Two-loop QCD amplitudes for $t\bar{t}γ$ production at hadron colliders

TL;DR

This work delivers a practical framework for two-loop QCD amplitudes in tt̄γ production, deriving IR poles with full $m_t$ dependence and approximating finite parts via a high-energy boosted mass-factorization approach. It introduces two complementary schemes (massless and semi-massive) to construct finite remainders, validates the IR structure against exact results, and provides detailed numeric studies including color-decomposed NNLO terms and exact $n_f^2$ contributions. The results form a crucial step toward NNLO predictions for tt̄γ production, enabling improved tests of top-quark interactions at hadron colliders. The study also identifies limitations related to power-suppressed terms and scalar-integral contributions, outlining future work on higher-power integrals and NLP factorization to enable full differential NNLO predictions.

Abstract

The associated production of a photon and a top-antitop quark pair ($t\bar{t}γ$) is important for measuring the top-quark charge and probing the top-photon interaction, and it requires improved theoretical predictions. We focus on the calculation of two-loop amplitudes for $t\bar{t}γ$ production at hadron colliders. The infrared singularities with full top-quark mass dependence are derived from universal anomalous dimensions combined with one-loop massive amplitudes expanded to higher orders in the dimensional regulator $ε$. The finite remainders are approximated in the high-energy boosted limit using the mass-factorization formula. To validate our approach, we compare approximate one-loop amplitudes up to $\mathcal{O}\left(ε^2\right)$, as well as the two-loop infrared poles, against our exact results. The results in this paper serve as an important step toward next-to-next-to-leading order predictions for $t\bar{t}γ$ production.

Figures (7)

• Figure 1: The squared amplitudes at $\mathcal{O}(\epsilon^0)$ for NLO and $\mathcal{O}(\epsilon^{-1})$ for NNLO in both the $q\bar{q}$ and $gg$ channels are shown as functions of the partonic center-of-mass energy $\sqrt{s_{12}}$. The other kinematic variables are fixed to $\theta_3 = 14\pi/29$, $\phi_3=34\pi/29$, $\theta_5=15\pi/29$ and $q_5=20\,q_{5,\rm max}/29$. Spin- and color-average factors of $1/36$ ($q\bar{q}$) and $1/256$ ($gg$) are included. The lower panel in each plot shows the ratios of the approximate results to the exact ones.
• Figure 2: The squared amplitudes at $\mathcal{O}(\epsilon^0)$ for NLO and $\mathcal{O}(\epsilon^{-1})$ for NNLO in both the $q\bar{q}$ and $gg$ channels, shown as functions of the angle parameter $\theta_3$. The other kinematic variables are fixed as $\sqrt{s_{12}}=5\TeV$, $\phi_3=34\pi/29$, $\theta_5=15\pi/29$ and $q_5=20\,q_{5,\rm max}/29$.
• Figure 3: The squared amplitudes as a function of the angle parameter $\theta_5$. The other kinematic variables are chosen as: $\sqrt{s_{12}}=5\TeV$, $\theta_3=14\pi/29$, $\phi_3=34\pi/29$ and $q_5=20\,q_{5,\rm max}/29$.
• Figure 4: Ratios of the approximate results in the massless scheme to the exact ones at various orders in $\epsilon$, as functions of $\sqrt{s_{12}}$. The phase-space parameters are the same as those in Fig. \ref{['fig:diffscheme']}.
• Figure 5: Ratios of the approximate results in the massless scheme to the exact ones at various orders in $\epsilon$, as functions of $\theta_3$. The other phase-space parameters are the same as those in Fig. \ref{['fig:diffschemetheta3s5']}.