Contribution of third generation quarks to two-loop helicity amplitudes for W boson pair production in gluon fusion

TL;DR

This work computes the contribution of the third-generation quarks to the two-loop helicity amplitudes for on-shell $W$-pair production in gluon fusion, a process enhanced by gluon luminosity at the LHC. The authors decompose the amplitude into tensor structures, apply a hybrid renormalisation scheme, and employ IBP reduction to reduce the problem to 334 master integrals across 26 integral families. Master integrals are evaluated numerically by solving differential equations in the top-quark mass using an imaginary-mass boundary condition, enabling high-precision results efficiently. The calculation yields one- and two-loop helicity amplitudes for specific phase-space points and provides grids and crosschecks against independent numerical tools, laying the groundwork to incorporate third-generation effects into NLO QCD descriptions of $W$-pair production and to extend the method to other massive-loop amplitudes. The approach offers a practical route for precise predictions where analytic solutions are intractable due to internal masses.

Abstract

We compute the contribution of third generation quarks ($t,\ b$) to the two-loop amplitude for on-shell $W$ boson pair production in gluon fusion $gg \to WW$. We present plots for the amplitude across partonic phase space as well as reference values for two kinematic points. The master integrals are efficiently evaluated by numerically solving a system of ordinary differential equations.

Figures (10)

• Figure 1: Representative two-loop Feynman diagrams for $gg \to WW$.
• Figure 2: The four basic topologies. The internal lines can be both massive and massless. The first three (a)--(c) are planar and can at most have 5 massive internal lines, while the nonplanar topology (d) can at most have 4 massive internal lines.
• Figure 3: Topologies of integral families. Solid and dashed lines correspond to massive and massless particles respectively. Internal massive particles have mass $m_{t}$ while external massive particles have mass $m_{W}$. All nine planar topologies and nonplanar no. 2 and 3 can be crossed ($p_1 \leftrightarrow p_2$) giving a total of 26 topologies.
• Figure 4: Topologies of boundary integrals. Solid and dashed lines correspond to massive and massless particles respectively. See appendix \ref{['sec:boundary']} for their explicit expressions.
• Figure 5: A typical master integral and its leading regions. Solid and dashed lines correspond to massive and massless particles respectively. Internal massive particles have mass $m_{t}$, while external massive particles have mass $m_{W}$.