Search for heavy neutrino in \(K^{+} \rightarrow \mu ^{+} \nu _{H}\) decay
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Abstract
A high statistics data sample of the \(K^{+}\rightarrow \mu ^{+}\nu _{\mu }\) decay was accumulated by the OKA experiment in 2012. The missing mass analysis was performed to search for the decay channel \(K^{+}\rightarrow \mu ^{+}\nu _{H}\) with a hypothetic stable heavy neutrino in the final state. The obtained missing mass spectrum does not show peaks that could be attributed to existence of stable heavy neutrinos in the mass range \((270< m_{\nu _{H}} < 375)\) MeV\(/c^{2}\). As a result, upper limits on the branching ratio and on the value of the mixing element \(U_{\mu H}^{2}\) are obtained.
1 Introduction
After the discovery of the Higgs boson there are no further guideline predictions from the Standard Model (SM) remained, hence, searches for a physics beyond the SM in a broad range of topics become an actual question. One of the promising research directions is inspired by observed neutrino oscillations [1, 2, 3, 4] which require non zero neutrino masses which, in turn, open possibility for existence of a set of heavy sterile neutrinos in one of the SM extensions – the Neutrino Minimal Standard Model (\(\nu \)MSM) [5, 6, 7]. Depending on the lifetime of those heavy neutrinos (\(\nu _{H}\)) experimental approaches can be divided into searches for possible decay products of \(\nu _{H}\), as, for example, reported in [8] or more recently in [9] and into searches for \(\nu _{H}\) with a long lifetime with the missing mass approach [10] and, recently, in two high statistics experiments [11, 12, 13], which reported upper limits on branching for \(\nu _{H}\) in a hundred MeV/c\(^2\) range. To contribute to the latter approach we analyzed a large data sample of \(K^{+} \rightarrow \mu ^{+} \nu \) recorded in 2012 by the OKA collaboration at IHEPProtvino to search for a possible process \(K^{+} \rightarrow \mu ^{+} \nu _{H}\), see Fig. 1, where \(\nu _{H}\) stands for one of the expected sterile \(\nu _{H}\) within \(\nu \)MSM and \(U_{\mu H}\) represents mixing element between SM muon neutrino and \(\nu _{H}\).
2 Separated kaon beam and OKA experiment
The OKA setup, Fig. 2, is a magnetic spectrometer complemented by electromagnetic and hadron calorimeters and a Decay Volume. The first magnet M\(_{1}\) with surrounding 1 mm pitch PC’s (BPC\(_{(1Y)}\), BPC\(_{(2Y,2X)}\), BPC\(_{(3X,3Y)}\), BPC\(_{(4X,4Y)}\) of \(\sim \)1500 channels in total [17, 18]) serves as a beam spectrometer. It is supplemented by two threshold Cherenkov counters Č\(_{1}\), Č\(_{2}\) for kaon identification and by beam trigger scintillation counters S\(_{(1)}\), S\(_{(2)}\), S\(_{(4)}\), each of \(200\times 200\times 1\) mm\(^{3}\), and a thicker one, \(60\times 85\times 6\) mm\(^{3}\), delivering the reference time, S\(_{(3)}\). The 11 m long Decay Volume (DV) filled with helium contains 11 rings of guard system (GS), which consists of 670 LeadScintillator sandwiches (20 layers of 1.5 mm: 5 mm each) with WLS readout grouped in 300 ADC channels. To supplement GS, a gamma detector (BGD, made of \(\sim \) 1050 \(5\times 5\times 42\) cm\(^3\) lead glass blocks [19]), located behind the DV is used as a veto at large angles, while low angle particles pass through a central opening. The wide aperture \(200\times 140\) cm\(^2\) magnet SM\(_{(SP40A)}\), with a field integral of \(\sim \) 1 Tm deflects charged particles in horizontal plane and serves as a spectrometer together with corresponding tracking chambers: 5k channels of 2 mm pitch PC’s (PC\(_{1,\ldots ,8}\)), 1k channels of 9 mm diameter straw tubes ST\(_{(1,2,3)}\) and 300 channels of 40 mm diameter drift tubes DT\(_{1,2}\). The matrix hodoscope HODO\(_{(matrix)}\) is composed of 252 \(12\times 12\times 1.5\) cm\(^3\) scintillator tiles with WLS+SiPM readout. It is used to improve time resolution and track matching. Two scintillator counters S\(_{bk1}\), S\(_{bk2}\) (80 and 90 mm in diameter with a thickness of 3.9 and 5 mm) serve to suppress undecayed beam particles at the trigger level. At the end of the OKA setup there are two calorimeters: electromagnetic (GAMS2000 of \(\sim \) 2300 \(3.8\times 3.8\times 45\) cm\(^3\) lead glass blocks [20]) and a hadron one (HCAL of 120 \(20\times 20\times 108\) cm\(^3\) ironscintillator sandwiches with WLS plates readout [21]) and, finally, four partially overlapping muon counters \(\mu \)C \((1\times 1\) m\(^2\) scintillators with WLS fibres readout) behind the HCAL. The data acquisition system of the OKA setup [22] works at \(\sim \) 25 kHz event rate with the mean event size of \(\sim \) 4 kByte.
3 Search for heavy neutrinos
A search for \(K^{+}\rightarrow \mu ^{+}\nu _{H}\) decay is done with the data set accumulated in November 2012 run with a 17.7 GeV/c beam momentum. Two prescaled triggers are used. The first one, with prescale factor of 1/10, selects beam kaons which decay inside the OKA setup: \( \mathtt{Tr}_{\mathtt{Kdecay}}=\mathtt{S}_{\mathtt{1}}{\cdot }\mathtt{S}_{\mathtt{2}}{\cdot } \mathtt{S}_{\mathtt{3}}{\cdot }{} \mathtt{S}_{\mathtt{4}}{\cdot }\check{\mathtt{C}}_{\mathtt{1}}{\cdot } \overline{\check{\mathtt{C}}}_{\mathtt{2}}{\cdot }\overline{\mathtt{S}}_{\mathtt{bk}}\). The second one, \(\mathtt{Tr_{K\rightarrow \mu X}=Tr_{Kdecay}{\cdot }{\mu }C}\), includes additionally muon counters \(\mu \)C and is prescaled by 1/4. The beam intensity (\(\mathtt{S}_{\mathtt{1}}\cdot \mathtt{S}_{\mathtt{2}}\cdot \mathtt{S}_{\mathtt{3}}\cdot \mathtt{S}_{\mathtt{4}}\)) is \(\sim 2\cdot 10^{6}\) per spill, the fraction of kaons in the beam is \(\sim \) 12.5%, i.e. the kaon intensity is \(\sim \) 250k/spill. The total number of kaons entering the DV corresponds to \(\sim 1.6\times 10^{10}\).
3.1 Event selection

events with single beam track and single secondary track are selected;

a single secondary track segment after the SM magnet is present in the event and it is well matched to the muon like signals in both GAMS2000 and HCAL calorimeters (i.e. one or two adjacent cells with the MIP energy deposition);

sufficient number of points on all the track segments is present to optimize the missing mass resolution;

the momentum of kaon is consistent with that delivered by beam settings of \(\approx 17.7\) GeV/c, while the required momentum of the secondary muon is below 16.4 GeV/c;

to ensure good decay vertex reconstruction and also to suppress events in which kaon decays after passing the DV, there is a requirement of 3 mrad minimal angle between the beam and the secondary track, and a requirement for the minimal distance between the beam and the secondary track to be \(< 1\) cm;

the decay vertex is inside DV, and is further restricted to be 3\(\sigma \) (of zvertex resolution) from the DV entrance and the position of Cutarget,^{2} which reduces the effective length of DV to 7.5 m;

other decay channels are suppressed by requiring the total energy deposition in GS and BGD to be below 50 MeV/c\(^2\) and 100 MeV/c\(^2\), respectively;

the total energy deposition in GAMS2000 and HCAL should be consistent with that of a single muon.
3.2 Signal and background studies
A signal from heavy neutrino in \(K^{+} \rightarrow \mu ^{+} \nu _{H}\) may show up itself as a peak in the missing mass distribution, \(m^2_{\nu } = m^2_{miss} = (p_{K}  p_{\mu })_{i} \cdot (p_{K}  p_{\mu })^{i}\), \(i = 1, 2, 3, 4\). We assume that heavy sterile neutrinos are stable.^{3} The investigation of possible signal and background contributions from kaon decays and from kaon scattering and interactions inside the OKA setup is done with a detailed GEANT3 simulation with the subsequent offline reconstruction and analysis. Different decay channels, simulated using MonteCarlo (MC) are weighted according to corresponding matrix elements and branching ratios [23]. The experimental data and main backgrounds, which survived the selections cuts are shown in the \((m^2_{miss};p_{\mu })\) plots of Fig. 3.
In the region of low \(m^{2}_{miss}\), both \(K^{+} \rightarrow \mu ^{+} \nu _{\mu }\) and \(K^{+} \rightarrow \mu ^{+} \nu _{\mu } \gamma \) dominate. In the region of \(m^{2}_{miss} > 0.05~\hbox {GeV}2\)/c\(^4\), the dominant contribution is given by \(K^{+}\rightarrow \pi ^{0}\mu ^{+}\nu _{\mu }\) decay channel, which can not be excluded by kinematic cuts without significant loss in acceptance due to its rather flat distribution in the region of interest (see Fig. 3d).
The \(K^{+}\rightarrow \pi ^{+}\pi ^{0}\) decay channel is suppressed by four orders of magnitude, but it is responsible for a small peak at \(m^{2}_{miss}\) around 0.1 GeV\(^2\)/c\(^4\), which should be taken into account. Therefore we limit our acceptance (see Fig. 3e) from the low muon momentum side by a smooth curve excluding the high event density spot from \(K^{+}\rightarrow \pi ^{+}\pi ^{0}\) at low \(p_{\mu }\approx 2\) GeV/c (see Fig. 3e), while at high values of the muon momentum (\(p_{\mu }\)) we additionally introduce a smooth upper limit on \(p_{\mu }\) to suppress a tail from badly reconstructed \(K^{+} \rightarrow \mu ^{+} \nu _{\mu }\) events. The contributions due to a misidentified electron from the \(K^{+} \rightarrow e^{+} \nu _{e} \pi ^{0}\) decay channel (Fig. 3f) and from processes when kaon either scatters or interacts while passing the setup (Fig. 3g) play a minor role.
3.3 Signal search with a subtraction of the MC simulated background
3.4 Upper limits with a common fit of signal and background
As an alternative, we do a series of fits of the initial \(m^{2}_{miss}\) distribution of Fig. 5 for \(m^{2}_{miss}>0.05~\hbox {GeV}^2/c^4\) with a sum of Gaussian signal for a given mass and all the MC background sources, again, with additional multiplication coefficients which now may be tuned at each mass point. Such a procedure is more signal favorable as compared to that of Sect. 3.3. The remaining part for the calculation of upper limits is done in the same way and the result is indicated in Fig. 9 by a thin smooth curve (b).
To confirm the obtained results we end up with a simplified approach, less dependent on MC, where the background contribution is taken from a 5th order polynomial approximation of the \(m^{2}_{miss}\) distribution of the experimental data within the range of \(m^{2}_{miss}>0.05~\hbox {GeV}^2/c^4\). Again we do a common fit for signal plus background, where we allow the tuning of the background parameters for each investigated position. In this case the obtained upper limit at 90% CL is shown with the dashed curve (c) in Fig. 9. This approach, however, is more sensitive to impurity of the signal, so as the final result curve (b) shall be used, as it uses realistic information about background shape and does not artificially suppresses signal (in contrast to approach a). A difference between these three approaches can be can be treated as a systematic error of the method.
3.5 Systematic uncertainties
To study the systematics, we select the method from Sect. 3.4 with MC background as a basis.
(I) The effects arising from selection cuts variation are estimated by performing over hundred standard analyses with the cuts randomly and uniformly changing in the region of \(\pm ~\sigma _{i}\) around nominal values, \(\sigma _{i}\) being the experimental in the ith variable, the main variables are: zvertex left and right cuts, muon selection in GAMS2000 and HCAL, while the thresholds in GS, BGD were varied within 20% of the nominal values. Also, the minimal number of points on the secondary track is changed by \(\pm ~1\) and, independently, its \(\chi ^2\) is relaxed by 25% during the set of tests. The result on relative efficiency variation is shown by the gray area in Fig. 7, while 10% truncated (from both sides of variation) area is shown for UL at Fig. 9 by pink vertically shaded area. The upper edge of this area corresponds to UL with effects of the cut variations taken into account.
(II) If the effect of discrepancy between the data and MC of Fig. 4 is interpreted as an extra data inefficiency in the region of interest, we get an additional correction factor which depends on \(m_{miss}\) and reaches \(\sim \) 15%. It is shown by the gray oblique shaded area in Fig. 9.
(III) A variation of the shape of the MC background was studied by exclusion of contributions from \(K^{+} \rightarrow e^{+} \nu _{\mu } \pi ^{0}\) (Fig. 3f) and from undecayed kaon (Fig. 3g). The effect is negligible.
(IV) A possible muon trigger inefficiency which is not described in MC is searched for by a separate analysis of two triggers, the effect can be neglected.
Hence, the upper edge of gray area (curve f of Fig. 9) corresponds to UL with systematics taken into account.
3.6 Evaluation of the \(U_{\mu H}^2\) mixing parameter upper limit
Finally, we obtain an upper limit on the mixing parameter \(U_{\mu H}^2\) between the muon neutrino and the heavy sterile neutrino \(\nu _{H}\), see Fig. 10.
Since the obtained upper limit on \(U_{\mu H}^{2}\) in the considered mass range does not exceed \(10^{5}\), the \(\nu _{H}\) mean life time is estimated to be greater than \(10^{6}\) sec, assuming it decays to SM particles [12, 13]. The corresponding \(\nu _{H}\) mean flight distance, estimated with a MC simulation for the \(\nu _{H}\) masses considered, ranges between 5–25 km. Hence the heavy neutrino in our case, indeed, can be regarded as stable particle.
4 Conclusions
The OKA 2012 data set was analyzed to search for heavy sterile neutrinos. A peak search method in the missing mass spectrum was used in the analysis. No signal is observed and the upper limit on the mixing between muon neutrino and a heavy sterile neutrino is set in the mass range 270–375 MeV/c\(^{2}\).
Footnotes
Notes
Acknowledgements
We express our gratitude to our colleagues in the accelerator department for the good performance of the U70 during data taking; to colleagues from the beam department for the stable operation of the 21K beam line, including RFdeflectors, and to colleagues from the engineering physics department for the operation of the cryogenic system of the RFdeflectors.
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