Transition from resonances to surface waves in elastic scattering

In this article, we study resonances and surface waves in π⁺–p scattering. We focus on the sequence whose spin-parity values are given by . A widely-held belief takes for granted that this sequence can be connected by a moving pole in the complex angular momentum (CAM)-plane, which gives rise to a linear trajectory of the form , which is the standard expression of the Regge pole trajectory. But the phenomenology shows that only the first few resonances lie on a trajectory of this type. For higher J^p this rule is violated and is substituted by the relation JkR, where k is the pion-nucleon c.m.s. momentum, and R1 fm. In this article we prove: (a) Starting from a non-relativistic model of the proton, regarded as composed by three quarks confined by harmonic potentials, we prove that the first three members of this π⁺–p resonance sequence can be associated with a vibrational spectrum of the proton generated by an algebra . Accordingly, these first three members of the sequence can be described by Regge poles and lie on a standard linear trajectory. (b) At higher energies the amplitudes are dominated by diffractive scattering, and the creeping waves play a dominant role. They can be described by a second class of poles, which can be called Sommerfeld’s poles, and lie on a line nearly parallel to the imaginary axis of the CAM-plane. (c) The Sommerfeld’s pole which is closest to the real axis of the CAM-plane is dominant at large angles, and describes in a proper way the backward diffractive peak in both the following cases: at fixed k, as a function of the scattering angle, and at fixed scattering angle θ=π, as a function of k. (d) The evolution of this pole, as a function of k, is given in first approximation by JkR.