Renormalization Group Flows for Track Function Moments

Track functions describe the collective effect of the fragmentation of quarks and gluons into charged hadrons, making them a key ingredient for jet substructure measurements at hadron colliders, where track-based measurements offer superior angular resolution. The first moment of the track function, describing the average energy deposited in charged particles, is a simple and well-studied object. However, measurements of higher-point correlations of energy flow necessitate a characterization of fluctuations in the hadronization process, described theoretically by higher moments of the track function. In this paper we derive the structure of the renormalization group (RG) evolution equations for track function moments. We show that energy conservation gives rise to a shift symmetry that allows the evolution equations to be written in terms of cumulants, $\kappa(N)$, and the difference between the first moment of quark and gluon track functions, $\Delta$. The uniqueness of the first three cumulants then fixes their all-order evolution to be DGLAP, up to corrections involving powers of $\Delta$, that are numerically suppressed by an effective order in the perturbative expansion for phenomenological track functions. However, at the fourth cumulant and beyond there is non-trivial RG mixing into products of cumulants such as $\kappa(4)$ into $\kappa(2)^2$. We analytically compute the evolution equations up to the sixth moment at $\mathcal{O}(\alpha_s^2)$, and study the associated RG flows. These results allow for the study of up to six-point correlations in energy flow using tracks, paving the way for precision jet substructure at the LHC.