Two-loop $gg \to Hg$ amplitude mediated by a nearly massless quark

TL;DR

This work computes the two-loop amplitude for $gg \to Hg$ in the limit of a nearly massless internal quark, using a differential-equation approach to master integrals with a controlled small-$m_b$ expansion. By projecting onto four form factors, performing IBP reductions, and solving a sparse system of differential equations, the authors obtain analytic master-integral solutions in terms of Goncharov polylogarithms, with integration constants fixed via boundary conditions and known massless limits. The resulting helicity amplitudes, including their soft and collinear limits and analytic continuation to Higgs production kinematics, provide a solid input for precision predictions of Higgs transverse momentum distributions and bottom/top interference effects. Ancillary files accompany the paper, containing the full analytic expressions and continued results for all helicity configurations. Overall, the method supplies a scalable framework for expanding loop amplitudes around nearly massless internal particles, with applications to high-$p_T$ Higgs phenomenology and Sudakov resummation.

Abstract

We analytically compute the two-loop scattering amplitude $gg \to Hg$ assuming that the mass of the quark, that mediates the ggH interaction, is vanishingly small. Our computation provides an important ingredient required to improve the theoretical description of the top-bottom interference effect in Higgs boson production in gluon fusion, and to elucidate its impact on the Higgs boson transverse momentum distribution.

Figures (4)

• Figure 1: Examples of two-loop Feynman diagrams that contribute to the process $gg\rightarrow Hg$.
• Figure 2: Seven-propagator Feynman integrals of the PL1 family, that appear in the form factors and require IBP reduction. The two integrals at the top are irreducible and correspond to two sectors in the family PL1 that contain master integrals with seven propagators. The integrals at the bottom correspond to reducible integrals. All momenta are incoming.
• Figure 3: Same as in Figure \ref{['fig::PL1']}, but for the integrals of the PL2 family.
• Figure 4: Same as in Figure \ref{['fig::PL1']} but for the integrals of the NPL family. Note that in this case all the three integrals are irreducible. The leftmost integral does not contribute to the form factors because of the color structure of the corresponding Feynman diagrams. For this reason we do not compute it.