Projected Energy Correlators: Two-Loop Jet Functions and NNLL Resummation

We present the next-to-next-to-leading logarithmic (NNLL) collinear resummation of projected $N$-point energy correlators (ENCs) up to $N=6$, matched to fixed-order predictions at NLO, in both electron-positron annihilation and Higgs decay to gluons. The key new ingredient is the two-loop jet function for $N=4,5,6$, which we compute semi-analytically using Integration-by-Parts and differential equations. We further include the leading non-perturbative corrections for ENCs, described by two universal soft matrix elements $\overline{\Omega}_{1q},\overline{\Omega}_{1g}$ of order $\Lambda_{\rm QCD}$, whose evolution is governed by anomalous dimensions for $(N-1)$-point correlators. The matched distributions are compared with parton-shower simulations from Pythia8 and Herwig7, and we study the sensitivity of both the absolute spectra and their ratios to the two-point energy correlator under variations of $\alpha_s$ and $\overline{\Omega}_{1q,1g}$. Our results show that higher-point projected energy correlators are now under quantitative control at NNLL accuracy, opening the door to future $\alpha_s$ extractions with complementary systematics.