Lagrangian with a charged, massive vector boson coupled to electromagnetism

[jaded2112]

Homework Statement

For this lagrangian, I am trying to find the constraints on the constants$\eta,\kappa, g,\lambda$.

Relevant Equations

$$
\textit{L}=\frac{-1}{4}\rho^\dagger_{\alpha\beta}\rho^{\alpha\beta} - m^2\rho^\dagger_{\alpha}\rho^{\alpha}-ig[(\rho^\dagger_{\alpha\beta}\rho^{\alpha} +\eta\rho_{\alpha\beta}\rho^{\dagger\alpha})A^{\beta}+\kappa\rho^{\dagger}_{\alpha}\rho_{\beta}\textit{F}^{\alpha\beta}+
$$ $$
\frac{\lambda}{m^2}\rho^{\dagger}_{\alpha\beta}\rho^{\beta}_{\sigma}\textit{F}^{\sigma\alpha}]
$$

I need to use hermiticity and electromagnetic gauge invariance to determine the constraints on the constants. Through hermiticity, i found that the coefficients need to be real. However, I am not sure how gauge invariance would come into the picture to give further contraints. I think the $\rho^{\alpha}$ field would go to $e^{i\theta(x)}\rho^{\alpha}$, and $\rho_{\alpha\beta} \rightarrow \partial_{\alpha}(e^{i\theta(x)}\rho_{\beta})-\partial_{\beta}(e^{i\theta(x)}\rho_{\alpha})$. So, if we consider the first two terms in the interaction part of the lagrangian, $-ig(\rho^\dagger_{\alpha\beta}\rho^{\alpha} +\eta\rho_{\alpha\beta}\rho^{\dagger\alpha})A^{\beta}$, and do a local $U(1)$ transformation, we get $$-ig [\rho^{\dagger}_{\alpha\beta}\rho^{\alpha} + \eta\rho_{\alpha\beta}\rho^{\dagger\alpha}+i(\rho^{\dagger}_{\alpha}\rho^{\alpha}-\eta\rho_{\alpha}\rho^{\dagger\alpha})\partial_{\beta}\theta + i\partial_{\alpha}\theta(\eta\rho_{\beta}\rho^{\dagger\alpha}-\rho^{\dagger}_\beta \rho^{\alpha})](A^{\beta}+\partial^{\beta}\theta) $$
After this, I am kind of stuck trying to figure out what eta would be. Any help is appreciated.

 

[vanhees71]

The usual idea is to write down a "free Lagrangian" of the charged fields, which in your case seems to be the Proca Lagrangian for a massive complex vector field. Then you make the symmetry under multiplication with a phase factor local by introducing the massless em. field as a "connection" to define a gauge-covariant derivative and then write such a covariant derivative instead of a usual partial derivative. This is called "minimal coupling" and is a successful heuristics that lead to the formulation of the Standard Model of elementary particle physics.