Measurement of the $K^+\rightarrow{\mu^+}{\nu_{\mu}}{\gamma}$ decay form factors in the OKA experiment

A precise measurement of the vector and axial-vector form factors difference $F_V-F_A$ in the $K^+\rightarrow{\mu^+}{\nu_{\mu}}{\gamma}$ decay is presented. About 95K events of $K^+\rightarrow{\mu^+}{\nu_{\mu}}{\gamma}$ are selected in the OKA experiment. The result is $F_V-F_A=0.134\pm0.021(stat)\pm0.027(syst)$. Both errors are smaller than in the previous $F_V-F_A$ measurements.


Introduction
Radiative kaon decays are sensitive to hadronic weak currents in low-energy region and provide a good testing for the chiral perturbation theory (χP T ). The amplitude of the K + → µ + ν µ γ decay includes two terms: internal bremsstrahlung (IB) and structure dependent term (SD) [1]. IB contains radiative corrections for K + → µ + ν µ decay. SD is sensitive to the electroweak structure of the kaon.
The differential decay rate can be written in terms of standard kinematic variables x = 2E * γ /M K and y = 2E * µ /M K [2], which are proportional to the photon E * γ and muon E * µ energy in the kaon rest frame (M K is the kaon mass). It includes IB, SD ± terms and their interference INT ± . The SD ± and INT ± contributions are determined by two form factors F V and F A .
The general formula for the decay rate is as follows: Here α is the fine structure constant, F K is K + decay constant (F K = 155.6 ± 0.4 M eV [3]) and Γ K µ2 is the K µ2 decay width. Fig. 1 shows the kinematic distribution for IB, INT − , INT + , SD − and SD + . The main goal of the analysis is to measure F V − F A by extracting the INT − term. Other terms are either suppressed by backgrounds or give negligible contribution to the total decay rate with respect to IB. In the lowest order of χP T O(p 4 ) F V and F A are constant and F V − F A = 0.052 [2]. The first measurement of F V − F A was made by the ISTRA+ experiment:

OKA detector and separated kaon beam
The OKA setup, Fig. 2, is a double magnetic spectrometer.
The OKA detector includes: • Beam spectrometer consisting of the magnet M2, 7 beam proportional chambers BPC, 4 beam scintillation counters S and 2 threshold Cherenkov countersČ 1,2 for the kaon identification; • 11 m long He filled decay volume DV with the guard system (GS) containing 670 Lead-Scintillator calorimetric modules 20×(5 mm Sc + 1.5 mm Pb) with WLS readout; • Main magnetic spectrometer on the basis of 200×140 cm 2 wide aperture magnet SP-40A with a field integral 1 Tm, complemented by 13 planes of proportional chambers (PC), straw (ST) and drift tubes (DT); • 2 gamma detectors: electromagnetic calorimeter GAMS-2000 and large angle detector EGS (EGS is used to supplement GS as a gamma veto at large angles); • Hadron calorimeter GDA-100 and 4 muon scintillation counters µC (marked as MC in Fig. 2) used for muon identification; • Pad (Matrix) Hodoscope MH for the trigger and track reconstruction. More details can be found in [5]. The data acquisition system of the OKA setup [6] operates at ∼ 25 kHz event rate with the mean event size of ∼ 4 kByte.
The OKA beam is a separated secondary beam of the U-70 Proton Synchrotron of NRC "Kurchatov Institute"-IHEP, Protvino [7]. RF-separation with the Panofsky scheme [8] is implemented. The beam contains up to 12.5% of kaons with an intensity of about 5 × 10 5 kaons per 3 sec U-70 spill. The beam momentum was 17.7 GeV/c during the data taking period used for the analysis (November 2012). The present study uses about 1/2 of the statistics collected in 2012, where 504M events were stored on tape.

Trigger streams and primary selection
The following trigger was used for the analysis: T GAM S = beam·Č 1 ·Č 2 · S bk · E GAM S , where beam = S 1 · S 2 · S 3 · S 4 is a coincidence of four beam scintillation counters,Č 1,2 -threshold Cherenkov counters (Č 1 selects pions,Č 2 -pions and kaons), S bk ("beam killer") -two scintillation counters on the beam axis after the magnet aimed to suppress undecayed beam particles. The analog amplitude sum in the GAMS-2000 is required be higher than E GAM S (E GAM S is chosen to be above the average MIP energy deposit). The 10 Figure 2: OKA setup. The particle beam goes from left to right.
times prescaled minimum bias trigger T kaon = beam·Č 1 ·Č 2 · S bk was used for the trigger efficiency measurement trig = (T GAM S ∩ T kaon )/T kaon (Fig. 3). This trigger efficiency was applied during the Monte Carlo (MC) simulation.
To select the decay channel the following requirements are applied: • 1 primary track; • 1 secondary track identified as muon in GAMS-2000, GDA-100 and µC; • 1 electromagnetic shower in GAMS-2000 with energy E tot > 1 GeV not associated with charged track; • GS energy deposition E GS < 10 M eV ; • EGS energy deposition E EGS < 100 M eV ; • Decay vertex inside the decay volume DV.

Event selection
The main background to the K + → µ + ν µ γ decay comes from 2 decay modes: K + → µ + ν µ π 0 (Kµ3) and K + → π + π 0 (K2π) with one γ lost from π 0 → γγ decay and π misidentified as µ. Additional contribution at y > 1 is given by the decay mode K + → µ + ν µ with an accidental γ. At low y values there is a small contribution from the K + → π + π − π + (K3π) decay. The MC simulation of the OKA setup is done within the GEANT3 framework [9]. Signal and background events are weighted according to corresponding matrix elements.
The K + → µ + ν µ γ event selection strategy is based on the ISTRA+ approach [4]. Signal extraction procedure starts with dividing all kinematic (x, y) region into strips in x with ∆x = 0.05 width. The following steps are implemented for each x-strip: • Fill the y plot.
• Select the signal region by a cut y min < y < y max and fill cos θ * µγ plot, where θ * µγ is an angle between µ and γ in the kaon rest frame. y min and y max are selected from the maximization of signal significance defined as S/ √ S + B where S is the signal and B is the background.
• Put a cut on cos θ * µγ to reject background and fill m k plot. m 2 k = (P µ + P ν + P γ ) 2 , where P µ , P ν , P γ are 4-momenta of decay particles in the laboratory frame, p ν = p K − p µ − p γ , E ν = | p ν |. m k peaks at the kaon mass for the signal.
• The last step is a simultaneous fit of all 3 histograms (y, cos θ * µγ , m k ) with the MINUIT tool [10] where the signal and backgrounds normalization factors are the fit parameters.
For the correct estimation of the statistical error σ exp , only the m k histogram is used. The MINOS program [10] is run once with the initial parameter values equal to those obtained in the simultaneous fit. Statistical errors were extracted from the MINOS output. Fig. 4 shows the selected kinematic region for the extraction of the INT − term. For the further analysis 10 x-strips were selected in the 0.1 < x < 0.6 region. The y-width varies from 0.12 to 0.30 inside x-strips.  The result of the simultaneous fit for the strip 2 (0.15 < x < 0.2) is shown in Fig. 5. Both signal and background shapes are taken from the MC simulation. The total normalization of the MC to data is made to the Kµ3 decay at y < 0.6, where the contribution of other backgrounds is very small. The relative normalization of other backgrounds is done according to their branching ratios. For the K + → µ + ν µ γ decay, only IB term is included in the simultaneous fit. The simultaneous fit gives a reasonable agreement between data and MC with χ 2 from 1.3 to 1.7 for different x-strips.

F V − F A calculation
For each x-strip the number of signal events N Data is extracted from the simultaneous fit and the IB event number N IB is obtained from MC. Their ratio is plotted as a function where p 0 is normalization factor, p 1 = F V − F A is the difference of vector and axial-vector form factors, ϕ IN T − (x) is the x-distribution for the reconstructed MC-signal events taken with the weights is a similar distribution for the same MC sample, but with the weights f IB (x true , y true ). Here x true , y true are "true" MC values of x and y.
The result of the fit is F V −F A = 0.134±0.021. The normalization factor is p 0 = 1.000±0.007. The total number of selected K + → µ + ν µ γ decay events is 95428 ± 309. In the next order χP T O(p 6 ) F V linearly depends on the momentum transfer q 2 [11] with the following parametrization [12]: The theoretical prediction is tested in three ways: • The final fit is performed with F V and F A fixed from χP T O(p 6 ) prediction: F V (0) = 0.082, F A = 0.034, λ = 0.4. This fit has bad compliance with χ 2 /N DF = 28.0/9.
• F V (0) is fixed from χP T O(p 6 ). F A and λ are used as fit parameters. (F V , λ) correlation is shown in Fig. 8. The theoretical prediction (red star) is slightly out of 3σ-ellipse.

Systematic errors
The obtained value of F V − F A depends on the width of x-strips, y and θ * µγ cuts and the fit procedure. The following sources of the systematic errors were investigated: • Non ideal description of signal and background by the MC.
For the estimation of this systematics, the statistical error in each bin of Fig. 6 was scaled by the factor χ 2 /N DF , where χ 2 /N DF is obtained from the simultaneous fit in each x-strip. A new fit of N Data /N IB with the same function p signal (x) gives the best description with χ 2 /N DF = 7.8/8 compared to χ 2 /N DF = 12.3/8 of the main fit. The new value of F V − F A = 0.138 is consistent with the main one but the fit error σ f it = 0.026 is larger. Assuming σ f it 2 = σ shape 2 + σ stat 2 the systematic error is σ shape = 0.015.
• The fit range in x (number of x-strips in the fit).
The ratio N Data /N IB was refitted by removing one or two bins on the left (right) edge. For the estimate of systematics the average difference between the new F V −F A values and the nominal one is taken. The error is negligible: σ x < 0.006.
• Width of x-strips. The F V − F A calculation is repeated for 2 different values of x-binning: ∆x = 0.035, ∆x = 0.07. The deviation of the new F V − F A value with respect to the main one gives σ ∆x = 0.011.
• y limits in x-strips.
The events inside FWHM of the y-distribution for the signal MC are selected. Such limits are tighter than those used in the main analysis. The difference between the new value and main one gives systematic error σ y = 0.008.
• Possible contribution of INT + . The INT + term is added to the final fit (see Section 5). The value |F V +F A | = 0.165±0.013 measured by the E787 experiment is used [13]. Two fits were repeated for the minimal (-0.178) and maximal (+0.178) possible values of F V + F A . The fitting function was: is the x-distribution similar one as ϕ IN T − (x). The maximal difference between obtained values of F V − F A and the main one measured in Section 5 is σ IN T + = 0.018.
Summing up quadratically all the systematic errors the total error is found to be 0.027.

Conclusion
The largest statistics of about 95K events of K + → µ + ν µ γ is collected by the OKA experiment. The INT − term is observed and F V − F A is measured: F V − F A = 0.134±0.021(stat)±0.027(syst). The result is 2.4σ above χP T O(p 4 ) prediction. A recent calculation in the framework of the gauged nonlocal effective chiral action (EχA) gives F V − F A = 0.081 [14]. The OKA result is 1.6σ above the EχA prediction.
The obtained value of F V − F A is in a reasonable agreement with a similar analysis of the ISTRA+ experiment: F V − F A = 0.21±0.04(stat)±0.04(syst) [4].