I have two question about axion strings:

- Why their appearance is connected with spontaneously broken symmetry? How to demonstrate that?
**Why they are stable topological configurations (look to the "Addition" text below)?**
- Why when we choose string located along $z-$axis and set solution for Peccei-Quinn scalar field $\varphi $ in a string-like form $\varphi = ve^{i\theta}$, where $v$ is VEV of $\varphi$, $\theta$ is axion, then we have
$$
[\partial_{x}, \partial_{y}]\theta = 2\pi \delta (x) \delta (y)?
$$
How to demonstrate that?

**Addition**

Let's assume axion "bare" lagrangian
$$
\tag 1 L = \frac{1}{2}|\partial_{\mu}\varphi |^{2} - \frac{\lambda}{4} (|\varphi |^{2} - v^{2})^{2}
$$
One of solution of corresponding e.o.m. is axion string - stable topological configuration. If string is located along z-axis and if it is static, then corresponding solution is simply ($\rho$ is polar radius, $\varphi$ corresponds to polar angle and, in fact, to axion)
$$
\varphi (x) = f(\rho ) e^{i n \varphi}, \quad f(0) = 0, \quad f(\infty ) = v,
$$
where $n$ is winding number.

Statement that configurations with different winding numbers are stable means that they are separated by infinite potential barriers. But I don't understand how $(1)$ creates barriers for different $n$.

**Addition 2**
Thank to the Meng Cheng comment. The first and the third questions are closed. Explicit proof of the statement of the third question:
$$
[\partial_{x}, \partial_{y}]e^{iarctg\left[\frac{y}{x}\right]} = i\partial_{x}\left[ \frac{x}{x^{2} + y^{2} + a^{2}}\right]_{\lim a \to 0} + i\partial_{y}\left[ \frac{y}{x^{2} + y^{2} + a^{2}}\right]_{\lim a \to 0} =
$$
$$
=i\left[\frac{2a^{2}}{(x^{2} + y^{2} + a^{2})^{2}}\right]_{\lim a \to 0} = 2 \pi i \left[\frac{a^{2}}{\pi}\frac{1}{(r^{2} + a^{2})} \right]_{\lim a =0} = 2 \pi i \delta_{a}(\mathbf r)
$$

This post imported from StackExchange Physics at 2015-06-30 15:36 (UTC), posted by SE-user Name YYY