Unitarity Triangle and Quadrangle

• Unitarity triangles and quadrangles are geometric representations that map flavor mixing parameters and CP-violating phases in both quark and lepton sectors.
• They provide a framework to extract measurable quantities such as the Jarlskog invariant and CP asymmetries from experimental data.
• Advanced constructs like the unitarity boomerang and quadrangle enable complete parameter determination and offer sensitive probes for new physics.

The unitarity triangle and quadrangle are geometric constructs in the complex plane that encode the structure of flavor mixing and CP violation in the Standard Model and its extensions. These constructions play a foundational role in both quark and lepton sectors, facilitating the visualization, extraction, and cross-verification of the physical parameters buried within mixing matrices. They also provide a powerful framework for testing the completeness and consistency of the Standard Model, especially with regard to the number of physical parameters (mixing angles and CP-violating phases), and act as sensitive tools for uncovering signs of new physics.

1. Unitarity Triangles: Formalism and Physical Significance

For any unitary 3×3 flavor mixing matrix $V$ (either $V_{\text{CKM}}$ or $U_{\text{PMNS}}$), unitarity implies orthogonality among its rows and columns, resulting in six independent relations of the form

$$\sum_k V_{ik} V^*_{jk} = 0, \quad \text{for } i \ne j,$$

each of which can be represented as a closed triangle in the complex plane by drawing the three complex vectors corresponding to the terms in the sum—these are the "unitarity triangles" (UTs).

Each triangle graphically encodes three of the four independent physical parameters of $V$ (three mixing angles and one CP-violating phase). The area of every UT is proportional to the Jarlskog invariant,

$J = c_1 c_2 c_3 s_1^2 s_2 s_3 \sin\delta,$

so all triangles of a unitary $V$ have equal area. The internal angles of these triangles correspond to physically measurable CP-violating combinations, such as $\alpha$, $\beta$, and $\gamma$ in the quark sector, or analogs in the lepton sector.

A concise table below summarizes core structural features:

Feature	Quark Sector (CKM)	Lepton Sector (PMNS)
Number of UTs	6 (from 6 orthogonality)	6 (Dirac, plus 3 "Majorana" UTs)
Area (CPV)	$J_{\text{CKM}}/2$	$J_{\text{PMNS}}/2$
Angles	$\alpha$, $\beta$, $\gamma$	Analogous quantities
Hierarchy	Strong	Weak ("nearly equilateral" UTs)

2. The Unitarity Boomerang and Quadrangle: Completing the Parameter Set

While a single unitarity triangle makes three physical quantities explicit (e.g., two internal angles and the area $J$), it does not display all four parameters required to specify a general $3\times3$ unitary matrix. The "unitarity boomerang" (Frampton et al., 2010, Li et al., 2010) overcomes this by combining two specially selected UTs sharing a common internal angle. This construction reveals all four parameters (three rotation angles plus the CP-violating phase), thus fully specifying $V$ using only geometrically accessible, rephasing-invariant quantities.

A typical boomerang is constructed as follows:

• Identify two UTs—UT(a) and UT(b)—which share a common internal angle.
• Each triangle contributes three independent quantities (side lengths and/or internal angles), but by sharing one angle, their union yields four independent, experimentally accessible parameters.
• In the original Kobayashi-Maskawa parameterization, the phase $\delta$ is given (up to small corrections) by $\delta \simeq \pi - \phi_2$, with $\phi_2$ the shared angle.

Quadrangles arise naturally in situations where more than three independent phases enter the structure of a mixing matrix. For a $4\times4$ unitary matrix (as in the presence of sterile neutrinos), the unitarity conditions involve sums over four terms, which in the complex plane define closed quadrangles rather than triangles (Verma et al., 2016). These "leptonic unitarity quadrangles" (LUQ) contain five independent geometrical parameters (four sides, four angles with two constraints), allowing them to encode all physical parameters of the extended mixing matrix, including the new phases introduced by sterile states.

3. Extraction and Parametrization-Dependence of Physical Parameters

The precise mapping from geometric properties (triangle or quadrangle side lengths and angles) to matrix parameters can depend on the adopted parametrization (e.g., Kobayashi-Maskawa, Chau-Keung, PDG). Explicit rephasing-invariant formulae (Yang, 25 Aug 2025, Yang, 15 Sep 2025) link the CP phase(s) in any parameterization directly to geometrically defined angles:

$$\delta^{(\alpha i)} = \arg\left[ \frac{ V_{\alpha 1} V_{\alpha 2} V_{\alpha 3} V_{1i} V_{2i} V_{3i} }{ V_{\alpha i}^3 \det V } \right].$$

Sum rules further relate different parametrization CP phases to the unitarity triangle angles, such as

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

showing that the phases are not just abstract parameters but correspond to explicit geometric angles on the complex plane (Yang, 15 Sep 2025).

In extended flavor spaces (e.g., 4 flavors), the LUQ's five independent geometric parameters completely determine the five physical parameters (four mixing angles, one CP phase), which can be mapped to oscillation phenomenology or other observables.

4. Experimental Determination and Global Analyses

The angles and sides of unitarity triangles (and quadrangles in the case of non-unitary mixing) are reconstructed from experimental data such as:

• CP asymmetries in $B$ meson decays (for $\beta$, $\gamma$ via tree and penguin processes) (Graham et al., 2011).
• Isospin analysis of $B \to \pi\pi$, $B \to \rho\rho$ and Dalitz-plot analyses for $\alpha$.
• Neutrino disappearance and appearance oscillation channels, mapped into triangles (or quadrangles if sterile states exist) for extraction of the Dirac and Majorana phases (Ellis et al., 2020, Xing et al., 2015).

Global fits constrain the apex of the unitarity triangle $(\bar\rho, \bar\eta)$, as in the UTfit approach (Bevan et al., 2014). Consistency among the different extractions, and sum rules relating various angles, provide overconstraints that test Standard Model unitarity.

A notable feature is that, due to ambiguities and experimental uncertainties, direct assignments of side lengths using measured $|V_{ij}|$ may sometimes fail the triangle inequality. Constrained fits—imposing area equality among triangles—are then used to enforce physical plausibility and test for consistency (Wysozka et al., 2022). Tensions (e.g., 2σ discrepancies in triangle closure) can drive further investigations.

5. Implications for CP Violation and Model Building

The geometric perspective on unitarity triangles and quadrangles underlies many phenomenological and theoretical insights:

• Origin of CP violation: The area ($\propto J$) of the triangle quantifies the magnitude of CP violation; a right unitarity triangle (α ≈ 90°) is closely tied to maximal or near-maximal CP-violating effects (Antusch et al., 2011, Antusch et al., 2013).
• Mass texture models: Discrete symmetries or specific mass matrix forms can enforce relations among the elements, naturally setting triangle angles (e.g., α ≈ 90°, β ≈ 22.6°) and illuminate possible solutions to the strong CP problem (Antusch et al., 2011, Antusch et al., 2013, Harrison et al., 30 Jan 2025).
• Lepton sector extensions: The analogy between the quark and lepton sectors allows the exploration of quark–lepton complementarity and the construction of geometric frameworks for the PMNS matrix, which also accommodates potential Majorana phases and lepton-number-violating phenomena (Xing et al., 2015).
• Beyond the Standard Model: Deviations from triangle closure, violations of area equality, or non-overlapping Jarlskog invariants across different unitarity triangles provide sensitive probes for new physics, including non-unitary mixing from sterile neutrinos or new CP-violating phases (Verma et al., 2016, Ellis et al., 2020).

6. Generalizations, Inverse Triangles, and Advanced Geometric Representations

Further sophistication arises in the geometric analysis by considering:

• Construction of alternative triangles (via inversion formulas, $U^\dagger = U^{-1}$), which leads to "inverse unitarity triangles" that encode higher-order invariants (with determinants). These add further constraints and rephasing-invariant measures, effectively linking the fifth-order invariants (with det $V$) to the usual triangle parameters (Yang, 15 Sep 2025).
• Organization of sum rules and rephasing invariants for a unified mapping between parameterization-dependent CP phases and geometric properties, thereby reducing redundancies and elucidating the true count of independent observables in flavor physics (Yang, 25 Aug 2025).
• Deformation of geometric objects (e.g., LUT → ELUT in matter, or the collapse to mass triangles/isosceles shapes under limiting conditions) as a diagnostic for emergent symmetries or experimental signatures (e.g., maximally violated CP in the lepton sector under μ–τ reflection symmetry) (He et al., 2016, Xing et al., 2015).

7. Experimental and Theoretical Trends

Advances in both theoretical formalism and experimental precision (e.g., improved measurement of $|V_{cb}|$ and $\gamma$ via rare kaon and $B$ decays (Buras, 2022)) are sharpening the utility of geometric unitarity representations. For instance, "boomerang plots" or $|V_{cb}|$–$\gamma$ plots are superior to conventional (ρ̄, η̄) unitarity triangle displays for directly correlating precise determinations and identifying new physics inconsistencies.

Future experiments (DUNE, T2HK, JUNO, upgraded LHCb and Belle-II) promise unprecedented constraints, and improved measurement-induced overconstraints may further test (or challenge) the unitarity paradigm. Discrepancies among various triangle constructions or quadrangle parameters would signal the presence of new dynamics, such as light sterile neutrino admixtures, additional generations, or non-standard interactions (Verma et al., 2016, Ellis et al., 2020, Masud et al., 2021).

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In summary, the unitarity triangle and quadrangle offer a powerful, rephasing-invariant, and geometric framework for elucidating and testing the structure of the Standard Model's flavor sector in both quark and lepton domains. They bridge the gap between abstract matrix elements and experimentally accessible CP-violating observables, act as a diagnostic for new sources of CP violation, and provide a unifying paradigm for interpreting experimental and theoretical progress across the entirety of flavor physics.