Reggeon quantum mechanics: a critical discussion

The quantum-mechanical problem of reggeon field theory in zero transverse dimensions is re-examined in order to set up a precise mathematical framework for the case μ=α(0)-1>0. The authors establish a Hamiltonian formulation in a Hilbert space for μ<0 and prove the equivalence of the related eigenvalue problem with a 'radial' Schroedinger-type equation in an L2(0, infinity) space. It is proved that the S-matrix and the pomeron Green functions, at fixed rapidity Y and triple-pomeron coupling lambda not equal to 0, have a spectral decomposition and are analytic in μ for -infinity<μ<+infinity. For μ>0, most of the qualitative results found by previous authors are confirmed and in particular the tunnelling shift [approximately exp(-μ2/2lambda2)] setting the scale for the asymptotic behaviour in Y. In the classical limit of lambda/μ small it is found that the action, for μ>0, develops a singularity in Y at some value Ysub(c). Arguements are given to show that for Y< approximately Ysub(c) perturbation theory breaks shown. Most of these results are shown to be stable against the addition of a small quartic coupling of the simplest type [lambda'(anti psipsi)2] up to the 'magic' value lambda'=lambda2/μ. The existence of a level crossing at this value is confirmed by an analytic continuation in lambda'. (Auth.)