Consider Yang-Mills theory (with possible including of fermions). It has a set $\{|n\rangle\}$ of vavua labeled by integer winding number $n$, defined as the Maurer-Cartan topological invariant: For the gauge element $g_{(n)}$ and corresponding unitary large gauge transformation $U(g_{(n)})$ we have

$$

|n\rangle = U(g_{(n)})|0\rangle, \quad n = \frac{i}{24\pi^{2}}\int \limits_{S^{3}} d^{3}\theta \epsilon^{ijk}\text{tr}\big[ g_{(n)}\partial_{i}g_{(n)}^{-1}g_{(n)}\partial_{k}g_{(n)}^{-1}g\partial_{k}g_{(n)}^{-1}\big]

$$

What is the most theory-independent argument that shows that the vacuum structure of the non-abelian gauge theory must correspond to the $\theta$-vacuum state

$$

|\theta\rangle = \sum_{n = -\infty}^{\infty}e^{in\theta}|n\rangle?

$$

**Examples of arguments which are not complete as for me**

*Argument 1*

Consider pure YM theory (without fermions). In order to argue why we have to use the $\theta$-vacuum as the ground state, people derive that the Hamiltonian $H$ is non-diagonal in the basis $\{|n\rangle\}$:

$$

\langle n|H|m\rangle \simeq e^{-\frac{8\pi^{2}}{g^{2}}|n - m|}

$$

alternatively, it is shown that vacuum **tunneling** is possible. This requires us to diagonalize the hamiltonian, and the $\theta$-vacuum basis is the diagonal basis.

*Argument 2*

The first argument, however, is valid only for pure Yang-Mills theory and breaks down when massless fermions are included, since massless fermions suppress the tunneling. People then use an argument based on the **cluster decomposition principle** (or CDP). A detailed argument is shown here. People introduce the conserved operator

$$

\tilde{Q}_{5} =\int d^{3}\mathbf r (J_{0,5} - 2K_{0}) ,

$$

where $K_{0}$ is defined as

$$

G_{\mu\nu,a}\tilde{G}^{\mu\nu}_{a} = 2\partial_{\mu}K^{\mu}.

$$

By using this charge, one shows that the VEV of non-zero 2c chirality operator $B(\mathbf x)$ (i.e., $[\tilde{Q}_{5}, \mathbf B(\mathbf x)] = 2c \mathbf B(\mathbf x)$) show that the VEV

$$

\langle n| B(\mathbf x)B^{\dagger}(0)|n\rangle

$$

doesn't satisfy the CDP

$$

\lim_{|\mathbf x| \to \infty}\langle n| B(\mathbf x)B^{\dagger}(0)|n\rangle = \lim_{|\mathbf x| \to \infty}\langle n|B(\mathbf x)|n\rangle \langle n|B(0)|n\rangle

$$

The $\theta$-vacuum is the solution of this problem.

But the details of this argument depend on the presence of fermions. More specifically, we introduce chirality and operates with the chirality operator $\tilde{Q}_{5}$.

**What do I want?**

I want some argument (possibly purely mathematical) which shows that we must choose the $\theta$-vacuum as the ground state of the YM theory (if it exists) independently of the precise content of the theory's fields (independent on the fact whether massless fermions are present).

Could You help?