The Next-to-Minimal Coleman-Weinberg Model

In the standard model (SM) the condition that the Higgs mass parameter vanishes is stable under radiative corrections and yields a theory that can be renormalized using dimensional regularization. Thus, this model allows to predict the Higgs boson mass. However, it is phenomenologically ruled out in its minimal version. Here, we present a phenomenologically viable, minimal extension which only includes an additional SM singlet and a U(1)X gauge symmetry. Phys. Lett B, to be published The particle content of the standard model of elementary particle physics (SM) is minimal in that it only contains particles that have already been observed plus one Higgs doublet needed to break the electroweak symmetry. The Lagrangian describing the interactions of the theory is obtained by forming all gauge-invariant and lorentz-invariant combinations of fields with dimensions less than four. The coefficients are in general arbitrary factors which have to be determined by experiment. In order to obtain a renormalizable theory none of these terms can be omitted as they are needed to cancel divergent contributions coming from quantum correction. The exception are terms whose absence enhances the symmetry of the theory. An example is chiral symmetry in the absence of a tree-level mass term for one or more fermions[1]. The potential of the SM contains only one parameter with dimension (mass) [and none with dimension (mass)], namely the Higgs mass parameter μ. From this mass term arises the most sever problem of the SM: the hierarchy problem[2]. Since the condition μ = 0 is not protected by any symmetry in the SM the large hierarchy between the electroweak scale and the Planck scale μ/MP ≃ 10 ≪ 1 can only be achieved by excessive fine-tuning. Rather than to explain the smallness of μ it is may be conceptually more convincing to assume μ = 0 altogether. The electroweak breaking in such a model can be achieved by a negative Higgs self-coupling at some low scale Λ due to the renormalization effects of the gauge couplings. In this model, with one parameter less than the SM ( i.e. μ = 0 or μ ≪ Λ) the Higgs mass (in units of the Higgs vacuum expectation value) is determined by the gauge and Yukawa couplings. This idea was first introduced by Coleman and Weinberg in ref. [4] were an upper limit on the Higgs mass (the CW bound) of mh ∼< 10 GeV and implicitly an upper limit on the top quark mass mt ∼< mZ was established. Unfortunately, both limits are by now in contradiction with experiment[5][6]. Nontheless, the study of models with particular conditions for the Higgs mass parameters is of continued interest[7][8] In this letter, we will present a simple extension of the CW model that is still phenomenologically viable. We assume that the Higgs mass parameter is generated dynamically as the vacuum expectation value (VEV) of a singlet field S. The tree-level potential of our model without mass terms can be written as V0 = λφ 2 (φφ) + λS 2 (SS) − λX(φφ)(SS) , (1) where φ denotes the SM Higgs doublet. This potential has an additional U(1)X symmetry that transforms S → exp(iα)S. By promoting this global symmetry to a local symmetry we introduce a gauge coupling gX that can trigger spontaneous symmetry breaking. In addition, we avoid the existence of a massless goldstone boson. The β functions are 16πβφ = 16π β φ + λ 2 X , Only in supersymmetric extensions of the SM can scalar mass terms be absent[3]