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 ${(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|)}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\theta}}$ ($0<\ensuremath{\theta}$) for large momentum transfer, we obtain for the asymptotic power behavior of the form factors of the two- and three-body bound states ${F}_{2}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}^{2})\ensuremath{\simeq}{(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}|)}^{\ensuremath{-}3\ensuremath{-}\ensuremath{\theta}}$ and ${F}_{3}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}^{2})\ensuremath{\simeq}{(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}|)}^{\ensuremath{-}6\ensuremath{-}2\ensuremath{\theta}}$, respectively. For separable potentials $V=g(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|)g(|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}}|)$ and $g(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|)\ensuremath{\simeq}{(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|)}^{\ensuremath{-}\frac{1}{2}\ensuremath{-}\ensuremath{\theta}}$ we find ${F}_{2}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}^{2})\ensuremath{\simeq}{(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}|)}^{\ensuremath{-}2\ensuremath{-}2\ensuremath{\theta}}$ ($0<\ensuremath{\theta}\ensuremath{\le}\frac{1}{2}$), ${F}_{2}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}^{2})\ensuremath{\simeq}{(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}|)}^{\ensuremath{-}2.5\ensuremath{-}\ensuremath{\theta}}$ ($\frac{1}{2}\ensuremath{\le}\ensuremath{\theta}$), and ${F}_{3}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}^{2})\ensuremath{\simeq}{(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}|)}^{\ensuremath{-}5\ensuremath{-}2\ensuremath{\theta}}$, 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. With an interaction which behaves as ${({k}^{2})}^{\ensuremath{-}\ensuremath{\theta}}$ ($0<\ensuremath{\theta}$) at large momentum transfer, we obtain ${F}_{2}({q}^{2})\ensuremath{\simeq}{({q}^{2})}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\theta}}$ and ${F}_{3}({q}^{2})\ensuremath{\simeq}{({q}^{2})}^{\ensuremath{-}2\ensuremath{-}2\ensuremath{\theta}}$.