Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops

TL;DR

The paper develops a contour-based, one-dimensional generalised power series method to solve planar two-loop master integrals with heavy-quark mass dependence, enabling high-precision QCD corrections for Higgs+jet production. By recasting multi-scale differential equations along a contour and solving via series expansions (including Frobenius-type handling for regular and singular points), the approach yields efficient analytic continuation across physical thresholds and rapid numerical evaluation across kinematic regions. The authors apply the method to a planar elliptic family (73 MIs: 65 canonical and 66–73 elliptic), demonstrating accurate analytic continuation above/below thresholds, detailed boundary-mcond transport, and practical timings (often around 1 second per integral above threshold) with errors down to $\mathcal{O}(10^{-32})$. They further show mapping to a finite region with $l,z$ variables and provide extensive numerical results, highlighting parallelizability and potential for broad applicability to Feynman integrals once differential equations are known.

Abstract

We obtain generalised power series expansions for a family of planar two-loop master integrals relevant for the QCD corrections to Higgs + jet production, with physical heavy-quark mass dependence. This is achieved by defining differential equations along contours connecting two fixed points, and by solving them in terms of one-dimensional generalised power series. The procedure is efficient and can be repeated in order to reach any point of the kinematic regions. The analytic continuation of the series is straightforward and we present new results below and above the physical thresholds. The method we use allows to compute the integrals in all kinematic regions with high precision. Performing a series expansion on a typical contour above the physical threshold takes on average $\mathcal{O}(1 \text{ second})$ per integral with worst relative error of $\mathcal{O}(10^{-32})$, on a single CPU core. After the series is found the numerical evaluation of the integrals in any point of the contour is virtually instant. Our approach is general and can be applied to Feynman integrals provided that a set of differential equations is available.

Figures (6)

• Figure 1: The four planar integral families contributing to two-loop $H{+}j$-production in QCD.
• Figure 2: Plots of the elliptic integral sectors, for $s_{13}=-1,\;p^2_4=\frac{13}{25},\;m^2=1$ as a function of $s_{12}$, at order $\epsilon^4$, obtained by series expanding along the contour $\gamma_{\text{thr}}$. The solid points represent values computed numerically with the software FIESTA Smirnov:2015mct.
• Figure 3: The physical region in the $l,z$ variables for fixed Higgs mass ($x_2=13/25$) and unit propagators mass (see text). The lines represent the singular points of the differential equations. The orange line corresponds to the physical threshold $x_{1}=4$.
• Figure 4: Plots along the contour $\gamma_{\infty}$, for values of the contour parameter $t\in[0,1]$. The solid points represent values computed numerically with the software FIESTA Smirnov:2015mct.
• Figure 5: Timings for the computation of the integrals up to and including a given sector at the point $s_{12}=5,s_{13}=-1,p_4^2=\frac{13}{15},m^2=1$ up to and including order $\epsilon^4$, on 1 CPU core. The sector id is defined in Eq. (\ref{['eq:secid']}) The point is reached by expanding along the contour $\gamma_{\text{thr}}$ defined in Section \ref{['sec:series sol DE A']}. The last two values on the right of the plot correspond to the elliptic sectors.