Two-loop helicity amplitudes for $gg \to ZZ$ with full top-quark mass effects

TL;DR

This work delivers a complete calculation of the two-loop QCD corrections to $gg\to ZZ$ with exact top-quark mass dependence by introducing a novel syzygy-based IBP reduction and a finite-basis integral framework. The authors develop an algorithm to construct finite linear combinations of divergent integrals, enabling stable numerical evaluation via sector decomposition, and implement a comprehensive reduction pipeline to master integrals without introducing propagator dots. They validate the approach against known large- and small-mass expansions and demonstrate robust numerical performance across phase space, highlighting scheme-dependent IR subtraction effects. The resulting amplitudes serve as a crucial building block for incorporating full top-mass effects in NLO cross sections for $ZZ$ production in gluon fusion and showcase a generalizable, automated methodology for multi-loop calculations.

Abstract

We calculate the two-loop QCD corrections to $gg \to ZZ$ involving a closed top-quark loop. We present a new method to systematically construct linear combinations of Feynman integrals with a convergent parametric representation, where we also allow for irreducible numerators, higher powers of propagators, dimensionally shifted integrals, and subsector integrals. The amplitude is expressed in terms of such finite integrals by employing syzygies derived with linear algebra and finite field techniques. Evaluating the amplitude using numerical integration, we find agreement with previous expansions in asymptotic limits and provide ab initio results also for intermediate partonic energies and non-central scattering at higher energies.

Figures (16)

• Figure 1: Example Feynman diagrams representing the two classes of diagrams
• Figure 2: Representative Feynman diagrams in class A with irreducible topologies. The number of master integrals in each topology are 3, 4, 3, 3, 5, 5, and 4 respectively
• Figure 3: Representative Feynman diagrams in class A with reducible topologies.
• Figure 4: Examples of divergent and finite integrals in the limit $\epsilon \to 0$ for a non-planar topology. Thick solid lines represent the top-quark while thick dashed lines represent Z-bosons. Topology (b) contains an irreducible numerator, where $k$ is the difference of the momenta of the edges marked by the thin dash lines.
• Figure 5: Integrals appearing in \ref{['eq:finitecomb1']}. $I_{1,1}$ is the corner integral of the topology under consideration. $I_{2,1}$ is a second integral in the topology, but with a numerator $(k^2-m_t^2)$, where $k$ is equal to the difference of the momenta of the edges marked by the thin dashed lines. Integrals $I_{3,1},I_{4,1},I_{5,1},I_{6,1},I_{7,1}$ belong to subtopologies. All integrals are defined in $d=4-2\epsilon$ dimensions.