Exact quark-mass dependence of the Higgs-gluon form factor at three loops in QCD

TL;DR

This work delivers an exact three-loop Higgs-gluon form factor in QCD with a single massive quark by numerically solving differential equations for master integrals, enabling precise quark-mass dependence and analytic expansions around the mass thresholds. It clarifies infrared renormalisation schemes and provides explicit finite-remainder results, validating them against existing approximants and giving high-precision large-mass, threshold, and high-energy expansions. The results directly improve the inclusion of top- and bottom-quark loop effects in Higgs hadroproduction at NNLO and establish the reliability of Padé-based approaches for certain mass regimes. Data and expansions are made available for phenomenology and further generalization to multiple massive quarks in future work.

Abstract

We determine the three-loop form factor parameterising the amplitude for the production of an off-shell Higgs boson in gluon fusion in QCD with a single massive quark. The result is obtained via a numerical solution of a system of differential equation for the occurring master integrals. The solution is also used to determine the high-energy and threshold expansions of the form factor. Our findings may be used for the evaluation of virtual corrections generated by top-quark and b-quark loops in Higgs boson hadroproduction cross sections at next-to-next-to-leading order.

Figures (8)

• Figure 1: Complete set of Feynman diagrams with two fermion loops contributing to the Higgs-gluon form factor at three-loop order. The fermion loop connected to the Higgs-boson line corresponds to a massive quark. The quark of the second fermion loop may be either massive or massless.
• Figure 2: Contours for the numerical solution of the differential equations for the master integrals. The points on the abscissa correspond to singularities of the differential equations. Every time a contour reaches the real axis, the interval between singularities is explored in both directions.
• Figure 3: Comparison of the three-loop coefficient of the finite remainder, Eq. \ref{['eq:finiteRemainderI']}, at $n_l = 5$, $L_\mu = 0$ (five massless quarks, renormalisation scale $\mu^2 = -s$), with the default Padé approximation, $[6,1]$, constructed in Ref. Davies:2019nhm (left panel) and improved to $[7,1]$ in Ref. Davies:2019roy (right panel), as function of $z = s/4M^2$ with $\sqrt{s}$ the center-of-mass energy of the Higgs boson and $M$ the mass of the single massive quark. The bands correspond to the uncertainty of the Padé approximations as estimated in Refs. Davies:2019nhm and Davies:2019roy. The lower plot shows the absolute difference between the approximation and the exact result. Also shown is the large-mass expansion (LME) of the three-loop coefficient of the finite remainder truncated at $\order{z^2}, \order{z^4}$ and $\order{z^{100}}$.
• Figure 4: Same as Fig. \ref{['fig:comparisonNL5']} but with $n_l = 0$.
• Figure 5: Relative difference, Eq. \ref{['eq:Delta']}, between the Padé approximation of the three-loop coefficient of the finite remainder $\mathcal{C}_I^{(2)}$ from Refs. Davies:2019nhm (left panel) and Davies:2019roy (right panel) and the exact result at $n_l = 0$, $L_\mu = 0$. $z \approx 8$ corresponds to a $\sqrt{s}$ = 1 TeV Higgs boson produced through a top-quark loop, whereas $z \approx 156$ corresponds to an on-shell Higgs boson produced through a b-quark loop.