Inverse Unitarity Triangles

• Inverse unitarity triangles are geometric constructs in the complex plane that encode the algebraic structure and CP phases of mixing matrices using inversion properties.
• They provide a unified representation of CP violation for both quark (CKM) and lepton (PMNS) sectors by employing higher-order rephasing invariants and sum rules.
• The framework connects geometric angles to CP phases through duality and precise sum rules, serving as a powerful tool for precision flavor physics and new physics searches.

Inverse unitarity triangles are geometric constructs in the complex plane that encode the algebraic structure and physical parameters of unitary mixing matrices, especially when derived from inversion properties, higher-order rephasing invariants, and generalized parameterizations. They provide alternative and sometimes more informative representations of CP violation, unitarity constraints, and new physics sensitivities in both the quark (CKM) and lepton (PMNS) sectors.

1. Definition and Algebraic Construction

A unitary matrix $U$ satisfies $U^\dagger = U^{-1}$, and its inverse can be written in terms of cofactors and determinant as:

$$U = \frac{1}{\det(U^*)} \begin{pmatrix} U^*_{22}U^*_{33}-U^*_{23}U^*_{32} & U^*_{23}U^*_{31}-U^*_{21}U^*_{33} & U^*_{21}U^*_{32}-U^*_{22}U^*_{31} \ U^*_{13}U^*_{32}-U^*_{12}U^*_{33} & U^*_{11}U^*_{33}-U^*_{13}U^*_{31} & U^*_{12}U^*_{31}-U^*_{11}U^*_{32} \ U^*_{12}U^*_{23}-U^*_{13}U^*_{22} & U^*_{13}U^*_{21}-U^*_{11}U^*_{23} & U^*_{11}U^*_{22}-U^*_{12}U^*_{21} \end{pmatrix}$$

(Yang, 15 Sep 2025). Each term in this matrix can be represented as a vector sum in the complex plane, $X + Y + Z = 0$, forming a triangle whose closure reflects the underlying unitarity.

Inverse unitarity triangles are thus constructed from sum relations among cofactors (not simply from row or column orthogonality conditions), and contain phase information from determinants and higher-order products of the mixing matrix elements. These triangles serve as duals or complements to the conventional unitarity triangles, encoding CP phases and mixing observables in geometric forms that highlight inversion symmetry.

2. Rephasing Invariant Formulations and Sum Rules

The foundation of this geometric representation is the use of rephasing-invariant quantities. For the CKM or PMNS matrix $V$, third-order rephasing invariants take the form

$$\chi_1 = \arg \left( \frac{V_{11}V_{22}V_{33}}{\det V} \right), \quad \psi_1 = \arg \left( - \frac{V_{11}V_{23}V_{32}}{\det V} \right), \text{etc.}$$

(Yang, 31 Aug 2025). These invariants allow both the nine angles of unitarity triangles and the nine CP phases from Euler-angle parameterizations $\delta^{(\alpha i)}$ to be written in a unified formalism.

Compact sum rules relate these CP phases and triangle angles:

$$\delta^{(\alpha, i+2)} - \delta^{(\alpha, i+1)} = \Phi_{\alpha-2, i} - \Phi_{\alpha-1, i}$$

with all indices modulo three (Yang, 25 Aug 2025). These sum rules generalize known relations such as $\delta_{PDG} + \delta_{KM} = \pi - \alpha + \gamma$ and reveal a deep connection—often called "duality"—between CP phases (derived from inverse invariants) and triangle geometry.

Two concise matrix equations organize all triangle angles ($\Phi$) and CP phases ($\Delta$):

$\Phi = \Psi - X, \qquad \Delta = \Pi' - \Psi - X,$

where $\Psi$ and $X$ are matrices of odd and even permutations of the third-order invariants, and $\Pi'$ encodes summed invariants (Yang, 31 Aug 2025). These relations underscore the equivalence between angles and CP phases, up to the structure imposed by third- and higher-order invariants.

3. Geometric Interpretation of CP Phases and Physical Angles

Inverse unitarity triangles provide a geometric picture where CP phases such as $\delta_{PDG}$ and $\delta_{KM}$ are represented as specific angles in the complex plane:

$$\delta_{PDG} = \arg \left( \frac{V_{11}V_{12}V_{23}V_{33}}{V_{13} \det V} \right), \quad \delta_{KM} = \pi - \arg \left( \frac{V_{12}V_{13}V_{21}V_{31}}{V_{11} \det V} \right)$$

(Yang, 15 Sep 2025). These geometric representations allow unambiguous identification of CP-violating phases, and their relation to interior angles of polygons or quadrangles (when two triangles are joined via sum rules).

For example, the sum rule

$\delta_{PDG} + \delta_{KM} = \pi - \alpha + \gamma$

can be interpreted via the construction of a quadrangle composed of a unitarity triangle and an alternative triangle obtained from the inversion formula. The internal angles (corresponding to $\alpha, \delta_{PDG}, \pi - \gamma, \delta_{KM}$) sum to $2\pi$, expressing the underlying relation in geometric terms.

4. Inverse Unitarity Triangles in the Lepton Sector and Non-unitarity

The concept extends naturally to the lepton sector, especially in the presence of new physics such as the inverse seesaw mechanism (González-Quiterio et al., 6 Sep 2024). In this scenario, the mixing matrix for light neutrinos becomes non-unitary,

$\mathcal{B}_n = (1 - \eta) \hat{V}^*,$

with $\eta = \frac{1}{2} m_D M^{-2} m_D^\dagger$. The closure conditions for the sum of the products of elements (e.g., $$\sum_{j=1}^{3} \mathcal{B}_{\alpha n_j} \mathcal{B}_{\beta n_j}^*$$) no longer yield exact triangles, leading to defective closure—graphically represented as a "gap" in the inverse unitarity triangle.

Experimental constraints (e.g., from $\mu \to e \gamma$ branching ratios) directly limit the magnitude of non-unitarity, with improved sensitivity expected from future experiments. The degree of closure defect in the inverse triangle correlates with the size of $\eta_{\alpha\beta}$, linking geometric features to measurable quantities.

5. Applications in Phenomenology and New Physics Searches

Inverse unitarity triangles and their related invariants offer powerful tools for phenomenological analyses:

• In the quark sector, inverse triangles capture all four independent parameters of the mixing matrix and serve as complete visualizations of mixing (Li et al., 2010, Frampton et al., 2010).
• Deviations from closure or consistency in the geometric invariants signal possible new physics, such as non-unitarity from active-sterile mixing or additional heavy states (Xing et al., 2020, González-Quiterio et al., 6 Sep 2024).
• In the lepton sector, the general framework enables constraints on astrophysical neutrino flavor ratios entirely from geometric unitarity bounds, independent of specific model details (Xu et al., 2014).
• In parameter space studies, duality and sum rules among inverse invariants allow for cross-checks and consistency tests among different CP phase definitions, vital for precision flavor physics (Yang, 25 Aug 2025, Yang, 31 Aug 2025, Yang, 15 Sep 2025).

6. Generalization and Unified Framework

Inverse unitarity triangles are not confined to specific parameterizations or sectoral applications—they emerge as a general feature of unitary matrices and their invariant algebra. By expressing both CP phases and unitarity triangle angles in terms of third-order rephasing invariants, the framework unifies the geometric and algebraic representations of flavor mixing. It accommodates all major parameterizations (PDG, KM, Chau-Keung, etc.), allows geometric interpretation of sum rules, and connects to higher-order invariants via duality (Yang, 31 Aug 2025).

This approach suggests that future explorations—especially around "inverse" constructions or precision measurements—will reveal deeper connections between underlying symmetries, geometric properties, and physical observables in both quark and lepton flavor physics.

7. Summary Table: Key Geometric Objects and Their Physical Significance

Object/Formula	Algebraic Content	Physical Interpretation
Inverse unitarity triangle	Sum of cofactors equals zero	Encodes full mixing & CP structure
CP phase $\delta^{(\alpha i)}$ (general)	$$\arg \left( \frac{ \prod V }{V_{\alpha i}^3 \det V} \right)$$	Geometric angle in inverse triangle
Sum rule (e.g. $\delta_{PDG} + \delta_{KM}$)	Relation among angles, CP phases, determinants	Geometric quadrangle in complex plane
Third-order invariants ($\chi_i$, $\psi_i$)	Product of 3 elements divided by $\det V$	Unified generator of angles, phases

Inverse unitarity triangles thus provide a rigorous and unified geometric-algebraic framework for analyzing mixing matrices, CP violation, and the search for new physics in flavor processes, linking high-order invariants and general parameterizations to experimental and theoretical insights.