Resolving the Proton Spin Crisis and Radius Puzzle: A Novel Mushroom-Shaped Proton Model ()
1. Introduction
The proton, a cornerstone of atomic structure, has fascinated physicists for over a century. Originally conceived as a simple, indivisible particle, the proton’s true complexity has gradually been unveiled through theoretical advancements and experimental discoveries. Today, it is understood that the proton is not a fundamental particle but rather a composite object made up of smaller constituents called quarks, bound together by the strong force, which is mediated by gluons.
The journey to understanding the proton’s inner structure began with the development of quantum mechanics and later quantum chromodynamics (QCD), which provided the framework for describing the interactions within the proton. The quark model, proposed independently by Murray Gell-Mann and George Zweig in 1964 [1] [2], revolutionized our understanding by suggesting that protons are composed of three quarks: two up quarks and one down quark. These quarks are held together by gluons, which are the force carriers of the strong interaction described by QCD.
Experimental evidence for the quark structure of protons came through deep inelastic scattering experiments conducted in the late 1960s and early 1970s at the Stanford Linear Accelerator Center (SLAC) [3].
In these experiments, electrons were fired at protons at high energies, and the scattering patterns revealed that the proton contained point-like constituents—quarks. These experiments provided the first direct evidence that protons were not fundamental particles but had an internal structure.
Further exploration of the proton’s inner structure has been conducted through high-energy particle collisions, such as those performed at CERN and Fermilab [4].
These experiments have revealed that when protons are smashed together at extremely high energies, they produce “jets” of particles. These jets are interpreted as the fragments of quarks and gluons, giving further insight into the proton’s quark-gluon structure [5].
Over time, several models have been developed to describe the distribution of quarks and gluons within the proton. The Parton model, introduced by Richard Feynman in the 1960s, treats quarks and gluons as point-like particles (partons) that carry fractions of the proton’s momentum. This model has been instrumental in interpreting the results of deep inelastic scattering and other high-energy experiments.
Another area of intense study has been the proton’s spin structure. Experiments have shown that the quarks’ spin contributes only about 30% to the total spin of the proton, suggesting that gluons and the orbital angular momentum of the quarks must play significant roles. This discovery has led to what is known as the “proton spin crisis”, challenging physicists to further investigate the sources of the proton’s spin [6].
The charge radius of the proton is another aspect of its internal structure that has puzzled scientists. Discrepancies in measurements obtained from electron-proton scattering versus those derived from the Lamb shift in muonic hydrogen have sparked what is now known as the “proton radius puzzle” [7]. These differing measurements suggest that our understanding of the proton’s spatial structure may need revision.
In this article, we delve into the inner structure of the proton, synthesizing the theoretical frameworks and experimental evidence that have shaped our current understanding. We will explore the quark-gluon dynamics within the proton, discuss the experimental techniques that have revealed its internal structure, and address and resolve the ongoing puzzles that continue to challenge physicists today.
2. The Quark Model and Quantum Chromodynamics (QCD)
The quark model is a foundational framework in particle physics that explains the composition of hadrons, including protons and neutrons, which are essential building blocks of atomic nuclei. According to this model, protons and neutrons, collectively known as nucleons, are composed of three quarks each. Specifically, a proton consists of two up quarks (u) and one down quark (d), while a neutron is made up of two down quarks and one up quark. These quarks are held together by the strong force, one of the four fundamental forces of nature, mediated by particles called gluons.
Gluons are massless elementary particles that act as the exchange particles, or force carriers, for the strong interaction, also known as the color force. In this context, “color” refers to a property used in Quantum Chromodynamics (QCD) to describe the charge carried by quarks and gluons, rather than a visual color. Quarks come in three “colors”—red, green, and blue—and gluons carry a combination of these colors, ensuring that quarks within a proton or neutron always combine to form a color-neutral (white) particle.
The self-interaction of gluons, a unique feature of QCD, leads to the non-linear and highly complex nature of the strong force. Unlike the electromagnetic force, which weakens with increasing distance, the strong force becomes stronger as quarks move farther apart, a phenomenon known as “asymptotic freedom”. This means that at very short distances, quarks behave almost as free particles, but as the distance increases, the force pulling them back together becomes stronger, effectively preventing quarks from being isolated.
Within the proton, quarks constantly interact via gluon exchange. These interactions are responsible for most of the proton’s mass. In fact, the majority of the proton’s mass does not come from the sum of the quark masses but from the energy associated with the strong force field created by the gluons. According to Einstein’s equation, E = mc2, this energy contributes significantly to the proton’s overall mass.
QCD also explains the dynamic and fluctuating nature of the proton’s internal structure. The proton is not a static entity; rather, it is a seething mass of quarks, antiquarks, and gluons. In addition to the three “valence” quarks that define the proton’s quantum numbers, there are numerous “sea” quarks and gluons that constantly pop in and out of existence due to quantum fluctuations. These sea quarks and gluons play a significant role in the proton’s properties, including its spin and charge distribution.
However, while QCD is the best-established theory describing the strong interaction, it has limitations that make it insufficient on its own for a complete understanding of the proton’s internal structure. QCD is a highly complex and non-linear theory. Calculations involving the strong force at low energies, where quarks are confined within protons, are notoriously difficult. These calculations often require sophisticated computational techniques like lattice QCD, which discretizes space-time into a grid. Despite these efforts, certain aspects of the proton’s structure remain challenging to calculate accurately.
Moreover, QCD, while mathematically rigorous, does not always provide an intuitive picture of how quarks and gluons give rise to the observable properties of the proton, such as its mass, spin, and charge radius. Complementary theories or models, such as the Quark Vortex Theory [8], offers alternative perspectives that might be easier to conceptualize and work with. The proton mass puzzle—the discrepancy between the proton’s actual mass and the sum of the masses of its constituent quarks—remains a significant issue. While QCD explains that most of the proton’s mass arises from the energy of the strong force, it does not fully resolve this puzzle in an easily accessible way, prompting the need for alternative models that might better capture the dynamics involved.
Furthermore, QCD is asymptotically free, meaning quarks behave as free particles at extremely high energies. However, in the strong coupling regime, where quarks are bound together in protons, the theory becomes exceedingly complex, and the confinement mechanism is not fully understood. This limitation suggests the potential value of additional theories that can address these aspects more directly.
Given these challenges, the development of complementary theories such as the Quark Vortex Theory is important. This theory offers a novel perspective on the proton’s internal structure, potentially providing a more intuitive and accessible explanation for its mass, spin, and charge distribution. As we continue to explore the fundamental nature of matter, having a diverse set of theoretical tools will be crucial in uncovering the deeper truths about quarks, gluons, and the very fabric of the universe.
3. The Quark Vortex Model: A New Approach
The Quark Vortex Model introduces a new way of thinking about elementary particle such as electrons [9] and quarks [10], viewing them not as point-like particles but as vortices within a superfluid vacuum. This superfluid vacuum, which pervades the universe, can be thought of as a quantum fluid that exist even before the big bang. In this model, quarks are stable, localized vortices within this quantum fluid, and their properties—such as mass, charge, and spin—are derived from the dynamics of these vortices.
Thus, the quark can be conceptualized as an irrotational, circular vortex within a frictionless superfluid medium, with concentric streamlines formed from the primordial vacuum during the Big Bang. This model envisions the quark as having concentric streamlines, where the rotational velocity of the superfluid is at its maximum at the center of the vortex and gradually decreases as one moves outward. This decrease in velocity continues until reaching the boundaries of the vortex, where the pressure gradient vanishes, and the flow becomes laminar and frictionless.
The frictionless nature of the flow implies that the vortex motion of the quark is indestructible and cannot be created anew, signifying a stable and persistent structure. For this stability to be maintained, the central region of the quark’s vortex, known as the negative suction point, must possess enough energy to accelerate virtual particles—such as virtual photons—up to the speed of light. If the energy in this central point is insufficient, the quark’s vortex structure would not remain stable, thereby undermining the quark’s integrity within this model.
4. The Role of Gluons in Quark Vortex Theory
In the context of the vortex model where quarks are conceptualized as circular vortices within a superfluid medium, gluons can be reinterpreted as the spiral arms of these quark vortices. Traditionally, gluons are understood as the force carriers responsible for the strong interaction that binds quarks together. However, within this vortex framework, gluons could be seen as natural extensions of the vortex’s rotational dynamics rather than as separate entities.
If the strong interaction is fully described by the fluid dynamics of the quark vortex, the necessity for gluons as distinct particles might be reconsidered. The interaction between quarks could result directly from the proximity and overlapping of their vortices, rendering the mediated exchange of particles, such as gluons, unnecessary.
The rotational motion within each vortex, combined with the associated pressure gradients and energy distribution, might inherently account for the attractive forces between quarks. This would eliminate the need for an additional particle like the gluon to mediate these forces. The spiral arms, conceptualized as gluon fields, would then play a crucial role in maintaining the stability and structure of the quark vortex. By connecting various parts of the vortex, these spiral arms contribute to the overall coherence and stability of the quark’s rotational motion, ensuring the persistence of the vortex.
In this model, the quark vortex is characterized by its frictionless and stable nature, a result of its inherent properties. The absence of friction means that the vortex motion is indestructible and does not require external forces—such as those traditionally provided by gluons—to sustain its stability. The internal dynamics of the vortex, particularly in terms of energy distribution, could be sufficient to maintain quark stability and govern interactions between quarks.
Moreover, as extensions of the vortex, gluons might also play a role in balancing rotational energy across the vortex, ensuring that the quark’s motion remains stable and that energy is not concentrated in one area, which could potentially destabilize the vortex. When quarks interact, their vortices may intertwine, with the spiral arms (or gluons) facilitating the exchange of forces and information between quarks, effectively linking their vortices in a dynamic and complex manner.
This vortex model offers a visual and dynamic way to understand gluons within a superfluid framework, providing a unique perspective on their role in quark interactions and the strong force. It could potentially unify the description of quarks and the forces between them into a single framework. If the vortex’s spiral arms naturally arise from the quark’s rotational dynamics, these features might provide a natural explanation for the strong force. In this view, what is traditionally understood as the strong force in Quantum Chromodynamics (QCD) could be an emergent property of the vortex dynamics, making gluons a redundant or unnecessary concept.
The quark vortex theory thus offers a simplified model of particle physics, reducing the number of fundamental particles and interactions that need to be considered. By explaining quark interactions through vortex dynamics alone, this theory avoids the complexities associated with gluon exchange, such as colour charge, confinement, and other intricate aspects of QCD.
In summary, the quark vortex theory potentially explains the strong interaction and quark confinement through the internal dynamics of quark vortices and their spiral arms, which correspond to the gluons. The theory relies on the inherent properties of the vortex, such as stability, energy distribution, and rotational dynamics, to fulfil the roles traditionally ascribed to gluons in the standard model. This approach might offer a more unified and simpler understanding of quark interactions within a superfluid framework.
5. The Structure of the Meson
The meson, a fundamental subatomic particle, is traditionally composed of a quark and an antiquark bound together by the strong force. Within the framework of Quantum Chromodynamics (QCD), this force is mediated by gluons. However, the Quark Vortex Theory provides a novel reinterpretation of meson structure, replacing the conventional quark-antiquark picture with the dynamics of counter-rotating vortex pairs within a superfluid vacuum.
In this theory, mesons are visualized as bound vortex pairs—two localized vortex structures rotating in opposite directions. This opposition in rotational motion creates a pressure gradient in the surrounding quantum fluid, which naturally pulls the vortices together, forming a stable—albeit temporary—meson structure (Figure 1).
Figure 1. An artistic illustration, where the upper part shows the quark and antiquark being drawn toward each other due to their opposite rotations. In the lower part of the illustration, the quark and antiquark are shown to experience repulsion due to the parallel alignment of their spiral arms.
This aspect of the interaction aligns with the traditional understanding that quarks are bound together by one of the strongest forces in nature. However, the quark vortex theory adds a new layer of interpretation by attributing this attraction to the fluid dynamics of the quark vortices.
While the attraction between opposite rotational vortices binds the quark and antiquark, the meson structure also exhibits inherent instability. In the lower part of the illustration, the quark and antiquark are shown to experience repulsion due to the parallel alignment of their spiral arms, which in the context of quark vortex theory, could be considered analogous to gluon fields.
This repulsive force arises when the vortex structures of the quark and antiquark align in parallel, leading to a conflict within the meson structure. This internal repulsion is a key factor contributing to the instability of mesons. Unlike protons and neutrons, which are stable due to their structure, mesons are known to have finite lifetimes. This instability ultimately results in the decay of mesons into other particles.
Another crucial aspect of meson instability is the fact that quarks and antiquarks are each other’s antiparticles. When they come into close contact, there is a tendency for them to annihilate each other. This annihilation process is a fundamental reason why mesons, despite being held together by the strong force, do not have indefinite lifespans. The interaction between the quark and antiquark vortices, coupled with their antiparticle nature, ensures that mesons eventually decay into other, more stable particles.
6. The Mathematical Framework for Calculating Strong Force
Understanding the proton’s properties through rigorous mathematical formalism is crucial for bridging theoretical predictions with empirical observations. By applying principles from quantum chromodynamics (QCD) and leveraging the novel concept of quarks as superfluid vortices, we aim to derive precise expressions for the strong force acting between quarks.
This subchapter presents derivations demonstrating how the Quark Vortex Theory provides a coherent framework for understanding the proton’s fundamental properties. The ability of this theory to accurately predict and describe the strong interaction, charge distribution, and mass of the proton will be examined in detail in a separate article, offering potential confirmation of the theory’s validity. Deviations or agreements with experimental data may shed light on the accuracy of this approach.
The attraction between two up quarks depends on the distance between them according to the equation:
where
Fstrong: is the strong force between the two quarks.
αs: is the strong coupling constant, a dimensionless number that characterizes the strength of the strong interaction [11].
ħ: is the reduced Planck’s constant.
c: is the speed of light in a vacuum.
r: is the distance between the two quarks.
This force arises from a Coulomb-like potential scaled by the SU(3) color group’s Casimir invariant CF = 4/3, a well-established result in QCD.
This is derived by differentiating the QCD Coulomb-like potential:
This equation can be understood as a formula that describes the energy and momentum in a system where strong interactions, akin to vortex dynamics, are
at play. By substituting
, the equation becomes:
In vortex dynamics, the energy related to a vortex filament is often connected to the circulation Γ and the radius of curvature r. The term
resembles the
momentum of a particle with wavelength r traveling at speed c, linking force to particle motion.
In our vortex-based approach, we reinterpret the force using vortex energy and geometry. We consider the Compton wavelength λ = 2πr as the circumference of a quark vortex, and relate Planck’s constant to vortex circulation as shown in a previous article [12]:
Substituting this into the QCD-like form and simplifying yields:
This formulation suggests that the strong force is proportional to the rest energy mc2 distributed along the vortex circumference, with the prefactor
reflecting a coupling between vacuum structure and confinement efficiency.
Notably, the strong force can be interpreted not merely as a direct interaction between particles, but as an emergent phenomenon arising from the pressure exerted by vacuum dynamics. In the context of vortex motion, energy confined within a finite radius creates a localized pressure field, which may drive the force responsible for quark confinement.
This concept connects naturally to the classical drag pressure formula used in fluid dynamics:
,
where:
P has the same numerical value as G, i.e., 6.67384 ± 0.00080 × 10−11 m3/kg∙s2 [13].
ρ is the vacuum density ρ ≈ 9.53 × 10−27 kg/m3 [8].
c = 3 × 108 m/s.
CD is the drag coefficient ≈ 0.1556.
Substituting these values, we find:
This value is numerically identical to the gravitational constant G, even though their physical dimensions differ. While pressure is defined as force per unit area, and G describes the gravitational interaction per unit mass and distance, the numerical equivalence suggests a deeper connection between the fabric of space and vacuum-mediated forces [13].
In this framework, the resistance offered by the vacuum to quark motion—modelled analogously to a drag force—is quantified by a dimensionless drag coefficient CD, while the effective efficiency of energy transfer during vortex confinement is represented by a factor η.
Empirically, if we take:
,
This result remarkably coincides with the known average value of the strong coupling constant αs ≈ 0.118 at the 1 GeV energy scale in quantum chromodynamics (QCD).
This value is consistent with CMS’s recent measurement from inclusive W and Z production at
and 8 TeV:
(CMS Collaboration, 2019) [14]
Rearranging gives:
This numerical concordance suggests that:
or equivalently:
This parallel supports the hypothesis that both the strong force and gravity may originate from the same underlying structure of the vacuum, governed by vortex dynamics and energy transfer processes. Such a unifying view opens the door to reinterpreting fundamental interactions as emergent phenomena shaped by the flow and resistance of the vacuum medium.
Building on this interpretation, we propose that the strong coupling constant αs itself may not be a fundamental constant, but rather an emergent parameter derived from vacuum drag properties.
This relation indicates that the strength of the strong interaction can be understood as a measure of how efficiently the vacuum medium transfers energy through drag-induced confinement. In this view, the vortex model replaces the color charge origin of αs in QCD with a more geometric and physical mechanism based on vortex structure, motion, and energy dissipation.
By grounding αs in fluid-like properties of the vacuum, we bridge high-energy particle physics with classical fluid dynamics, suggesting that confinement and the strong force may be macroscopic manifestations of vortex behavior in a structured vacuum.
The strong coupling constant αs is thus not treated as a fundamental constant, but rather as an emergent parameter that quantifies how efficiently vacuum drag—represented by CD is transformed into an attractive force between quarks within a confined superfluid-like vacuum.
By identifying this drag coefficient value with αs, the model proposes a direct link between the strong nuclear force and vacuum dynamics, hinting at a common mechanism underlying both strong and gravitational interactions. In this view, αs becomes a scaled measure of vacuum drag pressure, and the factor 4/3 arises naturally as the effective energy transfer ratio within the confined structure—defining the strength of interaction between quarks through the mechanics of vortex motion in the vacuum medium.
This equivalence implies a deep relationship between the strong force in quantum mechanics and gravitational effects, supporting the idea that these fundamental interactions may share a common underlying mechanism. By interpreting these forces and constants through the framework of vortex dynamics, we provide a bridge between quantum mechanics and cosmology, offering a unified perspective that connects the behavior of subatomic particles to large-scale universal forces.
Let us now compute the magnitude of this force for an up quark:
mu ≈ 4.1 × 10−30 kg.
r ≈ 0.87 × 10−15 m.
c = 3 × 108 m/s.
αs ≈ 0.118.
This result lies well within the expected magnitude of the strong force at femtometer scales and far exceeds the electromagnetic and gravitational forces acting at the same distance. This confirms the dominance of the strong interaction under confinement conditions, consistent with QCD predictions.
7. Resolving the Proton Radius Puzzle
The proton radius conflict is a significant issue in proton structure physics, arising from discrepancies in the measurements of the proton’s radius using electrons versus muons. Recent experimental investigations have deepened and simultaneously complicated our understanding of the proton’s structure. The proton, long considered a simple, point-like particle, has revealed unexpected complexities when its charge radius is examined with increasing precision.
A pivotal series of experiments conducted at the Paul Scherrer Institute in Switzerland have brought this issue to the forefront. Dr. Jan Bernauer and Dr. Randolph Pohl led independent research efforts aimed at measuring the proton’s charge radius using different methodologies, each providing crucial insights into the fundamental nature of the proton.
Dr. Bernauer’s approach relied on direct electron scattering from hydrogen nuclei, a method that has been a cornerstone of particle physics for decades. By bombarding hydrogen atoms with electrons and analyzing the scattering patterns, Bernauer’s team determined the proton’s charge radius to be 0.878 ± 0.005 fm [15] [16].
This precise measurement, while consistent with some earlier results, contributes to a body of work that both supports and challenges our current theoretical models.
Meanwhile, Dr. Pohl’s approach, using muons and examining the Lamb shift in muonic hydrogen, found a radius of 0.8409 ± 0.0004 fm [17].
This ~4% discrepancy between the two measurements, while seemingly small, is substantial in subatomic physics. It raises questions about the accuracy of existing models like GUT and Quantum Electrodynamics (QED). Physicists Ingo Sick and Dirk Trautmann suggest we might not fully grasp the implications of each experimental setup, indicating that our current models might need revisiting or refinement.
This divergence underscores a significant discrepancy, indicating that our understanding of proton structure is not yet complete and necessitates further investigation to reconcile these differences [18].
The discrepancy between these measurements and others using muonic hydrogen has sparked considerable debate within the scientific community, leading to what is known as the “proton radius puzzle”.
Our model, which conceptualizes the proton as a 3D mushroom-like structure formed by a specific spatial configuration of three quarks, offers a new interpretation: the radius that determines the force is not a static geometric boundary but a dynamic parameter defined by the interaction between quarks, particularly the strong force acting between the two up quarks. In this model, the observed radius corresponds to the distance at the widest part of the proton’s structure—where the up quarks interact via gluon-mediated forces. The relation between radius and force is given by:
where
αs is the strong coupling constant, 0.1183 ± 0.0009 [19].
ħ is the reduced Planck constant (1.0545718 × 10−34 Js).
c is the speed of light (2.998 × 108 m/s).
r is the distance between the quarks.
The force between quarks inside a meson can range from about 103 to 104 Newtons when quarks are close together. We’ll take a value somewhere in between:
This range is comparable to the forces found in other hadronic structures like protons and neutrons [20].
This calculation suggests that the radius of the proton is approximately 0.8409 × 10−15 m.
If we assume that the force between the quarks is 6.47 × 103 Newtons the which is also in the accepted range the radius of the proton will be as it was measured by Bernauer’s method 0.878 (fm).
This suggests that the measured radius depends on the magnitude of the strong force at the time of interaction, which itself may vary based on how the proton is probed—highlighting a dynamic aspect of the proton’s structure rather than a fixed geometric size.
It is unlikely that these discrepancies are related to variations in the strong force; rather, they may be due to the angle at which the proton is hit during measurement. The momentum and spatial distributions of quarks and gluons within the proton are correlated with its spin. For instance, the spatial distribution of quarks in a transversely polarized proton is found to be distorted in the transverse plane [21] [22].
This analysis reveals that the strong force equation is more than just a mathematical description of quark interactions; it embodies a deeper physical analogy, drawing parallels between quantum chromodynamics and vortex dynamics. By interpreting elementary particles as vortices, the equation not only accounts for the strong force between quarks but also offers a unified perspective where fundamental constants and force laws are expressions of underlying vortex structures. This approach enriches our understanding of the strong interaction, highlighting how the principles of vortex dynamics can illuminate the intricate forces at play within the quantum realm.
8. The Structure of Proton
The proton, a fundamental constituent of atomic nuclei, has long been a subject of intense study in particle physics. Although the proton is one of the most well-known particles in the universe, its internal structure remains a complex and evolving topic of research.
Protons are composed of three valence quarks—two up quarks and one down quark—held together by the strong force, mediated by gluons. This simple picture, however, belies the complex quantum dynamics within the proton. The sea of virtual quarks, antiquarks, and gluons that momentarily pop in and out of existence within the proton contribute to its properties, such as its mass and spin.
Recent experiments using high-energy particle colliders, such as the Large Hadron Collider (LHC) at CERN, have allowed physicists to probe the proton’s internal structure with unprecedented precision. These experiments have revealed that the distribution of quarks and gluons inside the proton is more intricate than previously thought. For instance, the LHCb experiment at CERN has provided evidence that the distribution of momentum among the quarks and gluons is not uniform but varies significantly, depending on the energy scale at which the proton is probed [23].
To explain the inherent complexity of the proton’s structure, quark vortex theory offers a new perspective.
According to this theory, the stabilization of particles like protons and neutrons, which are baryons, is rooted in the dynamics of quark vortices. While mesons, which consist of only two quarks, are inherently unstable, the introduction of a third quark in baryons like protons and neutrons leads to a more stable configuration (Figure 2).
Figure 2. The three quarks structure connected together, two up and one down (uud), the spiral arms prolongation of the quark vortex (gluons) that connect the three parts together are integral part of the quarks.
While mesons are inherently unstable, the introduction of a third quark, as seen in baryons like protons and neutrons, can stabilize the particle’s structure. This stabilization is achieved through the altered dynamics of the vortices when a third quark is present. The stability is related to two main aspects: the triangle structure and 3d structure.
In a triangle, any force applied at a vertex is distributed across the two adjacent sides, creating a balance of forces. This distribution ensures that the structure remains stable and does not collapse. For example, in engineering, triangles are used in trusses and frameworks because the forces are effectively spread out, minimizing the risk of deformation or collapse.
The second aspect is the formation of 3D structure.
The d-quark vortex rotates at speed of light, when it connects to the two up quarks they rotate at the same direction and at the same speed of the down quark creating 3D compact close structure resembling a mushroom (Figure 3).
Figure 3. Mushroom-like structure of the proton the lower Quark is at 90 degrees rotation plane relative to rotation plane of the quark and antiquark.
The cap of the proton is generally round and convex, like an umbrella or dome, and can be approximated as a hemisphere when calculating its volume. The stem is typically cylindrical, with a radius corresponding to the radius of the d-quark and a length equivalent to the quark’s diameter. This shape is particularly relevant when calculating the volume of the stem, which is crucial for determining the mass of the proton.
When viewed as a whole, the proton has a balanced and symmetrical appearance, with the cap sitting atop the stem.
The insights from quark vortex theory provide a compelling explanation for the stability of baryons compared to mesons. The theory suggests that the internal dynamics of quark vortices, particularly the influence of a third quark, are sufficient to maintain the stability of baryons without requiring additional forces or entities.
9. Internal Structure of the Proton
The study of the internal structure of the proton is fundamental to our understanding of particle physics, especially in the context of experiments conducted at the Large Hadron Collider (LHC). Central to this study are Parton Distribution Functions (PDFs), which describes how the proton’s momentum is distributed among its constituent quarks and gluons, collectively known as partons. Understanding PDFs is critical because they provide the necessary framework for predicting and interpreting the outcomes of high-energy collisions at the LHC, where protons interact with large momentum transfers [23].
Parton Distribution Functions are not just abstract mathematical constructs; they are pivotal to the analysis of experimental data in particle physics. When protons collide at high energies, as they do in the LHC, the interactions between the quarks and gluons within these protons determine the types of particles produced and the dynamics of their production. PDFs essentially encode the probability of finding a specific type of parton (such as a quark or a gluon) carrying a certain fraction of the proton’s momentum at a given energy scale. These functions are crucial for making accurate predictions in particle physics because they directly influence the cross-sections and other observable quantities in proton-proton collisions.
However, despite their importance, PDFs are not straightforward to calculate from first principles due to the complex, non-perturbative nature of Quantum Chromodynamics (QCD) at low energy scales. Instead, they are determined empirically through fits to experimental data, making them an essential bridge between theory and experiment. Accurate determination of PDFs is thus a cornerstone of modern particle physics, allowing scientists to make precise predictions about the outcomes of experiments and to search for new phenomena beyond the Standard Model.
Therefore, a good knowledge of the proton structure is essential for the accurate measurement of many physical quantities. ATLAS has well demonstrated the importance of this area for present and future work [24].
In this article, the traditional view of the proton’s internal structure, where partons are thought to be uniformly distributed within the proton, has been challenged by new theoretical model. The Mushroom Model, which is based on the Quark Vortex Model, offers a novel perspective on how quarks and gluons are distributed inside the proton. Unlike the traditional view, which tends to treat the proton as a relatively homogenous mixture of quarks and gluons, the Mushroom Model suggests that these partons may form vortex-like structures within the proton. These structures could create regions of varying density, resulting in a “mushroom” shape where denser regions are found around the edges and less dense areas are present at the core (Figure 4).
Figure 4. This image shows a vertically sliced mushroom, revealing an internal structure characterized by distinct radial layers, asymmetrical mass distribution, and a central hollow region.
The internal structure of protons and neutrons can be understood using an analogy to the density distribution seen in a mushroom. Just as a mushroom has a dense stem and a less dense cap, the density within protons and neutrons is not uniform.
The core, similar to the stem of a mushroom, is where the majority of the mass and energy is concentrated. This central region is often referred to as the “quark core”, where quarks are held together by the strong force mediated by gluons.
Surrounding this dense core is a more diffuse area, analogous to the cap of the mushroom. This region represents the outer parts of the proton or neutron, where the density is lower. Here, the gluon field and sea quarks (quark-antiquark pairs that are constantly being created and annihilated) contribute to the structure but with less intensity than in the core.
This density distribution reflects how the mass and energy are not evenly spread across the entire proton or neutron, but instead concentrated towards the center, diminishing towards the edges—just like the material density from the stem to the cap of a mushroom.
The implications of the Mushroom Model for our understanding of PDFs are profound. By providing a more detailed and dynamic picture of how partons are distributed within the proton, the model could lead to more accurate PDFs. This, in turn, would enhance the precision of predictions for high-energy collisions at the LHC. The traditional models may overlook subtle variations in parton densities, particularly at different momentum fractions and energy scales. The Mushroom Model, with its focus on quark vortices, offers a more nuanced understanding of these distributions, potentially revealing variations that could be critical for interpreting experimental data.
10. Proton Spin and Internal Vortex Geometry
The proton spin crisis emerged when experimental data revealed that quark spins account for only about 30% of the proton’s total spin. Current QCD-based explanations involve additional contributions from gluon spin and quark/gluon orbital angular momentum. In our vortex-based model, the proton is visualized as a geometrically asymmetric structure with a central vortex axis (like a stem) and spiraling components (like a cap).
11. The Proton Spin Crisis Resolved
For decades, physicists have sought to understand its properties, particularly its spin—a form of intrinsic angular momentum. However, in the late 1980s, the “proton spin crisis” emerged, shaking the foundations of particle physics. The crisis arose when it was discovered that the spin of the proton could not be fully explained by the spins of its constituent quarks, as was traditionally assumed.
In response to this conundrum, various theories have been proposed to account for the missing spin. One of the more intriguing ideas is the concept of a mushroom-shaped internal structure of the proton, which could provide a novel way to resolve the proton spin crisis.
In the conventional model, the proton is composed of three quarks—two up quarks and one down quark. These quarks are bound together by the strong force, mediated by particles called gluons. Quarks possess a property called spin, which is a form of intrinsic angular momentum. Initially, it was believed that the proton’s spin (which is 1/2) was simply the sum of the spins of its three quarks. However, experiments conducted in the late 1980s by the European Muon Collaboration (EMC) revealed that the quarks’ spins contribute only about 30% of the proton’s total spin. This unexpected result sparked the proton spin crisis, leading physicists to search for other sources of the proton’s spin.
Traditional models, which treated the proton as a relatively simple, spherical object, have struggled to provide a complete explanation. In response to this challenge, we propose a new theory: the proton may possess a mushroom-shaped internal structure, which could offer a fresh perspective on resolving the proton spin crisis.
The resulting spin of the proton is a complex interplay between the intrinsic spins of the quarks, their orbital angular momentum, and contributions from gluons.
The angular momentum of the proton indicates that there is a real internal rotation (spin) that confers upon it its rest mass. Angular momentum is a measure of the amount of rotation an object has, considering its mass, shape, and rotational velocity. Spin is a key component of angular momentum, and in the context of elementary particles, it is an intrinsic property that does not change regardless of the particle’s environment. Therefore, in quantum mechanics, angular momentum is quantized, meaning it can only take on certain discrete values.
In quantum mechanics, spin is a fundamental property of particles that can be represented by a vector. The length of this spin vector is measured in units of the reduced Planck constant, denoted as ħ. For quarks, a measurement of the spin vector component along any axis can yield only two possible values: +ħ1/2 or −ħ1/2. This quantization of spin values is why quarks are classified as spin-1/2 particles. Since the sign indicates the direction, we tend to call these “spin-up” and “spin-down”.
To resolve the proton spin crisis, the mushroom shape proton theory posits that the proton’s internal composition is asymmetric, with regions that resemble the cap and stem of a mushroom. These distinct regions contribute differently to the proton’s overall spin, offering a potential solution to the missing spin problem.
According to vortex theory an electron, an elementary particle, has a spin of 1/2, which reflects the fact that the core vortex of the electron must undergo two full rotations (720 degrees) to return to its original state [25].
The same thing takes place with the quarks. In order to maintain stable rotation, the vortex should complete 2πr, 360 degrees, which allows the minimal time needed to exist t = 2πr/c.
The two up quarks are spinning in opposite directions clockwise rotation and counterclockwise rotation in the y and x axis. While the d-quark has a clockwise. Rotation in z and x axis. Besides the intrinsic spins of the up quarks, they are rotating around in the same direction of the d-quark rotation creating a further angular momentum of the two up quarks and their spiral arms (gluons) (Figure 5).
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Figure 5. This diagram illustrates the proposed geometric and dynamic origin of proton spin according to the vortex-based model. The two up quarks exhibit opposite intrinsic spin orientations: one with spin +1/2ħ (clockwise, yellow arrow) and the other with spin −1/2ħ (counterclockwise, yellow arrow). The down quark (lower orange arrow) contributes a spin of +1/2ħ through its own clockwise vortex rotation. In addition to their intrinsic spins, the two up quarks rotate collectively around the axis of the down quark. This rotational motion represents the angular momentum carried by gluons, which mediate the strong force and bind the quarks. This contribution is illustrated by the upper orange arrow and corresponds to the gluon spin component. Altogether, the proton’s total spin emerges from the sum of quark intrinsic spins and the gluon-mediated orbital angular momentum, offering a physically coherent resolution to the proton spin puzzle.
The time required for each quark to complete a full vortex rotation is given by:
,
where r is the radius of rotation, and c is the speed of light. If the rotational radius of the up quarks and their associated gluon spirals is approximately half the radius of the proton (i.e., rq ≈ 0.435 × 10−15 m, then the time required for one full rotation around an up quark is approximately:
In contrast, assuming a proton radius of rp ≈ 0.87 × 10−15 m, the time required for a full rotation of the entire proton is:
This means that during each complete rotation of the proton, the internal up quark vortices perform approximately two full cycles. These internal cycles contribute additional angular momentum.
In this model, the proton’s total spin emerges from four distinct contributions:
1) +1/2ħ—Intrinsic clockwise spin of one up quark (yellow arrow).
2) –1/2ħ—Intrinsic counterclockwise spin of the second up quark (yellow arrow).
3) +1/2ħ—Intrinsic clockwise spin of the down quark (lower orange arrow).
4) +1/2ħ—Collective angular momentum of both up quarks rotating around the axis of the down quark (upper orange arrow), interpreted as the angular momentum carried by gluons.
The fourth component—often unaccounted for in traditional quark spin summations—represents the gluon spin or more precisely, the orbital angular momentum arising from the gluon-mediated coupling between the quarks. The spiral-shaped trajectories of the gluons act as binding filaments, adding coherent rotational motion around the proton’s central vortex axis.
Experimental data (EMC, COMPASS, RHIC) confirms that only ~30% of the proton’s spin arises from intrinsic quark spin. This vortex-based model aligns with QCD findings by attributing:
~30% to intrinsic quark spin.
~30%~40% to orbital angular momentum from the rotational motion of quarks (especially up quarks around the d-quark).
The remainder to gluon angular momentum—represented here as the spiraling co-rotation of the quarks mediated by gluons.
This composite picture resolves the proton spin crisis by integrating intrinsic, orbital, and gluon angular momentum into a unified geometric and dynamical model. The internal mushroom-shaped structure provides both the asymmetry and the rotational topology necessary to explain how the proton, a spin-1/2 particle, exhibits such a rich internal angular momentum structure.
12. Conclusions
The internal structure of the proton, once believed to be simple and spherical, reveals a far more intricate and dynamic composition. The introduction of the Quark Vortex Theory and the Mushroom Model provides a transformative framework that addresses critical challenges in modern particle physics—namely, the proton spin crisis, the proton radius puzzle, and—potentially—the proton mass puzzle, whose full resolution will be presented in a forthcoming article.
By modelling quarks as stable vortices within a frictionless superfluid vacuum and reconceptualizing gluons as spiral arms of these vortices, the theory naturally accounts for the proton’s spin, its asymmetric internal geometry, and charge distribution. The mushroom-shaped structure—resulting from the rotational alignment of the two up quarks around the down quark axis—introduces a topological and dynamic basis for understanding internal angular momentum and its manifestation as observed spin.
Moreover, this model interprets the gluon contribution not merely as field exchange but as a rotational, orbital angular momentum—integral to the proton’s observed properties. In this context, gluon dynamics are seen as intrinsic vortex behavior rather than external mediators, offering a more unified explanation of confinement, spin, and mass generation.
Although the Quark Vortex Theory qualitatively addresses the proton mass puzzle by attributing mass to the confined rotational energy of vortices, its quantitative formalism will be developed in detail in a future publication. There, we will explore how vortex curvature, drag, and superfluid tension contribute to the effective mass observed in nucleons.
To advance this model, further research should be done to:
Develop mathematical formalism to connect vortex curvature and superfluid drag with observable proton properties.
Perform numerical simulations to test predictions against lattice QCD and scattering data.
Conduct experimental validation through proton tomography, polarization observables, and muonic hydrogen studies.
Extend the framework to describe meson instability, nuclear binding, and multiquark systems.
Explore the connection between vacuum structure, energy quantization, and unification theories—linking vortex behavior to cosmological constants and gravitational phenomena.
In summary, the Quark Vortex Theory and Mushroom Model provide a rich, geometry-based view of subatomic structure—capable of reconciling discrepancies in spin, charge radius, and potentially mass. As theoretical tools and experimental precision evolve, these models may serve as a bridge between QCD and a more intuitive, unified understanding of matter and vacuum—marking a new chapter in the physics of the proton and beyond.