Spin effects on the high-energy deuteron cross-section defect: determining the spin-dependent nucleon amplitudes and testing the spin-dependent Glauber predictions at multi-GeV energies

Franco and Glauber's 1966 analysis expressed all deuteron cross sections in terms of the elementary neutron and proton elastic amplitudes and the deuteron ground-state wave function, and explained the cross-section defect (the failure of free-nucleon cross sections to add in the weakly bound deuteron) through single-double scattering interference. Its spin-dependent extension could not be applied quantitatively because the full set of spin-dependent nucleon scattering amplitudes at high energies was not measured, so spin effects on the defect were left unassessed. The formalism has since been completed and validated at intermediate energies, spin-averaged proton-proton (pp) and neutron-proton (np) amplitudes at 2-50 GeV/c are now data-fitted, and the unpolarized defect is reliably computed; however, the five complex spin-dependent nucleon-nucleon (NN) amplitudes above the phase-shift-analysis range (roughly 3 GeV for pp, 1.2 GeV for pn) remain undetermined by data, so the spin dependence of the deuteron total cross sections and of the cross-section defect at multi-GeV energies has never been measured or definitively computed. The surviving problem is to determine the spin-dependent NN amplitudes at multi-GeV energies (the antiproton-nucleon channels being an optional extension matching the source paper's $\bar{p}d$ application), and to compute with them the spin-dependent deuteron total cross sections and the spin dependence of the defect with a quantified error budget (frozen-nucleon approximation, double-scattering truncation, neglected commutators, short-range wave-function model dependence).

Background

The deuteron is the weakly bound proton-neutron system of nuclear physics (binding energy 2.2 MeV; the two nucleons are on average about 2 fm apart). Because free neutron targets do not exist, high-energy deuterium targets have long been used as neutron substitutes: one measures a cross section on deuterium and on hydrogen and obtains the neutron cross section by subtraction, assuming the deuteron cross section equals the sum of the free-proton and free-neutron values. Experiments from the 1950s onward showed that this additivity fails; for antinucleon beams the deviation reached 20-40% of the elementary antinucleon-nucleon cross sections. The deviation, the cross-section defect, $\delta\sigma = \sigma_p + \sigma_n - \sigma_d$ for total cross sections (with analogous definitions channel by channel), arises because the incident particle can scatter in sequence from both nucleons, so each nucleon partially shadows the other; at high energy, where the forward amplitudes are dominated by their positive imaginary parts, the defect is systematically positive.

Franco and Glauber (1966) gave the systematic high-energy treatment. In the Glauber (eikonal) approximation, the high-energy small-angle limit in which the incident particle moves along a straight line at fixed impact parameter $\mathbf{b}$ while the target nucleons stay frozen at their instantaneous positions during the short collision time, the scattering is described by profile functions $\Gamma(\mathbf{b})$ built from phase-shift functions accumulated along the path. Expanding for a two-nucleon target separates single scattering, double scattering, and their interference, so that deuteron cross sections become integrals over the elementary neutron and proton amplitudes $f_j(\mathbf{q})$ and the deuteron ground-state wave function $\phi(\mathbf{r})$ . The optical theorem, which fixes the total cross section from the imaginary part of the forward amplitude, then yields closed expressions for $\delta\sigma$ . Because double scattering is largest when the neutron and proton are close together, the defect weights the short-distance part of $\phi(\mathbf{r})$ heavily. Triple scattering (which requires a backward leg) and recoil energy transfer are neglected. For spin-dependent interactions, the elastic amplitude of two spin-1/2 particles constrained by charge symmetry and by rotational, reflection, and time-reversal invariance is a combination of five complex amplitudes $A, B, C, D, E$ multiplying spin-independent, single-spin, and double-spin operators; the paper expressed the deuteron total cross section in terms of these, replacing, in the spin-dependent double-scattering term, the ordered operator products by an anticommutator and thereby neglecting the commutator, which matters only when the two interaction regions overlap strongly. Applied to antiproton-deuteron data at 0.13-17.1 GeV, using Gaussian and grey-disk extrapolations for the small-momentum-transfer behavior of the input amplitudes and seven model deuteron wave functions, the unpolarized predictions agreed with the measurements.

The authors stated what was missing: the spin-dependent nucleon amplitudes at high energy were not measured completely enough to determine all components of the nucleon scattering matrices, so the spin-dependent formulas could not be applied quantitatively and spin effects on $\delta\sigma$ could not be assessed definitively; the short-range wave function was not accurately known; the validity of the approximations (frozen nucleons, neglected recoil and triple scattering, the anticommutator replacement) remained to be tested; and the off-forward amplitudes $f_j(\mathbf{q})$ at small but nonzero momentum transfer had to be extrapolated rather than measured.

Much of this has since been settled. Modern tools frame the current status. SAID, the partial-wave analysis database maintained at George Washington University, provides data-fitted nucleon-nucleon amplitude solutions, but only up to laboratory energies of about 3 GeV for proton-proton ( $pp$ ) and 1.2 GeV for proton-neutron ( $pn$ ) scattering. Above that range, Regge phenomenology, a relativistic framework describing amplitudes through the exchange of Regge trajectories with parametrized energy and momentum-transfer dependence, supplies model amplitudes that are not directly constrained by spin data. The Faddeev equations, the exact quantum-mechanical three-body scattering equations, provide benchmark calculations below roughly 1 GeV. The NICA Spin Physics Detector (SPD), under construction at the Joint Institute for Nuclear Research with polarized proton and deuteron beams, will provide double-polarized $pp$ , $pd$ , and $dd$ collisions at nucleon-nucleon center-of-mass energies from about 3.4 GeV at its first stage up to tens of GeV in the full program. Within this landscape, the refined spin-dependent Glauber model, which includes all ten helicity amplitudes (the amplitudes classified by the spin projections of the two nucleons along the collision axis) of $pp$ and $pn$ scattering, both the $S$ - and $D$ -wave components of the spin-1 deuteron ground state, and charge-exchange mechanisms, now reproduces exact Faddeev calculations and experimental data for proton-deuteron spin observables at intermediate energies near 1 GeV. Spin-averaged $pp$ and $np$ amplitudes at 2-50 GeV/c have been fitted to the accumulated small-|t| data explicitly for use in Glauber calculations, and the unpolarized defect machinery (the eclipse effect, the shadowing of one constituent nucleon by the other) reproduces measured deuteron-target reaction cross sections and underlies the standard geometric modeling of relativistic heavy-ion collisions.

What has not been settled is the spin-dependent sector at high energies. The five complex amplitudes per channel above the phase-shift-analysis range are still not determined by data; active practitioners state this explicitly in work from 2013 through 2025. Calculations at 3-50 GeV/c must assume the $pn$ amplitude equals the $pp$ one or use otherwise unconstrained Regge models, and no measurement of the spin-dependent deuteron total cross sections, of the vector and tensor polarization parts or projectile-helicity differences, or of the spin dependence of $\delta\sigma$ exists at multi-GeV energies.

Problem Statement

Complete the quantitative program of Franco and Glauber (1966) for the spin dependence of high-energy deuteron cross sections, which its authors could not carry out for lack of amplitude data. The components, which share one objective, a definitive, data-grounded assessment of spin effects on the high-energy deuteron cross-section defect, are:

(1) Input. Determine from experimental data the five complex spin-dependent elastic amplitudes of $pp$ and $pn$ scattering in the forward, small-momentum-transfer region, at laboratory energies above the reach of systematic phase-shift analyses (roughly above 3 GeV for $pp$ and above 1.2 GeV for $pn$ ); for antiproton beams, the corresponding $\bar{p}p$ and $\bar{p}n$ amplitudes are an optional extension matching the source paper's $\bar{p}d$ application. The determination must be genuinely data-driven: it must be constrained by measured spin-dependent observables.

(2) Prediction. With these amplitudes, compute the spin-dependent deuteron total cross sections at multi-GeV energies, the unpolarized part together with its vector- and tensor-polarization parts for the spin-1 deuteron and the projectile-helicity differences, and hence the spin dependence of the cross-section defect $\delta\sigma = \sigma_p + \sigma_n - \sigma_d$ . The calculation must use a realistic deuteron wave function including both $S$ - and $D$ -wave components and must quantify the uncertainty contributed by: the frozen-nucleon (eikonal) approximation and neglected recoil; the truncation of the multiple-scattering series at double scattering; the anticommutator treatment that neglects commutators of the spin-dependent interactions when they overlap; and the spread among modern short-range deuteron wave functions.

References

V. Franco and R. J. Glauber, "High-Energy Deuteron Cross Sections", Physical Review 142, 1195-1214 (1966). DOI: 10.1103/PhysRev.142.1195, https://doi.org/10.1103/PhysRev.142.1195 (source paper)
M. N. Platonova and V. I. Kukulin, "Refined Glauber model versus Faddeev calculations and experimental data for pd spin observables", Physical Review C 81, 014004 (2010); erratum: Physical Review C 94, 069902 (2016). DOI: 10.1103/PhysRevC.81.014004 and 10.1103/PhysRevC.94.069902, https://arxiv.org/abs/1612.08694 (arXiv:1612.08694, the preprint version posted 2016 with corrected formulas)
Yu. N. Uzikov and J. Haidenbauer, "Forward antiproton-deuteron elastic scattering and total spin-dependent antiproton-deuteron cross sections at intermediate energies", Physical Review C 79, 024617 (2009). DOI: 10.1103/PhysRevC.79.024617, https://arxiv.org/abs/0810.3997
Yu. N. Uzikov and J. Haidenbauer, "Elastic p̄d scattering and total p̄d cross sections", Physical Review C 87, 054003 (2013). DOI: 10.1103/PhysRevC.87.054003, https://arxiv.org/abs/1212.1640
Yu. N. Uzikov and J. Haidenbauer, "Elastic p̄d scattering and total p̄d cross sections reexamined", Physical Review C 88, 027001 (2013). DOI: 10.1103/PhysRevC.88.027001, https://arxiv.org/abs/1307.3415
Yu. N. Uzikov, J. Haidenbauer, A. Bazarova, and A. A. Temerbayev, "Spin observables of proton-deuteron elastic scattering at SPD NICA energies within the Glauber model and pN amplitudes" (2020). arXiv: 2011.04304, https://arxiv.org/abs/2011.04304; published in Physics of Particles and Nuclei 53, 419-425 (2022), DOI: 10.1134/S1063779622020848 (the journal version appears without J. Haidenbauer)
M. N. Platonova and Yu. N. Uzikov, "Time-reversal invariance violation effect in dd scattering", Chinese Physics C 49, 034108 (2025). DOI: 10.1088/1674-1137/ad9b9f, https://arxiv.org/abs/2410.03262
W. Horiuchi, Y. Suzuki, T. Uesaka, and M. Miwa, "Total reaction cross section on a deuteron target and the eclipse effect of the constituent neutron and proton", Physical Review C 102, 054601 (2020). DOI: 10.1103/PhysRevC.102.054601, https://arxiv.org/abs/2010.09956
V. Uzhinsky, A. Galoyan, Q. Hu, J. Ritman, and H. Xu, "Empirical Parameterization of Nucleon-Nucleon Elastic Scattering Amplitude at High Beam Momenta for Glauber Calculations and Monte Carlo Simulations", Physical Review C 94, 064003 (2016). DOI: 10.1103/PhysRevC.94.064003, https://arxiv.org/abs/1603.04731
M. L. Miller, K. Reygers, S. J. Sanders, and P. Steinberg, "Glauber Modeling in High Energy Nuclear Collisions", Annual Review of Nuclear and Particle Science 57, 205 (2007). DOI: 10.1146/annurev.nucl.57.090506.123020, https://arxiv.org/abs/nucl-ex/0701025
William P. Ford and J. W. Van Orden, "A Regge Model for Nucleon-Nucleon Spin Dependent Amplitudes", Physical Review C 87, 014004 (2013). DOI: 10.1103/PhysRevC.87.014004, https://arxiv.org/abs/1210.2648
V. V. Abramov, A. Aleshko, V. A. Baskov, et al., "Possible studies at the first stage of the NICA collider operation with polarized and unpolarized proton and deuteron beams", Physics of Particles and Nuclei 52, 1044-1119 (2021). DOI: 10.1134/S1063779621060022, https://arxiv.org/abs/2102.08477 (arXiv:2102.08477)
V. Baryshevsky, "High Energy Nuclear Optics of polarized nucleons and nuclei: research at complex Nuclotron M-NICA" (2026). arXiv: 2602.16470 [hep-ph], https://arxiv.org/abs/2602.16470 (review, submitted 18 February 2026)