This paper discusses a relation between the high-energy behaviour of the real part of the elastic scattering amplitude and the high-energy behaviour of its imaginary part. Using the PhragmenLindel of theorem we derive the formulae which express this relation. Under the assumption that the elastic amplitude becomes purely imaginary in the limit of infinite energy, these formulae suggest a simple fiveparameter description of the high-energy behaviour of: i) the total crosssections for the collisions A+B and A+B, where B is an antiparticle of B, and ii) the ratioX between the real and imaginary parts of the forward scattering amplitude for the corresponding elastic reactions. We proved that the existing data for nucleon-nucleon scattering can, in fact, be described in this way. We also checked that our results are consistent with the calculation based on the dispersion relations. The asymptotic formulae also show that:a) from the fact that the total cross-sections are going down with energy, it follows that the real part of the forward-scattering amplitude should be negative at high energies, andb) from the fact that the total pp cross-section changes with energy much more than the total pp cross-section, it follows that the absolute value of the ratioX should be smaller for pp scattering than for the pp scattering. We also considered the more general case when the ratio between the real and imaginary parts of the forward elastic amplitude tends to a nonvanishing limit at high energy. For nucleon-nucleon scattering the present day data suggest that the absolute value of the ratioX at infinite energy is smaller than 0.2.
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