Model-agnostic likelihood for the reinterpretation of the $B^+\to K^+ν\barν$ measurement at Belle II

TL;DR

This work addresses how to reinterpret the Belle II ${B^{+}\rightarrow K^{+}\nu\bar{\nu}}$ measurement beyond the SM by publishing a model-agnostic likelihood and supporting joint densities, enabling tests of arbitrary theories via kinematic reweighting. It demonstrates the reinterpretation within the Weak Effective Theory, deriving marginal posteriors for the Wilson-coefficient combinations $|C_{\mathrm{VL}}+C_{\mathrm{VR}}|$, $|C_{\mathrm{SL}}+C_{\mathrm{SR}}|$, and $|C_{\mathrm{TL}}|$, with a posterior mode at $(11.3,0.0,8.2)$ and credible intervals indicating a vector enhancement and a sizable tensor contribution. The analysis highlights the importance of incorporating differential $q^2$-dependent information and hadronic form-factor uncertainties, shows favorable model comparison for the WET scenario over background and SM hypotheses, and provides data and tooling (HEPData, pyhf, redist) to enable broad future reinterpretations. Overall, the study offers a rigorous, reusable framework to maximize the scientific impact of Belle II measurements in the NP landscape.

Abstract

We recently measured the branching fraction of the $B^{+}\rightarrow K^{+}ν\barν$ decay using 362fb$^{-1}$ of on-resonance $e^+e^-$ collision data under the assumption of Standard Model kinematics, providing the first evidence for this decay. To facilitate future reinterpretations and maximize the scientific impact of this measurement, we publicly release the full analysis likelihood along with all necessary material required for reinterpretation under arbitrary theoretical models sensitive to this measurement. In this work, we demonstrate how the measurement can be reinterpreted within the framework of the Weak Effective Theory. Using a kinematic reweighting technique in combination with the published likelihood, we derive marginal posterior distributions for the Wilson coefficients, construct credible intervals, and assess the goodness of fit to the Belle II data. For the Weak Effective Theory Wilson coefficients, the posterior mode of the magnitudes $|C_\mathrm{VL}+C_\mathrm{VR}|$, $|C_\mathrm{SL}+C_\mathrm{SR}|$, and $|C_\mathrm{TL}|$ corresponds to the point ${(11.3, 0.0, 8.2)}$. The respective 95\% credible intervals are $[1.9, 16.2]$, $[0.0, 15.4]$, and $[0.0, 11.2]$.

Figures (4)

• Figure 1: The ITA binned joint number densities. The horizontal axis corresponds to the generated $q^2$. The vertical axis represents the binning used in the ${B^{+}\rightarrow\xspace K^{+}\nu\bar{\nu}}$ analysis Belle-II:2023esi. The heatmap shows the weighted signal events.
• Figure 2: The ${B^{+}\rightarrow\xspace K^{+}\nu\bar{\nu}}$ differential branching fraction prediction from \ref{['eq:wet-width']}. Individual contributions are shown here with the combinations of vector, scalar and tensor Wilson coefficients set to unity, respectively. The uncertainties shown (bands) stem from the hadronic parameters.
• Figure 3: The marginalized posterior for the Wilson coefficients in \ref{['eq:wet-width']}. We adopt the convention that $C_{\mathrm{VL}}+C_{\mathrm{VR}}$, $C_{\mathrm{SL}}+C_{\mathrm{SR}}$ and $C_{\mathrm{TL}}$ are real valued. Diagonal and off-diagonal panels show the 1-dimensional and 2-dimensional sample density PDFs on a linear scale, respectively. The overall scale is omitted, as all relevant information is contained in the shape of the distribution. The contours indicate $68\%$ and $95\%$ credible intervals. The dashed black lines and cross mark the SM point; the dash-dotted yellow lines and cross indicate the posterior mode; dotted red lines mark the symmetry axes used for sample symmetrization.
• Figure 4: Observed and predicted best-fit yields in the highest-sensitivity bins of the analysis. These correspond to the $\eta(\mathrm{BDT}_2\xspace)>0.98$ region of the ITA. The signal is shown for the unconstrained ${B^{+}\rightarrow\xspace K^{+}\nu\bar{\nu}}$ SM (left) and the WET (right) predictions. The predicted background yields are shown individually for the neutral and charged $B\xspace$-meson decays, and the summed five continuum categories. Pulls are shown in the lower panels.