Integrated subtraction terms and finite remainders for arbitrary processes with massless partons at colliders in the nested soft-collinear subtraction scheme

TL;DR

The paper addresses infrared divergences in NNLO QCD for arbitrary massless parton processes at hadron and lepton colliders by advancing the nested soft-collinear subtraction scheme. It delivers a general derivation of the integrated subtraction terms and finite remainders, proving analytic cancellation of infrared poles in a process-independent manner and assembling a fully local, analytic NNLO framework. The master NNLO formula splits contributions into fully-resolved, single-unresolved, and double-unresolved sectors, with new features including complete initial-state clustering sums, full collinear anomalous dimensions, and integrated triple-collinear counterterms. The results enable automation of NNLO computations and are adaptable to lepton colliders, representing a significant step toward universal, high-precision QCD predictions for complex collider processes.

Abstract

We present integrated subtraction terms and finite remainders for arbitrary processes with massless partons at hadron and lepton colliders in the context of the nested soft-collinear subtraction scheme. These results provide the very last ingredients needed to make this scheme a fully local, analytic and process-independent framework for treating infrared singularities at next-to-next-to-leading order in perturbative QCD. The explicit infrared finiteness of all required contributions, as well as their process-independence, puts these results on par with subtraction schemes developed for next-to-leading order computations and opens up a clear path towards the automation of next-to-next-to-leading order computations in QCD.

Figures (2)

• Figure 1: Illustration of soft and collinear factorizations for real emissions and of virtual contributions. The unresolved parton $\mathfrak{m}$, shown in red, is either soft or collinear. The first pane illustrates the single-soft limit $E_\mathfrak{m} \to 0$, as defined in Eq. \ref{['eq_soft_limit_review']}. The second pane shows the final-state collinear splitting $[i\mathfrak{m}]^* \to i(z) + \mathfrak{m}(1-z)$, with $z=1-E_\mathfrak{m}/E_{[i\mathfrak{m}]}$, described in Eq. \ref{['eq_coll_limit_FS_review']}. The third pane depicts the initial-state collinear splitting $a \to [a\bar{\mathfrak{m}}]^* + \mathfrak{m}$, where $z=1-E_\mathfrak{m}/E_a$, from Eq. \ref{['eq_coll_limit_IS_review']}. Finally, the fourth pane represents the virtual contributions described in point vii).
• Figure 2: Examples of contributions to initial- and final-state collinear limits.