Asymptotic behavior of form factors for two- and three-body bound states

Abstract

The asymptotic power behavior of the electromagnetic form factors is examined for two-and three-body $s$-wave bound states, both relativistic and nonrelativistic. In the nonrelativistic case, we consider local and separable two-body potentials and we make use of the Faddeev equations in order to define the three-body bound states. For local potentials which behave as ${\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{k}}\mathrm{}\left|\right)\mathrm{}}^{-1-\theta }$ ($0<\mathrm{}\theta$) for large momentum transfer, we obtain for the asymptotic power behavior of the form factors of the two- and three-body bound states $F_2\left({\overrightarrow{\mathrm{q}}}^2\right)\simeq {\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{q}}\mathrm{}\left|\right)\mathrm{}}^{-3-\theta }$ and $F_3\left({\overrightarrow{\mathrm{q}}}^2\right)\simeq {\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{q}}\mathrm{}\left|\right)\mathrm{}}^{-6-2\mathrm{}\theta }$, respectively. For separable potentials $V=g\left(\right|\mathrm{}\overrightarrow{\mathrm{k}}\mathrm{}\left|\right)g\left(\right|\mathrm{}{\overrightarrow{\mathrm{k}}}^{\prime }\mathrm{}\left|\right)\mathrm{}$ and $g\left(\right|\mathrm{}\overrightarrow{\mathrm{k}}\mathrm{}\left|\right)\mathrm{}\simeq {\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{k}}\mathrm{}\left|\right)\mathrm{}}^{-\frac{1}{2}-\theta }$ we find $F_2\left({\overrightarrow{\mathrm{q}}}^2\right)\simeq {\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{q}}\mathrm{}\left|\right)\mathrm{}}^{-2-2\mathrm{}\theta }$ ($0<\mathrm{}\theta \le \frac{1}{2}$), $F_2\left({\overrightarrow{\mathrm{q}}}^2\right)\simeq {\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{q}}\mathrm{}\left|\right)\mathrm{}}^{-2.5-\theta }$ ($\frac{1}{2}\le \theta$), and $F_3\left({\overrightarrow{\mathrm{q}}}^2\right)\simeq {\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{q}}\mathrm{}\left|\right)\mathrm{}}^{-5-2\mathrm{}\theta }$, respectively. For the relativistic case, we consider the two- and three-body Bethe-Salpeter equation in the ladder approximation. We treat the spin-zero case only but we believe that our final conclusions will not be affected by the introduction of spin-$\frac{1}{2}$ particles....