Abstract
In this paper we give a variation of the gauge procedure which employs a scalar gauge field, $B (x)$, in addition to the usual vector gauge field, $A_\mu (x)$. We study this variant of the usual gauge procedure in the context of a complex scalar, matter field $\phi (x)$ with a U(1) symmetry. We will focus most on the case when $\phi$ develops a vacuum expectation value via spontaneous symmetry breaking. We find that under these conditions the scalar gauge field mixes with the Goldstone boson that arises from the breaking of a global symmetry. Some other interesting features of this scalar gauge model are: (i) The new gauge procedure gives rise to terms which violate C and CP symmetries. This may have have applications in cosmology or for CP violation in particle physics; (ii) the existence of mass terms in the Lagrangian which respect the new extended gauge symmetry. Thus one can have gauge field mass terms even in the absence of the usual Higgs mechanism; (iii) the emergence of a sine-Gordon potential for the scalar gauge field; (iv) a natural, axion-like suppression of the interaction strength of the scalar gauge boson.
Highlights
JHEP05(2014)096 where V (φ) is some scalar self interaction potential
We find that under these conditions the scalar gauge field mixes with the Goldstone boson that arises from the breaking of a global symmetry
We briefly review some previous work dealing with scalar gauge fields or other non-traditional ways of gauging a symmetry without a vector gauge boson
Summary
Starting with the same complex scalar field from (1.1) we gauge the phase symmetry of φ by introducing a real, scalar B(xμ) and two types of covariant derivatives as. Add a term like (Aμ − ∂μB)(Aμ − ∂μB) to the Lagrangian which is invariant with respect to the gauge field part only of the gauge transformation in (2.2). This condition ensures that the kinetic energy term for the scalar field φ has the standard form ∂μφ∂μφ∗ One could accomplish this as well by rescaling φ, but here we chose to accomplish this by placing conditions on the ci’s. If we assume spontaneous symmetry breaking (SSB) (i.e. the complex scalar field develops a vacuum expectation value like φ∗φ = ρ20) and if we choose to work in unitary gauge, such terms disappear. In the case of SSB where at low energies φ∗φ ≈ φ∗φ = ρ20 (i.e. φ∗φ is approximately constant), we see that the gauge field B is a physical field due to the way it appears in the physical unitary gauge We will discuss this point further below in section (3). The equation of motion for φ∗ is the complex conjugate of (2.9)
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