NNLL resummation for projected three-point energy correlator

The projected energy correlator measures the energy deposited in multiple detectors as a function of the largest angular distance x_L = (1 − cos χ_L)/2 between detectors. The collinear limit x_L → 0 of the projected energy correlator is particularly interesting for understanding the jet-substructures, while the large logarithms of x_L could potentially spoil the perturbation theory and must be resummed. As a necessary ingredient for its resummation at next-to-next-to-leading logarithmic (NNLL) accuracy, we calculate the two-loop jet functions for the projected three-point energy correlator (E3C), using direct integration method and the parameter space Integration-by-Part (IBP) method. We then present the NNLL resummation for e⁺e⁻ annihilation and an approximate NNLL resummation for pp → jj process, where the two-loop hard constant is estimated in the latter case. The convergence is improved and the hadronization effect in the collinear limit is suppressed when considering the ratio of E3C distribution to two-point energy-energy correlator (EEC). Our results show potential in precision determination of strong coupling constant using energy correlators from both e⁺e⁻ data and pp data.