Physlib.Particles.FlavorPhysics.CKMMatrix.Relations

For any equivalence class of CKM matrices $V$ under the phase-rephasing equivalence relation and for any row index $i \in \{0, 1, 2\}$, the sum of the squares of the absolute values of the elements in the $i$-th row is equal to 1. That is, $$|V_{i0}|^2 + |V_{i1}|^2 + |V_{i2}|^2 = 1$$ where $|V_{ij}|$ (denoted as `VAbs i j V`) represents the absolute value of the matrix element at row $i$ and column $j$.

For any CKM matrix $V$ (which is a $3 \times 3$ unitary matrix), the sum of the squares of the absolute values of the entries in its first row is equal to $1$: $$|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1$$ where $V_{ud}$, $V_{us}$, and $V_{ub}$ are the matrix elements representing transitions from the up ($u$) quark to the down ($d$), strange ($s$), and bottom ($b$) quarks respectively, and $| \cdot |$ denotes the complex norm (absolute value).

For any CKM matrix $V$, the sum of the squares of the absolute values of the entries in its second row is 1. Specifically, it holds that: $$|V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 = 1$$ where $V_{cd}$, $V_{cs}$, and $V_{cb}$ are the elements of the second row of $V$, and $|\cdot|$ denotes the complex norm.

For any CKM matrix $V$, which is represented as a $3 \times 3$ unitary matrix, the sum of the squares of the absolute values of the entries in its third row—denoted as $V_{td}, V_{ts},$ and $V_{tb}$—is equal to $1$. That is, $$|V_{td}|^2 + |V_{ts}|^2 + |V_{tb}|^2 = 1$$ where $|\cdot|$ denotes the norm of the complex matrix elements.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), the sum of the squared norms of the entries in its first row is equal to $1$: $$\text{normSq}(V_{ud}) + \text{normSq}(V_{us}) + \text{normSq}(V_{ub}) = 1$$ where $V_{ud}$, $V_{us}$, and $V_{ub}$ are the matrix elements representing transitions from the up ($u$) quark to the down ($d$), strange ($s$), and bottom ($b$) quarks respectively, and $\text{normSq}(z)$ denotes the squared absolute value $|z|^2$ of a complex number $z$.

For any CKM matrix $V$, the sum of the squared norms of the entries in its second row is equal to $1$. That is, $$|V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 = 1$$ where $V_{cd}$, $V_{cs}$, and $V_{cb}$ are the elements of the second row of $V$, and $|\cdot|^2$ denotes the squared norm of the complex matrix elements.

For any CKM matrix $V$, which is defined as a $3 \times 3$ unitary matrix, the sum of the squared magnitudes of the entries in its third row—denoted as $V_{td}, V_{ts},$ and $V_{tb}$—is equal to $1$. That is, $$|V_{td}|^2 + |V_{ts}|^2 + |V_{tb}|^2 = 1$$ where $|z|^2$ denotes the squared norm (`normSq`) of a complex number $z$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), the squared norms of the elements in its first row satisfy the following relation: $$|V_{ud}|^2 + |V_{us}|^2 = 1 - |V_{ub}|^2$$ where $V_{ud}$, $V_{us}$, and $V_{ub}$ are the matrix elements representing transitions from the up ($u$) quark to the down ($d$), strange ($s$), and bottom ($b$) quarks respectively, and $|z|^2$ denotes the squared norm (`normSq`) of a complex number $z$.

For any equivalence class of CKM matrices $V$ under the phase-rephasing equivalence relation, the absolute values of the matrix elements $|V_{ud}|$, $|V_{us}|$, and $|V_{ub}|$ satisfy the following relationship: $$|V_{ud}|^2 + |V_{us}|^2 = 1 - |V_{ub}|^2$$ where $|V_{ud}|$, $|V_{us}|$, and $|V_{ub}|$ denote the absolute values of the entries in the first row of the matrix corresponding to the up-quark transitions to the down, strange, and bottom quarks, respectively.

For any CKM matrix $V$, the condition that either $V_{ud} \neq 0$ or $V_{us} \neq 0$ is equivalent to the condition that $|V_{ub}| \neq 1$. Here, $V_{ud}$, $V_{us}$, and $V_{ub}$ are the matrix elements in the first row representing transitions from the up ($u$) quark to the down ($d$), strange ($s$), and bottom ($b$) quarks respectively, and $|\cdot|$ denotes the complex norm.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), if at least one of the matrix elements $V_{ud}$ or $V_{us}$ is non-zero, then the sum of their squared norms is also non-zero: $$|V_{ud}|^2 + |V_{us}|^2 \neq 0$$ where $V_{ud}$ and $V_{us}$ are the matrix elements representing transitions from the up ($u$) quark to the down ($d$) and strange ($s$) quarks respectively, and $|z|^2$ denotes the squared norm of a complex number $z$.

For an equivalence class of CKM matrices $\llbracket V \rrbracket$ under the phase-rephasing relation, if the absolute value of the matrix element $|V_{ub}|$ is not equal to $1$, then the sum of the squares of the absolute values of the matrix elements $|V_{ud}|$ and $|V_{us}|$ is non-zero: $$|V_{ud}|^2 + |V_{us}|^2 \neq 0$$ where $|V_{ud}|$, $|V_{us}|$, and $|V_{ub}|$ represent the absolute values of the entries in the first row (corresponding to the up quark transitions) of any representative matrix $V$ in the equivalence class.

For an equivalence class of CKM matrices $V$ (defined as $3 \times 3$ unitary matrices under phase-rephasing), if the absolute value of the matrix element $|V_{ub}|$ is not equal to $1$, then the square root of the sum of the squares of the absolute values of the matrix elements $|V_{ud}|$ and $|V_{us}|$ is non-zero: $$\sqrt{|V_{ud}|^2 + |V_{us}|^2} \neq 0$$ where $|V_{ud}|$, $|V_{us}|$, and $|V_{ub}|$ denote the absolute values of the entries in the first row of a representative matrix in the equivalence class $V$.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), if at least one of the matrix elements $V_{ud}$ or $V_{us}$ is non-zero, then the sum of their squared norms, treated as a complex number, is also non-zero: $$(|V_{ud}|^2 : \mathbb{C}) + |V_{us}|^2 \neq 0$$ where $V_{ud}$ and $V_{us}$ are the matrix elements representing transitions from the up ($u$) quark to the down ($d$) and strange ($s$) quarks respectively, and $|z|^2$ denotes the squared norm of a complex number $z$.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), if at least one of the matrix elements $V_{ud}$ or $V_{us}$ is non-zero, then the sum of the squares of their absolute values is non-zero in the complex numbers: $$(|V_{ud}|^2 : \mathbb{C}) + |V_{us}|^2 \neq 0$$ where $V_{ud}$ and $V_{us}$ are the matrix elements representing transitions from the up ($u$) quark to the down ($d$) and strange ($s$) quarks respectively, and $|V_{ij}|$ denotes the absolute value of the corresponding matrix entry.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), the inner product of the second row (indexed by the charm quark $c$) and the first row (indexed by the up quark $u$) is zero: $$V_{cd} \bar{V}_{ud} + V_{cs} \bar{V}_{us} + V_{cb} \bar{V}_{ub} = 0$$ where $V_{ij}$ denotes the matrix element corresponding to quarks $i$ and $j$, and $\bar{V}_{ij}$ denotes its complex conjugate.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), the elements of the first and third rows satisfy the orthogonality relation: $V_{td} V_{ud}^* + V_{ts} V_{us}^* + V_{tb} V_{ub}^* = 0$ where $V_{ij}$ represents the matrix element for the corresponding quark flavors $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$, and $V_{ij}^*$ denotes the complex conjugate.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), the following relationship holds between the elements of the first row (indexed by $u$) and the second row (indexed by $c$): $$V_{cd} \bar{V}_{ud} = -V_{cs} \bar{V}_{us} - V_{cb} \bar{V}_{ub}$$ where $V_{ij}$ denotes the matrix element for the corresponding up-type quark $i \in \{u, c, t\}$ and down-type quark $j \in \{d, s, b\}$, and $\bar{V}_{ij}$ denotes its complex conjugate.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), the following relationship holds between the elements of the first and second rows: $$V_{cs} \bar{V}_{us} = -V_{cd} \bar{V}_{ud} - V_{cb} \bar{V}_{ub}$$ where $V_{ij}$ represents the matrix element for the corresponding quark flavors $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$, and $\bar{V}_{ij}$ denotes the complex conjugate of the element.

For any equivalence class of CKM matrices $V$ (under the phase-rephasing equivalence relation) and for any row index $i \in \{0, 1, 2\}$, if the absolute value of the matrix element in the third column is $|V_{i2}| = 1$, then the absolute value of the matrix element in the first column is $|V_{i0}| = 0$.

For any equivalence class of CKM matrices $V$ under the phase-rephasing equivalence relation and for any row index $i \in \{0, 1, 2\}$, if the absolute value of the matrix element in the third column is equal to 1 ($|V_{i2}| = 1$), then the absolute value of the matrix element in the second column is zero ($|V_{i1}| = 0$).

Let $V$ be a $3 \times 3$ unitary matrix (a CKM matrix) with rows denoted by $\mathbf{V}_u, \mathbf{V}_c, \mathbf{V}_t$ and individual elements denoted by $V_{ij}$ for $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$. For any real number $\tau$, if the third row satisfies the relation $$\mathbf{V}_t = e^{i\tau} (\mathbf{V}_u^* \times \mathbf{V}_c^*),$$ where $\mathbf{V}^*$ denotes the component-wise complex conjugate and $\times$ denotes the three-dimensional cross product, then the complex conjugate of the element $V_{tb}$ is given by $$V_{tb}^* = e^{-i\tau} (V_{cs} V_{ud} - V_{us} V_{cd}).$$

Let $V$ be a $3 \times 3$ unitary matrix (a CKM matrix) with rows denoted by $\mathbf{V}_u, \mathbf{V}_c, \mathbf{V}_t$ and individual elements denoted by $V_{ij}$ for $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$. For any real number $\tau$, if the third row satisfies the cross-product relation $$\mathbf{V}_t = e^{i\tau} (\mathbf{V}_u^* \times \mathbf{V}_c^*),$$ where $\mathbf{V}^*$ denotes the component-wise complex conjugate and $\times$ denotes the three-dimensional cross product, then the following identity holds: $$e^{i\tau} V_{tb}^* V_{ud}^* = V_{cs} (|V_{ud}|^2 + |V_{us}|^2) + V_{cb} V_{ub}^* V_{us},$$ where $V_{ij}^*$ is the complex conjugate and $|V_{ij}|^2$ is the squared norm of the element.

Let $V$ be a $3 \times 3$ unitary matrix (a CKM matrix) with rows denoted by $\mathbf{V}_u, \mathbf{V}_c, \mathbf{V}_t$ and individual elements denoted by $V_{ij}$ for $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$. For any real number $\tau$, if the third row satisfies the cross-product relation $$\mathbf{V}_t = e^{i\tau} (\mathbf{V}_u^* \times \mathbf{V}_c^*),$$ where $\mathbf{V}^*$ denotes the component-wise complex conjugate and $\times$ denotes the three-dimensional cross product, then the following identity holds: $$e^{i\tau} V_{tb}^* V_{us}^* = -(V_{cd} (|V_{ud}|^2 + |V_{us}|^2) + V_{cb} V_{ub}^* V_{ud}),$$ where $V_{ij}^*$ is the complex conjugate and $|V_{ij}|^2$ is the squared norm of the element.

Let $V$ be a $3 \times 3$ unitary matrix (a CKM matrix) with rows denoted by $\mathbf{V}_u, \mathbf{V}_c, \mathbf{V}_t$ and individual elements denoted by $V_{ij}$ for $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$. If $V_{ud} \neq 0$ or $V_{us} \neq 0$, and for some real number $\tau$, the third row satisfies the cross-product relation $$\mathbf{V}_t = e^{i\tau} (\mathbf{V}_u^* \times \mathbf{V}_c^*),$$ where $\mathbf{V}^*$ denotes the component-wise complex conjugate and $\times$ denotes the three-dimensional cross product, then the matrix element $V_{cs}$ is given by: $$V_{cs} = \frac{-V_{ub}^* V_{us} V_{cb} + e^{i\tau} V_{tb}^* V_{ud}^*}{ |V_{ud}|^2 + |V_{us}|^2 }$$ where $V_{ij}^*$ denotes the complex conjugate and $|V_{ij}|^2$ denotes the squared norm of the element.

Let $V$ be a $3 \times 3$ unitary matrix (a CKM matrix) with rows denoted by $\mathbf{V}_u, \mathbf{V}_c, \mathbf{V}_t$ and individual elements denoted by $V_{ij}$ for $i \in \{u, c, t\}$ and $j \in \{d, s, b\}$. If $V_{ud} \neq 0$ or $V_{us} \neq 0$, and for some real number $\tau$, the third row satisfies the cross-product relation $$\mathbf{V}_t = e^{i\tau} (\mathbf{V}_u^* \times \mathbf{V}_c^*),$$ where $\mathbf{V}^*$ denotes the component-wise complex conjugate and $\times$ denotes the three-dimensional cross product, then the matrix element $V_{cd}$ is given by: $$V_{cd} = -\frac{V_{ub}^* V_{ud} V_{cb} + e^{i\tau} V_{tb}^* V_{us}^*}{ |V_{ud}|^2 + |V_{us}|^2 }$$ where $V_{ij}^*$ denotes the complex conjugate and $|V_{ij}|^2$ denotes the squared norm of the element.

For any indices $i, j \in \{0, 1, 2\}$ and any equivalence class of CKM matrices $V$ (under the phase-rephasing equivalence relation), the absolute value of the matrix entry $|V_{ij}|$ is non-negative, i.e., $|V_{ij}| \geq 0$.

For any indices $i, j \in \{0, 1, 2\}$ and any equivalence class of CKM matrices $V$ under the phase-rephasing equivalence relation, the absolute value of the matrix element $|V_{ij}|$ is less than or equal to 1, i.e., $|V_{ij}| \leq 1$.

For any equivalence class of CKM matrices $[V]$ (under phase rephasing) and any column index $j \in \{0, 1, 2\}$, the sum of the squares of the absolute values of the entries in the $j$-th column is equal to 1: $$|V_{0j}|^2 + |V_{1j}|^2 + |V_{2j}|^2 = 1$$ where $|V_{ij}|$ denotes the absolute value of the matrix element at row $i$ and column $j$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), the sum of the squares of the absolute values of the elements in the third column—corresponding to the $b$-quark transitions—is equal to 1: $$|V_{ub}|^2 + |V_{cb}|^2 + |V_{tb}|^2 = 1$$ where $V_{ub}, V_{cb}, V_{tb} \in \mathbb{C}$ are the matrix elements in the third column and $|\cdot|$ denotes the complex modulus.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), the sum of the squared norms of the elements in the third column—corresponding to the transitions to the $b$-quark—is equal to 1: $$|V_{ub}|^2 + |V_{cb}|^2 + |V_{tb}|^2 = 1$$ where $V_{ub}, V_{cb}, V_{tb} \in \mathbb{C}$ are the matrix elements in the third column and $|z|^2$ denotes the squared norm of a complex number $z$.

For any CKM matrix $V$ (which is a $3 \times 3$ unitary matrix), if the matrix elements $V_{ud}$ and $V_{us}$ are both equal to zero, then the matrix element $V_{cb}$ is also zero.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), if the matrix elements $V_{ud}$ and $V_{us}$ are both zero, then the absolute value of the matrix element $V_{cs}$ is given by $$|V_{cs}| = \sqrt{1 - |V_{cd}|^2}$$ where $|V_{cd}|$ is the absolute value of the charm-down matrix element.

For any equivalence class of CKM matrices $\llbracket V \rrbracket$ under the phase-rephasing relation, the magnitudes of the matrix elements $|V_{ub}|$, $|V_{cb}|$, and $|V_{tb}|$ satisfy the following relation: $$|V_{cb}|^2 + |V_{tb}|^2 = 1 - |V_{ub}|^2$$ where $|V_{ub}|$, $|V_{cb}|$, and $|V_{tb}|$ are the absolute values of the entries in the first, second, and third rows of the third column, respectively, of any representative $3 \times 3$ unitary matrix in the equivalence class.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), at least one of the matrix elements $V_{cb}$ or $V_{tb}$ is non-zero if and only if the absolute value of the matrix element $V_{ub}$ is not equal to 1: $$V_{cb} \neq 0 \lor V_{tb} \neq 0 \iff |V_{ub}| \neq 1$$ where $V_{ub}, V_{cb}, V_{tb} \in \mathbb{C}$ are the entries in the third column of the matrix, and $|\cdot|$ denotes the complex modulus.

For any equivalence class of CKM matrices $V$ and any column index $i \in \{0, 1, 2\}$, if the absolute value of the matrix element in the first row and $i$-th column is equal to 1 ($|V_{0i}| = 1$), then the absolute value of the matrix element in the second row and $i$-th column is equal to 0 ($|V_{1i}| = 0$).

Let $V$ be an equivalence class of CKM matrices under the phase-rephasing equivalence relation. For any column index $i \in \{0, 1, 2\}$, if the absolute value of the matrix element in the first row and $i$-th column is equal to 1 ($|V_{0i}| = 1$), then the absolute value of the matrix element in the third row and $i$-th column is equal to 0 ($|V_{2i}| = 0$).