$\nu$-point energy correletors with FastEEC: small-$x$ physics from LHC jets

In recent years, energy correlators have emerged as a powerful tool for studying jet substructure, with promising applications such as probing the hadronization transition, analyzing the quark-gluon plasma, and improving the precision of top quark mass measurements. The projected $N$-point correlator measures correlations between $N$ final-state particles by tracking the largest separation between them, showing a scaling behavior related to DGLAP splitting functions. These correlators can be analytically continued in $N$, commonly referred to as $\nu$-correlators, allowing access to non-integer moments of the splitting functions. Of particular interest is the $\nu \to 0$ limit, where the small momentum fraction behavior of the splitting functions requires resummation. Originally, the computational complexity of evaluating $\nu$-correlators for $M$ particles scaled as $2^{2M}$, making it impractical for real-world analyses. However, by using recursion, we reduce this to $M 2^M$, and through the FastEEC method of dynamically resolving subjets, $M$ is replaced by the number of subjets. This breakthrough enables, for the first time, the computation of $\nu$-correlators for LHC data. In practice, limiting the number of subjets to 16 is sufficient to achieve percent-level precision, which we validate using known integer-$\nu$ results and convergence tests for non-integer $\nu$. We have implemented this in an update to FastEEC and conducted an initial study of power-law scaling in the perturbative regime as a function of $\nu$, using CMS Open Data on jets. The results agree with DGLAP evolution, except at small $\nu$, where the anomalous dimension saturates to a value that matches the BFKL anomalous dimension.