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The massless two-loop two-point function

Isabella Bierenbaum, Stefan Weinzierl

TL;DR

This paper develops a convolution-based method to evaluate the massless master two-loop two-point integral with arbitrary propagator powers, enabling the Laurent expansion in $\varepsilon$ to arbitrary order. By factorizing the two-loop integral as a product of two primitive one-loop integrals through Mellin-Barnes representations, the authors show that the $\varepsilon$-expansion can be expressed entirely in terms of rational numbers and multiple zeta values. They prove, and demonstrate, that multiple zeta values suffice for the expansion, and discuss generalization to higher-loop topologies. The authors implement the approach computationally, achieving efficient symbolic expansion up to high weights and illustrating relevance for higher-order QCD calculations such as $\alpha_s^4$ corrections.

Abstract

We consider the massless two-loop two-point function with arbitrary powers of the propagators and derive a representation, from which we can obtain the Laurent expansion to any desired order in the dimensional regularization parameter eps. As a side product, we show that in the Laurent expansion of the two-loop integral only rational numbers and multiple zeta values occur. Our method of calculation obtains the two-loop integral as a convolution product of two primitive one-loop integrals. We comment on the generalization of this product structure to higher loop integrals.

The massless two-loop two-point function

TL;DR

This paper develops a convolution-based method to evaluate the massless master two-loop two-point integral with arbitrary propagator powers, enabling the Laurent expansion in to arbitrary order. By factorizing the two-loop integral as a product of two primitive one-loop integrals through Mellin-Barnes representations, the authors show that the -expansion can be expressed entirely in terms of rational numbers and multiple zeta values. They prove, and demonstrate, that multiple zeta values suffice for the expansion, and discuss generalization to higher-loop topologies. The authors implement the approach computationally, achieving efficient symbolic expansion up to high weights and illustrating relevance for higher-order QCD calculations such as corrections.

Abstract

We consider the massless two-loop two-point function with arbitrary powers of the propagators and derive a representation, from which we can obtain the Laurent expansion to any desired order in the dimensional regularization parameter eps. As a side product, we show that in the Laurent expansion of the two-loop integral only rational numbers and multiple zeta values occur. Our method of calculation obtains the two-loop integral as a convolution product of two primitive one-loop integrals. We comment on the generalization of this product structure to higher loop integrals.

Paper Structure

This paper contains 7 sections, 1 theorem, 76 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Review of known results for the two-loop integral
  3. The product structure of the two-loop integral
  4. Evaluation of the two-loop integral
  5. Results, checks and performance
  6. Summary and conclusions
  7. Integral transformations

Key Result

Theorem 4.1

Multiple zeta values are sufficient for the Laurent expansion of the two-loop integral $\hat{I}^{(2,5)}(m-\varepsilon,\nu_1,\nu_2,\nu_3,\nu_4,\nu_5)$, if all powers of the propagators are of the form $\nu_j=n_j+a_j\varepsilon$, where the $n_j$ are positive integers and the $a_j$ are non-negative rea

Figures (2)

  • Figure 1: Factorization in terms of diagrams: The two-loop two-point function on the l.h.s. is equal to the insertion of a one-loop three-point function into a one-loop two-point function. The insertion occurs at the shaded vertex.
  • Figure 2: Factorization at three loops: The first two topologies factorize into one-loop diagrams, whereas the last topology factorizes into a one-loop diagram and a two-loop diagram. A graph on the right side of the product operator $\star$ is inserted into the shaded vertex of the graph to the left of the product operator.

Theorems & Definitions (1)

  • Theorem 4.1