In the last parts, we have seen how a new parameter $\theta$ can emerge when we take a closer look at the structure of the QCD vacuum. Here we continue the standard story that was discussed in the first part.
The question, we would like to answer is: How can we incorporate the emergence of $\theta$ into our formalism?
The true QCD ground state: $|\theta\rangle$
Recall that, at least in the temporal gauge, we found a periodic structure of the QCD vacuum. The minima correspond to field configurations with different integer winding numbers. The basic idea is that the correct vacuum state is not one of the configurations with some definite winding number, but instead a superposition of all of them. This superposition emerges because the different degenerate ground states are connected by instanton processes. This simply means that the field can change from a configuration with some winding number $n$ into a configuration with a different winding number, through a tunnel process. This tunneling process through the potential barrier that separates the different ground states is what we call an instanton.
In addition, recall that we found a classification for all gauge transformations that satisfy $U (x) \to 1$ for $|x| \to \infty$. This classification made use of the label “winding number”. The thing is now that a ground state with definite winding number $n$ is not gauge invariant. A gauge transformation with winding number $n’$ changes the ground state $|n\rangle $ into $|n’\rangle $. Thus, if we want a gauge invariant ground state, we need to construct a superposition. This is another possibility to see why our true ground state is not one with a definite winding number, but a superposition.
We write this superposition as
$$ |\theta\rangle = \sum_{n=-\infty}^\infty e^{in\theta} |n\rangle $$
This superposition $ |\theta\rangle$ is known as the theta vacuum. This state only changes by a phase if we act on it with a gauge transformation. For example, when we act with a gauge transformation $g_1$ with winding number $1$ on it, we get
\begin{align}
g_1 |\theta\rangle &= g_1 \sum_{n=-\infty}^\infty e^{in\theta} |n \rangle \notag \\
&= g_1 \sum_{n=-\infty}^\infty e^{in\theta} |n \rangle \notag \\
&= \sum_{n=-\infty}^\infty e^{in\theta} g_1 |n \rangle \notag \\
&= \sum_{n=-\infty}^\infty e^{in\theta} |n +1 \rangle \notag \\
&= \sum_{n’=-\infty}^\infty e^{i(n’-1)\theta} |n’\rangle \notag \\
&= e^{-i\theta} \sum_{n’=-\infty}^\infty e^{i(n’)\theta} |n’\rangle \notag \\
&= e^{-i\theta} |\theta\rangle \notag \\
\end{align}
where we defined $n+1 = n’$ to bring the sum back to its original from. Thus, we can say that $\theta$ is an eigenstate of the operator $g_1$ with eigenvalue $e^{-i\theta}$. Moreover, since the Hamiltonian is invariant under gauge transformations, i.e. $[g_1,H]=0$ holds, this means that $|\theta\rangle$ can be simultaneously an energy eigenstate.
By invoking the analogy with an electron in a periodic potential, one can show that the energy of the ground state depends on $\theta$. In the analogous model, the different values of the Bloch momentum label different energy bands. The energy density of each such energy band is approximately $E_\theta / L = C- e^{-S_0} 2 B \cos(\theta) $, where $L$ denotes the “length of space”. An important observation here is that the band that corresponds to $\theta =0$ has the lowest energy density.
Another important observation is that the value of $\theta$ is fixed and cannot be changed. This can be seen as follows:
Consider a gauge invariant operator $B$. Gauge invariance means that $[g_1 ,B]=0$ holds. Therefore, we can compute
\begin{align}
0 &= \langle \theta | [g_1 ,B] |\theta’\rangle \notag \\
&= \langle \theta | g_1 B |\theta’\rangle – \langle \theta | B g_1 |\theta’\rangle \notag \\
&= e^{-i\theta} \langle \theta | B |\theta’\rangle – e^{-i\theta’} \langle \theta | B |\theta’\rangle \notag \\
&= (e^{-i\theta} – e^{-i\theta’})\langle \theta | B |\theta’\rangle .
\end{align}
This means that $ \langle \theta | B |\theta’\rangle =0$ unless $\theta = \theta ‘$. In other words, the value of $\theta $ cannot be changed by a gauge invariant operator! One such gauge invariant operator is, of course, the Hamiltonian $H$ and therefore our calculation shows that the value $\theta$ does not change as time moves on. Therefore $|\theta\rangle$ is really an energy eigenstate.
The true true QCD ground state: $|\phi\rangle$
We noted above, that $|\theta\rangle$ changes under gauge transformations by a phase: $g_1 |\theta\rangle= e^{-i\theta} |\theta\rangle $. Now want to get rid of the phase change $ e^{-i\theta}$ and use a ground state instead that is completely invariant under gauge transformations.
So, we first need to remember that our state $ | \theta\rangle$ describes a state of the gauge fields $A$. To emphasize this, we write now denote the ground state as $ | \theta [A]\rangle$.
We now recall the definition of the winding number of a field configuration $A$: $W[A]$, which was discussed in the first part.
Using $W[A]$, we can define a ground state that is completely unchanged by gauge transformations:
\begin{align}
|\phi\rangle &= e^{-iW[A]\theta} |\theta \rangle
\end{align}
A crucial property of $W[A]$ is that if we act on it with a gauge transformation of winding number $n$ it gets shifted by $n$:
$$ g_n W[A] g_n^{-1} = W[A] +n $$
This sounds, of course, reasonable, but can also be checked explicitly. (See, for example, Eq. 3.36-3.39 in “Topological investigations of quantized gauge theories” by R. Jackiw. The winding number $W[A]$ is a functional of the gauge field $A$. When we act on it with a gauge transformation, we therefore transform its argument, the gauge field: $g_n W[A] g_n^{-1} = W[A’] $, where $A’$ denotes the gauge transformed gauge field. We then put the gauge transformed gauge field into the formula for the winding number and then notice that this formula is not invariant. Instead we get an additional term. This additional term is exactly the winding number of the gauge transformation $g_n$.)
Now, to see that this new ground state is invariant under all gauge transformations, we act on it with a gauge transformation
\begin{align}
g_n |\phi\rangle &= g_n e^{-i\hat W\theta} |\theta \rangle \notag \\
&= g_n e^{-i\hat W\theta} g_n^{-1} g_n|\theta \rangle \notag \\
&= e^{-i (\hat W +n) \theta} g_n|\theta \rangle \notag \\
&= e^{-i (\hat W +n) \theta} e^{i n \theta} n|\theta \rangle \notag \\
&= e^{-i\hat W\theta} |\theta \rangle \\
&= |\phi\rangle \end{align}
Thus, have now found a truly gauge invariant ground state of the QCD gauge fields. With this construction at hand, we are finally ready to investigate the influence of the phase $\theta$ on the actual physics.
Physics in the $\theta$ vacuum
Now let’s consider the type of object that we usually consider in QFT, i.e. vacuum to vacuum transitions, but in the presence of the $\theta$ vacuum. As noted above, $|\theta \rangle$ is an eigenstate of $H$. In the path integral formalism and recalling that we work in Euclidean spacetime we have write
\begin{align}
e^{iE_\theta \tau} &= \langle \theta | e^{-H \tau} |\theta\rangle \notag \\
&= \mathcal{N} \int (dA_1) (dA_2)\int_{A_1}^{A_2}(DA) \langle \theta[A_2] | e^{-S_E[A]} |\theta \theta[A_1 ]\rangle,
\end{align}
where $S_E$ denotes the Euclidean action and $dA_1$ the integration over time-dependent functions $A_1(x)$. Using a “dilute gas” approximation for the occurrence of instantons, one can get an approximation for $E_\theta$ which is similar to the one quoted for the Bloch energy bands quoted above.
We now want to rewrite this expression in terms of the gauge invariant ground state $|\phi\rangle$. Using $|\phi\rangle = e^{-iW[A]\theta} |\theta \rangle $, we get $ e^{iW[A]\theta}|\phi\rangle = |\theta \rangle $ and therefore
\begin{align}
e^{iE_\theta \tau} &= \langle \theta | e^{-H \tau} |\theta\rangle \notag \\
&= \mathcal{N} \int (dA_1) (dA_2)\int_{A_1}^{A_2}(DA) \langle \phi[A_2] | e^{-iW[A_2]\theta} e^{-S_E[A]} e^{iW[A_2]\theta} |\phi[A_1]\rangle \notag \\
&= \mathcal{N} \int (dA_1) (dA_2)\int_{A_1}^{A_2}(DA) \langle \phi[A_2] | e^{-S_E[A]+ i(W[A_1]-W[A_2])\theta} |\phi[A_1]\rangle \notag \\
&= \mathcal{N} \int (dA_1) (dA_2)\int_{A_1}^{A_2}(DA) \langle \phi[A_2] | e^{-S_E[A]+ iW[A]\theta} |\phi[A_1]\rangle .
\end{align}
(For an alternative derivation see Eq. 10.70-10.72 and Eq. 10.98 in “Instantons and Solitons” by Rajaraman)
This means that the ingredient that we get if we want to take into account the complex structure of the QCD vacuum is simply a new term $ iW[A]\theta $ that gets added to the action.
In Minkowski space this simply means that we need to add a term
$$\Delta \mathcal L = \frac{\theta}{16 \pi^2} Tr[G_{\mu\nu} \tilde{G}^{\mu\nu}] $$
to the original Lagrangian.
To get a physical understanding of this term, we rewrite it in terms of the more familiar “color-E-field” and “color-B-field”, which are simply analogous to the electrical-E-field and the magnetic-B-field. In terms of these, we have
$$ Tr[G_{\mu\nu} \tilde{G}^{\mu\nu}] = Tr[4E_i B_i] ,$$
where the relationship between the field strength tensor $G_{\mu\nu}$ and the fields $E_i$ and $B_i$ is completely analogous to the definition in the electromagnetic theory.
Now there is something important, we can note immediately:
Under parity transformations, we have $E_i \to – E_i$ and $B_i \to B_i$.
Under a time-reversal transformation, we have $E_i \to E_i$ and $B_i \to – B_i$.
Thus, while the original Lagrangian $\mathcal L$ is invariant under both parity and time-reversal transformations, this new term is not!
This already hints towards dramatic physical implications of these new terms. However, to actually understand these implications, we need to “transport” $\theta$ into the fermion sector. This will be discussed in the next part. The “transport” of $\theta$ into the fermionic sector is possible through a so-called chiral rotation. Then, after performing this rotation, we can see that $\theta$ actually implies a non-zero electric dipole moment of the neutron $D_n$. This yields a direct possibility to measure $\theta$, because
$$ D_n\approx 5.2 \times 10 ^{-16} \theta \text { cm}. $$
This non-zero dipole moment is possible via a CP violating pion-nucleon coupling, which is directly related to $\theta$. In this sense, $\theta$ implies CP violation in QCD interactions!