NNLL Resummation for Projected Three-Point Energy Correlator

TL;DR

This paper develops NNLL resummation for the projected three-point energy correlator, calculating two-loop jet functions and applying the results to $e^+e^-$ and $pp$ processes, enhancing precision in jet substructure analysis.

Contribution

It introduces the first NNLL resummation for the projected three-point energy correlator, including two-loop jet function calculations using IBP and direct integration methods.

Findings

Improved convergence of the energy correlator distribution.

Suppressed hadronization effects in the collinear limit.

Potential for precise strong coupling constant determination.

Abstract

The projected energy correlator measures the energy deposited in multiple detectors as a function of the largest angular distance $x_L = (1 - \cos\chi_L)/2$ between detectors. The collinear limit $x_L\to 0$ 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 $pp\rightarrow jj$ 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.