In SU(N) Yang-Mills theory, there is a nonzero tunneling amplitude between different vacua $|0,n\rangle$ of the theory, labeled by Pontryagin index $n$, due to instanton effects. Therefore, the "true" vacuum of the Hilbert space is given by $$|\theta\rangle=\sum\limits_{n=-\infty}^{\infty}e^{in\theta}|0,n\rangle$$ called the $\theta-$vacuum.

In Baryogenesis, there is a violation of baryon number due to the anomaly $$\partial_\mu J^\mu_B=\frac{N_fg^2}{16\pi^2}F_{\mu\nu}^a \tilde{F}^{\mu\nu a}$$

For a sphaleron or instanton transition with $n=1$, it is said that when the vacuum changes from $n=1$ to $n=2$ (say, for example), the B-number violation is given by $\Delta B=2N_f$.

My questions are as follows.

Do the fermions, at any instant of time, live in a definite Yang-Mills vacuum labeled by a definite Pontryagin index $n$?

Since the true vacuum is the $\theta-$vacuum, which is the superposition given above, shouldn't the fermions at any instant live in $|\theta\rangle$ (because the vacua are not disjoint). If yes, what does it mean to say that fermions tunnel from one vacuum $|0,n_1\rangle$ to $|0,n_2\rangle$? If not, how can the Baryon number violate?

This post imported from StackExchange Physics at 2017-02-16 08:57 (UTC), posted by SE-user SRS