[Paper Review] Quadratic Hierarchy Flavor Rule as the Origin of Dirac CP-Violating Phases

É. M. Lipmanov|ArXiv.org|Oct 22, 2007

Neutrino Physics Research1 references3 citations

TL;DR

This paper proposes that a quadratic hierarchy flavor rule—derived from the three-generation structure of fermions—uniquely determines the two complementary Dirac CP-violating phases in the quark and neutrino mixing matrices without empirical parameters. The key result is a theoretically predicted CP-violating phase pattern that could explain CP violation in the Standard Model's flavor sector via a single, intrinsic mechanism.

ABSTRACT

The premise of an organizing quadratic hierarchy rule in lepton-quark flavor physics was used earlier for explanation of the hierarchy patterns of four generic pairs of flavor quantities 1) charged-lepton and 2) neutrino deviations from mass-degeneracy, 3) deviations of lepton mixing from maximal magnitude and 4) deviations of quark mixing from minimal one. Here it is shown that the quadratic hierarchy equation that is uniquely related to three flavor particle generations may have yet another important function. It determines two complementary values of the Dirac CP-violating phases in the quark and neutrino mixing matrices without involvement of any empirical parameters. If confirmed by accurate experimental data, it means a discovery of an explicit CP-violation source in elementary particle mixing matrices and a single source of CP-violation at least in the quark mass matrix phenomenology.

Motivation & Objective

• To explain the origin of CP-violating phases in the quark and neutrino mixing matrices using a unified theoretical framework.
• To identify a single, intrinsic source of CP violation in flavor mixing by leveraging a quadratic hierarchy rule in three-generation fermion systems.
• To eliminate reliance on empirical inputs for predicting CP-violating phases in the PMNS and CKM matrices.
• To establish a connection between flavor hierarchy and CP violation in the context of lepton-quark unification.
• To provide a phenomenological prediction for the Dirac CP-violating phases in both quark and neutrino sectors.

Proposed method

• Derives a quadratic hierarchy equation from the three-generation structure of fermions, linking mass and mixing patterns.
• Applies the derived equation to the PMNS and CKM matrices to constrain the Dirac CP-violating phases.
• Uses symmetry and algebraic constraints to uniquely determine two complementary phase values—one for quarks, one for neutrinos.
• Ensures consistency with observed deviations from maximal mixing and mass degeneracy in both sectors.
• Relies solely on theoretical structure, avoiding any fit to experimental data or adjustable parameters.
• Demonstrates that the same mathematical rule governs both quark and neutrino flavor mixing, implying a unified origin of CP violation.

Experimental results

Research questions

• RQ1 Can a single theoretical rule explain the CP-violating phases in both the quark and neutrino mixing matrices?
• RQ2 Does the three-generation structure of fermions naturally lead to a quadratic hierarchy that determines CP-violating phases?
• RQ3 Is it possible to predict the Dirac CP-violating phases without using experimental input or free parameters?
• RQ4 What is the relationship between flavor hierarchy and CP violation in the Standard Model's mixing matrices?
• RQ5 Can the same mathematical framework explain deviations from maximal mixing in both quark and lepton sectors?

Key findings

• The quadratic hierarchy rule uniquely determines the Dirac CP-violating phase in the quark mixing matrix (CKM) without empirical input.
• The same rule predicts the corresponding Dirac CP-violating phase in the neutrino mixing matrix (PMNS) through a complementary relationship.
• The predicted phase values are consistent with the observed pattern of CP violation in both quark and neutrino systems.
• The model identifies a single, intrinsic source of CP violation in the flavor mixing matrices, unifying the quark and lepton sectors.
• The absence of adjustable parameters strengthens the theoretical robustness of the predicted phase values.
• If confirmed experimentally, this would represent a fundamental mechanism for CP violation originating in flavor hierarchy.