Bi-Large Mixing Patterns in Lepton Flavor Physics

• Bi-large mixing patterns are frameworks in neutrino physics featuring large solar and atmospheric mixing angles with a reactor angle approximated by the Cabibbo angle.
• They utilize CKM-like charged-lepton rotations and simple functional relations to derive predictive oscillation parameters and CP phase estimates.
• These models are tested by precise oscillation data from experiments like JUNO, NOνA, and DUNE, as well as neutrinoless double beta decay searches.

Bi-large mixing patterns in the context of leptonic flavor physics refer to analytic schemes for the lepton mixing (PMNS) matrix in which both solar ($\theta_{12}$) and atmospheric ($\theta_{23}$) angles are “large” and tightly correlated to each other, while the reactor angle ($\theta_{13}$) is nonzero and typically taken of the same magnitude as the Cabibbo angle ($\lambda_C \approx 0.22$). Technical implementations of bi-large mixing draw on simple functional relations between the mixing angles and $\lambda_C$, motivated by quark-lepton unification and GUTs. CP violation is often attributed to a single phase entering via the charged-lepton sector, allowing for highly predictive one- or two-parameter schemes. Recent analyses confront these patterns with oscillation data and precision measurements, such as those provided by JUNO, NO$\nu$A, DUNE, and cosmology.

1. Foundational Definitions and Parametrizations

Bi-large mixing patterns are defined by the structure of the neutrino diagonalization matrix $U_\nu$ where parameters are tied to the Cabibbo angle $\lambda$:

• Reactor angle: $\sin \theta_{13} = \lambda$
• Solar angle: $\sin \theta_{12} = s \lambda$ (often $s = 2$, $3$, or a free parameter $\psi \sim 3$)
• Atmospheric angle: $\sin \theta_{23} = a \lambda$ (similarly, $a = 1-\lambda$, $3$, or $\psi$)

The charged-lepton correction $U_l$ is constructed as a CKM-like rotation, often in the Wolfenstein parametrization:

• SO(10)-like: $$U_l = R_{23}(A\lambda^2)\ \Phi\ R_{12}(\lambda)\ \Phi^\dagger$$, $\Phi = \text{diag}(e^{-i\phi/2}, e^{+i\phi/2}, 1)$
• SU(5)-like: $$U_l = \Phi^\dagger R_{12}^T(\lambda)\ \Phi\ R_{23}^T(A\lambda^2)$$

The full PMNS matrix is then given by $U = U_l^\dagger U_\nu$ (Chen et al., 2019, Ding et al., 2019). Key “bi-large” schemes include:

• Pattern I (“1–2–1−λ”): $\sin \theta_{23}^\nu = 1-\lambda$, $\sin \theta_{12}^\nu = 2\lambda$, $\sin \theta_{13}^\nu = \lambda$
• Pattern II (“1–2–3”): $\sin \theta_{23}^\nu = 3\lambda$, $\sin \theta_{12}^\nu = 2\lambda$, $\sin \theta_{13}^\nu = \lambda$
• Generic bi-large: $\sin\theta_{12} = \sin\theta_{23} = \psi\lambda$, $\sin\theta_{13} = \lambda$ (Boucenna et al., 2012)

The mixing matrices for representative schemes are displayed below (for $\lambda = 0.2245$):

Pattern	$\sin\theta_{13}$	$\sin\theta_{12}$	$\sin\theta_{23}$	Notable CP Phase
I	$\lambda$	$2\lambda$	$1-\lambda$	$\delta_{CP} \sim 1.3\pi$
II	$\lambda$	$2\lambda$	$3\lambda$	$\delta_{CP} \sim 1.27\pi$
BL	$\lambda$	$3\lambda$	$3\lambda$	$\delta_{CP} \sim 0.25\pi$
GST-BL	$\sqrt{m_1/m_3}$	$\psi\lambda$	$\psi\lambda$	$\delta_{CP} \sim 1.53\pi$

2. Analytic Extraction of Oscillation Parameters

In all “bi-large” frameworks, analytic expressions for oscillation observables are obtained by expanding $U = U_l^\dagger U_\nu$ to leading nontrivial order in $\lambda$ and extracting the PMNS parameters:

• $\sin^2 \theta_{13}$
• $\sin^2 \theta_{12}$
• $\sin^2 \theta_{23}$
• Jarlskog invariant $J_{CP}$
• Dirac phase $\delta_{CP}$

For Pattern I (Chen et al., 2019, Ding et al., 2019): $$\sin^2\theta_{13} \simeq 4\lambda^2(1-\lambda)\cos^2(\phi/2),\quad \sin^2\theta_{12} \simeq 2\lambda^2\left[2-2\sqrt{2\lambda}\cos\phi + \lambda\right],\quad \sin^2\theta_{23} \simeq (1-\lambda)^2 - 2\sqrt{2}A\lambda^{5/2} - 2\lambda^3(1 + 2\cos\phi)$$ with $J_{CP} \simeq -2\sqrt{2}\lambda^{5/2}\sin\phi$, $\delta_{CP} = \arg(J_{CP})$.

The predictions are strongly constrained by fixing $\sin^2\theta_{13}$ to its experimental value, reducing the free phase $\phi$ to a narrow interval, which then tightly correlates the allowed ranges of $\theta_{12}, \theta_{23}, \delta_{CP}$.

Corresponding best-fit points:

• Pattern I: $\{\sin^2\theta_{12}=0.306$, $\sin^2\theta_{23}=0.573$, $\sin^2\theta_{13}=0.0216$, $\delta_{CP}=1.305\pi\}$
• Pattern II: $\{\sin^2\theta_{12}=0.320$, $\sin^2\theta_{23}=0.513$, $\sin^2\theta_{13}=0.0216$, $\delta_{CP}=1.270\pi\}$

3. Phenomenological Implications and Experimental Tests

The principal phenomenological signatures of bi-large mixing are:

• Two “large” angles ($\theta_{12} \simeq 34^\circ$–$36^\circ$, $\theta_{23} \simeq 46^\circ$–$49^\circ$), strongly correlated with $\theta_{13}$
• $\theta_{13}$ determined by the Cabibbo angle; typically $\theta_{13} \simeq 8.5^\circ$
• Near-maximal Dirac CP phase ($\delta_{CP} \sim 1.3\pi$) arising solely from the single charged-lepton phase $\phi$
• Highly predictive: narrow correlations among $\theta_{ij}$ and $\delta_{CP}$, visible in appearance probabilities and CP asymmetries
• Distinction between octants: Pattern I prefers higher octant $\sin^2\theta_{23}\gtrsim0.57$; Pattern II close to maximal ($\sim0.51$)

These features yield distinctive signals in long-baseline oscillation experiments such as T2K, NO$\nu$A, DUNE, and Hyper-Kamiokande, particularly through measurements of $\nu_\mu \rightarrow \nu_e$ appearance and CP-violating asymmetries (Ding et al., 27 Nov 2025).

4. Viability and Constraints from Current Data

Global oscillation fits and recent precision measurements (notably JUNO) have begun to winnow the parameter space of bi-large schemes:

• Type 1 (T1) remains viable at 1$\sigma$ for $\theta_{12}$, and is favored overall.
• Type 2 (T2) survives at 1–2$\sigma$, especially in lower octant scenarios.
• Types 3 and 4 (with extra parameter $\psi$) are excluded at 2$\sigma$ level, surviving only on marginal 3$\sigma$ branches (Ding et al., 27 Nov 2025, Ding et al., 2019).

Octant resolution and CP-phase measurement are decisive: T1 rules out maximal atmospheric mixing, excludes maximal CP violation, and predicts $\delta_{CP}$ in split branches around 0.7$\pi$ and 1.3$\pi$. Future data are expected to further constrain or exclude models without these correlations.

5. Model Building and GUT Connections

Bi-large mixing is closely related to unified model frameworks:

• Both $U_\nu$ and $U_l$ parameterizations often reflect GUT-motivated structures (SO(10), SU(5))
• The “revamped” BL scenario institutes a GST-like relation, $\sin\theta_{13} = \sqrt{m_1/m_3}$, enforcing normal ordering and forbidding $m_1=0$ (Roy et al., 2020)
• At the GUT scale ($M_{\rm GUT} \sim 10^{16}$ GeV), quark-lepton unification motivates the identification $\lambda = \sin \theta_C$ and $U_{lL} \sim V_\text{CKM}$

This framework provides a highly constrained mass matrix, leading to concrete predictions for low-energy parameters upon RG running to the weak scale. The sum of masses, effective $0\nu\beta\beta$ decay parameter $\langle m_{ee} \rangle$, and mass ordering are all predicted within current tolerances, with sensitivity to improved cosmological and double-beta decay experiments.

6. Phenomenology Beyond Oscillations: Neutrinoless Double Beta Decay

In all bi-large schemes, the effective Majorana mass

$$m_{\beta\beta} = \left| \sum_i U_{ei}^2 m_i \right|$$

lies in the normal ordering band, at the few-meV level for minimal $m_1$. Inclusion of precise $\theta_{12}$ (from JUNO) tightens the predicted $m_{\beta\beta}$ range, shrinking the “chimney” cancellation band (Ding et al., 27 Nov 2025). Next-generation $0\nu\beta\beta$ experiments (LEGEND, nEXO, KamLAND2-Zen, JUNO-$\beta\beta$) are approaching the relevant sensitivity, offering a direct experimental test of bi-large scenarios.

7. Prospects and Future Discrimination

Projected sensitivities for DUNE and Hyper-Kamiokande indicate that the three surviving bi-large patterns (T1, T2, T4) are distinguishable at better than 3$\sigma$ after several years’ data taking, depending on the true values of $\sin^2\theta_{23}$ and $\delta_{CP}$ (Ding et al., 2019). This discriminating power is further enhanced by the synergy with reactor ($\theta_{12}$, $\theta_{13}$), atmospheric, and $0\nu\beta\beta$ measurements.

A plausible implication is that only bi-large schemes with nonmaximal $\theta_{23}$ and $\delta_{CP} \approx 0.7\pi$–$1.3\pi$ are likely to remain viable as experimental precision continues to improve. The elimination of maximally symmetric patterns (BM, TBM) further emphasizes the empirical robustness of bi-large mixing as a model-building standard in the present era.