# Higgs Mechanism and Lorentz transformation of currents

1. Mar 23, 2016

### CAF123

1. The problem statement, all variables and given/known data
Consider the Higgs mechanism lagrangian, $$\mathcal L = (D_{\mu} \phi)^* (D^{\mu} \phi) -\mu^2 (\phi^* \phi) - \lambda (\phi^* \phi)^2 - \frac{1}{4}F_{\mu \nu}F^{\mu \nu},$$ with $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$ and $D_{\mu} = \partial_{\mu} + iqA_{\mu}$. One can show that then $\partial_{\mu}F^{\mu \nu} = J^{\nu}$, where $J_{\nu} = iq ((D^{\nu} \phi)^* \phi - \phi^* D^{\nu} \phi)$. Following spontaneous symmetry breaking, the massless gauge field acquires mass and let's suppose it has the following oscillatory behaviour $$A_{\mu} = \cos (M t) \epsilon^{1}_{\mu}$$ where $\epsilon^1_{\mu} = (0,1,0,0)$ What is the four vector current and what is it if I apply a lorentz boost in the z direction?

2. Relevant equations

lorentz boost in z direction is $$t' = \gamma \left( t -\frac{vz}{c^2}\right)$$ and $z' = \gamma(z-vt)$

3. The attempt at a solution

I can evaluate the components of $A_{\mu}$ to get $A_{\mu} = (0,\cos (Mt), 0, 0)$ and then I could just evaluate the four current by evaluating each of its four components separately. When I am taking a lorentz transformation, would I just send $t \rightarrow t'$ and use the equation given above in relevant equations? Or would I also need to consider transformation of the differential operator as well?

Thanks!

2. Mar 28, 2016