O(4)Symmetry,NN¯Scattering, and the Regge Trajectory of the Pion

Abstract

The $N\overline{N}$ Bethe-Salpeter equation is studied exploiting the $O\left(4\right)\mathrm{}$ symmetry of the scattering amplitude at $t=0\mathrm{}$. Spinor $O\left(4\right)\mathrm{}$ spherical harmonics are developed and used in the projection of the Bethe-Salpeter equation both at $t=0\mathrm{}$ and for small finite $t$. With the choice of Toller quantum number $M=1\mathrm{}$ for the pion trajectory at $t=0\mathrm{}$, as seems to be implied by high-energy data, we find that the $M=1\mathrm{}$ trajectory must mix with another trajectory near $t=0\mathrm{}$ in order to produce a physical pion. In addition, $O\left(4\right)\mathrm{}$ expansions are given for the $N\overline{N}$ helicity amplitudes for both equal and unequal mass.