⚛️ phenomenology

Neutrino masses and mixing angles in the Plan B Model

This paper investigates the implications of the Plan B Model for neutrino physics, demonstrating that combining a Froggatt-Nielsen mechanism with a clockwork suppression allows the model to successfully fit current neutrino oscillation data using order-unity fundamental couplings while simultaneously addressing $b\rightarrow s \ell^+ \ell^-$ anomalies and quark mixing hierarchies.

Original authors: Ben Allanach, Prabhoda Chandra Sarjapur

Published 2026-09-02

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Ben Allanach, Prabhoda Chandra Sarjapur

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

Deep within the subatomic world, particles called neutrinos are everywhere, yet they remain among the most elusive and mysterious inhabitants of the universe. These ghostly particles, which pass through matter almost entirely unnoticed, have a tiny, almost imperceptible mass. For decades, physicists have struggled to explain why they are so light compared to other particles like electrons or quarks, and why they mix and change their identities as they travel through space. This mixing is described by a specific set of angles and mass differences that experiments have measured with increasing precision. Understanding the origin of these properties is crucial because it points toward new laws of physics that exist beyond our current best theory, the Standard Model. Recently, a specific theoretical framework known as the Plan B Model was proposed to solve a different puzzle involving the decay of heavy particles called B-mesons. This new paper takes that same model and asks a critical question: can it also explain the strange behavior of neutrinos?

The researchers, working from the University of Cambridge, began by examining the structure of the Plan B Model, which introduces a new force and a new type of particle to the Standard Model. This model was originally designed to fix discrepancies in how B-mesons decay, but it naturally includes right-handed neutrinos, a type of particle that has not yet been directly observed but is necessary for many theories of neutrino mass. In this framework, the model assigns specific "charges" to different families of particles, much like how different keys open different locks. These charges dictate how particles interact. When the researchers applied these rules to the neutrino sector, they found that the model naturally suggested a mechanism to create the tiny masses we observe, but with a catch: the initial calculations predicted masses that were far too heavy, billions of times heavier than what experiments actually see.

To bridge this gap, the team introduced a second layer of complexity known as the clockwork mechanism. Imagine a long chain of gears where turning the first gear only causes the last gear to move a tiny fraction of a turn; this is the essence of the clockwork idea. In the model, the researchers proposed that the right-handed neutrinos are not single particles but are part of a chain of connected fields. This arrangement creates an exponential suppression, effectively dialing down the predicted neutrino masses from the heavy, incorrect values to the incredibly light values observed in nature. By combining this clockwork suppression with the original model's rules, the team was able to construct a mathematical description of the neutrino mass matrix that could be tested against real-world data.

The team then performed a massive computational search, testing millions of possible combinations of the model's parameters to see which ones could reproduce the actual neutrino data collected by global experiments. They looked for solutions that matched the measured differences in neutrino masses and the angles that describe how they mix. Their analysis revealed that the model works remarkably well for both possible arrangements of neutrino masses, known as normal and inverted hierarchies. In the normal hierarchy, where one neutrino is significantly heavier than the other two, the model fits the data with a high degree of confidence. In the inverted hierarchy, where the two heavier neutrinos are nearly equal in mass, the fit is also successful, though it requires a slightly more specific arrangement of the underlying parameters.

Crucially, the researchers found that the model achieves these results without needing to fine-tune the numbers to an unnatural degree. In many theoretical models, getting the right answer requires setting the fundamental numbers to very specific, unlikely values. Here, the model works with fundamental numbers that are all of a similar, reasonable size, suggesting that the solution is robust and natural. The study also compared the two mass arrangements and found that the data slightly favors the normal hierarchy, a conclusion that aligns with current global experimental trends. Ultimately, this work demonstrates that the Plan B Model is not just a solution for B-meson decays but a comprehensive framework that can simultaneously explain the origin of neutrino masses and their mixing patterns, offering a coherent picture of physics that extends beyond our current understanding.

Technical Summary: Neutrino masses and mixing angles in the Plan B Model

Problem Statement
The Plan B Model was originally proposed to address tensions between Standard Model (SM) predictions and experimental measurements of $b \to s\ell^+\ell^-$ transitions (specifically $B$ -meson decays) while simultaneously offering a qualitative explanation for small off-diagonal entries in the quark mixing matrix. The model extends the SM gauge group by a spontaneously broken $U(1)_{3B_3 - L_e - 2L_\mu}$ symmetry with family-dependent fermion charges. While this framework successfully incorporates right-handed neutrinos (required for anomaly cancellation) and suppresses Charged Lepton Flavour Violation (CLFV), it faces a specific challenge in the neutrino sector. At the renormalisable level, the $U(1)_X$ charge assignments force the neutrino Yukawa and Majorana mass matrices to be diagonal (or nearly so), predicting two massless neutrinos and no mixing. This contradicts experimental neutrino oscillation data. Furthermore, even if off-diagonal terms are generated via the Froggatt-Nielsen (FN) mechanism, the resulting neutrino masses are typically too large (order GeV) compared to the sub-eV scale observed, given the TeV-scale breaking of the $U(1)_X$ symmetry.

Methodology
The authors investigate whether the Plan B Model can accommodate realistic neutrino masses and mixing angles by combining the Type-I seesaw mechanism with the FN mechanism and an additional "clockwork" suppression.

Model Setup: The model includes three right-handed neutrinos ( $\nu_R$ ) with $U(1)_X$ charges $(-1, -2, 0)$ . The FN mechanism generates effective Yukawa couplings suppressed by powers of $\epsilon = \langle \theta \rangle / \Lambda_{FN}$ , where $\theta$ is the flavon field and $\Lambda_{FN}$ is the scale of heavy vector-like fermions.
Mass Suppression: To resolve the discrepancy between the predicted GeV-scale neutrino masses and the observed sub-eV scale, the authors implement a clockwork mechanism. This involves promoting each $\nu_R$ into a chain of $K$ vector-like SM-singlet fermions. The SM Higgs couples only to the $k=0$ site, while Majorana masses are localized at the $k=K$ site. This setup generates an exponential suppression factor $q_Y^{-K}$ for the effective Dirac Yukawa couplings.
Analytic and Numerical Analysis:
- Normal Hierarchy (NH): The authors perform an analytic expansion of the light neutrino mass matrix $m_\nu$ in powers of $\epsilon$ . They identify that the leading-order term predicts a vanishing atmospheric mixing angle ( $\theta_{23} \approx 0$ ), which contradicts data. They argue that sub-leading terms involving ratios of $\mathcal{O}(1)$ coefficients can overcome this suppression.
- Inverted Hierarchy (IH): For the IH case, the authors show that specific texture zeros in the Majorana mass matrix (specifically $a_{11} = a_{13} = 0$ ) are required to achieve the necessary degeneracy between the two heaviest states. This is realized via an additional clockwork suppression for specific matrix entries.
- Bayesian Inference: The authors perform global Bayesian parameter scans using nested sampling (via the UltraNest package) to fit the model parameters ( $\epsilon$ , $\Lambda_{FN}$ , and dimensionless couplings $y_{ij}, a_{ij}$ ) against NuFIT 6.0 oscillation data. They utilize a "naturalness" criterion based on the spread ratio $r = |Y_{max}|/|Y_{min}|$ of the fundamental dimensionless couplings, defining "natural" as $r \leq 30$ .

Key Contributions and Results

Feasibility of the Mechanism: The study demonstrates that the Plan B Model can successfully reproduce the observed neutrino mass hierarchy and mixing angles. This requires the interplay of the FN mechanism (generating hierarchical textures) and the clockwork mechanism (providing the necessary overall mass suppression).
Parameter Space:
- For Normal Hierarchy (NH), the posterior distribution concentrates around $\epsilon \approx 0.42$ and a Majorana mass scale $\Lambda_{FN} \sim \mathcal{O}(10)$ TeV. The model fits the data with a global evidence of $\log Z_{NH} = -26.52$ .
- For Inverted Hierarchy (IH), the fit requires specific texture zeros ( $a_{11}=a_{13}=0$ ) and yields a global evidence of $\log Z_{IH} = -29.07$ .
Naturalness: The authors quantify the naturalness of the fit using the spread ratio $r$ $r$ .
- For NH, $P(r < 30) = 78.0 \pm 0.4\%$ .
- For IH, $P(r < 30) = 69 \pm 5\%$ .
- Both cases are deemed "reasonably natural," as the majority of the posterior volume satisfies the $r \leq 30$ criterion, comparable to a random distribution of $\mathcal{O}(1)$ couplings.
Hierarchy Preference: The Bayes factor between the two scenarios is $\exp(\log Z_{NH} - \log Z_{IH}) \approx 12.81$ , indicating strong evidence in favor of the Normal Hierarchy over the Inverted Hierarchy within the context of the Plan B Model.

Significance and Claims
The paper claims to demonstrate that the Plan B Model, originally motivated by $B$ -physics anomalies, is a viable framework for explaining neutrino phenomenology without fine-tuning. The model achieves three simultaneous goals:

Improving the global fit of the SM to $b \to s\ell^+\ell^-$ observables via a TeV-scale $Z'$ .
Explaining the smallness of neutrino masses and the structure of the PMNS matrix via the FN and clockwork mechanisms.
Suppressing Charged Lepton Flavour Violation (CLFV) through the specific texture of the charged lepton Yukawa matrix.

The authors conclude that the model works with order-unity fundamental dimensionless couplings, provided the additional suppression from the clockwork mechanism is present. They note that while the IH case is possible, the model statistically prefers the NH scenario, consistent with current experimental trends in global neutrino fits. The work does not propose new experimental searches but rather validates the consistency of the existing Plan B Model with neutrino data.

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