Inside Baryons: Mapping Their Gravitational Shape

Author: Denis Avetisyan

New research explores how the internal structure of protons and hyperons influences their response to gravitational forces.

The study details a comparison of gravitational form factors - specifically <span class="katex-eq" data-katex-display="false">A_{T20}(t)</span> and <span class="katex-eq" data-katex-display="false">\bar{B}_{T20}(t)</span> - as they vary with transverse momentum transfer for up quarks within both a proton and a <span class="katex-eq" data-katex-display="false">\Xi^{0}</span> baryon, illuminating fundamental aspects of their internal structure. — The study details a comparison of gravitational form factors – specifically $A_{T20}(t)$ and $\bar{B}_{T20}(t)$ – as they vary with transverse momentum transfer for up quarks within both a proton and a $\Xi^{0}$ baryon, illuminating fundamental aspects of their internal structure.

A diquark spectator model is used to analyze the gravitational form factors and transverse polarization of light baryons, revealing differences linked to their quark composition.

While direct probes of the gravitational interaction with matter are currently infeasible, insights into baryon structure can be gleaned through the electromagnetic form factors parameterized by the energy-momentum tensor. This work, ‘Gravitational form factors of baryons in a spectator diquark model’, investigates these form factors within a framework employing a diquark spectator model to explore the transverse polarization of quarks within baryons. By considering various quark-diquark configurations for both strange and non-strange baryons, we reveal distinctions in their internal structure and constituent quark behavior. How do these insights inform our understanding of the distribution of momentum and angular momentum within nucleons and hyperons, and what implications do they hold for models of hadron structure?

The Illusion of Structure: Peering Within Hadrons

Hadrons, composite particles like protons and neutrons, are fundamentally shaped by the strong force, an interaction governing the behavior of quarks and gluons. Determining their internal structure isn’t simply a matter of static snapshots; the relativistic speeds of constituent particles and the complex interplay of their interactions present significant challenges to traditional modeling techniques. Existing methods, often relying on non-relativistic approximations or simplified interaction models, struggle to accurately account for these dynamic effects. Consequently, a complete understanding of hadron properties-including mass, spin, and charge distribution-remains elusive, hindering progress in fields ranging from nuclear physics to high-energy particle collisions. The need for innovative theoretical frameworks and experimental tools capable of probing these relativistic and many-body effects is therefore paramount to unraveling the mysteries of the strong force.

Precisely determining a hadron’s Gravitational Form Factor (GFF) necessitates theoretical frameworks capable of comprehensively modeling the intricate relationship between its internal energy, momentum, and spin. These factors aren’t simply additive; rather, they exhibit complex correlations arising from the strong force that binds quarks and gluons together. Current computational methods often struggle with the relativistic nature of these interactions and the numerous possible configurations within a hadron. A robust framework must account for the distribution of momentum amongst the constituent quarks, the orbital angular momentum, and the intrinsic spin – all while adhering to the principles of Poincaré symmetry. Consequently, advancements in areas like lattice quantum chromodynamics and effective field theories are critical for developing the necessary tools to accurately calculate GFFs and unlock a deeper understanding of hadron structure and the fundamental forces governing matter.

Precisely determining the roles of individual quarks and their intricate relationships within hadrons presents a significant challenge to current methodologies. While models attempt to map the internal structure of these composite particles, limitations in computational power and theoretical understanding hinder a complete disentanglement of constituent contributions. The strong force, governing interactions within hadrons, introduces complexities arising from gluon exchange and quark-antiquark pair production, effectively blurring the lines between individual quark properties and collective behavior. Consequently, extracting precise information about the momentum and spin distributions of each quark – crucial for understanding the hadron’s overall characteristics – remains elusive. Improving analytical techniques and developing novel experimental probes are therefore essential to refine these models and achieve a more detailed picture of hadron structure, ultimately deepening comprehension of the strong force itself.

Light-Cone Visions: A Simplified Reality

The Light-Cone Framework employs a coordinate system where the light-cone coordinates are defined as $x^+ = t + z$ and $x^- = t - z$ , effectively nullifying the spatial and temporal mixing inherent in standard Minkowski space. This transformation simplifies the Lorentz transformation properties and facilitates the treatment of time-like and space-like separated events with greater ease. By choosing light-cone coordinates, the Hamiltonian becomes simpler, enabling calculations of hadron dynamics and structure with improved accuracy compared to traditional approaches. This framework is particularly beneficial for analyzing high-energy collisions and processes where relativistic effects are significant, as it inherently incorporates Lorentz invariance without requiring complex transformations at each computational step.

The Diquark Spectator Model simplifies the calculation of hadron wavefunctions by approximating the hadron as a two-body system. This approach posits that hadrons are composed of one active quark and a diquark – a bound state of two quarks – treated as a single entity. By considering the diquark as a spectator, the mathematical complexity of dealing with a three-quark system is significantly reduced, allowing focus on the interaction between the active quark and the diquark cluster. This simplification does not necessarily imply the diquark is a physically separate entity in all scenarios, but provides a useful framework for analyzing hadron structure and dynamics, particularly in regimes where relativistic effects are important.

The Diquark Spectator Model reduces the computational demands of describing hadron structure by approximating the full many-body wavefunction. Instead of explicitly calculating interactions between all constituent quarks, the model treats the hadron as a two-body system comprised of an “active” quark and a correlated “spectator” diquark. This simplification is valid because the diquark, formed by a tightly bound quark-quark pair, moves as a single entity relative to the active quark, effectively reducing the complexity from a three-body problem to an effectively two-body one. By focusing on the interaction between this active quark and the diquark cluster, the model retains the essential strong-force dynamics while significantly easing the mathematical burden of calculations, allowing for a more tractable approach to understanding hadron properties.

Deconstructing the Hadron: Mapping the Energy-Momentum Tensor

The calculation of Gravitational Form Factors (GFFs) relies on the Energy-Momentum Tensor (EMT) decomposition, which separates the total EMT of a hadron into contributions originating from its constituent quarks and diquarks. This approach treats the hadron as a composite system, summing the EMTs of its valence quarks and any relevant diquark correlations. Specifically, the EMT $T^{\mu\nu}$ is decomposed such that each quark $q_i$ and diquark $D_i$ contributes a term $T_q^{(i)\mu\nu}$ and $T_D^{(i)\mu\nu}$ respectively, allowing the GFFs – which relate the hadron’s internal structure to its gravitational interactions – to be calculated as moments of these individual contributions. This decomposition is fundamental to connecting the hadron’s observable properties to its underlying quark-diquark structure.

The Baryon-Quark-Diquark Vertex represents the interaction point within the baryon where the active quark, participating in the Gravitational Form Factor (GFF) calculation, exchanges momentum with the spectator diquark. This vertex is crucial as it defines the dynamics of the baryon’s internal structure and influences the calculated GFFs. Accurate modeling requires consideration of the relative momentum between the quark and diquark, the spin configurations, and the overall baryon wavefunction. The vertex is not a point-like interaction but rather a region where strong force interactions occur, necessitating a detailed understanding of the underlying quark-gluon dynamics to ensure a physically realistic decomposition of the Energy-Momentum Tensor.

Ultraviolet divergences arise in calculations involving the Baryon-Quark-Diquark Vertex due to the point-like nature of the interaction. To address this, a Dipolar Form Factor is introduced, effectively smearing the interaction and providing a momentum-space cutoff. This form factor, parameterized by a dipole function $\Lambda^2 / (q^2 + \Lambda^2)$ , where $q$ represents the momentum transfer and Λ is a regularization scale, suppresses high-momentum contributions. By introducing this scale-dependent attenuation, the integral representing the vertex remains finite and well-defined, preventing the appearance of unphysical, infinite results and maintaining the stability and physical relevance of the subsequent Gravitational Form Factor calculations.

The Echo of Flavor: Tensor Charges and Hadron Identities

Investigating the spin structure of hadrons requires detailed understanding of how quarks contribute to the particle’s angular momentum. Recent calculations have focused on Generalized Form Factors (GFFs) – a key tool for dissecting this internal structure – and a comparative analysis between the proton and the Ξ⁰ hyperon offers a unique window into the influence of quark flavor. By examining differences in these GFFs, researchers can map how the varying quark composition – the Ξ⁰ contains strange quarks absent in the proton – affects the distribution of momentum within the hadron. This approach highlights the subtle interplay between quark flavor and spin organization, revealing how the internal dynamics of a hadron are shaped by its fundamental constituents and offering insights into the complex relationship between a particle’s spin and its constituent quarks.

Calculations of Generalized Parton Distributions (GPDs) reveal nuanced distinctions between the internal momentum distribution of the Ξ⁰ hyperon and that of the proton. These GPDs, which describe the probability of finding a quark with a specific momentum fraction inside the hadron, demonstrate a markedly different structure for the Ξ⁰ compared to the proton. This suggests that the quarks within the Ξ⁰, possessing a strange quark in addition to up and down quarks, distribute momentum in a way that contrasts with the proton’s distribution. The observed variations in GPD shapes imply a different interplay between the constituent quarks’ momenta and their contributions to the overall spin of the Ξ⁰, offering a deeper understanding of the hadron’s internal dynamics and confirming the influence of quark flavor on its momentum configuration.

Calculations of generalized form factors (GFFs) reveal distinct behaviors between the proton and the Ξ⁰ hyperon concerning their internal momentum distribution. The GFF $A_{T20}(0)$ for the proton is determined to be 0.445, a value aligning closely with results from the Bogoliubov-Lebedev-Kroll-Quinn (BLFQ) approach, which estimates 0.480; in stark contrast, the Ξ⁰ exhibits a $A_{T20}(0)$ value of -0.106, demonstrating a complementary momentum profile. Further analysis of the $\bar{B}_{T20}(0)$ GFF yields a value of 1.057 for the proton, marginally exceeding estimations from BLFQ calculations, which suggests that the transverse momentum distribution within the proton requires continued refinement in theoretical models.

The internal architecture of hadrons, those particles composed of quarks and gluons, is increasingly understood through the study of their Tensor Charge – a fundamental property revealing how momentum and spin are distributed amongst the constituent quarks. Recent calculations focusing on the $Ξ⁰$ hyperon and comparing its characteristics to those of the proton demonstrate that this charge isn’t a uniform property. Instead, subtle differences in Generalized Form Factors (GFFs) suggest a complementary behavior between the two particles; while the proton exhibits a Tensor Charge value of approximately 0.445 for $A_{T20}(0)$ , the $Ξ⁰$ shows a value of -0.106. These findings indicate that quarks within the $Ξ⁰$ distribute transverse momentum and spin in a distinctly different manner than those within the proton, offering a novel probe into the complexities of hadron structure and refining the precision of theoretical models used to describe these fundamental building blocks of matter.

Beyond the Horizon: Expanding the Framework of Understanding

The established framework isn’t limited to a single hadron; its principles are readily adaptable to investigate the Generalized Form Factors (GFFs) of a wider range of composite particles. By systematically applying this approach to different hadron species – baryons beyond the proton, mesons, and potentially even exotic multi-quark states – researchers aim to build a comprehensive map of their internal three-dimensional structure. This extended exploration will reveal how the distribution of quarks and gluons dictates a hadron’s properties, moving beyond simple one-dimensional pictures. The resulting atlas of GFFs promises to unlock deeper insights into the strong force and the very architecture of matter, potentially resolving long-standing questions about hadron mass and spin composition, and providing crucial input for models of nuclear interactions.

A more complete understanding of hadron spin structure hinges on incorporating chiral-odd Generalized Parton Distributions (GPDs) into theoretical models. These GPDs describe the distribution of parton orbital angular momentum and their contributions to the spin of hadrons, going beyond the simple collinear picture. Specifically, they are crucial for accurately calculating the Tensor Anomalous Magnetic Moment – a subtle but significant property influencing how hadrons interact with magnetic fields. Current calculations often rely on simplified assumptions; by including chiral-odd GPDs, researchers can refine these calculations and potentially resolve discrepancies between theoretical predictions and experimental observations. This refinement promises a more nuanced picture of how spin emerges from the complex interplay of quarks and gluons within hadrons, furthering insights into the strong force and the fundamental building blocks of matter.

Investigations are now shifting toward a more nuanced understanding of how complex interactions-beyond the currently modeled strong force-influence generalized parton distributions within hadrons. This expansion aims to move beyond simplified scenarios and account for the intricate interplay of forces at play within nuclear matter. Researchers anticipate these refined models will have significant implications for nuclear physics, particularly in accurately describing the behavior of matter under extreme conditions, such as those found in neutron stars or heavy-ion collisions. Ultimately, this work seeks to constrain the equation of state of dense matter-a crucial factor in understanding the structure and stability of these exotic states-and provide more reliable predictions for phenomena observed in both terrestrial experiments and astrophysical observations.

The pursuit of gravitational form factors, as detailed in this study of baryon structure, feels akin to charting a course toward an event horizon. One attempts to define the boundaries of understanding, to map the internal composition of matter, yet the deeper the investigation, the more apparent the limitations become. As Simone de Beauvoir observed, “One is not born, but rather becomes, a woman.” Similarly, these baryons aren’t simply defined by their quark composition; their properties emerge from the complex interplay of forces, a becoming perpetually obscured by the very models used to describe them. The diquark spectator model offers a framework, a temporary scaffolding, but any attempt to fully grasp the transverse polarization or chiral-odd GPDs risks vanishing beyond the reach of observation, a reminder that any model is only an echo of the observable, and beyond the event horizon everything disappears.

What Lies Beyond the Horizon?

The calculation of gravitational form factors, even within the constrained framework of a diquark spectator model, reveals the inherent fragility of any attempt to map internal structure onto observed gravitational interactions. While the presented work offers a comparative analysis of the proton and Ξ⁰ hyperon, the very notion of ‘structure’ becomes increasingly suspect when considering the limits of quantum chromodynamics and the potential for unforeseen complexities at higher energy scales. Any extrapolation beyond the model’s inherent approximations – the fixed diquark mass, the simplified wave functions – risks venturing into a region where theoretical certainty dissolves.

Future investigations must address the model’s limitations, perhaps through incorporation of more sophisticated diquark representations or exploration of alternative spectator mechanisms. However, a more fundamental challenge lies in acknowledging the possibility that the observed differences in gravitational form factors may not stem from discernible internal constituents, but rather from subtle violations of fundamental symmetries or the emergence of entirely new physics. Numerical methods, coupled with rigorous analysis of Einstein equation stability, will be crucial, yet these techniques, like all tools, can only illuminate a finite portion of the theoretical landscape.

Ultimately, the pursuit of gravitational form factors serves not only to refine models of hadron structure, but also to remind one of the inherent limitations of knowledge. Each calculation, each refined parameter, is but a fleeting glimpse before the inevitable descent beyond the event horizon of our current understanding.

Original article: https://arxiv.org/pdf/2601.21430.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/