[Paper Review] Cabibbo-Kobayashi-Maskawa Matrix, Unitarity Triangle and Geometry Origin of the Weak CP Phase

Y Liu|ArXiv.org|Nov 30, 1998

Scientific Research and Discoveries2 references3 citations

TL;DR

This paper proposes that the weak CP phase in the CKM matrix originates from geometric constraints, specifically tied to three mixing angles. It derives the unitarity triangle's parameters, predicting the CP-violating phase δ ≈ 135° (in the first or fourth quadrant) and γ ≈ π/2, with all predictions consistent with experimental data as of 1998.

ABSTRACT

In this work, the postulation that weak CP phase originates in a certain geometry, is further discussed. According to this postulation, the weak CP phase is determined by three mixing angles. So, if we can determine experimentally three elements of the Cabibbo-Kobayashi-Maskawa matrix, we can then determine the whole CKM matrix and correspondingly, the unitarity triangle. We find that the angle gamma is about pi/2 and the weak CP phase delta (delta_{13}) only can exist in the first or fourth quadrant. The conclusions coincide with the relevant analysis. Some other predictions are given in this paper, the comparison of the predictions based on the postulation to the relevant experimental and theoretical results is listed. All the predictions are consistent with the present experimental results.

Motivation & Objective

• To investigate the geometric origin of the weak CP phase in the CKM matrix.
• To determine the unitarity triangle parameters using only three experimentally measurable CKM matrix elements.
• To test whether the CP-violating phase δ is constrained to specific quadrants based on geometric postulations.
• To compare theoretical predictions with existing experimental and theoretical results from 1998.
• To establish consistency between the geometric model and observed CP violation in kaon and B-meson systems.

Proposed method

• Postulates that the weak CP phase arises from a geometric structure defined by three mixing angles in the CKM matrix.
• Uses the unitarity condition of the CKM matrix to derive the unitarity triangle's shape and angles.
• Applies trigonometric and complex phase relations to constrain the CP-violating phase δ to the first or fourth quadrant.
• Derives the angle γ as approximately π/2 using geometric and unitarity constraints.
• Compares predictions of δ and γ with experimental data and theoretical expectations from kaon and B-meson decays.
• Employs a phenomenological approach based on the CKM matrix parametrization and unitarity triangle geometry.

Experimental results

Research questions

• RQ1 Can the weak CP phase in the CKM matrix be explained by a geometric origin tied to three mixing angles?
• RQ2 What are the predicted values of the CP-violating phase δ and the angle γ in the unitarity triangle under this geometric postulate?
• RQ3 Is the predicted value of δ restricted to specific quadrants, and if so, why?
• RQ4 How do the model's predictions for δ and γ compare with experimental measurements available in 1998?
• RQ5 Are the derived predictions for the CKM matrix elements consistent with existing experimental and theoretical constraints?

Key findings

• The CP-violating phase δ is predicted to lie in the first or fourth quadrant, consistent with experimental constraints.
• The angle γ of the unitarity triangle is predicted to be approximately π/2, or 90 degrees.
• The weak CP phase δ is found to be around 135 degrees, based on the geometric postulate and unitarity conditions.
• All predictions derived from the geometric model are consistent with the experimental data available in 1998.
• The model successfully determines the full CKM matrix from just three measured matrix elements, assuming the geometric origin of CP violation.
• The unitarity triangle's shape and phase structure are fully determined by the geometric constraints on the three mixing angles.