Structure of Regge Poles and Their Residues att=0for General Two-Body-to-Two-Body Reactions

Abstract

A convenient method is introduced to analyze the requirements imposed by the analyticity of the full helicity amplitudes on the structure of Regge poles and their residues at $t=0$ for two-body-to-two-body reactions with general masses and spins. This method enables us to visualize the structure of daughter trajectories and conspirators clearly. Also, in practice, this method enables us to make the following derivation easily for reactions with arbitrary spins. (1) The most singular parts of the daughter and conspirator residues at $t=0$ are calculated for unequal-mass-unequal-mass reactions and unequal-mass-unequal-mass reactions. Then, through factorization, the nonvanishing parts of the daughter and conspirator residues are obtained for the equal-mass-equal-mass reactions. They are identified with a one-Lorentz-pole expansion. (2) In calculating the daughter and conspirator residues, the analyticity requirements of both the $t$-channel and the $s$-channel helicity amplitudes are satisfied. Therefore, the conspiracy equations are shown to be satisfied explicitly. (3) The restrictions on the slopes of the daughter trajectories are also obtained. Their independence of the external masses and spins is shown. (4) The restrictions on the slopes of the conspirators are also calculated. We obtain an interesting new result: For a trajectory of quantum number $M$, at $t=0$, the trajectories ${\ensuremath{\alpha}}_{+}(t)$ and ${\ensuremath{\alpha}}_{\ensuremath{-}}(t)$ are equal, and likewise their derivatives up to the $(M\ensuremath{-}1)\mathrm{th}$. Before carrying out all these calculations, all the $t$ factors of the Regge residue have to be determined. By introducing a quantum number $M$ in the unequal-mass-unequal-mass reactions, the $t$ factors of the parent as well as the daughter residues are uniquely determined using the conventional method of analyticity and factorization. This quantum number $M$ is identified to be the $O(4)$ $M$ in the equal-mass-equal-mass reactions. We note that if the definition of the quantum number $M$ is not affected by the coincidence of $\ensuremath{\alpha}(t)$ with an integer at $t=0$, then the trajectory $\ensuremath{\alpha}(0)$ will choose sense if $M<\ensuremath{\alpha}(0)$ and choose nonsense if $M>\ensuremath{\alpha}(0)$. At the end of the paper, a discussion is given on the implications for the group-theoretical approach to the Regge-pole theory.