Spontaneous breaking of vector symmetries and the nondecoupling light Higgs particle.

Using a four fermion interaction Lagrangian, we demonstrate that the spontaneous breaking of vector symmetries requires the existence of a light (comparing with the heavy fermion mass) scalar particle and the low energy effective theory (the σ model) obtained after integrating out heavy fermion degrees of freedom is asymptotically a renormalizable one. When applying the idea to the electroweak symmetry breaking sector of the standard model, the Higgs particle’s mass is of the order of the electroweak scale. †) e-mail address: [email protected] PACS Number(s): 11.30.Qc, 12.60.Fr The spontaneous breaking of vector symmetries (SBVS) is an interesting subject in quantum field theory and also in particle physics, as long as it remains to be a possibility as a character of nature. In a previous paper [1] we have made an attempt in considering the breaking of the electroweak symmetry as a consequence of the SBVS between fermions with heavy bare masses. The motivation of such a consideration is to break the electroweak symmetry dynamically but with least influence on the low energy physics [1]. We modeled SBVS by a low energy effective Higgs–Yukawa interaction and, after integrating out the heavy fermion fields in the mean field approximation of the Higgs particle, estimated the low energy residual effects of these heavy fermions. It is shown that the heavy fermion fields are essentially decoupling at low energies, except that they can generate massless Goldstone excitations to be absorbed by the weak gauge fields. The low energy effective theory (the standard model) should therefore, be weakly interacting – a picture different from the technicolor models. Nothing can be said about the mass of the composite Higgs particle, within the context of the effective Higgs–Yukawa model in ref. [1]. A question then arise: Can it be as heavy as the heavy fermion mass? From general physical consideration we know that this should not be the case. Because if the Higgs particle’s mass is heavy enough, the remaining fields must be in a strongly interacting system because of the well known tree–level unitarity argument [2] – a result contrary to our motivation and the general expectation from the decoupling phenomena. The aim of this paper is to resolve this Higgs mass ambiguity, using a model of four fermion interactions with the dynamically generated SBVS. The SBVS through four fermion interactions was first analysed by Preskill and Weinberg [3] to study the possible violation of the ”persistent mass condition”. For a four fermion interaction with a global vector symmetry, there are primarily two scales, the cutoff scale Λ and the bare fermion mass M(M < Λ), and in addition, the interaction strength is characterized by a dimensionless coupling constant, G. Preskill and Weinberg have shown that, for a given cutoff Λ and a sufficiently large G, there exists a critical value Mc. When the fermion mass is below this critical point, M < Mc, the vector (isospin) symmetry is spontaneously broken down. As a consequence, there exist massless particles composed of massive constituents leading to a violation of the persistent mass condition. When M exceeds Mc, the system is in symmetry phase and the decoupling phenomenon occurs. The symmetry breaking is of second order and characterized by a new scale m (the fermion mass splitting) obtained after some fine tuning. For our purpose, the four fermion interaction Lagrangian can be written as L = Ψ̄ (i∂/−M) Ψ − G NcΛ [(Ψ̄ρ3Ψ ) + (Ψ̄ρ1~τΨ )] , (1) where Ψ = (ψ1, ψ2) T and ψ1,2 are SU(2) isospin doublets. The index i refers to the ”color” degree of freedom and runs over 1 to Nc. We assume Nc is large in the following. τi are generators of the SU(2) isospin group and ρi are Pauli matrices of One can add more terms and couplings, see for example ref. [1]. The present Lagrangian is the minimal one suitable for the discussion in the present paper.