A universal sum rule for hadronic elastic scattering with applications to the Tevatron, RHIC and LHC

Abstract

Under the assumption of total absorption and dominance of the imaginary part of the scattering amplitude, we present a sum rule for any hadronic elastic differential cross-section dσ dt : the dimensionless quantity 1 2 (dt) 1 π dσ dt → 1, at asymptotic energies. Experimental data from ISR and Tevatron confirm a trend towards its saturation and some estimates are presented for LHC. Its universality and further consequences for the nature of absorption in QCD based models for elastic and total cross-sections are explored.Ab initio calculations of hadronic elastic amplitudes and total cross-sections (in QCD) are presently difficult due to our meager understanding of "soft" physics, that is, the non-perturbative and confinement region of QCD. Hence, the need to invoke general principles such as analyticity and unitarity to obtain bounds and restrictions on these amplitudes [1][2][3]. Analyticity and unitarity are expected to hold for finite-ranged hadron dynamics, only massive hadrons being the bound states of quarks and glue. In the following, we find that, under rather mild assumptions, a universal behavior for all hadrons is likely to emerge at asymptotic energies. Consider the amplitude for an elastic process, be the square of the CM energy, t = (p a − p c ) 2 , be the momentum transfer and let us normalize the amplitude so that the differential and total cross-sections are given byTo enforce (direct or s-channel) unitarity and incorporate the knowledge that most of the hadronic scatterings at high energies are peaked in the forward direction, an eikonal formalism is convenient. The elastic amplitude may be expanded in the impact parameter (b-space) in the usual fashionin terms of the "partial b-wave" amplitudesFwhere the inelasticity factor η(s, b) lies between 0 and 1, and δ R (s, b) is the real part of the phase shift. The dimensionless "b-wave cross-sections" are given byandEqs.(3) show explicitly the maximum permissible rise for the different cross-sections due to unitarity. For complete absorption of "low" partial waves at asymptotic energies (which translates into η(s, b) → 0 for b → 0 and s → ∞), one obtains the geometric limit (including the contribution from shadow scattering):1

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“…Here we shall derive two asymptotic sum rules which are integrals over momentum transfer for the real and the imaginary parts of the elastic amplitude [293,294]. At high energies, ignoring all particle masses, let the complex elastic amplitude F(s, t) be normalised so that…”

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“…Since the phase φ = π, for the satisfaction of both ASR the model of Eq. (14) needs to be modified so as to include a real part in the leading amplitude i A(s), i.e. the leading term (at t ≈ 0) should acquire an extra phase.…”

The elastic differential pp cross-section at 13 TeV: an empirical model analysis

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