Adequate bases of phase space master integrals for $gg \to h$ at NNLO and beyond

TL;DR

The paper addresses the challenge of computing the gg → h cross section in the infinite top-mass limit with full x-dependence at NNLO and beyond. It employs Henn's canonical differential equations to construct a basis of master integrals that are pure functions, enabling straightforward epsilon expansions and weight-by-weight solutions, including for coupled systems. The authors develop a practical algorithm to identify and build canonical partners, demonstrate the method on NNLO phase-space topologies and a non-planar NNNLO topology, and verify results by transforming back to traditional reduction bases. This work provides explicit canonical bases and boundary-condition strategies that pave the way toward full N3LO predictions for Higgs production and offers a general framework for high-order multi-loop calculations.

Abstract

We study master integrals needed to compute the Higgs boson production cross section via gluon fusion in the infinite top quark mass limit, using a canonical form of differential equations for master integrals, recently identified by Henn, which makes their solution possible in a straightforward algebraic way. We apply the known criteria to derive such a suitable basis for all the phase space master integrals in afore mentioned process at next-to-next-to-leading order in QCD and demonstrate that the method is applicable to next-to-next-to-next-to-leading order as well by solving a non-planar topology. Furthermore, we discuss in great detail how to find an adequate basis using practical examples. Special emphasis is devoted to master integrals which are coupled by their differential equations.

Figures (6)

• Figure 1: The NLO topology $\text{TNLO}_2(a_1,a_2,a_3)$. The massive Higgs line is depicted by a double line, whereas the dashed line denotes the cut. Numbers in roman indicate the incoming and outgoing momenta $p_1$ and $p_2$, and numbers in italic label the propagators according to the corresponding indices.
• Figure 2: The NNLO topologies $\text{TT}X_c(a_1,a_2,a_3,a_4,a_5,a_6,a_7)$ involving our choice for a canonical basis. The subscript $c = 2, 3$ of topologies distinguishes two-particle cuts and three-particle cuts. The massive Higgs line is depicted by a double line, whereas dashed lines denote possible cuts. Numbers in roman indicate the incoming and outgoing momenta $p_1$ and $p_2$ and numbers in italic label the propagators according to the corresponding indices. In the text we define all integrals as single-cut integrals. For TTE and TTH, two cuts give the same contribution but only one of them is taken into account in the defnition of the corresponding master integrals.
• Figure 3: Two-particle cut diagrams appearing in our choice of canonical master integrals at NNLO.
• Figure 4: Three-particle cut diagrams appearing in our choice of canonical master integrals at NNLO.
• Figure 5: The sea snake topology $\text{TTS}_4(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9,a_{10},n_{11},n_{12})$. The notation is the same as in fig. \ref{['fig:NNLOtops']}. The indices $n_{11}$ and $n_{12}$ denote irreducible scalar products which appear in the numerator and are always less than or equal to zero.