Asymptotic behavior of angular integrals in the massless limit

We investigate the small-mass asymptotics of a class of massive $d$-dimensional angular integrals. These integrals arise in a wide range of perturbative quantum field theory calculations. We derive expressions characterizing their behavior in the vicinity of the massless limit for all cases with up to two denominators. The results established in this work are applicable to phase-space calculations where an integration over virtuality including the massless limit is required.

Figure 1

Sketch illustrating the splitting of the massive one-denominator integral in Eq. (9). The figure shows the slice $x_0=1$ of Minkowski space, the blue circle indicates the intersection with the light-cone where $v^2=0$.

Figure 2

Sketch illustrating the splitting of the single-massive two-denominator integral in Eq. (14). The figure shows the slice $x_0=1$ of Minkowski space, the blue circle indicates the intersection with the light cone where $v^2=0$.

Figure 3

Sketch illustrating the splitting of the double-massive two-denominator integral in Eq. (18). The figure shows the slice $x_0=1$ of Minkowski space, the blue circle indicates the intersection with the light cone where $v^2=0$.