Space--time analysis of high-energy scattering

We present a phenomenological space-time analysis of high-energy scattering. For a two-body process a + b → c + d, the off-shell amplitude A(ν, κ, κ′, t) [ν ≡ (pₐ + p_c)⋅p, p ≡ p_b + p_d, κ = (pₐ + p_c)², t = Δ² = (p_c − pₐ)², κ′ = (pₐ + p_c)⋅Δ = p_c² − pₐ² ≡ κ_c − κₐ] is written in terms of its Lehmann-Symanzik-Zimmermann correlation function ℱ(z⋅p, z⋅Δ, z², t). The properties of A(ν, κ, …) in the R-limit (ν → ∞, with κ, κ′, and t fixed) and in the A-limit (ν → ∞, $\omega \equiv -\frac{\kappa }{2\nu }$, κ′, and t fixed) are translated into space-time properties of ℱ(z). By combined use of the assumed Regge ν^α(t) R-limit behavior, scaling ν^{ρ−2}F(ω) A-limit behavior, as well as of the assumed Regge small-ω A-limit behavior ${lim}_{\omega \to 0}{lim}_{A}A(\nu ,\kappa ,\dots )={lim}_{\kappa \to \infty }{lim}_{R}A(\nu ,\kappa ,\dots )$, we determine the form of ℱ(z) in the interesting space-time regions to be (z⋅p)^α(t) g(z⋅Δ, z², t) as (z⋅p) → ∞ and f_ρ(z⋅p, z⋅Δ, t)(z²)^{-ρ} near the light cone (LC) z² = 0, with f_ρ(λ, 0, t) ∼ λ^α(t) as λ → ∞. This latter property of f_ρ is shown to uniquely correspond to having quark-hadron amplitudes with Regge behavior, at least in quark models for light-cone operator-product expansions. The large-z² behavior of ℱ(z) is determined from the assumed Regge behavior of the pole-dominance approximation to A(ν, κ, …), and we again find a (z⋅p)^α(t) large-(z⋅p) behavior of ℱ(z) in this region. The emerging picture is simply summarized by setting ℱ(z) ≅ f_ρ(z⋅p, 0, t) g(z⋅Δ, z², t) with g(z⋅Δ, z², t) ∼ (z²)^{-ρ} near the LC. The only important space-time regions for high-energy scattering are the regions z⋅p ∼ O(ν), which is shown to determine the moving Regge poles, and z² ≅ 0, which is shown to give rise to a fixed-pole behavior if and only if α(t) < ρ − 2. We also consider the high-energy limit of semiweak exclusive processes, and show that, contrary to existing claims, such processes need not scale.