 {\bf 1} & 0 \\
\end{array}
\right),
\end{align}
and write down a $U(2N_f)$ breaking term,
\begin{align}
 {\rm Tr} \left(
\xi^{-1} X \xi X
\right).
\end{align}
This term gives a mass to $\tilde \pi$ while leaving $\pi$ massless.

In the low-energy limit where $\tilde\pi^a$ are decoupled, one can set
$\tilde \pi^a = 0$, and the WZ term among pions vanishes. However, one
can trace the existence of the WZ term by turning on external gauge
fields.
Let us introduce, the background gauge fields for the unbroken $U(N_f)$
global symmetry $A^\mu$, as the one which couples to the
$U(N_f)$ vector current, $\bar \psi \gamma^\mu T^a \psi + \bar {\tilde
\psi} \gamma^\mu T^a {\tilde \psi}$. The $U(1)$ part is the baryon
number. The equation of motion for ${b}$ now gives, ${b} =
A + \cdots$, and thus we have
\begin{align}
 {S}_{\rm WZ}& =  \pm {N \over 4 \pi} \int_{M^3} {\rm Tr} \left(
 A dA + {2 \over 3} A^3
 \right) + \cdots.
\label{eq:WZext}
\end{align}
In particular, we have a term
\begin{align}
 \pm {1 \over 4 \pi} \int_{M^3} B {\rm Tr} (dA),
\end{align}
where $B$ is the baryon number normalized such that the quarks have the
charge $1/N$.
The above term gives the baryon number $(B=1)$ for a monopole that has
the unit magnetic charge of the $U(1)$ subgroup of $U(N_f)$. For
example, the monopole made of the (11)~component of $b$ has the baryon
number $B=1$.


\section{QCD$_4$ on a circle}
\label{sec:qcd4}

We discuss the four-dimensional QCD compactified on $S^1$ and look for a
relation to the phase transition in QCD$_3$.
We start with the action of $SU(N)$ gauge theory coupled with massless
$N_f$ Dirac fermions, $\Psi_i$, $(i=1,\cdots,N_f)$, 
with the assumption $N > N_f$
on $M^3 \times S^1$,
\begin{align}
 S = \int_{M^3 \times S^1} d^4 x
\biggl[
&
-{1 \over 2 g_4^2} {\rm Tr} \left(f_{M N} f^{M N} \right)
+ {\theta (x_3) \over 32 \pi^2} \epsilon_{MNPQ} 
{\rm Tr} \left( f^{MN} f^{PQ} \right)
\nonumber \\
&
+ i \bar \Psi_i \gamma^M (\partial_M - i a_M) \Psi_i
\nonumber \\
&
- \partial_M \alpha_L^i (x_3) \bar \Psi_i \gamma^M P_L \Psi_i
- \partial_M \alpha_R^i (x_3) \bar \Psi_i \gamma^M P_R \Psi_i
\bigg].
\label{eq:qcd}
\end{align}
The periodic boundary conditions are imposed on gauge fields. The
Lorentz indices, $M$, $N$, $P$, $Q$, run from $0$ to $3$, where $x_3$ is
the $S^1$ direction. $P_{L,R}$ are projection operators of chirality,
$P_{L,R} = (1 \mp \gamma_5)/2$.
We introduced $x_3$ dependent background fields, $\theta (x_3)$ and
$\alpha^i_{L,R} (x_3)$.
The boundary condition of the quarks are
\begin{align}
 \Psi_i (x_3 + 2 \pi R) = e^{i \nu} \Psi_i (x_3),
\end{align}
where $0 \leq \nu < 2 \pi$.


The $S^1$ compactification requires that $\theta$ and $\alpha$ are also
valued on $S^1$ that allows 
\begin{align}
 \int_{S^1} d\theta = 2 \pi k, \quad
 \int_{S^1} d \alpha_{L,R}^i = 2 \pi m^i_{L,R},
 \label{eq:periodicities}
\end{align}
where $k$ and $m^i_{L,R}$ are integers.
The integral on $S^1$ should be properly defined as in
Ref.~\cite{Cordova:2019jnf} so that the partition function does not
depend on the coordinate system on $S^1$. (See
Appendix~\ref{sec:windingTheta} for the definition.)
There is a redundancy due to the anomalous chiral symmetry, $\Psi_{L(R)i} \to
e^{i \beta^i_{L(R)}} \Psi_{L(R)i}$,
\begin{align}
 \theta (x_3) & \to \theta (x_3) - \sum_i (\beta^i_R (x_3) - \beta^i_L (x_3)), \quad
 \alpha^i_{L,R} (x_3) \to \alpha^i_{L,R} (x_3) + \beta^i_{L,R} (x_3),
\label{eq:redundancy}
\end{align}
where 
\begin{align}
 \int_{S^1} d\beta_{L,R}^i  = 2 p \pi,
\end{align}
with $p$ integers to maintain the boundary conditions.


In the following discussion, we are particularly interested in the theory with
\begin{align}
k = N_f, \quad m_{L,R}^i = 0.
\label{eq:param}
\end{align}
since the vacuum structure looks the same as the three dimensional case
discussed in the previous section.
By the chiral rotations, this theory is equivalent to, for example,
\begin{align}
k = 0, \quad m_R^i = 1, \quad m_L^i = 0.
\label{eq:thetazero}
\end{align}
The physics should depend on the combination:
\begin{align}
\bar \theta (x_3) =  \theta (x_3) + \sum_i (\alpha^i_R (x_3) - \alpha^i_L (x_3)),
 \quad \bar k& = k + \sum_i (m_R^i - m_L^i).
 \label{eq:combi}
\end{align}

We discuss the phase structure of the theory as a function of the radius $R$.
The dynamical scale $\Lambda_4$ is defined as
\begin{align}
 \Lambda_4^b& = \Lambda^b e^{-{8 \pi^2}/g_4^2 (\Lambda)},\quad
b = {11 \over 3} N - {2 \over 3} N_f,
\end{align}
where $\Lambda$ is an arbitrarily high scale. 
For $\Lambda_4 R \gg 1$, the low energy dynamics is described by hadrons
on an $S^1$ compactified background.
In the other limit, $\Lambda_4 R \ll 1$, the low energy description is a
three dimensional gauge theory on $M^3$ via the KK
decomposition.  The gauge coupling constant in the three dimensional
effective theory is given by
\begin{align}
 {1 \over g_3^2}& = {2 \pi R \over g_4^2 (1/R)},
\end{align}
and the dynamical scale in the three dimensional theory is defined as
\begin{align}
 \Lambda_3& = {g_3^2 N \over 8 \pi} = {g_4^2 (1/R) N \over 16 \pi^2 R}
= {1 \over R} \left(
- {2 b \over N} \log (\Lambda_4 R)
\right)^{-1}.
\end{align}
One can see that for a small enough $\Lambda_4 R$, there is an energy
gap between the dynamical scale $\Lambda_3$ and the mass of the first KK
mode, $1/R$.