Search for $T$-violating transverse muon polarization in the $K^{+}\to {\pi }^0{\mu }^{+}\nu$ decay

At the 12-GeV proton synchrotron of KEK, an experiment (KEK-PS-E246) was performed to measure the transverse muon polarization ($P_T$) in $K^{+}\to {\pi }^0{\mu }^{+}\nu$ decays as a direct search of violation of time reversal invariance ($T$-violation). $P_T$ is a very sensitive probe of $CP$ violation beyond the standard model. A stopped positive kaon beam was used. An elaborate high-acceptance detector, in conjunction with a 12-sector superconducting toroidal spectrometer, was utilized. The stopped kaon method provided several advantages in performing a high-precision experiment. The decay was identified by detecting a ${\mu }^{+}$ with the spectrometer and a ${\pi }^0$ with a CsI(Tl) calorimeter surrounding the target as two photons as well as one photon with high energy. For the $P_T$ measurement, ${\pi }^0$ events in the forward (fwd) and backward (bwd) regions in the calorimeter were relevant. Systematic errors were greatly suppressed by exploiting the rotational and fwd-bwd symmetry of the system. Polarization measurement was done in terms of asymmetry measurement of decay positrons of stopped muons in the polarimeter. A longitudinal field method was applied for the $P_T$ polarization component. In the data analysis a novel technique of the two-analysis method was employed. Two independent analyses were performed following their own policy and criteria and good events were selected. Then, the two analyses were combined. A data quality check was carefully done by looking at the null asymmetry $A_0$, the sensitivity function $A_N$, the kinematical attenuation coefficient, and the decay-plane distributions. The asymmetry measurement for finite-size muon stoppers was performed “differentially” using the position information by the chamber in front of the polarimeter. From the measurements during 1996–2000, we accumulated $11.8\times {10}^6$ good events after the two-analysis combination, and deduced $P_T=-0.0017\pm 0.0023(\mathrm{stat})\pm 0.0011(\mathrm{syst})$, corresponding to the $T$-violating parameter $\mathrm{Im}\xi =-0.0053\pm 0.0071(\mathrm{stat})\pm 0.0036(\mathrm{syst})$. From these results the upper limits of $|P_T|<0.0050$ (90% C.L.) and $|\mathrm{Im}\xi |<0.016$ (90% C.L.) were deduced. Systematic errors were carefully studied and estimated. Almost all of the errors were canceled by the 12-sector summation and/or fwd-bwd subtraction, and the total systematic error was concluded to be as a half of the statistical error. The present result set constraints on model parameters of several theoretical models. For the three-Higgs-doublet model, e.g., a limit $|\mathrm{Im}({\alpha }_1{\gamma }_1^{*})|<544(M_{H_1}/\mathrm{GeV}{)}^2$ was obtained. Implications for several other models are discussed.

Figure 1

The intensity distribution (a), and the three components of the muon polarization in the $K_{\mu 3}$ decay of (b) longitudinal component $P_L$, (c) normal component $P_N$, and (d) transverse component $P_T/\mathrm{Im}\xi$ as defined by Eq. (8). Here and in the Monte Carlo simulation later on, $\mathrm{Re}\xi =-0.35$ was assumed, which was the evaluation value of the Particle Data Group at the start of the present experiment [29].

Figure 2

Radiative corrections in the $K_{\mu 3}$ decay [21].

Figure 3

Experimental setup: (a) cross section side view, (b) end view, and (c) schematic view of one sector of the polarimeter. For details see [26].

Figure 4

$T$-violating azimuthal polarization $P_T$ for the cases of the ${\pi }^0$ going in the forward (fwd) and backward (bwd) directions.

Figure 5

Beam Cherenkov counter multiplicity spectrum: (a) kaon ring spectrum at kaon incident, (b) pion ring at kaon incident, (c) kaon ring at pion incident, and (d) pion ring at pion incident.

Figure 6

Kaon stopping distribution in the target: $\rho (y,z)$ (up), $\rho (x,z)$ (middle), $\rho (x,y)$ (lower left), and the $z$ projection (lower right) in the target with an array diameter of 93 mm and a fiducial $z$ length of 20 cm.

Figure 7

Muon stopper configuration and magnetic field distribution. The direction of the flux is only schematically illustrated.

Figure 8

Figure-of-Merit (FoM) distribution of $P_T$ measurement on the Dalitz plane. The FoM is defined as $P_T\times \sqrt{N}$. The contour is the linear step from 0 to 6.

Figure 9

Detector acceptance for $2\gamma$ events (left) and $1\gamma$ events (right) of the detector on the Dalitz plane. The height of the contour is in relative scale.

Figure 10

Kaon decay time distribution measured as a difference between the target fibers timing and the Fid-counter decay particle timing for the total events of $2\gamma$ and $1\gamma$. In the forward ${\pi }^0$ spectrum, one sees a peak at the prompt timing coming from $K^{+}$ decay-in-flight which was removed in the analysis.

Figure 11

A typical charged particle momentum spectrum of all the $2\gamma$ and $1\gamma$ events. The bump below $190\mathrm{MeV}/c$ containing $K_{\mu 3}$ and $K_{e3}$ was accepted in the analyses.

Figure 12

TOF mass spectrum of charged particles in the momentum range of 100 to $190\mathrm{MeV}/c$ in the analysis A1. The positrons from $K_{e3}$ could be rejected.

Figure 13

Incident muon $y$ distribution in the stopper measured by the C4 chamber (top), and the intrinsic geometrical asymmetry for those muons (bottom). The real stopping distribution has a smearing due to beam divergence and multiple scattering. The geometrical asymmetry was measured as the null asymmetry $A_0$ by accepting all ${\pi }^0$ directions.

Figure 14

Typical performance of the CsI(Tl) ${\pi }^0$ detection. (a) two photon invariant mass spectrum of $K_{\mu 3}$, (b) cluster timing distribution also of $K_{\mu 3}$ events showing the time resolution, (c) opening angle distribution of $K_{\pi 2}$ showing the angular resolution, and (d) two photon and one photon energy spectra of $K_{\mu 3}$ events.

Figure 15

$\chi 2$ distribution of charged particle tracking in the A1 analysis

Figure 16

Track consistency check at ring (top) and target (bottom).

Figure 17

Missing mass distribution for $K_{\mu 3}$ and $K_{\pi 2}$ events.

Figure 18

${\chi }^2$ distribution of charged particle tracking in the A2 analysis.

Figure 19

Opening angle ${\theta }_{\mu \pi (\gamma )}$ distribution of the $K_{\mu 3}$ decay region ($p<190\mathrm{MeV}$) for ${\chi }^2<10$ and ${\chi }^2>20$ of $2\gamma$ events. It is seen that $K_{\pi 2}$ background with back-to-back correlation is concentrated in the large ${\chi }^2$ region.

Figure 20

Positron time spectra for 6 data sets of common, A1-uncommon, and A2-uncommon for $2\gamma$ and $1\gamma$ events

Figure 21

Correlation plot of the two photon invariant mass $M_{\gamma \gamma }$ between the A1 and A2 analyses for $2\gamma$ events. The selected region in each analysis is indicated with the vertical lines for A1 and horizontal lines for A2.

Figure 22

Distribution of decay planes relative to the polarimeter axis for the common data. The solid lines are for fwd and the dotted lines are for bwd.

Figure 23

Distribution of decay planes relative to the polarimeter axis for the A1-uncommon data. The solid lines are for fwd and the dotted lines are for bwd.

Figure 24

Distribution of decay planes relative to the polarimeter axis for the A2-uncommon data. The solid lines are for fwd and the dotted lines are for bwd.

Figure 25

Cross section of the muon polarimeter of one sector at certain radial position $r$ with the tilted positron counters. The $y$ coordinate was measured by the C4 chamber.

Figure 26

Analyzing power function $\alpha (y)$ deduced from $A_N(y)$ of all the data. It was absolutely calibrated. The center of the stopper is at $y=18.5$. The sides of $y\le 18$ and $y\ge 19$ were symmetrized.

Figure 27

$A_T$ dependence on $\mathrm{C}4\mathrm{\text{−}}y$ (black dots) plotted together with $A_N$ (open circles) for the 6 data sets

Figure 28

$P_T$ dependence on $\mathrm{C}4\mathrm{\text{−}}y$. Black dots are $2\gamma$ events and open circles are $1\gamma$ events. The center of the stopper is at $y=18.5$.

Figure 29

Ideogram of $\mathrm{Im}\xi$ for the 18 data sets. A statistical distribution with ${\chi }^2/\mathrm{dof}=0.78$ is observed. The black dots are for data set I, the open circles are for II, and the stars are for III. The hatched stripe indicates $\mathrm{Im}\xi =-0.0053\pm 0.0071$.

Figure 30

$P_T$ dependence on muon energy for $2\gamma$ and $1\gamma$ events.

Figure 31

$P_T$ dependence on photon energy. For $2\gamma$ events the sum energy (namely, the ${\pi }^0$ energy) was taken. For $1\gamma$ events the energy of the observed photon was taken.

Figure 32

$P_T$ of each gap.

Figure 33

$A_T$ dependence on the average of the attenuation factor which is defined by Eq. (19) and is nearly the cosine of the ${\pi }^0$ polar angle ${\theta }_{{\pi }^0(\gamma )}$. The final result of the present experiment corresponds to the ratio between the regions of $>0.342$ and $<-0.342$.