Using an effective-range-type expansion in the $j$ plane for the $N$ and $D$ functions, the $t$-channel partial-wave amplitude $A(j,t)\mathrm{} (=\frac{N}{D})$ is expressed as $\frac{[F_1+F_2(j-{\alpha }_c)+G_1{(j-{\alpha }_c)}^{\frac{1}{2}}]}{[j-{\alpha }_0+\epsilon {(j-{\alpha }_c)}^{\frac{1}{2}}]}$, where a square-root singularity is assumed for simplicity. For ${\alpha }_0$ linear in $t$ the following results are obtained: (1) The Regge poles (= ${\alpha }_{\pm }$) are complex below a certain $t$ value. (ii) The amplitude $A(s,t)\mathrm{}$ in the scattering region is of the form $a_{+}\mathrm{}(s,t)\mathrm{}{s}^{{\alpha }_{+}}+a_{-}\mathrm{}(s,t)\mathrm{}{s}^{{\alpha }_{-}}$, with $a_{\pm }\mathrm{}(s,t)\mathrm{}$ expressible as a sum of two terms. The first term is one-half the residue of the complex pole, whether the pole be on the physical or unphysical sheet, the second term is a series involving the product $({\alpha }_{\pm }-{\alpha }_c)\ln s$. It is found by explicit calculation that if $\epsilon$ is small ($\approx 0.1$) then only one or two terms of the series are important up to quite high $s\left(\approx 200 {\mathrm{BeV}}^2\right)$. Only at asymptotic $s$ will the series sum up to give the tip of the cut contribution, $\frac{s^{{\alpha }_c}}{{\left(\ln s\right)}^{\frac{3}{2}}}$. At presently available energies, therefore, the $s$ dependence is largely given by $s^{{\alpha }_{\pm }}$. The results remain unchanged if the pole is on the real axis ($\epsilon =0\mathrm{}$). (iii) At the $t$ value where the poles collide, the $s$ dependence is of a typical double pole from $s^{\alpha }\ln s$. (iv) It is observed that the strength of the cut is manifested through $\left|\frac{F_2}{F_1}\right|$ and $\left|\frac{G_1}{F_1}\right|$ as well as through $\epsilon$. (v) For small, fixed $\epsilon$ the $F_2$ term plays a crucial role in shifting the zeros in $t$ of $A(s,t)\mathrm{}$ from their simple pole values. (vi) The sign of the cut is intimately connected with the phase of the complex residues for $t\le 0$, the width of the $t$ channel resonances, and with the question of determining the sheet on which the poles are located for $t\le 0$. (vii) Finally, $A(s,t)\mathrm{}$ is in general not factorizable but can be written as a sum of (complex conjugate) factorized quantities.

Close

PrevNext

View figure in article