Projected transverse momentum resummation in top-antitop pair production at LHC

The transverse momentum distribution of the $t\bar{t}$ system is of both experimental and theoretical interest. In the presence of the azimuthally asymmetric divergences, pursuing the resummation at the high logarithmic precision is rather demanding in general. In this paper, we propose the projected transverse momentum spectrum $\mathrm{d}\sigma_{t\bar{t}}/\mathrm{d} q_{\tau}$, which is derived from the classical $\vec{q}_{\mathrm{T}}$ spectrum by integrating out the rejection component $q_{\tau_{\perp}}$ with respect to a reference unit vector $\vec{\tau}$, to serve as an alternative resolution to curb those asymmetric influences, in additional to the azimuthally averaged case $\mathrm{d}\sigma_{t\bar{t}}/\mathrm{d} |\vec{q}_{\mathrm{T}}|$. In the context of the effective field theories, SCET$_{\mathrm{II}}$ and HQET, we will demonstrate that in spite of the $q_{\tau_{\perp}}$ integrations, the leading asymptotic terms of $\mathrm{d}\sigma_{t\bar{t}}/\mathrm{d} q_{\tau}$ still observe the factorization pattern in terms of the hard, beam, and soft functions in the vicinity of $ q_{\tau}=0$~GeV. With the help of the renormalisation group equation techniques, we evaluate the resummed spectra $\mathrm{d}\sigma_{t\bar{t}}/\mathrm{d} q_{\mathrm{T,in}}$, $\mathrm{d}\sigma_{t\bar{t}}/\mathrm{d} q_{\mathrm{T,out}}$, and $\mathrm{d}\sigma_{t\bar{t}}/\mathrm{d} \Delta\Phi_{t\bar{t}}$ at NLL+NLO, NNLL+NNLO, and approximate NNLL'+NNLO accuracy. This is the first time the azimuthal spectrum $\Delta\Phi_{t\bar{t}}$ is appraised at or beyond the NNLL level including a consistent treatment of both beam collinear and soft radiation.