Living Without Supersymmetry -- the Conformal Alternative and a Dynamical Higgs Boson

Abstract

We show that key results of supersymmetry can be achieved via conformal symmetry. We propose that the Higgs boson be a dynamical bound state rather than an elementary scalar, so that there is no quadratic divergence self-energy problem for it and no need to invoke supersymmetry to resolve it. We study a conformal invariant theory of interacting fermions and gauge bosons, in which there is scaling with anomalous dimensions and dynamical symmetry breaking, with the dynamical dimension of $\bar{\psi}\psi$ being reduced from 3 to 2. With this reduction we augment the theory with a then renormalizable 4-fermion interaction with dynamical dimension equal to 4. We reinterpret the theory as a renormalizable version of the Nambu-Jona-Lasinio (NJL) model, with the gauge theory sector with its now massive fermion being the mean field and the 4-fermion interaction being the residual interaction. It is this residual interaction that generates dynamical Goldstone and Higgs states, states that, as noted by Baker and Johnson, the gauge theory sector itself does not possess. The Higgs boson is found to be a narrow resonance just above threshold. We couple the theory to conformal gravity, with the interplay between conformal gravity and the 4-fermion interaction taking care of the vacuum energy problem. With conformal gravity being a consistent quantum gravity theory there is no need for string theory with its supersymmetric underpinnings. With conformal gravity fits to galactic rotation curves and the accelerating universe not needing dark matter, there is no need to introduce supersymmetry for either the vacuum energy problem or to provide a potential dark matter candidate. We propose that it is conformal symmetry rather than supersymmetry that is fundamental, with the theory of nature being a locally conformal, locally gauge invariant, non-Abelian NJL theory.