How does the axion solution actually work?

[ingenue]

I'm just reading Srednicki's QFT book, where the author gave an exercise on this axion. He posed the Lagrangian as $$(\partial_\mu a)^2+(\theta+a/f)F\tilde{F}$$, then he said we can use the shifting symmetry of a to kill the theta term.

But even if we do that, we're left with the term $$aF\tilde{F}/f$$, and because a is a field, we can not simply discard it. So why is this a resolution to the strong CP problem? I'm probably being dumb here, but please kindly explain this to me. thx

 

[Haelfix]

By introducing the axion 'a' field, the 2nd term there in the lagrangian is now automatically and naturally made to be very small and very nearly CP invariant (it would be zero/invariant exactly where it not for the weak interaction and b/c the axion field is not quite a goldstone boson, but instead gets a mass from instanton effects, so eg its vev gets shifted by a tiny amount).

All observables in the theory now no longer depend on just theta or a, but rather the combination, which is small.

This kind of explains away the theta vacuum finetuning problem, at the cost of introducing a new degree of freedom and a puzzle as to why we haven't found the stupid thing yet.

 

[ingenue]

By introducing the axion 'a' field, the 2nd term there in the lagrangian is now automatically and naturally made to be very small and very nearly CP invariant (it would be zero/invariant exactly where it not for the weak interaction and b/c the axion field is not quite a goldstone boson, but instead gets a mass from instanton effects, so eg its vev gets shifted by a tiny amount).

All observables in the theory now no longer depend on just theta or a, but rather the combination, which is small.

This kind of explains away the theta vacuum finetuning problem, at the cost of introducing a new degree of freedom and a puzzle as to why we haven't found the stupid thing yet.

So are you saying that because $$f$$ is very large, the term $$aF\tilde{F}/f$$ is very small, so that we no longer worry the CP violating effect caused by it?

 

[Haelfix]

The point is that we want a method to dynamically relax that theta term into something small. By introducing this new pseudo goldstone field, note that it induces that 2nd term in the lagrangian, which looks like a potential term for the new effective field theory. Note that it is periodic.

Since this is by hypothesis a spontaneously broken symmetry, we want to minimize the potential with respect to < a > which implies that the minimum occurs when (theta + <a>/f) = 0 (This is a saddle point, and the minimum where the field settles in the limit where the weak interaction is ignored.. If the weak interaction is taken into account, the above is not quite zero but still remains smallish).

This of course solves the problem, b/c as you said the scale of f is large and thus theta is made to be small.

 

[Haelfix]

As an aside.. There are really only three possible solutions to the strong CP problem.

The first is that the theta term is anthropically finetuned for the existence of life.
The second is that CP is spontaneously broken. This is equivalent to the proposition that one of the quark masses vanishes identically, in which case you can always rotate the theta angle away by a field redefinition (in other words, all theta vacuums are equivalent).
The final solution is really having a chiral symmetry that enforces that theta limits to zero (or is small), which is this Axion solution (and its many variants, which all basically follow a similar formula where you spontaneously break a PQ symmetry and dynamically relax theta to a minimum)