Numerical evaluation of two-loop QCD helicity amplitudes for $gg\to t \bar{t} g$ at leading colour

TL;DR

This work benchmarks the two-loop finite remainders for $gg\to t\bar{t} g$ in the leading-colour approximation and introduces a novel over-complete basis of special functions to express master integrals, enabling analytic UV/IR pole cancellation even in the presence of elliptic integrals. It combines four-dimensional projection for helicity amplitudes with finite-field IBP reductions and a non-canonical differential-equation framework to construct and numerically evaluate the finite remainders, emphasizing a minimal set of non-polylogarithmic functions that appear in the finite part. The authors provide benchmark numerical results for the helicity sub-amplitudes, validate their approach via gauge- and pole-consistency checks, and demonstrate significant simplifications and performance gains, outlining a viable path toward full NNLO predictions for $t\bar{t}$+jet processes. Their methodology broadens the practical computation of two-loop amplitudes with elliptic master integrals and offers a scalable framework for future phenomenological applications at the LHC.

Abstract

We present the first benchmark evaluation of the two-loop finite remainders for the production of a top-quark pair in association with a jet at hadron colliders in the gluon channel. We work in the leading colour approximation, and perform the numerical evaluation in the physical phase space. To achieve this result, we develop a new method for expressing the master integrals in terms of a (over-complete) basis of special functions that enables the infrared and ultraviolet poles to be cancelled analytically despite the presence of elliptic Feynman integrals. The special function basis makes it manifest that the elliptic functions appear solely in the finite remainder, and can be evaluated numerically through generalised series expansions. The helicity amplitudes are constructed using four dimensional projectors combined with finite-field techniques to perform integration-by-parts reduction, mapping to special functions and Laurent expansion in the dimensional regularisation parameter.

Figures (6)

• Figure 1: Examples of Feynman diagrams with one and two mass-counterterm insertions, together with the terms of \ref{['eq:Amren']} they contribute to. Thick lines denote massive particles, and crossed circles indicate the counterterm insertions.
• Figure 2: Integral families relevant for the leading-colour two-loop $gg\rightarrow t\bar{t}g$ amplitude. The hexagon-triangle (${\rm HT}$) families, in the second row, can be reduced to pentagon-box (${\rm PB}$) families, in the first row, except for 4 master integrals of ${\rm HT}_B$ which can instead be expressed as products of one-loop integrals. Four more families, $\mathrm{PB}_{A}'$, $\mathrm{PB}_{B}'$, $\mathrm{HT}_{A}'$ and $\mathrm{HT}_{B}'$, are obtained by exchanging $1 \leftrightarrow 2$ and $3 \leftrightarrow 5$ in $\mathrm{PB}_{A}$, $\mathrm{PB}_{B}$, $\mathrm{HT}_{A}$ and $\mathrm{HT}_{B}$, respectively.
• Figure 3: Graphs representing the sectors of the family ${\rm PB}_B$ which contain master integrals that we did not express in terms of iterated integrals at order $\epsilon^4$. The external momenta are all outgoing, the thick line denotes the top, and the arrows indicate the momentum's direction. The sub-captions list the relevant master integrals. They are defined by multiplying the scalar propagators associated with the graphs by the following numerators under the integral sign: N_{15} = \epsilon^4 \, d_{15}^2 \, (k_2 + p_2)^2 + \ldots \,,N_{19} = \epsilon^4 \, d_{45} \, \mathrm{tr}\bigl(\gamma_5 \slashed{p}_3 (\slashed{k}_1-\slashed{p}_2-\slashed{p}_3) \slashed{p}_4 \slashed{p}_2 \bigr) \,,N_{20} = \epsilon^4 \, d_{45} \, \mathrm{tr}\bigl(\slashed{p}_3 (\slashed{k}_1-\slashed{p}_2-\slashed{p}_3) \slashed{p}_4 \slashed{p}_2 \bigr) \,, \qquad N_{35} = \epsilon^4 \, d_{15} (d_{12} + m_t^2) \,,N_{36} = 2 \, \epsilon^4 \, \sqrt{(d_{15}-d_{34})^2-2 d_{34} m_t^2} \, (k_1 \cdot p_1) \,,N_{37} = 2 \, \epsilon^4 (2 \epsilon - 1) \, d_{15} \, (k_2 \cdot p_2) \,,where the ellipsis denotes sub-sector terms. We recall that the sector in figure (b) contains the nested square root, whereas the sector in figure (c) involves elliptic functions.
• Figure 4: Matrix plot displaying the non-zero entries of the connection matrix in \ref{['eq:DEfuncs']}. Blue dots indicate non-zero entries involving logarithmic one-forms, while the red entries contain also non-logarithmic special functions. The solid black lines separate subsets of functions with different transcendental weight, as shown on the left, with $4^*$ denoting the non-polylogarithmic special functions.
• Figure 5: Distributions of the evaluation time per segment in the solution of the DEs for the MIs of each family with DiffExp.