Physlib.Particles.NeutrinoPhysics.Basic

Given a function $\theta: \{0, 1, 2\} \to \mathbb{R}$ that maps each index to a real-valued phase, `diagPhase` defines a $3 \times 3$ diagonal complex matrix. For any indices $i, j \in \{0, 1, 2\}$, the entry of the matrix is given by $e^{i \theta_i}$ if $i = j$, and is $0$ if $i \neq j$.

The diagonal phase matrix $\text{diag}(e^{i\theta_0}, e^{i\theta_1}, e^{i\theta_2})$ is equal to the identity matrix when all the phases $\theta_j$ are zero.

Let $\text{diagPhase}(\theta)$ be the $3 \times 3$ diagonal complex matrix defined by a phase function $\theta: \{0, 1, 2\} \to \mathbb{R}$, where the diagonal entries are given by $e^{i\theta_j}$ for $j \in \{0, 1, 2\}$. The matrix $\text{diagPhase}(0)$, where $0$ is the zero element of the function space, is equal to the matrix $\text{diagPhase}(\theta)$ where $\theta$ is the constant function $\theta(j) = 0$ for all $j$.

For any function $\theta: \{0, 1, 2\} \to \mathbb{R}$ representing a set of three real phases, the Hermitian conjugate (conjugate transpose) of the diagonal phase matrix $\text{diag}(e^{i\theta_0}, e^{i\theta_1}, e^{i\theta_2})$ is equal to the diagonal phase matrix with negated phases, $\text{diag}(e^{-i\theta_0}, e^{-i\theta_1}, e^{-i\theta_2})$. That is, $(\text{diagPhase } \theta)^\dagger = \text{diagPhase } (-\theta)$.

For any two functions $\theta, \phi: \{0, 1, 2\} \to \mathbb{R}$ representing sets of phase angles, the product of the diagonal phase matrices $\text{diagPhase}(\theta)$ and $\text{diagPhase}(\phi)$ is equal to the diagonal phase matrix $\text{diagPhase}(\theta + \phi)$. Here, $\text{diagPhase}(\theta)$ denotes the $3 \times 3$ complex diagonal matrix $\text{diag}(e^{i\theta_0}, e^{i\theta_1}, e^{i\theta_2})$.

For any vector of phases $\theta: \{0, 1, 2\} \to \mathbb{R}$, `diagPhase_unitary` defines a $3 \times 3$ diagonal unitary matrix in $U(3)$ whose entries are given by $e^{i \theta_j}$ for $j \in \{0, 1, 2\}$. Specifically, it is the matrix $$\begin{pmatrix} e^{i \theta_0} & 0 & 0 \\ 0 & e^{i \theta_1} & 0 \\ 0 & 0 & e^{i \theta_2} \end{pmatrix}$$ viewed as an element of the unitary group $U(3)$.

For any function $\theta: \{0, 1, 2\} \to \mathbb{R}$ representing a set of three phase angles, the matrix underlying the unitary group element $\text{diagPhase\_unitary}(\theta)$ is equal to the diagonal complex matrix $\text{diagPhase}(\theta)$, whose entries are $e^{i\theta_j}$ for $j \in \{0, 1, 2\}$.

Given three real numbers $a, b, c \in \mathbb{R}$, the lepton phase shift matrix is the $3 \times 3$ diagonal complex matrix defined by: $$\text{diag}(e^{ia}, e^{ib}, e^{ic}) = \begin{pmatrix} e^{ia} & 0 & 0 \\ 0 & e^{ib} & 0 \\ 0 & 0 & e^{ic} \end{pmatrix}$$ This matrix represents the phase shift freedom of the charged lepton sector in the context of the PMNS (Pontecorvo–Maki–Nakagawa–Sakata) framework.

Given three real numbers $d, e, f \in \mathbb{R}$, the neutrino phase shift matrix is defined as the $3 \times 3$ diagonal complex matrix: $$\text{diag}(e^{id}, e^{ie}, e^{if}) = \begin{pmatrix} e^{id} & 0 & 0 \\ 0 & e^{ie} & 0 \\ 0 & 0 & e^{if} \end{pmatrix}$$ This matrix represents the phase shift freedom of the neutrino sector, assuming neutrinos are Dirac particles.

Given two real parameters $\alpha_1$ and $\alpha_2$, the Majorana phase matrix is defined as the $3 \times 3$ diagonal complex matrix: $$\text{diag}(1, e^{i \alpha_1/2}, e^{i \alpha_2/2})$$ In the context of neutrino physics, if neutrinos are Majorana particles, this matrix represents the physical phase shifts that cannot be absorbed into the definition of the neutrino fields.

Two $3 \times 3$ complex unitary matrices $U, V \in U(3)$ satisfy the Dirac PMNS equivalence relation if there exist real-valued phase functions $\theta, \phi: \{0, 1, 2\} \to \mathbb{R}$ such that $$U = \text{diag}(e^{i\theta_0}, e^{i\theta_1}, e^{i\theta_2}) \cdot V \cdot \text{diag}(e^{i\phi_0}, e^{i\phi_1}, e^{i\phi_2})$$ where $\text{diag}(e^{i\alpha_k})$ denotes the diagonal matrix with entries $e^{i\alpha_0}, e^{i\alpha_1}, e^{i\alpha_2}$. This relation represents the equivalence of mixing matrices under the rephasing of lepton fields in the Dirac neutrino case.

The Dirac PMNS equivalence relation is reflexive: for any $3 \times 3$ complex unitary matrix $U \in U(3)$, $U$ is Dirac PMNS equivalent to itself. Two $3 \times 3$ complex unitary matrices $U, V \in U(3)$ satisfy the Dirac PMNS equivalence relation if there exist real-valued phase functions $\theta, \phi: \{0, 1, 2\} \to \mathbb{R}$ such that $$U = \text{diag}(e^{i\theta_0}, e^{i\theta_1}, e^{i\theta_2}) \cdot V \cdot \text{diag}(e^{i\phi_0}, e^{i\phi_1}, e^{i\phi_2})$$ where $\text{diag}(e^{i\alpha_k})$ denotes the diagonal matrix with entries $e^{i\alpha_0}, e^{i\alpha_1}, e^{i\alpha_2}$. This relation represents the physical equivalence of mixing matrices under the rephasing of lepton fields in the Dirac neutrino case.

For any $3 \times 3$ complex unitary matrices $U, V \in U(3)$, if $U$ is Dirac PMNS equivalent to $V$, then $V$ is Dirac PMNS equivalent to $U$. Two $3 \times 3$ complex unitary matrices $U, V \in U(3)$ satisfy the Dirac PMNS equivalence relation if there exist real-valued phase functions $\theta, \phi: \{0, 1, 2\} \to \mathbb{R}$ such that $$U = \text{diag}(e^{i\theta_0}, e^{i\theta_1}, e^{i\theta_2}) \cdot V \cdot \text{diag}(e^{i\phi_0}, e^{i\phi_1}, e^{i\phi_2})$$ where $\text{diag}(e^{i\alpha_k})$ denotes the diagonal matrix with entries $e^{i\alpha_0}, e^{i\alpha_1}, e^{i\alpha_2}$. This relation represents the physical equivalence of mixing matrices under the rephasing of lepton fields in the Dirac neutrino case.

For any $3 \times 3$ complex unitary matrices $U, V, W \in U(3)$, if $U$ is Dirac PMNS equivalent to $V$ and $V$ is Dirac PMNS equivalent to $W$, then $U$ is Dirac PMNS equivalent to $W$. Two matrices $A, B \in U(3)$ satisfy the Dirac PMNS equivalence relation if there exist real-valued phase functions $\theta, \phi: \{0, 1, 2\} \to \mathbb{R}$ such that $$A = \text{diag}(e^{i\theta_0}, e^{i\theta_1}, e^{i\theta_2}) \cdot B \cdot \text{diag}(e^{i\phi_0}, e^{i\phi_1}, e^{i\phi_2})$$ where $\text{diag}(e^{i\alpha_k})$ denotes the diagonal matrix with entries $e^{i\alpha_0}, e^{i\alpha_1}, e^{i\alpha_2}$.