Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations

TL;DR

This work develops a complete framework for angular integrals with three denominators in $d=4-2\varepsilon$, combining integration-by-parts, mass-reduction via two-point splitting, a dimensional-shift identity, and differential equations to obtain the $\varepsilon$-expansion up to order $\varepsilon$ for arbitrary integer denominators and masses. A key advance is the appearance of a term proportional to a Euclidean Gram determinant in the $\varepsilon$-expansion, with its coefficient expressed as a sum of Clausen functions, linking Euclidean, spherical, and hyperbolic geometry. The pole and finite parts can be reconstructed from known two-denominator integrals, while the genuine three-denominator contribution at order $\varepsilon$ is encoded in the Clausen-function geometry; results have direct applications to multi-particle phase-space calculations. The paper provides explicit master-integral results (massless, single-, double-, and triple-massive) and a Mathematica implementation, and confirms parallel independent work with consistent numerical checks.

Abstract

Angular integrals arise in a wide range of perturbative quantum field theory calculations. In this work we investigate angular integrals with three denominators in $d=4-2\varepsilon$ dimensions. We derive integration-by-parts relations for this class of integrals, leading to explicit recursion relations and a reduction to a small set of master integrals. Using a differential equation approach we establish results up to order $\varepsilon$ for general integer exponents and masses. Here, reduction identities for the number of masses, known results for two-denominator integrals, and a general dimensional-shift identity for angular integrals considerably reduce the required amount of work. For the first time we find for angular integrals a term contributing proportional to a Euclidean Gram determinant in the $\varepsilon$-expansion. This coefficient is expressed as a sum of Clausen functions with intriguing connections to Euclidean, spherical, and hyperbolic geometry. The results of this manuscript are applicable to phase-space calculations with multiple observed final-state particles.

Figures (4)

• Figure 1: This flowchart provides an overview of the calculation of the $\varepsilon$-expansion of the general three-denominator angular integral $I_{j_1j_2 j_3}^{(m)}$. In a first step, recursion relations (see app. \ref{['app: Recursion relations']}) derived from IBP relations (see sec. \ref{['sec: IBP relations']}) are used for a reduction to the master integral $I_{1,1,1}^{(m)}$. In a second step, the double- and triple-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. \ref{['sec: Reduction of masses']}). 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 pole and finite part in terms of known two-denominator integrals (see sec. \ref{['sec: Dimensional shift identity']}). In a final step, the order $\varepsilon$ contribution is calculated by applying the method of differential equations -- requiring suitable differential operators for angular integrals (see sec. \ref{['sec: Differential operators for angular integrals']}) and again making use of the recursion relations -- to the massless and single-massive master integral in $d=6-2\varepsilon$ dimensions (see sec. \ref{['sec: Differential equations']} and \ref{['sec: Integration for master integrals']}). Results are found in sec. \ref{['sec: Results']}. Image created with JaxoDraw Binosi:2003.
• Figure 2: Illustration of the initial condition and the integration path for the massless three-denominator angular integral depending on vectors $v_{1,2,3}$. The plot shows the slice $x_0=1$ of $d$-dimensional Minkowski space. The blue unit sphere depicts the intersection with the light-cone, where $v^2=0$. The symmetric configuration showcased by the solid arrows is used as the boundary value. In this configuration $I_{1,1,1}^{(0)}(1,1,1)=2\pi\mathrm{C}$, where $\mathrm{C}$ denotes Catalan's constant. The dashed black lines showcase the integration path. First, we rotate $v_1$ along path $(a)$, which is in the direction of $\partial/\partial v_{12}$. The axis of rotation is along $v_3$. Second, we rotate $v_1$ along path $(b)$, which is in the direction of $\partial/\partial v_{13}$. The axis of rotation is along $v_2$. Third, we rotate $v_2$ along path $(c)$, which is in the direction of $\partial/\partial v_{23}$, reaching the final configuration with general $v_{12}$, $v_{13}$, $v_{23}$. The axis of rotation is the rotated $v_1$ shown in dashed red. Note that the volume of the tetrahedron spanned by the vectors $v_{1,2,3}$ in this subspace is proportional to $\sqrt{Y_{123}^{(0)}}$ .
• Figure 3: Visualization of the geometrical quantities appearing in the result \ref{['eq: Geometrical interpretation of massless integral']} for the massless integral in six dimensions. Analogous to figure \ref{['fig:Boundary value']}, the picture shows the slice $x_0=1$ of six-dimensional Minkowski space, specifically the three-dimensional subspace defined by $\vec{v}_{1}$, $\vec{v}_{2}$, $\vec{v}_{3}$. These vectors span a tetrahedron with volume $\mathbb{V}(\vec{v}_1,\vec{v}_2,\vec{v}_3)$ shaded in gray. Note that the factor of $6$ in eq. \ref{['eq: Geometrical interpretation of massless integral']} is not from $d=6$ but originates from eq. \ref{['eq: Y123 geometrically']}. The dihedral angle between the faces that meet at $\vec{v}_i$ is denoted by $\phi_i$, i.e. e.g. $\phi_1$ is the angle between the face spanned by $(\vec{v}_1,\vec{v}_2)$ and $(\vec{v}_1,\vec{v}_3)$. They contribute with $\mathrm{Cl}_2(2\phi_i)$ in eq. \ref{['eq: Geometrical interpretation of massless integral']}. The $\Omega$ are solid angles, i.e. areas of triangles on the the unit sphere. The red area $\Omega_{\bar{1}23}$ has corners at $(-\vec{v}_1,\vec{v}_2,\vec{v}_3)$, the yellow area $\Omega_{1\bar{2}3}$ has corners at $(\vec{v}_1,-\vec{v}_2,\vec{v}_3)$, and the blue area $\Omega_{\bar{1}23}$ has corners at $(\vec{v}_1,\vec{v}_2,-\vec{v}_3)$. They contribute with $\mathrm{Cl}_2(\Omega)$ in eq. \ref{['eq: Geometrical interpretation of massless integral']}. The black area $\Omega_{123}$ is the solid angle covered by $(\vec{v}_1,\vec{v}_2,\vec{v}_3)$ and contributes negatively with $-\mathrm{Cl}_2(\Omega)$ in eq. \ref{['eq: Geometrical interpretation of massless integral']}.
• Figure 4: The left panel shows a plot of the Clausen function on the interval $[0,2\pi]$. The right panel displays the combination $\mathrm{Cl}_2(2\arctan x)$ as it appears in the results from section \ref{['sec: Results']}.