On multi-propagator angular integrals

TL;DR

The paper develops a loop-inspired toolkit for multi-propagator angular integrals, introducing a Lee-Pomeransky–like Euler representation, recursion-based IBP reductions, and a dimensional-shift/recurrence framework that collapses large-scale master sets to a small number of branch integrals. It yields explicit new results for the four-denominator case with arbitrary masses and an all-order $\\varepsilon$-expansion for the massless three-denominator integral, including soft-log resummation. The methods expose a decomposition into branch integrals that suppresses the original high scale count from $n(n-1)/2+m$ down to $n+1$, enabling systematic, all-order analyses via differential equations and polylogarithmic structures. The work strengthens the bridge between loop-integral techniques and phase-space integrals, with potential for practical use in high-multiplicity QCD calculations and related phenomenology.

Abstract

We study multi-propagator angular integrals, a class of phase-space integrals relevant to processes with multiple observed final states and a test-bed for transferring loop-integral technology to phase space integrals without reversed unitarity. We present an Euler integral representation similar to Lee-Pomeransky representation and explicitly describe a recursive IBP reduction and dimensional shift relations for the general case of $n$ denominators. On the level of master integrals, applying a differential equation approach, we explicitly calculate the previously unknown angular integrals with four denominators for any number of masses to finite order in $\varepsilon$. Extending the idea of dimensional recurrence, we explore the decomposition of angular integrals into branch integrals reducing the number of scales in the master integrals from $(n+1)n/2$ to $n+1$. To showcase the potential of this method, we calculate the massless three denominator integral to establish all-order results in $\varepsilon$ including a resummation of soft logarithms.

Figures (2)

• Figure 1: This flowchart provides an overview of the calculation of the $\varepsilon$-expansion of the general four-denominator angular integral $I_{j_1j_2 j_3 j_4}^{(m)}$. In a first step, recursion relations derived from IBP relations (see sec. \ref{['sec: IBP relations']}) are used for a reduction to the master integral $I_{1111}^{(m)}$. In a second step, the double-, triple-, and quadruple-massive integrals are expressed in terms of massless and single-massive ones through mass reduction formulae derived from the two-point splitting lemma (see sec. 4 of Haug:2024yfi for details). In a third step, a combination of a dimensional shift identity, relating integrals in $d$ and $d+2$ dimensions, with the recursion relations allows for the determination of the pole part and some finite contributions in terms of known three denominator integrals (see sec. \ref{['sec: Dimensional shift relation']}). In a final step, the order $\varepsilon^0$ contribution is calculated by applying the method of differential equations --- requiring suitable differential operators for angular integrals (see sec. 5 of Haug:2024yfi) and again making use of the recursion relations --- to the massless and single-massive master integral in $d=6-2\varepsilon$ dimensions. Graphic created with JaxoDraw Binosi:2003.
• Figure 2: Illustration of the decomposition of $I_{111}$ into branches, each associated with a permutation $\sigma\in S_3$. The horizontal $\oplus$-sum over branch integrals $\mathcal{B}_3(v_{\sigma(1)},v_{\sigma(2)},v_{\sigma(3)})$ is performed according to eq. \ref{['eq: Branch splitting']} with prefactors $x_{2,2}\!\left(v_{\sigma(1)},v_{\sigma(2)}\right)x_{3,3}\!\left(v_{\sigma(1)},v_{\sigma(2)},v_{\sigma(3)}\right)$ inherited from each branch. Vertically growing the branches starting from the 'roots' $\mathcal{B}_1(v_i)$ along the arrows by including a new vector $v_j$ is done with eq. \ref{['eq: Branch iteration']}. Here, the construction of $\mathcal{B}_{i+1}$ involves summation over $\mathcal{B}_{i}$ in shifted dimensions.