Relation between the Eikonal and Mandelstam Representations

The even and odd parts $A^{\pm }\mathrm{}(s,t)\mathrm{}$ of a scattering amplitude having Mandelstam singularity structure are explicit functions of the transverse momentum $P_t=ksin\theta$. It is shown that $A^{+}\mathrm{}(s,t)\mathrm{}$ and $\frac{A^{-}\mathrm{}(s,t)\mathrm{}}{cos\theta }$ can always be put in an eikonal (impact parameter) representation which is valid at all energies and angles, such that the new representation for the full amplitude $A(s,t)\mathrm{}$ agrees with the usual eikonal approximation in the limit of small angles and high energies. We also establish a relation between the asymptotic growth of the $t$- and $u$-channel absorptive parts and the nature of the singularity of the eikonal function $\chi \mathrm{}(b,s)\mathrm{}$ for zero impact parameter. Moreover, if the function $e^{i\chi \mathrm{}(b,s)\mathrm{}}\sim \frac{1}{b^{\alpha \mathrm{}\left(s\right)+2\mathrm{}}}$ with $\alpha \mathrm{}\left(s\right)>-2\mathrm{}$, we will have a family of moving Regge poles in the complex $l$ plane at $l=\alpha \mathrm{}\left(s\right)-2n$ ($n=0, 1, \cdots$) giving us a connection between the positions of a family of the Regge poles in the complex $l$ plane and the nature of the singularity of the eikonal function for zero impact parameter.