Search for heavy neutral lepton production in $K^+$ decays to positrons

A search for heavy neutral lepton ($N$) production in $K^+\to e^+N$ decays using the data sample collected by the NA62 experiment at CERN in 2017--2018 is reported. Upper limits of the extended neutrino mixing matrix element $|U_{e4}|^2$ are established at the level of $10^{-9}$ over most of the accessible heavy neutral lepton mass range 144--462 MeV/$c^2$, with the assumption that the lifetime exceeds 50 ns. These limits improve significantly upon those of previous production and decay searches. The $|U_{e4}|^2$ range favoured by Big Bang Nucleosynthesis is excluded up to a mass of about 340 MeV/$c^2$.


Introduction
All Standard Model (SM) fermions except neutrinos are known to exhibit right-handed chirality.The existence of right-handed neutrinos, or heavy neutral leptons (HNLs), is hypothesised in many SM extensions in order to generate non-zero masses of the SM neutrinos via the seesaw mechanism [1].For example, the Neutrino Minimal Standard Model [2] simultaneously accounts for dark matter, baryogenesis, and neutrino masses and oscillations by postulating two HNLs in the MeV-GeV mass range and a third HNL, a dark matter candidate, at the keV mass scale.
Mixing between HNLs (also denoted N below) and active neutrinos gives rise to HNL production in decays of SM particles and decays of HNLs into SM particles.Both classes of processes can in principle be detected experimentally.The expected branching fraction of the K + → + N decay ( = e, µ) is [3] where B(K + → + ν) is the measured branching fraction of the SM leptonic decay, |U 4 | 2 is the mixing parameter, m N is the HNL mass, and ρ (m N ) is a kinematic factor: with x = (m /m K ) 2 , y = (m N /m K ) 2 and λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ac).By definition, ρ (0) = 1.The product B(K over most of the allowed m N range; it drops to zero at the kinematic limit m N = m K − m and, in the positron case, reduces to B(K + → e + ν) = 1.582(7) × 10 −5 [4] for m N → 0 due to helicity suppression.
A search for K + → e + N decays in the HNL mass range 144-462 MeV/c 2 using the data collected by the NA62 experiment at CERN in 2017-2018 is reported.The results, which assume the HNL lifetime exceeds 50 ns, are presented as upper limits of |U e4 | 2 at 90% CL for a number of mass hypotheses.

Beam, detector and data sample
The layout of the NA62 beamline and detector [6] is shown schematically in Fig. 1.An unseparated secondary beam of π + (70%), protons (23%) and K + (6%) is created by directing 400 GeV/c protons extracted from the CERN SPS onto a beryllium target in spills of 3 s effective duration.The central beam momentum is 75 GeV/c, with a momentum spread of 1% (rms).
Beam kaons are tagged with 70 ps time resolution by a differential Cherenkov counter (KTAG) using nitrogen gas at 1.75 bar pressure contained in a 5 m long vessel as radiator.Beam particle positions, momenta and times (to better than 100 ps resolution) are measured by a silicon pixel spectrometer consisting of three stations (GTK1,2,3) and four dipole magnets.A muon scraper (SCR) is installed between GTK1 and GTK2.A 1.2 m thick steel collimator (COL) with a central aperture of 76 × 40 mm 2 and outer dimensions of 1.7 × 1.8 m 2 is placed upstream of GTK3 to absorb hadrons from upstream K + decays (a variable aperture collimator of 0.15 × 0.15 m 2 outer dimensions was used up to early 2018).Inelastic interactions of beam particles in GTK3 are detected by an array of scintillator hodoscopes (CHANTI) located just after GTK3.The beam is delivered into a vacuum tank evacuated to a pressure of 10 −6 mbar, which contains a 75 m long fiducial decay volume (FV) starting 2.6 m downstream of GTK3.The beam divergence at the FV entrance is 0.11 mrad (rms) in both horizontal and vertical planes.Downstream of the FV, undecayed beam particles continue their path in vacuum.
Momenta of charged particles produced by K + decays in the FV are measured by a magnetic spectrometer (STRAW) located in the vacuum tank downstream of the FV.The spectrometer consists of four tracking chambers made of straw tubes, and a dipole magnet (M) located between the second and third chambers and providing a horizontal momentum kick of 270 MeV/c.The momentum resolution achieved is σ p /p = (0.30⊕0.005•p)%,where the momentum p is expressed in GeV/c.
A ring-imaging Cherenkov detector (RICH), consisting of a 17.5 m long vessel filled with neon at atmospheric pressure (with a Cherenkov threshold for muons of 9.5 GeV/c), is used for the identification of charged particles and time measurements with 70 ps precision (for positrons).Two scintillator hodoscopes (CHOD, which include a matrix of tiles and two orthogonal planes of slabs, arranged in four quadrants) downstream of the RICH provide trigger signals and time measurements with 200 ps precision.
A 27X 0 thick quasi-homogeneous liquid krypton (LKr) electromagnetic calorimeter is used for particle identification and photon detection.The calorimeter has an active volume of 7 m 3 , is segmented in the transverse direction into 13248 projective cells of approximately 2×2 cm 2 , and provides an energy resolution σ E /E = (4.8/√ E ⊕ 11/E ⊕ 0.9)%, where E is expressed in GeV.To achieve hermetic acceptance for photons emitted in K + decays in the FV at angles up to 50 mrad to the beam axis, the LKr calorimeter is supplemented by annular lead glass detectors (LAV) installed in 12 positions in and downstream of the vacuum tank, and two lead/scintillator sampling calorimeters (IRC, SAC) located close to the beam axis.An iron/scintillator sampling hadronic calorimeter formed of two modules (MUV1,2) and a muon detector (MUV3) consisting of 148 scintillator tiles located behind an 80 cm thick iron wall are used for particle identification.
The data sample used for the analysis is obtained from 0.79 × 10 6 SPS spills recorded during 360 days of operation in 2017-2018, at a typical beam intensity of 2.2 × 10 12 protons per spill corresponding to a mean beam particle rate at the FV entrance of 500 MHz, and a mean K + decay rate in the FV of 3.7 MHz.The trigger line used for the K + → πν ν measurement [7], consisting of both hardware (L0) and software (L1) stages, is used for the analysis.Overall trigger efficiency for single positrons with momenta below 30 GeV/c is measured to be (90 ± 4)% using data samples.

Event selection
Assuming HNL lifetime exceeding 50 ns, and considering that the HNL produced in K + → e + N decays would be boosted by a Lorentz factor of O(100), HNL decays in flight into SM particles in the 156 m long volume between the start of the FV and the last detector (SAC) can be neglected (Section 5).Therefore the K + → e + N decay is characterized by a single positron in the final state, similarly to the SM K + → e + ν decay.The principal selection criteria follow.
• A positron track reconstructed in the STRAW spectrometer with momentum in the range 5-30 GeV/c is required.The momentum is restricted to this range, because the L0 trigger required that the total energy deposited in the LKr not exceed 30 GeV.The track's trajectory through the STRAW chambers and its extrapolation to the LKr calorimeter, CHOD and MUV3 should be within the fiducial geometrical acceptance of these detectors.
The positron time is evaluated as the mean time of the RICH signals spatially associated with the track.
• Backgrounds due to particle misidentification are suppressed to a negligible level by applying the following particle identification criteria to the single track: the ratio of energy, E, deposited in the LKr calorimeter to momentum, p, measured by the STRAW spectrometer is required to be 0.92 < E/p < 1.08; a particle identification algorithm based on the RICH signal pattern within 3 ns of the positron RICH time is applied; no signal in the MUV3 detector spatially consistent with the projected track impact point and within 4 ns of the positron time is allowed.
• Backgrounds from beam pion decays (mainly π + → e + ν, and to a lesser extent π + → µ + ν followed by muon decay µ + → e + ν ν) are suppressed by requiring a kaon signal in the KTAG detector within 1 ns of the positron time.
• Identification of the K + track in the GTK relies on the time difference, ∆t GK , between a GTK track and the KTAG signal, and spatial compatibility of the GTK and STRAW tracks quantified by the distance, d, of closest approach.A discriminant D(∆t GK , d) is defined using the ∆t GK and d distributions measured with K + → π + π + π − decays.Among GTK tracks with |∆t GK | < 0.5 ns, the track of the parent kaon is assumed to be the one giving the D value most consistent with a K + → e + decay.It is also required that d < 4 mm to reduce the background from K + → µ + ν decays in the FV followed by muon decay µ + → e + ν ν.The decay vertex is defined as the point of closest approach of the GTK and STRAW tracks, taking into account the stray magnetic field in the vacuum tank.
• Background from K + → µ + ν decays upstream of GTK3 followed by muon decays in the FV arises from pileup in the GTK, and is suppressed by geometrical conditions.The reconstructed K + decay vertex is required to be located in the FV at a minimum distance from the start of the FV, varying from 2 m to 12 m depending on the angle between the K + momentum in the laboratory frame and the positron momentum in the K + rest frame.
• Backgrounds from multi-body K + decays are suppressed by veto conditions.The positron track must not form vertices with any other STRAW track.LKr energy deposition clusters not spatially compatible with the positron track within 8 ns of the positron time are not allowed.Activity in the large-angle (LAV) and small-angle (SAC, IRC) photon veto detectors within 3 ns of the positron time is forbidden.Activity in the CHANTI detector within 4 ns of the positron time is not allowed; more than two signals in the CHOD tiles within 6 ns of the positron time and not spatially associated with the positron are also not allowed.Data loss due to the veto conditions from accidental activity ("random veto") averaged over the sample is measured to be about 30%.The squared missing mass is computed as m 2 miss = (P K − P e ) 2 , where P K and P e are the kaon and positron 4-momenta, obtained from the 3-momenta measured by the GTK and STRAW detectors and using the K + and e + mass hypotheses.
Simulation of particle interactions with the detector and its response is performed with a Monte-Carlo simulation package based on the Geant4 toolkit [8].The m 2 miss spectra of the events selected from data and simulated samples, and their ratio, are displayed in Fig. 2. The signal from the SM leptonic decay K + → e + ν is observed as a peak at m 2 miss = 0 with a resolution of 1.7 × 10 −3 GeV 2 /c 4 .The simulation is tuned to reproduce this resolution to a 1% relative precision.The SM signal region is defined in terms of the reconstructed squared missing mass as |m 2 miss | < 0.01 GeV 2 /c 4 .

Measurement principle
A peak-search procedure measures the K + → e + N decay rate with respect to the K + → e + ν rate for an assumed HNL mass m N .This approach benefits from first-order cancellations of residual detector inefficiencies not fully accounted for in simulations, as well as trigger inefficiencies and random veto losses, common to signal and normalization modes.The expected number of K + → e + N signal events N S can be written as where the branching fraction B SES (K + → e + N ) and the mixing parameter |U e4 | 2 SES corresponding to the observation of one signal event, the single event sensitivity (SES), are defined as where N K is the number of K + decays in the FV, A N is the signal selection acceptance, and the kinematic factor ρ e (m N ) is defined in Eq. ( 2).
The number of K + decays in the FV is evaluated using the number of K + → e + ν candidates reconstructed in the data sample.Data losses due to trigger efficiencies and random vetoes are included in the N K definition, which makes the value of N K specific to this analysis.The dominant background due to K + → µ + ν decay followed by µ + → e + ν ν decay (0.08% in relative terms) is taken into account.Other backgrounds, including the contribution from K + → µ + ν decay with a misidentified muon, are negligible.
The number of K + decays is computed as where N SM = 3.495 × 10 6 is the number of selected data events in the SM signal region; A e = (6.27± 0.02 stat ) × 10 −2 and A µ = (1.24± 0.19 stat ) × 10 −9 are the acceptances of the selection for the K + → e + ν decay and the K + → µ + ν decay (followed by muon decay) evaluated with simulations; and B(K + → e + ν) = (1.582± 0.007) × 10 −5 and B(K + → µ + ν) = 0.6356 ± 0.0011 are the branching fractions of these decays [4].The uncertainty quoted in N K is due to the precision of the external input B(K + → e + ν) and the statistical and systematic accuracy of the simulation.The systematic uncertainty is evaluated by varying the selection criteria.

Background evaluation with simulations
The search procedure is based on a data-driven estimation of the background to K + → e + N decays, which is valid in the absence of peaking signal-like background structures in the reconstructed mass spectrum.Simulations are used to understand the background qualitatively, to optimize the event selection, and to justify the search procedure.The main background to K + → e + N and K + → e + ν decays comes from K + → µ + ν decay followed by muon decay µ + → e + ν ν.This background is reduced by the requirement of spatial compatibility between the positron and kaon tracks, and its ultimate level is limited by the resolution on the track directions provided by the STRAW and GTK spectrometers.The background from K + → µ + ν decays upstream of the vacuum tank is 60 times smaller than that from decays in the vacuum tank.
Backgrounds from beam pion decays π + → e + ν and π + → µ + ν decays, followed by muon decay, arise from π + misidentification in the KTAG due to the presence of an in-time beam kaon not decaying in the FV.Considering the beam rate, the kaon fraction in the beam, and the KTAG-RICH timing conditions used, the beam pion misidentification probability due to pileup is about 6%.The probability of beam pion identification as a kaon in the KTAG in the absence of pileup is negligible.The beam pion decay background populates the missing mass region m 2 miss > 0.13 GeV 2 /c 4 , because of the 30 GeV/c upper limit imposed on the e + momentum.The reconstructed m 2 miss spectrum of data events is described by simulations to a few percent relative precision (Fig. 2).The accuracy of the simulation is limited by systematic effects, such as in the modelling of the LAV detector response to soft radiative photons.

Search procedure
The K + → e + N process is investigated in 264 mass hypotheses, m N , within the HNL search region between 144 and 462 MeV/c 2 .The distances between adjacent mass hypotheses are equal to the mass resolution σ m shown in Fig. 3 (left), rounded to 0.1 MeV/c 2 .This mass resolution is three times better than that of the 2015 data sample collected without the GTK spectrometer [9].Event selection requires that |m miss − m N | < 1.5σ m for each mass hypothesis, where m miss is the reconstructed missing mass.In each HNL mass hypothesis, sidebands are defined in the reconstructed missing mass spectrum as 1.5σ m < |m miss − m N | < 11.25σ m , additionally requiring the missing mass to be within the range 122-465 MeV/c 2 .The number of expected background events, N exp , within the ±1.5σ m signal window is evaluated with a second-order polynomial fit to the sideband data of the m miss spectrum, where the bin size is 0.75σ m .The uncertainty, δN exp , in the number of expected background events includes statistical and systematic components.The former comes from the statistical errors in the fit parameters, and the latter is evaluated as the difference between N exp obtained from fits using second and third order polynomials.The dominant contribution to δN exp is statistical, except near the boundaries of the HNL search region where the systematic uncertainty is comparable.Further systematic errors due to possible HNL signals in the sidebands are found to be negligible; this check is made assuming |U e4 | 2 to be equal to the expected sensitivity of the analysis.The ratio δN exp /N exp is typically 0.2-0.3%, but reaches a few percent close to the limits of the search region.
An auxiliary selection with a tighter maximum positron momentum requirement of 20 GeV/c is used to achieve a locally smooth m miss spectrum of background events in the sidebands for mass hypotheses in the range 356-382 MeV/c 2 .The beam pion decay background threshold in the m 2 miss spectrum is thereby shifted from 0.14 GeV 2 /c 4 (Fig. 2) to 0.165 GeV 2 /c 4 .The signal selection acceptances, A N , for the standard and the auxiliary selections as functions of m N obtained with simulations assuming infinite HNL lifetime are displayed in Fig. 3 (right).The acceptance for a mean lifetime of 50 ns (considering decays to detectable particles) is lower than shown by O(1%) in relative terms, making the results of the search valid for lifetimes in excess of 50 ns.Acceptances for lifetimes of 5 (1) ns decrease by factors of up to 2 (10), depending on the HNL mass.
The simulation of the missing mass resolution outside the K + → e + ν peak is validated with a sample of fully reconstructed K + → π + π + π − decays by studying the resolution of ∆m 3π = (P K − P 3 ) 2 − (P 1 + P 2 ) 2 , where P i (i = 1, 2, 3) and P K are the reconstructed pion and kaon 4-momenta.The resolution on ∆m 3π can be measured for both data and simulated samples because the true value of this quantity is always zero.The resolution varies in the range 0.8-1.3MeV/c 2 depending on (P 1 + P 2 ) 2 , and the simulation agrees with the data to better than 5%, validating the signal acceptance estimates to 2% relative precision.

Results
The number of observed events, N obs , within the signal window, the number of expected background events, N exp , and its uncertainty, δN exp , are used to compute the local signal significance for each mass hypothesis The quantities N obs , N exp , and δN exp are used to evaluate the upper limit at 90% CL of the number of K + → e + N decays, N S , in each HNL mass hypothesis using the CL S method [10].The values of N obs , the obtained upper limits of N S , and the expected ±1σ and ±2σ bands of variation of N S in the background-only hypothesis are shown in Fig. 4 (left).
Upper limits at 90% CL of the branching fraction B(K + → e + N ) and the mixing parameter |U e4 | 2 are obtained from those of N S according to Eq. (3), using the single event sensitivities B SES (K + → e + N ) and |U e4 | 2 SES shown as functions of the HNL mass in Fig. 4 (right).The obtained limits of B(K + → e + N ) are displayed in Fig. 5.The obtained limits of |U e4 | 2 , together with the limits from previous HNL production searches in K + [9,11] and π + [12,13]   the 30-470 MeV/c 2 mass range, and the Big Bang Nucleosynthesis (BBN) constraint [14], are shown in Fig. 6.The reported result improves the existing limits of |U e4 | 2 obtained in production searches over the whole mass range considered.Comparison with HNL decay searches is available in Refs.[1,15].The obtained limits of |U e4 | 2 improve over the decay searches [16,17] in the mass region below 400 MeV/c 2 .

Summary and outlook
A search for HNL production in K + → e + N decays has been performed with the data set collected by the NA62 experiment at CERN in 2017-2018.Upper limits of the decay branching fraction and the mixing parameter |U e4 | 2 have been established at the 10 −9 level over most of the HNL mass range 144-462 MeV/c 2 with the assumption of mean lifetime exceeding 50 ns.These limits become weaker by factors of up to 2 (10) for lifetimes of 5 (1) ns.The |U e4 | 2 results are significantly better than previous limits obtained from HNL production and decay searches [1], and other experimental constraints [15,18].The values of |U e4 | 2 favoured by the BBN constraint [14] are excluded for HNL masses up to about 340 MeV/c 2 .
An improvement in sensitivity of this analysis in terms of |U e4 | 2 can only be expected with future NA62 data.Considering the background conditions, the sensitivity to |U e4 | 2 is proportional to δN exp /N exp , and improves as the relative statistical uncertainty in N exp decreases with sample size as 1/ √ N K .HNL masses below 144 MeV/c 2 , not accessible to this analysis due to the shape of the background mass spectrum, can be probed via the K + → π 0 e + N decay.The data set currently available for this search, collected with pre-scaled trigger lines, corresponds to N K ≈ 3 × 10 10 .The single event sensitivity in the mass range of interest estimated using the formalism of Ref. [5] is |U e4 | 2 SES ≈ 10 −8 ; the search is expected to be limited by the K + → π 0 e + νγ background.[11], NA62 (2015 data) [9]; and π + → e + N decays: TRIUMF [12], PIENU [13].The lower boundary on |U e4 | 2 imposed by the BBN constraint [14] is shown by a dashed line.

Figure 1 :
Figure 1: Schematic side view of the NA62 beamline and detector.

Figure 2 :
Figure 2: Left: reconstructed squared missing mass (m 2 miss ) distributions for data and simulated events.Decays to muons contribute via muon decays in flight.The boundaries of the SM signal region (upper arrows) and the HNL search region (lower arrows) are defined in Sections 2 and 5 of the text.Right: the ratio of data and simulated m 2 miss spectra.The statistical uncertainties shown are dominated by those of the simulated spectra.

Figure 3 :
Figure 3: HNL mass resolution σ m (left) and acceptances A N of the standard and auxiliary selections (right) evaluated from simulations as functions of the HNL mass.Above the mass of 420 MeV/c 2 , the two selections have equal acceptance as the momentum of the positron produced in the K + → e + N decay is always below 20 GeV/c.The boundaries of the HNL search region are indicated by vertical arrows.

Figure 4 :
Figure 4: Left: the values of N obs , the obtained upper limits at 90% CL of the numbers of K + → e + N events, and the expected ±1σ and ±2σ bands in the background-only hypothesis for each HNL mass value considered.Right: single event sensitivities B SES (K + → e + N ) (dashed line) and |U e4 | 2 SES (solid line) as functions of the HNL mass.Note the reduced sensitivity in the region of 356-382 MeV/c 2 in which the auxiliary selection is used.The boundaries of the HNL search region are indicated by vertical arrows.

Figure 5 :
Figure 5: The obtained upper limits at 90% CL of B(K + → e + N ) and the expected ±1σ and ±2σ bands in the background-only hypothesis for each HNL mass value considered.