[Paper Review] What Physics Does The Charged Lepton Mass Relation Tell Us?

Yoshio Koide|arXiv (Cornell University)|Sep 3, 2018

Computational Physics and Python Applications4 citations

TL;DR

This paper investigates why the charged lepton mass relation $ K = \frac{m_e + m_μ + m_τ}{(\sqrt{m_e} + \sqrt{m_μ} + \sqrt{m_τ})^2} = \frac{2}{3} $ holds with exceptional accuracy using pole masses, despite quantum corrections predicting it should only hold for running masses. It reviews Sumino's mechanism and a modified model with inverted family gauge boson mass hierarchy to cancel logarithmic divergences, and shows that the relation can be naturally preserved in a supersymmetric framework, resolving a long-standing theoretical puzzle.

ABSTRACT

The charged lepton mass relation $K \equiv (m_e +m_μ+m_τ)/(\sqrt{m_e} %+\sqrt{m_μ} +\sqrt{m_τ})^2= 2/3 $ is excellently satisfied by observed masses (pole masses). However, the formula $K=2/3$ should be never satisfied with the observed charged lepton masses. We will review a mechanism by proposed by Sumino and recent related topics.

Motivation & Objective

• To resolve the theoretical puzzle of why the charged lepton mass relation $ K = \frac{2}{3} $ holds with high precision using pole masses, despite quantum corrections predicting it should apply only to running masses.
• To examine the limitations of Sumino's original mechanism, which relies on anomaly-non-invariant $ (\mathbf{3}, \mathbf{3}^*) $ family symmetry assignments and leads to forbidden $ \Delta N_{\text{family}} = 2 $ decays.
• To propose and analyze a modified Sumino model with $ (\mathbf{3}, \mathbf{3}) $ assignment and inverted mass hierarchy $ M_{ii}^2 \propto m_i^{-1} $, enabling cancellation of logarithmic divergences without anomalies.
• To explore the viability of the $ K $ and $ \kappa $ relations in a supersymmetric framework, where no vertex corrections disturb the relations, thus naturally preserving them.
• To assess phenomenological implications, including low-scale family gauge boson masses and signals in $ \mu $-to-$ e $ conversion and LHC production.

Proposed method

• Derives the mass relation $ K = \frac{2}{3} $ from a scalar nonet $ \Phi $ with vacuum expectation value $ \langle\Phi\rangle = v_0 \text{diag}(z_1, z_2, z_3) $, assuming a potential with $ \mu^2 $, $ \lambda $, and $ \lambda' $ couplings.
• Imposes the condition $ \partial V / \partial \Phi = 0 $, leading to two constraints: $ \mu^2 + \lambda[\Phi\Phi] + \lambda'[\Phi]^2 = 0 $ and $ [\Phi\Phi] - \frac{2}{3}[\Phi]^2 = 0 $, the latter yielding $ K = \frac{2}{3} $.
• Applies the QED radiative correction formula $ m_i(\mu) = m_i^{\text{pole}} \left\{1 - \frac{\alpha(\mu)}{\pi} \left(1 + \frac{3}{4} \log \frac{\mu^2}{(m_i^{\text{pole}})^2} \right) \right\} $, showing that pole masses violate $ K = \frac{2}{3} $ unless corrected.
• Introduces Sumino's mechanism using U(3) family gauge bosons $ A_i^j $ with masses $ M_{ij}^2 \propto (m_i + m_j) $, where $ \log M_{ii}^2 $ terms cancel $ \log m_i^2 $ divergences via opposite-sign couplings to $ e_L $ and $ e_R $.
• Proposes a modified model with $ (e_L, e_R) = (\mathbf{3}, \mathbf{3}) $ and inverted hierarchy $ M_{ii}^2 \propto m_i^{-1} $, so that $ \log M_{ii}^2 \propto -\log m_i $, enabling cancellation without anomaly issues.
• Re-derives the $ K $ and $ \kappa $ relations in a supersymmetric framework, where no vertex corrections exist, thus preserving the relations exactly at all energy scales.

Experimental results

Research questions

• RQ1 Why does the charged lepton mass relation $ K = \frac{2}{3} $ hold so precisely with pole masses, despite quantum corrections that should break this relation?
• RQ2 What is the origin of the apparent coincidence between the observed pole masses and the theoretical $ K = \frac{2}{3} $ relation, given that the formula is derived for running masses?
• RQ3 How can the logarithmic divergences in the QED correction to the lepton mass be canceled to preserve the $ K $-relation in a non-SUSY framework?
• RQ4 Can a viable model be constructed that cancels the $ \log m_i $ terms without introducing anomalies or forbidden $ \Delta N_{\text{family}} = 2 $ decays?
• RQ5 Does the $ K $ and $ \kappa $ relation remain intact in a supersymmetric model, and if so, why is this significant for the stability of the relation?

Key findings

• The observed value of $ K(m_{ei}^{\text{obs}}) = \frac{2}{3} \times (0.999989 \pm 0.000014) $ is consistent with the theoretical prediction to within 1.4 × 10⁻⁵, indicating a non-accidental physical origin.
• Using pole masses, the relation yields $ K(m_{ei}^{\text{run}}) = \frac{2}{3} \times (1.00189 \pm 0.00002) $ at $ \mu = m_Z $, showing a significant deviation that signals a theoretical problem.
• Sumino's mechanism cancels the $ \log m_i $ divergence via $ \log M_{ii} $ terms, but requires $ (e_L, e_R) = (\mathbf{3}, \mathbf{3}^*) $, leading to anomalies and forbidden decays.
• The modified Sumino model with $ (e_L, e_R) = (\mathbf{3}, \mathbf{3}) $ and $ M_{ii}^2 \propto m_i^{-1} $ achieves the same cancellation without anomalies, allowing for a viable phenomenology.
• In the modified model, the lightest family gauge boson $ A_3^3 $ can have a mass of a few TeV if quark family numbers are inverted, weakening constraints from $ K^0 $-$ \bar{K}^0 $ and $ D^0 $-$ \bar{D}^0 $ mixing.
• Very recently, the $ K $ and $ \kappa $ relations were successfully re-derived in a supersymmetric framework, where no vertex corrections exist, thus naturally preserving the relations at all energy scales.