Reggeon field theory for α(0) > 1

We obtain the asymptotic behaviour of the scattering amplitude when the pomeron has intercept α(0) larger than one. The reggeon field theory is studied by introducing a lattice in impact parameter space. Use is made of a previous result showing asymptotically the dynamics is controlled at each lattice site (α′ = 0 case) by a two-level structure. This leads to a non-Hermitean Hamiltonian expressed in terms of spin operators in which the intersite interaction terms is proportional to the pomeron slope α′. The spectrum of such a system shows a degenerate ground state for α(0) > α_c >~ 1 and a continuum with vanishing excitation gap at α(0) = α_c. The vacuum does not change structure at the critical value. The critically is shown by an order parameter which is given by the matrix element of a field operator between the vacuum and its degenerate companion. The nature of this critical phenomenon is better understood by continuously transforming the Hamiltonian into that of an Ising model with a transverse field which shows a well-known second-order phase transition. By defining the S-matrix so as to preserve the formal perturbation expansion, we find that for α(0) > α_c, the zero gap state contributes a non-trivial asymptotic constant. The final asymptotic picture is that of a gray disc expanding like log s, so that the total cross section behaves as (log s)². For α(0) < α_C, the vacuum is non-degenerate and correspondingly the total cross section drops to zero as an inverse power of s.