production from the vacuum of hadronic states with those quantum numbers.

For example, with $J$ the electromagnetic hadronic current of light quarks, $$J^{\mu}(x)=\frac{2}{3}\bar{u}(x)\gamma^{\mu}u(x)-
\frac{1}{3}\bar{d}(x)\gamma^{\mu}d(x)-
\frac{1}{3}\bar{s}(x)\gamma^{\mu}s(x)$$ the relation to the total $e^+ e^-$ annihilation cross–section into hadrons is $$\sigma(q^2)_{e^+ e^- {\rightarrow}\mbox{\rm hadrons}}=\frac{4\pi^2 \alpha}{q^2}
\,e^2 \frac{1}{\pi}{\mbox{\rm Im}}\Pi_{\mbox{\tiny\rm EM}}(q^2),$$ and $$\label{eq:spechad}
\begin{split} 
-3\theta(q)q^2 &\,\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{\mbox{\tiny\rm
EM}}(q^2)= \\
&\sum_{\Gamma}\langle 0\vert J^{\mu}_{\mbox{\tiny\rm EM}}(0)\vert
\Gamma\rangle
\langle\Gamma\vert J_{\mu,\mbox{\tiny\rm EM}}(0)^{\dagger}\vert 0\rangle
(2\pi)^{3}\delta^{(4)}(q-p_{\Gamma}),
\end{split}$$ where the sum is extended to all possible physical states, [*on–shell states*]{}, with an integration over their corresponding phase space understood. A pictorial representation of these equations is shown in Fig. \[fig:Fig2a\] below.

The lowest possible state in this case is a two–pion state. The function $\Pi_{\mbox{\tiny\rm EM}}(q^2)$ is analytic in the complex $q^2$–plane but for a cut in the real axis: $4m_{\pi}^2\le q^2\le\infty$, as illustrated in Fig. \[fig:Fig2b\].

I want to show first how these properties appear explicitly in a simple case which you should be familiar with, and then we shall discuss a general proof.

### Vacuum Polarization due to Massive Fermions {#subsec:vpmf}

Let us consider QED with a massive fermion. We want to calculate the lowest order vacuum polarization contribution from this fermion, i.e. the one–loop Feynman diagram in Fig. \[fig:Fig2c\].

Combining 