Properties of the elastic scattering amplitude at high energies

Abstract

It is shown that the imaginary part of the scattering amplitude A1(t,s) is positive and has all positive derivatives with respect to the momentum transfer t in the non-physical region up to its first singularity given by the Landau curve t = t1(s). Hence for the Regge pole with largest l we have dl/dt > 0 in this interval. The dependence of l on t is studied when t → 4μ2. It appears that l(t) = R(t)+const · (4μ2−t)λ+12ifλ+12 is not an integer, and l(t) = R1(t)+const · (t−4μ2)λ+1 ln(4μ2−t) if it is; λ = l(4μ2), and the functions R and R1 have no singularity when t → 4μ2. It is proved that l(t) goes into the upper half-plane when t>4μ2. All these results are obtained without the assumption of the existence of a Hamiltonian. Different possibilities are discussed for the dependence l(t) when t→∞. It is analysed whether the poles corresponding to the “elementary particles” are continuous functions of l. In the Appendix it is shown that in calculating the spectral density π(s,t) the conditions s(t−4μ2) <μ4 is the condition of neglecting all singularities in the l plane except the pole with the maximum Re l.