, integral representations, and generalized hypergeometric function expressions, valid to ${\rm O}(\varepsilon)$ in $d=4-2\varepsilon$. In the equal-mass limit, the results are recast in terms of elliptic sums and elliptic integrals, revealing the elliptic nature of these diagrams beyond polylogarithms. The findings deliver practical analytic tools for NRQCD matching, parapositronium decays, and near-threshold $t\bar t$ production, and establish a foundation for further exploration of elliptic polylogarithms and related special functions in multi-loop computations.

Abstract

In this paper we consider two-loop two-, three- and four-point diagrams with elliptic structure in the case of two different masses $m$ and $M$. The latter diagrams generally arise within NRQCD matching procedures and are relevant for parapositronium decay and top pair production at threshold. We present the obtained results in several different representations: series solution with binomial coefficients, integral representation and representation in terms of generalized hypergeometric functions. The results are valid up to $\mathcal{O}(\varepsilon)$ terms in $d=4-2\varepsilon$ space-time dimensions. In the limit of equal masses $m=M$ the obtained results are written in terms of elliptic constants with explicit series representation.

Two-loop diagrams in non-relativistic QCD with elliptics

TL;DR

This work analyzes two-loop NRQCD-relevant diagrams with elliptic structure arising from two masses and under threshold kinematics. It develops a master-parameter approach that reduces two-loop diagrams to integrals over one-loop building blocks with an -dependent effective mass, and provides three distinct representations: a mass-ratio expansion in , integral representations, and generalized hypergeometric function expressions, valid to in . In the equal-mass limit, the results are recast in terms of elliptic sums and elliptic integrals, revealing the elliptic nature of these diagrams beyond polylogarithms. The findings deliver practical analytic tools for NRQCD matching, parapositronium decays, and near-threshold production, and establish a foundation for further exploration of elliptic polylogarithms and related special functions in multi-loop computations.

Abstract

In this paper we consider two-loop two-, three- and four-point diagrams with elliptic structure in the case of two different masses and . The latter diagrams generally arise within NRQCD matching procedures and are relevant for parapositronium decay and top pair production at threshold. We present the obtained results in several different representations: series solution with binomial coefficients, integral representation and representation in terms of generalized hypergeometric functions. The results are valid up to terms in space-time dimensions. In the limit of equal masses the obtained results are written in terms of elliptic constants with explicit series representation.

Paper Structure

This paper contains 13 sections, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Relation between one- and two-loop diagrams
  3. One-loop integrals
  4. Two-point case
  5. Three-point case
  6. Four-point case
  7. Two-loop diagrams
  8. Series representations
  9. Integral representations
  10. Representations in terms of generalized hypergeometric functions
  11. Equal-mass case
  12. Conclusions
  13. Frobenius solution for a system of differential equations

Figures (2)

  • Figure 1: (a) Two-loop nonplanar diagram contributing to NNLO matching in nonrelativistic field theory and (b) diagram obtained from it by contracting the massless-propagator lines. An additional mass $M$ is introduced to simplify the actual calculations (see discussion in the text).
  • Figure 2: Typical one-loop diagrams resulting from the IBP reduction of the integral in Eq. (\ref{['Iabcd']}). The thin and thick solid lines represent scalar propagators with masses $m$ and $M$, respectively. In the case of the box diagram (c), also the line numbering is indicated.