Possible Electric Charge Nonconservation and Dequantization in Su(2)× U(1) Models with Hard Symmetry Breaking

We study a novel type of extensions of the Standard Model which include a hard mass term for the U(1) gauge field and, optionally, the additional scalar multiplets spontaneously violating the electric charge conservation. Contrary to the case of abelian massive electrodynamics, in these theories the massiveness of photon necessarily implies non-conservation (and also dequantization) of the electric charge (even in the absence of spontaneous breakdown of the electromagnetic symmetry). On the other hand, unexpectedly, there exist models with charge non-conservation where it is possible to keep the photon mass zero (at least, at the tree level). Typeset using REVTEX 1 1. In the past, there have been many papers exploring the possibility that the photon may have non-zero mass [1]. At first, these works were made in the context of massive electrodynamics, which is an abelian U(1) theory with the added photon mass term 1 2 m2A2μ. The characteristic feature of such a theory is that the conservation of the electric charge is not violated by the presence of the photon mass term. The reason is that the photon mass term violates the local gauge invariance but not the global one. Thus, massive electrodynamics suggests that it it is possible to have a massive photon along with the exact conservation of the electric charge. Later, there emerged another class of theories with massive photon: non-abelian gauge theories with electric charge non-conservation [2–9]. The primary emphasis of these works was not on the massiveness of photon , but on the study of the possible electric charge nonconservation in gauge theories. Yet the massiveness of photon appeared to be an automatic consequence of the violation of the electric charge conservation. One of the important discoveries made in those works was the close relation between the two ideas: electric charge (non)conservation and electric charge (de)quantization [2,4,10]. Given all previous works, one important question still remains unexplored: is it possible to have massive photon and exact electric charge conservation in realistic theories? Of course, it is possible within U(1) massive electrodynamics, but it is not a realistic theory. What we are interested in is this: can the Standard SU(2) × U(1) model be extended or modified in such a way as to have both the massive photon and the exact electric charge conservation simultaneously. At first sight all we have to do is to give a hard mass to the U(1) gauge boson Bμ. That would presumably make photon massive without spoiling the electric charge conservation. However, we will show that it turns out not to be the case. A related but different question that we are going to consider is this: can we have a massless photon in a theory with electric charge non-conservation? One reason to ask this question is the very stringent experimental bound on the photon mass: mγ < 10 −24 GeV or even 10 GeV [11]. This bound places a very tight constraints on any theory with electric 2 charge non-conservation and one is naturally curious how to evade it. Naively, one might think that the answer to the above question is negative. Yet in this work we will construct examples of realistic SU(2)×U(1) models in which the photon mass is zero at the tree level but the electric charge is not conserved. 2.Let us first consider a model which differs from the Standard Model only in one point: its lagrangian contains a mass term for the U(1) gauge field B (before spontaneous symmetry breaking): L = L0 + 1 2 mB μ, (1) where L0 is the Standard Model lagrangian. After spontaneous symmetry breaking, we diagonalize the gauge boson mass matrix and obtain the physical fields A and Z which we identify with the photon and Z-boson: A3μ = Zμ cos θ ′ + Aμ sin θ ′ (2) Bμ = Aμ cos θ ′ − Zμ sin θ. (3) where the mixing angle sin θ is different from the Weinberg angle of the Standard Model (sin θ): sin θ = sin θ + m M Z (1− e 2 sin θ + e − sin θ) ≈ sin θ + 0.64 2