6
$\begingroup$

In the $U(1)$-Peccei-Quinn-Symmetry an axion field $a$ is introduced in order to solve the strong CP-Problem. It is said, that below a certain scale $f_{a}$ this symmetry is broken and you are able to write the field as $a = \langle a\rangle +\,a_\mathrm{phy}$ where $\langle a\rangle$ is the VEV and $a_\mathrm{phy}$ denotes the physical axion field, that is associated with the axion particle.

The VEV is $\langle a\rangle = -\theta f_{a}$ and so the CP-Violating term vanishes, since: $\mathcal{L} \supset (a/f_{a} + \theta) G \tilde{G}$. This is what I've understood so far. However, what I am wondering about now is: Why doesn't the interaction term $\mathcal{L} \supset a_\mathrm{phy} \,G\,\tilde{G}$ violate CP-Symmetry? The only possibility I see is if $a_\mathrm{phy}$ also transforms under CP-Symmetry but I wasn't able to find anything about that.

What I'm also not able to understand is: Why is the symmetry breaking scale parameter $f_a$ also called the decay constant of the axion? I'm not sure why these two are equivalent in this context.

$\endgroup$
1
$\begingroup$

The Lagrangian is as I understand an effective Lagrangian. The Lagrangian for the $a$ field is $$ \frac{1}{2}\partial_\mu a\partial^\mu a + F\left(\frac{a}{f}\right)Tr(E_{QCD}\cdot B_{QCD}) $$ This function is to lowest order $F(x) = \frac{1}{2}x^2$. The $F\tilde F = Tr(E_{QCD}\cdot B_{QCD})$ is then a mass source term, with the trace over the color indices. This is a mass term. This compensates for the $\theta$ angle for $CP$ violations and so $$ F\left(\frac{a}{f}\right) \rightarrow F\left(\frac{a}{f}\right) + \theta $$ We now have the $CP$ violating angle $\theta$ and the mass of this scalar field, called the axion, cancelling each other. In this way the effective Lagrangian remains $CP$.

$\endgroup$
  • $\begingroup$ I don't understand how that means the axion is a pseudo-scalar. $\endgroup$ – Nanashi No Gombe Feb 11 at 13:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.