[Paper Review] Towards the saturation of the Froissart bound

Joachim Kupsch|ArXiv.org|Jan 31, 2008

Elasticity and Material Modeling39 references3 citations

TL;DR

This paper investigates pion-pion scattering amplitudes that satisfy Mandelstam analyticity, crossing symmetry, and inelastic unitarity constraints, aiming to saturate the Froissart bound on total cross sections. Using Regge-type amplitudes with crossing cuts and a modified Khuri representation, the study constructs solutions that achieve a total cross section scaling as $(\log s)^{-3}$, approaching saturation of the Froissart bound, though elastic unitarity remains unfulfilled in the optimal construction.

ABSTRACT

It is the aim of this paper to summarize results about the construction of amplitudes, which rigorously satisfy Mandelstam analyticity, crossing symmetry, and (at least partly) the constraints imposed by elastic and inelastic unitarity. The results are discussed under particular emphasis of a strong increase of the absorptive part of the forward amplitude and the saturation of the Froissart bound.

Motivation & Objective

• To explore whether the Froissart bound on total cross sections can be saturated under rigorous constraints of analyticity, crossing symmetry, and unitarity.
• To analyze the role of elastic unitarity in limiting the growth of the absorptive part of the forward amplitude.
• To construct amplitudes that satisfy Mandelstam analyticity, crossing symmetry, and inelastic unitarity, while optimizing the high-energy behavior of the total cross section.
• To identify the obstacles posed by elastic unitarity in achieving tighter bounds or saturation of the Froissart bound.

Proposed method

• Uses Mandelstam spectral integrals with one subtraction to represent amplitudes and applies a non-linear fixed-point mapping to enforce elastic unitarity.
• Employs a Regge-type ansatz with leading trajectories having intercept $\alpha(0) \leq 1$ and logarithmic corrections to model high-energy behavior.
• Applies the Khuri representation and Watson-Sommerfeld transform to generalize fixed-point mappings to Regge amplitudes, enabling inclusion of elastic unitarity.
• Utilizes Mellin-Barnes integral representations to express the double spectral function and unitarity conditions in terms of holomorphic functions.
• Introduces a ghost-killing factor $\alpha(s)$ in the Khuri representation to cancel spurious poles and ensure analyticity.
• Applies linearization techniques to quadratic unitarity inequalities to derive constraints on the absorptive part and spectral functions.

Experimental results

Research questions

• RQ1 Can the Froissart bound be saturated by scattering amplitudes that rigorously satisfy Mandelstam analyticity, crossing symmetry, and inelastic unitarity?
• RQ2 What is the maximal possible growth of the total cross section under these constraints, and how does elastic unitarity affect this bound?
• RQ3 Why does the inclusion of elastic unitarity prevent further improvement beyond $\sigma_{\text{tot}} \sim (\log s)^{-3}$ in existing constructions?
• RQ4 How do Regge poles and cuts contribute to the absorptive part and total cross section in the high-energy limit?
• RQ5 Can the Khuri representation be extended to include both Regge poles and elastic unitarity while preserving analyticity and crossing symmetry?

Key findings

• The optimal construction satisfying Mandelstam analyticity, crossing symmetry, and inelastic unitarity yields a total cross section scaling as $\sigma_{\text{tot}}(s) \sim (\log s)^{-3}$ at high energies.
• Amplitudes constructed with Regge poles and crossing cuts can saturate the Froissart bound, but only when elastic unitarity is not imposed.
• Elastic unitarity imposes severe constraints that prevent further increase in the absorptive part beyond the $ (\log s)^{-3} $ behavior.
• The use of a ghost-killing factor $\alpha(s)$ in the Khuri representation successfully cancels kinematic poles and ensures analyticity in the required strip.
• The Mellin-Barnes representation of the unitarity integral allows for a precise analytic continuation and derivation of the partial wave amplitude structure.
• The function $\phi(s-i0,\alpha(s+i0))$ is shown to be Hölder continuous, supporting the regularity of the solution under the derived constraints.