Precise Measurement of the $K_{e2}/K_{\mu 2}$ Branching Ratio and New Physics Search with a Stopped $K^+$ Beam Experiment

The J-PARC TREK/E36 experiment with a stopped $K^+$ beam is designed to provide a more precise measurement of the branching ratio $R_K=\Gamma(K^+\rightarrow e^+\nu)/\Gamma(K^+\rightarrow \mu^+\nu)$ than previous in-flight $K^+$ decay experiments. $R_K$ is very precisely predicted by the Standard Model (SM) with an uncertainty of $4\times10^{-4}$ and any deviation from this prediction would very clearly indicate the existence of new physics beyond the SM. Additionally, the experiment is searching for light neutral bosons ($A^{'}$), which could be associated with dark matter or explain the $g_{\mu}$-2 anomaly and the proton radius puzzle. In the experiment, a $K^+$ beam was stopped by a scintillating fiber target, and charged decay products were momentum analyzed and tracked by a 12-sector superconducting toroidal magnetic spectrometer and multi-wire proportional chambers (MWPCs) combined with a photon calorimeter with a large solid angle (75\% of $4\pi$) and 3 different particle identification systems. In this talk, the status of the $R_K$ and $A^{'}$ analyses is presented, and the MWPC calibration and tracking by a Kalman filter are reported. This work has been supported by awards DE-SC0003884 and DE-SC0013941 in U.S., NSERC in Canada, and Kaken-hi in Japan.