Two-loop integrals for $t\bar{t}γ$ production at hadron colliders in the leading-colour approximation

Introduction

Top-quark production processes play a prominent role in the precision programme of the Large Hadron Collider (LHC). Among these, top-quark pair production in association with a photon ($t\bar{t}\gamma$) offers direct access to the electromagnetic properties of the top quark and therefore constitutes a powerful probe of the Standard Model (SM) as well as of possible new-physics effects. Indeed, $t\bar{t}\gamma$ production enables a direct measurement of the top-quark electric charge [1] and probes the structure of the top–photon coupling [2]. Furthermore, any deviation of the measurements from the SM prediction could be a hint of a new exotic top-like particle or of an anomalous electric dipole moment of the top quark [3, 4, 5, 6, 7], as studied in the context of effective field theories [8]. Although the cross section for $t\bar{t}\gamma$ production at the LHC is much smaller than that for top-quark pair production alone, the growing amount of data has enabled the ATLAS and CMS collaborations to carry out extensive measurements of both the inclusive cross section and differential distributions for this process, with recent results published in refs. [9, 10]. As the precision of experimental measurements at the LHC increases, more accurate theoretical predictions for this process are required [11].

Fixed-order NLO QCD and EW predictions have so far been computed [12, 13, 14, 15], and the so-called complete NLO predictions are also available [16]. Considerable effort has gone into a more realistic description of this process, taking into account the effects of top-quark decays. NLO QCD theoretical predictions have been combined with a parton shower [17]. Furthermore, NLO predictions have been provided in the narrow-width approximation [18, 19, 20] and with full off-shell effects [21, 22, 23]. More recently, approximate NNLO results have become available. One approach incorporates second-order soft-gluon corrections into the complete NLO result [24], while another [25] employs the massification procedure [26] together with a semi-massive scheme [27], in which all loop integrals entering the amplitudes are treated as massless. The first steps towards an exact NNLO QCD calculation have also been taken with the computation of the one-loop amplitudes through $\mathcal{O}(\varepsilon^2)$ [28]. The main bottleneck in reaching NNLO QCD accuracy for $pp \to t\bar{t}\gamma$ is the computation of the two-loop five-particle scattering amplitudes that enter the double-virtual corrections. This is particularly challenging because of the large number of kinematic variables and the complicated analytic structures arising from the multiple massive internal propagators in the loop integrals.

Thanks to recent advances in both integral reduction and the computation of loop integrals, two-loop $2\to3$ amplitudes with massive internal propagators have come within reach [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Only in the past year have the first NNLO QCD results making use of two-loop amplitudes with exact mass dependence appeared, for $t\bar{t}+$jet [34, 35], $t\bar{t}W$ [38, 39] and the quark-initiated channel of $t\bar{t}H$ [40] production. These calculations have been performed in the generalised leading-colour approximation, in which the two-loop finite remainders are expanded in the number of colours and of closed massless fermion loops, and only the leading terms are retained. This approximation simplifies the calculation considerably, while still yielding a significant improvement in the precision of the resulting theoretical predictions.

In the case of $t\bar{t}\gamma$ production, the leading-colour two-loop amplitudes involve the three integral families already encountered in $t\bar{t}+$jet production, together with three additional families that contribute at leading colour because the photon carries no colour charge. Here, by leading colour we mean the terms of the two-loop amplitudes that dominate when the number of colours, $N_c$, and the number of light-quark flavours, $n_l$, are large. The amplitudes also contain terms of order $N_c \sum_f Q_f$, where the sum runs over the light-quark flavours and $Q_f$ is the fractional electric charge of flavour $f$. Although in some cases these terms have been found to be comparable in size to the leading-colour ones (see e.g. ref. [41]), they involve non-planar integrals that are much more challenging to compute, and we neglect them here.

The three new planar families are considerably more complicated than those relevant to $t\bar{t}+$jet production. With up to three massive internal propagators, they exhibit a much richer analytic structure, in which several elliptic curves and nested square roots appear simultaneously. To compute the integrals in these families, we follow the strategy developed in refs. [31, 42] and successfully applied to $t\bar{t}+$jet and $t\bar{t}W$ production. This strategy builds on the methods [43, 44, 45, 46, 47] that underlie the rapid progress in two-loop $2\to3$ amplitudes with massless internal propagators over the past few years. Specifically, we construct integral bases that satisfy algebraic differential equations of a form that allows us to express the solution in terms of a set of special functions, tailored to simplify the future computation of the two-loop leading-colour amplitudes. We evaluate these special functions efficiently by numerically integrating the differential equations they satisfy [48, 49, 50, 35].

This article is organised as follows. In § 2, we introduce our notation and the relevant integral families. In § 3, we describe the structure of the differential equations and discuss how we obtained them, with emphasis on the sectors that involve elliptic curves or nested square roots. § 4 illustrates how we expressed the solution to the differential equations in terms of special functions. In § 5, we present our strategy for the evaluation of the special functions. We summarise our results and give an outlook in § 6. Appendix § 7 displays the five-point sectors of the three families studied here, and lists the corresponding master integrals in our bases.

Notation

We generated the Feynman diagrams for the two-loop partonic amplitudes contributing to $t\bar{t}\gamma$ production at hadron colliders with QGRAF [51], and retained only the leading terms in the expansion in the number of colours and of light-quark flavours circulating in the loops. The Feynman integrals that appear can be grouped into six planar families. Three of these families — $\mathrm{PB}_A$, $\mathrm{PB}_B$ and $\mathrm{PB}_C$, shown in Figure 1 — are also needed for the production of a top-quark pair in association with a jet in the same approximation, and were computed in refs. [29, 31, 42, 33]. Here we focus on the remaining three families: $\mathrm{PB}_D$, $\mathrm{PB}_E$ and $\mathrm{PB}_F$, depicted in Figure 2. In addition to these six pentagon-box topologies, there are also hexagon-triangle topologies, which can be reduced either to the pentagon-boxes shown in fig:ttj_families,fig:tta_families or to products of one-loop integrals. We follow the conventions of ref. [31], so that the results for all $t\bar{t}\gamma$ families can be combined straightforwardly.

We consider scattering processes of the form

$$\begin{equation} P(-p_4)\,P'(-p_5) \to t(p_1)\,\bar{t}(p_2)\,\gamma(p_3)\, , \tag*{(1)} \end{equation}$$

where the incoming partons $P$ and $P'$ can be either two gluons or a quark–antiquark pair. The external momenta $p_i$ are four-dimensional and all outgoing, so that momentum conservation reads

$$\begin{equation} p_1 + p_2 + p_3 + p_4 + p_5 = 0 \, . \tag*{(2)} \end{equation}$$

The on-shell conditions are given by

$$\begin{equation} p_1^2 = p_2^2 = m_t^2 \,, \qquad \qquad p_3^2 = p_4^2 = p_5^2 = 0 \, , \tag*{(3)} \end{equation}$$

where $m_t$ is the top-quark mass. There are six independent Lorentz invariants, which we choose as

$$\begin{equation} \vec{x} = \left(d_{12}, d_{23}, d_{34}, d_{45}, d_{15}, m_t^2 \right) \, , \tag*{(4)} \end{equation}$$

with $d_{ij} = p_i \cdot p_j$. In the following, we will occasionally use the other scalar products to simplify the expressions. For convenience, we write them here in terms of $\vec{x}$:

$$\begin{equation} \begin{split} d_{13} &= d_{45} - d_{12} - d_{23} - m_t^2 \,,\\ d_{14} &= d_{23} - d_{15} - d_{45} \,, \\ d_{24} &= d_{15} - d_{23} - d_{34} \,, \\ d_{25} &= d_{34} - d_{12} - d_{15} - m_t^2 \,, \\ d_{35} &= d_{12} - d_{34} - d_{45} + m_t^2 \,. \end{split} \tag*{(5)} \end{equation}$$

The physical region of interest for $t\bar{t}\gamma$ production is the $s_{45}$ channel. The conditions for the external momenta to be real and correspond to positive energies and physical scattering angles are given by the following inequalities in the scalar invariants,

$$\begin{equation} \begin{aligned} \quad d_{12}>0 \,, \qquad d_{23}>0 \,, \qquad d_{34}<0\,, \qquad d_{45}>0 \,, \qquad d_{15}<0 \,, \\ d_{13}>0 \,, \qquad d_{14}<0 \,, \qquad d_{24}<0 \,, \qquad d_{25}<0 \,, \qquad d_{35}<0 \,, \end{aligned} \tag*{(6)} \end{equation}$$

supplemented by $m_t^2 >0$ and the Gram-determinant constraints

$$\begin{equation} \mathbb{G}(p_i,p_j)<0 \,, \quad \mathbb{G}(p_i,p_j,p_k)>0 \,, \quad \mathbb{G}(p_i,p_j,p_k,p_l)<0 \,, \tag*{(7)} \end{equation}$$

where $i$, $j$, $k$ and $l$ take distinct values in $\{1,2,3,4,5\}$. In (7), $\mathbb{G}$ denotes the Gram determinant, defined for arbitrary four-vectors $a_i$ and $b_i$ as

$$\begin{equation} \mathbb{G}(a_1,\ldots,a_n ) = \mathbb{G} \begin{pmatrix} a_1,\ldots,a_n \\ a_1,\ldots,a_n \end{pmatrix}\,, \qquad \quad \mathbb{G}\begin{pmatrix} a_1,\ldots,a_n \\ b_1,\ldots,b_n \end{pmatrix} = {\text{det}}\left(a_i\cdot b_j \right) \,. \tag*{(8)} \end{equation}$$

The integrals of each family $\mathrm{F} \in \{ {\mathrm{PB}_D}, {\mathrm{PB}_E}, {\mathrm{PB}_F}\}$ take the form

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