⚛️ phenomenology

Quantum entanglement within quarkonium

This paper utilizes a nonperturbative light-front Hamiltonian framework to demonstrate that the von Neumann entanglement entropy of quark-antiquark pairs in heavy quarkonium serves as a novel probe of nonperturbative structure, directly linking quantum information measures to transverse momentum dependent parton distributions and revealing significant sensitivity to meson polarization.

Original authors: Wenyu Zhang, Yiyu Zhou, Yang Li, Qun Wang

Published 2026-07-28

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CC BY 4.0

Original authors: Wenyu Zhang, Yiyu Zhou, Yang Li, Qun Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

Imagine the universe is built from tiny, invisible Lego bricks called quarks. These bricks snap together to form protons and neutrons, the core of every atom in your body. But here's the weird part: you can never pull a single quark out to look at it on its own. They are stuck together by a super-strong glue called the "strong force," which acts like an unbreakable rubber band. If you try to pull them apart, the energy snaps the band and creates new quarks instead. This means the inside of a particle is a chaotic, buzzing dance floor where quarks and their anti-matter partners (antiquarks) are constantly interacting.

For decades, scientists have tried to map this dance floor, treating the particles like a cloud of probability—essentially asking, "Where is the quark likely to be?" But there's a deeper layer to this story. Quantum mechanics tells us that these particles aren't just independent dancers; they are "entangled." Think of entanglement as a magical connection where two particles share a single secret. If you know everything about one, you instantly know something about the other, no matter how far apart they are. In the chaotic world of particle physics, this entanglement isn't just a spooky trick; it's a fundamental part of how matter holds itself together. Understanding this "quantum handshake" could help us decode the hidden rules of the universe, revealing why particles have mass and how they spin.

Now, enter a team of researchers who decided to peek behind the curtain of this quantum dance. They focused on a special, heavy type of particle called "quarkonium," which is like a heavy-duty version of a proton, made of a heavy quark and its heavy antiquark partner. Using a sophisticated computer simulation method called "Basis Light-Front Quantization" (BLFQ)—which is like taking a high-speed, 3D snapshot of the particle's internal wave—they calculated something called "entanglement entropy." In simple terms, this is a score that measures how much information is "lost" when you look at just one part of the pair (the quark) while ignoring the other (the antiquark). If the score is high, the two are deeply entangled; if it's low, they are more independent.

The team discovered that this "entanglement score" isn't just a random number; it's directly linked to how the particles are spinning and moving. For the simplest, non-spinning particles (spin-0), the score is basically a measure of how spread out the particle's momentum is, similar to a standard probability map. However, things get much more interesting with spinning particles (spin-1). The researchers found that the entanglement score changes dramatically depending on how the particle is oriented. It's as if the quantum connection between the quark and antiquark tightens or loosens based on the direction the particle is pointing.

In their simulations, they looked at specific heavy particles like charmonium (made of charm quarks) and bottomonium (made of bottom quarks). They found that for particles spinning in one direction (labeled as $m_J = 0$ ), the entanglement score was different than when they were spinning in another direction ( $m_J = 1$ ). For example, in certain charmonium states, the $m_J = 0$ state had a higher entanglement score than the $m_J = 1$ state, but this pattern flipped for other types of particles. This suggests that the "quantum glue" holding these heavy particles together is incredibly sensitive to their polarization, or spin alignment.

The researchers also had to solve a tricky math problem involving "infrared parameters," which are like background noise in their calculations. By comparing their results to a different mathematical representation (using harmonic oscillators, think of them as vibrating springs), they were able to pin down the exact value of this noise, ensuring their final scores were accurate. They found that the entanglement entropy didn't seem to care much about how "excited" the particle was (whether it was in a basic state or a higher energy level), but it cared a lot about its spin.

Ultimately, this work suggests that entanglement entropy is a powerful new tool for looking inside particles. It connects the abstract world of quantum information theory with the concrete measurements of how particles move and spin. While the study was a simulation and not a direct experiment in a collider, it provides a clear roadmap for future experiments. The authors suggest that if we can measure these spin-dependent patterns in real-world collisions, we might be able to use entanglement as a new kind of microscope to see the non-perturbative, messy, and beautiful inner workings of the strong force that builds our universe.

Technical Summary: Quantum Entanglement within Quarkonium

Problem and Motivation
Understanding the internal structure of hadrons remains a central challenge in nuclear and particle physics, particularly regarding non-perturbative Quantum Chromodynamics (QCD) phenomena such as color confinement and the emergence of hadronic mass. While traditional approaches treat parton distributions as probabilistic densities, the underlying quantum nature of these constituents suggests a need for frameworks that capture the complete picture of hadronic wave functions, including quantum correlations. Recent developments in quantum information science (QIS) have established connections between entanglement entropy and scattering cross-sections, as well as the saturation of gluon densities. However, a direct, first-principles derivation of partonic entanglement from hadronic wave functions, specifically incorporating spin degrees of freedom, has been largely unexplored. This paper addresses the gap in understanding how spin polarization shapes the entanglement structure between constituent quarks and antiquarks in heavy quarkonium systems.

Methodology
The authors employ a non-perturbative light-front Hamiltonian framework to investigate quark-antiquark entanglement in heavy quarkonia (charmonium and bottomonium). The methodology proceeds as follows:

Formalism Construction: The total Hilbert space is decomposed into quark, antiquark, and gluon subspaces. For a pure bipartite state, the entanglement entropy is defined as the von Neumann entropy of the reduced density matrix obtained by tracing out the antiquark and gluon degrees of freedom.
Light-Front Wave Functions (LFWFs): The study utilizes valence-sector LFWFs derived from the Basis Light-Front Quantization (BLFQ) approach. The effective Hamiltonian includes light-front kinetic energy, a soft-wall AdS/QCD confining potential, and a one-gluon exchange interaction.
Spin Dependence: The formalism explicitly incorporates spin degrees of freedom. For spin-0 mesons, the reduced density matrix is diagonalized to relate entanglement entropy to the Shannon entropy of unpolarized Transverse Momentum Dependent (TMD) distributions. For spin-1 mesons, the authors derive explicit expressions linking the entropy eigenvalues to polarized and tensor-polarized TMDs, accounting for vector and tensor polarization states ( $|m_J=0\rangle$ and $|m_J=\pm 1\rangle$ ).
Infrared (IR) Parameter Resolution: A significant technical component involves resolving the IR divergence inherent in momentum-space entropy calculations. The authors match the momentum-space entropy to a finite result obtained in a harmonic oscillator representation (using a narrow wavepacket limit). This matching uniquely determines the IR cutoff and the transverse box-regularization parameter consistent with the BLFQ basis truncation.

Key Contributions and Results

Spin-0 vs. Spin-1 Entropy: For spin-0 quarkonia, the entanglement entropy is shown to reduce to the Shannon entropy of the unpolarized TMD ( $f_1$ ), up to constant contributions from color and spin. For spin-1 quarkonia, the entropy exhibits a nontrivial dependence on the meson's polarization state, explicitly connecting to polarized ( $g_1$ ) and tensor-polarized ( $f_{1LL}, f_{1LT}, f_{1TT}$ ) TMDs.
Numerical Evaluation: Using BLFQ-derived wave functions, the authors evaluated the entanglement entropy for various charmonium and bottomonium states.
- Polarization Sensitivity: A pronounced sensitivity to the polarization of vector mesons was observed. Specifically, for $1^{--}$ and $1^{++}$ quarkonia, the $|m_J=0\rangle$ state possesses a larger entanglement entropy than the $|m_J=1\rangle$ state, whereas the opposite trend is observed for $1^{+-}$ quarkonia.
- Excitation Patterns: The entropy does not exhibit a clear systematic correlation with the radial or orbital excitation of the state. However, the transverse entropy density shows peak structures whose multiplicity reflects the excitation pattern.
- Rotation Dependence: By rotating the polarization states about a transverse axis, the authors found that while entanglement entropy varies with the rotation angle $\theta$ , this variation is subleading compared to the difference between the $|m_J=0\rangle$ and $|m_J=1\rangle$ states. Heavier systems (bottomonium) show smaller variations with $\theta$ compared to charmonium, suggesting a suppression of spatial-symmetry breaking in heavier quarkonia.
IR Parameter Fixing: The study successfully determined the IR cutoff ( $\Lambda_{IR}$ ) and transverse box size ( $L_\perp$ ) by matching the momentum-space entropy to the harmonic oscillator representation in the narrow wavepacket limit, ensuring finite and scheme-independent results.

Significance and Claims
The paper establishes entanglement entropy as a novel probe of non-perturbative quarkonium structure, forging a direct link between quantum information measures and partonic observables (specifically TMDs). The authors claim that this work extends previous investigations by explicitly resolving the spin structure, revealing how spin polarization modifies the relationship between partonic entanglement and TMDs.

The significance lies in providing a controlled, QCD-motivated setting to study how quantum entanglement encodes the dynamics of bound states. The results suggest that entanglement entropy could serve as a new observable for future experimental facilities, such as Electron-Ion Colliders (EICs), particularly through angular correlations in vector meson production. The authors note that while the current framework is limited to the valence sector, it lays the groundwork for future extensions to full QCD, including higher Fock sectors and dynamical gluons, though they acknowledge that such extensions introduce challenges regarding gauge invariance and the definition of the reduced density matrix in the presence of sea quarks and gluons.

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