The Collinear Limit of the Energy-Energy Correlator

The energy-energy-correlator (EEC) observable in $e^+e^-$ annihilation measures the energy deposited in two detectors as a function of the angle between the detectors. The collinear limit, where the angle between the two detectors approaches zero, is of particular interest for describing the substructure of jets produced at hadron colliders as well as in $e^+e^-$ annihilation. We derive a factorization formula for the leading power asymptotic behavior in the collinear limit of a generic quantum field theory, which allows for the resummation of logarithmically enhanced terms to all orders by renormalization group evolution. The relevant anomalous dimensions are expressed in terms of the timelike data of the theory, in particular the moments of the timelike splitting functions, which are known to high perturbative orders. We relate the small angle and back-to-back limits to each other via the total cross section and an integral over intermediate angles. This relation provides us with the initial conditions for quark and gluon jet functions at order $alpha_s^2$. In QCD and in $mathcal{N}=1$ super-Yang-Mills theory, we then perform the resummation to next-to-next-to-leading logarithm, improving previous calculations by two perturbative orders. We highlight the important role played by the non-vanishing $beta$ function in these theories, which while subdominant for Higgs decays to gluons, dominates the behavior of the EEC in the collinear limit for $e^+e^-$ annihilation, and in $mathcal{N}=1$ super-Yang-Mills theory. In conformally invariant $mathcal{N}=4$ super-Yang-Mills theory, reciprocity between timelike and spacelike evolution can be used to express our factorization formula as a power law with exponent equal to the spacelike twist-two spin-three anomalous dimensions, thus providing a connection between timelike and spacelike approaches.