Physlib.Particles.FlavorPhysics.CKMMatrix.StandardParameterization.Basic

Given four real parameters $\theta_{12}, \theta_{13}, \theta_{23}$, and $\delta_{13}$, this definition constructs the standard parameterization of the Cabibbo–Kobayashi–Maskawa (CKM) matrix as a $3 \times 3$ complex matrix $V$: \[ V = \begin{pmatrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i \delta_{13}} \\ -s_{12} c_{23} - c_{12} s_{13} s_{23} e^{i \delta_{13}} & c_{12} c_{23} - s_{12} s_{13} s_{23} e^{i \delta_{13}} & s_{23} c_{13} \\ s_{12} s_{23} - c_{12} s_{13} c_{23} e^{i \delta_{13}} & -c_{12} s_{23} - s_{12} s_{13} c_{23} e^{i \delta_{13}} & c_{23} c_{13} \end{pmatrix} \] where $c_{ij} = \cos \theta_{ij}$, $s_{ij} = \sin \theta_{ij}$, and $i$ denotes the imaginary unit. This matrix represents the mixing between the three generations of quarks in the Standard Model.

For any real parameters $\theta_{12}, \theta_{13}, \theta_{23}$, and $\delta_{13}$, the standard $3 \times 3$ CKM matrix $V$, defined by the parameterization: \[ V = \begin{pmatrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i \delta_{13}} \\ -s_{12} c_{23} - c_{12} s_{13} s_{23} e^{i \delta_{13}} & c_{12} c_{23} - s_{12} s_{13} s_{23} e^{i \delta_{13}} & s_{23} c_{13} \\ s_{12} s_{23} - c_{12} s_{13} c_{23} e^{i \delta_{13}} & -c_{12} s_{23} - s_{12} s_{13} c_{23} e^{i \delta_{13}} & c_{23} c_{13} \end{pmatrix} \] where $c_{ij} = \cos \theta_{ij}$ and $s_{ij} = \sin \theta_{ij}$, is a unitary matrix. This is expressed by the relation $V^\dagger V = I$, where $V^\dagger$ denotes the conjugate transpose (Hermitian conjugate) of $V$ and $I$ is the $3 \times 3$ identity matrix.

Given four real parameters $\theta_{12}$, $\theta_{13}$, $\theta_{23}$, and $\delta_{13}$, this definition constructs a CKM matrix (a $3 \times 3$ unitary matrix) using the standard parameterization: \[ V = \begin{pmatrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i \delta_{13}} \\ -s_{12} c_{23} - c_{12} s_{13} s_{23} e^{i \delta_{13}} & c_{12} c_{23} - s_{12} s_{13} s_{23} e^{i \delta_{13}} & s_{23} c_{13} \\ s_{12} s_{23} - c_{12} s_{13} c_{23} e^{i \delta_{13}} & -c_{12} s_{23} - s_{12} s_{13} c_{23} e^{i \delta_{13}} & c_{23} c_{13} \end{pmatrix} \] where $c_{ij} = \cos \theta_{ij}$, $s_{ij} = \sin \theta_{ij}$, and $i$ denotes the imaginary unit. This matrix represents the mixing between the three generations of quarks in the Standard Model.

For any real parameters $\theta_{12}, \theta_{13}, \theta_{23}$, and $\delta_{13}$, let $V$ be the CKM matrix in the standard parameterization. Let $\mathbf{v}_u$, $\mathbf{v}_c$, and $\mathbf{v}_t$ denote the three row vectors of $V$ corresponding to the up, charm, and top quarks, respectively. Then the row vector $\mathbf{v}_t$ is equal to the cross product of the complex conjugates of $\mathbf{v}_u$ and $\mathbf{v}_c$: \[ \mathbf{v}_t = \mathbf{v}_u^* \times \mathbf{v}_c^* \] where $\mathbf{v}^*$ denotes the element-wise complex conjugate and $\times$ denotes the standard three-dimensional cross product.

For any real parameters $\theta_{12}, \theta_{13}, \theta_{23}$, and $\delta_{13}$ such that $\sin \theta_{12} \geq 0$, $\cos \theta_{13} \geq 0$, $\sin \theta_{23} \geq 0$, and $\cos \theta_{12} \geq 0$, let $V$ be the CKM matrix constructed using the standard parameterization. Then the quantity $V_{us}V_{ub}V_{cd}\text{Sq}$ evaluated for the phase-equivalence class $\llbracket V \rrbracket$ is equal to $$\sin^2 \theta_{12} \cos^2 \theta_{13} \sin^2 \theta_{13} \sin^2 \theta_{23}$$ where the equivalence class $\llbracket V \rrbracket$ is defined by the rephasing of fermion fields.

For real parameters $\theta_{12}, \theta_{13}, \theta_{23}$, and $\delta_{13}$ such that $\sin \theta_{12} \geq 0$, $\cos \theta_{13} \geq 0$, $\sin \theta_{23} \geq 0$, and $\cos \theta_{12} \geq 0$, let $V$ be the CKM matrix constructed using the standard parameterization. Then the quantity $\text{mulExp}\delta_{13}$ evaluated for the phase-equivalence class $\llbracket V \rrbracket$ is equal to: $$\sin \theta_{12} \cos^2 \theta_{13} \sin \theta_{23} \sin \theta_{13} \cos \theta_{12} \cos \theta_{23} e^{i \delta_{13}}$$ where $i$ is the imaginary unit and the equivalence class $\llbracket V \rrbracket$ is defined by the rephasing of fermion fields.