Geometric IR subtraction for real radiation

TL;DR

The paper tackles infrared divergences in real radiation at NLO/NNLO by introducing a geometric, local slicing-subtraction framework that leverages soft/collinear factorisation and normal coordinates to produce universal, analytically integrable counter-terms. It demonstrates a concrete path to a fully local integrand subtraction method and provides general pole formulas for final-state radiation in Yang–Mills theory with arbitrary multiplicities, expressed through $Γ$-functions and ${}_pF_q$ hypergeometric functions. Key contributions include explicit counter-terms at NLO and NNLO, a master-integral reduction approach via IBP and reverse unitarity, and a nontrivial NNLO check in Higgs decays to gluons. The approach promises clearer, more locality-preserving subtraction terms with favorable analytic structure, potentially enabling more efficient and systematic higher-order QCD calculations for complex final states.

Abstract

A scheme is proposed for the subtraction of soft and collinear divergences present in massless real emission phase space integrals. The scheme is based on a local slicing procedure which utilises the soft and collinear factorisation properties of amplitudes to produce universal counter-terms whose analytic integration is relatively simple. We propose that this scheme can be promoted to a fully local subtraction method. As a first application the scheme is applied to establish a general pole formula for final state real radiation at NLO and NNLO in Yang Mills theory for arbitrary multiplicities. All required counter-terms are evaluated to all orders in the dimensional regulator in terms of $Γ$ - and ${}_pF_q$ hypergeometric - functions. As a proof of principle the poles in the dimensional regulator of the $H\to gggg$ double real emission contribution to the $H\to gg$ decay rate are reproduced.

Figures (7)

• Figure 1: The grey triangular surface $Q^2=s_{12}+s_{13}+s_{23}$ represents the physical phase space. The fat blue lines labelled by $C_{12}$ and $C_{23}$ show the locations of collinear singularities, while the small red circle labelled by $S_2$ shows the location of the soft singularity.
• Figure 2: The triangular surface $Q^2=s_{12}+s_{13}+s_{23}$ is split into singular and finite regions. The red region is the soft region $S_2$. Collinear regions $C_{12}$ and $C_{23}$ are indicated in blue. The finite region $F$ is shown in grey. The soft-collinear overlap is just visible where the blue bands intersect with the red triangular region.
• Figure 3: This figure shows $\Delta I(\lambda)$ on the left and also separately the $\epsilon^0$ coefficients of $I_F,I_{\mathrm{Singular}}$ and their sum on the right in the slicing method. In both figures $I_F$ is evaluated numerically with the CUBA implementation of the Vegas algorithm using $10^8$ points for each value of $\lambda$.
• Figure 4: This figure shows $\Delta I(\lambda)$ on the left and also separately the $\epsilon^0$ coefficients of $I_F,I_{\mathrm{Singular}}$ and their sum on the right in the subtraction method. In both figures $I_F$ is evaluated numerically with the CUBA implementation of the Vegas algorithm using $10^8$ points for each value of $\lambda$.
• Figure 5: The picture illustrates the identities $S_A\cap S_B\subset S_{AB}$ (right) and $C_{AB}\cap C_{AD}\subset C_{ABD}$ (left).