Infrared singularities of multileg amplitudes with a massive particle at three loops

TL;DR

The paper tackles the problem of determining the complete three-loop soft anomalous dimension for multileg QCD amplitudes with a single massive colored particle and any number of massless partons. It introduces a regulated four-Wilson-line correlator and employs the Method of Regions to perform a lightcone expansion, solving the resulting region integrals with differential equations to obtain a uniform weight-five generalized polylogarithm representation. The central achievement is the analytic computation of the quadrupole function ${\cal F}_{1,3}^{(3)}$, expressed as an antisymmetric combination of weight-five GPLs in variables $x$, $z$, and $\bar{z}$, and validated against the massless limit and collinear factorization constraints across all three channels. This result completes the three-loop infrared structure relevant for heavy-quark processes, enabling precision resummations and laying the groundwork for extensions to more massive legs or higher-loop orders.

Abstract

We determine the complete three-loop QCD soft anomalous dimension for multileg amplitudes involving a single massive coloured particle and any number of massless ones. This is achieved by applying a novel strategy based on a lightcone expansion of correlators of semi-infinite Wilson lines using the method of regions. The resulting region integrals depend exclusively on rescaling-invariant ratios that remain finite in the limit. We evaluate these integrals using differential equation techniques. The result is written in terms of uniform weight five generalised polylogarithms of a twelve letter alphabet in three variables, and is compatible with the massless limit as well as with two- and three-particle collinear factorization.

Figures (5)

• Figure 1: Representative three-loop diagrams contributing to the quadrupole structure. The thick line corresponds to the timelike Wilson line with velocity $\beta_Q$, while the thin lines to Wilson lines with nearly lightlike velocities. The two leftmost diagrams depict connected webs, $W_{1111}$, the middle one is a representative diagram of the $W_{1112}$ web and the two rightmost ones belong to multiple gluon-exchange webs, $W_{1122}$ and $W_{1113}$, respectively.
• Figure 2: The diagrams with two three-gluon vertices contributing to the three channels $P$. From left to right, the diagrams are associated with the colour factor $\mathbf{T}_{uv;w Q}$ where the triplet of indices $(u,v;w)$ is in $P$.
• Figure 3: Permutation relations between the three functions in the sum over $P=\{(i,k;j),(j,k;i),(i,j;k)\}$ in Eq. \ref{['cstructure']}. The two functions at the bottom of the triangle are related by a sequence of two permutations, where the role $T_{i\leftrightarrow j}$ is to generate an extra minus sign.
• Figure 4: Permutation relations between Euclidean domains.
• Figure 5: Permutation relations between physical domains.