Renormalization‑Group Flow Constraints on Charged‑Lepton Mass Ratios from a Unified UV Boundary Condition

In the paper's words

The hierarchy of charged‑lepton masses (m_e, m_μ, m_τ) lacks a dynamical explanation within the Standard Model, limiting the predictive power of grand‑unified model building. A concrete gap is the absence of a demonstrated mechanism by which a simple ultraviolet (UV) boundary condition on Yukawa couplings can generate the observed ratios after renormalization‑group (RG) evolution. We adopt the minimal hypothesis that at a putative grand‑unification scale M_rm GUT=10¹6 GeV all three charged‑lepton Yukawa couplings are equal, y_e(M_rm GUT)=y_μ(M_rm GUT)=y_τ(M_rm GUT)≡ y₀.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The integration yields a universal suppression factor $S=\exp[-0.235925]\approx0.7899$, implying identical Yukawas at $M_Z$ and therefore a predicted mass ratio $m_\mu/m_e=1$.

   from Abstract

2. We obtain a concrete numerical prediction $m_\mu/m_e=1$ and compare it with the experimental ratio $206.768$ (Section 5).

   from 1. Introduction

3. The analytic RG integration predicts that a universal UV Yukawa texture yields identical charged‑lepton masses at the electroweak scale, i.e. $$ \boxed{\displaystyle \frac{m_\mu}{m_e}=1 } . \tag{20} $$ The observed ratio is $206.768$, a deviation of more than two orders of magnitude.

   from 5. Results

4. The universal suppression factor $S=0.788999$ reduces all three Yukawas by the same amount, confirming that gauge‑coupling effects alone cannot generate the hierarchy.

   from 5. Results

5. Neglected Yukawa self‑terms : For the tau lepton the term $\frac{3}{2}y_\tau^2$ is not negligible; its inclusion would introduce a mild non‑universality but not enough to generate a factor of $200$.

   from 6. Discussion

6. If a future full two‑loop numerical integration were to produce a universal factor $S\approx0.005$, the predicted ratio would approach the observed value; however, such a drastic change would require gauge couplings far outside the experimentally measured range, indicating a breakdown of the SM RG framework.

   from 6. Discussion

How it is done

1. Counting RG flows as solitons interpolating between fixed points provides a topological classification of possible trajectories and highlights that distinct “sectors’’ of flows can be distinguished by invariants such as the $\mu$‑theorem [6].

   from 2. Background and Related Work

2. We adopt the minimal hypothesis (H1): at a UV scale $M_{\rm GUT}=10^{16}\,\text{GeV}$ the three charged‑lepton Yukawa couplings are exactly equal, $$ y_e(M_{\rm GUT})=y_\mu(M_{\rm GUT})=y_\tau(M_{\rm GUT})\equiv y_0 . \tag{1} $$ All other SM parameters (gauge couplings, Higgs quartic, etc.) are taken from standard two‑loop fits at $M_{\rm GUT}$, but for the present analytic treatment we retain only the one‑loop contributions to the charged‑lepton Yukawas.

   from 3. Methods

3. We now evaluate the numerical value of $S$ step by step, explicitly showing every arithmetic operation.

   from 4. Analysis

4. If a future full two‑loop numerical integration were to produce a universal factor $S\approx0.005$, the predicted ratio would approach the observed value; however, such a drastic change would require gauge couplings far outside the experimentally measured range, indicating a breakdown of the SM RG framework.

   from 6. Discussion

What would falsify it

1. The principal result therefore falsifies the hypothesis that a universal UV Yukawa texture can reproduce the lepton hierarchy at one‑loop order, redirecting future model‑building efforts toward non‑trivial UV textures or higher‑order RG effects.

   from Abstract

2. The charged‑lepton mass hierarchy—$m_e\approx0.511\,\text{MeV}$, $m_\mu\approx105.66\,\text{MeV}$, $m_\tau\approx1.777\,\text{GeV}$—poses a concrete real‑world problem for particle‑physics phenomenology because it prevents a first‑principles prediction of basic observables such as decay rates, magnetic moments, and lepton‑flavour‑violating amplitudes.

   from 1. Introduction

3. Experimental programmes at intensity‑frontier facilities (e.g., Mu2e, COMET) allocate billions of dollars to search for processes whose rates depend sensitively on the underlying Yukawa structure; the inability to predict the hierarchy therefore inflates systematic uncertainties and hampers the design of optimal detector configurations.

   from 1. Introduction

Where the claim stops

1. One‑loop approximation : Higher‑order gauge and Yukawa contributions (two‑loop terms, threshold corrections) are omitted, potentially altering the suppression factor.

   from 6. Discussion

2. Averaged gauge couplings : Replacing the full running by arithmetic means introduces a systematic error; a full numerical integration would refine $S$ but cannot change the qualitative conclusion that the factor is common to all three leptons.

   from 6. Discussion

3. If a future full two‑loop numerical integration were to produce a universal factor $S\approx0.005$, the predicted ratio would approach the observed value; however, such a drastic change would require gauge couplings far outside the experimentally measured range, indicating a breakdown of the SM RG framework.

   from 6. Discussion

Where to read

1. Abstract172 words · 1 min
2. 1. Introduction279 words · 1 min
3. 2. Background and Related Work446 words · 2 min
4. 3. Methods305 words · 1 min
5. 4. Analysis154 words · 1 min
6. 5. Results141 words · 1 min
7. 6. Discussion254 words · 1 min
8. 7. Practical Implications and Engineering Pathways299 words · 1 min
9. 8. Conclusion122 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report154 words · 1 min
12. Appendix B. Claim attribution118 words · 1 min