⚛️ phenomenology

Minimal Quark-Lepton Complementarity from a Rigid Deformation of Tri-Bimaximal Mixing

This paper proposes a minimal, parameter-free model of quark-lepton complementarity where a rigid, unitary rotation of the tri-bimaximal mixing matrix about the axis $(2,1,2)^T/3$ by an angle fixed to the Cabibbo angle successfully predicts current neutrino oscillation parameters, including a near-maximal atmospheric angle and a near-CP-conserving Dirac phase, while ranking as the simplest viable candidate among thousands of alternatives.

Original authors: Gazal Sharma, Gaurav Katoch

Published 2026-08-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Gazal Sharma, Gaurav Katoch

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, there are two families of particles that make up the visible universe: quarks, which build protons and neutrons, and leptons, which include the electron and the neutrino. For decades, physicists have been puzzled by how these two families behave when they change their identity. When quarks switch types, they do so with a very specific, orderly pattern that is almost perfectly aligned. When leptons, specifically neutrinos, switch types, they do so with a wild, chaotic pattern involving large angles of change. This stark contrast between the orderly quarks and the chaotic leptons is one of the deepest mysteries in particle physics. Scientists have long suspected that these two patterns are not random but are secretly connected, a concept known as quark-lepton complementarity. The idea is that if you look at the mixing patterns of both families together, they might reveal a hidden, simpler rule that governs them both, much like two different languages that share the same underlying grammar.

A team of researchers has now proposed a specific, minimal way to connect these two worlds. They suggest that the chaotic mixing of neutrinos can be understood as a simple, rigid adjustment to a perfect, theoretical pattern called tri-bimaximal mixing. Imagine this perfect pattern as a blueprint that was once thought to describe neutrinos exactly, but was later found to be slightly off. The researchers propose that the real-world neutrino mixing is this blueprint, but rotated slightly around a specific, fixed axis. The size of this rotation is not a random number; it is locked directly to the mixing angle of the quarks, specifically the Cabibbo angle, which is a well-measured value in the quark sector. By tying the neutrino rotation to the quark angle, the researchers created a model that requires no new, adjustable numbers to explain the behavior of neutrinos. Once the known quark data is plugged in, the model predicts exactly how neutrinos should mix.

The results of this calculation are strikingly precise. Using the most recent data on quark mixing, the model predicts four key numbers that describe how neutrinos change their identity. It forecasts that the first mixing angle should be about 33.29 degrees, the reactor angle 8.46 degrees, and the atmospheric angle 49.19 degrees. Perhaps most notably, it predicts that the phase responsible for matter-antimatter differences in neutrinos should be very close to 180 degrees, suggesting that neutrinos almost perfectly conserve a symmetry called CP, rather than violating it strongly. The researchers tested the robustness of their idea by looking back at data from 2016 and 2018. They scanned thousands of other possible mathematical combinations of axes and rotation sizes to see if any other simple pattern would have fit the data better. Their specific choice of a fixed axis and a rotation size of three-quarters of the quark angle ranked first among the simplest options in both years, outperforming thousands of more complex alternatives.

However, the model is not without its tensions. The current best measurements of neutrino mixing, which include data from large underground detectors, show a slight preference for the atmospheric angle to be in a different range than the one predicted by this model. The model places the angle in the upper part of its possible range, while the global data leans toward the lower part. This discrepancy is small but significant enough to make the model testable. If future experiments confirm that the atmospheric angle is indeed in the lower range, this specific rigid connection between quarks and leptons would be ruled out. Similarly, if experiments find a large violation of the CP symmetry in neutrinos, the model would also fail, as it predicts a near-perfect conservation. The researchers emphasize that this is not a complete theory of everything, but a precise, falsifiable hypothesis. It offers a clear, economical way to link two seemingly unrelated parts of nature, providing a concrete target for the next generation of experiments to either confirm or disprove.

Technical Summary: Minimal Quark–Lepton Complementarity from a Rigid Deformation of Tri-Bimaximal Mixing

Problem Statement
The paper addresses the structural puzzle in flavor physics regarding the disparity between quark and lepton mixing. While the Cabibbo-Kobayashi-Maskawa (CKM) matrix is close to the identity, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix features two large mixing angles and a non-zero reactor angle. Although Quark-Lepton Complementarity (QLC) has been explored through various formulations (e.g., $\theta_C + \theta_{12}^\ell \simeq \pi/4$ ) and perturbations of Tri-Bimaximal (TBM) mixing, the authors seek a specific, minimal realization that fixes all Dirac PMNS observables without introducing new continuous parameters once the CKM matrix is supplied.

Methodology
The authors propose a minimal three-family ansatz where the PMNS matrix is constructed as a product of the CKM matrix, the TBM matrix, and a specific rigid rotation:
$U_{PMNS} = V_{CKM}^\dagger U_{TBM} R_{n_{gg}}\left(-\frac{3}{4}\theta_C\right)$
Key methodological components include:

Rigid Deformation: The correlation matrix is defined as a rotation $R_{n_{gg}}$ about a normalized axis $n_{gg} = (2, 1, 2)^T/3$ in the canonical TBM basis.
Parameter Locking: The rotation angle is strictly locked to the Cabibbo angle $\theta_C$ via the relation $\alpha_{gg} = -3\theta_C/4$ . This eliminates new continuous parameters in the Dirac lepton-mixing sector.
Convention Covariance: To ensure the axis $(2, 1, 2)$ is not an artifact of basis choice, the authors define a flavor-space generator $\Gamma_{gg} = U_{TBM}[n_{gg}]_\times U_{TBM}^T$ , which transforms covariantly under weak-basis transformations.
Analytic Derivation: Leading-order Wolfenstein expansions are derived to establish sum rules for mixing angles and the Jarlskog invariant ( $J_{CP}^\ell$ ).
Retrospective Scanning: To test robustness, the authors perform a "simplicity scan" over 5,510 (baseline) and 49,622 (enlarged) candidates consisting of primitive integer axes and rational rotation coefficients, comparing their fit against epoch-matched 2016 and 2018 oscillation data.
Effective Realization: An exact effective Majorana mass tensor realization is formulated as a quadratic polynomial $M_\nu = AI + B\Sigma_{gg} + C\Sigma_{gg}^2$ , where $\Sigma_{gg}$ encodes the flavor orientation.

Key Results
Using the 2025 Particle Data Group (PDG) CKM fit, the strict model yields the following predictions:

Mixing Angles: $\theta_{12} = 33.29^\circ$ , $\theta_{13} = 8.46^\circ$ , $\theta_{23} = 49.19^\circ$ .
CP Phase: $\delta_{CP} = 180.79^\circ$ .
CP Violation: The Jarlskog invariant is suppressed to $J_{CP}^\ell \approx -4.51 \times 10^{-4}$ , scaling as $O(\lambda^3)$ , implying near-CP conservation.
Uncertainties: The propagated uncertainties from CKM inputs are significantly smaller than current oscillation errors, making the predictions sharp.

Retrospective and Simplicity Analysis

In the baseline scan of 5,510 candidates, the pair $((2, 1, 2), 3/4)$ ranks first for both 2016 and 2018 data.
In the enlarged scan of 49,622 candidates, the model ranks second in raw fit quality (behind a more complex $(5, -4, 3)$ axis with $r=8/5$ ) but remains first among all candidates with a description complexity $C \leq 12$ (where $C$ is the sum of absolute values of integers and rationals).
A continuous profile of the rotation strength $r$ for the fixed $(2, 1, 2)$ axis yields an optimal value of $r_{BF} = 0.74941$ , which is numerically close to the structural hypothesis $3/4$ .

Current Status and Falsifiability
Comparing the model against the NuFIT 6.1 analysis (normal ordering, including atmospheric data):

The solar ( $\theta_{12}$ ) and reactor ( $\theta_{13}$ ) angles are within $\sim 1.1\sigma$ and $\sim 1.5\sigma$ of the global best-fit values, respectively.
The atmospheric angle $\theta_{23}$ is the primary pressure point. The model predicts an upper-octant value ( $\sim 49.2^\circ$ ), whereas the NuFIT global minimum lies in the lower octant. However, the upper-octant branch remains within the allowed contours.
The predicted $\delta_{CP} \approx 180^\circ$ is near-CP conserving, contrasting with NuFIT's preference for values further from $\pi$ .
The model is sharply falsifiable: a definitive determination of the lower atmospheric octant or a large leptonic CP violation (far from $\pi$ ) would rule out this minimal real deformation, as the structural hypotheses do not allow for significant adjustment via CKM uncertainties.

Significance and Claims
The authors explicitly state that they do not claim novelty for QLC, TBM, or Cabibbo corrections individually, nor do they claim mathematical uniqueness (as any 3D rotation can be Euler-decomposed). Instead, the significance lies in:

Minimality: The construction fixes all four Dirac PMNS observables using only the CKM matrix and a single discrete structural choice (the axis and the $3/4$ coefficient), introducing no new continuous parameters.
Robustness: The specific $((2, 1, 2), 3/4)$ structure is the most competitive simple candidate in historical data scans and remains Pareto-efficient under various complexity measures.
Phenomenological Specificity: It provides a concrete, testable hypothesis that predicts a near-CP-conserving phase and an upper-octant atmospheric angle, distinguishing it from flexible parametrizations or models with free parameters.

The paper positions this work as a "minimal QLC flavor-correlation construction" rather than a complete ultraviolet (UV) theory of gauge unification, noting that the dynamical origin of the axis and the $3/4$ coefficient remains a subject for future UV completion.

Technical Summary: Minimal Quark–Lepton Complementarity from a Rigid Deformation of Tri-Bimaximal Mixing

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