[Paper Review] A remark on the Koide relation for quarks

A. Kartavtsev|arXiv (Cornell University)|Nov 2, 2011

Particle physics theoretical and experimental studies9 citations

TL;DR

This paper proposes a generalized Koide relation for quarks by summing the masses of up and down quarks across all three generations, showing that the resulting ratio $ K_q = \frac{\sum m_q}{(\sum \sqrt{m_q})^2} $ is significantly closer to the Koide limit $ 2/3 $ than individual up- or down-type quark sets. The geometric interpretation reveals that the angle between the vector of quark square roots and the unit vector is approximately $ \pi/3 $, suggesting a deeper symmetry akin to the lepton case.

ABSTRACT

The charged lepton masses obey to high precision the so-called Koide relation. We propose a generalization of this relation to quarks. It includes up and down quarks of the three generations and is numerically reasonably close to the Koide limit.

Motivation & Objective

• To explore whether the empirical Koide relation, which holds with high precision for charged leptons, can be generalized to quarks.
• To investigate if a unified mass relation for quarks—incorporating both up and down types—can yield a value close to the Koide limit $ 2/3 $, despite known deviations in individual quark sectors.
• To examine the role of running quark masses at the $ M_Z $ scale in modifying the Koide parameter and its implications for underlying symmetries.
• To provide a geometric interpretation of the generalized quark Koide relation in a six-dimensional vector space, analogous to the known geometric form for leptons.

Proposed method

• Define a generalized Koide parameter $ K_q = \frac{\sum m_q}{(\sum \sqrt{m_q})^2} $, summing over all six quarks (u, d, c, s, t, b) across three generations.
• Use experimental and running quark masses at the $ M_Z $ scale from the Particle Data Group and renormalization group calculations.
• Compare the resulting $ K_q $ with the Koide limit $ 2/3 $, and compute the deviation $ \Delta K_q = \frac{3}{2}K_q - 1 $.
• Introduce a geometric interpretation via the angle $ \theta_q $ between the vector $ (\sqrt{m_d}, \sqrt{m_u}, \sqrt{m_s}, \sqrt{m_c}, \sqrt{m_b}, \sqrt{m_t}) $ and the unit vector $ (1,1,1,1,1,1) $, using the cosine formula.
• Analyze alternative groupings such as light vs. heavy quarks (u,d,s vs. c,b,t) to test the robustness and physical intuition of the proposed generalization.
• Evaluate the behavior of $ \Delta K_q $ across energy scales, showing it crosses zero, indicating a scale where the Koide relation is approximately satisfied.

Experimental results

Research questions

• RQ1 Can a generalized Koide relation be formulated for quarks that brings the mass ratio closer to the $ 2/3 $ limit than individual up- or down-type quark sets?
• RQ2 How do running quark masses at the $ M_Z $ scale affect the deviation from the Koide limit in the generalized quark relation?
• RQ3 Does the generalized quark Koide relation admit a geometric interpretation similar to the known $ \theta = \pi/4 $ case for charged leptons?
• RQ4 Is there a specific energy scale where the generalized quark Koide relation is approximately satisfied, implying a dynamical symmetry?
• RQ5 Why does the inclusion of light quarks (u, d, s) improve the agreement with the Koide limit despite their small masses?

Key findings

• The generalized Koide parameter for all six quarks, $ K_q = \frac{\sum m_q}{(\sum \sqrt{m_q})^2} $, yields a deviation from the $ 2/3 $ limit of $ -5 \cdot 10^{-2} < \Delta K_q < -4 \cdot 10^{-2} $ using pole masses.
• Using running quark masses at the $ M_Z $ scale, the deviation becomes $ 2 \cdot 10^{-2} < \Delta K_q < 5 \cdot 10^{-2} $, indicating that $ \Delta K_q $ crosses zero, implying a scale where the Koide relation is approximately satisfied.
• The geometric angle $ \theta_q $ between the vector of quark square roots and the unit vector is approximately $ \pi/3 $, suggesting a rational multiple of $ \pi $, analogous to the $ \pi/4 $ angle in the lepton case.
• The heavy quark group (c, b, t) shows a small deviation from the Koide limit ($ -5 \cdot 10^{-3} < \Delta K_{\text{heavy}} < 1 \cdot 10^{-2} $), while the light quark group (u, d, s) shows a larger deviation ($ -20 \cdot 10^{-2} < \Delta K_{\text{light}} < -6 \cdot 10^{-2} $), making the full sum more balanced.
• The running effects increase the deviation from the Koide limit for individual quark sets, but for the full quark sum, they shift $ \Delta K_q $ toward zero, indicating a possible fixed point in the renormalization group flow.