  We compute, to the first non-trivial order in the $\epsilon$-expansion of a
perturbed scalar field theory, the anomalous dimensions of an infinite class of
primary operators with arbitrary spin $\ell=0,1,..$, including as a particular
case the weakly broken higher-spin currents, using only constraints from
conformal symmetry. Following the bootstrap philosophy, no reference is made to
any Lagrangian, equations of motion or coupling constants. Even the space
dimensions d are left free. The interaction is implicitly turned on through the
local operators by letting them acquire anomalous dimensions. When matching
certain four-point and five-point functions with the corresponding quantities
of the free field theory in the $\epsilon\to 0$ limit, no free parameter
remains. It turns out that only the expected discrete d values are permitted
and the ensuing anomalous dimensions reproduce known results for the weakly
broken higher-spin currents and provide new results for the other spinning
operators.
