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Renormalization group evolution equations describing the scale dependence of quantities in QCD play a central role in the interpretation of experimental data. Arguably the most important evolution equations for collider physics applications are the DGLAP equations, which describe the evolution of a quark or gluon fragmenting into hadrons, with only a single hadron identified at a time. Their spacelike counterparts have been widely applied to observables in hadron­-hadron collisions, DIS, etc. In recent years, the study of the correlations of energy flow within jets, motivated by high energy jets in hadron colliders, has come to play a central role at collider experiments, providing innovative advances in studying the Standard Model. This necessitates an understanding of correlations, going beyond the standard DGLAP paradigm. In this work, a general renormalization group equation describing the collinear dynamics that account for correlations in the fragmentation is derived at next­-to­-leading order (NLO), by analytically calculating a certain class of jet functions. The work also illustrates how to reduce the equation to the DGLAP equation and equations of multi­-hadron fragmentation functions. 

The aforementioned general collinear evolution equation is obtained through a universal non-perturbative function, the so­-called "track function". This function provides a field­-theoretic approach to calculating track­-based observables. Thus, it is of phenomenological interest: jets consist of collimated sprays of hadrons, and jet substructure measurements strongly urge to resolve small-angle information where working exclusively with tracks (charged particles) can help. In this work, based on the track function formalism, the two­- to six­-point energy correlators on tracks within a jet are analytically computed at NLL, and their ability to image the confinement transition is studied with Pythia. Additionally, the work has recently achieved the NNLL resummation (three-loop DGLAP evolution) for the track energy-energy correlation. The results are crucial for the theoretical interpretation of recent (and future) experimental measurements of energy correlators.