Impact Parametrization of the Scattering Amplitude

We study the new impact-parameter representation using $J_0\left(kb sin\vartheta \right)$ instead of the conventional $J_0\left(2\mathrm{kb} sin\frac{1}{2}\vartheta \right)$. We give group-theoretic arguments for this representation in the case of unequal-mass spinless scattering. Assuming full Mandelstam analyticity for the scattering amplitude, we describe the analytic properties of the impact parameter amplitude $A^{(\pm )}\mathrm{}(k,b)\mathrm{}$. For convenience, we have set the masses equal in the discussion of analyticity. The singularity structure of $A^{(\pm )}\mathrm{}(k,b)\mathrm{}$ governs the analyticity in the $l$-plane of $a_l^{}{}_{}{}^{(\pm )}\mathrm{}\left(k\right)\mathrm{}$, the direct $s$-channel partial-wave amplitude. We study conditions for there to be infinitely many one, or no (moving) Regge poles. Using Serber's optical-potential model, we find infinite families of poles of increasing order in the angular momentum plane. In connection with the Freedman and Wang amplitude, we also state conditions for there to be infinite, one, or no (moving) Lorentz poles. As an example, we quote the Coulomb potential which leads to an infinite number of Lorentz poles. We remark on the actual connection with the eikonal approximation, as well as on earlier work on the impact-parameter representation involving $J_0\left(2\mathrm{kb} sin\frac{1}{2}\vartheta \right)$.