Physlib.Particles.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $s_{12}$, which corresponds to $\sin \theta_{12}$ in the standard parameterization. It is defined as the ratio of the absolute value of the matrix element $|V_{us}|$ to the square root of the sum of the squares of the absolute values $|V_{ud}|$ and $|V_{us}|$: $$s_{12} = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$$ where $|V_{ud}|$ and $|V_{us}|$ are the magnitudes of the entries in the first row corresponding to the transitions from the up quark to the down and strange quarks, respectively.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $s_{13}$, which corresponds to $\sin \theta_{13}$ in the standard parameterization. It is defined as the absolute value of the matrix element in the first row and third column, $|V_{ub}|$, for any representative matrix $V$ in the equivalence class.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $s_{23}$, which corresponds to $\sin \theta_{23}$ in the standard parameterization. It is defined as: \[ s_{23} = \begin{cases} |V_{cd}| & \text{if } |V_{ub}| = 1 \\ \frac{|V_{cb}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}} & \text{if } |V_{ub}| \neq 1 \end{cases} \] where $|V_{ij}|$ denotes the magnitude of the CKM matrix element corresponding to quarks $i$ and $j$. This value is independent of the choice of representative $V \in [V]$.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $\theta_{12} \in \mathbb{R}$, which represents the mixing angle between the first and second generations in the standard parameterization. It is defined as: \[ \theta_{12} = \arcsin(s_{12}) \] where $s_{12}$ is the parameter calculated from the magnitudes of the matrix elements of $V$ as $s_{12} = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $\theta_{13} \in \mathbb{R}$, which represents the mixing angle between the first and third generations in the standard parameterization. It is defined as: \[ \theta_{13} = \arcsin(s_{13}) \] where $s_{13}$ is the sine of the mixing angle, calculated as the magnitude of the matrix element in the first row and third column, $|V_{13}|$ (or $|V_{ub}|$).

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $\theta_{23} \in \mathbb{R}$, which represents the mixing angle between the second and third generations in the standard parameterization. It is defined as: \[ \theta_{23} = \arcsin(s_{23}) \] where $s_{23}$ is the sine of the mixing angle calculated from the magnitudes of the matrix elements of $V$.

For an equivalence class of a CKM matrix $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $c_{12}$, defined as the cosine of the mixing angle $\theta_{12}$ in the standard parameterization: \[ c_{12} = \cos(\theta_{12}) \] where $\theta_{12}$ is the mixing angle between the first and second generations.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $c_{13}$, which is defined as the cosine of the mixing angle $\theta_{13}$ in the standard parameterization: \[ c_{13} = \cos \theta_{13} \] where $\theta_{13}$ is the mixing angle between the first and third generations of quarks.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the real number $c_{23}$, which is the cosine of the mixing angle $\theta_{23}$ in the standard parameterization. It is defined as: \[ c_{23} = \cos \theta_{23} \] where $\theta_{23}$ is the mixing angle between the second and third generations of quarks.

For an equivalence class $[V]$ of CKM matrices (defined as $3 \times 3$ unitary matrices modulo diagonal phase rephasing), this function returns the real-valued CP-violating phase $\delta_{13}$ used in the standard parameterization. Mathematically, it is defined as the argument of the complex invariant: $$\delta_{13} = \arg \left( V_{us} V_{cb} V_{ub}^* V_{cs}^* + \frac{|V_{us}|^2 |V_{ub}|^2 |V_{cb}|^2}{|V_{ud}|^2 + |V_{us}|^2} \right)$$ where $V_{ij}$ are the elements of the representative matrix $V$.

For any equivalence class of CKM matrices $V$ under the phase-rephasing equivalence relation, the parameter $s_{12} = \sin \theta_{12}$ is non-negative, i.e., $s_{12} \geq 0$. This parameter is defined as $$s_{12} = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$$ where $|V_{ud}|$ and $|V_{us}|$ are the absolute values of the entries in the first row of the CKM matrix corresponding to transitions from the up quark to the down and strange quarks, respectively.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the parameter $s_{13} = \sin \theta_{13}$ is non-negative, i.e., $s_{13} \geq 0$. This parameter is defined as the absolute value of the matrix element in the first row and third column, $|V_{ub}|$.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the parameter $s_{23} = \sin \theta_{23}$ is non-negative, i.e., $s_{23} \geq 0$. This parameter is defined as: \[ s_{23} = \begin{cases} |V_{cd}| & \text{if } |V_{ub}| = 1 \\ \frac{|V_{cb}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}} & \text{if } |V_{ub}| \neq 1 \end{cases} \] where $|V_{ij}|$ represents the magnitude of the CKM matrix element corresponding to quarks $i$ and $j$.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the parameter $s_{12} = \sin \theta_{12}$ satisfies $s_{12} \leq 1$. The parameter $s_{12}$ is defined as $$s_{12} = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$$ where $|V_{ud}|$ and $|V_{us}|$ are the absolute values of the matrix elements corresponding to the transitions from the up quark to the down and strange quarks, respectively.

For any equivalence class of CKM matrices $V$ under the phase-rephasing equivalence relation, the parameter $s_{13} = \sin \theta_{13}$ (defined as the absolute value of the matrix element $|V_{13}|$) satisfies $s_{13} \leq 1$.

For any CKM matrix (viewed as an equivalence class under the phase-rephasing equivalence relation), the parameter $s_{23} = \sin \theta_{23}$ satisfies the inequality $s_{23} \leq 1$.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the mixing angle $\theta_{12}$ and the parameter $s_{12}$ satisfy the relation: $$\sin \theta_{12} = s_{12}$$ where $s_{12}$ is defined in terms of the CKM matrix elements as $s_{12} = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$, and $\theta_{12}$ is defined as $\arcsin(s_{12})$.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the mixing angle $\theta_{13}$ and the parameter $s_{13}$ satisfy the relation: $$\sin \theta_{13} = s_{13}$$ where $s_{13}$ is defined as the absolute value of the CKM matrix element $|V_{13}|$ (also denoted $|V_{ub}|$), and the angle $\theta_{13}$ is defined as $\arcsin(s_{13})$.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the sine of the mixing angle $\theta_{23}$ is equal to the parameter $s_{23}$, where $\theta_{23} = \arcsin(s_{23})$. That is, $$\sin \theta_{23} = s_{23}$$ where $s_{23}$ is defined based on the magnitudes of the CKM matrix elements $|V_{ij}|$.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the complex sine of the mixing angle $\theta_{12}$ is equal to the parameter $s_{12}$: $$\sin \theta_{12} = s_{12}$$ where $s_{12}$ is the real parameter defined by the magnitudes of the matrix elements as $s_{12} = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$, and $\theta_{12}$ is the mixing angle defined as $\theta_{12} = \arcsin(s_{12})$.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the complex sine of the mixing angle $\theta_{13}$ is equal to the parameter $s_{13}$ (defined as the magnitude of the matrix element $|V_{13}|$). That is, $$\sin(\theta_{13}) = s_{13}$$ where $\sin$ is the sine function for complex numbers.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the complex sine of the mixing angle $\theta_{23}$ is equal to the real parameter $s_{23}$: $$\sin \theta_{23} = s_{23}$$ where $s_{23}$ is the parameter defined by the magnitudes of the matrix elements $|V_{ij}|$, and $\theta_{23}$ is the mixing angle defined as $\theta_{23} = \arcsin(s_{23})$.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the absolute value of the complex sine of the mixing angle $\theta_{12}$ is equal to the real sine of $\theta_{12}$: $$|\sin(\theta_{12})| = \sin(\theta_{12})$$ where $\theta_{12}$ is the real mixing angle defined as $\theta_{12} = \arcsin(s_{12})$, and $s_{12}$ is the non-negative parameter defined by the magnitudes of the matrix elements $|V_{ud}|$ and $|V_{us}|$ as: $$s_{12} = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$$

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the complex absolute value (norm) of the sine of the mixing angle $\theta_{13}$ is equal to the real sine of $\theta_{13}$. That is, $$|\sin \theta_{13}| = \sin \theta_{13}$$ where $\theta_{13}$ is the mixing angle associated with the equivalence class $[V]$.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the absolute value of the complex sine of the mixing angle $\theta_{23}$ is equal to the real sine of $\theta_{23}$: $$|\sin \theta_{23}| = \sin \theta_{23}$$ where $\theta_{23}$ is the mixing angle defined as $\theta_{23} = \arcsin(s_{23})$, and $s_{23}$ is the non-negative real parameter representing the sine of the mixing angle in the standard parameterization.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, if the magnitude of the matrix element $|V_{ub}|$ is equal to $1$, then the parameter $s_{12}$ (which represents $\sin \theta_{12}$ in the standard parameterization) is equal to $0$. The parameter $s_{12}$ is defined as the ratio of the absolute value $|V_{us}|$ to the square root of the sum of the squares of the absolute values $|V_{ud}|$ and $|V_{us}|$: $$s_{12} = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$$ where $|V_{ud}|$ and $|V_{us}|$ are the magnitudes of the entries in the first row corresponding to transitions from the $u$ quark to the $d$ and $s$ quarks, respectively.

Let $[V]$ be an equivalence class of CKM matrices under the phase-rephasing equivalence relation. If the absolute value of the matrix element $|V_{ub}|$ is equal to $1$, then the parameter $s_{13}$ in the standard parameterization (which represents $\sin \theta_{13}$) is equal to $1$.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, if the magnitude of the matrix element $|V_{ub}|$ is equal to $1$, then the parameter $s_{23}$ (which represents $\sin \theta_{23}$ in the standard parameterization) is equal to the magnitude of the matrix element $|V_{cd}|$.

Let $[V]$ be an equivalence class of CKM matrices under the phase-rephasing equivalence relation. If the magnitude of the matrix element $|V_{ub}|$ is not equal to $1$, then the standard parameter $s_{23}$ (representing $\sin \theta_{23}$) is given by: $$s_{23} = \frac{|V_{cb}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$$ where $|V_{ij}|$ denotes the absolute value of the matrix element for the corresponding quarks.

Let $[V]$ be an equivalence class of CKM matrices under the phase-rephasing equivalence relation. For the standard parameterization of the CKM matrix, the cosine of the mixing angle $\theta_{12}$ is equal to the parameter $c_{12}$: \[ \cos(\theta_{12}) = c_{12} \] where $\theta_{12}$ is the mixing angle between the first and second generations of quarks.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the cosine of the mixing angle $\theta_{13}$ is equal to the parameter $C_{13}$: \[ \cos(\theta_{13}) = C_{13} \] where $\theta_{13}$ is the mixing angle between the first and third generations of quarks and the CKM matrices are $3 \times 3$ unitary matrices. Two matrices $U, V$ are equivalent if there exist diagonal phase matrices $P$ and $Q$ such that $U = PVQ$.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the cosine of the mixing angle $\theta_{23}$ is equal to the parameter $C_{23}$: \[ \cos(\theta_{23}) = C_{23} \] where the CKM matrices are $3 \times 3$ unitary matrices and the equivalence relation $\sim$ is defined by $U \sim V$ if $U = P V Q$ for some diagonal phase matrices $P$ and $Q$.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the absolute value of the cosine of the mixing angle $\theta_{12}$ is equal to the cosine of $\theta_{12}$: \[ |\cos(\theta_{12})| = \cos(\theta_{12}) \] where $\theta_{12}$ is the mixing angle between the first and second quark generations in the standard parameterization.

For any equivalence class $[V]$ of a CKM matrix under the phase-rephasing equivalence relation, let $\theta_{13}$ be the mixing angle between the first and third generations in the standard parameterization. The absolute value of the cosine of this angle satisfies: \[ |\cos \theta_{13}| = \cos \theta_{13} \] This identity reflects that $\cos \theta_{13}$ is non-negative, consistent with the definition of $\theta_{13}$ as $\arcsin(|V_{13}|)$ where $\theta_{13} \in [-\pi/2, \pi/2]$.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, let $\theta_{23}$ be the mixing angle between the second and third quark generations. The magnitude of the cosine of this angle is equal to the cosine itself: \[ |\cos \theta_{23}| = \cos \theta_{23} \] This implies that $\cos \theta_{23} \geq 0$ for CKM matrices in the standard parameterization.

For any equivalence class $[V]$ of a CKM matrix under the phase-rephasing equivalence relation, let $s_{12}$ and $c_{12}$ be the real parameters representing $\sin \theta_{12}$ and $\cos \theta_{12}$ in the standard parameterization. Then, these parameters satisfy the identity: $$s_{12}^2 + c_{12}^2 = 1$$

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the parameters $s_{13}$ and $c_{13}$ satisfy the identity \[ s_{13}^2 + c_{13}^2 = 1 \] where $s_{13}$ and $c_{13}$ are defined as $\sin \theta_{13}$ and $\cos \theta_{13}$ respectively, and $\theta_{13}$ is the mixing angle between the first and third generations of quarks.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, let $s_{23}$ and $c_{23}$ be the real-valued parameters representing $\sin \theta_{23}$ and $\cos \theta_{23}$ respectively in the standard parameterization. These parameters satisfy the identity $$s_{23}^2 + c_{23}^2 = 1$$ where $\theta_{23}$ is the mixing angle between the second and third quark generations.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, if the magnitude of the matrix element $|V_{ub}|$ (the entry in the first row and third column) is equal to $1$, then the parameter $c_{12}$, which represents $\cos \theta_{12}$ in the standard parameterization, is equal to $1$.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, if the absolute value of the matrix element $|V_{ub}|$ is equal to $1$, then the parameter $c_{13}$ (which represents $\cos \theta_{13}$ in the standard parameterization) is equal to $0$.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, if the magnitude of the matrix element $|V_{ub}|$ is not equal to $1$, then the parameter $c_{12} = \cos \theta_{12}$ in the standard parameterization is given by: $$c_{12} = \frac{|V_{ud}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$$ where $|V_{ud}|$ and $|V_{us}|$ are the magnitudes of the matrix elements corresponding to the $u \to d$ and $u \to s$ quark transitions, respectively.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the parameter $c_{13}$, which represents $\cos \theta_{13}$ in the standard parameterization, is equal to the square root of the sum of the squares of the absolute values of the matrix elements $|V_{ud}|$ and $|V_{us}|$: $$c_{13} = \sqrt{|V_{ud}|^2 + |V_{us}|^2}$$ where $|V_{ud}|$ and $|V_{us}|$ are the magnitudes of the entries in the first row and the first and second columns, respectively.

Let $[V]$ be an equivalence class of CKM matrices under the phase-rephasing equivalence relation. If the magnitude of the matrix element $|V_{ub}|$ is not equal to $1$, then the standard parameter $c_{23}$ (representing $\cos \theta_{23}$) is given by: $$c_{23} = \frac{|V_{tb}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}$$ where $|V_{tb}|$, $|V_{ud}|$, and $|V_{us}|$ denote the absolute values of the entries in the CKM matrix corresponding to the specified quark transitions.

For any equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, the absolute value of the matrix element corresponding to the transition from the up quark to the down quark, $|V_{ud}|$, is equal to the product of the standard parameters $c_{12}$ and $c_{13}$: $$|V_{ud}| = c_{12} c_{13}$$ where $c_{12} = \cos \theta_{12}$ is the cosine of the mixing angle between the first and second generations, and $c_{13} = \cos \theta_{13}$ is the cosine of the mixing angle between the first and third generations.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the absolute value of the matrix element corresponding to the transition from the up quark to the strange quark, $|V_{us}|$, is equal to the product of the standard parameters $s_{12}$ and $c_{13}$: $$|V_{us}| = s_{12} c_{13}$$ where $s_{12} = \sin \theta_{12}$ is the sine of the mixing angle between the first and second generations, and $c_{13} = \cos \theta_{13}$ is the cosine of the mixing angle between the first and third generations.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the absolute value of the matrix element $|V_{ub}|$ (the entry in the first row and third column) is equal to the standard parameter $s_{13}$, which corresponds to $\sin \theta_{13}$ in the standard parameterization.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the magnitude of the matrix element $|V_{cb}|$ (the entry corresponding to the charm and bottom quarks) is equal to the product of the standard parameters $s_{23}$ and $c_{13}$: $$|V_{cb}| = s_{23} \cdot c_{13}$$ where $s_{23} = \sin \theta_{23}$ and $c_{13} = \cos \theta_{13}$ in the standard parameterization of the CKM matrix.

For any equivalence class $[V]$ of CKM matrices under the phase-rephasing equivalence relation, the magnitude of the matrix element $|V_{tb}|$ (corresponding to the transition between the top quark and the bottom quark) is equal to the product of the standard parameters $c_{23}$ and $c_{13}$: $$|V_{tb}| = c_{23} \cdot c_{13}$$ where $c_{23} = \cos \theta_{23}$ and $c_{13} = \cos \theta_{13}$ represent the cosines of the mixing angles between the second and third generations, and the first and third generations of quarks, respectively, in the standard parameterization.

For any equivalence class $[V]$ of CKM matrices (represented by $3 \times 3$ unitary matrices under the phase-rephasing equivalence relation), if the mixing angle $\theta_{13}$ satisfies $\cos \theta_{13} = 0$, then the absolute value of the matrix element $|V_{ub}|$ (the entry in the first row and third column) is equal to $1$.

For any CKM matrix $V$, let $\llbracket V \rrbracket$ be its phase-rephasing equivalence class. The magnitudes of the CKM matrix elements $|V_{ud}|$, $|V_{ub}|$, $|V_{us}|$, $|V_{cb}|$, and $|V_{tb}|$, and the standard mixing angles $\theta_{12}$, $\theta_{13}$, and $\theta_{23}$ evaluated on $\llbracket V \rrbracket$ satisfy the following equivalence: $$|V_{ud}| = 0 \lor |V_{ub}| = 0 \lor |V_{us}| = 0 \lor |V_{cb}| = 0 \lor |V_{tb}| = 0 \iff \cos \theta_{12} = 0 \lor \cos \theta_{13} = 0 \lor \cos \theta_{23} = 0 \lor \sin \theta_{12} = 0 \lor \sin \theta_{13} = 0 \lor \sin \theta_{23} = 0$$

For any CKM matrix $V$ and any real phase $\delta_{13}$, let $\theta_{12}, \theta_{13}$, and $\theta_{23}$ be the mixing angles of the phase-rephasing equivalence class $\llbracket V \rrbracket$. If we construct a new CKM matrix using the standard parameterization with these extracted angles and the phase $\delta_{13}$, then the complex invariant $\text{mulExp}\delta_{13}$ of the resulting equivalence class is given by: $$\text{mulExp}\delta_{13} \left( \llbracket \text{standParam}(\theta_{12}, \theta_{13}, \theta_{23}, \delta_{13}) \rrbracket \right) = \sin \theta_{12} \cos^2 \theta_{13} \sin \theta_{23} \sin \theta_{13} \cos \theta_{12} \cos \theta_{23} e^{i \delta_{13}}$$ where $i$ is the imaginary unit.

For any CKM matrix $V$ and any real phase $\delta_{13}$, let $\theta_{12}, \theta_{13}$, and $\theta_{23}$ be the mixing angles extracted from the phase-rephasing equivalence class $\llbracket V \rrbracket$. The complex invariant $\text{mulExp}\delta_{13}$ evaluated on the equivalence class of the standard parameterization constructed with these angles and $\delta_{13}$ is equal to zero if and only if at least one of the matrix element magnitudes $|V_{ud}|$, $|V_{ub}|$, $|V_{us}|$, $|V_{cb}|$, or $|V_{tb}|$ of the original equivalence class $\llbracket V \rrbracket$ is zero. Formally, this is expressed as: $$\text{mulExp}\delta_{13}(\llbracket \text{standParam}(\theta_{12}, \theta_{13}, \theta_{23}, \delta_{13}) \rrbracket) = 0 \iff |V_{ud}| = 0 \lor |V_{ub}| = 0 \lor |V_{us}| = 0 \lor |V_{cb}| = 0 \lor |V_{tb}| = 0$$ where $\text{mulExp}\delta_{13}$ is defined as the complex invariant $J_{\mathbb{C}} + \frac{|V_{us}|^2 |V_{ub}|^2 |V_{cb}|^2}{|V_{ud}|^2 + |V_{us}|^2}$.

For any CKM matrix $V$ and any real phase $\delta_{13}$, let $\theta_{12}, \theta_{13}$, and $\theta_{23}$ be the mixing angles of the phase-rephasing equivalence class $\llbracket V \rrbracket$. Let $V_{std}$ be the CKM matrix constructed using the standard parameterization with these extracted angles and the phase $\delta_{13}$. The absolute value (norm) of the complex invariant $\text{mulExp}\delta_{13}$ of the equivalence class $\llbracket V_{std} \rrbracket$ is: \[ |\text{mulExp}\delta_{13}(\llbracket V_{std} \rrbracket)| = \sin \theta_{12} \cos^2 \theta_{13} \sin \theta_{23} \sin \theta_{13} \cos \theta_{12} \cos \theta_{23} \] where the complex invariant is defined as $\text{mulExp}\delta_{13} = J_{\mathbb{C}} + \frac{|V_{us}|^2 |V_{ub}|^2 |V_{cb}|^2}{|V_{ud}|^2 + |V_{us}|^2}$, and $J_{\mathbb{C}}$ is the complex Jarlskog invariant.

For any CKM matrix $V$ and real phase $\delta_{13}$, let $\theta_{12}, \theta_{13}$, and $\theta_{23}$ be the mixing angles extracted from the phase-rephasing equivalence class $\llbracket V \rrbracket$. Let $V_{std}$ be the CKM matrix constructed using the standard parameterization with these extracted angles and the phase $\delta_{13}$. If the complex invariant of the resulting equivalence class, defined as $\text{mulExp}\delta_{13}(\llbracket V_{std} \rrbracket) = J_{\mathbb{C}} + \frac{|V_{us}|^2 |V_{ub}|^2 |V_{cb}|^2}{|V_{ud}|^2 + |V_{us}|^2}$, is non-zero, then: $$e^{i \arg(\text{mulExp}\delta_{13}(\llbracket V_{std} \rrbracket))} = e^{i \delta_{13}}$$ where $\arg(z)$ denotes the argument of the complex number $z$ and $i$ is the imaginary unit.

Let $V$ be a CKM matrix (a $3 \times 3$ complex unitary matrix), and let $\theta_{12}$, $\theta_{13}$, and $\theta_{23}$ be the mixing angles extracted from its equivalence class under phase rephasing. If $\cos \theta_{13} = 0$, then for any real phase $\delta_{13}$, the matrix $V(\theta_{12}, \theta_{13}, \theta_{23}, \delta_{13})$ constructed via the standard CKM parameterization is equivalent to the matrix $V(\theta_{12}, \theta_{13}, \theta_{23}, 0)$. Two matrices $U$ and $V$ are considered equivalent ($U \approx V$) if there exist diagonal phase matrices $P_L, P_R \in U(3)$ such that $U = P_L V P_R$, representing a rephasing of the fermion fields.

Let $V$ be a CKM matrix (a $3 \times 3$ unitary matrix), and let $\theta_{12}$, $\theta_{13}$, and $\theta_{23}$ be the mixing angles extracted from its equivalence class under phase rephasing. If $\cos \theta_{12} = 0$, then for any real phase $\delta_{13}$, the matrix $V(\theta_{12}, \theta_{13}, \theta_{23}, \delta_{13})$ constructed via the standard CKM parameterization is equivalent to the matrix $V(\theta_{12}, \theta_{13}, \theta_{23}, 0)$. Two matrices $U$ and $V$ are considered equivalent ($U \approx V$) if there exist diagonal phase matrices $P_L, P_R \in U(3)$ such that $U = P_L V P_R$, representing a rephasing of the fermion fields.