How to derive the Goldstone-Wilczek current?

Question

I am reading this [Goldstone-Wilczek's celebrated] paper [1] 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 current expectation 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 in a field-theoretic way. In particular I have two questions:

1. How to get the Feynman diagram in Fig.3? (I am aware of the similar question in [here][2] but it does not answer my question, and the Feynman diagram below is from that thread) (For clarity, the curly line is a current, the solid line is a fermion and the dash-line is  a scalar)
2. How to calculate this Feynman diagram?

I have looked on the internet and at quite a few references, and could not figure it out. Any help is appreciated.

  [1]: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.47.986
  [2]: http://physics.stackexchange.com/questions/116931/tadpole-diagram-and-vacuum

Answer 1

It is interesting to view your question above not only in light of the Goldstone-Wilczek (G-W) approach (G-W has provided a method for computing the fermion charge induced by a classical profile), but also by computing  $1/2$-fermion charge found by [Jackiw-Rebbi](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.13.3398) using G-W method. For simplicity, let us consider the 1+1D case, and let us consider the $Z_2$ domain wall and the $1/2$-charge found by Jackiw-Rebbi. The construction, valid for 1+1D systems, works as follows.

Consider a Lagrangian describing spinless fermions $\psi(x)$ coupled to a classical background profile $\lambda(x)$
via a term $\lambda\,\psi^{\dagger}\sigma_{3} \psi$. In the high temperature phase, the v.e.v. of $\lambda$ is zero and no mass is generated
for the fermions. In the low temperature phase, the $\lambda$ acquires two degenerate vacuum values $\pm  \langle \lambda \rangle$ 
that are related by a ${Z}_2$ symmetry. Generically we have
$$
\langle \lambda \rangle\,\cos\big( \phi(x) - \theta(x) \big),
$$
where we use the bosonization dictionary 
$
\psi^{\dagger}\sigma_{3} \psi
\rightarrow
\cos(\phi(x))
$
and a phase change $\Delta\theta = \pi$ captures the existence of a domain wall
separating regions with opposite values of the v.e.v. of $\lambda$.
From the fact that the fermion density 
$$
\rho(x)
=
\psi^{\dagger}(x)\psi(x) 
= 
\frac{1}{2\pi} \partial_{x} \phi(x)
,$$ 
and the current 
$$J^\mu=\psi^{\dagger}\gamma^\mu\psi =\frac{1}{2\pi}\epsilon^{\mu \nu} \partial_\nu \phi,$$
it follows that the induced charge $Q_{\text{dw}}$ on the kink by a domain wall is
$$
Q_{\text{dw}} 
= 
\int^{x_0 + \varepsilon}_{x_0 - \varepsilon}\,dx\,\rho(x)
=
\int^{x_0 + \varepsilon}_{x_0 - \varepsilon}\,dx\,\frac{1}{2\pi} \partial_{x} \phi(x)
=
\frac{1}{2\pi} \pi = 
\frac{1}{2},
$$
where $x_0$ denotes the center of the domain wall.

You can try to extend to other dimensions, but then you may need to be careful and you may not be able to use the bosonization. 

See more details of the derivation here in the [page 13](https://arxiv.org/abs/1403.5256) of [this paper](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.195134).

Comments

Hi Idear,

Thanks for the reply. Indeed G-W's current becomes very clear in the language of bosonization. It is possible as well to come to the same formula via dimension reduction, see for example http://journals.aps.org/prb/abstract/10.1103/PhysRevB.78.195424. Basically the angle $\theta$ in G-W's current correspond to the gauge field in the Quantum Hall response, which is kind of an "extension" to higher dimensions. 

But I am just curious how G-W come up with this diagram in the first place? Why we have to insert five scalar fields in the diagram, not one or two for example? 

It seems related to the soliton background, which I could not find a suitable reference... Any idea?

---

1) Yes, the dimensional reduction may work.

2) They Feynman diagram shall come up in any number of scaler field background insertion. The 5 dashed lines are just schematic. You can consider just 1 dashed line is good enough. I remember that I compute it before directly through this Feynman diagram, but it is a bit long calculation. I have to dig it out, it is somewhere in my files. In any case of two approaches, the result shall be the same.

---

That would make much more sense, I was struggling to understand why five insertion gives only one derivative of the scalar field... Having say that, it will be great if you could sketch your calculation here, just briefly. Thank you.