Two-Photon Mechanism of Particle Production by High-Energy Colliding Beams

Abstract

We report on the calculation of the cross sections for the production of positive-charge-conjugation states such as ${\pi }^0$, $\eta$, $e^{+}{e}^{-}$, ${\mu }^{+}{\mu }^{-}$, and ${\pi }^{+}{\pi }^{-}$ by a two-photon mechanism in $e^{-}{e}^{+}$ and $e^{-}{e}^{-}$ collisions. We give the precise relationship of the process $e+e\to e+e+X (X \mathrm{is}\mathrm{any} C=+ \mathrm{state})$ to the corresponding two-photon annihilation process $\gamma +\gamma \to X$, as well as a careful derivation of the equivalent-photon approximation. In the case of the ${\pi }^0$ production, we have found that, for the beam energy $E$ in the 1-3 GeV range, the exact total cross section is 20-30% larger than the one calculated previously in the equivalent-photon approximation. However, the introduction of form factors cuts down the exact total cross section, reducing it to within 10% of the equivalent-photon-approximation result. For $\eta$ production the agreement is even better. Thus it appears that the use of the equivalent-photon approximation is justified in most cases. We discuss detailed angular distributions in this approximation for the case of ${\pi }^{+}{\pi }^{-}$ production. One important problem which cannot be adequately studied in the equivalent-photon approximation is the degree of noncoplanarity of the ${\pi }^{+}{\pi }^{-}$ (and the ${\mu }^{+}{\mu }^{-}$) pair. We have studied this problem using the exact formula and found that, for $E=1\mathrm{}$ GeV, typically 40-50% of pion pairs will be produced with the noncoplanarity angle greater than 12°. We discuss the general structure of the $\gamma +\gamma \to {\pi }^{+}+{\pi }^{-}$ amplitude as well as a simple model incorporating the $\sigma$ meson. We also give a rough estimate of multihadron production cross sections.