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Jets and their substructure play a central role in many analyses at the LargeHadron Collider (LHC). To improve the precision of measurements, as well as toenable measurement of jet substructure at increasingly small angular scales,tracking information is often used due to its superior angular resolution androbustness to pile-up. Calculations of track-based observables involvenon-perturbative track functions, that absorb infrared divergences inperturbative calculations and describe the transition to charged hadrons. Theinfrared divergences are directly related to the renormalization groupevolution (RGE), and can be systematically computed in perturbation theory.Unlike the standard DGLAP evolution, the RGE of the track functions isnon-linear, encoding correlations in the fragmentation process. We compute thenext-to-leading order (NLO) evolution of the track functions, which involves inits kernel the full $1\rightarrow3$ splitting function. We discuss in detailhow how we implement the evolution equation numerically, and illustrate thesize of the NLO corrections. We also show that our equation can be viewed as amaster equation for collinear evolution at NLO, by illustrating that byintegrating out specific terms, one can derive the evolution for any $N$-hadronfragmentation function. Our results provide a crucial ingredient for obtainingtrack-based predictions for generic measurements at the LHC, and for improvingthe description of the collinear dynamics of jets.

Multi-Collinear Splitting Kernels for Track Function Evolution