Energy dependence and dynamical basis of the proton matter-distribution parameters in the Chou–Yang geometrical model of elastic $pp$/$p\bar{p}$ scattering

In their 1990 geometrical-model analysis of elastic $pp$ and $p\bar{p}$ scattering from the CERN ISR to the Tevatron, T. T. Chou and C. N. Yang described the proton's hadronic matter distribution by a dipole form factor with an energy-dependent range parameter: the opacity normalization $K$ and the inverse squared range $m^{-2}$ were determined through fits to elastic $pp$ data at $\sqrt{s}=23.5$ GeV and $p\bar{p}$ data at 546 GeV, parametrized linearly in $\ln s$, and the Tevatron prediction was obtained by extrapolation. This formulation leaves two issues open: (i) the true energy dependence of $K$ and $m^{-2}$, to be determined from accurate elastic data at Tevatron and higher energies (such data now exist at 1.8/1.96 TeV and at LHC energies of 7, 8 and 13 TeV); and (ii) the absence of any theoretical justification or physical interpretation for the dipole hadronic form factor, for the functional forms of the opacity $\Omega(b)$, and for the implied exponential density $\rho(r)=(m^{3}/8\pi)\exp(-mr)$. Both cores remain open: later work reformulated the energy-dependence question (generalized geometrical scaling, eikonal-zero models) and continues Chou–Yang extractions with TOTEM data, but no work to date establishes data-supported $K(s)$, $m^{-2}(s)$ trends in the original parametrization, and the dynamical status of the dipole hadronic matter form factor — whose heuristic origin is the Wu–Yang identification with the electromagnetic form factor — remains unresolved.

Background

Elastic scattering — the process in which two hadrons collide and emerge with their quantum numbers unchanged and no new particles produced — contributes roughly 18–28% of the total interaction rate at high energy and is measured through the differential cross section $d\sigma/dt$ as a function of the squared four-momentum transfer $t$. The machines relevant here are the CERN Intersecting Storage Rings (ISR, $pp$ collisions up to $\sqrt{s}=62$ GeV, where $s$ is the squared center-of-mass energy), the CERN SPS $p\bar{p}$ Collider (up to about 630 GeV), the Fermilab Tevatron $p\bar{p}$ collider (1.8–1.96 TeV), and, later, the LHC with the TOTEM elastic-scattering program (7, 8 and 13 TeV).

A standard tool for these reactions is the impact-parameter representation: the scattering amplitude is written as a two-dimensional Fourier transform over the impact parameter $b$, the transverse separation of the two hadrons' centers. In the eikonal approximation (appropriate for high-energy, small-angle scattering), the amplitude is $a(s,t)\propto\int d^{2}b\,e^{iq\cdot b}[1-\exp(-\Omega(s,b)/2)]$, where the opacity $\Omega(s,b)$ is a real positive function measuring the absorptive strength of the interaction at separation $b$; $\Omega\to\infty$ corresponds to a completely black disk.

The geometrical model of Chou and Yang (1968) treats each hadron as an extended cloud of hadronic matter with density $\rho(\mathbf{r})$. By Lorentz contraction, each proton sees the other as a two-dimensional compressed density in the plane transverse to the collision axis, $D(x,y)=\int\rho(x,y,z)dz$, and the opacity is taken proportional to the overlap (convolution) of the two compressed densities at separation $b$. The unknown hadronic matter form factor was identified, via the Wu–Yang conjecture (a 1965 heuristic speculation), with the measured electromagnetic form factor of the proton. The standard empirical parametrization of the proton's electric (Sachs) form factor is the dipole form $G_{D}(q)=[1+q^{2}/\mu^{2}]^{-2}$ with $\mu^{2}=0.71$ GeV$^{2}$; in the hadronic context the scale $m$ (with $\mu\to m$) is a range parameter of the matter distribution, and in the 1990 parametrization it is energy-dependent, so the identification with the electromagnetic scale is heuristic rather than strict. For two protons, the convolution of two dipole densities gives $\Omega(b)=K\,(mb)^{3}K_{3}(mb)/8$, where $K_{3}$ is a modified Bessel function and $K$ an energy-dependent normalization. In the classic fixed-dipole version of the model the normalization is determined from the experimental value of the measured total cross section $\sigma_{\rm tot}$ at each energy (Ávila and Menon, 2008); in the 1990 parametrization, by contrast, $K(s)$ and $m^{-2}(s)$ are jointly determined from fits to elastic differential-cross-section data, with $\sigma_{\rm tot}$ entering as a forward constraint through the optical theorem. The three-dimensional density whose Fourier transform is the dipole form factor is the exponential $\rho(r)=(m^{3}/8\pi)\exp(-mr)$, with rms radius $\sqrt{12}/m$.

In Chou and Yang (Phys. Lett. B 244, 113 (1990)), this scheme fits the elastic $pp$ (ISR, $\sqrt{s}=23.5$ GeV) and $p\bar{p}$ (SPS Collider, 546 GeV) data with a dipole matter form factor whose range parameter depends on energy; the parameters $K$ and $m^{-2}$ are parametrized linearly in $\ln s$ (Beggio–Menon–Valin 2000, 'following the authors'), and the Tevatron prediction is obtained by extrapolation. The paper notes three limitations: no cogent theoretical justification exists for the dipole form of the hadronic form factor or for the functional forms of $\Omega(b)$ proposed in the literature; the true energy dependence of $K$ and $m^{-2}$ must await accurate elastic data at Tevatron and higher energies; and no physical interpretation is claimed for the implied exponential density $\rho(r)=(m^{3}/8\pi)\exp(-mr)$.

Later literature established several relevant constraints. Zero-free dipole eikonals describe $d\sigma/dt$ only at small momentum transfer ($|t|\lesssim 2$ GeV$^{2}$) and predict spurious secondary dips at larger $|t|$, while eikonal models with a zero (such as the Bourrely–Soffer–Wu impact-parameter picture) describe the data better (Ávila and Menon, 2008). The Wu–Yang identification of the hadronic and electromagnetic form factors remains unproven within phenomenology. A nonperturbative QCD construction based on the stochastic vacuum model (Dosch, Ferreira and Krämer, 1994) derives an eikonal from gauge-field vacuum correlators and relates hadronic sizes to scattering observables, but it yields a different opacity structure and does not derive the dipole hadronic matter form factor. The generalized geometrical scaling of Kawasaki, Maehara and Yonezawa (1993) reformulates the energy-dependence question at the level of a scaling variable and holds to about 2% up to $\sqrt{s}=550$ GeV.

Problem Statement

Within the Chou–Yang geometrical model as formulated by Chou and Yang (1990) — elastic $pp$/$p\bar{p}$ scattering described by an eikonal whose opacity is the convolution of two dipole-form-factor hadronic matter distributions with parameters $K$ (opacity normalization) and $m^{-2}$ (inverse squared range) — resolve the open issues this formulation leaves:

(a) Empirical: determine, from accurate elastic differential-cross-section data at Tevatron and higher energies (measurements now exist at 1.8 and 1.96 TeV and at LHC energies of 7, 8 and 13 TeV), the true energy dependence of $K$ and $m^{-2}$, replacing the assumed linear-in-$\ln s$ parametrization, and establish the data-supported trends of these parameters — including whether the assumed dipole form factor and opacity forms remain adequate where the data test them.

(b) Theoretical: provide a derivation from underlying dynamics (QCD), or a principled physical interpretation (the latter including the implied three-dimensional exponential density $\rho(r)=(m^{3}/8\pi)\exp(-mr)$), for the dipole parametrization of the hadronic matter form factor and for the functional forms of the opacity $\Omega(b)$ — or demonstrate that these forms are inadequate and must be replaced.

These two components are distinct open issues of the same formulation, and both are mandatory for resolution: they may be reported and evaluated separately, but neither alone resolves the task. Within this split, (a)'s adequacy question is empirical and is settled by the elastic data, while (b) requires a derivation, a principled physical interpretation, or a demonstration that the dipole and opacity forms must be replaced; a data-driven finding that these forms are inadequate therefore satisfies (a) but does not by itself satisfy (b).

References

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