Search for the decay $B^+ \rightarrow K^+\tau^+\tau^-$ using data from the Belle and Belle II experiments

Using data from the Belle and Belle II experiments, this paper reports the first search for the rare decay $B^+ \rightarrow K^+\tau^+\tau^-$, finding no significant excess and setting an improved upper limit on its branching fraction of $0.56 \times 10^{-3}$ at the 90% confidence level.

The Gist

The Great Particle Detective Hunt: Chasing a Ghostly Decay

Imagine the universe as a giant, high-speed racetrack where tiny particles zoom around at nearly the speed of light. In this paper, a massive team of scientists (the Belle and Belle II collaborations) acted like elite detectives, searching for a very specific, incredibly rare "crime scene" that has never been caught before.

Here is the story of their hunt, explained in plain English.

1. The Setting: A Cosmic Collision Course

The scientists used two giant particle accelerators (KEKB and SuperKEKB) in Japan. Think of these as massive circular racetracks where they smash electrons and positrons (anti-electrons) together.

When these particles collide, they create a burst of energy that briefly turns into a heavy, unstable particle called the $\Upsilon(4S)$. This particle is like a fragile egg that almost immediately cracks open into two smaller particles: a $B^+$ meson and a $B^-$ meson.

The team collected data from 1.2 billion of these collisions. That's a lot of eggs to crack!

2. The Mystery: A "Ghostly" Decay

The scientists were looking for a specific, rare event: a $B^+$ meson turning into a $K^+$ (a kaon) and a pair of $\tau$ (tau) leptons.

Why is this exciting?

• The Standard Model (The Rulebook): According to our current best understanding of physics (the Standard Model), this decay should happen very rarely—about 1 or 2 times in every 10 million tries.
• The New Physics (The Plot Twist): If "New Physics" exists (particles or forces we haven't discovered yet), this decay could happen 1,000 times more often than the Rulebook predicts. Finding it would be like finding a fingerprint that proves a new criminal gang is operating in town.

3. The Challenge: The Invisible Ghosts

The biggest problem with this search is that the $\tau$ particles are like ghosts.

• When a $\tau$ particle decays, it turns into other particles, but it also spits out neutrinos.
• Neutrinos are like invisible ghosts; they pass through everything without leaving a trace. Detectors cannot see them.
• Because these ghosts escape, the scientists can't simply look at the final pieces and say, "Aha! This is the decay we wanted!" The energy balance doesn't add up because the ghosts took some energy with them.

4. The Detective Trick: The "Shadow" Method

Since they can't see the signal directly, the scientists used a clever trick called "Full Reconstruction."

Imagine you are at a party where two identical twins (the $B^+$ and $B^-$ mesons) are born at the same time.

1. The Shadow: The scientists focus on one twin (the $B^-$). They catch every single piece of it as it falls apart. Because they know exactly how much energy and momentum the twin had at birth, they can calculate exactly what the other twin (the $B^+$) must have had.
2. The Search: Now that they know what the "signal" twin should look like, they look at the rest of the party. They are hunting for a specific combination: a Kaon and two Leptons (electrons or muons).
3. The Energy Leak: Since the signal twin is supposed to produce invisible ghosts (neutrinos), there should be a "missing energy" gap. The scientists looked for events where the energy in the room didn't quite add up, specifically looking for a tiny amount of "extra" energy left over in the detectors that shouldn't be there if the decay happened normally.

5. The Filter: Sifting Through the Noise

The problem is that for every one "ghostly" signal event, there are 100 million boring, common events that look similar. It's like trying to find one specific grain of sand on a beach while a hurricane is blowing.

To solve this, they built a series of filters:

• The Mass Check: They checked the weight of the particles to make sure they weren't just common background noise.
• The "No Extra Stuff" Rule: They looked for events where the energy in the room was almost perfect, with just a tiny, specific amount of "leakage" (the neutrinos).
• The Sideband Method: They looked at "neighborhoods" of data where they knew the signal couldn't exist. By counting how many "fake" events were in those neighborhoods, they could mathematically predict how many fake events would be in the "signal neighborhood."

6. The Verdict: No Ghosts Found (Yet)

After analyzing all 1.2 billion collisions, the scientists looked at their "signal neighborhood."

• What they expected: Based on the background noise, they expected to see about 14 events in the Belle detector and 3.5 in the Belle II detector.
• What they found: They found 11 and 6 events, respectively.

The Result: The numbers matched the background noise perfectly. There was no "ghost" signal. No evidence of New Physics was found in this specific decay.

7. The Takeaway: A Tighter Net

Even though they didn't find the new physics they were hoping for, this is a huge success.

• The Limit: They set a new, stricter rule: "If this decay happens, it must happen less than 0.56 times in every 1,000 tries."
• The Improvement: This limit is four times better than the previous best limit set by the BABAR experiment years ago.

In simple terms: They didn't find the criminal, but they tightened the net so much that if the criminal is out there, they are now much harder to hide. This forces theorists to rethink their ideas about what "New Physics" could look like, narrowing down the possibilities for the next generation of experiments.

Technical Summary

1. Problem and Motivation

• Physics Goal: The primary objective is to search for the flavor-changing neutral current (FCNC) decay $B^+ \to K^+ \tau^+ \tau^-$. In the Standard Model (SM), FCNCs are loop-suppressed and highly rare. The SM branching fraction is predicted to be $(1.0–2.0) \times 10^{-7}$ (excluding contributions from $B^+ \to K^+ [c\bar{c} \to] \tau^+ \tau^-$).
• New Physics Potential: Many Beyond-the-Standard-Model (BSM) scenarios proposed to explain existing anomalies in $B \to D^{(*)} \tau \nu$ and $B \to K \nu \bar{\nu}$ decays predict a significant enhancement of the $B^+ \to K^+ \tau^+ \tau^-$ rate, potentially up to $10^3$ times the SM prediction. This brings the rate within the reach of current experimental sensitivity.
• Current Status: The only previous search was conducted by the BABAR experiment, which set a 90% confidence level (CL) upper limit of $2.25 \times 10^{-3}$. This paper aims to improve upon this limit using the larger datasets from the Belle and Belle II experiments.

2. Methodology

The analysis utilizes a dataset corresponding to $1.2 \times 10^9$ $\Upsilon(4S)$ mesons produced in $e^+e^-$ collisions at the KEKB and SuperKEKB colliders. The data is split between the Belle ($772 \times 10^6$ decays) and Belle II ($387 \times 10^6$ decays) experiments.

Event Reconstruction Strategy

• Full Event Interpretation (FEI): To overcome the challenge of multiple undetected neutrinos from $\tau$ decays (which prevent full kinematic reconstruction of the signal side), the analysis employs the "tag-side" technique. The partner $B^-$ meson is fully reconstructed using the FEI algorithm in its hadronic decay modes.
• Signal Side Selection: The signal $B^+$ candidate is reconstructed from:
  • A charged kaon ($K^+$) with charge opposite to the tag-side $B^-$.
  • Two oppositely charged leptons ($\ell^+ \ell^-$), where $\ell \in \{e, \mu\}$. The analysis focuses on the leptonic decays of the $\tau$ leptons ($\tau \to \ell \nu_\ell \bar{\nu}_\tau$).
  • Combinations analyzed: $e^+e^-$, $e^\pm\mu^\mp$, and $\mu^+\mu^-$.

Kinematic and Topological Cuts

• Tag-Side Constraints: The partner $B$ must satisfy $M_{bc} > 5.27$ GeV/$c^2$ and $-0.15 < \Delta E < 0.10$ GeV to ensure correct reconstruction.
• Background Suppression:
  • $D$-meson suppression: A cut on the opposite-charge kaon-lepton mass, $m(K^+\ell^-) > 1.9$ GeV/$c^2$, is applied. This removes the dominant background from $B^+ \to D^0(\to K^-\ell^+\nu)\ell^+\nu$ decays, reducing background by ~99% while retaining ~20% of the signal.
  • $J/\psi$ and $\psi(2S)$ suppression: The dilepton mass is restricted to $< 3.0$ GeV/$c^2$ to reject $B^+ \to J/\psi K^+$, and $q^2 > 14.18$ GeV$^2/c^4$ is imposed to suppress $B^+ \to \psi(2S) K^+$.
  • $\pi^0$ Veto: Events with unassociated $\pi^0$ candidates are rejected to suppress backgrounds with extra neutral particles.

Signal Extraction Variable

• $E_{extra}$: The primary discriminant is the "extra energy" ($E_{extra}$), defined as the sum of energies of all neutral clusters in the calorimeter not used in the reconstruction of the $\Upsilon(4S)$ or the $B$ mesons.
• Signal Characteristic: Since the signal involves multiple neutrinos, the visible energy in the event is lower than the beam energy. Consequently, signal events cluster at very low $E_{extra}$ values, whereas background events (which often have extra hadrons or photons) populate higher $E_{extra}$ regions.
• Background Estimation: The background yield in the signal region ($E_{extra} < 100$ MeV for Belle, $< 250$ MeV for Belle II) is estimated by extrapolating from three independent sidebands:
  1. Low $q^2$ ($< 14.18$ GeV$^2/c^4$).
  2. Tag-side $M_{bc}$ sideband ($5.20 < M_{bc} < 5.27$ GeV/$c^2$).
  3. High $E_{extra}$ region.
  • A linear scaling fit is applied to correct simulation shapes to match data in these sidebands before extrapolation.

3. Key Contributions

• Combined Analysis: This is the first combined analysis of Belle and Belle II data for this specific decay channel, leveraging the full statistical power of both experiments.
• Optimized Selection: The introduction of the $m(K^+\ell^-) > 1.9$ GeV/$c^2$ cut significantly improved the signal-to-background ratio, a critical step given the overwhelming background from semileptonic $B$ decays.
• Data-Driven Background Correction: The use of sideband data to correct simulation shapes and normalizations ensures robust background estimation, minimizing reliance on pure Monte Carlo predictions for the signal region.
• Systematic Control: Comprehensive systematic uncertainties were evaluated, including PID efficiencies, tracking, $\pi^0$ veto efficiency, and form-factor modeling.

4. Results

• Observation: No significant excess of events was observed over the expected background in the signal region.
  • Belle: Observed 11 events vs. expected $14.1 \pm 1.6$ (stat) $\pm 1.9$ (syst).
  • Belle II: Observed 6 events vs. expected $3.5 \pm 0.7$ (stat) $\pm 0.9$ (syst).
• Branching Fraction Limit: The combined result sets a 90% CL upper limit on the branching fraction:
  $$\mathcal{B}(B^+ \to K^+ \tau^+ \tau^-) < 0.56 \times 10^{-3}$$
• Improvement: This limit is a factor of four more stringent than the previous BABAR limit ($2.25 \times 10^{-3}$).
• Measured Value: The measured branching fraction is consistent with zero: $(0.04^{+0.27+0.17}_{-0.23-0.18}) \times 10^{-3}$.

5. Significance

• Constraint on BSM Physics: The result significantly constrains the parameter space of various New Physics models that attempt to explain the $B$-physics anomalies. While the limit is still orders of magnitude above the SM prediction ($10^{-7}$), it rules out the most extreme enhancement scenarios (factors of $10^3$) proposed in some theoretical frameworks.
• Experimental Benchmark: This analysis demonstrates the capability of the Belle II experiment (and the combined Belle/Belle II dataset) to perform complex searches involving multiple neutrinos and missing energy, setting a new standard for sensitivity in rare $B$ decays.
• Future Prospects: The methodology established here, particularly the use of $E_{extra}$ and the specific kinematic cuts, provides a robust framework for future searches with the full Belle II dataset, which will eventually reach sensitivities closer to the SM prediction.