The Fourier-Bessel expansion of the scattering amplitude at zero momentum transfer for particles with unequal mass

Abstract

We investigate the problem of whether a Fourier-Bessel type expansion can be valid at zero-momentum transfer for the unequal-mass scattering amplitude, as a substitute for the Regge-Watson-Sommerfeld representation. It is shown that the results of a previous paper, giving the solution of the same problem for amplitudes satisfying an unsubtracted dispersion relation can be extended to a class of asymptotically growing scattering amplitudes. In the second part of the paper a Lorentz-pole expansion is outlined, which is valid at arbitrary momentum transfer. The expansion coefficients of this last expansion are expressed by means of the discontinuities of the amplitudes. Finally, a comparison is made between the Fourier-Bessel expansion and the Lorentz expansion. We argue that, at least as far as the unequal-mass formulas are concerned, it is highly unnatural to assume that a pomeron with intercept α(0) = 1 belongs purely to the four-vector representation of the Lorentz group. Rather, it is a mixed object belonging to the four-dimensional and the accompanying infinite-dimensional ones, both characterized by the same Casimir operator eigenvalues.