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Spatial imaging of polarized deuterons at the Electron-Ion Collider

This paper investigates diffractive vector meson production in electron-polarized deuteron collisions at the Electron-Ion Collider, demonstrating how the deuteron's polarization state influences azimuthal angular distributions, effective radius structure, and gluon saturation effects.

Original authors: Heikki Mäntysaari, Farid Salazar, Björn Schenke, Chun Shen, Wenbin Zhao

Published 2026-10-02

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CC0 1.0

Original authors: Heikki Mäntysaari, Farid Salazar, Björn Schenke, Chun Shen, Wenbin Zhao

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the heart of matter, protons and neutrons are not solid, unchanging spheres. Instead, they are dynamic clouds of even smaller particles called quarks and gluons, buzzing with activity. When scientists smash particles together at incredibly high speeds, they can peel back the layers of these clouds to see how the pieces are arranged. For decades, researchers have used electron beams to map the insides of protons, creating a detailed picture of their internal structure. However, the next step in this exploration involves looking at the simplest atomic nucleus of all: the deuteron. A deuteron is a fragile pair made of just one proton and one neutron bound together. While scientists understand how these two particles sit together at low energies, they know very little about how the invisible sea of gluons inside the deuteron is distributed, especially when the nucleus is spinning or "polarized." Understanding this arrangement is crucial because it serves as a baseline for studying much heavier, more complex nuclei, helping physicists refine their models of how the universe holds itself together.

A team of theoretical physicists has now proposed a way to take a three-dimensional snapshot of this polarized deuteron using a future machine called the Electron-Ion Collider. In their study, they simulated collisions where a beam of electrons strikes a deuteron that has been aligned in a specific direction. When an electron hits the deuteron, it can knock out a heavy particle called a vector meson, which acts like a probe revealing the shape of the target it just hit. The researchers found that the shape of the deuteron, as seen through these collisions, changes depending on how the deuteron is spinning and the angle at which the new particle flies away. If the deuteron is spinning sideways relative to the beam, the size of the nucleus that the electron "sees" appears to shrink and grow as the angle of the collision changes. This creates a distinct pattern in the data, much like how a spinning top looks different from the side than it does from the front.

The study reveals that this effect is not uniform; it depends heavily on the specific orientation of the deuteron's spin. When the deuteron spins in one direction, the probability of finding the particles inside is concentrated in a way that makes the nucleus look larger from certain angles and smaller from others. When it spins in a different direction, this pattern flips. By measuring the angles at which the vector mesons are produced, scientists can extract a mathematical description of these shapes. The researchers calculated that these angular patterns would show up clearly in the data, allowing future experiments to distinguish between different spin states with high precision. This is significant because it offers a direct way to image the spatial distribution of the gluons inside the nucleus, confirming whether the heavy gluons follow the same layout as the lighter protons and neutrons.

Beyond just mapping the shape, the researchers also looked for signs of a phenomenon called gluon saturation. This occurs when the density of gluons becomes so high that they begin to merge and interact with each other, effectively capping the number of particles that can exist in a small space. The simulations suggest that this saturation effect is slightly stronger when the deuteron is spinning in a specific alignment compared to when it is spinning sideways. This happens because the aligned deuteron presents a denser, more compact target to the incoming electron, forcing the gluons to crowd together more tightly. While the difference is small, it is measurable and provides a new tool for testing theories about how matter behaves under extreme conditions.

The paper concludes that these specific measurements are entirely within reach of the upcoming Electron-Ion Collider. By analyzing the angles and energies of the particles produced in these collisions, physicists will be able to reconstruct the three-dimensional wave function of the polarized deuteron. This work does not just confirm what is already known; it opens a new window into the quantum world, turning abstract equations into a tangible picture of how the building blocks of the universe are arranged when they are set in motion. The findings suggest that with the right experimental setup, we can finally see the hidden geometry of the simplest atomic nucleus, providing a clearer foundation for understanding the complex behavior of all matter.

Technical Summary: Spatial Imaging of Polarized Deuterons at the Electron-Ion Collider

Problem Statement
While the internal quark and gluon structure of protons has been extensively mapped via Deep Inelastic Scattering (DIS) at facilities like HERA, the spatial distribution of small- $x$ gluons within the deuteron remains poorly understood. Although the deuteron's structure at large momentum fractions ( $x$ ) is well-characterized, new measurements are required to constrain models for deuteron-gold collisions at the Relativistic Heavy Ion Collider (RHIC) and to probe the fundamental small- $x$ gluon structure. Furthermore, while polarization offers a unique handle to control the orientation of the deuteron, the specific impact of the polarization-dependent nuclear wave function on exclusive vector meson production in electron-deuteron ( $e+d$ ) collisions has not been fully explored in the context of the future Electron-Ion Collider (EIC).

Methodology
The authors employ the Color Glass Condensate (CGC) effective theory to study diffractive vector meson production ( $J/\psi$ and $\rho$ ) in virtual photon-deuteron collisions at small $x$ . The core of the methodology involves:

Polarization-Dependent Initial Conditions: The study implements the polarization-dependent squared wave function of the deuteron into the initial conditions for color-charged correlators. The nucleon positions are sampled according to the probability distributions derived from the Reid93 nucleon-nucleon potential, distinguishing between transverse polarization states ( $j_z = \pm 1$ and $j_z = 0$ ) and longitudinal polarization.
Scattering Amplitude Calculation: Using the dipole picture, the amplitude for exclusive vector meson production is calculated. This involves the splitting of a virtual photon into a quark-antiquark pair, eikonal scattering off the small- $x$ gluon field (described by Wilson line correlators), and recombination into the vector meson.
Cross-Section Analysis: The authors compute both coherent (target remains in the same quantum state) and incoherent (target state changes) differential cross-sections with respect to the momentum transfer squared ( $|t|$ ) and the azimuthal angle ( $\Phi$ ) of the produced meson relative to the deuteron polarization axis.
Fourier Decomposition and Radius Extraction: The azimuthal dependence of the cross-section is analyzed via Fourier decomposition to extract coefficients ( $a_n$ ). Additionally, an effective transverse radius is extracted by fitting the coherent $|t|$ -spectra to a Gaussian form factor model within the range $0 < |t| < 0.3$ GeV $^2$ .

Key Contributions

Azimuthal Asymmetries from Wave Function Structure: The paper proposes and calculates a mechanism where the polarization-dependent nuclear wave function of the deuteron generates azimuthal asymmetries in the produced vector meson. Specifically, the anisotropy in the nucleon positions relative to the polarization vector translates into angular modulations of the cross-section.
Differentiation of Polarization States: The study demonstrates that the Fourier coefficients ( $a_2, a_4$ ) and the effective radius exhibit distinct behaviors for transverse polarization states ( $j_z = \pm 1$ vs. $j_z = 0$ ) and longitudinal polarization.
Saturation Probes: The work utilizes the ratio of coherent photoproduction cross-sections between longitudinal and transverse polarizations to probe gluon saturation effects, linking the geometric size of the polarized deuteron to the effective gluon density.

Results

Angular Dependence of Cross-Sections: For transversely polarized deuterons with $j_z = \pm 1$ , the $|t|$ distribution flattens as the angle $\Phi$ increases from 0 to $\pi/2$ . Conversely, for $j_z = 0$ , the slope increases with $\Phi$ . Longitudinally polarized deuterons show no such angular dependence in the transverse plane projection.
Fourier Coefficients: The second-order Fourier coefficient ( $a_2$ ) shows opposite signs for $j_z = 0$ and $j_z = \pm 1$ at low $|t|$ , dominated by the $\cos(2\Phi)$ mode. As $|t|$ increases, higher-order modes ( $\cos(4\Phi)$ ) become significant, and sign changes in $a_2$ occur at different $|t|$ values for the different $j_z$ states.
Effective Radius: The extracted effective transverse radius ( $R$ ) for transverse polarization varies with $\Phi$ . For $j_z = \pm 1$ , $R$ peaks at $\Phi = 0$ and minimizes at $\Phi = \pm \pi/2$ . For $j_z = 0$ , $R$ dips at $\Phi = 0$ and peaks at $\Phi = \pi/2$ . No $\Phi$ dependence is observed for longitudinal polarization.
Saturation Effects: The ratio of coherent $\rho$ meson cross-sections (longitudinal to transverse) at $|t|=0$ is consistently below unity, indicating stronger gluon saturation effects for longitudinally polarized deuterons. This is attributed to the smaller effective area (higher gluon density) of the longitudinally polarized state. The suppression is slightly more pronounced at smaller $x_P$ values but remains within a 2% magnitude difference due to the deuteron's two-nucleon structure.

Significance
The paper claims that these small- $x$ observables will be accessible at the future Electron-Ion Collider (EIC). The primary significance lies in the ability of exclusive vector meson production to "image" the spatial structure of the polarized deuteron wave function. By measuring the azimuthal angular dependence of the cross-section and the effective radius, experimentalists can directly verify the spatial distribution of small- $x$ gluons and determine if it follows the spatial distribution of nucleons at large $x$ . Furthermore, the study suggests that polarized targets offer a new handle for understanding gluon saturation dynamics, with the potential for enhanced signals if similar studies are extended to larger polarized nuclei. The work provides a theoretical baseline for interpreting future EIC data regarding the internal geometry and gluon dynamics of the deuteron.

Technical Summary: Spatial Imaging of Polarized Deuterons at the Electron-Ion Collider

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