Two-loop leading colour QCD helicity amplitudes for top quark pair production in the gluon fusion channel

TL;DR

This work delivers analytic, two-loop leading-colour QCD helicity amplitudes for top-quark pair production in gluon fusion, explicitly including heavy-fermion loops that introduce elliptic master integrals. It employs a massive spinor-helicity formalism with a rational kinematics parametrisation based on momentum twistors and leverages finite-field arithmetic and differential equations to obtain compact amplitudes and complete master integral sets. Notably, two new elliptic master integrals, $\mathcal{J}_{21}$ and $\mathcal{J}_{22}$, are computed, with the finite remainders expressed via iterated integrals over elliptic curves; numerical evaluation in both Euclidean and physical regions is discussed. The results enable precise predictions for $t\bar{t}$ production in the gluon-fusion channel and lay groundwork for handling mass dependence and elliptic sectors in future multi-loop calculations.

Abstract

We present a complete set of analytic helicity amplitudes for top quark pair production via gluon fusion at two-loops in QCD. For the first time, we include corrections due to massive fermion loops which give rise to integrals over elliptic curves. We present the results of the missing master integrals needed to compute the amplitude and obtain an analytic form for the finite remainders in terms of iterated integrals using rationalised kinematics and finite field sampling. We also study the numerical evaluation of the iterated integrals.

Figures (11)

• Figure 1: Sample Feynman diagrams corresponding to various internal flavour contributions at one loop as specified in Eq. \ref{['eq:flamplitude1loop']}. Red lines, black spiral lines and black lines represent massless quarks, gluons and top quarks, respectively.
• Figure 2: Sample Feynman diagrams corresponding to various internal flavour contributions at two loops as specified in Eq. \ref{['eq:flamplitude2loop']}. Red lines, black spiral lines and black lines represent massless quarks, gluons and top quarks, respectively.
• Figure 3: Sample Feynman diagrams with mass counterterm insertions at tree level and one loop that contribute in Eqs. \ref{['eq:mren_1loop']} and \ref{['eq:mren_2loop']}. Black spiral lines and black lines represent gluons and top quarks, respectively. Circled crosses indicate the mass counterterm insertions.
• Figure 4: Maximal topologies for the $N_c^2$ part of planar two-loop $\bar{t}tgg$ amplitude. Black-solid lines represent top quarks, red-dashed lines represent gluons.
• Figure 5: Master integral topologies with maximal number of propagators (7) that appear in the leading colour 2-loop amplitude for $\bar{t}tgg$. Black-solid lines represent massive particles, red-dashed lines represent massless particles.