UV properties of Galileons: Spectral Densities

We propose a picture for the UV properties of Galileon field theories. We conjecture that Galileons, and all theories incorporating the Vainshtein mechanism, fall into Jaffe's class of `non-localizable' field theories characterized by an exponential growth in their Kallen-Lehmann spectral densities. Similar properties have been argued to arise for Little String Theories and M-theory. For such theories, the notion of micro-causality and the time ordering used to define the S-matrix and correlation functions must be modified, and we give a Lorentz invariant prescription for how this can be achieved. In common with General Relativity (GR), the scattering amplitudes for Galileons are no longer expected to satisfy polynomial boundedness away from the forward scattering or fixed physical momentum transfer limits. This is a reflection of the fact that these theories are fundamentally gravitational and not local field theories. We attribute this to the existence of a locality bound for Galileons, analogous to the Giddings-Lippert locality bound for GR. We utilize the recently developed Galileon duality to define a UV finite, Lorentz invariant, quantization of a specific Galileon theory for which the energy of all states are positive definite. We perform an explicit computation of the Wightman functions for this theory, and demonstrate the exponential growth associated with the locality bound. In analogy with GR, the bound is correlated with the absence of Galileon Duality (i.e. Diffeomorphism) invariant local observables. We argue that these theories can be quantized in a manner which preserves Lorentz invariance and macro-causality and that the latter ensures that the superluminalities found in the low energy effective theory are absent in the full theory.