Upper bounds on the Elastic differential cross section

Abstract

We derive improved axiomatic upper bounds on the elastic unpolarized differential cross section, dσ/dΩ, at high energies for the scattering of particles with arbitrary spin. We prove that dσ/dω ≤ _s→∝σel/4φ {(L+1)²[P_L(cos θ)]² + sin²θ[P_L'(cos θ)]²} with L=½ √S/t₀In s/σ_el), where s and t are, respectively, the squares of the c.m. energy and the c.m. momentum transfer; θ is the c.m. scattering angle. Further √t₀ , is the mass of the lowest mass state that couples to the crossed t-channel (being equal to twice the pion mass for ππ and πN scattering). This result has the following important consequences: 1. (i) for forward scattering, (dσ/dt) t-0 ≤ _s→∝ σel/4t₀[ In (s/σ_el)]²2. (ii) for fixed θ ≢ 0, π (dσ/dω (s;cosθ) ≤_s→∝ 1/4φ²√s/t₀ σel/sinθ In) s/σel and 3. (iii) for fixed negative t,dσ/dω (s;t) ≤_s→∝ 1/8φ√t₀ s/√ −tσel In (s/σel)We also give an upper bound on differential cross section involving the diffraction peak width but not having an explicit ln s dependence.