A Note on Spontaneously Broken Lorentz Invariance

We consider a relativistic effective field theory of vector boson whose vacuum exhibits spontaneous breaking of Lorentz invariance. We argue that a simple model of this type, considered recently by Kraus and Tomboulis, is obstructed from having a consistent ultraviolet completion according to the diagnostic recently suggested by Adams, Arkani-Hamed, Dubovsky, Nicolis, and Rattazzi. Relativistic quantum field theory, formally defined as Lorentz invariant ultraviolet (UV) fixed point and a subsequent renormalization group flow, is an elegant formalism successfully capturing the physics of elementary particles. As a quantum theory, they rest on the foundation of unitary evolution of states in the Hilbert space. States themselves form a representation of the Lorentz group, and the dynamics dictates the spectrum of states and their interactions. Gauge principle is a critical ingredient for keeping unitarity and Lorentz invariance mutually compatible in models including states of spin one (or higher). When combined with the requirement that the UV fixed point is a weakly coupled field theory with a Lagrangian formulation, the list of possible models is incredibly short. They essentially consist solely of asymptotically free gauge theories minimally coupled to matter fields which are spin zero or spin half. It is therefore quite remarkable that QCD is, and that the Standard Model admits a UV completion of, a model of this type. A closely related and familiar formalism in a theorist’s toolbox is the effective field theory (reviewed, e.g., in [2–4]). Roughly speaking, effective field theory is the result of flowing down the renormalization group from the UV to some scale of interest from the point of view of a probe or a physical process. It is not possible to track the flow of all the parameters of the renormalization group in closed analytic form. Nonetheless, one can draw useful conclusions about the strength of physical effects through systematic analysis of the dimensions of operators in the effective action, energy scales, and symmetries, within the framework of effective field theory. A productive and frequently adopted attitude in the use of effective field theory technique is to not dwell on the UV completion. By following this dictum, model builders in cosmology, astrophysics, and particle physics are free to introduce wide range of exotic models and to study their implications. In light of the restrictiveness imposed by the consistency, however, there always persist some degree of doubt that a given effective field theory model might not admit a consistent UV completion. This is especially the case for exotic models exhibiting features such as the spontaneous breaking of Lorentz invariance. It is not difficult to imagine that effective field theories arising from a consistent UV complete theories are somehow restricted, e.g., in the numerical values of the coefficients of various operators in the effective action. These conditions have not been explored systematically simply because they are difficult to derive from first principles. Along this line of thought, a simple connection relating positivity of forward scattering amplitude, possibility of superluminal propagation in a non-trivial background, and a sign of certain irrelevant operators in the effective field theory, was pointed out recently by Adams, An example of embedding of the Standard Model in a conformal UV fixed point was described in [1].