⚛️ phenomenology

Constraining the new contributions to electron $g-2$ in a radiative neutrino mass model

This paper demonstrates that a radiative neutrino mass model with TeV-scale scalar leptoquarks can resolve the electron $g-2$ anomaly under inverted neutrino mass ordering while predicting negligible effects on the muon $g-2$ and testable lepton-flavor-violating tau decay rates, all while satisfying stringent constraints from $\mu \to e\gamma$ and neutrino oscillation data.

Original authors: Bayu Dirgantara, J. Julio

Published 2026-02-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bayu Dirgantara, J. Julio

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

Imagine the Standard Model of physics as a giant, incredibly detailed recipe book for how the universe works. For decades, this book has been perfect, but recently, scientists have found a few "typos" in the ingredients list. Two of the biggest mysteries are:

The Ghostly Neutrinos: We know these tiny particles have mass, but the recipe book says they should be weightless.
The Wobbly Spinning Tops: Electrons and muons (heavy cousins of electrons) act like spinning tops. The recipe predicts exactly how fast they should wobble (their "magnetic moment"), but real-world experiments show they are wobbling slightly differently than expected.

This paper is like a team of chefs trying to fix the recipe book. They propose a new set of ingredients (particles called Leptoquarks) that can explain both the ghostly neutrinos and the wobbly tops at the same time.

Here is the story of their discovery, broken down into simple concepts:

1. The New Ingredients: Leptoquarks

The scientists introduce two new particles, named S and R. Think of them as "hybrid fruits" that can turn a quark (the building block of protons) into a lepton (like an electron).

Why they are special: These particles are heavy (about 1,500 times heavier than a proton) and exist at a scale we might be able to test soon with giant particle colliders.
The Magic Trick: They can flip the "handedness" of particles (chirality). Imagine a left-handed glove suddenly turning into a right-handed one. This flip is crucial because it amplifies the effect on how electrons and muons wobble, potentially fixing the "typos" in the recipe.

2. The Tightrope Walk: Solving Two Problems at Once

The team faces a tricky balancing act.

The Goal: They want to fix the electron's wobble and the muon's wobble using these new particles.
The Trap: If they use the same "flavor" of particle to fix both, it creates a dangerous side effect: a process called $\mu \to e\gamma$ (a muon turning into an electron and a photon). This is like trying to fix a leak in the kitchen sink by flooding the whole house. Experiments have never seen this happen, so the "leak" must be tiny.

The Solution: The authors use a clever "decoupling" strategy.

They design the recipe so that the Electron gets its help from a "Charm" quark (a medium-heavy ingredient).
The Muon gets its help from a "Top" quark (the heaviest ingredient).
Because they use different ingredients for different tasks, they avoid the dangerous side effect. It's like using a wrench for the sink and a screwdriver for the faucet; they don't interfere with each other.

3. The Neutrino Puzzle: The Two-Loop Secret

To make the neutrinos have mass, the model uses a "loop" mechanism. Imagine a particle going around a track to gain weight.

One-Loop: The particle goes around the track once.
Two-Loop: The particle goes around, stops, and goes around again.

The paper's big discovery is that both loops are necessary. You can't just use one; you need the combination of the single lap and the double lap to match the actual data we have on how neutrinos oscillate (change flavors).

The Consequence: This requirement is very strict. It acts like a sieve, filtering out most of the possible settings for the new particles.

4. The Big Result: Who Wins?

Because the "sieve" (neutrino data) is so strict, the model has to make a hard choice:

The Muon: The model predicts that the new particles cannot fix the muon's wobble significantly. The muon's "wobble" remains consistent with the Standard Model (or the new lattice QCD calculations). The "new physics" correction is too small to matter.
The Electron: However, the model can fix the electron's wobble! But there's a catch. It only works if the neutrinos follow a specific pattern called "Inverted Ordering" (where the lightest neutrino is actually the heaviest of the light ones, a bit like a pyramid that is upside down).
- If the neutrinos are "Inverted," the model perfectly explains the electron's weird wobble observed in Rubidium atom experiments.
- If the neutrinos are "Normal," the model fails to explain the electron.

5. The Future: A Testable Prediction

The best part about this theory is that it's not just math; it's testable.

The Smoking Gun: The model predicts that Tau particles (another heavy cousin of the electron) should occasionally decay into electrons and photons ( $\tau \to e\gamma$ ) or three electrons ( $\tau \to 3e$ ).
The Timeline: These decays are predicted to happen at a rate that is just barely below what our current detectors can see. The next generation of experiments (like those at the Belle II lab or future colliders) should be able to spot them.
- If they find these decays: The theory is a winner.
- If they don't: The theory is likely wrong.

Summary Analogy

Imagine you are trying to fix a car that has two problems: the engine is making a weird noise (Muon g-2), and the radio is staticky (Electron g-2).

Most mechanics say, "You need a new engine part."
This paper says, "No, we need a specific type of hybrid gear (Leptoquark)."
They show that if you use this gear, you can fix the radio (Electron) perfectly, but the engine noise (Muon) stays mostly the same because the gear is tuned to a specific setting (Inverted Neutrino Ordering).
The only way to prove they are right is to look under the hood and see if the car starts making a new, specific clicking sound (Tau decays) that only this gear would cause.

In short: This paper offers a unified, testable explanation for neutrino masses and electron anomalies, but it demands that neutrinos are arranged in a specific "inverted" way and predicts that future experiments will soon see new particle decays.

1. Problem Statement

The paper addresses two major anomalies in particle physics:

Neutrino Mass Origin: The Standard Model (SM) cannot explain the non-zero mass of neutrinos. While the seesaw mechanism is a popular explanation, it requires physics at extremely high energy scales ( $\sim 10^{14}$ GeV), making it untestable. Radiative neutrino mass models offer a testable alternative at the TeV scale.
Lepton Anomalous Magnetic Moments ( $g-2$ ):
- Muon ( $a_\mu$ ): Recent lattice QCD calculations suggest no significant deviation from the SM, unlike earlier data-driven results.
- Electron ( $a_e$ ): There is a $5\sigma$ discrepancy in the determination of the fine-structure constant ( $\alpha_{em}$ ) between Cesium (Cs) and Rubidium (Rb) experiments. The Rb measurement implies a positive deviation ( $\delta a_e \approx +3.5 \times 10^{-13}$ ), suggesting new physics, while the Cs measurement implies a negative deviation.

The Challenge: Constructing a model that simultaneously explains neutrino masses and the electron $g-2$ anomaly (specifically the Rb result) without violating stringent constraints from lepton-flavor-violating (LFV) processes like $\mu \to e\gamma$ .

2. Methodology and Model Framework

The authors analyze a specific radiative neutrino mass model featuring two scalar leptoquarks (LQs) at the TeV scale:

$S(3, 1, -1/3)$ : A singlet under $SU(2)_L$ .
$R(3, 2, 1/6)$ : A doublet under $SU(2)_L$ , containing components $R_{2/3}$ and $R_{-1/3}$ .

Key Theoretical Features:

Neutrino Mass Generation: The model generates neutrino masses via both one-loop and two-loop diagrams.
- One-loop: Involves $S$ and down-type quarks.
- Two-loop: Involves $S$ , $R$ , $W$ bosons, and up-type quarks.
- Crucially, these contributions arise from different sets of couplings and cannot be treated as simple higher-order corrections to one another; both must be present to fit data.
Chiral Enhancement: The scalar LQ $S$ couples to both left- and right-handed up-type quarks. This allows for internal chiral enhancement in the calculation of lepton $g-2$ , significantly boosting the contribution if heavy quarks (top or charm) run in the loop.
Flavor Textures: To avoid the stringent $\mu \to e\gamma$ $μ \to e γ$ constraint while enhancing $a_e$ $a_{e}$ and $a_\mu$ $a_{μ}$ separately, the authors employ specific Yukawa coupling textures (TX 1 and TX 2).
- These textures decouple the electron and muon sectors.
- The electron $g-2$ is enhanced by the charm quark mass.
- The muon $g-2$ is enhanced by the top quark mass.
- This separation prevents the dangerous chirally-enhanced $\mu \to e\gamma$ decay, which typically arises when both moments are induced by the same heavy quark.

3. Key Contributions and Analysis

The paper performs a comprehensive phenomenological analysis with the following steps:

Neutrino Mass Fitting:
- The authors derive the neutrino mass matrix $M_\nu$ , which is a rank-2 matrix, implying one massless neutrino.
- They demonstrate that fitting neutrino oscillation data (mass splittings and mixing angles) requires the simultaneous presence of both one-loop and two-loop contributions. If one is omitted, the model fails to reproduce the observed mixing patterns or requires non-perturbative couplings.
- This requirement severely constrains the parameter space, particularly the ratios of Yukawa couplings.
Constraints on $g-2$ :
- Muon ( $a_\mu$ ): Due to the tight constraints from neutrino data on the relevant couplings (specifically those involving the top quark), the new physics contribution to the muon $g-2$ is found to be negligible ( $\delta a_\mu \lesssim 10^{-11}$ ). This is consistent with the recent Theory Initiative (TI) lattice results showing no significant anomaly.
- Electron ( $a_e$ ): The model can resolve the electron $g-2$ $g - 2$ discrepancy implied by the Rubidium experiment ( $\delta a_e \approx +3.5 \times 10^{-13}$ $δ a_{e} \approx + 3.5 \times 1 0^{- 13}$ ).
  - This resolution is only possible within 2 $\sigma$ uncertainty if the neutrino mass ordering is Inverted (IO).
  - For Normal Ordering (NO), the constraints from neutrino data are too strong, ruling out the region required to explain the Rb anomaly.
  - The model naturally predicts a positive $\delta a_e$ , aligning with the Rb result and excluding the Cesium result.
LFV and Other Constraints:
- $\mu \to e\gamma$ : The specific texture choices ensure this decay is suppressed (GIM-like cancellation or vanishing couplings), satisfying the bound $BR < 1.5 \times 10^{-13}$ .
- Tau Decays: The model predicts observable rates for LFV tau decays, specifically $\tau \to e\gamma$ and $\tau \to 3e$ , which are close to current experimental limits but within the sensitivity of next-generation experiments.
- Perturbativity: Renormalization Group Equation (RGE) analysis shows the model remains perturbative up to $\sim 100$ TeV.

4. Results

Neutrino Sector: The model successfully fits neutrino oscillation data with a massless lightest neutrino. It necessitates a specific interplay between one-loop and two-loop terms, constraining the parameter $\epsilon$ (ratio of loop integrals) and Yukawa couplings.
Muon $g-2$ : The new physics contribution is negligible ( $\delta a_\mu \approx 0$ ), consistent with the lack of a muon anomaly in recent lattice QCD interpretations.
Electron $g-2$ : The model can explain the Rb anomaly ( $\delta a_e > 0$ ) within 2 $\sigma$ , but only under the condition of Inverted Mass Ordering and specific values for the CP-violating phase $\delta$ .
Predictions:
- $\tau \to e\gamma$ and $\tau \to 3e$ : Branching ratios are predicted to be near current limits ( $BR(\tau \to e\gamma) \gtrsim 6 \times 10^{-9}$ ), making them prime targets for future experiments like Belle II.
- $\mu \to e$ conversion: Suppressed due to the texture choices, remaining well below future sensitivities.

5. Significance

This paper provides a unified framework that links the origin of neutrino masses to the electron $g-2$ anomaly. Its primary significance lies in:

Resolving the Electron Anomaly: It offers a concrete theoretical explanation for the Rubidium-based electron $g-2$ discrepancy while remaining consistent with the lack of a muon anomaly.
Discriminating Neutrino Ordering: The model makes a sharp prediction: the electron $g-2$ anomaly can only be explained if the neutrino mass ordering is Inverted. Future determination of the mass ordering will effectively validate or falsify this specific scenario.
Testability: Unlike high-scale seesaw models, this TeV-scale model is directly testable. The predicted LFV tau decays ( $\tau \to e\gamma, \tau \to 3e$ ) are within reach of upcoming experiments, providing a clear path for experimental verification.
Texture Strategy: It demonstrates the efficacy of using flavor textures to decouple lepton sectors, allowing for large chiral enhancements in $g-2$ while evading the most dangerous LFV constraints.

In conclusion, the authors show that a radiative neutrino mass model with TeV-scale leptoquarks can simultaneously explain neutrino oscillation data and the electron $g-2$ anomaly (under inverted ordering), while predicting observable signals in tau decays.