Observed model-independent upper limits at 90 % CL of $\mathcal{B}(K^{+}\rightarrow\pi^{+}X)$ as function of $X$ mass, for several $X$ lifetime hypotheses, assuming $X$ decays to visible SM particles.

Di-muon mass spectrum of selected $K^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ events from 2017–2018 data.

Expected and observed model-independent upper limits at 90 % CL for $\mathcal{B}(K^{+}\rightarrow\pi^{+}X)\times\mathcal{B}(X\rightarrow\mu^{+}\mu^{-})$ as a function of $m_{X}$ for $\tau_{X}=0$.

Observed model-independent upper limits at 90 % CL for $\mathcal{B}(K^{+}\rightarrow\pi^{+}X)\times\mathcal{B}(X\rightarrow\mu^{+}\mu^{-})$ for several $\tau_{X}$ values.

Branching ratio of the $K^{+}\rightarrow\pi^{+}A^{\prime}$ decay divided by the kinetic mixing coupling squared, $\varepsilon^{2}$, as a function of $m_{A^{\prime}}$ for the BC2 model with dark photon $A^{\prime}$.

Excluded region, at 90 % CL, of the parameter space $(m_{A^{\prime}},\varepsilon)$ for a dark photon $A^{\prime}$, decaying invisibly, in the BC2 model for the NA62 $K^{+}\rightarrow\pi^{+}X_{\rm inv}$ search.

Excluded region, at 90 % CL, of the parameter space $(m_{A^{\prime}},\varepsilon)$ for a dark photon $A^{\prime}$, decaying invisibly, in the BC2 model for the NA62 $\pi^{0}\rightarrow{\rm inv}$ search.

Branching ratio of the $K^{+}\rightarrow\pi^{+}S$ decay divided by $\text{sin}^{2}\theta$, as a function of $m_{S}$.

Excluded regions, at 90 % CL, of the parameter space $(m_{S},\rm{sin}^{2}\theta)$ for a dark scalar $S$, in the BC4-inv model for the NA62 $K^{+}\rightarrow\pi^{+}X_{\rm inv}$ search.

Excluded regions, at 90 % CL, of the parameter space $(m_{S},\rm{sin}^{2}\theta)$ for a dark scalar $S$, in the BC4-inv model for the NA62 $\pi^{0}\rightarrow{\rm inv}$ search.

Excluded regions, at 90 % CL, of the parameter space $(m_{S},\rm{sin}^{2}\theta$) for a dar scalar S, in the BC4 model, from the NA62 $K^{+}\rightarrow\pi^{+}X_{\rm inv}$ search.

Excluded regions, at 90 % CL, of the parameter space $(m_{S},\rm{sin}^{2}\theta$) for a dar scalar S, in the BC4 model, from the NA62 $K^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ study.

Branching ratio of the $K^{+}\rightarrow\pi^{+}a$ decay divided by $(C_{ff}/\Lambda)^{2}$, as a function of $m_{a}$, assuming Λ = 1 TeV.

Excluded regions, at 90 % CL, of the parameter space $(m_{a}, C_{ff}/\Lambda)$ for an ALP $a$, in the BC10-inv model, evaluated assuming $\Lambda = 1$ TeV, for the NA62 $K^{+}\rightarrow\pi^{+}X_{\rm inv}$ search.

Excluded regions, at 90 % CL, of the parameter space $(m_{a}, C_{ff}/\Lambda)$ for an ALP $a$, in the BC10-inv model, evaluated assuming $\Lambda = 1$ TeV, for the NA62 $\pi^{0}\rightarrow\rm{inv}$ search.

Excluded regions, at 90 % CL, of the parameter space $(m_{a}, C_{ff}/\lambda)$ for an ALP $a$, in the BC10 model, evaluated assuming $\Lambda = 1$ TeV for the NA62 $K^{+}\rightarrow\pi^{+}X_{\rm inv}$ search.

Excluded regions, at 90 % CL, of the parameter space $(m_{a}, C_{ff}/\lambda)$ for an ALP $a$, in the BC10 model, evaluated assuming $\Lambda = 1$ TeV for the NA62 $\pi^{0}\rightarrow\rm{inv}$ search.

Excluded regions, at 90 % CL, of the parameter space $(m_{a}, C_{ff}/\lambda)$ for an ALP $a$, in the BC10 model, evaluated assuming $\Lambda = 1$ TeV for the NA62 $K^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ study.

Branching ratio of the $K^{+}\rightarrow\pi^{+}a$ decay divided by $(C_{GG}/\Lambda)^{2}$, as a function of $m_{a}$, assuming $\Lambda = 1$ TeV.

Excluded regions, at 90 % CL, of the parameter space $(m_{a}, C_{GG}/\Lambda)$ for an ALP $a$, of the BC11 model, evaluated assuming $\Lambda = 1$ TeV, from the NA62 search for $K^{+}\rightarrow\pi^{+}X_{\rm inv}$.

Excluded regions, at 90 % CL, of the parameter space $(m_{a}, C_{GG}/\Lambda)$ for an ALP $a$, of the BC11 model, evaluated assuming $\Lambda = 1$ TeV, from the NA62 search for $\pi^{0}\rightarrow\rm{inv}$.

Excluded regions, at 90 % CL, of the parameter space $(m_{a}, C_{GG}/\Lambda)$ for an ALP $a$, of the BC11 model, evaluated assuming $\Lambda = 1$ TeV, from the NA62 $K^{+}\rightarrow\pi^{+}\gamma\gamma$ study.

90% CL upper limit in axion-like particle coupling to SM fermions at UV scale Lambda=1TeV vs mass parameter space for di-lepton final states.

90% CL upper limit in axion-like particle coupling to SM gluons at UV scale Lambda=1TeV vs mass parameter space for hadronic final states.

Number of expected $a \to K^+K^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $C_{bs}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\eta$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $C_{bs}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\gamma$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $C_{bs}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $C_{bs}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $C_{bs}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\pi^0$ assuming production coupling $\theta_{\pi^0} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\pi^0$ assuming production coupling $\theta_{\pi^0} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to K^+K^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\pi^0$ assuming production coupling $\theta_{\pi^0} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\eta$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\pi^0$ assuming production coupling $\theta_{\pi^0} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\gamma$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\pi^0$ assuming production coupling $\theta_{\pi^0} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\pi^0$ assuming production coupling $\theta_{\pi^0} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\pi^0$ assuming production coupling $\theta_{\pi^0} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta$ assuming production coupling $\theta_{\eta} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta$ assuming production coupling $\theta_{\eta} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to K^+K^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta$ assuming production coupling $\theta_{\eta} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\eta$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta$ assuming production coupling $\theta_{\eta} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\gamma$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta$ assuming production coupling $\theta_{\eta} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta$ assuming production coupling $\theta_{\eta} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta$ assuming production coupling $\theta_{\eta} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta^\prime$ assuming production coupling $\theta_{\eta^\prime} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta^\prime$ assuming production coupling $\theta_{\eta^\prime} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to K^+K^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta^\prime$ assuming production coupling $\theta_{\eta^\prime} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\eta$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta^\prime$ assuming production coupling $\theta_{\eta^\prime} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\gamma$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta^\prime$ assuming production coupling $\theta_{\eta^\prime} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta^\prime$ assuming production coupling $\theta_{\eta^\prime} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in mixing with $\eta^\prime$ assuming production coupling $\theta_{\eta^\prime} = 1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for Primakoff production assuming production coupling $C_{\gamma\gamma}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for Primakoff production assuming production coupling $C_{\gamma\gamma}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to K^+K^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for Primakoff production assuming production coupling $C_{\gamma\gamma}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\eta$ events for BR = 1 in the plane (mass, width) after full selection, for Primakoff production assuming production coupling $C_{\gamma\gamma}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\gamma$ events for BR = 1 in the plane (mass, width) after full selection, for Primakoff production assuming production coupling $C_{\gamma\gamma}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for Primakoff production assuming production coupling $C_{\gamma\gamma}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $a \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for Primakoff production assuming production coupling $C_{\gamma\gamma}/\Lambda=1\,\mathrm{GeV}^{-1}$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to K^+K^-$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to K^+K^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for mixing production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for mixing production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to K^+K^-$ events for BR = 1 in the plane (mass, width) after full selection, for mixing production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to K^+K^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for mixing production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-$ events for BR = 1 in the plane (mass, width) after full selection, for mixing production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for mixing production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for mixing production assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in light meson decays assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in light meson decays assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in light meson decays assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in light meson decays assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $A^\prime \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for production in light meson decays assuming production coupling $\varepsilon=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to K^+K^-$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to \pi^+\pi^-$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to \pi^+\pi^-\pi^0\pi^0$ events for BR = 1 in the plane (mass, width) after full selection, for Bremsstrahlung production assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to e^+e^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to \mu^+\mu^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to K^+K^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

Number of expected $S \to \pi^+\pi^-$ events for BR = 1 in the plane (mass, width) after full selection, for production in B meson decays assuming production coupling $\mathrm{sin}^2\theta=1$. Acceptance for particles that reach the FV and decay therein and mass resolution.

90% CL upper limit in dark photon coupling vs mass parameter space for the $A^\prime \rightarrow l^+l^-$ analysis.

90% CL upper limit in $BR(B \rightarrow K a, a \rightarrow e^+e^-)$ vs lifetime $\tau$ parameter space.

90% CL upper limit in $BR(B \rightarrow K a, a \rightarrow e^+e^-)$ vs lifetime $\tau$ parameter space.

Upper limits at 90% CL of $B(K^+\to\pi^+a)\times B(a\to\gamma\gamma)$ in the prompt ALP decay assumption.

See caption of Figure 9.

See caption of Figure 9.

See caption of Figure 9.

See caption of Figure 9.

See caption of Figure 9.

See caption of Figure 9.