We consider the conformal higher spin (CHS) theory in d = 4 that contains the
s = 1 Maxwell vector, s = 2 Weyl graviton and their higher spin s = 3, 4, . . . counterparts
with higher-derivative s kinetic terms. The interacting action for such theory can be
found as the coefficient of the logarithmically divergent part in the induced action for
sources coupled to higher spin currents in a free complex scalar field model. We explicitly
determine some cubic and quartic interaction vertices in the CHS action from scalar loop
integrals. We then compute the simplest tree-level 4-particle scattering amplitudes 11→11,
22→22 and 11→22 and find that after summing up all the intermediate CHS exchanges they
vanish. This generalises the vanishing of the scattering amplitude for external conformal
scalars interacting via the exchange of all CHS fields found earlier in arXiv:1512.08896.
This vanishing should generalise to all scattering amplitudes in the CHS theory and as
in the conformal scalar scattering case should be a consequence of the underlying infinite
dimensional higher spin symmetry that extends the standard conformal symmetry.