First look at the evaluation of three-loop non-planar Feynman diagrams for Higgs plus jet production

TL;DR

This work tackles the computation of three-loop Feynman integrals for Higgs plus jet production with one off-shell leg, a key input for high-precision QCD predictions. It develops canonical differential equations and obtains analytic solutions in terms of generalised polylogarithms up to transcendental weight six, including both planar and non-planar ladder topologies. A key finding is the emergence of new symbol alphabet letters in non-planar cases and a counterexample to adjacency relations, challenging prevailing assumptions about the function space and cluster algebra structures. The results are validated numerically against pySecDec and feyntrop, and the analytic solutions are provided for use in future bootstrap approaches and phenomenological applications.

Abstract

We present new computations for Feynman integrals relevant to Higgs plus jet production at three loops, including first results for a non-planar class of integrals. The results are expressed in terms of generalised polylogarithms up to transcendental weight six. We also provide the full canonical differential equations, which allows us to make structural observations on the answer. In particular, we find a counterexample to previously conjectured adjacency relations, for a planar integral of the tennis-court type. Additionally, for a non-planar triple ladder diagram, we find two novel alphabet letters. This information may be useful for future bootstrap approaches.

Figures (4)

• Figure 1: The planar and non-planar integral families considered in this paper. Here $k_{ij}\equiv k_i-k_j$. Thin lines indicate one-shell momenta, whilst thick ones indicate off-shell ones. The labelling of the integral families follows the convention of Ref. Henn:2020lye.
• Figure 2: Singular configurations present in the analytic evaluation of our integral families. The shaded region corresponds to the Euclidean region, where all integrals are real-valued. The same is also true for all GPLs without dependence on letters $\alpha_7$ and $\alpha_8$, which is the case for integral families A, B2, E1, and E2. Our GPL representation for family B1 is manifestly real-valued in region I only (but can be analytically continued to other regions).
• Figure 3: Integrals $f_{\text{B1}}^{41}$ and $f_{\text{B1}}^{67}$ that depend on the letters $\alpha_7$ and $\alpha_8$ of alphabet \ref{['eq:alphabet_st']}, respectively. The two integrals are related by the symmetry $p_1 \leftrightarrow p_2$.
• Figure 4: Integrals $f^{110}_{\text{E1}}$ and $f^{127}_{\text{E1}}$ that violate the adjacency conditions $\tilde{A}_4\cdot \tilde{A}_5=\tilde{A}_5\cdot \tilde{A}_4=0$. The two integrals are related by the symmetry $p_2 \leftrightarrow p_3$.