The origin of fermion mass hierarchies remains one of the unresolved problems in fundamental physics. While the Standard Model successfully parameterizes particle masses through Yukawa couplings, it does not predict their numerical values. In this work, we present a semi-empirical relation connecting the electron rest mass to the proton mass and the fine-structure constant . The proposed relation involves a leading scaling with a small corrective term proportional to . Using CODATA 2022 recommended values, the expression reproduces the experimentally measured electron mass within relative deviation of order . The relation is not derived from first principles but is motivated by known mass-ratio regularities involving powers of . Possible interpretations, limitations, and directions for theoretical justification are discussed.
Table of Contents
1. Introduction
2. Background and Motivation
3. Proposed Semi-Empirical Relation
4. Numerical Evaluation Using CODATA 2022 Values
5. Comparison with Simpler Approximations
6. Interpretation and Limitations
7. Relation to Other Leptonic Mass Ratios
8. Possible Theoretical Contexts
9. Future Directions
10. Conclusions
Research Objectives and Topics
This work aims to propose and evaluate a semi-empirical mathematical relation that connects the electron rest mass to the proton mass and the fine-structure constant. The research seeks to document a specific numerical regularity in fermion mass hierarchies rather than claiming a derivation from first principles.
- Analysis of fermion mass hierarchy challenges in the Standard Model.
- Evaluation of empirical scaling relations involving the fine-structure constant.
- Quantitative assessment of the proposed mass formula using CODATA 2022 constants.
- Comparison of the model's predictive accuracy against established approximations.
- Discussion on potential phenomenological origins and future theoretical investigations.
Excerpt from the Book
6. Interpretation and Limitations
It is important to emphasize that Eq. (1) is not derived from quantum field theory. The appearance of may be suggestive of higher-order radiative effects, but no explicit diagrammatic or renormalization-group argument is presented here.
Similarly, the fractional power should be regarded as a phenomenological correction rather than a theoretically motivated term. Fractional powers of coupling constants do not normally arise in perturbation theory, though they may appear in effective or non-perturbative contexts.
Summary of Chapters
1. Introduction: Discusses the longstanding puzzle of elementary particle masses and the lack of a fundamental explanation for fermion mass hierarchies within the Standard Model.
2. Background and Motivation: Reviews historical attempts to use empirical relations involving the fine-structure constant to approximate particle mass ratios.
3. Proposed Semi-Empirical Relation: Introduces the specific mathematical formula linking the electron mass, proton mass, and the fine-structure constant.
4. Numerical Evaluation Using CODATA 2022 Values: Performs a rigorous calculation of the proposed formula using current standardized physical constants.
5. Comparison with Simpler Approximations: Provides a comparative table showing that the new relation yields significantly higher numerical accuracy than previous estimates.
6. Interpretation and Limitations: Clarifies that the relation is phenomenological and lacks a direct basis in quantum field theory.
7. Relation to Other Leptonic Mass Ratios: Examines whether the scaling logic can be extended to other particles like the muon.
8. Possible Theoretical Contexts: Proposes potential areas, such as effective field theories or dimensional transmutation, where such relations might emerge.
9. Future Directions: Outlines necessary next steps, including testing against updated data and exploring non-perturbative models.
10. Conclusions: Summarizes the findings, reiterating that the formula is a useful phenomenological observation for future research into mass hierarchies.
Keywords
Electron mass, Proton mass, Fine-structure constant, Fermion mass hierarchy, Standard Model, Yukawa couplings, Particle physics, Semi-empirical relation, Numerical regularity, CODATA, Quantum field theory, Phenomenological correction, Leptonic mass, Scaling laws, Dimensional transmutation.
Frequently Asked Questions
What is the primary focus of this research?
The work focuses on identifying and testing a semi-empirical mathematical relationship between the electron mass, the proton mass, and the fine-structure constant to better understand fermion mass hierarchies.
What are the central thematic fields?
The paper bridges theoretical particle physics, precision numerical evaluation using CODATA standards, and phenomenological modeling of mass ratios.
What is the core research objective?
The goal is to document a specific numerical regularity that allows for a highly accurate approximation of the electron mass, which currently lacks an explanation from first principles.
Which methodology is employed?
The authors utilize an empirical approach, constructing a formula based on scaling factors and fractional powers, followed by a numerical verification against high-precision experimental CODATA 2022 data.
What is covered in the main body?
The main body details the formula construction, the computational verification, a comparative analysis with existing simple approximations, and a cautious discussion of the theoretical limitations.
Which keywords define this work?
Key terms include fermion mass hierarchy, electron-proton mass ratio, fine-structure constant, and phenomenological scaling.
How does the proposed formula compare to simpler models?
The proposed relation, denoted as Eq. (1), achieves a relative deviation of approximately 0.0003%, offering a significant improvement over traditional simple approximations that show deviations of over 2%.
Does the formula represent a new fundamental theory?
No, the authors explicitly state that the relation is not derived from quantum field theory and should be regarded as a numerical regularity or a phenomenological correction rather than a fundamental law.
Can this relation be applied to other particles?
The authors discuss potential extensions to other leptons like the muon, though they note that these applications are currently tentative and require further systematic study.
- Quote paper
- Fazal Rehman (Author), Derivation of a Semi-Empirical Relation Between the Electron Mass, Proton Mass, and the Fine-Structure Constant, Munich, GRIN Verlag, https://www.grin.com/document/1692650
