Consider a Yang-Mills theory, possibly including fermions. It has many possible vacua $\{|n\rangle\}$ labelled 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 which shows that the vacuum 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 for me

  1. Consider a pure YM theory without fermions. In order to argue why we have to use the $\theta$-vacuum as the ground state, people show that the Hamiltonian $H$ is non-diagonal in the basis $\{|n\rangle\}$: $$ \langle n|H|m\rangle \simeq e^{-\frac{8\pi^{2}}{g^{2}}|n - m|} $$ and therefore, vacuum tunnelling is possible. This requires us to diagonalize the Hamiltonian, and the $\theta$-vacuum basis is the diagonal basis.

But this argument works fine only when semiclassical approximation is valid, and also only if massless fermions are not included.

  1. 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 tunnelling. People then use an argument based on the cluster decomposition principle (or CDP). The detailed argument is shown e.g. in "The structure of the gauge theory vacuum" by Callan Jr. One introduces a conserved operator $$ \tilde{Q}_{5} =\int d^{3}\mathbf r (J_{0,5} - 2K_{0}) , $$ where $K_{0}$ is defined as $$ G_{\mu\nu,a}\tilde{G}^{\mu\nu}_{a} = 2\partial_{\mu}K^{\mu}, $$ and by using this charge 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 this argument (its detailed part) depends on presence of fermions. Precisely, we introduce the chirality and operates with 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 from the precise field content, in particular independent of the presence of massless fermions. Is there such an argument?

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    $\begingroup$ Doesn't Srednicki prove this (non-perturbatively and without mentioning fermions) in the paragraph in the middle of pg. 593, just by requiring the Hamiltonian to be gauge-invariant? $\endgroup$ – tparker Nov 1 '17 at 18:44

Consider euclidean Yang Mills on $S^4$ or $R^3\times S_1$. The euclidean partition function is a sum over different topological sectors labeled by $$ q = \frac{1}{32\pi^2}\int d^4 x \, G_{\mu\nu}^a\tilde{G}^{\mu\nu\, a} $$ Now perform a Fourier transform with respect to $q$, that is determine the partition function $Z(\theta)=\sum_q e^{iq\theta}Z_q$. If we wish to study the ground state ($T=0$) we let the size of the $S_1$ go to infinity, The partition function will now sample the ground state wave function. We can define the averages of any operator as $\langle\theta |O|\theta\rangle\equiv \frac{1}{Z(\theta)}\sum_q e^{iq\theta}O_q$, which effectively defines $|\theta\rangle$.

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    $\begingroup$ This seems to give a definition of the $\lvert \theta\rangle$, but not an argument why they should be the "true" vacua rather than the $\lvert q\rangle$. Could you elaborate a bit on that? $\endgroup$ – ACuriousMind Aug 8 '17 at 21:39
  • $\begingroup$ I am simply using the fact that the partition function of QCD involves a sum over different topological sectors. The semi-classical approximation is sufficient to establish the existence of these sectors, even if the it is not quantitatively reliable to compute their contribution. $\endgroup$ – Thomas Aug 10 '17 at 2:38
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    $\begingroup$ I'm not doubting the existence/definition given here, but the question doesn't merely ask how to define the $\lvert \theta \rangle$ - it already writes $\lvert \theta \rangle = \sum_n \mathrm{e}^{\mathrm{i}n\theta}\lvert n\rangle$ - but why these are the "true" vacua one should consider as the ground state. You've given no argument why the $Z(\theta)$ are any "better" at being vacuum functionals than the $Z_q$, unless I'm missing something here that you consider obvious. $\endgroup$ – ACuriousMind Aug 10 '17 at 9:54
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    $\begingroup$ The partition function is automatically a sum over all topological sectors. Nature may have chosen to weight them with a phase factor $e^{i\theta q}$, but we have no evidence that $\theta\neq 0$. However, even for $\theta=0$ we have no choice but to sum over all fields, and therefore all $q$. $\endgroup$ – Thomas Aug 10 '17 at 17:59
  • $\begingroup$ I see. What then has your mention of the Fourier transform to do with it? Shouldn't the argument rather be that the total partition function is of the form $\sum_q f(q) Z_q)$ and that a variant of cluster decomposition forces $f(q)$ to be an exponential phase factor through $f(q_1 + q_2) = f(q_1)f(q_2)$? $\endgroup$ – ACuriousMind Aug 10 '17 at 18:14

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