Complex Angular Momentum in Spinor Bethe-Salpeter Equation

A model of fermion-antifermion scattering, mediated by pseudoscalar neutral bosons, is described by the corresponding spinorial Bethe-Salpeter equation in the ladder approximation. The decomposition of the equation into partial waves by means of fusion amplitudes and conventional spherical harmonics is discussed in detail; various important symmetries of the resulting kernel and Born amplitudes are pointed out. The resulting set of coupled equations is continued into the complex angular momentum ($J$) plane, and it is shown that Fredholm theory is inapplicable for any $J$. The equations are solved in a weak coupling, low-energy limit by an iterative scheme. The resulting solutions exhibit a cut of the square-root type extending along the real axis to $\frac{G}{2\pi }$ ($G=\mathrm{coupling}\mathrm{constant}$) in the right-hand plane; other cuts and poles prevent the extension of our solutions into the left-hand $J$ plane. The dominance of the cut is used to extract the large momentum transfer limit and obtain certain results for the high-energy limit in the cross channel. The total cross section for fermion-antifermion annihilation is extracted by means of the optical theorem and is found to exhibit an energy dependence of the form $t^{\left(\frac{G}{2\pi }\right)-1\mathrm{}}{\left[\ln t\right]}^{-\frac{3}{2}}$, where $t$ is the c.m. energy squared.