Collider experiments often exploit information about the quantum numbers of final state hadrons to maximize their sensitivity, with applications ranging from the use of tracking information (electric charge) for precision jet substructure measurement
s, to flavor tagging for nucleon structure studies. For such measurements perturbative calculations in terms of quarks and gluons are insufficient, and non-perturbative track functions describing the energy fraction of a quark or gluon converted into a subset of hadrons (e.g. charged hadrons), must be incorporated. Unlike fragmentation functions, track functions describe correlations between hadrons, and therefore satisfy complicated non-linear evolution equations whose structure has so far eluded calculation beyond the leading order. In this Letter we develop an understanding of track functions, and their interplay with energy flow observables, beyond the leading order, allowing them to be used in state-of-the-art perturbative calculations for the first time. We identify a shift symmetry in the evolution of their moments that fixes their structure, and we explicitly compute the evolution of the first three moments at next-to-leading order, allowing for the description of up to three-point energy correlations. We then calculate the two-point energy correlator on charged particles at $O(alpha_s^2)$, illustrating explicitly that infrared singularities in perturbation theory are absorbed by moments of the track functions, and also highlighting how these moments seamlessly interplay with modern techniques for perturbative calculations. Our results extend the boundaries of traditional perturbative QCD, enabling precision perturbative predictions for energy flow observables sensitive to the quantum numbers of hadronic states.
We apply results proved in [Li19] to the linear order expansions of non-trivial free homogeneous structures and the universal n-linear order for $ngeq 2$, and prove the simplicity of their automorphism groups.
