Search for the $\boldsymbol{B^0 \to K^0_{\rm S} \tau^+ \tau^-}$ decay

Technical Summary: Search for the $B^0 \to K^0_S \tau^+ \tau^-$ Decay

Problem and Motivation
The paper addresses the search for the rare semileptonic decay $B^0 \to K^0_S \tau^+ \tau^-$. In the Standard Model (SM), this flavor-changing neutral-current (FCNC) process involving third-generation quarks and leptons is predicted to be extremely rare, with a branching fraction of $\mathcal{O}(10^{-7})$. However, recent measurements in related transitions ($b \to s\nu\bar{\nu}$, $b \to s\mu^+\mu^-$, and $b \to c\tau^-\bar{\nu}_\tau$) show discrepancies with SM predictions. These discrepancies suggest the potential existence of New Physics (BSM) mediators, such as leptoquarks or $Z'$ bosons, which could enhance $b \to s\tau^+\tau^-$ rates by up to three orders of magnitude. While previous searches have constrained vector meson final states ($B^+ \to K^+\tau^+\tau^-$ and $B^0 \to K^{*0}\tau^+\tau^-$), pseudoscalar meson decays remain less explored. This work presents the first search for the $B^0 \to K^0_S \tau^+ \tau^-$ decay mode.

Methodology
The analysis utilizes data collected by the Belle and Belle II experiments at the KEKB and SuperKEKB $e^+e^-$ colliders, respectively. The datasets correspond to integrated luminosities of 711 fb$^{-1}$ and 365 fb$^{-1}$, comprising a total of approximately 1.16 billion $\Upsilon(4S)$ events.

• Event Selection: The analysis employs a "tagging" technique where one $B$ meson ($\bar{B}^0_{\text{tag}}$) is fully reconstructed in a hadronic decay mode using the Full Event Interpretation (FEI) algorithm. This constrains the kinematics of the signal $B^0$ meson ($B^0_{\text{sig}}$). The signal decay $B^0_{\text{sig}} \to K^0_S \tau^+ \tau^-$ is reconstructed from the remaining particles.

  • The $K^0_S$ is reconstructed via $K^0_S \to \pi^+\pi^-$.
  • The $\tau^+\tau^-$ pair is reconstructed in five mutually exclusive categories based on the decay modes of the $\tau$ leptons into final states with a single charged particle: $\ell\ell$ (leptonic-leptonic), $\rho\ell$, $\ell^-h^+$, $\ell^+h^-$, and no-$\ell$ (hadronic-hadronic).
  • Kinematic constraints, including the beam-constrained mass ($M_{bc}$) and energy difference ($\Delta E$) for the tag side, and the reconstructed $q^2$ (invariant mass squared of the $\tau^+\tau^-$ system) for the signal side, are applied.

• Background Suppression: A Boosted Decision Tree (BDT) classifier is trained separately for each of the five categories and for both detectors. Input variables include kinematic properties of the signal and tag sides (e.g., $q^2_{\text{rec}}$, thrust angles, vertex separations) and "Rest of Event" (ROE) variables (e.g., energy sum in the electromagnetic calorimeter, missing energy/momentum). The BDT output is transformed into a uniformly distributed observable $O'$ for signal events.

• Signal Extraction: A simultaneous binned maximum likelihood fit is performed on the $O'$ distribution across all five categories and both datasets. The fit templates for signal, $B\bar{B}$ background, and $q\bar{q}$ background are derived from Monte Carlo simulations, corrected for efficiency differences and normalized to the data luminosity. Systematic uncertainties are incorporated as nuisance parameters in the likelihood function.

Key Contributions

1. First Search: This is the first experimental search for the $B^0 \to K^0_S \tau^+ \tau^-$ decay.
2. Combined Dataset: The analysis combines the full Belle dataset (1999–2010) and Belle II dataset (2019–2022), significantly increasing the statistical power compared to previous single-experiment searches.
3. Multivariate Analysis: The use of category-specific BDT classifiers allows for optimized background suppression across diverse final-state topologies (leptonic vs. hadronic $\tau$ decays).
4. Isospin Combination: The results are combined with the recent measurement of the isospin-partner decay $B^+ \to K^+\tau^+\tau^-$ to derive a constraint on the isospin-averaged branching fraction $B(B \to K\tau^+\tau^-)$.

Results

• Observation: No significant signal is observed in the data. The measured branching fraction is consistent with zero:
  $$B(B^0 \to K^0_S \tau^+ \tau^-) = (1.2 \pm 3.3 \text{ (stat)} \pm 2.9 \text{ (syst)}) \times 10^{-4}$$
• Upper Limits:
  • For the neutral mode alone, an upper limit is set at the 90% confidence level (CL):
    $$B(B^0 \to K^0_S \tau^+ \tau^-) < 8.3 \times 10^{-4}$$
  • Combining this result with the $B^+ \to K^+\tau^+\tau^-$ measurement under the assumption of isospin symmetry yields a combined upper limit:
    $$B(B \to K\tau^+\tau^-) < 5.4 \times 10^{-4} \quad (90\% \text{ CL})$$
  • The combination improves the observed upper limit by approximately 5% relative to the limit derived from the charged mode alone.

Significance
The paper claims that this result provides the most stringent constraint to date on $b \to s\tau^+\tau^-$ transitions involving a pseudoscalar $K$ meson. While the observed limit is still orders of magnitude above the SM prediction ($\sim 10^{-7}$), it establishes a baseline for future sensitivity improvements. The result is particularly valuable for testing BSM scenarios that predict enhancements in these decays, as the sensitivity to different mediator classes depends on the spin-parity of the hadronic final state. The analysis demonstrates the capability of the Belle and Belle II collaborations to probe rare $\tau$-inclusive $B$ decays despite the challenging background environment and the presence of multiple neutrinos.