The Energy-Energy Correlation in the back-to-back limit at N$^3$LO and N$^3$LL$^\prime$

Abstract

We present the analytic formula for the Energy-Energy Correlation (EEC) in electron-positron annihilation computed in perturbative QCD to next-to-next-to-next-to-leading order (N$^3$LO) in the back-to-back limit. In particular, we consider the EEC arising from the annihilation of an electron-positron pair into a virtual photon as well as a Higgs boson and their subsequent inclusive decay into hadrons. Our computation is based on a factorization theorem of the EEC formulated within Soft-Collinear Effective Theory (SCET) for the back-to-back limit. We obtain the last missing ingredient for our computation - the jet function - from a recent calculation of the transverse-momentum dependent fragmentation function (TMDFF) at N$^3$LO. We combine the newly obtained N$^3$LO jet function with the well known hard and soft function to predict the EEC in the back-to-back limit. The leading transcendental contribution of our analytic formula agrees with previously obtained results in $\mathcal{N} = 4$ supersymmetric Yang-Mills theory. We obtain the $N=2$ Mellin moment of the bulk region of the EEC using momentum sum rules. Finally, we obtain the first resummation of the EEC in the back-to-back limit at N$^3$LL$^\prime$ accuracy, resulting in a factor of $\sim 4$ reduction of uncertainties in the peak region compared to N$^3$LL predictions.