⚛️ phenomenology

Mass, forces, and the dark sector in a Kaluza-Klein model with 6 compact dimensions

This paper proposes a unified 10-dimensional Kaluza-Klein model that successfully calculates the masses of 102 hadrons with high accuracy, predicts 31 additional masses, and offers testable candidates for dark matter and dark energy by incorporating strong, electromagnetic, and weak interactions alongside gravity.

Original authors: Manfred M. Vieten

Published 2026-10-19

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Manfred M. Vieten

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, two towering theories dominate our understanding of the universe, yet they speak different languages and refuse to shake hands. On one side stands the Standard Model, a highly successful framework that explains how tiny particles interact through forces like electricity, magnetism, and the nuclear glue that holds atoms together. It treats these forces as exchanges of invisible messenger particles. On the other side sits General Relativity, Einstein's masterpiece, which describes gravity not as a force but as the bending of space and time itself. For decades, physicists have struggled to merge these two ideas into a single, unified description of reality. The Standard Model works brilliantly for the very small but ignores gravity, while General Relativity rules the cosmos but breaks down when applied to the subatomic world. This disconnect leaves a gap in our knowledge, particularly regarding the mysterious "dark sector" of the universe—dark matter and dark energy—which makes up most of the cosmos but remains invisible to our current instruments.

A researcher at the University of Konstanz, Manfred M. Vieten, has proposed a new way to bridge this gap by revisiting an old idea called Kaluza-Klein theory. Originally conceived nearly a century ago, this theory suggested that if we added extra, hidden dimensions to our familiar four-dimensional world (three of space and one of time), the forces of nature might simply be the result of how space curves within those extra dimensions. Vieten's work takes this concept and expands it into a ten-dimensional universe. In this model, our visible world is just a slice of a much larger structure, and the six extra dimensions are curled up so tightly that we cannot see them. The paper argues that the particles we know—electrons, protons, and neutrons—are not fundamental points but rather stable vibrations or "excitations" traveling through these hidden, cylindrical dimensions. By treating all forces, including the strong nuclear force and the weak force, as geometric curvatures in this ten-dimensional space, the researcher attempts to calculate the masses of these particles from first principles, without relying on the complex quantum mechanics usually required.

The core of this study is a bold attempt to calculate the masses of hundreds of different particles, known as hadrons, which are made of quarks. In the standard view, calculating these masses is incredibly difficult and often requires approximations. Vieten's approach, however, treats the particle's mass as a direct result of its energy moving through these compact dimensions. The researcher built a mathematical framework where the size of the hidden dimensions and the strength of the forces are linked. By using the known masses of a few well-measured particles—such as the electron, the proton, and the neutron—the model determines the specific values for the hidden dimensions' size and the strength of the interactions. Once these values are set, the model is used to predict the masses of 102 different hadrons. The results are striking: when compared to experimental data, the calculated masses match the observed values with remarkable precision. In 70% of the cases, the difference between the prediction and the measurement is less than 5%. In 89% of cases, the error is under 10%, and in every single case, the error remains below 20%. The correlation between the predicted and measured values is so strong that the statistical likelihood of this happening by chance is virtually zero.

Beyond simply getting the numbers right, the model offers a new perspective on how particles behave when they collide. The research suggests that when an electron and a positron (its antimatter counterpart) crash into each other at high speeds, they do not simply bounce off or instantly vanish. Instead, the model predicts that they slow down dramatically as they approach a specific distance, coming to a near halt before merging or transforming. This behavior implies a repulsive force at very close range that is not accounted for in standard theories. Furthermore, the study identifies a potential candidate for dark matter: a type of "sterile boson." These are particles that have mass but do not interact with light or the nuclear forces, meaning they would be invisible to our detectors and could only be felt through their gravity. The model predicts specific masses for these particles, suggesting they could be created in high-energy collisions if the energy is tuned correctly.

The paper also touches on the nature of dark energy, the mysterious force driving the expansion of the universe. The researcher proposes that the asymmetry in how electric and weak forces act on different types of quarks inside a proton or neutron could cause these particles to jitter or move in a random, erratic pattern. Over cosmic scales, this collective jittering of matter could mimic the effect of dark energy, pushing galaxy clusters apart. If this connection holds true, it would mean that dark energy is not a constant property of empty space but is instead linked to the density of ordinary matter, potentially weakening as the universe evolves and more matter transforms into dark matter.

While the results are compelling, the paper presents this as a theoretical framework that requires further testing. The researcher outlines specific experiments that could confirm or refute these ideas. One proposal involves looking for the "sterile bosons" in particle colliders by searching for missing energy at specific mass values predicted by the model. Another suggestion is to observe the behavior of ultra-cold neutrons to see if they exhibit the predicted random motion. The study also notes that the model predicts the existence of massive photons for certain generations of particles, though these have not yet been detected. The author acknowledges that while the mathematical fit to existing data is excellent, the model relies on several assumptions about the geometry of the hidden dimensions and the nature of the forces within them.

Ultimately, this work offers a unified geometric picture of the universe where mass, force, and the hidden dimensions of space are inextricably linked. It suggests that the heavy mass of a proton is not an intrinsic property but a consequence of how its internal components vibrate through the fabric of a ten-dimensional spacetime. By successfully predicting the masses of over a hundred particles and offering testable ideas for dark matter and dark energy, the study provides a fresh, geometric lens through which to view the fundamental building blocks of nature. Whether this specific ten-dimensional model is the final answer remains to be seen, but it demonstrates that looking at the universe through the geometry of extra dimensions can yield precise and surprising insights into the nature of reality.

Technical Summary: Mass, Forces, and the Dark Sector in a Kaluza-Klein Model with 6 Compact Dimensions

Problem Statement
The Standard Model of particle physics, while highly successful in describing fundamental interactions via boson exchange, leaves significant gaps. These include the inability to fully explain dark matter and dark energy, the exclusion of gravity, and only partial success in calculating meson and baryon masses. Furthermore, the Standard Model relies on quantum mechanics and complex fields, whereas gravity is described by the classical geometry of General Relativity. This paper addresses the challenge of reconciling these frameworks by proposing a unified physical model that attributes all forces—including strong, electromagnetic, and weak interactions—to the curvature of spacetime, extending the Kaluza-Klein concept beyond the traditional five dimensions.

Methodology
The author formulates a classical field theory within a 10-dimensional spacetime consisting of one temporal dimension, three large-scale spatial dimensions, and six compactified cylindrical dimensions. Unlike the Standard Model, which uses complex wave functions and virtual boson exchange, this model describes particles as stable excitations (deformations) of the 10-dimensional spacetime traveling at the speed of light.

Geometric Framework: The 10D spacetime is embedded in a 20-dimensional flat "container" space. The metric tensor for the physical space is derived from this embedding. The six compact dimensions are assigned specific roles: dimensions 4–6 for the strong interaction, dimension 7 for electromagnetism, and dimensions 8–9 for the weak interaction.
Force Generation: Forces arise from the curvature of these specific manifolds. The model employs a generalized geodesic equation rather than potential energy gradients to calculate forces. The metric tensors for strong, electric, and weak interactions depend on the normalized separation ( $z$ ), normalized velocity ( $\beta$ ), and the derivatives of velocity with respect to $z$ .
Mass Derivation: Particle mass is derived from the total energy of excitations in the compact dimensions, comprising rest energy, kinetic energy, potential energy, and contributions from magnetic fields and dipole moments. The model introduces "flavor constants" ( $A_i$ ) to account for standing waves and specific excitation configurations.
Parameter Determination: Key parameters, including the compactification radius ( $\rho$ ), weak coupling ( $\alpha_w$ ), strong coupling ( $\alpha_s$ ), and the antineutrino/neutrino density ratio ( $\delta$ ), are determined by fitting the model to empirical data: the electron $g$ -factor, charged lepton masses, and the masses of the proton, neutron, and specific mesons ( $\pi, \rho, \phi, \psi, \Upsilon$ ).

Key Contributions

Unified Geometric Description: The paper presents a framework where strong, electromagnetic, and weak forces are manifestations of spacetime curvature in distinct compactified manifolds, treating them as independent interactions that couple to specific charges.
Mass Calculation without Free Parameters: Once the initial parameters are fixed using a small set of known masses, the model calculates the masses of 102 hadrons (scalar and vector mesons, spin-1/2 and spin-3/2 baryons) without introducing further free parameters.
Neutrino Screening Effect: The model incorporates a screening effect caused by the ambient density of neutrinos and antineutrinos, introducing a density ratio parameter ( $\delta$ ) that influences the radii of weak interaction cylinders and the masses of weakly interacting particles.
Sterile Bosons and Dark Matter: The framework predicts the existence of "sterile bosons"—spin-zero particles with mass but no strong, electromagnetic, or weak charges. These particles interact solely via gravity, positioning them as candidates for dark matter.
Force Asymmetry and Dark Energy: The model identifies asymmetries in electric and weak forces acting on up and down quarks. It proposes that these asymmetries induce a "random walk" motion in nucleons, which the author suggests could be the microscopic origin of dark energy.

Results

Hadron Masses: The model successfully calculates the masses of 102 hadrons. Comparison with 71 experimentally measured masses shows a correlation coefficient of $r = 0.997$ ( $p < 10^{-75}$ ). Relative errors are below 0.05 for 70% of the cases, below 0.1 for 89%, and below 0.2 for 100%.
Predicted Values:
- Compactification Radius: $\rho \approx 9.40 \times 10^{-16}$ m.
- Strong Coupling: $\alpha_s \approx 0.535$ .
- Gluon Mass: The model predicts a massive gluon with $m_{gluon} \approx 449.2$ MeV/ $c^2$ , which is consistent with experimental upper bounds derived from jet data.
- Neutrino Masses: The model derives non-zero masses for second and third-generation neutrinos ( $m_{\nu\mu} \approx 0.0086$ eV/ $c^2$ , $m_{\nu\tau} \approx 0.0501$ eV/ $c^2$ ) based on flavor constants and the density ratio $\delta$ .
- Sterile Boson Masses: A table of predicted masses for sterile bosons is provided, ranging from ~225 MeV/ $c^2$ to ~4717 MeV/ $c^2$ , depending on the flavor constants.
Collision Dynamics: The model predicts that in $e^+e^-$ collisions, particles decelerate significantly at a normalized separation of $z \approx 1$ , potentially leading to particle transmutation. Conversely, like-charge lepton collisions ( $e^+e^+$ or $e^-e^-$ ) exhibit a repulsive barrier preventing approach closer than $z \approx 9$ .

Significance and Claims
The paper claims to offer a testable alternative to the Standard Model that integrates gravity and the other fundamental forces through a geometric, 10-dimensional Kaluza-Klein framework. Its significance lies in:

Providing a unified geometric explanation for mass generation and force dynamics without relying on virtual boson exchange.
Offering specific, quantitative predictions for hadron spectroscopy that align closely with experimental data.
Proposing a mechanism for dark matter (sterile bosons) and a potential link between force asymmetries and dark energy (nucleon random walk).
Suggesting that the universe may be evolving toward a state where dark matter dominates, potentially leading to a cyclic universe scenario.

The author proposes three specific avenues for experimental validation:

Searching for missing mass resonances in high-energy collisions to detect sterile bosons.
Investigating the "random walk" behavior of nucleons, potentially observable in ultra-cold polarized neutrons.
Observing a decreasing rate of cosmic acceleration (dark energy) as matter converts to dark matter, consistent with recent DESI data.

The paper concludes that while the model successfully reproduces known masses and offers new theoretical insights, further theoretical refinement and dedicated experimental investigations are necessary to fully validate its capabilities and limitations.

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