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Consistent gauge interaction involving dynamical coupling and anomalous current

We show a possible way to construct a consistent formalism where the effective electric charge can change with space and time without destroying the invariance. In the previous work [1][2] we took the gauge coupling to be of the form $g(ϕ)j_μ(A^μ +\partial^μB)$ where $B$ is an auxiliary field, $ ϕ$ is a scalar field and the current $j_μ$ is the Dirac current. This term produces a constraint $ (\partial_μϕ) j^μ=0 $ which can be related to M.I.T bag model by boundary condition. In this paper we show that when we use the term $ g(ϕ)j_μ(A^μ - \partial^μ(\frac{1}{\square}\partial_ρA^ρ)) $, instead of the auxiliary field $ B $, there is a possibility to produce a theory with dynamical coupling constant, which does not produce any constraint or confinement. The coupling $ j_μ^{A}(A^μ - \partial^μ(\frac{1}{\square}\partial_ρA^ρ)) $ where $ j_μ^{A} $ is an anomalous current also discussed.