[Paper Review] Constructing the CKM and PMNS matrices from mixing data

Gulsheen Ahuja, Manmohan Gupta+1 more|arXiv (Cornell University)|Nov 27, 2006

Fault Detection and Control Systems3 citations

TL;DR

This paper constructs the CKM and PMNS mixing matrices using the latest experimental data, minimizing theoretical assumptions and NP effects. It explores CKM matrix reconstruction from tree-level elements and unitarity triangle constraints, while for PMNS, it varies $s_{13}$ and the Dirac CP phase $\delta$, yielding $J_l = 0.0192 \pm 0.0079$, indicating potential CP violation in the leptonic sector with $\delta \in 15^\circ - 165^\circ$. The results provide data-driven templates for GUT-scale mass matrix models.

ABSTRACT

The CKM and PMNS matrices have been constructed based on the latest measurements, largely free from theoretical inputs as well as likely NP effects in the case of the former. To facilitate the construction of the CKM matrix in the PDG representation as well as in view of the comparatively large error in the measured value of the CP violating phase δ, the possibility of its construction from the tree level measured CKM elements has also been explored using the unitarity triangle. In view of the persistent difference between the |V_{ub}| exclusive and inclusive values, we have carried out separate analyses corresponding to these. The PMNS matrix has been constructed by incorporating the constraints due to solar and atmospheric neutrinos as well as by giving full variation to the Dirac-like CP violating phase δand considering different values of s_{13} having implications for different models of lepton mass matrices. Taking clue from quark mixing phenomenology, an analogous analysis of the leptonic unitarity triangle allows an estimate of the likely presence of CP violation in the leptonic sector.

Motivation & Objective

• To construct the CKM and PMNS matrices using only the latest experimental measurements, minimizing theoretical inputs and NP effects.
• To explore the feasibility of reconstructing the CKM matrix from tree-level measured elements $V_{us}$, $V_{cb}$, and $\delta$, given the large uncertainty in $\delta$.
• To analyze the PMNS matrix under varying $s_{13}$ values and full variation of the Dirac CP phase $\delta$, reflecting different lepton mass matrix models.
• To estimate the likely range of the Jarlskog invariant $J_l$ in the leptonic sector using the unitarity triangle area, as a proxy for CP violation.
• To provide data-based mixing matrices as a foundation for formulating GUT-scale mass matrices, free from NP and theoretical bias.

Proposed method

• Uses the PDG parametrization for both CKM and PMNS matrices, ensuring unitarity is built-in and angles are directly linked to measured elements.
• Constructs the CKM matrix using $V_{us}$, $V_{cb}$, $|V_{ub}|$ (exclusive and inclusive), and $\delta$ from data, applying unitarity triangle constraints.
• For the PMNS matrix, incorporates oscillation data for solar and atmospheric neutrinos, with $s_{13}$ varied across values relevant to different lepton mass models.
• Performs a Gaussian error propagation on matrix elements to estimate the uncertainty in the Jarlskog invariant $J_l$ in the leptonic sector.
• Calculates $J_l$ as the area of the leptonic unitarity triangle using the first two rows of the PMNS matrix, with $\delta$ and $s_{13}$ fully varied.
• Uses 3$\sigma$ confidence level inputs to derive robust estimates of $J_l$ and $\delta$ ranges, ensuring reliability against current experimental uncertainties.

Experimental results

Research questions

• RQ1 Can the CKM matrix be reliably reconstructed from tree-level measured elements and unitarity triangle constraints, despite the large uncertainty in $\delta$?
• RQ2 How do exclusive and inclusive $|V_{ub}|$ measurements affect the construction of the CKM matrix?
• RQ3 What is the impact of varying $s_{13}$ on the structure of the PMNS matrix, and how does this affect predictions for lepton mass models?
• RQ4 What is the most probable range of the Jarlskog invariant $J_l$ in the leptonic sector, given current uncertainties in $s_{13}$ and $\delta$?
• RQ5 Can a data-based PMNS matrix be constructed that is compatible with recent theoretical constructions like those by Bjorken et al. and Giunti?

Key findings

• The CKM matrix was constructed using $V_{us}$, $V_{cb}$, $|V_{ub}|$ (exclusive and inclusive), and $\delta$, yielding two distinct matrices with varying $|V_{ub}|$ values.
• The PMNS matrix was constructed for three different $s_{13}$ values, showing that $s_{13}$ significantly affects the matrix elements, especially $U_{e2}$, $U_{\mu2}$, and $U_{\tau2}$.
• The Jarlskog invariant in the leptonic sector was estimated as $J_l = 0.0192 \pm 0.0079$, indicating a non-zero, likely CP-violating phase in the lepton sector.
• The CP-violating phase $\delta$ is estimated to lie in the range $15^\circ$ to $165^\circ$, suggesting a non-trivial CP-violating phase is possible.
• The constructed PMNS matrices are fully compatible with recent theoretical models, including those by Bjorken et al. and Giunti, supporting their phenomenological viability.
• The analysis demonstrates that data-driven mixing matrices, especially with minimal theoretical input, provide robust templates for GUT-scale mass matrix model building.