s-dependence, sign of the cut, and factorization in a t-channel partial-wave amplitude with Regge cuts

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) (=N/D) is expressed as $\frac{[F_1+F_2(j-{\alpha }_c)+G_1{(j-{\alpha }_c)}^{\frac{1}{2}}]}{[j-{\alpha }_0+ϵ{(j-{\alpha }_c)}^{\frac{1}{2}}]}$, where a square-root singularity is assumed for simplicity. For α₀ 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) in the scattering region is of the form $a_{+}(s,t)s^{{\alpha }_{+}}+a_{-}(s,t)s^{{\alpha }_{-}}$, with $a_{\pm }(s,t)$ 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 $({\alpha }_{\pm }-{\alpha }_c)\ln s$. It is found by explicit calculation that if ε is small (≈0.1) then only one or two terms of the series are important up to quite high s(≈200 BeV²). 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 (ε=0). (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 ε. (v) For small, fixed ε the F₂ 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≤0, the width of the t channel resonances, and with the question of determining the sheet on which the poles are located for t≤0. (vii) Finally, A(s,t) is in general not factorizable but can be written as a sum of (complex conjugate) factorized quantities.