O(4), Off-ShellNN¯Scattering, and Massless Pions

An $O\left(4\right)\mathrm{}$ formalism for off-shell $N\overline{N}$ scattering at $t=0\mathrm{}$ is developed which diagonalizes the Bethe-Salpeter equation into $O\left(4\right)\mathrm{}$ components. The $N\overline{N}$ scattering potential due to single pseudoscalar meson exchange is explicitly reduced into its $O\left(4\right)\mathrm{}$ components. Symmetry properties of the $O\left(4\right)\mathrm{}$ amplitudes due to the requirements of $C$ and $P$ invariance are exhibited. The off-shell $N\overline{N}$ scattering amplitude at $t=0\mathrm{}$ is found to have six distinct types of Lorentz poles, in contrast to the three types found by Freedman and Wang for the on-shell problem. It is shown that massless pions of the type found by Nambu and Jona-Lasinio have the definite $O\left(4\right)\mathrm{}$ quantum numbers $n=0\mathrm{}$, $M=0\mathrm{}$. It is further argued that $M=1\mathrm{}$ massless pions are impossible and that Regge trajectories belonging to $M=1\mathrm{}$ Lorentz poles choose nonsense when $j$ goes through zero.