I am reading this Goldstone-Wilczek celebrated's paper on fractional quantum number. In particular they derived for the following Dirac Lagrangian ($\phi_1$ and $\phi_2$ are scalar fields)
$$\mathscr{L}=i\bar{\psi}\gamma^\mu\partial_\mu\psi+g\bar{\psi}(\phi_1+i\gamma_5\phi_2)\psi$$
that the expectation value of current reads ($\theta=\tan^{-1}(\phi_2/\phi_1)$)
$$\langle j^\mu\rangle=\frac{1}{2\pi}\epsilon^{\mu\nu}\partial_\nu\theta$$,
which is equation (2). It is possible to understand it via dimensional reduction or bosonization. But I would like to understand it from a field-theoretic way. In particular I have two questions:
- How to get the Feynman diagram in Fig.3? (I am aware the similar question in here but it does not answer my question, and the Feynman diagram below is from that thread) (For clarity, curly line is current, solid line is fermion and dash-line is scalar)
- How to calculate this Feynman diagram?
I have look on the internet and quite a few references and could not figure it out. Any helps are appreciated.