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Two-loop leading-colour QCD helicity amplitudes for Higgs boson production in association with a bottom-quark pair at the LHC

Simon Badger, Heribertus Bayu Hartanto, Jakub Kryś, Simone Zoia

TL;DR

This work delivers the first complete analytic two-loop QCD helicity amplitudes for Higgs production in association with a bottom-quark pair in the leading-color, five-flavour framework, using massless bottom quarks with finite Yukawa coupling. The authors combine a Feynman-diagram approach with finite-field sampling, IBP reduction to uniform-weight master integrals, and a novel finite-remainder basis to express results as analytic rational coefficients times functions with well-controlled analytic structure. They introduce a tailored basis of special functions to simplify the finite remainders, enable robust numerical evaluation via generalized power-series, and verify the results through multiple independent checks including scale dependence, parity relations, and direct squared-amplitude comparisons. The methodology yields compact, cross-validated expressions across the full phase space and establishes a framework applicable to other 2→3 processes with off-shell legs, with ready-to-use ancillary data for phenomenology. This advances precision predictions for bbH at the LHC and informs future higher-order QCD calculations in multi-leg, multi-scale observables.

Abstract

We compute the two-loop QCD helicity amplitudes for the production of a Higgs boson in association with a bottom-quark pair at a hadron collider. We take the approximations of leading colour and work in the five flavour scheme, where the bottom quarks are massless while the Yukawa coupling is non-zero. We extract analytic expressions from multiple numerical evaluations over finite fields and present the results in terms of an independent set of special functions that can be reliably evaluated over the full phase space.

Two-loop leading-colour QCD helicity amplitudes for Higgs boson production in association with a bottom-quark pair at the LHC

TL;DR

This work delivers the first complete analytic two-loop QCD helicity amplitudes for Higgs production in association with a bottom-quark pair in the leading-color, five-flavour framework, using massless bottom quarks with finite Yukawa coupling. The authors combine a Feynman-diagram approach with finite-field sampling, IBP reduction to uniform-weight master integrals, and a novel finite-remainder basis to express results as analytic rational coefficients times functions with well-controlled analytic structure. They introduce a tailored basis of special functions to simplify the finite remainders, enable robust numerical evaluation via generalized power-series, and verify the results through multiple independent checks including scale dependence, parity relations, and direct squared-amplitude comparisons. The methodology yields compact, cross-validated expressions across the full phase space and establishes a framework applicable to other 2→3 processes with off-shell legs, with ready-to-use ancillary data for phenomenology. This advances precision predictions for bbH at the LHC and informs future higher-order QCD calculations in multi-leg, multi-scale observables.

Abstract

We compute the two-loop QCD helicity amplitudes for the production of a Higgs boson in association with a bottom-quark pair at a hadron collider. We take the approximations of leading colour and work in the five flavour scheme, where the bottom quarks are massless while the Yukawa coupling is non-zero. We extract analytic expressions from multiple numerical evaluations over finite fields and present the results in terms of an independent set of special functions that can be reliably evaluated over the full phase space.

Paper Structure

This paper contains 14 sections, 48 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Structure of the $pp\to b\bar{b}H$ Amplitudes at Leading Colour
  3. Tree-Level Amplitudes
  4. Amplitude Reduction and Reconstruction
  5. A Custom Basis of Special Functions for the Finite Remainders
  6. Further Validation
  7. Direct computation of the squared finite remainders
  8. Renormalisation scale dependence
  9. Relation between $+++-$ and $++-+$
  10. Convergence near spurious poles
  11. Results
  12. Conclusions
  13. Renormalisation Constants
  14. One-Loop Results

Figures (3)

  • Figure 1: Sample Feynman diagrams corresponding to the various closed fermion loop contributions at one loop as specified in Eq. \ref{['eq:NfDecomposition1L']}. Black-dashed, red, black-spiralled and black lines represent Higgs bosons, bottom quarks, gluons and light quarks (bottom quarks included), respectively.
  • Figure 2: Sample Feynman diagrams corresponding to the various closed fermion loop contributions at two loops as specified in Eq. \ref{['eq:NfDecomposition2L']}. Black-dashed, red, black-spiralled and black lines represent Higgs bosons, bottom quarks, gluons and light quarks (bottom quarks included), respectively.
  • Figure 3: Reduced squared finite remainders $\mathcal{H}^{(L)}$ at tree level, one and two loops evaluated on the one-dimensional phase space slice defined in Eq. \ref{['eq:parametrisation']}, as functions of the variable $y_2$, for the channels defined in Eq. \ref{['eq:channel_definition']}.