Question

Challenge question:

Using a 1D Albelian U(1) transformation to spontaneously break symmetry.

Answer 1

Using a 1D Albelian U(1) transformation to spontaneously break symmetry.

Consider the following Lagrangian (density):
\[\mathcal{L} = (\partial^\mu + ieA^\mu)\phi^*(\partial_\mu-ieA_\mu)-m^2\phi^*\phi-\lambda(\phi^*\phi)^2-\frac14F_{\mu\nu}F^{\mu\nu}\]
for a complex field \(\phi = \frac{1}{\sqrt{2}}(\phi_1+i\phi_2)\) which has a non-zero vacuum expectation value \(v\) that can be found by minimising the potential \(V(\phi)=\frac12 m^2\phi^2+\frac14\lambda\phi^4\)

Find \(v\) and use the following transformation that introduces a small deviation from the minimum of the potential:
\[\phi\to\sqrt\frac12(v+h(x))e^{i\theta(x)/v}\]\[A_\mu \to A_\mu+\frac{1}{ev}\partial_\mu\theta\]
to derive a new version of the Lagrangian that includes a massive scalar boson and a massive vector gauge boson.