Figure 1

Unitarity triangles in the $\overline{\rho }-\overline{\eta }$ plane. Two sides of the triangle can be expressed by the CKM matrix elements $|V_{td}|/A{\lambda }^3$ and $|V_{ub}/V_{cb}|/\lambda$, respectively, where $A$ and $\lambda$ are parameters in the Wolfenstein parametrization.Reuse & Permissions

Figure 2

Unitarity triangle determined by $B$ and $K$ decays. The parameters $\overline{\rho }$ and $\overline{\eta }$ can be determined in two ways: the angle $\beta$ from the $CP$-violating asymmetry in the decay $B_d^0\to J/\psi K_s^0$, and from the length of the side from $\Delta M_{B_s}/\Delta M_{B_d}$ in $B-\overline{B}$ mixing; the height of the triangle from $\mathcal{B}(K_L^0\to {\pi }^0\nu \overline{\nu })$ and the radius of a circle centered at $({\overline{\rho }}_0,0)$ from $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$.Reuse & Permissions

Figure 3

Second-order weak processes that contribute to the $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ branching ratio: the Box diagram (upper) and two “$Z$-penguin” diagrams (bottom).Reuse & Permissions

Figure 4

Momentum spectra (in $\mathrm{MeV}/c$) of charged particles from $K^{+}$ decays in the rest frame. The values in the parentheses represent the branching ratios of the decay modes [18]. The hatched spectrum shows the ${\pi }^{+}$ momentum from $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ decay assuming the $V-A$ interaction.Reuse & Permissions

Figure 5

Schematic side (a) and end (b) views of the upper half of the E949 detector. Illustrated in this figure, an incoming $K^{+}$ that traverses all the beam instruments, stops in the target and undergoes the decay $K^{+}\to {\pi }^{+}{\pi }^0$. The outgoing ${\pi }^{+}$ and one photon from ${\pi }^0\to \gamma \gamma$ are also shown. The detector elements and acronyms are described in detail in the text.Reuse & Permissions

Figure 6

Schematic end view of the inner RSSC. The tubes run through the beam direction.Reuse & Permissions

Figure 7

Radiation length with (solid curve) and without (dotted curve) the BVL as a function of the cosine of the polar angle. These curves account for the contribution from RS, BV, EC and BVL. Other photon veto counters are not accounted for in this plot.Reuse & Permissions

Figure 8

Range in plastic scintillator (cm) versus the momentum ($\mathrm{MeV}/c$) of the charged particles for events that passed the $\pi \nu \overline{\nu }(1)$ trigger. These data represent $\sim 0.3\%$ of $N_K$.Reuse & Permissions

Figure 9

Schematic diagrams of the single beam background: (1) single beam $K^{+}$ background and (2) single beam ${\pi }^{+}$ background. The various detector elements and acronyms are described in the text.Reuse & Permissions

Figure 10

Schematic diagrams of the double beam background: (1) double beam $K^{+}-K^{+}$ background and (2) double beam $K^{+}-{\pi }^{+}$ background. The various detector elements and acronyms are described in the text.Reuse & Permissions

Figure 11

Schematic diagram of the charge exchange interaction background. The various detector elements and acronyms are described in detail in the text.Reuse & Permissions

Figure 12

Pictorial explanation of the bifurcation method. Background level in region A can be estimated from the observed number of events in the other regions, if CUT1 and CUT2 are uncorrelated.Reuse & Permissions

Figure 13

Illustration of the range stack track fitting in $x-y$ plane. All the relevant definitions are given in this plot. The arc represents the extrapolation of the fitted UTC track.Reuse & Permissions

Figure 14

The momentum of charged particles versus $t_{C_K}-t_{rs}$ and $t_{C_{\pi }}-t_{rs}$. The $K\mu 2$ monitor was used to represent the single beam $K^{+}$ events. The data plots indicate beam background contamination at beam time ($t_{C_K}$ or $t_{C_{\pi }}$) close to $t_{rs}$ in the $\pi \nu \overline{\nu }(1)$ trigger sample. The arrows indicate the rejected timing regions. The statistics in these plots account for about 0.3% of $N_K$.Reuse & Permissions

Figure 15

The energy loss in the B4 hodoscope for beam $K^{+}$ (solid line) and beam ${\pi }^{+}$ (dashed line). The $K^{+}$’s were the $K_{\mu 2}$ peak events in the $K\mu 2$ monitor sample, while the ${\pi }^{+}$’s were the scattered ${\pi }^{+}$’s in the skim 6 sample. The arrow indicates the cut position.Reuse & Permissions

Figure 16

Distribution of the time difference, $t_{\pi }-t_K$ measured by the target, for the $K_{\mu 2}$ peak events (solid line) from the $K\mu 2$ monitor sample and the scattered ${\pi }^{+}$’s (dashed line) from the skim 6 sample. Beam ${\pi }^{+}$’s can be assigned a time $t_K$ by the target fiber reconstruction due to misidentification.Reuse & Permissions

Figure 17

Beam likelihood for $K^{+}$ decays at rest (solid line) from the $K\mu 2$ monitor sample and scattered ${\pi }^{+}$’s (dashed line) from the skim 6 sample. The energy loss of $K^{+}$ in the target was required to be greater than 25 MeV and the number of $K^{+}$ fibers was required to be greater than 2 before evaluating the beam likelihood.Reuse & Permissions

Figure 18

Distributions of the maximum energy deviation for ${\pi }^{+}$’s (solid line) and ${\mu }^{+}$’s (dashed line) in RS. The cut positions are indicated by the arrows.Reuse & Permissions

Figure 19

Distributions of the probability of energy loss consistency in the RS $dE/dx$ measurement for ${\pi }^{+}$’s (solid line) and ${\mu }^{+}$’s (dashed line). The arrow indicates the cut position.Reuse & Permissions

Figure 20

Likelihood distributions of the RS energy measurement for ${\pi }^{+}$’s (solid line) and ${\mu }^{+}$’s (dashed line). The arrow indicates the cut position.Reuse & Permissions

Figure 21

Distributions of the range deviation in the RS for ${\pi }^{+}$’s (solid line) ${\mu }^{+}$’s (dashed line) tracks. The arrow indicates the cut position.Reuse & Permissions

Figure 22

Results of fits with the single- and double-pulse hypotheses to the TD pulse shapes in the stopping counter. The solid curve is for the double-pulse hypothesis and the dashed curve is for the single-pulse hypothesis.Reuse & Permissions

Figure 23

Schematic view of signal (left) and background (right) of the ${\mu }^{+}\to e^{+}$ decay. Each rectangle represents an RS counter. The central counter represents the stopping counter and the shaded rectangles represent hit counters. The arrow indicates the possible direction of the positron or charged track producing the hits.Reuse & Permissions

Figure 24

Distributions of the input variables for the neural net function in ${\pi }^{+}\to {\mu }^{+}{\nu }_{\mu }$ induced (solid lines) and ${\mu }^{+}\to e^{+}{\nu }_e{\overline{\nu }}_{\mu }$ induced (dashed lines) decays. The energy of the second pulse (a), time of the second pulse (b), log of the product of the single-pulse fit ${\chi }^2$’s for both ends (c), $z$ position difference between the first and second pulses obtained from the energy ratio of both ends (d), log of the product of the ${\chi }^2$ ratios of single- to double-pulse fits for both ends (e) and time difference of the second pulses in both ends (f).Reuse & Permissions

Figure 25

Distributions of the outputs of the NN function for ${\pi }^{+}$ (solid line) and ${\mu }^{+}$ (dashed line) events. Events with an output of the NN function less than 0.76 were rejected.Reuse & Permissions

Figure 26

Rejection of the NN ${\pi }^{+}\to {\mu }^{+}$ decay cut as a function of the acceptance. The intersection of the vertical and horizontal dashed lines shows the rejection and acceptance at the nominal cut position.Reuse & Permissions

Figure 27

The timing resolution as a function of visible energy in various PV counters.Reuse & Permissions

Figure 28

Illustration of the optimization process to determine the photon veto parameters.Reuse & Permissions

Figure 29

Offline rejection of the photon veto cuts against the $K_{\pi 2}$ background as a function of the acceptance. The crossing point of the vertical and horizontal lines shows the rejection and acceptance at the cut position.Reuse & Permissions

Figure 30

The expected kinematic background events in the $1/3$ normalization branch (histogram with left axis) as a function of $K_{\pi 2}$ kinematic cut (Edev) position for the $K_{\pi 2}$ background (a), maximum momentum cut position for the $K_{\mu 2}$ range tail background (b) and range-momentum cut (${\chi }_{rm}$) position for the ${\mu }^{+}$ band background (c). Also shown is the relative acceptance change as the cuts (curve with right axis).Reuse & Permissions

Figure 31

Pictorial explanation of the correlation study using the events near, but outside the signal region. The vertical axis “$a$” and the horizontal axis “$b$” are for the loosening factors, while the $1\times 1$ black region represents the standard signal region.Reuse & Permissions

Figure 32

Measurement of the $K^{+}\to {\pi }^{+}{\pi }^0$ branching ratio as a function of run number in E949.Reuse & Permissions

Figure 33

Range versus kinetic energy of the events satisfying all of the cuts, except for the phase space cuts on both the range and energy. The plots are shown separately for E949 only (left) and E787 plus E949 results. The rectangle represents the signal region defined in E787 (dashed lines) and E949 (solid lines). Events around $E=108\mathrm{MeV}$ were due to $K_{\pi 2}$, which were not removed by the photon veto cuts. The light points in the right-hand plot represent the expected distribution of $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ events from simulation.Reuse & Permissions

Figure 34

Reconstruction of the candidate event (end view). The clusters of squares indicate both the $K^{+}$ track and the ${\pi }^{+}$ track in the target. The hit IC sector is shown next to the ${\pi }^{+}$ cluster in the target. The curve is the result of the UTC track fit. The circles along the track are the hits in the UTC. The radius of each circle gives the drift distance. The RS and RSSC hit layers are shown outside the UTC. Also displayed are the TD data in the ${\pi }^{+}$ stopping counter, the reconstruction in the target and the CCD data in the $K^{+}$ stopping fiber. In the fit to the TD pulse shape, the ${\pi }^{+}$ pulse (dashed line) and the ${\mu }^{+}$ pulse (dotted line) are shown separately. No obvious ${\pi }^{+}$ pulse was observed in the CCD pulse shape for the $K^{+}$ stopping fiber.Reuse & Permissions

Figure 35

The expected relative kinematic background rate in the normalization branch versus the relative acceptance for the $K_{\pi 2}$ background (a), the $K_{\mu 2}$ range tail background (b) and the ${\mu }^{+}$ band background (c). Both background rates and acceptance curves were normalized to one at the cut positions.Reuse & Permissions

Figure 36

The expected relative beam background rate versus the relative acceptance for the single-beam and CEX backgrounds (a) and the double-beam background (b). Both background rates and acceptance curves were normalized to one at the cut positions.Reuse & Permissions

Figure 37

The allowed regions in the ${\lambda }_t$ plane allowed by the combined E787 and E949 determination of the $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ branching ratio (gray), $B^{+}\to {\tau }^{+}\nu$ (blue) and $B$-mixing measurements (yellow). The regions outside the lighter (darker) shading have $\mathrm{C}.\mathrm{L}.>0.95$ (0.68). The area excluded by $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ at $\mathrm{C}.\mathrm{L}.>0.95$ is indicated by the gray line. The red-shaded region is allowed by the combination of these measurements and the small black region denotes the region allowed by all CKM-related measurements as evaluated by the CKMfitter Group [82].Reuse & Permissions