Searches for charged-lepton-flavor violation in $\chi_{bJ}(1P)$ decays

Using 158 million $\Upsilon(2S)$ decays collected by the Belle detector, this paper reports the first searches for charged-lepton-flavor violation in $\chi_{bJ}(1P)$ decays, finding no significant signals and setting upper limits on the relevant branching fractions at the $10^{-6}$to$10^{-5}$ level.

The Gist

Imagine the universe as a giant, high-speed racetrack where tiny particles zoom around at nearly the speed of light. At the KEKB collider in Japan, scientists crash these particles together to see what happens when they smash. The Belle II experiment is like a massive, ultra-high-definition security camera system watching these crashes, looking for anything weird or unexpected.

This specific paper is a report from that camera system, and here is the story it tells, broken down into simple terms:

1. The Goal: Catching a "Shape-Shifter"

In the Standard Model (the rulebook of physics), particles have a "flavor," like a specific uniform. Electrons wear blue, muons wear red, and taus wear green. The rulebook says these uniforms never change. An electron can never turn into a muon.

However, physicists suspect there are hidden rules (New Physics) that might allow these particles to swap uniforms. This is called Charged-Lepton-Flavor Violation (CLFV). Finding a particle that changes its uniform would be like seeing a blue-shirted player suddenly turn into a red-shirted player in the middle of a soccer game. It would prove the rulebook is incomplete and that there are new, undiscovered forces at play.

2. The Stage: The "Bottomonium" Family

The scientists didn't just look at any particle; they looked at a specific family called $\chi_{bJ}(1P)$.

• The Analogy: Imagine the $\Upsilon(2S)$ particle as a heavy, excited parent. When it calms down, it sheds a photon (a particle of light) and becomes a slightly lighter, excited child called $\chi_{bJ}$.
• There are three "siblings" in this family: $\chi_{b0}$, $\chi_{b1}$, and $\chi_{b2}$. They are like a scalar (spin-0), an axial-vector (spin-1), and a tensor (spin-2) version of the same family.
• The scientists wanted to see if any of these siblings could decay (break apart) into two different particles, like an electron and a muon ($e^\pm \mu^\mp$), or an electron and a tau ($e^\pm \tau^\mp$).

3. The Hunt: Sifting Through the Noise

The team used data from 158 million collisions. That's a lot of data!

• The Challenge: It's like trying to find a single specific needle in a haystack the size of a mountain. Most of the time, the particles break apart in boring, predictable ways.
• The Strategy: They built a "filter" (a set of computer rules) to ignore the boring stuff. They looked for events where:
  1. A photon was emitted (the parent calming down).
  2. Two charged tracks appeared (the children).
  3. One track was an electron and the other was a muon (or a tau).
  4. The energy and momentum added up perfectly, like a balanced scale.

They also used "control modes" (safe, known decays) to make sure their camera and filters were working correctly. It's like calibrating a metal detector with a known coin before searching for gold.

4. The Result: The Great Silence

After analyzing the mountain of data, the result was: Nothing.

• They didn't find a single case where an electron turned into a muon or a tau in these specific decays.
• The Good News: Even though they didn't find the "shape-shifter," they learned something very important. They set a speed limit on how often this could happen.
• They calculated that if this flavor-changing magic does happen, it must be incredibly rare—less than 1 in a million (for electron-muon) or 1 in 100,000 (for electron-tau) times.

5. Why This Matters: Closing the Door

Think of the Standard Model as a house with many locked doors. Scientists have been trying to open them to find "New Physics."

• This paper tried to open a specific, previously unexplored door (the scalar sector of the $\chi_{b0}$ particle).
• Since they found nothing, they effectively locked that door tighter. They told the universe: "If there is a secret passage here, it's so narrow that we can't see it with our current tools."
• They also translated their findings into mathematical constraints (Wilson coefficients), which are like "fences" that tell theorists exactly how big their new theories can be. If a theory predicts a violation larger than their fence, that theory is now likely wrong.

Summary

The Belle II team acted like cosmic detectives, searching for a magical particle transformation that shouldn't exist. They looked at 158 million crashes, used sophisticated filters to ignore the noise, and found zero evidence of the transformation.

While they didn't find the "smoking gun" of new physics, they successfully narrowed the search area. They proved that if this flavor-changing magic exists, it is hiding in a very deep, very dark corner of the universe, and we will need even more powerful tools to find it next time.

Technical Summary

Here is a detailed technical summary of the Belle II Preprint 2026-005, titled "Searches for charged-lepton-flavor violation in $\chi_{bJ}(1P)$ decays."

1. Problem and Motivation

Charged-Lepton-Flavor Violation (CLFV) is a phenomenon strictly forbidden in the Standard Model (SM) but predicted by various Beyond-the-Standard-Model (BSM) theories. While extensive searches have been conducted for CLFV in $\tau$ decays, mesons, and the Higgs boson, this paper addresses a specific gap: CLFV in the decays of bottomonium P-wave states ($\chi_{bJ}(1P)$).

• Unique Probes: The $\chi_{b0}(1P)$ (scalar), $\chi_{b1}(1P)$ (axial-vector), and $\chi_{b2}(1P)$ (tensor) states offer unique low-energy probes for CLFV in different spin sectors.
  • $\chi_{b0}$ decays provide a probe for scalar operators, complementary to Higgs searches.
  • $\chi_{b1}$ and $\chi_{b2}$ offer the first searches for CLFV in axial-vector and tensor particles, respectively.
• Target Channels: The study searches for decays of the form $\chi_{bJ}(1P) \to \ell_1^\pm \ell_2^\mp$ where $\ell_1 \neq \ell_2$ (specifically $e^\pm\mu^\mp$, $e^\pm\tau^\mp$, and $\mu^\pm\tau^\mp$).

2. Methodology

Data Sample and Simulation

• Dataset: The analysis utilizes 158 million $\Upsilon(2S)$ mesons collected by the Belle detector at the KEKB collider ($e^+e^-$ collisions at $\sqrt{s} = 10.023$ GeV), corresponding to an integrated luminosity of $24.7 \text{ fb}^{-1}$.
• Production Mechanism: The $\chi_{bJ}(1P)$ states are produced via the radiative transition $\Upsilon(2S) \to \gamma \chi_{bJ}(1P)$.
• Simulation:
  • Signal and control modes are simulated using EVTGEN (decay models) and PYTHIA (background processes like $e^+e^- \to \ell^+\ell^-$, $q\bar{q}$, $\gamma\gamma$).
  • Detector response is modeled with GEANT3.
  • Final-state radiation is handled by PHOTOS.

Event Selection and Reconstruction

• Control Modes: To validate the analysis and estimate efficiencies, the study uses cascade decays $\chi_{bJ}(1P) \to \gamma \Upsilon(1S)$ followed by $\Upsilon(1S) \to \ell^+\ell^-$ ($\ell=e, \mu$). These are chosen because SM direct decays to same-flavor pairs are unmeasurably small ($O(10^{-20})$–$O(10^{-9})$).
• Signal Selection:
  • Tracks: Events must have exactly two oppositely charged tracks.
  • Particle ID: One track identified as an electron ($e$-likelihood $>0.5$) and the other as a muon ($\mu$-likelihood $>0.9$).
  • Kinematics: Track momentum $> 200$ MeV; impact parameter constraints applied.
  • $\tau$ Reconstruction: For $\chi_{bJ} \to \ell\tau$ searches, $\tau$ decays are reconstructed via $\tau \to \mu \bar{\nu}_\mu \nu_\tau$ or $\tau \to e \bar{\nu}_e \nu_\tau$.
  • Photon Veto: To suppress radiative Bhabha scattering ($e^+e^- \to \gamma e^+e^-$), specific angular and momentum cuts are applied (e.g., rejecting events with high-momentum electrons or specific photon angles).
  • Kinematic Fit: A four-constraint (4C) kinematic fit is applied to enforce energy-momentum conservation.
• Recoil Mass: The primary observable is the recoil mass against the photon(s).
  • $M_{\text{recoil}}^{\gamma_1}$ is used to identify the $\chi_{bJ}(1P)$ state.
  • $M_{\text{recoil}}^{\gamma_1 \ell_1}$ is used for $\tau$ modes to constrain the $\tau$ mass.

Statistical Analysis

• Fitting: Unbinned maximum-likelihood fits are performed on the recoil mass distributions.
  • Signal Model: Sum of Gaussian and bifurcated Gaussian functions.
  • Background Model: Exponential and constant PDFs.
• Limit Setting: Since no significant signal was observed, upper limits on branching fractions are calculated at the 90% confidence level (CL) using the $CL_s$ method.
• Systematics: Uncertainties include track finding, lepton ID, photon detection, trigger efficiency, simulation statistics, and branching fraction inputs. Total systematic uncertainties range from ~10% to ~20% depending on the channel and spin state.

3. Key Contributions

1. First Searches: This is the first experimental search for CLFV in $\chi_{bJ}(1P)$ decays for all three spin states ($J=0, 1, 2$) and all three lepton flavor combinations ($e\mu, e\tau, \mu\tau$).
2. Control Mode Validation: The paper successfully measures the product of branching fractions for the control modes $\Upsilon(2S) \to \gamma \chi_{bJ} \to \gamma \gamma \ell^+\ell^-$, validating the reconstruction efficiency and background modeling. The results agree with world averages.
3. Wilson Coefficient Constraints: For the scalar sector ($\chi_{b0}$), the results are translated into bounds on the Wilson coefficients ($C_{SL}, C_{SR}$) of effective scalar operators mediating CLFV, providing constraints on the energy scale ($\Lambda$) of new physics.

4. Results

No significant signal was observed in any of the nine search channels. The results are summarized as follows:

• Branching Fraction Limits:
  • $e^\pm \mu^\mp$ modes: Upper limits set at the level of $10^{-6}$.
    • Example: $\mathcal{B}(\chi_{b0} \to e^\pm \mu^\mp) < 4.0 \times 10^{-6}$.
  • $e^\pm \tau^\mp$ and $\mu^\pm \tau^\mp$ modes: Upper limits set at the level of $10^{-5}$.
    • Example: $\mathcal{B}(\chi_{b0} \to e^\pm \tau^\mp) < 41.0 \times 10^{-6}$.
• Wilson Coefficients:
  • The non-observation of $\chi_{b0} \to \ell_1 \ell_2$ decays excludes specific regions in the parameter space of the Wilson coefficients $C_{SL}/\Lambda^2$ and $C_{SR}/\Lambda^2$. The excluded regions are visualized as circles in Figure 4 of the paper.

5. Significance

• New Physics Sensitivity: By probing the scalar sector via $\chi_{b0}$, this study complements direct Higgs boson searches, offering a distinct low-energy window into BSM physics.
• Spin-Dependent Constraints: The simultaneous search across $J=0, 1, 2$ allows for the differentiation of new physics models based on the spin structure of the mediating operators (scalar vs. axial-vector vs. tensor).
• Benchmark for Future Experiments: These results establish the first experimental benchmarks for CLFV in bottomonium P-wave states, guiding future high-luminosity searches at Belle II and other facilities.
• Theoretical Impact: The derived bounds on Wilson coefficients provide direct constraints on effective field theory (EFT) models predicting CLFV, ruling out specific parameter spaces for scalar-mediated interactions.

In conclusion, this paper represents a milestone in flavor physics by extending the search for lepton flavor violation into the bottomonium P-wave sector, setting stringent limits that constrain theoretical models of new physics.