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  • Open Access

Analytical Soft Functions for Heavy-Quark Final States at Hadron Colliders

Ze Long Liu* and Pier Francesco Monni†

  • Theoretical Physics Division, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China and CERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland

  • *Contact author: liuzelong@ihep.ac.cn
  • †Contact author: pier.monni@cern.ch
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Phys. Rev. Lett. 136, 231902 – Published 10 June, 2026

DOI: https://doi.org/10.1103/2ksm-243x

Abstract

We present the first computation of the complete two-loop, fully differential soft function describing the production of a heavy-quark pair in association with a color-singlet system at hadron colliders. This result constitutes one of the most complex soft functions known to date and it is obtained in closed analytic form for generic multidimensional kinematics. This allows us to obtain novel analytic results for the transverse-momentum-dependent and threshold soft functions in this class of processes. We further obtain a decomposition of the soft function into dipole and tripole color correlators, thereby supplying essential building blocks for processes involving a heavy-quark pair produced together with additional light jets at both hadron and lepton colliders. These results represent a key ingredient for advancing precision predictions for heavy-quark physics at the LHC.

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Phys. Rev. Lett. 136, 231902– Published 10 June, 2026
  • Received 23 December 2025
  • Revised 19 March 2026
  • Accepted 20 April 2026
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