Light-cone behavior of the pion Bethe-Salpeter wave function in the ladder model

The Bethe-Salpeter wave function $\chi (q^{\nu }+P^{\nu }, q^{\nu })$ for two spin-½ quarks bound by the exchange of a scalar meson is examined in the ladder model. We seek the behavior of $\chi$ as the squared momentum, ${\mathrm{}(q+P)\mathrm{}}^2$, on one leg becomes infinite while the squared momentum, $q^2$, on the other leg remains fixed. This behavior is investigated by making a Wick rotation, expanding $\chi$ in partial-wave amplitudes ${\chi }_J^i\left(q^2\right)$ of the group O(4), and then looking for the rightmost poles of ${\chi }_J^i\left(q^2\right)$ in the complex $J$ plane. Our results verify (in the ladder model) the useful hypothesis that the locations of these poles are independent of $q^2$ and can thus be computed in the $q^2\to \infty$ limit by using conformal invariance.