Physlib.Particles.FlavorPhysics.CKMMatrix.Invariants

For a CKM matrix $V$, the complex Jarlskog invariant is defined as the complex number $V_{us} V_{cb} V_{ub}^* V_{cs}^*$, where $V_{ij}$ denotes the entry of the matrix $V$ corresponding to row $i$ and column $j$, and $V^*$ denotes the complex conjugate.

Let $V$ and $U$ be CKM matrices, represented as $3 \times 3$ complex unitary matrices. If $V$ and $U$ are equivalent under the phase rephasing relation ($V \approx U$), then their complex Jarlskog invariants are equal: $$\text{jarlskog}\mathbb{C}_{CKM}(V) = \text{jarlskog}\mathbb{C}_{CKM}(U)$$ The complex Jarlskog invariant for a CKM matrix $V$ is defined as $V_{us} V_{cb} V_{ub}^* V_{cs}^*$, where $V_{ij}$ are the matrix entries and $V^*$ denotes the complex conjugate.

The complex Jarlskog invariant for an equivalence class $[V]$ of CKM matrices is the complex number defined by $$\text{jarlskog}\mathbb{C}([V]) = V_{us} V_{cb} V_{ub}^* V_{cs}^*$$ where $V$ is any representative $3 \times 3$ unitary matrix of the equivalence class, $V_{ij}$ represents the entry at row $i$ and column $j$, and $V^*$ denotes the complex conjugate. This value is well-defined on the quotient space because the underlying expression $V_{us} V_{cb} V_{ub}^* V_{cs}^*$ is invariant under the phase rephasing equivalence relation $V \approx U$.

For an equivalence class $[V]$ of CKM matrices in the quotient space of $3 \times 3$ unitary matrices under phase-rephasing, this function calculates the real-valued invariant defined as: $$\frac{|V_{us}|^2 |V_{ub}|^2 |V_{cb}|^2}{|V_{ud}|^2 + |V_{us}|^2}$$ where $|V_{ud}|, |V_{us}|, |V_{ub}|,$ and $|V_{cb}|$ are the magnitudes (absolute values) of the matrix elements corresponding to the transitions between the specified quarks (up, charm, down, strange, and bottom).

For an equivalence class $[V]$ of CKM matrices in the quotient space of $3 \times 3$ unitary matrices under phase-rephasing, this function defines a complex-valued invariant given by the sum of the complex Jarlskog invariant and a specific real-valued invariant: $$\text{jarlskog}_{\mathbb{C}}([V]) + \frac{|V_{us}|^2 |V_{ub}|^2 |V_{cb}|^2}{|V_{ud}|^2 + |V_{us}|^2}$$ where $\text{jarlskog}_{\mathbb{C}}([V]) = V_{us} V_{cb} V_{ub}^* V_{cs}^*$ and $|V_{ij}|$ are the magnitudes of the CKM matrix elements. The complex argument of this resulting value corresponds to the CP-violating phase $\delta_{13}$ in the standard parameterization of the CKM matrix.