# Axion strings and spontaneously broken symmetry

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I have two question about axion strings:

1. Why their appearance is connected with spontaneously broken symmetry? How to demonstrate that?
2. Why they are stable topological configurations (look to the "Addition" text below)?
3. 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?

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
What's an "axion string"? Googling gives me only "axions in string theory" stuff.

This post imported from StackExchange Physics at 2015-06-30 15:36 (UTC), posted by SE-user ACuriousMind
@ACuriousMind : string is topologically stable solution of equation of motion for case of one space dimension in a case of spontaneously broken continuous symmetry. As for axion string, some outlook is here: arxiv.org/pdf/hep-ph/9807374v2.pdf . However, I don't understand explicitly how spontaneously broken symmetry causes existence of string as topologically stable solution.

This post imported from StackExchange Physics at 2015-06-30 15:36 (UTC), posted by SE-user Name YYY
Sounds like what you are talking about is just vortex lines in a superfluid.

This post imported from StackExchange Physics at 2015-06-30 15:36 (UTC), posted by SE-user Meng Cheng
@MengCheng : maybe you're right, and this case is very similar to superconductors. But I still have problem with understanding its topological nature.

This post imported from StackExchange Physics at 2015-06-30 15:36 (UTC), posted by SE-user Name YYY
@ACuriousMind : section II in an article.

This post imported from StackExchange Physics at 2015-06-30 15:36 (UTC), posted by SE-user Name YYY
@MengCheng : thank you! And the last question, if you please: we add boundary conditions for $\varphi = f(r)e^{i\theta}$ such that $f(0) = 0, f(\infty ) = v$ to avoid bad behaviour of $\varphi$ at $r = 0$ and to satisfy the minimum condition of potential?
@NameYYY That is exactly what happens for vortices: the amplitude of the order parameter is suppressed at the core (so there is an energy associated), and far away from the core the amplitude is just $v$. I think in my previous comments I was abusing $v$ for $f$, so all I mean by $v=0$ is that the amplitude has to vanish. Sorry for the confusion.
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