Four-Dimensional Symmetry

Abstract

We study the symmetry of scattering amplitudes at vanishing momentum transfer. We show that the little group of the Poincaré group corresponding to vanishing four-momentum (isomorphic to the homogeneous Lorentz group, or, by analytic continuation, to the four-dimensional rotation group) is in general not a symmetry of the scattering amplitude. However, the spectrum of the amplitude is classified according to the larger symmetry. Regge poles occur in families; the members of the family follow the first one in integer steps at vanishing momentum transfer and are classified according to a new quantum number derived from the higher symmetry. We derive a one-parameter "mass formula" describing the deviation of the slopes of Regge trajectories from the value required by the higher symmetry. The theory is applied to the problem of the high-energy scattering of particles of arbitrary mass, and leads to an unambiguous asymptotic expression for the scattering amplitude. We analyze the implications of the new symmetry for the spectrum of hadrons. The predicted new Regge trajectories lead to observable particles and resonances; their coupling strength is determined by the symmetry in terms of the parameters of the "parent" trajectories. In particular we assign the $N^{\prime }\mathrm{}\left(1400\right)\mathrm{}$ resonance to the second "daughter" of the $N_{\alpha }$ trajectory; the symmetry breaking turns out to be small, and the decay width of $N^{\prime }\mathrm{}\left(1400\right)\mathrm{}$ is computed in satisfactory agreement with the experimental result.