Physlib.Particles.FlavorPhysics.CKMMatrix.Rows

For a given CKM matrix $V$, the function returns its first row, which corresponds to the up quark $u$. This row is represented as a vector in $\mathbb{C}^3$ consisting of the elements $(V_{ud}, V_{us}, V_{ub})$.

For a CKM matrix $V$, the notation $[V]_u$ denotes its first row, which corresponds to the up quark ($u$). This row is represented as a vector in $\mathbb{C}^3$ consisting of the components $(V_{ud}, V_{us}, V_{ub})$.

Given a CKM matrix $V$, this function represents the row associated with the charm quark. It maps the indices $\{0, 1, 2\}$ (corresponding to the down, strange, and bottom quarks) to the complex matrix elements $(V_{cd}, V_{cs}, V_{cb})$.

For a CKM matrix $V$, the notation $[V]_c$ denotes its second row, which corresponds to the charm quark. This row is represented as a function from the index set $\{0, 1, 2\}$ to the complex numbers $\mathbb{C}$, representing the matrix elements $(V_{cd}, V_{cs}, V_{cb})$.

Given a CKM matrix $V \in U(3)$, the function returns the third row of the matrix, which corresponds to the top quark ($t$). This row is represented as a vector in $\mathbb{C}^3$ with components $(V_{td}, V_{ts}, V_{tb})$.

Given a CKM matrix $V$, the notation $[V]_t$ represents the vector in $\mathbb{C}^3$ corresponding to the third row of the matrix, which is associated with the top quark ($t$).

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), let $[V]_u$ denote its first row, which corresponds to the up quark. The Hermitian inner product of this row with itself is equal to 1, i.e., $\overline{[V]_u} \cdot [V]_u = 1$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), let $[V]_c$ denote its row corresponding to the charm quark. The Hermitian inner product of this row with itself is equal to 1, i.e., $\overline{[V]_c} \cdot [V]_c = 1$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), let $[V]_t$ denote its third row, which corresponds to the top quark ($t$). The Hermitian inner product of this row with itself is equal to 1, i.e., $\overline{[V]_t} \cdot [V]_t = 1$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), the row corresponding to the up quark $[V]_u$ and the row corresponding to the charm quark $[V]_c$ are orthogonal. Specifically, their Hermitian inner product is zero: $$\overline{[V]_u} \cdot [V]_c = 0$$ where $[V]_u$ is the first row of $V$ and $[V]_c$ is the second row of $V$.

For any CKM matrix $V$, the up-quark row $[V]_u$ and the top-quark row $[V]_t$ are orthogonal. This is expressed as the Hermitian inner product of the two rows being zero: $$\overline{[V]_u} \cdot [V]_t = 0$$ where $[V]_u$ is the first row of $V$ and $[V]_t$ is the third row of $V$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), the charm-quark row $[V]_c$ and the up-quark row $[V]_u$ are orthogonal. This means that their Hermitian inner product is zero: $$\overline{[V]_c} \cdot [V]_u = 0$$ where $[V]_c$ is the row vector $(V_{cd}, V_{cs}, V_{cb})$ and $[V]_u$ is the row vector $(V_{ud}, V_{us}, V_{ub})$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), the charm-quark row $[V]_c$ and the top-quark row $[V]_t$ are orthogonal. This is expressed as the Hermitian inner product of the two rows being zero: $$\overline{[V]_c} \cdot [V]_t = 0$$ where $[V]_c$ is the second row of the matrix containing the elements $(V_{cd}, V_{cs}, V_{cb})$ and $[V]_t$ is the third row containing $(V_{td}, V_{ts}, V_{tb})$.

For any CKM matrix $V$, the top-quark row $[V]_t$ and the up-quark row $[V]_u$ are orthogonal, meaning their Hermitian inner product is zero: $\overline{[V]_t} \cdot [V]_u = 0$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), the top-quark row $[V]_t$ and the charm-quark row $[V]_c$ are orthogonal. This is expressed as the Hermitian inner product of the two rows being zero: $$\overline{[V]_t} \cdot [V]_c = 0$$ where $[V]_t$ is the third row of the matrix containing the elements $(V_{td}, V_{ts}, V_{tb})$ and $[V]_c$ is the second row containing the elements $(V_{cd}, V_{cs}, V_{cb})$.

For any CKM matrix $V$, let $[V]_u$ be the row vector representing the up quark couplings and $[V]_c$ be the row vector representing the charm quark couplings. Then the complex conjugate of the cross product of the complex conjugated rows is equal to the cross product of the original rows: $$\overline{\overline{[V]_u} \times \overline{[V]_c}} = [V]_u \times [V]_c$$ where $\overline{\mathbf{z}}$ denotes the complex conjugate of the vector $\mathbf{z}$ and $\times$ denotes the vector cross product in $\mathbb{C}^3$.

For any CKM matrix $V$, let $[V]_c$ be the row vector representing the charm quark couplings and $[V]_t$ be the row vector representing the top quark couplings. Then the complex conjugate of the cross product of the complex conjugated rows is equal to the cross product of the original rows: $$\overline{\overline{[V]_c} \times \overline{[V]_t}} = [V]_c \times [V]_t$$ where $\overline{z}$ denotes the complex conjugate of $z$ and $\times$ denotes the vector cross product in $\mathbb{C}^3$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), let $[V]_u$ be the row vector representing the up-quark couplings and $[V]_c$ be the row vector representing the charm-quark couplings. The vector formed by the cross product of the complex conjugated rows, $\overline{[V]_u} \times \overline{[V]_c}$, is normalized to 1. Specifically, the Hermitian inner product of this cross product with itself is equal to 1: $$\overline{(\overline{[V]_u} \times \overline{[V]_c})} \cdot (\overline{[V]_u} \times \overline{[V]_c}) = 1$$ where $\overline{\mathbf{z}}$ denotes the complex conjugate of the vector $\mathbf{z}$ and $\times$ denotes the vector cross product in $\mathbb{C}^3$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix), let $[V]_c$ be the row vector representing the charm-quark couplings and $[V]_t$ be the row vector representing the top-quark couplings. The vector formed by the cross product of the complex conjugated rows, $\overline{[V]_c} \times \overline{[V]_t}$, is normalized to 1. Specifically, the Hermitian inner product of this cross product with itself is equal to 1: $$\overline{(\overline{[V]_c} \times \overline{[V]_t})} \cdot (\overline{[V]_c} \times \overline{[V]_t}) = 1$$ where $\overline{\mathbf{z}}$ denotes the complex conjugate of the vector $\mathbf{z}$, $\times$ denotes the vector cross product in $\mathbb{C}^3$, and $\cdot$ denotes the Hermitian inner product.

For a given CKM matrix $V \in U(3)$, this function maps each index $i \in \{0, 1, 2\}$ to its corresponding row in the matrix. Specifically, it maps the index $0$ to the up-quark row $[V]_u = (V_{ud}, V_{us}, V_{ub})$, the index $1$ to the charm-quark row $[V]_c = (V_{cd}, V_{cs}, V_{cb})$, and the index $2$ to the top-quark row $[V]_t = (V_{td}, V_{ts}, V_{tb})$. The result is a mapping where each row is represented as a vector in $\mathbb{C}^3$.

For any CKM matrix $V$ (represented as a $3 \times 3$ unitary matrix over $\mathbb{C}$), the collection of its row vectors $\{[V]_u, [V]_c, [V]_t\}$ is linearly independent over the complex numbers $\mathbb{C}$.

For a given CKM matrix $V \in U(3)$, this definition constructs a basis for the vector space $\mathbb{C}^3$ (represented as the space of functions $\text{Fin } 3 \to \mathbb{C}$) consisting of the three row vectors of $V$. Because the rows of a unitary matrix are linearly independent and the dimension of the space is 3, these vectors $\{[V]_u, [V]_c, [V]_t\}$ form a basis for $\mathbb{C}^3$ over the complex numbers $\mathbb{C}$.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), let $[V]_u, [V]_c,$ and $[V]_t$ denote the row vectors corresponding to the up, charm, and top quarks, respectively. There exists a real phase $\kappa \in \mathbb{R}$ such that the up-quark row is equal to the cross product of the complex-conjugated charm and top rows, multiplied by a phase factor $e^{i\kappa}$: $$[V]_u = e^{i\kappa} (\overline{[V]_c} \times \overline{[V]_t})$$ where $\overline{\mathbf{z}}$ denotes the element-wise complex conjugate of vector $\mathbf{z}$ and $\times$ denotes the standard vector cross product in $\mathbb{C}^3$.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), let $[V]_u$, $[V]_c$, and $[V]_t$ denote the row vectors corresponding to the up, charm, and top quarks, respectively. There exists a real number $\tau$ such that the top-quark row is equal to the phase-shifted cross product of the complex conjugates of the up-quark and charm-quark rows: $$[V]_t = e^{i\tau} (\overline{[V]_u} \times \overline{[V]_c})$$ where $\overline{\mathbf{v}}$ denotes the element-wise complex conjugate of the vector $\mathbf{v}$ and $\times$ denotes the vector cross product in $\mathbb{C}^3$.

For any two CKM matrices $U$ and $V$ (represented as $3 \times 3$ unitary matrices), if their up-quark rows are equal ($[U]_u = [V]_u$), their charm-quark rows are equal ($[U]_c = [V]_c$), and their top-quark rows are equal ($[U]_t = [V]_t$), then the matrices themselves are equal ($U = V$).

Given a CKM matrix $V \in U(3)$ and six real phase parameters $a, b, c, d, e, f \in \mathbb{R}$, let $V'$ be the phase-shifted matrix defined by $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ This function calculates the cross product of the complex conjugates of the $u$-row and the $c$-row of $V'$. The result is a vector in $\mathbb{C}^3$ given by $(\vec{V'}_u)^* \times (\vec{V'}_c)^*$, where $\vec{V'}_u$ and $\vec{V'}_c$ are the first and second rows of the transformed matrix, respectively.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $V'$ be the matrix obtained by applying the phase shift transformation $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ for real parameters $a, b, c, d, e, f \in \mathbb{R}$. Let $\vec{V}_u, \vec{V}_c \in \mathbb{C}^3$ be the rows of $V$ corresponding to the up and charm quarks, and $\vec{V'}_u, \vec{V'}_c \in \mathbb{C}^3$ be the corresponding rows of the transformed matrix $V'$. The first component (index 0) of the cross product of their complex conjugates satisfies: $$((\vec{V'}_u)^* \times (\vec{V'}_c)^*)_0 = e^{-i(a + b + e + f)} ((\vec{V}_u)^* \times (\vec{V}_c)^*)_0$$ where $i$ is the imaginary unit.

For any $3 \times 3$ unitary CKM matrix $V$ and real parameters $a, b, c, d, e, f \in \mathbb{R}$, let $V'$ be the phase-shifted matrix obtained by $V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$. Let $\vec{V}_u$ and $\vec{V}_c$ be the first and second rows of $V$ (the $u$ and $c$ rows), and $\vec{V'}_u$ and $\vec{V'}_c$ be the corresponding rows of $V'$. The second component (index 1) of the cross product of the complex conjugates of these rows satisfies: $$[(\vec{V'}_u)^* \times (\vec{V'}_c)^*]_1 = e^{-i(a + b + d + f)} [(\vec{V}_u)^* \times (\vec{V}_c)^*]_1$$ where $i$ is the imaginary unit and $(\cdot)^*$ denotes complex conjugation.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $a, b, c, d, e, f \in \mathbb{R}$ be real phase parameters. Let $V'$ be the phase-shifted matrix defined by: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The third component (at index 2) of the cross product of the complex conjugates of the $u$-row and the $c$-row of the transformed matrix $V'$ satisfies: $$[(\vec{V'}_u)^* \times (\vec{V'}_c)^*]_2 = e^{i(-a-b-d-e)} [(\vec{V}_u)^* \times (\vec{V}_c)^*]_2$$ where $i$ is the imaginary unit, $\vec{V}_u$ and $\vec{V}_c$ are the first and second rows of the matrix $V$, respectively, and $\times$ denotes the 3D cross product.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $a, b, c \in \mathbb{R}$ be phase parameters. Suppose we apply a phase shift transformation to $V$ using only the left-hand diagonal matrix (setting the right-hand parameters $d, e, f$ to $0$): $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V$$ Then the $u$-row (the first row) of the resulting matrix $V'$, denoted as $[V']_u$, is equal to the $u$-row of the original matrix $V$ multiplied by the phase factor $e^{ia}$: $$[V']_u = e^{ia} [V]_u$$ where $i$ is the imaginary unit.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $a, b, c \in \mathbb{R}$ be phase parameters. Let $V'$ be the matrix obtained by applying the phase shift transformation with zero phases on the right-hand side: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V$$ The charm row of the resulting matrix, denoted $[V']_c$, is equal to the charm row of the original matrix, $[V]_c$, scaled by the phase factor $e^{ib}$: $$[V']_c = e^{ib} [V]_c$$ where $i$ is the imaginary unit.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $a, b, c \in \mathbb{R}$ be real phase parameters. Let $V'$ be the matrix obtained by applying a phase shift transformation to $V$ with only row phases: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V$$ Then the top quark row (the third row) of the resulting matrix, denoted $[V']_t$, is equal to the original top quark row $[V]_t$ scaled by the phase $e^{ic}$: $$[V']_t = e^{ic} [V]_t$$ where $i$ is the imaginary unit.