High-energy color-singlet--color-singlet scattering in non-Abelian gauge theories

In the quark model, hadrons are thought to be singlet states with respect to "color" SU(3). Therefore, in this paper, we study singlet-singlet scattering in a non-Abelian gauge theory in which color SU(3) is the gauge symmetry. We avoid the difficult bound-state problem by representing the hadron as a scalar field $\phi$, a color singlet, which interacts with the quarks through an effective coupling: $G\phi \Sigma {i=1\mathrm{}}^3{\overline{\psi }}_i{\psi }_i$. We calculate the high-energy ($s\to \infty$, $t\le 0$ fixed) $\phi \phi$ scattering amplitude to sixth order in the quark-gluon coupling constant $g$. The calculation is done by using the "infinite-momentum technique" as developed by Chang and Ma. To justify this technique we also calculate high-energy fermion-fermion scattering in a non-Abelian gauge theory using a more rigorous method. We compare our result with an "infinite-momentum technique" calculation done by McCoy and Wu and a similar calculation done by Tyburski. The $\phi \phi$ scattering amplitude is infrared finite. The total cross section for high-energy $\phi \phi$ scattering is found to be ${\sigma }_{\phi \phi } = 64g^4{G}^4{\left(2\mathrm{}\pi \right)}^{-12\mathrm{}}{\pi }^7$ [$N_1\mathrm{}\left(b\right)-\left(\frac{3g^2}{8{\pi }^2}\right)\mathrm{lns}{N}_2\mathrm{}\left(b\right)\mathrm{}$], where $N_1\mathrm{}\left(b\right)\mathrm{}$ and $N_2\mathrm{}\left(b\right)\mathrm{}$ are positive functions depending only on $b = \frac{{({\mu }^2-4\mathrm{}{m}^2)}^{\frac{1}{2}}}{\mu }$, where $\mu$ is the hadron mass and $m$ is the quark mass.