The $q_{\mathrm{T}}$ and $\Delta\phi_{t\bar{t}}$ spectra in top-antitop hadroproduction at NNLL+NNLO: the interplay of soft-collinear resummation and Coulomb singularities

In this paper, we present the resummation-improved differential transverse momentum and azimuthal decorrelation cross sections, $d\sigma_{t\bar{t}}/dq_{\mathrm{T}}$ and $d\sigma_{t\bar{t}}/d\Delta\phi_{t\bar{t}}$, in top-antitop pair production at the LHC. Our calculation is based on the observation that both cross sections are dominated by topologies where the top-quark pair is well separated, expressed in their relative velocity $\beta_{t\bar{t}}\sim\mathcal{O}(1)$, at colliding energies of $\sqrt{s}=13\,\text{TeV}$ or higher. Therefore, the asymptotic behaviour in the limits $q_{\mathrm{T}}\to0$ and $\Delta\phi_{t\bar{t}}\to0$ can mostly be captured by the soft and collinear resummation in the HQET$+$SCET framework. Nevertheless, starting at NNLL, Coulomb singularities emerge in the threshold regime, $\beta_{t\bar{t}}\to0$, in both the hard sector and its evolution kernels, leading to unphysical results upon integration over the entire $\beta_{t\bar{t}}$ range. To this end, two prescriptions, dubbed the D- and R-prescription, are introduced to regularise these Coulomb singularities. They embody two fundamentally different methods to truncate the threshold enhanced terms, rendering their contribution finite. In the absence of a combined threshold and small-transverse-momentum resummation, we present a quantitative assessment of the ambiguity introduced by the choice of prescription, itself a test of the sensitivity of our calculation to such threshold enhancements, for both the $d\sigma_{t\bar{t}}/dq_{\mathrm{T}}$ and $d\sigma_{t\bar{t}}/d\Delta\phi_{t\bar{t}}$ spectra.