This paper fills an important technical gap in precision calculations for heavy‑quark pair production at hadron colliders. The authors derive and implement the missing subtraction pieces for the quark–quark (qq), different‑flavour quark–quark (qq′) and quark–gluon (qg) initial‑state channels through next‑to‑next‑to‑leading order (NNLO) in Quantum Chromodynamics (QCD). They put these results into the NNLOJET framework and show analytic and numerical checks that the pieces behave as required.

The team worked inside the antenna subtraction formalism. Antenna subtraction builds local counterterms that mimic the singular behaviour of extra radiation so that each part of the NNLO calculation can be made finite and integrated numerically. Building on earlier results for massive particles, the authors complete the NNLO treatment of the qq and qq′ channels by deriving the integrated subtraction contributions needed at the real–virtual and double‑virtual levels. They also present, for the first time, the NNLO antenna subtraction terms for the qg channel keeping full colour, meaning they keep the complete dependence on the QCD colour charges rather than using simplifying approximations.

The calculation required several new technical ingredients. In particular, the authors derived massive soft factors — functions that describe low‑energy radiation off heavy quarks — and computed convolutions that involve integrated massive antenna functions. These additions extend the existing library of integrated antenna terms so it can handle processes with massive fermions such as top quarks. The work is implemented in full within the NNLOJET code base used for parton‑level event generation.

Why this matters: top‑quark pair production is a cornerstone measurement at the Large Hadron Collider. Experimental measurements are now at the few‑percent level. To match that precision, theory predictions must include NNLO QCD effects and be fully differential so that experimental cuts and distributions can be modeled. By completing the qq, qq′ and qg channel pieces and providing validated subtraction terms for massive quarks, this paper moves the antenna subtraction programme closer to providing such precision predictions for heavy‑quark pair production.