Evaluating two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence

TL;DR

This work delivers a comprehensive analytic treatment of a non-planar two-loop master-integral family (F) relevant to Higgs+jet production with full heavy-quark mass dependence. By casting the polylogarithmic sectors into a canonical differential-equation form and constructing a real, symmetric alphabet, the authors obtain weight-2 closed forms and one-fold integral representations for weights 3 and 4, while handling two elliptic sectors through generalized power-series expansions and numerical series. Boundary terms are computed via expansions by regions to bootstrap the full solution, and a series-contour approach enables precise numerical results across all kinematic regions, including near thresholds. The methodology yields a practical, efficient route to mass-dependent NLO/NNLO predictions for Higgs+jet observables and sets the stage for addressing the remaining non-planar G-family in future work.

Abstract

We present the analytic computation of a family of non-planar master integrals which contribute to the two-loop scattering amplitudes for Higgs plus one jet production, with full heavy-quark mass dependence. These are relevant for the NNLO corrections to inclusive Higgs production and for the NLO corrections to Higgs production in association with a jet, in QCD. The computation of the integrals is performed with the method of differential equations. We provide a choice of basis for the polylogarithmic sectors, that puts the system of differential equations in canonical form. Solutions up to weight 2 are provided in terms of logarithms and dilogarithms, and 1-fold integral solutions are provided at weight 3 and 4. There are two elliptic sectors in the family, which are computed by solving their associated set of differential equations in terms of generalized power series. The resulting series may be truncated to obtain numerical results with high precision. The series solution renders the analytic continuation to the physical region straightforward. Moreover, we show how the series expansion method can be used to obtain accurate numerical results for all the master integrals of the family in all kinematic regions.

Figures (5)

• Figure 1: The seven integral families contributing to two-loop $H{+}j$-production in QCD.
• Figure 2: The integral family F with momenta and propagator labels.
• Figure 3: The interval along which we plot the basis $B$, is covered by three expansions obtained by patching and analytical continuation.
• Figure 4: Plot of the integrals $B^{(4)}_{72}$ and $B^{(4)}_{73}$. Note the singular behaviour at $s=4m^2, \ (m=1)$. The solid points represent values computed numerically with the software FIESTA Smirnov:2015mct.
• Figure 5: The 73 master integrals. Shown on the figure is the sector, i.e. the set of propagators, to which the master integrals belong. Higher powers of propagators, numerators, or prefactors are not shown. External momenta are labelled using $p_{ij} = p_1{+}p_j$ and $p_4 = p_1{+}p_2{+}p_3$. Masses (internal as well as external) are indicated with a thicker line.