Theta Vacuum of Yang-Mills theory and Baryon number violation

Question

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.

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

2. 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

Answer 1

There is a definite fermion number only before and after the transition, as one can see from your equation for divergence of baryon current. Analogously, the system is in the definite vacuum state only at $t= \pm \infty$.

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

Comments

If the system is in $\theta$-vacuum, which is an eigenstate of the Hamiltonian, there shouldn't be any jump. Am I wrong?

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

---

@SRS Yes, I think so.

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

---

Do you agree or you think I'm wrong? I didn't get your point. If the system is in $\theta-$vacuum, why should we talk about, for example, evolution from $n=1$ to $n=2$?

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

---

@SRS As far as I understood, you considered transition from vacuum labeled by $n_1$ to some other one with $n_2$. Am I right? In any case, there is a very detailed discussion of the subject in Ch. 23 of Weinberg. Perhaps you should consult it?

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