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