High precision measurements of electroweak observables represent powerful tests of the Standard Model (SM) to obtain hints of new physics [1]. The K + → l + ν l (K l2 ) decay, which is one of the simplest decays among the K + decay channels, is a clean and sensitive channel to perform such tests. Lepton universality signifies identical coupling constants for the three lepton generations, and it is a basic assumption in the SM. Although each K l2 decay width can be described using the K l2 hadronic form factor with a few percent accuracy, this form factor can be canceled out by forming the ratio of the electronic K + → e + ν (K e2 ) and muonic K + → μ + ν (K μ2 ) decay channels (R K ).
In the R K determination, the radiative K + → e + νγ decay, which is the K e2 decay accompanied with photon emission, has to be taken into account. There are two K e2γ processes [2,3]: the internal bremsstrahlung (IB) process, K IB e2γ , mostly with low-energy photon emission, and the structure dependent (SD) process, K SD e2γ , with high-energy photon emission roughly in the same and oppo-site directions of the e + motions, respectively. In order to compare the experimental value with the SM prediction, the IB process has to be included in the K e2 sample (K e2(γ ) = K e2 + K IB e2γ ) because it is impossible to experimentally separate the IB process from the K e2 decay. The SM prediction, R SM K = (2.477 ± 0.001) × 10 -5 , can be calculated with excellent accuracy [4][5][6][7], and this makes it possible to search for new physics effects by a precise R K measurement [8,9]. On the other hand, the SD process, which has a large hadronic uncertainty, is regarded as a background for R K and has to be subtracted from the observed e + events. The K e2(γ ) branching ratio is strongly suppressed down to ∼10 -5 due to the helicity suppression mechanism of the weak charged current. The SD process is not subject to the above helicity suppression, and the K SD e2γ branching ratio is comparable to that of K e2(γ ) . The SD process is sensitive to the electroweak structure of the kaon and has been the subject of extensive theoretical studies [2,3,10,11].
In the NA62 experiment [9], which produced the result with the smallest uncertainty, in-flight kaon decays in a 74 GeV/c beam with a momentum spread of ±1.4 GeV/c (rms) were used, and the decay particle momentum region from 15 to 65 GeV/c was investigated. The K μ2 and K SD e2γ decays were the main background sources in the K e2(γ ) sample. On the other hand, low energy kaons from φ → K + K -decays were used in the preceding KLOE experiment [8], and the experimental result was dominated by the statistical uncertainty. The NA62 and KLOE results were obtained to be R K = (2.488 ± 0.007 stat ± 0.007 sys ) × 10 -5 and R K = (2.493 ± 0.025 stat ± 0.019 sys ) × 10 -5 , respectively, and both are consistent with the SM prediction within the uncertainties. It should be noted that the branching ratio for K SD e2γ reported by the KLOE group was used in the NA62 analysis, and this SD contribution was subtracted from the observed K e2(γ ) samples. Therefore, an experimental check of the SD branching ratio with a systematically different approach from KLOE is important. In this letter, we present a new measurement of the branching ratio of the K SD e2γ decay relative to that of the K e2(γ ) decay, Br(K SD e2γ )/Br(K e2(γ ) ), performed with the J-PARC E36 experiment, which is also aiming at testing lepton universality violation with a precise R K measurement [12,13].
In contrast to the previous R K measurements, the E36 experiment used a stopped K + beam in conjunction with a 12-sector iron-core superconducting toroidal spectrometer [14] and a highly segmented CsI(Tl) calorimeter [15]. Schematic cross sectional side and end views of the detector configuration are shown in Fig. 1. Because of the rotational symmetry of the 12 identical gaps in the spectrometer and the large directional acceptance of the π 0 detector, spectra distortions due to detector acceptance are cancelled and systematic uncertainties are greatly suppressed [16]. The experimental apparatus was originally constructed for the KEK-PS E246/E470 experiments: a search for T -violating muon polarization in K + → π 0 μ + ν decay [17] and spectroscopic studies of various K + decay channels [18].
The experiment was performed in 2015 at the J-PARC Hadron Experimental Facility using a 780 MeV/c separated K + beam provided by the K1.1BR beam line [19]. A K /π ratio of ∼1 was obtained by means of an electrostatic separator system. The K + beam was discriminated from pion background by a Fitch-type Cherenkov counter (BC) [20]. An efficiency of more than 99% with a small π mis-trigger probability of <1% was achieved for the K + identification. The typical K + beam intensity was 1.0 ×10 6 in a 2-s spill duration and 6-s repetition rate. In total 4.5 × 10 5 spills were used for the physics production runs. The kaons were slowed down by a degrader and stopped in an active target (TGT), which consisted of 256 3.1×3.1 mm 2 thin scintillating bars of 20-cm length forming a cylindrical bundle with a 5.6-cm diameter, located at the center of the detector assembly. The K + stopping efficiency was typically ∼0.25 relative to BC K + triggering, and the K + stopping profile had a round shape with a Gaussian-like distribution with σ z of ∼4 cm in the beam direction.
e2γ and K e2(γ ) candidates were identified by analyzing the e + momentum (p) with the 12-sector spectrometer taken under the same trigger and DAQ conditions and, in addition, detecting the photon in the CsI(Tl) calorimeter for K SD e2γ . The trigger condition for event readout was a hit in both TOF counters and in TGT in addition to the K + beam particle identification by BC. The data were collected at a central magnetic field of B = 1.5 T, which was optimized for the positron momenta in the region of 220-250 MeV/c. Charged particles from TGT were tracked and momentum-analyzed by reconstructing the particle trajectory using multi-wire proportional chambers (MWPCs) located at the entrance (C2) and exit (C3 and C4) of the magnet gap, as well as by TGT and a spiral fiber tracker (SFT) made of scintillating fiber bundles [21,22] surrounding TGT. The momentum was corrected for the energy loss in TGT assuming that all particles were muons, therefore the π + and e + momenta after the correction were slightly shifted from their true values. The momentum spectrum before imposing the PID analysis is shown in Fig. 2 (a). Two peaks due to the K μ2 and K + → π + π 0 (K π 2 ) decays are clearly visible, although the K π 2 events are reduced due to the lower spectrometer acceptance.
The momentum resolution was σ p = 2.0 MeV/c at 236 MeV/c. The K + decay time, defined as the time of the e + signal at the TOF1 counter, was required to be more than 1.5 ns later than the K + arrival time determined by BC. Small time-of-flight corrections from BC to TGT, and TGT to TOF1 were accounted for on average. The fraction of in-flight K + decays and any other prompt backgrounds were suppressed down to 0.1%.
Particle identification (PID) of μ + , π + , and e + was carried out in each of the 12 sectors using three independent PID systemsan aerogel Cherenkov counter (AC) [23], a lead-glass Cherenkov counter (PGC) [24], and by measuring the time-of-flight (TOF) between the TOF1 and TOF2 plastic scintillation counters with timing resolutions of 250 ps and 100 ps, respectively. The AC and TOF1 surrounded TGT while TOF2 was located about 90 cm behind C4 resulting in a typical path length of 2.7 m from TOF1. The PGC was placed just after TOF2 at the end of the spectrometer. Fig. 3 shows the e + efficiency (solid/red) and the μ + rejection probability (dotted/black) at the momentum of 247 MeV/c and 236 MeV/c, respectively, as functions of the (a) AC, (b) PGC, and (c) M 2 TOF cut points. The e + efficiency for each PID system was obtained by preselecting e + from the K + → π 0 e + ν (K e3 ) and in-flight K e3 decays for the momentum region higher than the K e3 endpoint momentum (228 MeV/c) by using tighter PID conditions than nominally for the other two PID systems. The μ + rejection probability was determined using μ + s from the K μ2 decays. Positrons were selected by setting thresholds for AC and PGC at channel 100 and 140, respectively. Also, the mass-squared of the charged particle (M 2 TOF ) obtained from the TOF, momentum, and path length was required to be M 2 TOF < 4000 (MeV 2 /c 4 ). For the AC, PGC, and M 2 TOF efficiency determinations, the cut points of (PGC > 150, M 2 TOF < The peak structure due to the predominant K μ2 and K π 2 decays is seen at 236 MeV/c and 205 MeV/c, respectively, in (a). The K π 2 decay is reduced due to the momentum acceptance of the spectrometer. The K e2(γ ) and K e3 decays, as well as the remaining K μ2 events due to μ + mis-identification are presented in (b). The K e2(γ ) peak is observed with a tail structure in the lower momentum region due to the emission of internal and external bremsstrahlung before entering the spectrometer. The momentum in (b) was scaled so that the K e2(γ ) peak position is at 247 MeV/c, and consequently the K μ2 peak position appears at 233 MeV/c. The e + momentum below 225 MeV/c is not usable for the K SD e2γ and K e2(γ ) decays due to the high K e3 contribution. The K SD + e2γ , K e2(γ ) , K e3 and K μ2 decays determined by simulation calculations are also shown in (b). 3000 MeV 2 /c 4 ), (AC > 290, M 2 TOF < 3000 MeV 2 /c 4 ), and (AC > 290, PGC > 150) were used, and the μ + impurity was estimated to be less than 0.5%. These positron selection cuts, as shown in Fig. 3, were chosen to remove most of the K μ2 backgrounds with a μ + rejection probability of (99.934±0.002 stat )%, while maintaining a reasonable e + efficiency of (75.2±0.4 stat )%. This was determined to minimize the total uncertainty in the K SD e2γ branching ratio measurement from the K μ2 subtraction. Since the pulse height of the PGC counter increased with increasing e + momentum and the path length depended on the charged particle momentum, this introduced a momentum dependence in the PGC and M 2 TOF detection efficiency. On the other hand, the AC efficiency was nearly constant in the observed momentum region. The momentum dependence of the three PID elements was measured from 200 to 250 MeV/c in order to correct for this effect. The black/solid line in Fig. 2 (b) shows the charged-particle momentum spectrum with the positron PID condition applied and without constraints from the CsI(Tl).
The K e2(γ ) , K SD e2γ , and K e3 decays, as well as the remaining K μ2 events due to μ + mis-identification are observed, and the momentum was slightly scaled so that the K e2(γ ) peak position is at 247 MeV/c. The K e2(γ ) peak has a tail structure in the lower momentum region due to the emission of internal and external bremsstrahlung before entering the spectrometer. The contribution of K e2(γ ) events with high energy bremsstrahlung emission was outside of the spectrometer acceptance. By applying variable cut conditions to suppress K μ2 , it could be confirmed that there was no K e3 tail beyond 230 MeV/c.
The photon detector, a barrel of 768 CsI(Tl) crystals, covered ∼70% of the total solid angle [16]. There were 12 holes for outgoing charged particles to enter the spectrometer and 2 holes for the beam entrance and exit. Each crystal has a length of 25 cm and covers 7.5 • in both the polar and azimuthal direction. The photon energy and hit position were obtained by summing the energy deposits and calculating the energy-weighted centroid of participating crystals in the Moliere spread. To read out the CsI(Tl) calorimeter, VF48 Flash ADCs [25] were employed to record the waveform data in order to resolve pulse-pileup events with high efficiency.
The hardware threshold was set at ∼17 MeV to limit the event size. The CsI(Tl) energy and timing resolutions of a single module at 105 MeV were σ E /E ≈ 2.6% and σ t = 10.7 ns [26], respectively, and the position resolution was obtained as σ pos = 7.6 mm. Accidental backgrounds were reduced by choosing a timing window of ±50 ns. In addition, some of the photons that passed through the holes in the CsI(Tl) calorimeter into the spectrometer sectors were detected by gap shower counters (GSC), which are sandwich detectors of plastic scintillators and lead plates that will allow us to perform a supplemental K SD e2γ study by detecting the radiative photons.
In the present study, the SD branching ratio, Br(K SD e2γ ) normalized to that of K e2(γ ) decay, Br(K e2(γ ) ), was determined from the ratio of the K SD e2γ and K e2(γ ) yields, corrected for the detector acceptance as
where N is the number of the accepted events and R is the ratio of the overall acceptances for K e2(γ ) and K SD e2γ , respectively, obtained by a Monte Carlo simulation. In contrast to the previous KLOE experiment which determined Br(K SD e2γ ) relative to Br(K μ2 ), the present experiment was able to disentangle both the number of N(K e2(γ ) ) and N(K SD e2γ ) events directly from the charged particle momentum spectra. The spectrum in Fig. 2 (b) was decomposed by simulating the spectrum of each contributing process and fitting the linear combination to the measured spectrum. To further constrain Br(K SD e2γ ), separate spectra were obtained for events with 1 and 2 photons detected in the CsI(Tl) calorimeter and for events without conditions on the number of photons; these were fit simultaneously with the ratio Br(K SD e2γ /Br(K e2(γ ) ) and the yields of K SD e2γ and K μ2 decay as free parameters. The fit makes use of the R values from the MC simulation. Our method has the following advantages : (1) charged particles from the K e2(γ ) and K SD e2γ decays are e + with similar momenta, and the PID efficiency up to a small p dependence cancels out; (2) since the K e2(γ ) decay produces a peak at 247 MeV/c in the momentum spectrum, as shown in Fig. 2 (b), the K e2(γ ) yield can be accurately determined, and, at the same time, the K e2(γ ) events are largely suppressed by requiring a photon hit in the CsI(Tl) calorimeter for the K SD e2γ selection; (3) the CsI(Tl) acceptance can be determined using the two photons from the K π 2 decay; (4) other systematic uncertainties from imperfect reproducibility of the experimental conditions such as tracker inefficiencies, detector misalignment, DAQ deadtime, etc. are also cancelled out in the ratio determination.
The detector acceptance for the K SD e2γ decays was calculated by a GEANT4-based Monte Carlo simulation assuming the theoretical scheme of vector and axial-vector transitions [2,3]. The simulation data were generated assuming the Dalitz density given as
where G F is the Fermi constant, α is the fine structure constant, m K is the kaon mass, and θ c is the Cabibbo angle. The form factors
V and A represent the vector and axial-vector transitions, respectively. The kinematical density distribution for both helicity terms f SD + and f SD -can be described as
, (c) is the opening angle between the π + and π 0 , and (d) is the invariant mass M γ γ . The black dots are the experimental data. The contribution from the π 0 decay and events with at least one of the two clusters being accidental are shown as the dotted (blue) and dashed (green) histograms, respectively, and the solid (red)
histogram is obtained by summing the two components.
A was constant, according to the Chiral Perturbation Theory (ChPT) model at O(pfoot_1 ) [10,11]. The λ and A/V 0 parameters were taken to be λ = 0.3 ± 0.1 and A/V 0 = 0.4 ± 0.1,foot_0 respectively, which is the current theoretically conceivable range of ChPT O(p 6 ) model calculations [3]. The K e2(γ ) decay with the IB component, calculated including re-summation of the decay probability for multiple photon emission [27], and the K μ2 decay were also generated using the same simulation code.
Since the CsI(Tl) calorimeter surrounded the beam axis, it was exposed to a high rate of scattered beam particles and accidental backgrounds in the calorimeter contributed to the raw K SD + e2γ event samples. This accidental background was included in the simulation in order to reproduce the actual experimental conditions. We used the experimental background events and merged them with the simulation data as follows. Since the K μ2 decays with p μ = 236 MeV/c did not have accompanying photons, the CsI(Tl) signals which coincide with the K μ2 decays within the timing window of ±50 ns can be treated as pure accidental backgrounds. The fraction of the radiative K + → μ + νγ decay is negligibly small and causes no effect in this background study. The K μ2 events were selected only by the momentum and PID analyses, and these CsI(Tl) signals were merged with the simulation data of the K SD + e2γ , K e2(γ ) , and K μ2 decays. It should be noted that the ratio of the single-cluster and zero-cluster K μ2 events was = (18.85 ± 0.03 stat )%.
The validity of this simulation method was checked using two photons (E γ 1 > E γ 2 ) from the π 0 decay in K + → π + π 0 tagged by the π + with 200 < p π < 210 MeV/c and the photon energy higher than 21 MeV. Also, events with large shower leakage from the calorimeter were rejected by requiring E γ 1 + E γ 2 > 120 MeV. Fig. 4 shows the experimental spectra (dots) of (a) E γ 1 , (b) E γ 2 , (c) opening angle between the two photons, and (d) invariant mass (M γ γ ), together with the simulation data. The contribution from the π 0 decay and events with at least one accidental background hit in the two clusters are shown as the dotted (blue) and dashed (green) histograms, respectively. The solid (red) histogram is obtained by summing the two components and normalizing to the experimental yield. The results of the simulation are in good agreement with the experimental data, which indicates a good understanding of the photon measurement by the CsI(Tl) calorimeter. Also, the detection efficiencies of all CsI(Tl) modules were determined using the K π 2 events. Using the information of the π + and one of the two photons, the second photon energy and direction were calculated, and the existence of the actual photon cluster was checked.
In order to relatively enhance the K SD + e2γ events and suppress the K μ2 background events, photon hits in the CsI(Tl) calorimeter were required. Due to pile-up of the accidental backgrounds in the CsI(Tl), the accepted K SD + e2γ events included 2-cluster events in the calorimeter with a ratio of the probability obtained using K μ2 events compared with 1-cluster events. Since event loss in the 1-Fig. 5. Charged-particle momentum spectra with requiring (a) one photon cluster and (b) two photon clusters in coincidence with the e + track, and (c) charged particles without any CsI(Tl) constraint. The dots (black) are the experimental data. The solid (green), dashed (blue), and dashed-dotted (magenta) lines are the K SD + e2γ , K e2(γ ) , and K μ2 decays, respectively, determined by simulation calculations. The thick-red lines are the fitted results obtained by adding all the decay contributions. The events are shown only for the fitted momentum range.
Results of the individual counts N, acceptance ratio R , and Br(K SD + e2γ )/Br(K e2(γ ) ) values with statistical uncertainties obtained by simultaneously fitting the events with 1-cluster, 2-cluster, and without any CsI(Tl) constraint for Prun and Crun. An error-weighted average of the Prun and Crun results was adopted as the final result. Also, 1.14 ± 0.07 1.0 ± 0.2 1 .12 ± 0.07 cluster data and the appearance of the 2-cluster events were taken into account in the simulation, the K SD + e2γ branching ratio can be derived by comparing the experimental data with the simulation for both the 1-cluster and 2-cluster events simultaneously.
The K SD + e2γ decays with 1-cluster in the CsI(Tl) were obtained using the following procedure. The photon energy and the opening angle between the e + and γ were required to be E γ > 21 MeV and cos θ eγ < -0.8. This E γ cut point was a little higher than the hardware threshold to remove effects from small gain variations of each CsI(Tl) module. Assuming the K + → e + νγ decay kinematics, the missing-mass-squared was calculated as 2 where p is the momentum vector. The accepted interval was imposed to be -4000 < M 2 miss < 8000 MeV 2 /c 4 . The momentum spectrum is shown in Fig. 5 (a) indicated by the dots. Here, a small contribution from K μ2 with an accidental hit remained after the K SD + e2γ selection cuts. On the other hand, the K e2(γ ) events with an accidental hit were efficiently removed by the K SD + e2γ selection cuts, and the fraction is negligibly small. The decays in the 2-cluster data were selected in a similar manner. If one of the two clusters satisfied the conditions for the 1-cluster analysis, the event was adopted as a K SD + e2γ decay and the associated CsI(Tl) cluster was chosen as the true photon event, as shown in Fig. 5 (b). It should be noted that the K μ2 surviving fraction relative to the K SD + e2γ yield in the 2-cluster data is approximately twice that observed in the 1-cluster data because there are two photon candidates in the 2-cluster analysis.
The Br(K SD + e2γ )/Br(K e2(γ ) ) value was obtained to be 1.14 ± 0.07 for the Prun data set (as defined below; see Table 1) by simultaneously fitting the momentum spectra of the events with 1-cluster, 2-cluster, and without any CsI(Tl) constraint using the simulation data of the K SD + e2γ , K e2(γ ) , and K μ2 decays, as shown in Fig. 5 (a)(b)(c). Here, the value of obtained with the K μ2 events was used as a constraint in the fit. The solid (green), dotted (blue), and dashed-dotted (magenta) lines are the decomposed K SD + e2γ , K e2(γ ) , and K μ2 events. The thick-red line is the fit result obtained by adding all the decay contributions. The fitting regions of p > 230, 232, and 240 MeV/c for the events with 1-cluster, 2cluster, without any CsI(Tl) constraint, respectively, were chosen to reduce the effects from the K μ2 subtraction to minimize the uncertainty of K SD + e2γ by eliminating most of the K μ2 events. Note that it is very difficult to reproduce these surviving K μ2 events after the PID selection and the M 2 miss , cosθ eγ , E γ cuts by the simulation. The Br(K SD + e2γ )/Br(K e2(γ ) ) result as well as the accepted K SD + e2γ and K e2(γ ) yields used in the fitting and the associated R values are given in Table 1 under the heading "Prun" (physics run), along with the statistical uncertainties from the fits. The statistical uncertainty of R obtained from the MC calculation was less than 10 -3 .
The events in Fig. 5 (a) were used for an event selection validity check. Fig. 6 shows the distribution of (a) E γ , (b) cosθ eγ , and (c) M 2 miss . The K μ2 background fraction in Fig. 6 was successfully sup-Fig. 6. The K SD + e2γ spectra with the 1-cluster requirement: (a) E γ , (b) cosθ eγ , and (c) M 2 miss . The K SD + e2γ events were selected by imposing p > 230 MeV/c, -4000 < M 2 miss < 8000 MeV 2 /c 4 , and cosθ eγ < -0.8 to suppress the K e3 and K μ2 contributions. The black dots are the experimental data. The solid (green) and dashed-dotted (magenta) histograms are the simulation data of K SD + e2γ and K μ2 with accidental backgrounds, respectively. The thick-red line is the total simulation result obtained by adding each component. pressed down to ∼2% of the K SD + e2γ yield in the fitted momentum range. The experimental data (dots) are in good agreement with the simulation (thick-solid/red), indicating a correct understanding of the K SD + e2γ acceptance. The decomposed K SD + e2γ (solid/green) and K μ2 (dashed-dotted/magenta) contributions are also shown.
In this experimental study, one of the key issues is the treatment of the accidental background in the CsI(Tl) calorimeter and the K μ2 background that survives after the PID analysis. In order to validate this analysis method, the data taken during the commissioning runs dedicated to K + beam and PID detector tuning were used as systematic-control data (Crun), in which the amount of K μ2 background was larger. As a result, the surviving K μ2 fraction in the Crun data was a factor of ∼3 higher than in the Prun data. These data samples were independently analyzed using the same analysis codes adopted for the Prun data. The e + momentum spectra were obtained using the same PID condition for events with the 1-cluster, 2-cluster, and without any CsI(Tl) constraint, as shown in Fig. 7 (a), (b), and (c), respectively, indicated by the dots. The Br(K SD + e2γ )/Br(K e2(γ ) ) ratio was derived to be 1.0 ± 0.2, which is consistent with the result using the Prun data in spite of the larger number of K μ2 background events. The solid (green), dashed (blue), and dashed-dotted (magenta) lines in Fig. 7 are the K SD + e2γ , K e2(γ ) , and K μ2 decays, respectively, obtained from the simulation. The thick-red line is the fit result obtained by adding all the decay contributions. The details of the analysis result are summarized in Table 1. In addition to the Crun analysis described above, a separate study was performed with the Prun data. The cuts were tightened to remove most of the K μ2 background events and relaxed to accept the genuine K SD + e2γ events with higher efficiency,
The difference of the A/V 0 value obtained by the O(p 4 ) and O(p
) calculations is adopted as a systematic uncertainty.