CERN Accelerating science

Particle identification performance as a function of momentum, using information from LKr, MUV1,2 and MUV3 (left), and from RICH (right). The $\pi^{+}$ identification efficiency is shown as blue circles (left vertical axis) and the probability of misidentifiaction of a $\mu^{+}$ as a $\pi^{+}$ is shown as red squares (right vertical axis).

Particle identification performance as a function of momentum, using information from LKr, MUV1,2 and MUV3 (left), and from RICH (right). The $\pi^{+}$ identification efficiency is shown as blue circles (left vertical axis) and the probability of misidentifiaction of a $\mu^{+}$ as a $\pi^{+}$ is shown as red squares (right vertical axis).

Left: $\pi^{0}$ rejection inefficiency as a function of the $\pi^+$ momentum. Right: definitions of kinematic regions in the $(p_{\pi^{+}},m_{\rm miss}^{2})$ plane. Region CR3D is the same as the signal region in this projection, but contains events outside the 3-dimensional signal regions definition.

Left: $\pi^{0}$ rejection inefficiency as a function of the $\pi^+$ momentum. Right: definitions of kinematic regions in the $(p_{\pi^{+}},m_{\rm miss}^{2})$ plane. Region CR3D is the same as the signal region in this projection, but contains events outside the 3-dimensional signal regions definition.

Left: selection acceptances, $A_{\pi\pi}$ and $A_{\pi\nu\bar{\nu}}$, as functions of $\pi^{+}$ candidate momentum displayed as blue squares and red circles, respectively. Right: random veto efficiency as a function of the instantaneous beam intensity.

Left: selection acceptances, $A_{\pi\pi}$ and $A_{\pi\nu\bar{\nu}}$, as functions of $\pi^{+}$ candidate momentum displayed as blue squares and red circles, respectively. Right: random veto efficiency as a function of the instantaneous beam intensity.

Left: trigger efficiency ratio $\varepsilon_\text{trig}$ as a function of the of $\pi^+$ momentum. Right: expected and observed numbers of $K^{+} \rightarrow \mu^{+}\nu$ events in trigger efficiency validation samples, as a function of the $\mu^+$ momentum.

Left: trigger efficiency ratio $\varepsilon_\text{trig}$ as a function of the of $\pi^+$ momentum. Right: expected and observed numbers of $K^{+} \rightarrow \mu^{+}\nu$ events in trigger efficiency validation samples, as a function of the $\mu^+$ momentum.

Left: distribution of $m_{\rm miss}^{2}$ of the kinematic tails data control sample for $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays. The colours show the contributions from the kinematic regions defined in figure~\ref{fig:pi0Rejection}-right. The average value of $f_\text{kin}(K^{+}\rightarrow\pi^{+}\pi^{0})$ is calculated as the ratio between the red area and the yellow area (and similarly for the control regions). Right: $f_\text{kin}(K^{+}\rightarrow\pi^{+}\pi^{0})$ as a function of the $\pi^+$ momentum, evaluated for signal regions (red) and for control regions (by exchanging the signal region with the corresponding control region).

Left: distribution of $m_{\rm miss}^{2}$ of the kinematic tails data control sample for $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays. The colours show the contributions from the kinematic regions defined in figure~\ref{fig:pi0Rejection}-right. The average value of $f_\text{kin}(K^{+}\rightarrow\pi^{+}\pi^{0})$ is calculated as the ratio between the red area and the yellow area (and similarly for the control regions). Right: $f_\text{kin}(K^{+}\rightarrow\pi^{+}\pi^{0})$ as a function of the $\pi^+$ momentum, evaluated for signal regions (red) and for control regions (by exchanging the signal region with the corresponding control region).

Left: event distribution in the $(m_{\rm miss}^{2},m_{\rm miss,\mu\nu\gamma}^{2})$ plane for a minimum bias data control sample with MUV3 associated signals and no calorimetric BDT condition applied. The $K^{+}\rightarrow\mu^{+}\nu\gamma$ candidates are clearly visible as a horizontal line around $m_{\rm miss,\mu\nu\gamma}^{2}=0$ extending towards high $m_{\rm miss}^{2}$ values including R2. Right: $K^{+}\rightarrow\mu^{+}\nu\gamma$ background validation samples, four bins in sidebands of the calorimetric BDT pion probability. Expectations include contributions from $K^{+}\rightarrow\mu^{+}\nu\gamma$ ($K_{\mu2\gamma}$), $K^{+}\rightarrow\mu^{+}\nu$ ($K_{\mu2}$) and upstream (section~\ref{sec:UpstreamBackground}) events.

Left: event distribution in the $(m_{\rm miss}^{2},m_{\rm miss,\mu\nu\gamma}^{2})$ plane for a minimum bias data control sample with MUV3 associated signals and no calorimetric BDT condition applied. The $K^{+}\rightarrow\mu^{+}\nu\gamma$ candidates are clearly visible as a horizontal line around $m_{\rm miss,\mu\nu\gamma}^{2}=0$ extending towards high $m_{\rm miss}^{2}$ values including R2. Right: $K^{+}\rightarrow\mu^{+}\nu\gamma$ background validation samples, four bins in sidebands of the calorimetric BDT pion probability. Expectations include contributions from $K^{+}\rightarrow\mu^{+}\nu\gamma$ ($K_{\mu2\gamma}$), $K^{+}\rightarrow\mu^{+}\nu$ ($K_{\mu2}$) and upstream (section~\ref{sec:UpstreamBackground}) events.

Left: expected and observed numbers of events in the control regions shown in figure~\ref{fig:pi0Rejection}-right. The global $p$-value of the comparison is 0.80, the lowest single-region $p$-value is 0.24. Right: expected and observed numbers of events in the upstream validation samples. The first bin is the signal region and therefore only the expectation is displayed. The global $p$-value is 0.97, the lowest single-sample $p$-value is 0.14.

Left: expected and observed numbers of events in the control regions shown in figure~\ref{fig:pi0Rejection}-right. The global $p$-value of the comparison is 0.80, the lowest single-region $p$-value is 0.24. Right: expected and observed numbers of events in the upstream validation samples. The first bin is the signal region and therefore only the expectation is displayed. The global $p$-value is 0.97, the lowest single-sample $p$-value is 0.14.

Left: distribution of the CDA variable for events in the upstream reference sample (URS). Right: matching probability $P_{\text{match}}$ measured in the $K^{+}\rightarrow\pi^{+}\pi^{0}$ normalisation sample in bins of $\Delta T_{\text{match}}$ and $N_{\text{GTK}}$.

Left: distribution of the CDA variable for events in the upstream reference sample (URS). Right: matching probability $P_{\text{match}}$ measured in the $K^{+}\rightarrow\pi^{+}\pi^{0}$ normalisation sample in bins of $\Delta T_{\text{match}}$ and $N_{\text{GTK}}$.

Left: distribution of the observed data events satisfying the signal selection criteria in the $(p_{\pi^{+}},m_{\rm miss}^{2})$ plane. Events in the background, control and signal regions are shown by small grey, small black and large black markers, respectively. Right: $m_{\rm miss}^{2}$ projection including SM signal~\cite{Buras:2015qea} (assuming $\mathcal{B}_{\pi\nu\bar{\nu}}^{\text{SM}} = 8.4\times10^{-11}$) and background expectations from $K^{+}\to\pi^{+}\pi^{0}$ ($K_{2\pi}$), $K^{+}\to\mu^+\nu$ ($K_{\mu2}$), $K^{+}\to\mu^+\nu\gamma$ ($K_{\mu2\gamma}$), $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ ($K_{3\pi}$), $K^+\to\pi^+\pi^-e^+\nu$ ($K_{e4}$) and upstream. The total expected background and its uncertainty is shown by the black line and hatched bars, respectively. In the signal region R1, events in the momentum range $35$--$45\,\text{GeV}/c$ are excluded.

Left: distribution of the observed data events satisfying the signal selection criteria in the $(p_{\pi^{+}},m_{\rm miss}^{2})$ plane. Events in the background, control and signal regions are shown by small grey, small black and large black markers, respectively. Right: $m_{\rm miss}^{2}$ projection including SM signal~\cite{Buras:2015qea} (assuming $\mathcal{B}_{\pi\nu\bar{\nu}}^{\text{SM}} = 8.4\times10^{-11}$) and background expectations from $K^{+}\to\pi^{+}\pi^{0}$ ($K_{2\pi}$), $K^{+}\to\mu^+\nu$ ($K_{\mu2}$), $K^{+}\to\mu^+\nu\gamma$ ($K_{\mu2\gamma}$), $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ ($K_{3\pi}$), $K^+\to\pi^+\pi^-e^+\nu$ ($K_{e4}$) and upstream. The total expected background and its uncertainty is shown by the black line and hatched bars, respectively. In the signal region R1, events in the momentum range $35$--$45\,\text{GeV}/c$ are excluded.

Left: test statistic $q$ as a function of the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ branching ratio for 2021--2022 data. Right: numbers of expected and observed events in the six categories used for the statistical analysis of 2021--2022 data. The background expectation is shown in blue, while the signal (using the measured value of the branching ratio) plus background expectation is shown in green.

Left: test statistic $q$ as a function of the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ branching ratio for 2021--2022 data. Right: numbers of expected and observed events in the six categories used for the statistical analysis of 2021--2022 data. The background expectation is shown in blue, while the signal (using the measured value of the branching ratio) plus background expectation is shown in green.

Left: test statistic $q$ as a function of the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ branching ratio for 2016--2022 data. Right: numbers of expected and observed events in the 15 categories used for the statistical analysis of 2016--2022 data (table~\ref{tab:StatTreatmentInputTable}). The background expectation is shown in blue, while the signal (using the measured value of the branching ratio) plus background expectation is shown in green.

Left: test statistic $q$ as a function of the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ branching ratio for 2016--2022 data. Right: numbers of expected and observed events in the 15 categories used for the statistical analysis of 2016--2022 data (table~\ref{tab:StatTreatmentInputTable}). The background expectation is shown in blue, while the signal (using the measured value of the branching ratio) plus background expectation is shown in green.

Left: summary of $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ branching ratio measurements from the BNL E787 and E949 experiments~\cite{BNL-E949:2009dza}, and the NA62 experiment using the 2016--2018~\cite{PnnRun1Paper}, 2021--2022 (equation~\ref{eqn:BR2122}) and 2016--2022 (equation~\ref{eqn:BR1622}) data. Statistical and total uncertainties are shown by thinner and thicker vertical bars, respectively. These are compared to the two recent SM predictions~\cite{Buras:2022wpw,DAmbrosio:2022kvb}. Right: global status of the $K\rightarrow\pi\nu\bar{\nu}$ decay modes, showing the most stringent $\mathcal{B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})$ upper limit~\cite{KOTO:2024zbl}, the Grossman-Nir bound~\cite{Grossman:1997sk,PDG}, the two recent SM predictions~\cite{Buras:2022wpw,DAmbrosio:2022kvb}, and the $\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})$ result from the combined 2016--2022 NA62 dataset (the $1\,\sigma$ and $2\,\sigma$ ranges are displayed in darker and lighter shaded areas, respectively).

Left: summary of $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ branching ratio measurements from the BNL E787 and E949 experiments~\cite{BNL-E949:2009dza}, and the NA62 experiment using the 2016--2018~\cite{PnnRun1Paper}, 2021--2022 (equation~\ref{eqn:BR2122}) and 2016--2022 (equation~\ref{eqn:BR1622}) data. Statistical and total uncertainties are shown by thinner and thicker vertical bars, respectively. These are compared to the two recent SM predictions~\cite{Buras:2022wpw,DAmbrosio:2022kvb}. Right: global status of the $K\rightarrow\pi\nu\bar{\nu}$ decay modes, showing the most stringent $\mathcal{B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})$ upper limit~\cite{KOTO:2024zbl}, the Grossman-Nir bound~\cite{Grossman:1997sk,PDG}, the two recent SM predictions~\cite{Buras:2022wpw,DAmbrosio:2022kvb}, and the $\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})$ result from the combined 2016--2022 NA62 dataset (the $1\,\sigma$ and $2\,\sigma$ ranges are displayed in darker and lighter shaded areas, respectively).