+ 0.79 F(s) \nonumber \\
\langle n_{pn}^- \rangle &=& -0.14 + 0.81 F(s) \nonumber \\
\langle n_{nn}^- \rangle &=& +0.35 + 0.77 F(s) \, .
\label{paras}\end{aligned}$$ The function $F$ was introduced by Fermi [@fermi], $$F(s) = \frac{(\sqrt{s}-2m_N)^{3/4}}{s^{1/8}}\, ,
\label{ferf}$$ with $s$ measured in GeV$^2$. The parametrizations in (\[paras\]) describe the data rather well except in the threshold region. We approximate the inelastic nucleon–nucleon cross section $\sigma_{\rm in}$ by the inelastic proton–proton cross section. A convenient parametrization is given in [@topo], $$\sigma_{\rm in} = 30.9 - 28.9\,p_L^{-2.46} - 0.835\,\ln{p_L}
+0.192\,\ln^2{p_L}\, ,
\label{sin}$$ where $p_L$ is the laboratory momentum in GeV/c and the cross section is in mb. This parametrization is good for $p_L > 0.968$ GeV/c.

The pion production cross sections, as described above, are displayed in Figure 3. They have the right shapes and the right orders of magnitude compared to data [@topo]. However, direct comparison is not possible. First of all, data generally does not exist for final states with $\pi^+$, $\pi^-$, and $\pi^0$. Usually, exclusive experiments can only measure charged mesons or neutral mesons, not both. Secondly, we have not been so sophisticated as to include vector mesons, the $\eta$ meson, and kaons. For our purpose such sophistication 