[Paper Review] An exact CKM matrix related to the approximate Wolfenstein form

Jihn E. Kim, Min‐Seok Seo|arXiv (Cornell University)|May 17, 2011

Particle physics theoretical and experimental studies7 citations

TL;DR

This paper proposes an exact parametrization of the CKM quark mixing matrix that replaces the approximate Wolfenstein form, based on the hierarchical structure θ₂,₃ ≈ O(θ₁²). By placing the weak CP phase δ explicitly in the (31) element as sinθ₁sinθ₂e^{iδ} and redefining the unitarity triangle with δ at the origin, the model provides a direct geometric link between the phase and CP violation, offering a clearer framework for connecting CP violation to Yukawa textures and new physics beyond the SM.

ABSTRACT

Noting the hierarchy between three mixing angles, $θ_{2,3}={\cal O}(θ_1^2)$, we present an exact form of the quark mixing matrix, replacing Wolfenstein's approximate form. In addition, we suggest to rotate the unitarity triangle, using the weak CP phase convention where the phase is located at the (31) element $\sinθ_1\sinθ_2 e^{iδ}$ while the (13) element $\sinθ_1\sinθ_3$ is real. For the $(ab)$ unitarity triangle, the base line (x-axis) is defined from the product of the first row elements, $V_{1a}V^*_{1b}$, and the angle between two sides at the origin is defined to be the phase $δ$. This is a useful definition since every Jarlskog triangle has the angle $δ$ at the origin, defined directly from the unitarity condition. It is argued that $δ$ represents the barometer of the weak CP violation, which can be used to relate it to possible Yukawa textures.

Motivation & Objective

• To develop an exact form of the CKM matrix that better reflects the observed hierarchy in quark mixing angles (θ₂,₃ ≈ O(θ₁²)).
• To redefine the unitarity triangle so that the weak CP phase δ is directly visible as the angle at the origin, improving geometric clarity.
• To provide a parametrization where vanishing of a single parameter (κ_b or κ_t) leads to vanishing of either the (13) or (31) CKM element, signaling no CP violation.
• To connect the CP-violating phase δ to the underlying Yukawa texture structure via explicit mass matrix constructions.

Proposed method

• Introduces an exact CKM matrix parametrization using three mixing angles and a single CP-violating phase δ, replacing the Wolfenstein expansion.
• Redefines the unitarity triangle with the (1a)(1b) product as the base and δ as the angle at the origin, ensuring δ is directly measurable from the matrix.
• Uses the Jarlskog invariant J_Jkg as a measure of CP violation, derived from the commutator of up- and down-type quark mass matrices.
• Constructs explicit weak eigenstate mass matrices Ŵ^(u) and Ŵ^(d) via bi-unitary transformations from diagonal mass matrices, with parameters scaled by powers of λ.
• Derives the CKM matrix as V_KS = L^(u)L^(d)†, linking it to the Yukawa textures through the transformation matrices.
• Considers two cases (R = 1 and R = L) to explore different mass matrix textures, revealing patterns of zeros that may signal underlying symmetries.

Experimental results

Research questions

• RQ1 Can an exact parametrization of the CKM matrix be constructed that better reflects the observed hierarchy of mixing angles compared to the Wolfenstein approximation?
• RQ2 How can the unitarity triangle be redefined so that the weak CP phase δ is directly visible as the angle at the origin, improving interpretability?
• RQ3 Does a parametrization exist where the vanishing of a single parameter leads to the vanishing of either the (13) or (31) CKM element, thereby signaling the absence of CP violation?
• RQ4 Can the CP-violating phase δ be systematically linked to the structure of the Yukawa textures through explicit mass matrix constructions?

Key findings

• The proposed exact CKM matrix replaces the Wolfenstein form with a parametrization where the CP phase δ is explicitly located in the (31) element as sinθ₁sinθ₂e^{iδ}, ensuring direct geometric interpretation.
• The unitarity triangle is redefined so that δ is the angle at the origin between the sides defined by V₁aV*₁b, making δ directly observable from the matrix elements.
• The Jarlskog invariant J_Jkg is shown to be proportional to λ⁶ times the product of κ_b and κ_t, confirming that CP violation arises at O(λ⁶), consistent with the Wolfenstein expansion.
• Explicit constructions of weak eigenstate mass matrices (Eqs. 15 and 16) reveal patterns of zeros (six in Eq. 15, four in Eq. 16), suggesting potential underlying symmetries in the Yukawa textures.
• The model shows that vanishing of either κ_b or κ_t leads to the vanishing of the (13) or (31) element, respectively, and thus to no CP violation, validating the parametrization as a sensitive probe of CP violation.
• The framework provides a clear path to linking the origin of CP violation to specific Yukawa texture structures in the Standard Model, potentially guiding new physics models.