Moving Branch Points injPlane and Regge-Pole Unitarity Conditions

Abstract

Moving branch points in the $j$ plane are investigated on the basis of analysis of multiparticle terms of the unitarity condition in the $t$ channel. A definite assumption about the form of an analytic continuation of these terms into complex $j$ is used. It is shown that in this case in the $j$ plane there arise branch points of the partial amplitude $f_j\mathrm{}\left(t\right)\mathrm{}$ corresponding to the production thresholds of two or more Regge poles with relative orbital momentum equal to - 1. In the case of two zero-spin particles in the intermediate state, the partial wave has a singularity at negative integral values of the orbital momentum. Azimov has found that such singularities shift to the right if the particles in the intermediate state have nonzero spin. The branch points in the $j$ plane result from the extension of this shift throughout the Regge trajectory. This mechanism of emergence of branch points has been indicated by Mandelstam for the case of Feynman diagrams of a certain class. The presence of these branch points at $j=j_n\mathrm{}\left(t\right)\mathrm{}$ where $j_n\mathrm{}\left(t\right)=n\alpha \left(\frac{t}{n^2}\right)-n+1\mathrm{}$ changes essentially the analytic properties of $f_j\mathrm{}\left(t\right)\mathrm{}$ in the $t$ plane, leading to the emergence in the $t$ plane of branch points at $t=t_n\mathrm{}\left(j\right)\mathrm{}$, where $t_n\mathrm{}\left(j\right)\mathrm{}$ is the solution of the equation $j=j_n\mathrm{}\left(t\right)\mathrm{}$. The discontinuity ${{\delta }_t}^{\mathrm{}\left(n\right)\mathrm{}}{f}_j\mathrm{}\left(t\right)\mathrm{}$ of the amplitude $f_j\mathrm{}\left(t\right)\mathrm{}$ on the singularity $t=t_n\mathrm{}\left(j\right)\mathrm{}$ corresponding to the $n$-Regge-pole production threshold (Regge-pole unitarity conditions) is calculated. It is shown that this discontinuity has a form similar to the conventional unitarity condition. ${{\delta }_t}^{\mathrm{}\left(n\right)\mathrm{}}{f}_j\mathrm{}\left(t\right)=\left(\frac{1}{2i}\right)\left[f_j\right(t+i\epsilon )-f_j(t-i\epsilon \left)\right]$ being given by the product of the amplitudes $N_{j,n}$ of production of $n$ Regge poles determined above and under a cut made in the $t$ plane from the point $t=t_n\mathrm{}\left(j\right)\mathrm{}$. The discontinuity

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