Physlib.Particles.FlavorPhysics.CKMMatrix.PhaseFreedom

Let $V$ be a $3 \times 3$ unitary CKM matrix. Let $V'$ be the matrix obtained by applying row phase shifts $(u, c, t)$ and column phase shifts $(d, s, b)$ to $V$ such that $$V' = \begin{pmatrix} e^{iu} & 0 & 0 \\ 0 & e^{ic} & 0 \\ 0 & 0 & e^{it} \end{pmatrix} V \begin{pmatrix} e^{id} & 0 & 0 \\ 0 & e^{is} & 0 \\ 0 & 0 & e^{ib} \end{pmatrix}$$ where $u, c, t, d, s, b \in \mathbb{R}$. If the phase parameters $u$ and $d$ satisfy the condition $u + d = -\arg(V_{ud})$, then the $ud$-entry (the element at row 0 and column 0) of the transformed matrix $V'$ is real and equal to the absolute value $|V_{ud}|$.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $u, c, t, d, s, b \in \mathbb{R}$ be real phase parameters. If the sum of the phase parameters $u$ and $s$ satisfies the condition $u + s = -\arg(V_{us})$, then the entry in the first row and second column of the phase-shifted matrix $V' = \text{diag}(e^{iu}, e^{ic}, e^{it}) \cdot V \cdot \text{diag}(e^{id}, e^{is}, e^{ib})$ is equal to the absolute value $|V_{us}|$. Here $\arg(V_{us})$ denotes the phase of the complex matrix element $V_{us}$ in the first row and second column.

Let $V$ be a CKM matrix (a $3 \times 3$ unitary matrix). Suppose we apply a phase shift transformation to $V$ using six real parameters $u, c, t, d, s, b \in \mathbb{R}$ to obtain a new matrix $V'$: $$V' = \begin{pmatrix} e^{iu} & 0 & 0 \\ 0 & e^{ic} & 0 \\ 0 & 0 & e^{it} \end{pmatrix} V \begin{pmatrix} e^{id} & 0 & 0 \\ 0 & e^{is} & 0 \\ 0 & 0 & e^{ib} \end{pmatrix}$$ If the parameters $u$ and $b$ satisfy the condition $u + b = -\arg(V_{ub})$, where $V_{ub}$ is the matrix element in the first row and third column, then the corresponding element of the transformed matrix is real and equal to the absolute value of the original element, i.e., $V'_{ub} = |V_{ub}|$.

Let $V$ be a $3 \times 3$ unitary CKM matrix. Consider a phase shift transformation of $V$ parameterized by six real numbers $u, c, t$ (corresponding to rows) and $d, s, b$ (corresponding to columns), resulting in a new matrix $V'$ defined as: $$V' = \begin{pmatrix} e^{iu} & 0 & 0 \\ 0 & e^{ic} & 0 \\ 0 & 0 & e^{it} \end{pmatrix} V \begin{pmatrix} e^{id} & 0 & 0 \\ 0 & e^{is} & 0 \\ 0 & 0 & e^{ib} \end{pmatrix}$$ If the phase parameters $c$ and $s$ satisfy the condition $c + s = -\arg(V_{cs})$, where $V_{cs}$ is the entry of $V$ at row index 1 and column index 1, then the entry $V'_{cs}$ of the transformed matrix is equal to the absolute value of the original entry, i.e., $V'_{cs} = |V_{cs}|$.

Let $V$ be a $3 \times 3$ unitary CKM matrix. Let $V'$ be the matrix obtained by applying phase shifts to the rows and columns of $V$ with real parameters $u, c, t$ (for the rows) and $d, s, b$ (for the columns), defined by $V' = \text{diag}(e^{iu}, e^{ic}, e^{it}) V \text{diag}(e^{id}, e^{is}, e^{ib})$. If the phase parameters $c$ and $b$ satisfy the condition $c + b = -\arg(V_{cb})$, then the entry in the second row and third column of the transformed matrix is real and equal to the magnitude of the original entry, i.e., $V'_{cb} = |V_{cb}|$.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $u, c, t, d, s, b \in \mathbb{R}$ be real phase parameters. Let $V'$ be the transformed matrix obtained by: $$V' = \text{diag}(e^{iu}, e^{ic}, e^{it}) \cdot V \cdot \text{diag}(e^{id}, e^{is}, e^{ib})$$ If the parameters $t$ and $b$ satisfy the condition $t + b = -\arg(V_{tb})$, where $\arg(V_{tb})$ is the phase of the entry in the third row and third column, then the $tb$ entry of the transformed matrix is real and equal to the absolute value of the original entry, i.e., $V'_{tb} = |V_{tb}|$.

Let $V$ be a $3 \times 3$ unitary CKM matrix. For any six real phase parameters $u, c, t, d, s, b \in \mathbb{R}$, let $V'$ be the transformed CKM matrix obtained by: $$V' = \text{diag}(e^{iu}, e^{ic}, e^{it}) \cdot V \cdot \text{diag}(e^{id}, e^{is}, e^{ib})$$ where $i$ is the imaginary unit. If the phase parameters $c$ and $d$ satisfy the condition $c + d = \pi - \arg(V_{cd})$, then the entry of the transformed matrix in the charm row and down column satisfies: $$V'_{cd} = -|V_{cd}|$$ where $\arg(V_{cd})$ is the phase of the original matrix element $V_{cd}$ and $|V_{cd}|$ is its absolute value.

Let $V$ be a CKM matrix, which is a $3 \times 3$ unitary matrix. Let $\mathbf{V}_u, \mathbf{V}_c, \mathbf{V}_t \in \mathbb{C}^3$ denote the first, second, and third rows of $V$ respectively (corresponding to the up, charm, and top quarks). Suppose there exists a real number $\tau$ such that the rows satisfy the relation $e^{i\tau} (\mathbf{V}_u^* \times \mathbf{V}_c^*) = \mathbf{V}_t$, where $\mathbf{V}^*$ denotes the component-wise complex conjugate and $\times$ denotes the cross product in $\mathbb{C}^3$. Given six real phase parameters $u, c, t, d, s, b \in \mathbb{R}$, let $V'$ be the transformed CKM matrix obtained by: $$V' = \text{diag}(e^{iu}, e^{ic}, e^{it}) \cdot V \cdot \text{diag}(e^{id}, e^{is}, e^{ib})$$ If the parameters satisfy the condition $\tau = -(u + c + t + d + s + b)$, then the rows of the transformed matrix $V'$ satisfy: $$\mathbf{V}'_t = (\mathbf{V}'_u)^* \times (\mathbf{V}'_c)^*$$ where $i$ is the imaginary unit.

For a CKM matrix $U$, which is a $3 \times 3$ complex unitary matrix with entries $V_{ij}$ ($i \in \{u, c, t\}$, $j \in \{d, s, b\}$), this proposition holds if the matrix elements $V_{ud}$, $V_{us}$, $V_{cb}$, and $V_{tb}$ are equal to their absolute values (and are thus non-negative real numbers), and the third row vector $\mathbf{V}_t$ is equal to the cross product of the complex conjugates of the first row $\mathbf{V}_u$ and the second row $\mathbf{V}_c$, expressed as: $$V_{ud} = |V_{ud}|, \quad V_{us} = |V_{us}|, \quad V_{cb} = |V_{cb}|, \quad V_{tb} = |V_{tb}|$$ and $$\mathbf{V}_t = \mathbf{V}_u^* \times \mathbf{V}_c^*$$ where $\mathbf{V}^*$ denotes the element-wise complex conjugate and $\times$ denotes the standard cross product in $\mathbb{C}^3$.

For a CKM matrix $U$, represented as a $3 \times 3$ complex unitary matrix with entries $V_{ij}$ ($i \in \{u, c, t\}$, $j \in \{d, s, b\}$), this proposition holds if: 1. The matrix elements $V_{ud}$, $V_{us}$, and $V_{cb}$ are $0$. 2. The matrix element $V_{ub}$ is equal to $1$. 3. The third row vector $\mathbf{V}_t$ is equal to the cross product of the complex conjugates of the first row $\mathbf{V}_u$ and the second row $\mathbf{V}_c$: $$\mathbf{V}_t = \mathbf{V}_u^* \times \mathbf{V}_c^*$$ where $\mathbf{V}^*$ denotes the element-wise complex conjugate and $\times$ denotes the standard cross product in $\mathbb{C}^3$. 4. The matrix elements $V_{cd}$ and $V_{cs}$ are real and satisfy $V_{cd} = -|V_{cd}|$ and $V_{cs} = \sqrt{1 - |V_{cd}|^2}$, where $|V_{cd}|$ is the absolute value of the $V_{cd}$ entry.

Let $V$ be a CKM matrix, represented as a $3 \times 3$ complex unitary matrix, with entries $V_{ij}$ where $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$. Suppose $a, b, c, d, e, f$, and $\tau$ are real parameters satisfying the following system of linear equations based on the arguments (phases) of specific matrix elements: 1. $a + d = -\arg(V_{ud})$ 2. $a + e = -\arg(V_{us})$ 3. $b + f = -\arg(V_{cb})$ 4. $c + f = -\arg(V_{tb})$ 5. $\tau = -(a + b + c + d + e + f)$ Then the variables $b, c, d, e,$ and $f$ are uniquely determined by $a, \tau,$ and the phases of $V$ as follows: - $b = -\tau + \arg(V_{ud}) + \arg(V_{us}) + \arg(V_{tb}) + a$ - $c = -\tau + \arg(V_{cb}) + \arg(V_{ud}) + \arg(V_{us}) + a$ - $d = -\arg(V_{ud}) - a$ - $e = -\arg(V_{us}) - a$ - $f = \tau - \arg(V_{ud}) - \arg(V_{us}) - \arg(V_{cb}) - \arg(V_{tb}) - a$

Let $V$ be a CKM matrix, and let $a, b, c, d, e, f$ be real numbers representing phase shifts. If these parameters satisfy the system of equations: 1. $a + f = -\arg V_{ub}$ 2. $a + b + c + d + e + f = 0$ 3. $b + d = \pi - \arg V_{cd}$ 4. $b + e = -\arg V_{cs}$ Then the parameters are related by: - $c = \arg V_{cd} + \arg V_{cs} + \arg V_{ub} + b - \pi$ - $d = \pi - \arg V_{cd} - b$ - $e = -\arg V_{cs} - b$ - $f = -\arg V_{ub} - a$

For any $3 \times 3$ unitary CKM matrix $V$, there exists a CKM matrix $U$ that is equivalent to $V$ under phase rephasing ($V \approx U$) and satisfies the condition `FstRowThdColRealCond`. Two matrices $V, U$ are equivalent if there exist real parameters $a, b, c, d, e, f \in \mathbb{R}$ such that $$U = \text{diag}(e^{ia}, e^{ib}, e^{ic}) V \text{diag}(e^{id}, e^{ie}, e^{if}).$$ The condition `FstRowThdColRealCond` is satisfied for $U$ if the matrix elements $V_{ud}, V_{us}, V_{cb}$, and $V_{tb}$ are non-negative real numbers (i.e., $V_{ij} = |V_{ij}|$) and the third row vector $\mathbf{V}_t$ is equal to the cross product of the complex conjugates of the first row $\mathbf{V}_u$ and the second row $\mathbf{V}_c$: $$\mathbf{V}_t = \mathbf{V}_u^* \times \mathbf{V}_c^*.$$

Let $V$ be a $3 \times 3$ unitary CKM matrix that satisfies the condition `FstRowThdColRealCond` (meaning $V_{ud}$, $V_{us}$, $V_{cb}$, and $V_{tb}$ are non-negative real numbers and the row vectors satisfy $\mathbf{V}_t = \mathbf{V}_u^* \times \mathbf{V}_c^*$). If $V_{ud} = 0$ and $V_{us} = 0$ (which, for a unitary matrix, implies $|V_{ub}| = 1$), then there exists a CKM matrix $U$ equivalent to $V$ under phase rephasing ($V \approx U$) such that $U$ satisfies the `ubOnePhaseCond`. Specifically, $U$ satisfies the following: 1. $U_{ud} = 0$, $U_{us} = 0$, and $U_{cb} = 0$. 2. $U_{ub} = 1$. 3. The third row vector $\mathbf{U}_t$ is the cross product of the complex conjugates of the first two rows: $\mathbf{U}_t = \mathbf{U}_u^* \times \mathbf{U}_c^*$. 4. The remaining charm-row elements are real and satisfy $U_{cd} = -|U_{cd}|$ and $U_{cs} = \sqrt{1 - |U_{cd}|^2}$.

Let $V$ be a $3 \times 3$ unitary CKM matrix. Suppose $V$ satisfies the condition `FstRowThdColRealCond`, which implies that the matrix elements $V_{ud}$, $V_{us}$, $V_{cb}$, and $V_{tb}$ are non-negative real numbers, and the third row vector is given by the cross product of the complex conjugates of the first two rows, $\mathbf{V}_t = \mathbf{V}_u^* \times \mathbf{V}_c^*$. If $V_{ud} \neq 0$ or $V_{us} \neq 0$, then the matrix element $V_{cd}$ is given by: $$V_{cd} = -\frac{|V_{tb}| |V_{us}| + |V_{ub}| |V_{ud}| |V_{cb}| e^{-i \arg(V_{ub})}}{|V_{ud}|^2 + |V_{us}|^2}$$ where $|V_{ij}|$ denotes the absolute value of the entry in the $i$-th row and $j$-th column, and $\arg(V_{ub})$ is the phase of the element $V_{ub}$.

Let $V$ be a $3 \times 3$ unitary CKM matrix with entries $V_{ij}$ for $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$. Suppose that $V_{ud} \neq 0$ or $V_{us} \neq 0$. If $V$ satisfies the condition `FstRowThdColRealCond` (meaning that the matrix elements $V_{ud}, V_{us}, V_{cb}$, and $V_{tb}$ are non-negative real numbers and the third row vector $\mathbf{V}_t$ is the cross product of the complex conjugates of the first two rows, $\mathbf{V}_t = \mathbf{V}_u^* \times \mathbf{V}_c^*$), then the matrix element $V_{cs}$ is given by: $$V_{cs} = \frac{|V_{tb}| |V_{ud}| - |V_{ub}| |V_{us}| |V_{cb}| e^{-i \arg V_{ub}}}{|V_{ud}|^2 + |V_{us}|^2}$$ where $|V_{ij}|$ denotes the magnitude of the matrix element and $\arg V_{ub}$ denotes the phase of $V_{ub}$.