Decays of Scalar and Pseudoscalar Higgs Bosons into Fermions: Two-loop QCD Corrections to the Higgs-Quark-Antiquark Amplitude

TL;DR

The authors compute the neutral Higgs to heavy-quark amplitude at two-loop order in QCD for a general state with both scalar and pseudoscalar couplings, providing scalar and pseudoscalar form factors $F_S(s)$ and $F_P(s)$ valid for arbitrary $s$ and $m$. They employ on-shell renormalization for the heavy quark and $\overline{\text{MS}}$ for the coupling, and express results in terms of harmonic polylogarithms up to weight 4, with analytic continuation to the timelike region and detailed threshold and high-energy ($s\gg m^2$) expansions. The work includes spacelike results, threshold expansions (via $\beta=\sqrt{1-4m^2/s}$), and asymptotic expansions in $r=s/m^2$, highlighting the IR structure and potential implications for differential decay descriptions and CP studies of neutral Higgs bosons. This provides essential building blocks for precise predictions of $h\to Q\overline{Q}X$ and related processes, including differential distributions and CP-odd observables, at NNLO in QCD.

Abstract

As a first step in the aim of arriving at a differential description of neutral Higgs boson decays into heavy quarks, $h \to Q {\bar Q}X$, to second order in the QCD coupling $α_S$, we have computed the $hQ{\bar Q}$ amplitude at the two-loop level in QCD for a general neutral Higgs boson which has both scalar and pseudoscalar couplings to quarks. This amplitude is given in terms of a scalar and a pseudoscalar vertex form factor, for which we present closed analytic expressions in terms of one-dimensional harmonic polylogarithms of maximum weight 4. The results hold for arbitrary four-momentum squared, $q^2$, of the Higgs boson and of the heavy quark mass, $m$. Moreover we derive the approximate expressions of these form factors near threshold and in the asymptotic regime $m^2/q^2 \ll 1$.

Figures (2)

• Figure 1: Tree-level and one-loop diagrams that contribute to the heavy-quark scalar and pseudoscalar form factors (\ref{['decomp']}). The curly line represents a gluon, the double-line denotes the heavy quark of mass $m$. The external quarks are on their mass-shell: $p_1^2 = p_2^2 = m^{2}$.
• Figure 2: Two-loop vertex diagrams involved in the calculation of the form factors at order ${\mathcal{O}}(\alpha_{S}^{2})$. The notation is as in Fig. 1. The single straight and dotted lines represent massless quarks and ghosts, respectively. The external quarks are on their mass-shell, $p_1^2 = p_2^2 = m^{2}$. Crossed diagrams are not drawn.