due to the condition of fixed net-charge, with a finite charge bath composed of $\pi^+$ and unobserved $\pi^-$. We again split the single particle partition function for $\pi^-$ into an observed, $z^-_{\Omega}\left(\phi_{N_{\Omega}},\phi_Q\right)$, and an unobserved part, $z^-_{\bar \Omega} \left(\phi_Q\right)$, $$\begin{aligned}
z^-\left(\phi_{N_{\Omega}},\phi_Q\right) = z^-_{\Omega}
\left(\phi_{N_{\Omega}},\phi_Q\right) + z^-_{\bar \Omega} 
\left(\phi_Q\right) =  \frac{gV}{\left(
    2\pi\right)^3} \int \limits_{\Omega} d^3 p ~ 
e^{-\frac{\varepsilon}{T}}  e^{-i \phi_Q} e^{i \phi_{N_{\Omega}}} +
\frac{gV}{\left( 2\pi\right)^3} 
\int \limits_{\bar \Omega} d^3 p ~ e^{-\frac{\varepsilon}{T}} e^{-i \phi_Q},\end{aligned}$$ while we do not want to measure $\pi^+$ and $\pi^0$, and thus: $$\begin{aligned}
z^+ \left(\phi_Q\right) ~=~  \frac{gV}{\left(
    2\pi\right)^3} \int  d^3 p ~ e^{-\frac{\varepsilon}{T}} e^{+ i \phi_Q}~,
\qquad \qquad \textrm{and} \qquad \qquad 
z^0 ~=~  \frac{gV}{\left(
    2\pi\right)^3} \int  d^3 p ~ e^{-\frac{\varepsilon}{T}} .\end{aligned}$$ The normalization of the CE multiplicity distribution is given by the CE system partition function $Z_{ce}$, i.e. the number of all micro states with fixed charge Q, $Z^{ce} = I_Q\left(2z \right) \exp(z^0)$, where $I_Q$ is the modified Bessel function. The multiplicity distribution of $N_{\Omega}$ in a momentum space segment $\Omega$, while charge $Q$ is globally conserved, can be obtained from Fourier integration of the generalized GCE partition function $\mathcal{Z} \left( \phi_{N_{\Omega}}, \phi_Q \right) = \exp \left[ z^-_{\Omega}
  \left( \phi_{N_{\Omega}},  \phi_Q \right)~+~ z^-_{\bar{ \Omega}}
  \left( \phi_Q \right) 