  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.
