  We study implications of exact conformal invariance of scalar quantum field
theories at the critical point in non-integer dimensions for the evolution
kernels of the light-ray operators in physical (integer) dimensions. We
demonstrate that all constraints due the conformal symmetry are encoded in the
form of the generators of the collinear sl(2) subgroup. Two of them, S_- and
S_0, can be fixed at all loops in terms of the evolution kernel, while the
generator of special conformal transformations, S_+, receives nontrivial
corrections which can be calculated order by order in perturbation theory.
Provided that the generator S_+ is known at the k-1 loop order, one can fix the
evolution kernel in physical dimension to the k-loop accuracy up to terms that
are invariant with respect to the tree-level generators. The invariant parts
can easily be restored from the anomalous dimensions. The method is illustrated
on two examples: The O(n)-symmetric phi^4 theory in d=4 to the three-loop
accuracy, and the su(n) matrix phi^3 theory in d=6 to the two-loop accuracy. We
expect that the same technique can be used in gauge theories e.g. in QCD.
