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

TL;DR

This paper reports a precise measurement of the difference between vector and axial-vector form factors in the $K^+ ightarrow \mu^+ u_\mu \gamma$ decay, using about 95,000 events from the OKA experiment, with improved accuracy over previous results.

Contribution

The study provides the most precise measurement to date of the $F_V - F_A$ form factor difference in this decay mode, reducing uncertainties compared to prior measurements.

Findings

Measured $F_V - F_A = 0.134 \, extpm \, 0.021 (stat) \, extpm \, 0.027 (syst)$.

Selected about 95,000 decay events in the OKA experiment.

Achieved smaller errors than previous measurements.

Abstract

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.

Full text

Measurement of the $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$

decay form factors in the OKA experiment

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

The OKA collaboration

V.I. Kravtsov, V.A. Duk111Present Address: University of Birmingham, Birmingham, United Kingdom, S.N. Filippov, E.N. Gushchin, A.A. Khudyakov, Yu.G. Kudenko222Also at Moscow Institute of Physics and Technology, Moscow Region, Russia333Also at NRNU Moscow Engineering Physics Institute (MEPhI), Moscow, Russia, A.Yu. Polyarush

(INR RAS, Moscow, Russia),

S.A. Akimenko, A.V. Artamonov, A.M. Blik, V.S. Burtovoy, S.V. Donskov, A.P. Filin, A.M. Gorin, A.V. Inyakin, G.V. Khaustov, S.A. Kholodenko, V.N. Kolosov, V.F. Kurshetsov, V.A. Lishin, M.V. Medynsky, Yu.V. Mikhailov, V.F. Obraztsov, V.A. Polyakov, A.V. Popov, V.I. Romanovsky, V.I. Rykalin, A.S. Sadovsky, V.D. Samoilenko, V.K. Semenov444Deceased, M.M. Shapkin, O.V. Stenyakin, O.G. Tchikilev, V.A. Uvarov, O.P. Yushchenko

(NRC "Kurchatov Institute"-IHEP, Protvino, Russia),

V.N. Bychkov, G.D. Kekelidze, V.M. Lysan, B.Zh. Zalikhanov

(JINR, Dubna, Russia)

Abstract. A precise measurement of the vector and axial-vector form factors difference $\displaystyle F_{V}-F_{A}$ in the $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$ decay is presented. About 95K events of $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$ are selected in the OKA experiment. The result is $\displaystyle F_{V}-F_{A}=0.134\pm 0.021(stat)\pm 0.027(syst)$. Both errors are smaller than in the previous $\displaystyle F_{V}-F_{A}$ measurements.

1 Introduction

Radiative kaon decays are sensitive to hadronic weak currents in low-energy region and provide a good testing for the chiral perturbation theory ($\displaystyle{\chi}PT$). The amplitude of the $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$ decay includes two terms: internal bremsstrahlung (IB) and structure dependent term (SD) [1]. IB contains radiative corrections for $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}$ decay. SD is sensitive to the electroweak structure of the kaon.

The differential decay rate can be written in terms of standard kinematic variables

$\displaystyle x=2E^{\ast}_{\gamma}/M_{K}$ and $\displaystyle y=2E^{\ast}_{\mu}/M_{K}$ [2], which are proportional to the photon $\displaystyle E^{\ast}_{\gamma}$ and muon $\displaystyle E^{\ast}_{\mu}$ energy in the kaon rest frame ($\displaystyle M_{K}$ is the kaon mass). It includes IB, SD± terms and their interference INT±. The SD± and INT± contributions are determined by two form factors $\displaystyle F_{V}$ and $\displaystyle F_{A}$.

The general formula for the decay rate is as follows:

$\displaystyle\frac{d\Gamma}{dxdy}=A_{IB}f_{IB}(x,y)+A_{SD}[(F_{V}+F_{A})^{2}f_{SD^{+}}(x,y)+(F_{V}-F_{A})^{2}f_{SD^{-}}(x,y)]$

          $\displaystyle-A_{INT}[(F_{V}+F_{A})f_{INT^{+}}(x,y)+(F_{V}-F_{A})f_{INT^{-}}(x,y)]$,

where

$\displaystyle f_{IB}(x,y)=[\frac{1-y+r}{x^{2}(x+y-1-r)}][x^{2}+2(1-x)(1-r)-\frac{2xr(1-r)}{x+y-1-r}]$,

$\displaystyle f_{SD^{+}}(x,y)=[x+y-1-r][(1-x)(1-y)+r]$,

$\displaystyle f_{SD^{-}}(x,y)=[1-y+r][(x+y-1)(1-x)-r]$,

$\displaystyle f_{INT^{+}}(x,y)=[\frac{1-y+r}{x(x+y-1-r)}][(1-x)(1-x-y)+r]$,

$\displaystyle f_{INT^{-}}(x,y)=[\frac{1-y+r}{x(x+y-1-r)}][x^{2}-(1-x)(1-x-y)-r]$,

and $\displaystyle r=[\frac{M_{\mu}}{M_{K}}]^{2}$, $\displaystyle A_{IB}=\Gamma_{K_{\mu 2}}\frac{\alpha}{2\pi}\frac{1}{(1-r)^{2}}$, $\displaystyle A_{SD}=\Gamma_{K_{\mu 2}}\frac{\alpha}{8\pi}\frac{1}{r(1-r)^{2}}[\frac{M_{K}}{F_{K}}]^{2}$,

$\displaystyle A_{INT}=\Gamma_{K_{\mu 2}}\frac{\alpha}{2\pi}\frac{1}{(1-r)^{2}}[\frac{M_{K}}{F_{K}}]$. Here $\displaystyle\alpha$ is the fine structure constant, $\displaystyle F_{K}$ is $\displaystyle K^{+}$ decay constant ($\displaystyle F_{K}=155.6\pm 0.4\>MeV$ [3]) and $\displaystyle\Gamma_{K_{\mu 2}}$ is the $\displaystyle K_{\mu 2}$ decay width.

Fig. 1 shows the kinematic distribution for IB, INT-, INT+, SD- and SD+. The main goal of the analysis is to measure $\displaystyle{F_{V}}-{F_{A}}$ 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 $\displaystyle{\chi}PT\>O(p^{4})$ $\displaystyle F_{V}$ and $\displaystyle F_{A}$ are constant and $\displaystyle{F_{V}}-{F_{A}}=0.052$ [2]. The first measurement of $\displaystyle{F_{V}}-{F_{A}}$ was made by the ISTRA+ experiment: $\displaystyle F_{V}-F_{A}=0.21{\pm}0.04(stat){\pm}0.04(syst)$ [4].

2 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 counters $\displaystyle\check{C}_{1,2}$ for the kaon identification;

• •

11 m long He filled decay volume DV with the guard system (GS) containing 670 Lead-Scintillator calorimetric modules 20$\displaystyle\times$(5 mm Sc + 1.5 mm Pb) with WLS readout;

• •

Main magnetic spectrometer on the basis of $\displaystyle 200{\times}140$ cm2 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 $\displaystyle\mu$C (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 $\displaystyle\sim{25}$ kHz event rate with the mean event size of $\displaystyle\sim{4}$ 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 $\displaystyle 12.5\%$ of kaons with an intensity of about $\displaystyle 5\times{10^{5}}$ 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.

3 Trigger streams and primary selection

The following trigger was used for the analysis: $\displaystyle T_{GAMS}=beam\cdotp\overline{\check{C_{1}}}\cdotp\check{C_{2}}\cdotp\overline{S_{bk}}\cdotp{E_{GAMS}}$, where $\displaystyle beam={S_{1}}\cdotp{S_{2}}\cdotp{S_{3}}\cdotp{S_{4}}$ is a coincidence of four beam scintillation counters, $\displaystyle\check{C}_{1,2}$ - threshold Cherenkov counters ($\displaystyle\check{C_{1}}$ selects pions, $\displaystyle\check{C_{2}}$ - pions and kaons), $\displaystyle S_{bk}$ (’’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 $\displaystyle E_{GAMS}$ ($\displaystyle E_{GAMS}$ is chosen to be above the average MIP energy deposit). The 10 times prescaled minimum bias trigger $\displaystyle T_{kaon}=beam\cdotp\overline{\check{C_{1}}}\cdotp\check{C_{2}}\cdotp\overline{S_{bk}}$ was used for the trigger efficiency measurement $\displaystyle\epsilon_{trig}={(T_{GAMS}\cap{T_{kaon}})}/{T_{kaon}}$ (Fig. 4). 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 $\displaystyle\mu$C;

• •

1 electromagnetic shower in GAMS-2000 with energy $\displaystyle E_{tot}>1\>GeV$ not associated with charged track;

• •

GS energy deposition $\displaystyle E_{GS}<10\>MeV$;

• •

EGS energy deposition $\displaystyle E_{EGS}<100\>MeV$;

• •

Decay vertex inside the decay volume DV.

4 Event selection

The main background to the $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$ decay comes from 2 decay modes: $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\pi^{0}}$ ($\displaystyle K\mu 3$) and $\displaystyle K^{+}\rightarrow{\pi^{+}}{\pi^{0}}$ ($\displaystyle K2\pi$) with one $\displaystyle\gamma$ lost from $\displaystyle\pi^{0}\rightarrow{\gamma\gamma}$ decay and $\displaystyle\pi$ misidentified as $\displaystyle\mu$. Additional contribution at $\displaystyle y>1$ is given by the decay mode $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}$ with an accidental $\displaystyle\gamma$. At low y values there is a small contribution from the $\displaystyle K^{+}\rightarrow{\pi^{+}}{\pi^{-}}{\pi^{+}}$ ($\displaystyle K3\pi$) 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 $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$ event selection strategy is based on the ISTRA+ approach [4]. Signal extraction procedure starts with dividing all kinematic ($\displaystyle x,\>y$) region into strips in $\displaystyle x$ with $\displaystyle\Delta{x}=0.05$ width. The following steps are implemented for each $\displaystyle x$-strip:

• •

Fill the $\displaystyle y$ plot.

• •

Select the signal region by a cut $\displaystyle y_{min}<y<y_{max}$ and fill $\displaystyle\cos{\theta^{\ast}_{\mu\gamma}}$ plot, where $\displaystyle\theta^{\ast}_{\mu\gamma}$ is an angle between $\displaystyle\mu$ and $\displaystyle\gamma$ in the kaon rest frame. $\displaystyle y_{min}$ and $\displaystyle y_{max}$ are selected from the maximization of signal significance defined as $\displaystyle S/\sqrt{S+B}$ where $\displaystyle S$ is the signal and $\displaystyle B$ is the background.

• •

Put a cut on $\displaystyle\cos{\theta^{\ast}_{\mu\gamma}}$ to reject background and fill $\displaystyle m_{k}$ plot. $\displaystyle m^{2}_{k}=(P_{\mu}+P_{\nu}+P_{\gamma})^{2}$, where $\displaystyle P_{\mu}$,$\displaystyle\>P_{\nu}$,$\displaystyle\>P_{\gamma}$ are 4-momenta of decay particles in the laboratory frame,

$\displaystyle\vec{p}_{\nu}=\vec{p}_{K}-\vec{p}_{\mu}-\vec{p}_{\gamma}$, $\displaystyle E_{\nu}=|\vec{p}_{\nu}|$. $\displaystyle m_{k}$ peaks at the kaon mass for the signal.

• •

The last step is a simultaneous fit of all 3 histograms ($\displaystyle y,\cos{\theta^{\ast}_{\mu\gamma}},m_{k}$) with the

MINUIT tool [10] where the signal and backgrounds normalization factors are the fit parameters.

For the correct estimation of the statistical error $\displaystyle\sigma_{exp}$, only the $\displaystyle m_{k}$ 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 $\displaystyle x$-strips were selected in the $\displaystyle 0.1<x<0.6$ region. The $\displaystyle y$-width varies from 0.12 to 0.30 inside $\displaystyle x$-strips.

The result of the simultaneous fit for the strip 2 ($\displaystyle 0.15<x<0.2$) 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 $\displaystyle K\mu 3$ decay at $\displaystyle y<0.6$, where the contribution of other backgrounds is very small. The relative normalization of other backgrounds is done according to their branching ratios. For the $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$ decay, only IB term is included in the simultaneous fit. The simultaneous fit gives a reasonable agreement between data and MC with $\displaystyle\chi^{2}$ from 1.3 to 1.7 for different $\displaystyle x$-strips.

5 $\displaystyle F_{V}-F_{A}$ calculation

For each $\displaystyle x$-strip the number of signal events $\displaystyle N_{Data}$ is extracted from the simultaneous fit and the IB event number $\displaystyle N_{IB}$ is obtained from MC. Their ratio is plotted as a function of $\displaystyle x$ (Fig. 6). For the signal containing IB only this ratio would be equal to 1. It is the case for small $\displaystyle x$, when the IB is dominating and INT- is negligible. For large $\displaystyle x$ the INT- term gives significant negative contribution resulting in smaller values of $\displaystyle N_{Data}/N_{IB}$.

The $\displaystyle N_{Data}/N_{IB}$ distribution is fitted with a function $\displaystyle p_{signal}(x)=p_{0}(1+p_{1}({{\varphi}_{INT^{-}}}(x)/{{\varphi}_{IB}}(x)))$, where $\displaystyle p_{0}$ is normalization factor, $\displaystyle p_{1}=F_{V}-F_{A}$ is the difference of vector and axial-vector form factors, $\displaystyle{{\varphi}_{INT^{-}}}(x)$ is the $\displaystyle x$-distribution for the reconstructed MC-signal events taken with the weights $\displaystyle({M_{K}}/{F_{K}}){f_{INT^{-}}}(x_{true},y_{true})$, $\displaystyle{{\varphi}_{IB}}(x)$ is a similar distribution for the same MC sample, but with the weights $\displaystyle{f_{IB}}(x_{true},y_{true})$. Here $\displaystyle x_{true}$, $\displaystyle y_{true}$ are "true" MC values of $\displaystyle x$ and $\displaystyle y$.

The result of the fit is $\displaystyle F_{V}-F_{A}=0.134{\pm}0.021$. The normalization factor is $\displaystyle p_{0}=1.000{\pm}0.007$. The total number of selected $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$ decay events is $\displaystyle 95428\pm 309$.

In the next order $\displaystyle{\chi}PT\>O(p^{6})$ $\displaystyle F_{V}$ linearly depends on the momentum transfer $\displaystyle q^{2}$ [11] with the following parametrization [12]: $\displaystyle{F_{V}}={F_{V}(0)}(1+{\lambda}(1-x))$, $\displaystyle{F_{A}}=const$. The theoretical prediction is tested in three ways:

• •

The final fit is performed with $\displaystyle F_{V}$ and $\displaystyle F_{A}$ fixed from $\displaystyle{\chi}PT\>O(p^{6})$ prediction: $\displaystyle F_{V}(0)=0.082$, $\displaystyle F_{A}=0.034$, $\displaystyle{\lambda}=0.4$. This fit has bad compliance with $\displaystyle{\chi^{2}}/NDF=28.0/9$.

• •

$\displaystyle F_{V}(0)$ and $\displaystyle F_{A}=0.034$ are taken from $\displaystyle{\chi}PT\>O(p^{6})$, $\displaystyle\lambda$ is a fit parameter. It gives $\displaystyle{\lambda}=2.28\pm 0.53$ with $\displaystyle{\chi^{2}}/NDF=15.8/8$ (Fig. 8).

• •

$\displaystyle F_{V}(0)$ is fixed from $\displaystyle{\chi}PT\>O(p^{6})$. $\displaystyle F_{A}$ and $\displaystyle\lambda$ are used as fit parameters. ($\displaystyle F_{V},\>\lambda$) correlation is shown in Fig. 8. The theoretical prediction (red star) is slightly out of $\displaystyle 3\sigma$-ellipse.

6 Systematic errors

The obtained value of $\displaystyle F_{V}-F_{A}$ depends on the width of $\displaystyle x$-strips, $\displaystyle y$ and $\displaystyle\theta^{\ast}_{\mu\gamma}$ 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 $\displaystyle\sqrt{{\chi^{2}}/NDF}$, where $\displaystyle{\chi^{2}}/NDF$ is obtained from the simultaneous fit in each $\displaystyle x$-strip. A new fit of $\displaystyle N_{Data}/N_{IB}$ with the same function $\displaystyle p_{signal}(x)$ gives the best description with $\displaystyle{\chi^{2}}/NDF=7.8/8$ compared to $\displaystyle{\chi^{2}}/NDF=12.3/8$ of the main fit. The new value of $\displaystyle F_{V}-F_{A}=0.138$ is consistent with the main one but the fit error $\displaystyle\sigma_{fit}=0.026$ is larger. Assuming $\displaystyle{\sigma_{fit}}^{2}={\sigma_{shape}}^{2}+{\sigma_{stat}}^{2}$ the systematic error is $\displaystyle\sigma_{shape}=0.015$.

• •

The fit range in $\displaystyle x$ (number of $\displaystyle x$-strips in the fit).

The ratio $\displaystyle N_{Data}/N_{IB}$ was refitted by removing one or two bins on the left (right) edge. For the estimate of systematics the average difference between the new $\displaystyle F_{V}-F_{A}$ values and the nominal one is taken. The error is negligible: $\displaystyle\sigma_{x}<0.006$.

• •

Width of $\displaystyle x$-strips.

The $\displaystyle F_{V}-F_{A}$ calculation is repeated for 2 different values of $\displaystyle x$-binning: $\displaystyle\Delta{x}=0.035$, $\displaystyle\Delta{x}=0.07$. The deviation of the new $\displaystyle F_{V}-F_{A}$ value with respect to the main one gives $\displaystyle\sigma_{\Delta{x}}=0.011$.

• •

$\displaystyle y$ limits in $\displaystyle x$-strips.

The events inside FWHM of the $\displaystyle y$-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 $\displaystyle\sigma_{y}=0.008$.

• •

Possible contribution of INT+.

The INT+ term is added to the final fit (see Section 5). The value $\displaystyle|F_{V}+F_{A}|=0.165{\pm}0.013$ measured by the E787 experiment is used [13]. Two fits were repeated for the minimal (-0.178) and maximal (+0.178) possible values of $\displaystyle F_{V}+F_{A}$. The fitting function was:

$\displaystyle p_{signal}(x)=p_{0}(1+({F_{V}+F_{A}})({{\varphi}_{INT^{+}}}(x)/{{\varphi}_{IB}}(x))+({F_{V}-F_{A}})({{\varphi}_{INT^{-}}}(x)/{{\varphi}_{IB}}(x)))$,

where $\displaystyle{{\varphi}_{INT^{+}}}(x)$ is the $\displaystyle x$-distribution similar one as $\displaystyle{{\varphi}_{INT^{-}}}(x)$. The maximal difference between obtained values of $\displaystyle F_{V}-F_{A}$ and the main one measured in Section 5 is $\displaystyle\sigma_{INT^{+}}=0.018$.

Summing up quadratically all the systematic errors the total error is found to be 0.027.

7 Conclusion

The largest statistics of about 95K events of $\displaystyle K^{+}\rightarrow{\mu^{+}}{\nu_{\mu}}{\gamma}$ is collected by the OKA experiment. The INT- term is observed and $\displaystyle F_{V}-F_{A}$ is measured:

$\displaystyle F_{V}-F_{A}=0.134{\pm}0.021(stat){\pm}0.027(syst)$.

The result is $\displaystyle 2.4\sigma$ above $\displaystyle{\chi}PT\>O(p^{4})$ prediction.

A recent calculation in the framework of the gauged nonlocal effective chiral action ($\displaystyle E{\chi}A$) gives $\displaystyle F_{V}-F_{A}=0.081$ [14]. The OKA result is $\displaystyle 1.6\sigma$ above the $\displaystyle E{\chi}A$ prediction.

The obtained value of $\displaystyle F_{V}-F_{A}$ is in a reasonable agreement with a similar analysis of the ISTRA+ experiment: $\displaystyle F_{V}-F_{A}=0.21{\pm}0.04(stat){\pm}0.04(syst)$ [4].

8 Acknowledgements

The authors express their gratitude to IHEP colleagues from the accelerator department for the good performance of the U-70 during data taking; from the beam department for the stable operation of the 21K beam line, including RF-deflectors, and from the engineering physics department for the operation of the cryogenic system of the RF-deflectors.

The work is partially supported by the Russian Fund for Basic Research,

grant N18-02-00179A.

9 References