$CP$ Invariance of Yang-Mills Vacua in Electroweak Theory

Question

It is well know that quantum Yang-Mills theory has a periodic vacuum structure. Consider electroweak theory. For a single generation of fermions, the theory is CP invariant. I would like to know if the periodic vacua of the theory are also CP invariant.

One expects the trivial vacuum with topological charge $n=0$ to be CP invariant, where the topological charge, defined as

\begin{equation} n= \int d^4x \mathcal{P}(x), \end{equation} is the integral of the Pontryagin density $\mathcal{P}(x)$ over the spacetime manifold. However, since $\mathcal{P}(x) \sim tr\left( F \tilde{F} \right)$ is odd under CP, one generally expects gauge configurations that have non-zero topological charge to be odd under CP (?), and this includes the non-trivial vacua.

On the other hand, electroweak theory is different from QCD in that there is no explicit theta angle term in the action, so it seems to me that, as opposed to QCD, there should be no way for us to physically distinguish between the vacua and therefore they should all have the same properties under C, P and T (note however that $\textit{changing}$ vacua via sphaleron and instanton processes does have physical significance, but that is not the main concern of the question).

What is the resolution of this apparent paradox?

Will the pure gauge configurations that can be connected to the trivial vacuum via large gauge transformations (and hence have non-zero topological charge) be even under CP or odd?

This post imported from StackExchange Physics at 2019-07-25 09:06 (UTC), posted by SE-user Optimus Prime

Answer 1

There is an explicit $\theta$ term in the electroweak Lagrangian. However, the value of $\theta$ is basis-dependent: the chiral anomaly means that certain redefinitions of the fermion fields can shift $\theta$---see, e.g., here, or section 29.5 of Schwartz's Quantum Field Theory and the Standard Model. In the electroweak theory we can choose a basis in which $\theta = 0$, and since physics is basis-independent this $\theta = 0$ theory will make the same predictions as the theory we started with. So the theta angle term is there, it just happens to vanish in a certain choice of coordinates.

This post imported from StackExchange Physics at 2019-07-25 09:06 (UTC), posted by SE-user John Dougherty

Comments

Dear @John, I'm familiar with both references you mentioned : ). I know a bit about how the chiral anomaly can be used to "rotate away" the vacuum angle. Though frankly, I don't see how this answers the question. What am I missing?

This post imported from StackExchange Physics at 2019-07-25 09:06 (UTC), posted by SE-user Optimus Prime

---

Ah, sorry for telling you what you already knew! I'm sorry if I misunderstood, but I interpreted the question as a request to resolve a paradox. The conflict is between (i) the fact that individual classical configurations can break CP symmetry and (ii) the fact that the EW Lagrangian is CP-symmetric. My answer was offering a reason why (i) and (ii) don't actually conflict, due to the chiral anomaly.

This post imported from StackExchange Physics at 2019-07-25 09:06 (UTC), posted by SE-user John Dougherty

---

Hmm, I think we're still not on the same page : ) Perhaps the first few sentences in the question were misleading. I merely mentioned CP invariance of the theory to avoid complications due to mixing and all that. To state it better, the conflict is in deriving contradictory CP transformation properties for the non-trivial vacua.

This post imported from StackExchange Physics at 2019-07-25 09:06 (UTC), posted by SE-user Optimus Prime