Dispersion Relations and High Energy Limits in Quantum Field Theory

Abstract

The lower and upper bounds of the high energy limit of pion-nucleon forward scattering amplitudes are investigated. The former is studied using available experimental data and found that at least one subtraction is necessary in the dispersion relations. The latter can be determined under an additional requirement that the forward amplitude has no zero in the complex energy plane. Some discussions are given to the problems on the high-energy behaviours in quantum field theories. It has been conjectured that the total cross section for the pion-nucleon scat­ tering approaches a constant value and does not damp so rapidly at high energy limit. This conjecture comes from the experimental facts which show that at very high energies the cross section keeps a constant value, about 30mb. 1 l The disper­ sion relations for pion-nucleon forward scattering amplitudes given by Goldberger 2 l are derived from this supposition. However, no theoretical foundation which ex­ cludes the possibility of damping of cross section at high energy limit or excludes the validity of unsubtracted dispersion relations has been given except the work of Arnowitt and Feldman. 3 l In this paper we will show using available experimental data that at least one subtraction is necessary in the dispersion relations for pion-nucleon forward scat­ tering amplitudes irrespective of the detailed structures of interactions. This is the same result as obtained by Arnowitt and Feldman who showed this, under the as­ sumption of the existence of I.e/ term in the interaction Lagrangian, by construct­ ing unsubtracted dispersion relation in the form derived by Goldberger and finally by ·leading to the inconsistency. However, their method seems to be incomplete and inconclusive. This point will be discussed in § 4. In the same section, some words shall be added to the work of Kallen 4 l on the magnitudes of the renor­ malization constants. For the determination of the upper bound of high energy limit of total cross