Critical analysis of derivative dispersion relations at high energies

Abstract

We discuss some formal and fundamental aspects related with the replacement of integral dispersion relations by derivative forms, and their practical uses in high energy elastic hadron scattering, in particular $pp$ and $\bar{p}p$ scattering. Starting with integral relations with one subtraction and considering parametrizations for the total cross sections belonging to the class of entire functions in the logarithm of the energy, a series of results is deduced and our main conclusions are the following: (1) except for the subtraction constant, the derivative forms do not depend on any additional free parameter; (2) the only approximation in going from integral to derivative relations (at high energies) concerns to assume as zero the lower limit in the integral form; (3) the previous approximation and the subtraction constant affect the fit results at both low and high energies and therefore, the subtraction constant can not be disregarded; (4) from a practical point of view, for single-pole Pomeron and secondary reggeons parametrizations and center-of-mass energies above 5 GeV, the derivative relations with the subtraction constant as a free fit parameter are completely equivalent to the integral forms with finite (non-zero) lower limit. A detailed review on the conditions of validity and assumptions related with the replacement of integral by derivative relations is also presented and discussed.

Revised version, 30 pages, 16 eps-figures, elsart.cls (included), to appear in Nucl Phys. A

References in corpus (4)

• Benchmarks for the Forward Observables at RHIC, the Tevatron-run II and the LHC
• The hard pomeron in soft data
• Analytic models and forward scattering from accelerator to cosmic-ray energies
• Extrema bounds for the soft Pomeron intercept

Cited by in corpus (20)

• Aspects of a dynamical gluon mass approach to elastic hadron scattering at LHC
• Soft Pomerons and the Forward LHC Data
• On the rise of proton-proton cross-sections at high energies
• A study on analytic parametrizations for proton-proton cross-sections and asymptotia
• Forward Elastic Scattering and Pomeron Models
• Phenomenological analysis connecting proton-proton and antiproton-proton elastic scattering
• Non-perturbative QCD effects in forward scattering at LHC
• Leading components in forward elastic hadron scattering: Derivative dispersion relations and asymptotic uniqueness
• Representation of Integral Dispersion Relations by Local Forms
• Phenomenological Analysis of $pp$ and $\bar{p}p$ Elastic Scattering Based on Theoretical Bounds in High-Energy Physics
• Total Hadronic Cross Section Data and the Froissart-Martin Bound
• An updated analysis on the rise of the hadronic total cross-section at the LHC energy region
• Bounds on the rise of total cross section from LHC7 and LHC8 data
• Testing the black disk limit in $pp$ collisions at very high energy
• Structure of Forward pp and pp Elastic Amplitudes at Low Energies
• Applicability of a Representation for the Martin's Real-Part Formula in Model-Independent Analyses
• Exact treatment of dispersion relations in pp and pp elastic scattering
• Derivative Dispersion Relations
• Optical Theorem, Crossing Property and Derivative Dispersion Relations: Implications on the Asymptotic Behavior of ${\sigma }_{tot}(s)$ and $\rho (s)$
• The ratio ${\rho }_{\bar{p}p}^{pp}(s)$ in Froissaron and Maximal Odderon approach