Multiplicities in Deep Hadron-Hadron Scattering

We derive an expression for the average multiplicity for an inclusive experiment $a+b\to c+X$, where $a$, $b$, and $c$ are hadrons and $c$ is detected with a large transverse momentum. If $E_c$ is the energy of $c$ in the center-of-mass system of $a$ and $b$, we find ${〈n(s,E_c)〉}_{\mathrm{deep}}\sim (C_{e^{+}{e}^{-}}+C_x)\ln {E_c}^2+C_h\ln {(\frac{\sqrt{s}}{2E_c}-1)}^2$ as $s$ and the transverse momentum of $c$ get large. If $E_c>\frac{\sqrt{s}}{4}$, the last term is absent. $C_{e^{+}{e}^{-}}$ and $C_h$ are related to the average multiplicities in $e^{+}{e}^{-}$ annihilation and small-angle hadron-hadron scattering, respectively, while $C_x$ is determined. by the final hadron density in the hole plateau region. We also apply our method to the semi-inclusive reaction $a+b\to c+d+X$, where $c$ and $d$ both have large $p_{\perp }$. Finally, we speculate about the value of $C_x$, and about nonasymptotic relations between multiplicities in various reactions.