  The method of large spin perturbation theory allows to analyse conformal
field theories (CFT) by turning the crossing equations into an algebraic
problem. We apply this method to a generic CFT with weakly broken higher spin
(HS) symmetry, to the first non-trivial order in the breaking parameter. We
show that the spectrum of broken currents, for any value of the spin, follows
from crossing symmetry. After discussing a generic model of a single scalar
field, we focus on vector models with $O(N)$ global symmetry. We rediscover the
spectrum of several models, including the $O(N)$ Wilson-Fisher model around
four dimensions, the large $O(N)$ model in $2<d<4$ and cubic models around six
dimensions, not necessarily unitary. We also discuss models where the
fundamental field is not part of the spectrum. Examples of this are weakly
coupled gauge theories and our method gives an on-shell gauge invariant way to
study them. At first order in the coupling constant we show that again the
spectrum follows from crossing symmetry, to all values of the spin. Our method
provides an alternative to usual perturbation theory without any reference to a
Lagrangian.
