We study the relation between the dilaton action and sigma models for the Goldstone bosons of the spontaneous breaking of the conformal group. We argue that the relation requires that the sigma model is diffeomorphism invariant. The origin of the WZW
terms for the dilaton is clarified and it is shown that in this approach the dilaton WZW term is necessarily accompanied by a Weyl invariant term proposed before from holographic considerations.
We argue that when conformal symmetry is spontaneously broken the trace anomalies in the broken and unbroken phases are matched. This puts strong constraints on the various couplings of the dilaton. Using the uniqueness of the effective action for th
e Goldstone supermultiplet for broken ${cal N}=1$ superconformal symmetry the dilaton effective action is calculated.
We consider gluon and gluino scattering amplitudes in large N beta-deformed N=4 SYM with real beta. A direct inspection of the planar diagrams shows that the scattering amplitudes to all orders in perturbation theory are the same as in the undeformed
N=4 SYM theory. Using the dual sigma-model description, we find the same equality at strong coupling to all orders in the sigma-model loop expansion. Finally, we show that the same analysis holds for gluon scattering amplitudes in a three-parameter deformation of planar N=4 SYM that breaks all the supersymmetry.
