Local form factor subtraction for three-loop QCD corrections to electroweak production in quark-antiquark annihilation

TL;DR

The paper develops a local subtraction framework for three-loop QCD corrections to electroweak production in quark–antiquark annihilation, centring on constructing finite integrands via pointwise infrared and ultraviolet counterterms guided by Ward identities. It derives two-loop Ward identities for the qqg sector and introduces local modifications to the gluon triangle, together with shift-integrable counterterms to preserve local factorisation across collinear regions; it also reveals non-factorising loop polarisation effects arising from ghost contributions at this order. The work further details ultraviolet subtractions and the handling of shift mismatches in one- and two-loop subgraphs, laying the groundwork for extending finite-integrand methods to general three-loop electroweak amplitudes. These developments are a crucial step toward enabling precise N3LO predictions for multi-boson processes at colliders and advancing numerical approaches to high-loop QCD calculations.

Abstract

We extend a local subtraction framework to three-loop QCD corrections for the production of multiple electroweak bosons in quark-antiquark annihilation. We derive two-loop Ward identities that ensure the factorisation of most collinear singularities from the hard-scattering process in the sum over integrands. Infrared and ultraviolet singularities are removed point-by-point in loop momentum space using a minimal set of counterterms, which can be integrated analytically in terms of known master integrals. Additional counterterms eliminate non-factorising terms arising from loop momentum shifts and one-loop corrections to the gluon three-point function. We identify previously unknown non-factorising loop polarisation effects in the single-collinear regions, which pose challenges for local integrability and require further investigation. The techniques presented here are a first crucial step in formulating a systematic approach for constructing finite integrands for general electroweak amplitudes at three-loop order.

Figures (12)

• Figure 1: Modified Feynman rules used to show local collinear factorisation in the region where the external gluon with virtual momentum $k$ becomes collinear to an external antiquark with momentum $p_2$, using the diagrammatic notation of ref. Anastasiou:2022eym. The multiplicative term $2\eta_2^\nu/d_2$ associated to the collinear approximation (c.f. eq. \ref{['eq:g_k_p2']}) is not shown, and will be implicit throughout this paper.
• Figure 2: A graphical representation of the abelian Ward identity for quark lines, eq. \ref{['eq:qq_Ward']}. A ghost ending at a quark line indicates an insertion of the collinear momentum $k$ at the vertex, where the adjacent quark propagator is cancelled, denoted by a cross.
• Figure 3: A pictorial representation of the QCD Ward identity for the three-gluon vertex, eq. \ref{['eq:gg_Ward']}. Ghost lines ending at a gluon line indicate an insertion of the momentum $k$ at the vertex, where the adjacent gluon line is cancelled.
• Figure 4: A diagrammatic version of the QCD Ward identity for the four-gluon vertex, eq. \ref{['eq:4g_ward']}. For consistency with the modified ghost-gluon-gluon vertex we have introduced a dummy Lorentz index $\delta$.
• Figure 5: Loop momentum assignment for the three-loop amplitude.