hard}_{\rm in}}{ds} &=& \pi R^2 \gamma^2 n_0^2 \,
\sigma_{\rm in}(s)
\int_0^{t_0} dt \int_{-vt}^{vt} dz
\int_0^1 \frac{dx_1}{x_1} \int_0^1 \frac{dx_2}{x_2} \,
v_{12} \, \delta(s-s_{12}) \nonumber \\
&\,& \sum_{N_1 = 0}^{\infty} \sum_{N_2 = 0}^{\infty} H(x_1,N_1)\,
S(N_1,d_1(z,t))\,H(x_2,N_2)\,S(N_2,d_2(z,t)) \, .
\label{dncoll}\end{aligned}$$ Here $\sigma_{\rm in}$ is the inelastic nucleon-nucleon cross section, and $\sqrt{s_{12}}$ is the total energy in the nucleon-nucleon collision where the nucleons have momentum fractions $x_1$ and $x_2$.

Meson Production Cross Sections
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A phase transition to quark–gluon plasma will become thermodynamically favorable if the energy density is large enough. The corresponding phase boundary in the temperature/chemical potential plane was explored in [@us]. Until now we have only selected nucleon-nucleon scatterings in which the total available energy $\sqrt{s}$ is large. In addition, we need to specify what fraction of this energy goes into meson production. In this section we estimate the pion number distribution function $P_n (s)$, which is the probability of producing $n$ pions in a nucleon–nucleon collision in free space. The pion number distribution function is linked to the cross section $\sigma_n$ for producing $n$ pions by $$P_n (s) = \sigma_n (s)/ \sigma_{\rm in}(s) \,.
\label{topo}$$ Given $P_n (s)$ we can estimate the number of nucleon-nucleon collisions that would lead to the 