The complete set of two-loop master integrals for Higgs + jet production in QCD

TL;DR

The paper completes the two-loop non-planar master integrals for Higgs plus jet production in full QCD by formulating differential equations for a canonical-like basis and solving them via one-dimensional generalized power-series along kinematic contours. It identifies and treats elliptic sectors alongside polylogarithmic ones, with careful boundary conditions from a heavy-mass limit and a robust contour-patching strategy to ensure convergence across the physical region. The approach is validated through extensive top- and bottom-quark results and cross-checked against established numerical tools, confirming very high precision and practical runtimes. Overall, the work provides a complete, high-precision numerical framework for NLO Higgs+jet corrections and related NNLO contributions in the full theory, with broad applicability and public implementations.

Abstract

In this paper we complete the computation of the two-loop master integrals relevant for Higgs plus one jet production initiated in arXiv:1609.06685, arXiv:1907.13156, arXiv:1907.13234. We compute the integrals by defining differential equations along contours in the kinematic space, and by solving them in terms of one-dimensional generalized power series. This method allows for the efficient evaluation of the integrals in all kinematic regions, with high numerical precision. We show the generality of our approach by considering both the top- and the bottom-quark contributions. This work along with arXiv:1609.06685, arXiv:1907.13156, arXiv:1907.13234 provides the full set of master integrals relevant for the NLO corrections to Higgs plus one jet production, and for the real-virtual contributions to the NNLO corrections to inclusive Higgs production in QCD in the full theory.

Figures (5)

• Figure 1: The integral family with momenta and propagator labels.
• Figure 2: The 84 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 $p_{ij} = p_i{+}p_j$ and $p_4 = p_1{+}p_2{+}p_3$. Masses (internal as well as external) are indicated with a thicker line.
• Figure 3: These figures illustrate subdivisions of an expansion in the unit interval [-1,1] with singularities at $-1, 0$ and $1$, in terms of additional expansions, such that each expansion can be matched to the next one at a fixed fraction of the distance to its nearest singularities. The numbers on top are the matching points between neighbouring expansions, while the numbers at the bottom indicate the expansions points for (a) $k = 2$: Moving at most half the distance to the nearest singularity, (b) $k = 3$: Moving at most one-third the distance to the nearest singularity.
• Figure 4: Depiction of lines along which we produce samples in the physical region of the top. The black lines $\vec{0}\rightarrow \left(1/(n+1), n/(n+1)\right) \rightarrow \left(1/(n+1), 1/(n+1)\right)$ are computed first to obtain boundary values for $n$ horizontal lines, depicted in grey. The horizontal lines are themselves used to produce $n$ evenly spaced samples, denoted by blue dots. The particle production threshold $s = 4m^2$ is depicted by a dashed red line. Depicted is the case with $n = 10$. The actual plots are produced with $n = 100$.
• Figure 5: On the left (resp. right) are shown the real (blue) and imaginary (orange) part of the integrals of the top sector of Family G in the case of a virtual bottom (resp. top) quark running in the loop.