Physlib.Particles.FlavorPhysics.CKMMatrix.Basic

Given three real numbers $a, b, c \in \mathbb{R}$, the phase shift matrix is the $3 \times 3$ complex diagonal matrix defined by $$\text{diag}(e^{ia}, e^{ib}, e^{ic}) = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix}$$ where $i$ denotes the imaginary unit.

The phase shift matrix evaluated with zero phases $a = 0, b = 0, c = 0$ is the identity matrix: $$\text{phaseShiftMatrix}(0, 0, 0) = 1$$

For any real numbers $a, b$, and $c$, the conjugate transpose (denoted by $H$) of the phase shift matrix $$\text{diag}(e^{ia}, e^{ib}, e^{ic}) = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix}$$ is equal to the phase shift matrix with negated phases, $\text{diag}(e^{-ia}, e^{-ib}, e^{-ic})$.

For any real numbers $a, b, c, d, e, f \in \mathbb{R}$, the product of two phase shift matrices is equal to the phase shift matrix whose phases are the sums of the corresponding phases: $$\text{phaseShiftMatrix}(a, b, c) \cdot \text{phaseShiftMatrix}(d, e, f) = \text{phaseShiftMatrix}(a + d, b + e, c + f)$$ where $\text{phaseShiftMatrix}(x, y, z)$ is the $3 \times 3$ complex diagonal matrix $\text{diag}(e^{ix}, e^{iy}, e^{iz})$.

For three real numbers $a, b, c \in \mathbb{R}$, this definition constructs a $3 \times 3$ complex unitary matrix in $U(3)$ given by the diagonal matrix $$\text{diag}(e^{ia}, e^{ib}, e^{ic}) = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix}$$ where $i$ is the imaginary unit.

For any real numbers $a, b, c \in \mathbb{R}$, the underlying matrix of the unitary phase-shift element $\text{phaseShift}(a, b, c)$ is the phase-shift matrix $\text{phaseShiftMatrix}(a, b, c)$. That is, $$\text{phaseShift}(a, b, c) = \text{phaseShiftMatrix}(a, b, c) = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix}$$ where the equality represents the coercion of the element from the unitary group $U(3)$ to its $3 \times 3$ complex matrix representation.

Two $3 \times 3$ complex unitary matrices $U, V \in U(3)$ satisfy the phase shift relation if there exist real numbers $a, b, c, e, f, g \in \mathbb{R}$ such that $$U = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix} V \begin{pmatrix} e^{ie} & 0 & 0 \\ 0 & e^{if} & 0 \\ 0 & 0 & e^{ig} \end{pmatrix}$$ In the context of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, this relation defines the equivalence of matrices that are related by the rephasing of up-type and down-type quark fields.

For any $3 \times 3$ complex unitary matrix $U \in U(3)$, $U$ satisfies the phase shift relation with itself. This means there exist real numbers $a, b, c, e, f, g \in \mathbb{R}$ such that $$U = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix} U \begin{pmatrix} e^{ie} & 0 & 0 \\ 0 & e^{if} & 0 \\ 0 & 0 & e^{ig} \end{pmatrix}$$ In the context of the CKM matrix, this expresses the reflexivity of the equivalence relation defined by the rephasing of quark fields.

For any $3 \times 3$ complex unitary matrices $U, V \in U(3)$, if $U$ and $V$ satisfy the phase shift relation, then $V$ and $U$ also satisfy the phase shift relation. This relation holds if there exist real numbers $a, b, c, e, f, g \in \mathbb{R}$ such that $$U = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix} V \begin{pmatrix} e^{ie} & 0 & 0 \\ 0 & e^{if} & 0 \\ 0 & 0 & e^{ig} \end{pmatrix}$$

Let $U, V, W \in U(3)$ be $3 \times 3$ complex unitary matrices. If $U$ and $V$ satisfy the phase shift relation, and $V$ and $W$ satisfy the phase shift relation, then $U$ and $W$ also satisfy the phase shift relation. Two matrices $X, Y \in U(3)$ are said to satisfy the phase shift relation if there exist real numbers $a, b, c, e, f, g \in \mathbb{R}$ such that $$X = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix} Y \begin{pmatrix} e^{ie} & 0 & 0 \\ 0 & e^{if} & 0 \\ 0 & 0 & e^{ig} \end{pmatrix}$$ This theorem expresses the transitivity of the equivalence relation used to define the Cabibbo-Kobayashi-Maskawa (CKM) matrix through the rephasing of quark fields.

The phase shift relation on the set of $3 \times 3$ complex unitary matrices $U(3)$ is an equivalence relation. Two matrices $U, V \in U(3)$ satisfy this relation if there exist real numbers $a, b, c, e, f, g \in \mathbb{R}$ such that $$U = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix} V \begin{pmatrix} e^{ie} & 0 & 0 \\ 0 & e^{if} & 0 \\ 0 & 0 & e^{ig} \end{pmatrix}$$ This implies the relation is reflexive, symmetric, and transitive.

The type `CKMMatrix` represents the set of all $3 \times 3$ unitary matrices with entries in the complex numbers $\mathbb{C}$, corresponding to the unitary group $U(3)$.

For any two CKM matrices $U$ and $V$, if their underlying $3 \times 3$ unitary matrices are equal, then $U$ and $V$ are equal as CKM matrices.

For a CKM matrix $V$, the notation $[V]_{ud}$ refers to the entry in the first row and first column of the matrix, representing the $ud$ element $V_{ud}$.

For a $3 \times 3$ CKM matrix $V$, the notation $[V]_{us}$ denotes the matrix element at row index 0 and column index 1, which represents the $V_{us}$ component corresponding to the transition between the $u$ (up) and $s$ (strange) quarks.

The notation $[V]_{ub}$ represents the entry of a $3 \times 3$ CKM matrix $V$ at row index 0 and column index 2. This corresponds to the matrix element associated with the transition between the up quark ($u$) and the bottom quark ($b$).

For a CKM matrix $V$, the notation $[V]_{cd}$ denotes the complex matrix element $V_{cd}$ located at row index $1$ (corresponding to the charm quark $c$) and column index $0$ (corresponding to the down quark $d$).

For a CKM matrix $V$, the notation $[V]_{cs}$ denotes the matrix element $V_{cs}$ corresponding to the charm quark row and the strange quark column (indices $(1, 1)$ in a 0-indexed $3 \times 3$ matrix).

The notation $[V]_{cb}$ denotes the element of a CKM matrix $V$ located at the second row and third column, representing the mixing between the charm ($c$) and bottom ($b$) quarks. In terms of matrix indices, this corresponds to the entry $V_{1,2}$ (using 0-based indexing).

The notation $[V]_{td}$ denotes the entry in the third row and first column of a CKM matrix $V$, representing the transition amplitude between the top ($t$) and down ($d$) quarks.

The notation $[V]_{ts}$ denotes the element of a CKM matrix $V$ located at the third row (index 2) and the second column (index 1), which corresponds to the coupling between the top ($t$) and strange ($s$) quarks.

The notation $[V]_{tb}$ denotes the element of the CKM matrix $V$ located at the third row and third column, which corresponds to the transition between the top ($t$) and bottom ($b$) quarks.

The setoid structure on the type of CKM matrices (the group of $3 \times 3$ unitary matrices $U(3)$) defined by the phase shift equivalence relation. Two matrices $U, V \in U(3)$ are considered equivalent, $U \approx V$, if there exist real parameters $a, b, c, d, e, f \in \mathbb{R}$ such that $$U = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix} V \begin{pmatrix} e^{id} & 0 & 0 \\ 0 & e^{ie} & 0 \\ 0 & 0 & e^{if} \end{pmatrix}$$ where $i$ is the imaginary unit. This relation identifies CKM matrices that differ only by the rephasing of fermion fields.

Given a $3 \times 3$ unitary CKM matrix $V$ and six real parameters $a, b, c, d, e, f \in \mathbb{R}$, this function returns the matrix obtained by multiplying $V$ on the left and right by diagonal phase matrices: $$\text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ where $i$ is the imaginary unit. Physically, this represents the transformation of the CKM matrix $V$ resulting from shifting the phases of the fermion fields.

For any $3 \times 3$ unitary CKM matrix $V$ and any six real parameters $a, b, c, d, e, f \in \mathbb{R}$, the matrix $V$ is equivalent to the phase-shifted matrix $V'$ defined by $$V' = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix} V \begin{pmatrix} e^{id} & 0 & 0 \\ 0 & e^{ie} & 0 \\ 0 & 0 & e^{if} \end{pmatrix}$$ under the rephasing equivalence relation $\approx$.

For any $3 \times 3$ unitary CKM matrix $V$ and real phase parameters $a, b, c, d, e, f \in \mathbb{R}$, let $V'$ be the matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ Then the $ud$ component (the entry at row 0 and column 0) of the resulting matrix satisfies: $$[V']_{ud} = e^{i(a + d)} [V]_{ud}$$ where $i$ is the imaginary unit and $[V]_{ud}$ denotes the $(0,0)$ element of the matrix $V$.

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 matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The $us$ component (the entry at row index 0 and column index 1) of the resulting matrix satisfies: $$V'_{us} = e^{i(a+e)} V_{us}$$ where $i$ is the imaginary unit.

Let $V$ be a $3 \times 3$ unitary CKM matrix and $a, b, c, d, e, f \in \mathbb{R}$ be real phase parameters. Let $V'$ be the matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The $ub$ component (the entry at row index 0 and column index 2) of the resulting matrix satisfies: $$V'_{ub} = e^{i(a+f)} V_{ub}$$ where $i$ is the imaginary unit.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $a, b, c, d, e, f \in \mathbb{R}$ be phase parameters. Let $V'$ be the matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The $cd$ component of the resulting matrix (the entry at row index 1 and column index 0) satisfies: $$V'_{cd} = e^{i(b + d)} V_{cd}$$ where $i$ is the imaginary unit.

Let $V$ be a $3 \times 3$ unitary CKM matrix and $a, b, c, d, e, f \in \mathbb{R}$ be phase parameters. Let $V'$ be the matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The $cs$ component (the entry at row index 1 and column index 1) of the resulting matrix satisfies: $$V'_{cs} = e^{i(b+e)} V_{cs}$$ where $i$ is the imaginary unit.

Let $V$ be a $3 \times 3$ unitary CKM matrix and let $a, b, c, d, e, f \in \mathbb{R}$ be phase parameters. Let $V'$ be the matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The $cb$ component of the resulting matrix (the entry at row index 1 and column index 2) satisfies: $$V'_{cb} = e^{i(b + f)} V_{cb}$$ where $i$ is the imaginary unit.

Let $V$ be a $3 \times 3$ unitary CKM matrix and $a, b, c, d, e, f \in \mathbb{R}$ be phase parameters. Let $V'$ be the matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The $td$ component (the entry at row index 2 and column index 0) of the resulting matrix satisfies: $$V'_{td} = e^{i(c+d)} V_{td}$$ where $i$ is the imaginary unit.

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 matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The $ts$ component (the entry at row index 2 and column index 1) of the resulting matrix satisfies: $$V'_{ts} = e^{i(c+e)} V_{ts}$$ where $i$ is the imaginary unit.

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 matrix obtained by applying the phase shift transformation: $$V' = \text{diag}(e^{ia}, e^{ib}, e^{ic}) \cdot V \cdot \text{diag}(e^{id}, e^{ie}, e^{if})$$ The $tb$ component (the entry at row index 2 and column index 2) of the resulting matrix satisfies: $$V'_{tb} = e^{i(c+f)} V_{tb}$$ where $i$ is the imaginary unit.

For a $3 \times 3$ unitary complex matrix $V$ and indices $i, j \in \{0, 1, 2\}$, this function returns the absolute value $|V_{ij}|$ of the matrix element at row $i$ and column $j$.

Let $V$ and $U$ be two CKM matrices (represented as $3 \times 3$ unitary matrices). If $V$ and $U$ are equivalent under phase shifts ($V \approx U$), then the absolute values of their matrix elements are equal. That is, for any indices $i, j \in \{0, 1, 2\}$: $$|V_{ij}| = |U_{ij}|$$ where $V_{ij}$ and $U_{ij}$ denote the entries of the matrices $V$ and $U$ respectively.

Given indices $i, j \in \{0, 1, 2\}$, this function maps an equivalence class of CKM matrices $\llbracket V \rrbracket$ (under the phase-rephasing equivalence relation) to the absolute value $|V_{ij}|$ of the matrix element at row $i$ and column $j$ for any representative matrix $V \in \llbracket V \rrbracket$. This value is well-defined on the quotient space because the absolute values of the matrix entries are invariant under row and column phase shifts.

Given an equivalence class $[V]$ of CKM matrices (the quotient of $3 \times 3$ unitary matrices under phase-rephasing), this function returns the absolute value of the matrix element corresponding to the transition between the up quark and the down quark, denoted $|V_{ud}|$. Mathematically, for any representative matrix $V \in [V]$, it returns $|V_{0,0}|$.

For an equivalence class of CKM matrices $\llbracket V \rrbracket$ under phase-rephasing, this function returns the absolute value of the matrix element $|V_{us}|$, corresponding to the entry in the first row (up quark, $u$) and the second column (strange quark, $s$). This is defined by applying the absolute value function $V_{Abs}$ to the indices $i=0$ and $j=1$.

For an equivalence class of CKM matrices $\llbracket V \rrbracket$ in the quotient space of $3 \times 3$ unitary matrices under phase-rephasing, this definition corresponds to the absolute value of the matrix element $|V_{ub}|$. It is defined as the absolute value of the entry in the first row ($i=0$, representing the $u$ quark) and third column ($j=2$, representing the $b$ quark) of any representative matrix $V$ belonging to the equivalence class.

Given an equivalence class of CKM matrices $[V]$ (under the phase-rephasing equivalence relation), this function returns the absolute value of the matrix element corresponding to the charm ($c$) and down ($d$) quarks, denoted as $|V_{cd}|$. This value is independent of the choice of representative $V \in [V]$.

Given an equivalence class of CKM matrices $\llbracket V \rrbracket$ under the phase-rephasing equivalence relation, this function returns the absolute value $|V_{cs}|$ of the matrix element corresponding to the charm ($c$) and strange ($s$) quarks, which resides at row index 1 and column index 1.

Given an equivalence class of CKM matrices $\llbracket V \rrbracket$ under the phase-rephasing relation, this function returns the magnitude (absolute value) $|V_{cb}|$ of the matrix element corresponding to the charm quark ($c$) and the bottom quark ($b$). This is defined as the absolute value of the entry in the second row ($i=1$) and third column ($j=2$) of any representative unitary matrix $V \in \llbracket V \rrbracket$.

This function returns the absolute value $|V_{td}|$ of the entry in the third row (corresponding to the top quark) and first column (corresponding to the down quark) of a representative matrix $V$ from an equivalence class of CKM matrices under the phase-rephasing equivalence relation.

For an equivalence class of CKM matrices $[V]$ under the phase-rephasing equivalence relation, this function returns the absolute value of the matrix element corresponding to the top quark $t$ and the strange quark $s$, denoted as $|V_{ts}|$. This value is independent of the choice of representative $V$ because phase shifts do not affect the magnitude of matrix entries.

For an equivalence class of CKM matrices $\llbracket V \rrbracket$ under the phase-rephasing relation, this function returns the absolute value of the $tb$ matrix element, $|V_{tb}|$. This corresponds to the entry in the third row ($i=2$) and third column ($j=2$) of any representative $3 \times 3$ unitary matrix $V \in \llbracket V \rrbracket$.

For a CKM matrix $V$, this definition represents the complex-valued ratio of the matrix elements $V_{ub}$ and $V_{ud}$, defined as $\frac{V_{ub}}{V_{ud}}$.

For a CKM matrix $V$, the notation $[V]ub|ud$ represents the ratio of the $ub$ element to the $ud$ element, which is the complex value $\frac{V_{ub}}{V_{ud}}$.

For a given CKM matrix $V$, this function calculates the ratio of the matrix element $V_{us}$ to the matrix element $V_{ud}$, expressed as the complex number $V_{us} / V_{ud}$.

The notation $[V]_{us|ud}$ represents the ratio of the matrix elements $V_{us}$ and $V_{ud}$ of a CKM matrix $V$, given by the expression $V_{us} / V_{ud}$.

Given a CKM matrix $V$, this function calculates the complex-valued ratio of the matrix element $V_{ud}$ to the matrix element $V_{us}$, represented as $\frac{V_{ud}}{V_{us}}$. These elements correspond to the transitions between the up quark and the down and strange quarks, respectively.

For a CKM matrix $V$, the notation $[V]_{ud|us}$ represents the ratio of the matrix element $V_{ud}$ to the matrix element $V_{us}$, where $V_{ud}$ and $V_{us}$ correspond to the up-down and up-strange quark mixing entries respectively.

For a CKM matrix $V$, this function computes the ratio of the matrix element $V_{ub}$ to the matrix element $V_{us}$, returning the result as a complex number $\mathbb{C}$.

For a CKM matrix $V$, the notation $[V]ub|us$ represents the ratio of the matrix element $V_{ub}$ to the matrix element $V_{us}$.

For a given CKM matrix $V$ (a $3 \times 3$ unitary matrix), this function calculates the ratio of the matrix element $V_{cd}$ to the matrix element $V_{cb}$, where $c$ denotes the charm quark row, and $d$ and $b$ denote the down and bottom quark columns, respectively. The result is a complex number $\mathbb{C}$ given by $\frac{V_{cd}}{V_{cb}}$.

For a given CKM matrix $V$, the notation $[V]_{cd|cb}$ represents the ratio of the matrix element $V_{cd}$ to the matrix element $V_{cb}$, where $c$ denotes the charm quark row and $d, b$ denote the down and bottom quark columns, respectively.

For any CKM matrix $V$ (a $3 \times 3$ unitary matrix), if the matrix element $V_{cb}$ (representing the charm-bottom quark transition) is non-zero, then the matrix element $V_{cd}$ (representing the charm-down quark transition) is equal to the product of the ratio $\frac{V_{cd}}{V_{cb}}$ and the element $V_{cb}$, i.e., $V_{cd} = \frac{V_{cd}}{V_{cb}} \cdot V_{cb}$.

For a CKM matrix $V$, this function returns the ratio $V_{cs} / V_{cb}$, where $V_{cs}$ and $V_{cb}$ are the complex matrix elements corresponding to the charm-strange and charm-bottom quark transitions, respectively.

For a CKM matrix $V$, the notation $[V]_{cs|cb}$ represents the ratio of the matrix element $V_{cs}$ to the matrix element $V_{cb}$, where $V_{cs}$ and $V_{cb}$ are the elements corresponding to the charm-strange and charm-bottom quark transitions respectively.

Let $V$ be a CKM matrix. If the matrix element $V_{cb}$ is non-zero, then the element $V_{cs}$ is equal to the product of the ratio $R_{cs/cb}$ and $V_{cb}$, where $R_{cs/cb}$ is defined as the ratio of the charm-strange element to the charm-bottom element ($V_{cs} / V_{cb}$).