Towards establishing the second $b$-flavored CKM unitarity triangle

TL;DR

This paper explores the subtle differences between two $b$-flavored CKM unitarity triangles, proposing a new method to test the CKM model's consistency and investigate potential new physics through their geometric relationship.

Contribution

It introduces a generalized Wolfenstein parametrization to analyze the second $b$-flavored unitarity triangle and discusses experimental prospects for its establishment.

Findings

A new geometric relation places the apexes of the two triangles on the same circular arc.

Differences between the apexes are of order $\\lambda^2$, with $\lambda \simeq 0.22$.

Shape insensitivity to two-loop RG effects up to order $\\lambda^4$.

Abstract

Some fine differences between the twin $b$-flavored unitarity triangles are calculated by means of a generalized Wolfenstein parametrization of the CKM matrix, and a possibility of experimentally establishing the second triangle is briefly discussed. We find that the apexes of these two triangles, characterized respectively by $(\overline{\rho}, \overline{\eta})$ and $(\widetilde{\rho}, \widetilde{\eta})$, are located on the same circular arc in the complex plane. This observation provides us with a new way to test consistency of the CKM picture of CP violation in the quark sector and probe possible new physics. The differences between the apexes (i.e., $\widetilde{\rho} - \overline{\rho}$ and $\widetilde{\eta} - \overline{\eta}$) are found to be of ${\cal O}(\lambda^2)$ with $\lambda \simeq 0.22$ being the Wolfenstein expansion parameter, and the shapes of these two triangles are found to be insensitive to the two-loop renormalization-group-equation running effects up to the accuracy of ${\cal O}\left(\lambda^4\right)$.