  The cross-couplings among several Weyl gravitons (described in the free limit
by a sum of linearized Weyl actions) in the presence of a scalar field are
studied with the help of the deformation theory based on local BRST cohomology.
Under the hypotheses of locality, smoothness of the interactions in the
coupling constant, Poincare invariance, (background) Lorentz invariance, and
the preservation of the number of derivatives on each field, together with the
supplementary assumption that the internal metric defined by the sum of Weyl
lagrangians is positively defined, we prove that there are no consistent
cross-interactions among different Weyl gravitons in the presence of a scalar
field. The couplings of a single Weyl graviton to a scalar field are also
discussed.
