Completing the multiple diffraction theory of high-energy elastic $p\bar{p}$ and $pp$ scattering: the averaged parton-parton amplitude beyond the forward cone, its $t$-dependent phase, and the energy-dependent transverse interaction radius

Glauber and Velasco (Phys. Lett. B 147, 380, 1984) described elastic proton-antiproton scattering at $\sqrt{s}=546$ GeV by multiple diffraction theory: the proton and antiproton are treated as clusters of partons, and the elastic amplitude is the eikonal transform of an opacity built, in the leading binary-collision approximation, from the proton electric form factor $G_p(t)$ and an averaged parton-parton amplitude $f(t)$. Fits to the then-preliminary UA4 forward data gave $f(t)/f(0)=(1-at)^{-1/2}$ with $a\approx1.8\ (\mathrm{GeV}/c)^{-2}$, a small real component of the opacity ($\rho\approx0.14$), $\sigma_{\mathrm{tot}}\approx62$ mb, and an rms transverse interaction radius of about 0.47 fm for the elementary interaction. The analysis left open the behavior of $f(t)$ beyond the forward cone selected by $G_p^2(t)$, the role of the $t$-dependent complex phase of $f(t)$, the higher-order opacity corrections that enter when the parton clusters become opaque to individual partons, form-factor systematics, the energy dependence of the transverse radius, and the use of incomplete large-$|t|$ data. Later literature resolved the data-side items (final UA4 cross sections up to $|t|=1.55\ \mathrm{GeV}^2$, measured $\rho$ at 546 GeV, improved form factors and forward-quantity methodology) and partially developed the framework (elementary-amplitude studies, dispersion-relation real parts, eikonal extraction in momentum-transfer space, quark-diquark descendants at LHC energies), but a complete multiple-diffraction determination of the averaged parton-parton amplitude — its large-$|t|$ behavior together with its $t$-dependent phase, the quantitative effect of the higher-order opacity corrections, and the systematic energy dependence of the interaction radius across ISR, SPS, Tevatron and LHC energies — remains unpublished; the problem is recorded as open.

Background

In high-energy collisions of hadrons, a sizeable fraction of the total cross section is elastic scattering — the process $p\bar{p}\to p\bar{p}$ or $pp\to pp$ in which the hadrons emerge intact and only their momenta are deflected. The central observable is the differential cross section $d\sigma/dt$ as a function of the squared four-momentum transfer $t$ ($t<0$): inside the forward diffraction cone ($|t|\lesssim1\ \mathrm{GeV}^2$) it falls roughly exponentially, and at larger $|t|$ it develops a characteristic dip-and-bump structure. The optical theorem of scattering theory relates the forward elastic amplitude at $t=0$ to the total cross section $\sigma_{\mathrm{tot}}$, and the ratio $\rho=\mathrm{Re}\,F(0)/\mathrm{Im}\,F(0)$ of the real to the imaginary part of the forward amplitude is extracted from the interference between the hadronic amplitude and the precisely known Coulomb amplitude at very small $|t|$ (the Coulomb-nuclear interference, CNI, region).

Elastic scattering at small momentum transfer is naturally described by eikonal (profile-function) methods, introduced by Glauber for the scattering of composite particles and known in this context as multiple diffraction theory. The amplitude is written as

$$F(t)=i\int_0^\infty J_0(b\sqrt{-t})\,\{1-\exp[-\Omega(b)]\}\,b\,db,$$

where $J_0$ is the zeroth-order Bessel function, $b$ is the impact parameter (the transverse separation of the two projectile trajectories), and $\Omega(b)$ is the opacity — a generally complex function encoding the probability that the projectiles interact at separation $b$; the factor $1-\exp[-\Omega]$ resums the series of repeated elementary collisions. The geometrical model of Chou and Yang identifies the opacity with the overlap of the matter distributions of the two hadrons, so that hadron shape directly controls the scattering (see T. T. Chou and C. N. Yang, Phys. Lett. B 244 (1990) 113 for the energy-dependent formulation). The proton electric form factor $G_p(t)$, measured in electron-proton scattering, encodes the spatial distribution of charge — and, in parton language, of partons — inside the proton.

Glauber and Velasco (1984) applied this framework to the elastic $p\bar{p}$ data recorded at $\sqrt{s}=546$ GeV by the UA4 experiment at the CERN SPS $p\bar{p}$ collider (the Sp$\bar{p}$S; $\sqrt{s}$ denotes the center-of-mass energy, and the ISR, CERN's earlier proton-proton collider, reached only $\sqrt{s}=63$ GeV). They pictured the proton and antiproton as clusters of partons (the proton's constituents as probed in deep-inelastic scattering; related 'valon' pictures treat the proton as three dressed quarks). In the leading, binary-collision approximation, the opacity is the convolution of the squared proton form factor $G_p^2(t)$ — each elementary collision requires partons to be found in both clusters — with an averaged parton-parton amplitude $f(t)$, the effective interaction of one parton of one proton with one parton of the other. Fits to the then-available forward UA4 data, using empirical small-$|t|$ form-factor parametrizations derived from electron scattering (the Felst polynomial fit, and the four-pole BSWW fit of the Borkowski et al. Mainz analysis), determined the single-parameter form $f(t)/f(0)=(1-at)^{-1/2}$ with $a\approx1.8\ (\mathrm{GeV}/c)^{-2}$, a small real component of the opacity (corresponding to $\rho\approx0.14$), $\sigma_{\mathrm{tot}}\approx62$ mb, and — interpreting the falloff parameter diffractively through $\langle r^2\rangle=3a$ — an rms transverse interaction radius of about 0.47 fm for the averaged parton-parton interaction. The pure Chou-Yang choice $f(t)/f(0)=1$ with purely absorptive interactions could not reproduce the data, so the elementary interaction must have nontrivial size and phase structure; the fits also reproduced the observed rapid change of the logarithmic slope of $d\sigma/dt$ near the forward direction as a form-factor effect.

The 1984 analysis was, by construction, confined to the forward cone — the form factor $G_p^2(t)$ suppresses sensitivity to larger $|t|$ — and to preliminary UA4 data. The limitations explicitly stated in the analysis define the present problem: (i) the behavior of the averaged parton-parton amplitude at larger $|t|$ could not be determined, the analysis permitting examination of $f(t)/f(0)$ only within the cone defined by the proton form factor; (ii) the expected $t$-dependent complex phase factor of $f(t)$ was omitted entirely, since for plausible phases its influence would be felt appreciably only at the larger momentum transfers where the then-preliminary data offered little means of discriminating between different choices; (iii) only the leading binary-collision term of the opacity expansion was used, the correction terms that become relevant when either parton cluster is too opaque to an average parton in the other being deferred until the parameters warrant them; (iv) the conclusions relied on the available proton form-factor parametrizations (Felst versus BSWW), whose residual differences at very small $|t|$ limited the precision of the inferred slope changes and parameters; (v) the inferred transverse interaction radius was believed to be an energy-dependent parameter, with only indications of significantly smaller values from ISR $pp$ studies and no systematic determination of its energy dependence performed; and (vi) the UA4 data at $-t\gtrsim0.5\ (\mathrm{GeV}/c)^2$ were incomplete and preliminary, preventing discrimination among parametrizations and the detection of phase effects at larger momentum transfers.

Problem Statement

Within the multiple diffraction (Glauber-Velasco) framework — in which the elastic $p\bar{p}$ or $pp$ amplitude is the eikonal transform of an opacity built from the proton form factor and an averaged parton-parton amplitude $f(t)$ — carry the 1984 analysis beyond the forward cone to completion. Specifically, determine from the elastic scattering data record now available (CERN ISR $pp$ and SPS $p\bar{p}$ data including the final UA4 measurements, Tevatron $p\bar{p}$ data, and LHC $pp$ data, together with the Coulomb-nuclear-interference measurements of $\rho$): (1) the averaged parton-parton amplitude $f(t)$ at larger $|t|$, where the data now include the kink/shoulder region at 546 GeV and the dip-bump regions at Tevatron and LHC energies; (2) whether $f(t)$ and the elastic amplitude carry a $t$-dependent complex phase, and its phenomenological role at larger momentum transfers, where the phase influence that the 1984 fits could not discriminate is now testable against the modern data; (3) the size and effect of the higher-order, beyond-binary-collision opacity corrections that enter when the parton clusters become sufficiently opaque to individual partons; and (4) the transverse interaction radius implied by the fitted $t$-dependence of the elementary amplitude, in particular its systematic energy dependence across ISR, SPS, Tevatron, and LHC energies. These four points form one linked objective: a complete, internally consistent multiple-diffraction determination of the effective parton-parton interaction — its transverse size, its phase, and the corrections to the opacity expansion — and of the energy dependence of the interaction geometry, superseding the 1984 forward-cone analysis and testing whether the parton-cluster multiple-diffraction picture remains adequate when confronted with the full modern data record (including the non-exponential low-$|t|$ behavior and the measured forward-phase data). All four points must be determined jointly and consistently over the full data record (CERN ISR $pp$, SPS $p\bar{p}$ including the final UA4 measurements, Tevatron $p\bar{p}$, and LHC $pp$, together with the Coulomb-nuclear-interference measurements of $\rho$); a determination restricted to a single energy or a single observable, such as an SPS-only fit of $f(t)$, is a partial contribution and does not by itself resolve the objective.

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