# Theta-vacuum in the gauge field theory

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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?

I want some argument (possibly purely mathematical) which shows that we must choose the θ-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).

Is there a reason making you believe there exists one?

You can set it to zero. The term proportional to $\theta$ is a topological term and won't affect the equations of motion. This amounts into setting the second Chern class of the Yang-Mills bundle equal to zero. You can work with arbitrary $\theta$ as well which is better if you want to understand say magnetic-electric duality.  But in general really specify your theory you usually choose a Chern character $(r(E), c_1(E), ch_2(E))$.

@conformak_gk, that's not OP's question, even with $\theta=0$ it is still a $\theta$-vacuum instead of a $|n\rangle$-vacuum, i.e. $\theta=0$ is still not a vacuum of CS charge eigenstate.

I second the first comment by @JiaYiyang : Why would an argument not depending on chiral fermions be expected to exist? A careful account with chiral fermions considered is in arxiv:0907.2522 and the authors stress (p. 3)

the interplay between the topology of the gauge group and the chiral transformations [...] gives rise to a non-trivial vacuum structure is clearly displayed under general assumptions.

Curiously, I was just this morning having a discussion about this with Eduardo Fradkin. We wondered if there is any evidence (beyond  Haldane's observation about $\theta0,\pi$ relating to integeger and half integer spin chains)  of the effect of $\theta$  on any physical system. In particlular is there any *non-handwaving*  calculation of  the renormalization group flow of $\theta$?

@MichaelStone I'm not an expert on this, but I think recently 1703.00501 showed $\theta=\pi$ QCD has a quite distinct phase structure compared to $\theta=0$ QCD. As for RG flow, I remember in $N=2$ SUSY people calculated the flow of the complex coupling $\tau=\frac{i\theta}{2\pi}+\frac{4\pi}{g^2}$, which should a fortiori give you the flow of $\theta$.

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