S-dependence, sign of the cut, and factorization in at-channel partial-wave amplitude with Regge cuts

Abstract

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