Author: Denis Avetisyan
New research connects the entanglement of partons within a proton to a classical description of their momentum distribution, offering a richer understanding of hadronic entropy.
This study compares the Kharzeev-Levin entanglement entropy with a semiclassical Wehrl entropy derived from phase-space distributions to reveal the contribution of transverse momentum information to the overall entropy content of partons at small-$x$.
While entanglement entropy offers a compelling link between hadronic multiplicity and underlying quantum correlations, its longitudinal nature provides an incomplete picture of the proton’s internal structure. This work, ‘QCD Wehrl and entanglement entropies in a gluon spectator model at small-$x$’, explores a semiclassical approach using Wehrl entropy, constructed from phase-space distributions, to characterize parton distributions within the proton. We demonstrate that transverse momentum information, captured within the Wehrl entropy, provides a more comprehensive description of hadronic entropy than purely longitudinal measures. Does this framework offer a pathway toward reconciling quantum and classical descriptions of strong interaction dynamics within hadrons?
Emergent Order: Unveiling the Hadron’s Hidden Structure
Hadrons, such as protons and neutrons, are not fundamental particles but composite structures built from more elementary constituents called partons – primarily quarks and gluons. Understanding the internal architecture of these hadrons necessitates investigating the intricate quantum correlations that bind these partons together. Unlike simple arrangements, the partons exist in a dynamic, relativistic state governed by the strong force, leading to complex entanglement patterns. These correlations aren’t merely a structural detail; they fundamentally dictate how hadrons interact in high-energy collisions, influencing scattering cross-sections and the production of new particles. Consequently, accurately characterizing these quantum relationships is crucial for refining theoretical models and predicting experimental outcomes in particle physics, representing a significant frontier in understanding matter at its most fundamental level.
Predicting the outcomes of high-energy particle collisions demands a precise understanding of the internal structure of hadrons – protons and neutrons. However, conventional models often fall short because they struggle to fully account for the complex quantum correlations between the constituent partons – quarks and gluons – that comprise these particles. These correlations aren’t simply a matter of individual parton properties; rather, they represent a fundamentally interconnected system where the state of one parton instantaneously influences others, regardless of distance. This inability to accurately capture this interconnectedness introduces significant uncertainties into collision predictions, particularly at the extreme energies probed by modern particle accelerators. Consequently, refining theoretical frameworks to better represent these intra-hadron correlations is crucial for advancing the field of high-energy physics and interpreting experimental results with greater confidence.
Entanglement entropy, a measure of quantum correlation exceeding classical correlations, has become increasingly vital in characterizing the complex internal structure of hadrons – particles like protons and neutrons. While theoretically elegant, directly calculating this entropy for partonic systems proves remarkably difficult. The sheer number of interacting partons – quarks and gluons – within a hadron, coupled with the relativistic nature of these interactions, creates a computational bottleneck. Existing methods often rely on approximations or simplified models, potentially obscuring the true extent of entanglement and hindering accurate predictions of hadron behavior in high-energy collisions. Researchers are actively developing novel theoretical frameworks and computational techniques, including lattice gauge theory and effective field theories, to circumvent these challenges and unlock a more complete understanding of entanglement’s role in the quantum world of strong interactions.
Recognizing the difficulty in directly calculating entanglement entropy within the complex environment of hadrons, researchers are actively constructing theoretical models that connect this fundamental quantum property to experimentally accessible parton distribution functions. These models posit that the degree of entanglement between partons-quarks and gluons-influences the probability of finding a particular parton carrying a certain fraction of the hadron’s momentum. By linking entanglement to these observable distributions, scientists aim to indirectly probe the quantum correlations inside hadrons and refine predictions for high-energy collisions. This approach promises a pathway to understanding how entanglement contributes to the overall structure and behavior of matter at its most fundamental level, potentially revealing new insights into the strong force that binds quarks and gluons together.
A Measure of Complexity: The Kharzeev-Levin Model
The Kharzeev-Levin model establishes a quantifiable link between a hadron’s entanglement entropy, denoted as S, and its effective number of degrees of freedom, N_{eff}. The model proposes the relationship S = \ln N_{eff}, suggesting that entanglement entropy directly reflects the complexity of the hadron’s internal structure. Specifically, a higher entanglement entropy indicates a greater number of independently fluctuating partonic constituents within the hadron, effectively characterizing the hadron’s internal many-body quantum state. This formulation allows for a calculation of N_{eff} based on measurable quantities related to entanglement, providing a pathway to assess the complexity of strong interaction systems.
The Kharzeev-Levin model leverages experimentally determined Parton Distribution Functions (PDFs) to quantitatively estimate entanglement entropy within hadrons. Specifically, the model employs the probability density functions – typically represented as f(x, Q^2), where x is the parton momentum fraction and Q^2 is the transverse momentum squared – to calculate the entanglement entropy between the target hadron and the produced particles in a high-energy collision. This calculation involves integrating the PDF over the appropriate kinematic range, effectively summing the contributions from all possible parton momentum fractions. The resulting value provides a measure of the entanglement generated during the interaction, directly linking theoretical predictions to observable quantities derived from experimental data such as deep inelastic scattering and proton-proton collisions.
The Kharzeev-Levin model establishes a connection between the entanglement entropy of a hadron and its internal quantum structure as described by Parton Distribution Functions (PDFs). PDFs represent the probability of finding a parton with a specific momentum fraction within the hadron, effectively mapping the hadron’s momentum space. By relating entanglement entropy-a measure of quantum correlations-to these PDFs, the model proposes that the distribution of partons directly reflects the degree of quantum entanglement within the hadron. Consequently, analysis of experimentally determined PDFs can provide information about the non-perturbative aspects of Quantum Chromodynamics (QCD) and the correlations between partons, offering a pathway to understanding the complex internal dynamics of hadrons beyond simple constituent counting.
The Kharzeev-Levin model introduces a novel framework for investigating strong interactions by relating entanglement entropy to the properties of partonic systems within hadrons. Traditionally, studies of strong interactions have relied on perturbative QCD and phenomenological models; however, this approach leverages concepts from quantum information theory to provide complementary insights. By treating hadrons as complex quantum systems and utilizing Parton Distribution Functions (PDFs) to characterize their internal structure, the model allows for the calculation of entanglement entropy-a measure of quantum correlations-directly from experimentally accessible data. This connection enables researchers to explore the dynamics of partons, including their correlations and interactions, through the lens of quantum entanglement, potentially revealing new aspects of confinement and hadronization processes.
Mapping the Phase Space: The Wigner Distribution
The Wigner distribution function, denoted as W(x, p), offers a phase-space representation of parton distribution within a hadron. It describes the probability density of finding a parton with momentum p at a given position x inside the hadron. Unlike simple parton distribution functions (PDFs) which only provide the probability density of finding a parton with a given momentum fraction, the Wigner distribution includes both position and momentum information, allowing for the study of correlations between these variables. This is achieved through a Fourier transform relating the Wigner function to the operator expectation value of a Wilson line, effectively encoding both coordinate and momentum space information into a single function. The Wigner distribution is quasi-probability distribution, meaning it can, in certain cases, take on negative values, which are indicators of non-classical correlations within the hadron.
The Wigner distribution functions as a generating function for a variety of important parton distributions used in high-energy physics. Specifically, integrating the Wigner distribution over all momenta transverse to a chosen direction yields the traditional Parton Distribution Functions (PDFs), representing the probability of finding a parton with a certain fraction of the hadron’s longitudinal momentum. Integrating over the longitudinal momentum, while keeping a fixed transverse momentum, produces Transverse Momentum Distributions (TMDs), crucial for understanding particle production in hadronic collisions. Furthermore, performing a Fourier transform with respect to the transverse momentum yields Generalized Parton Distributions (GPDs), which describe the amplitude for a parton to carry momentum between two different longitudinal positions within the hadron and are key to understanding the hadron’s structure and dynamics.
The Light-Front Spectator Model (LFS) facilitates Wigner distribution calculations by treating the hadron as a system evolving in light-front time, where the internal dynamics are largely unaffected by the observation time. This allows for a separation of longitudinal and transverse degrees of freedom; the longitudinal component dictates the evolution, while the transverse component-the “spectator” component-remains relatively static and defines the parton’s transverse momentum. Within the LFS, the Wigner distribution W(x, \vec{k}_\perp) is expressed as a sum over all possible Fock states of the hadron, weighted by their probability amplitudes and incorporating the momentum fractions x and transverse momenta \vec{k}_\perp of the constituent partons. This framework directly connects the Wigner distribution to the underlying hadron’s wavefunction and allows for the calculation of various parton distribution functions based on the model’s assumptions regarding the hadron’s internal structure.
The Wigner distribution, representing the parton’s quantum mechanical phase-space density, can be computationally accessed through the Light-Front Spectator Model combined with the Soft-Wall AdS/QCD correspondence. This approach leverages the holographic principle to map strong interactions in Quantum Chromodynamics (QCD) to a gravitational dual in Anti-de Sitter (AdS) space. Specifically, the Soft-Wall model provides a simplified AdS/QCD background enabling analytical and numerical calculations of hadron wavefunctions. These wavefunctions are then used within the Light-Front Spectator Model to project out the Wigner distribution W(x, \vec{k}), where x represents the longitudinal momentum fraction and \vec{k} the transverse momentum of the parton. This method allows for investigations of non-perturbative QCD phenomena and provides a pathway to compute parton distribution functions directly from the underlying theory.
Collective Behavior: The Color Glass Condensate
The Color Glass Condensate (CGC) provides a theoretical framework for understanding the behavior of hadrons and nuclei when probed at extremely high energies. As energy increases in particle collisions, the strong force causes the internal constituents – partons like quarks and gluons – to proliferate. The CGC addresses this by positing that at very small momentum fractions – meaning partons carrying only a tiny fraction of the total momentum – these constituents become densely packed. This dense packing leads to strong interactions and correlations between the partons, effectively creating a “glassy” state. Crucially, the CGC doesn’t treat these partons as individual, independent entities, but rather as a collective, highly interacting system, allowing physicists to model the evolution of particle interactions at energies achievable in modern colliders and to explain phenomena like the unexpectedly small scattering cross-sections observed in deep inelastic scattering experiments. This framework is essential for describing the initial stages of heavy-ion collisions, where the creation of a quark-gluon plasma is thought to occur.
The Balitsky-Kovchegov (BK) equation represents a significant advancement in understanding how gluon distributions evolve within the Color Glass Condensate. This equation doesn’t simply track the number of gluons; it explicitly accounts for the complex interplay of multiple interactions between them. As energy increases, gluons proliferate, and their interactions become increasingly important, leading to a non-linear evolution described by the BK equation. This non-linearity is crucial because it captures the effects of gluon splitting and merging, as well as the screening and saturation of gluon density. Effectively, the BK equation predicts how the strong force field generated by a hadron or nucleus changes with energy, allowing physicists to model the behavior of matter at extremely high energies and densities, such as those found in heavy-ion collisions. \frac{\partial \mathcal{S}}{\partial Y} = \sigma_g \mathcal{S} + \frac{1}{2} \sigma_g \left( \mathcal{S}^2 - \mathcal{S} \right) represents a simplified form of the BK equation, where \mathcal{S} represents the scattering amplitude, and Y denotes rapidity.
The transition from the relatively sparse distribution of gluons within a hadron to a dense, overlapping system is fundamentally captured by the saturation scale, denoted as Q_s. This scale isn’t a fixed constant, but rather a dynamic quantity that increases with energy and/or atomic number, signaling the point where gluon-gluon interactions become significant enough to modify the parton distribution. Below Q_s, the system behaves as a dilute gas of partons, where linear evolution equations are sufficient. However, as energies increase and Q_s is exceeded, the system enters a regime of gluon saturation, where non-linear effects – such as gluon recombination – become crucial for accurately describing the gluon distribution and the overall dynamics of strong interactions. Consequently, Q_s serves as a critical parameter in Color Glass Condensate calculations, defining the boundary between perturbative and non-perturbative regimes and influencing predictions for observables in heavy-ion collisions and deep inelastic scattering.
The advancements represented by the Color Glass Condensate framework and related theoretical tools offer a crucial window into the fundamental nature of the strong force, one of the four fundamental forces governing the universe. By detailing the behavior of quarks and gluons-the constituents of matter-at extraordinarily high energies, these developments illuminate how matter behaves under conditions mimicking those present in the very early universe or within the cores of neutron stars. The ability to model the dense partonic systems described by the CGC allows physicists to probe the limits of Quantum Chromodynamics (QCD), the theory of the strong force, and potentially reveal new states of matter. Furthermore, understanding these extreme energy interactions is paramount to interpreting the results of experiments at particle colliders like the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC), where scientists recreate these conditions to study the properties of matter at its most fundamental level.
Towards a Complete Picture: Refining the Framework
The challenge of quantifying information loss within quantum systems finds a compelling approach in the Wehrl entropy, a tool defined directly within the system’s phase space. Unlike traditional entropy measures, the Wehrl entropy provides a semiclassical framework for assessing how information is obscured as a quantum system evolves, effectively bridging the gap between quantum mechanics and classical statistical mechanics. Crucially, this entropy isn’t merely a theoretical construct; it possesses a demonstrable connection to the Husimi distribution, a quasi-probability distribution offering a phase-space representation of quantum states. By leveraging this link, researchers gain a means to visualize and analyze information loss in a way that’s intuitively connected to classical concepts of phase space density and statistical uncertainty, offering a novel perspective on the fundamental limits of knowledge within quantum systems.
Investigating the connection between Wehrl entropy and the Wigner distribution function promises a deeper understanding of hadronic structure at the quantum level. The Wigner distribution, a quasiprobability distribution in phase space, offers a way to visualize quantum states, and its relationship to Wehrl entropy-a measure of information loss-could reveal how partons, the fundamental constituents of hadrons, are distributed and interact. Current research suggests that analyzing this interplay might expose subtle correlations and dynamical features within hadrons that are not captured by traditional approaches focusing solely on entanglement. By refining the connection between these two theoretical tools, scientists hope to map the complex quantum landscape of hadrons with greater precision, potentially unveiling new insights into their internal dynamics and ultimately, the strong force that binds quarks and gluons together. This pursuit offers a path toward a more complete and nuanced picture of matter at its most fundamental level.
A comprehensive understanding of the internal structure of hadrons necessitates moving beyond simplified descriptions of parton distributions; generalized distributions, such as Generalized Transverse Momentum Distributions (GTMDs), offer a powerful framework for achieving this. These distributions correlate the longitudinal momentum and transverse position of partons within the hadron, providing a more complete picture of their dynamics than traditional parton distribution functions. By accessing information about both the longitudinal and transverse degrees of freedom, GTMDs enable researchers to probe the intricate interplay between parton momentum, spin, and spatial arrangement. This detailed insight is crucial for unraveling the complexities of hadron structure and for accurately modeling high-energy collisions, ultimately paving the way for a more refined understanding of quantum chromodynamics and the strong force.
This research demonstrates a consistent distinction between Wehrl entropy and entanglement entropy within hadronic systems, revealing that Wehrl entropy consistently registers a higher value. This difference isn’t merely numerical; it underscores the critical role of transverse degrees of freedom in fully characterizing the entropy of hadrons. While entanglement entropy focuses on correlations along the longitudinal direction, Wehrl entropy, defined in phase space, inherently incorporates information about particle motion in all directions. Consequently, the observed disparity suggests that a complete understanding of hadronic entropy necessitates accounting for these transverse momenta, broadening the scope beyond purely longitudinal considerations and offering a more comprehensive picture of information content within these complex quantum systems.
Recent calculations reveal a striking difference in the behavior of Wehrl entropy and entanglement entropy when considering changes in the virtuality, Q^2. While entanglement entropy consistently increases with higher Q^2 values – indicating growing correlations at shorter distance scales – Wehrl entropy demonstrates remarkable stability, remaining largely independent of these changes. This suggests that the information loss quantified by Wehrl entropy, which considers transverse degrees of freedom in phase space, is less sensitive to the resolution scale probed by Q^2 than the entanglement captured by traditional entanglement entropy. The consistent value of Wehrl entropy, even as entanglement entropy rises, highlights the crucial role of transverse momentum in defining hadronic entropy and provides a new perspective on the internal structure of hadrons at varying energy scales.
To accurately quantify information loss within quantum systems – particularly in the complex realm of hadron structure – researchers employed a Gaussian smearing technique during calculations. This method, characterized by a width of 1/2, serves a crucial dual purpose: it guarantees the positivity of the resulting distributions, preventing non-physical negative probabilities, and simultaneously respects the fundamental limits imposed by the Heisenberg uncertainty principle. By effectively ‘blurring’ the phase-space representation, the technique avoids artificially sharp features that would violate this principle and ensures a minimal, yet valid, resolution. This careful approach allows for a robust and physically meaningful calculation of quantities like Wehrl entropy, providing insights into the quantum structure of hadrons without being hampered by mathematical artifacts or inconsistencies with established quantum mechanical principles.
The study delves into the entropy of partons, effectively mapping the internal complexity of a proton. This resonates with the notion that order doesn’t require central planning, but arises from the interplay of local rules. Just as a coral reef’s ecosystem emerges from individual polyp interactions, the hadronic entropy detailed in the paper isn’t a pre-ordained structure, but a consequence of the phase-space distributions and transverse momentum information of its constituent gluons. As Paul Feyerabend observed, “Anything goes,” suggesting that multiple, even seemingly contradictory, approaches can contribute to understanding complex systems – a principle clearly demonstrated by comparing Kharzeev-Levin entanglement entropy with Wehrl entropy to gain a more complete picture of hadronic structure.
The Road Ahead
The correspondence explored within this work-between entanglement entropies and phase-space distributions-suggests a broader principle at play. The observed connection isn’t a demonstration of control, but rather an emergent property. Attempts to dictate hadronic entropy from first principles may prove perpetually elusive; instead, the rules governing parton interactions at local transverse momentum scales appear sufficient to generate the observed global behavior. The semi-classical Wehrl entropy offers a useful descriptive tool, yet its limitations-a reliance on classical notions of phase space-hint at underlying complexities.
Future investigations will likely benefit from relaxing the assumptions inherent in spectator models. A more complete picture demands consideration of genuinely multi-particle correlations, moving beyond simplified representations of gluon saturation. The emphasis should not be on finding the fundamental entropy, but on understanding how entropy arises from the interplay of local rules.
Ultimately, the question isn’t whether these various entropy measures perfectly align, but whether their discrepancies reveal previously unappreciated facets of strong field dynamics. The search for a singular, all-encompassing definition may be a misguided pursuit; perhaps the richness of hadronic structure is best reflected in a multiplicity of perspectives, each valid within its own domain of applicability.
Original article: https://arxiv.org/pdf/2512.24855.pdf
Contact the author: https://www.linkedin.com/in/avetisyan/
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2026-01-04 16:38