Test of Lepton Flavour Universality in K+ -->l+nu Decays

A precision test of lepton flavour universality has been performed by measuring the ratio RK of kaon leptonic decay rates K+ -->e+nu and K+ -->mu+nu in a sample of 59813 reconstructed K+ -->e+nu candidates with (8.71 +- 0.24)% background contamination. The result RK = (2.487 +- 0.013) * 10^{-5} is in agreement with the Standard Model expectation.


Introduction
In the Standard Model (SM) the decays of pseudoscalar mesons to light leptons are helicity suppressed. In particular, the SM width of P ± → ℓ ± ν decays with P = π, K, D, B (denoted P ℓ2 in the following) is where G F is the Fermi constant, M P and M ℓ are meson and lepton masses, f P is the decay constant, and V qq′ is the corresponding Cabibbo-Kobayashi-Maskawa matrix element. Although the SM predictions for the P ℓ2 decay rates are limited by hadronic uncertainties, their specific ratios do not depend on f P and can be computed very precisely. In particular, the SM prediction for the ratio R K = Γ(K e2 )/Γ(K µ2 ) of kaon leptonic decay widths inclusive of internal bremsstrahlung (IB) radiation is [1] where δR QED = (−3.79 ± 0.04)% is an electromagnetic correction due to the IB and structuredependent effects. Within certain two Higgs doublet models (2HDM), including the minimal supersymmetric model (MSSM), R K is sensitive to lepton flavour violating (LFV) effects appearing at the oneloop level via the charged Higgs boson (H ± ) exchange [2,3], representing a unique probe into mixing in the right-handed slepton sector [4]. The dominant contribution due to the LFV coupling of the H ± is where tan β is the ratio of the two Higgs vacuum expectation values, and |∆ 31 R | is the mixing parameter between the superpartners of the right-handed leptons, which can reach ∼ 10 −3 . This can enhance R K by O(1%) without contradicting any experimental constraints known at present, including upper bounds on the LFV decays τ → eX with X = η, γ, µμ. On the other hand, R K is sensitive to the neutrino mixing parameters within the SM extension involving a fourth generation [5].
The first measurements of R K were performed in the 1970s [6,7,8]; the current PDG world average [9] is based on a more precise recent result [10] R K = (2.493 ± 0.031) × 10 −5 . A new measurement of R K based on a part of the data sample collected by the NA62 experiment at CERN in 2007 is reported in this letter. The analyzed K e2 sample is ∼ 4 times larger than the total world sample, allowing a measurement of R K with a precision well below 1%.

Beam, detector and data taking
The beam line and setup of the NA48/2 experiment [11,12] have been used for the NA62 data taking in 2007. Experimental conditions have been optimized for the K e2 /K µ2 measurement.

Kaon beam
The beam line was originally designed to deliver simultaneous unseparated K + and K − beams derived from the primary 400 GeV/c protons extracted from the CERN SPS. In 2007, the muon sweeping system was optimized for the positive beam, and the sample used for the present analysis was collected with the K + beam only. Positively charged particles within a narrow momentum band with a central momentum of 74.0 GeV/c and a spread of 1.4 GeV/c (rms) are selected by the first two magnets in a four dipole achromat and by momentum-defining slits incorporated into a 3.2 m thick copper/iron proton beam dump, which also blocks the negatively-charged particles. The beam subsequently passes through acceptance-defining and cleaning collimators and a set of four quadrupoles of alternating polarity, as well as muon sweeping magnets, before entering the fiducial decay volume contained in a 114 m long cylindrical vacuum tank with a diameter of 1.92 m upstream, increasing to 2.4 m downstream.
With about 1.8 × 10 12 primary protons incident on the target per SPS pulse of about 4.8 s duration repeating every 14.4 or 16.8 s, the secondary beam flux at the entrance of the decay volume is 2.5 × 10 7 particles per pulse. The fractions of K + , π + , p + , e + and µ + in the secondary beam are 0.05, 0.63, 0.21, 0.10 and 0.01, respectively. The fraction of those beam kaons decaying in the vacuum tank at nominal momentum is 18%. The beam transverse size at the entrance to the decay volume is δx = δy = 4 mm (rms), and its horizontal and vertical angular divergences are about 20 µrad (rms). The beam central momentum, transverse position at the entrance to the vacuum tank and direction varied slowly over time with respect to the nominal ones in the ranges of ∼ 0.1 GeV/c 2 , ∼ 1 mm and ∼ 10 µrad, respectively.
The beam line also transmits certain off-momentum charged kaons and pions punching through the proton beam dump with a suppression factor of ∼ 10 −3 . However the subsequent decays of these particles do not contribute to the signal region of the present analysis.

Detector
The charged particle properties are measured in a magnetic spectrometer, housed in a tank filled with helium at nearly atmospheric pressure, placed after the decay volume and separated from the vacuum by a thin (∼ 0.4% radiation lengths X 0 ) Kevlar R window. The spectrometer comprises four drift chambers (DCHs) [13], two upstream and two downstream of a dipole magnet which gives a horizontal transverse momentum kick of 265 MeV/c to singly-charged particles. Each DCH is composed of eight planes of sense wires, and provides a spatial resolution of 90 µm in each projection. The measured momentum resolution is σ p /p = 0.48% ⊕ 0.009% · p, where p is expressed in GeV/c. A plastic scintillator hodoscope (HOD) producing fast trigger signals and providing precise time measurements of charged particles is placed after the spectrometer. It consists of a plane of vertical strips, followed by a similar plane of horizontal strips (128 counters in total). Both planes have regular octagonal shapes and a central hole for the passage of the beam.
The HOD is followed by a quasi-homogeneous liquid krypton electromagnetic calorimeter (LKr) [14] used for lepton identification and as a photon veto in the present analysis. The LKr is 127 cm (or 27X 0 ) thick along the beam, with projective readout consisting of copper/beryllium ribbons extending from the front to the back of the detector. The 13248 readout cells have a transverse size of approximately 2×2 cm 2 each and have no longitudinal segmentation. The energy resolution is σ E /E = 0.032/ √ E ⊕ 0.09/E ⊕ 0.0042 (E in GeV). The spatial resolution for the transverse coordinates x and y of an isolated electromagnetic shower is σ x = σ y = 0.42/ √ E ⊕ 0.06 cm (E in GeV). An aluminium beam pipe of 158 mm outer diameter and 1.1 mm thickness traversing the centres of all detector elements allows the undecayed beam particles to continue their path in vacuum. The outer transverse sizes of the subdetectors are about 2.4 m.

Trigger and data acquisition
A minimum bias trigger configuration has been employed, resulting in high efficiency. The K e2 trigger condition consists of coincidences of signals in the two HOD planes (the Q 1 signal), loose lower and upper limits on DCH hit multiplicity (the 1-track signal), and LKr energy deposit (E LKr ) of at least 10 GeV. The K µ2 trigger condition requires a coincidence of the Q 1 and 1-track signals downscaled by a factor D = 150. The non-downscaled K µ2 trigger rate is 0.5 MHz, and is dominated by beam halo muons; the K e2 trigger rate is about 10 kHz. Downscaled control samples based on trigger signals from the DCHs, HOD and LKr have been collected to monitor the performance of the main trigger signals. The data taking took place during four months starting in June 2007. About 40% of the 350k recorded good SPS spills are used for the present analysis.

Analysis strategy
The analysis strategy is based on counting the numbers of reconstructed K e2 and K µ2 candidates collected concurrently. Therefore the analysis does not rely on an absolute beam flux measurement, and several systematic effects (due to beam simulation, accidental activity, charged track reconstruction, Q 1 trigger efficiency, and time-dependent effects) cancel at first order.
Due to the significant dependence of acceptance and background on lepton momentum, the R K measurement is performed independently in 10 momentum bins covering a range from 13 to 65 GeV/c. The lowest momentum bin spans 7 GeV/c, while the others are 5 GeV/c wide. The selection criteria have been optimized separately in each momentum bin. The data samples in the momentum bins are statistically independent, however the systematic errors are partially correlated. The ratio R K in each bin is computed as where N (K ℓ2 ) are the numbers of selected K ℓ2 candidates (ℓ = e, µ), N B (K ℓ2 ) are the numbers of background events, A(K µ2 )/A(K e2 ) is the ratio of the geometric acceptances (referred to as the acceptance correction in the following), f ℓ are the lepton identification efficiencies, ǫ(K ℓ2 ) are the trigger efficiencies, f LKr is the global efficiency of the LKr readout (which affects only the K e2 selection), and D = 150 is the K µ2 trigger downscaling factor. To evaluate the acceptance correction and the geometric parts of the acceptances for background processes entering the computation of N B (K ℓ2 ), a detailed Monte Carlo (MC) simulation based on Geant3 [15] is used. It includes a description, with time variations, of the beam line optics, the full detector geometry, materials, magnetic fields, local inefficiencies of DCH wires, and inactive LKr cells (0.8% of channels). Particle identification, trigger and readout efficiencies are measured directly from data.

Data analysis 3.1 Event reconstruction and selection
Charged particle trajectories are reconstructed from hits and drift times in the spectrometer. Track momenta are evaluated using a detailed magnetic field map. Fine calibrations of spectrometer field integral and DCH alignment are performed by monitoring the mean reconstructed K + → π + π + π − invariant mass, and the missing mass in K µ2 decays.
Clusters of energy deposition in the LKr are found by locating the maxima in the digitized pulses from individual cells in both space and time and accumulating the energy within a radius of approximately 11 cm. Shower energies are corrected for energy outside the cluster boundary, energy lost in inactive cells and cluster energy sharing. The energy response has been calibrated with samples of positrons from K + → π 0 e + ν decays.
Due to the topological similarity of K e2 and K µ2 decays, a large part of the selection is common for the two decay modes, which leads to significant cancellations of the related systematic uncertainties. The main selection criteria are listed below.
• Exactly one reconstructed charged particle track (lepton candidate) geometrically consistent with originating from a kaon decay is required. The electric charge of the track must be positive.
• The extrapolated track impact points in the DCHs, HOD and LKr must be within their geometrical acceptances. The LKr acceptance condition includes appropriate separations from the detector edges and inactive cells.
• The reconstructed track momentum must be in the range 13 to 65 GeV/c. The lower limit ensures high efficiency of the E LKr > 10 GeV trigger condition. Above the upper limit, the analysis is affected by large uncertainties due to background subtraction.
• No LKr clusters with energy E > E veto = 2 GeV and in time with the track are allowed, unless they can be associated to the track via direct energy deposition or bremsstrahlung. (Most clusters due to bremsstrahlung in front of the spectrometer magnet are resolved from those directly deposited by the track). This requirement provides a photon veto for suppression of backgrounds from K + → e + νγ, K + → π 0 e + ν, and K + → π + π 0 decays. However the veto is not hermetic due to the beam pipe and the limited transverse size of the LKr.
• The decay vertex is reconstructed as the point of closest approach of the lepton candidate track extrapolated upstream, and the kaon beam axis. The measured stray magnetic field in the vacuum tank is taken into account. The position of the kaon beam axis is monitored with a sample of fully reconstructed K + → π + π + π − decays.
• The distance from the kaon decay vertex to the beginning of the vacuum tank is required to exceed a minimum value ranging from 8 m at low lepton momentum to 43 m at high momentum, which removes the bulk of the beam halo background (discussed in Section 3.2.4).
• For further suppression of the beam halo and several other backgrounds, the reconstructed closest distance of approach of the track to the beam axis must not exceed 3.5 cm.
The following two main criteria are used to distinguish K e2 from K µ2 decays.
• The kinematic identification of K e2 (K µ2 ) decays is based on constraining the reconstructed squared missing mass in the positron (muon) hypothesis: Here P K is the average kaon four-momentum (monitored in time with K + → π + π + π − decays), and P ℓ is the reconstructed lepton four-momentum (under the positron or muon mass hypothesis). The limits M 2 1 and M 2 2 have been optimized for each lepton momentum bin, taking into account the M 2 miss (ℓ) resolution (which varies from 0.0025 (GeV/c 2 ) 2 at mid track momentum to 0.005 (GeV/c 2 ) 2 at low and high track momentum), the radiative mass tails, and the background conditions. M 2 1 varies between 0.013 and 0.016 (GeV/c 2 ) 2 and M 2 2 between 0.010 and 0.013 (GeV/c 2 ) 2 . The kinematic separation of K e2 and K µ2 decays is illustrated in Fig. 1a. • The lepton identification is based on the ratio E/p of energy deposition in the LKr to momentum measured by the spectrometer. Charged particles with (E/p) min < E/p < 1.1, where (E/p) min = 0.95 for p > 25 GeV/c and (E/p) min = 0.9 otherwise, are identified as positrons. At low lepton momenta, the background from particle mis-identification is negligible. For p > 25 GeV/c, the larger (E/p) min limit minimises the net uncertainty from K µ2 background subtraction and particle mis-identification inefficiency. Charged particles with E/p < 0.85 are classified as muons. The data E/p spectra of positrons and muons are shown in Fig. 1c.

The K e2 sample
The number of K e2 candidates in the signal region is N (K e2 ) = 59813. The sources of background in the K e2 sample are discussed below.

K µ2 background
Kinematic separation of K e2 from K µ2 decays is achievable at low lepton momentum only (p 35 GeV/c), as shown in Figs. 1a and 1b. At high lepton momentum, the K µ2 decay with a mis-identified muon (E/p > 0.95, see Fig. 1c) is the largest background source. The dominant process leading to mis-identification of the muon as a positron is 'catastrophic' bremsstrahlung in or in front of the LKr leading to significant energy deposit in the LKr. Mis-identification due to accidental LKr clusters associated with the muon track is negligible, as concluded from a study of the sidebands of track-cluster time difference and distance distributions.
The muon mis-identification probability P µe has been measured as a function of momentum. To collect a muon sample free from the typical ∼ 10 −4 positron contamination due to µ → e decays, a 9.2X 0 thick lead (Pb) wall covering ∼ 20% of the geometric acceptance was installed approximately 1.2 m in front of the LKr calorimeter (between the two HOD planes) during a dedicated period of data taking with K + and K − beams. The K e2 sample collected with the However, muon passage through the Pb wall affects the measured P Pb µe via two principal effects: 1) ionization energy loss in Pb decreases P µe and dominates at low momentum; 2) bremsstrahlung in Pb increases P µe and dominates at high momentum. To evaluate the correction factor f Pb = P µe /P Pb µe , a dedicated MC simulation based on Geant4 (version 9.2) [16] has been developed to describe the propagation of muons downstream from the last DCH, involving all electromagnetic processes including muon bremsstrahlung [17].
The measurements of P Pb µe in momentum bins compared with the results of the MC simulation and the correction factors f Pb obtained from simulation, along with the estimated systematic uncertainties of the simulated values, are shown in Fig. 2. The relative systematic uncertainties on P µe and P Pb µe obtained by simulation have been estimated to be 10%, and are mainly due to the simulation of cluster reconstruction and energy calibration. However the error of the ratio f Pb = P µe /P Pb µe is significantly smaller (δf Pb /f Pb = 2%) due to cancellation of the main systematic effects. The measured P Pb µe is in agreement with the simulation within their uncertainties. The positive correlation between the reconstructed M 2 miss (e) and E/p, which are both computed using the reconstructed track momentum, leads to an apparent dependence of P µe on M 2 miss (e). This effect is significant for intermediate lepton momenta where the K µ2 background comes from events with underestimated M 2 miss (e) and a smaller muon mis-identification probability (see Fig. 1a). This correlation has been taken into account.
The K µ2 background contamination integrated over lepton momentum has been computed to be (6.11 ± 0.22)% using the measured P Pb µe corrected by f Pb . The quoted error comes from the limited size of the data sample used to measure P Pb µe (0.16%), the uncertainty δf Pb (0.12%), and the model-dependence of the correction for the M 2 miss (e) vs E/p correlation (0.08%). The first error component is uncorrelated between the lepton momentum bins, while the others are fully correlated.
As a stability check, the evaluation of P µe has been performed with an additional requirement that the energy deposit in the HOD counters downstream from the Pb wall is small (limited to the equivalent of 1.5 to 3 minimum ionizing particles), which strongly suppresses muons undergoing bremsstrahlung in the Pb wall. The stability of P µe is consistent with the assigned uncertainty δf Pb . Additionally, a stability check of R K with respect to variation of (E/p) min for lepton momentum p > 25 GeV/c in the range from 0.90 to 0.97 has been performed. The observed relative stability of R K within ±0.2%, although the K µ2 background varies from 17% to 3%, is consistent with the uncertainty assigned to the K µ2 background.
The K µ2 decay also contributes to background via µ + → e + νν decays in flight. Energetic forward secondary positrons compatible with K e2 kinematics and topology are suppressed by muon polarisation effects [18]. Radiative corrections to the muon decay [19] lead to a further ∼ 10% relative background suppression. This background contamination has been estimated to be (0.27 ± 0.04)%, where the dominant uncertainty is due to the simulated statistics.
3.2.2 K + → e + νγ background R K is defined to be fully inclusive of internal bremsstrahlung (IB) radiation [1]. The structuredependent (SD) K + → e + νγ process [20,21] may lead to a K e2 signature if the positron is energetic and the photon is undetected. In particular, the SD + component with positive photon helicity peaks at high positron momentum in the K + rest frame (E * e ≈ M K /2) and has a similar branching ratio to K e2 . The background due to K + → e + νγ (SD − ) decay with negative photon helicity peaking at E * e ≈ M K /4 and the interference between the IB and SD processes are negligible.
The SD + background contribution has been estimated by MC simulation as (1.07 ± 0.05)%, using a recent measurement of the K + → e + νγ (SD + ) differential decay rate [10]. The quoted uncertainty is due to the limited precision on the form factors and decay rate, and is therefore correlated between lepton momentum bins. A stability check of R K against variation of the E veto limit in a wide range has been performed. While the K + → e + νγ (SD + ) background is enhanced by a factor of 4.5 for E veto = 14 GeV with respect to E veto = 2 GeV, R K remains stable within ±0.1%, which is consistent with the above uncertainty.
Kinematically, K + → π 0 e + ν and K + → π + π 0 decays can be reconstructed with low missing mass in the K e2 signal region, either because the charged track undergoes a large multiple scatter or because the kaon is in the high momentum tail of the beam distribution. The systematic uncertainties due to subtraction of these backgrounds have been estimated as 50% of the contributions themselves, due to the limited precision of the simulation of the kaon momentumdistribution tails. The backgrounds are at a level below 0.1%.

Beam halo background
As no tracking is available in the beam region to tag an incoming kaon, beam halo muons can become a source of background to K e2 decays in case of µ + → e + ν eνµ decay or muon misidentification as a positron. The choice of the signal region in terms of the longitudinal position Table 1: Summary of backgrounds in the K e2 sample.
The halo background has been measured directly by reconstructing the K + e2 candidates from one control data sample collected with the K − beam transmitted by the beam line and the K + beam (but not its halo) blocked, and another control data sample collected with both K + and K − beams blocked. The control samples are normalised to the data in the region −0.3 < M 2 miss (µ) < −0.1 (GeV/c 2 ) 2 populated mainly by beam halo events. The 'cross-talk' probability to reconstruct a K + e2 candidate due to a K − decay with e + emission ( is at the level of ∼ 10 −4 and is taken into account. The halo background rate and kinematical distribution are qualitatively reproduced by a simulation of the beam line. The background contamination has been estimated to be (1.16 ± 0.06)%, where the error comes from the limited size of the control samples (uncorrelated between lepton momentum bins) and the normalisation uncertainty due to decays of beam kaons and pions upstream of the decay volume (correlated between momentum bins).

Summary of backgrounds in the K e2 sample
Backgrounds integrated over lepton momentum are summarised in Table 1. The total background contamination is (8.71 ± 0.24)%; its uncertainty is smaller than the relative statistical uncertainty of 0.43%. The M 2 miss (e) and lepton momentum distributions of K e2 candidates and backgrounds are shown in Fig. 3.

The K µ2 sample
The number of K µ2 candidates collected with a trigger chain involving downscaling by a factor of 150 is N (K µ2 ) = 1.803 × 10 7 . The only significant background source in the K µ2 sample is the beam halo. Its contribution is mainly at low muon momentum, and has been measured to be (0.38 ± 0.01)% using the same technique as for the K e2 sample. The M 2 miss (µ) and muon momentum spectra of K µ2 candidates and the halo background are presented in Fig. 4.

Geometrical acceptance correction
The ratio of geometric acceptances A(K µ2 )/A(K e2 ) in each lepton momentum bin has been evaluated with MC simulation. The radiative K + → e + νγ (IB) process, which is responsible for the loss of about 5% of the K e2 acceptance by increasing the reconstructed M 2 miss (e), is taken into account following [20], with higher order corrections according to [22,23].   Figure 4: (a) Reconstructed squared missing mass M 2 miss (µ) distribution of the K µ2 candidates compared with the sum of normalised estimated signal and background components. The deficit of reconstructed MC events in the region of the K + → π + π 0 peak is due to the limited precision of the beam simulation, and is mostly outside the signal region. (b) Lepton momentum distributions of the K µ2 candidates and the beam halo background (the latter is scaled for visibility). The acceptance correction is strongly influenced by bremsstrahlung suffered by the positron in the material upstream of the spectrometer magnet (Kevlar window, helium, DCHs). This results in an almost momentum-independent loss of K e2 acceptance of about 6%, mainly by increasing the reconstructed M 2 miss (e). The relevant material thickness has been measured by studying the spectra and rates of bremsstrahlung photons produced by low intensity 25 GeV/c and 40 GeV/c electron and positron beams steered into the DCH acceptance, using special data samples collected in the same setup by the NA48/2 experiment in 2004 and 2006. Using these measurements, the material thickness during the 2007 run has been estimated to be (1.56 ± 0.03)%X 0 . The quoted uncertainty is dominated by the limited knowledge of helium purity in the spectrometer tank; its measured purity of (92 ± 4)% corresponds to a thickness of (0.26 ± 0.03)%X 0 . This translates into a systematic uncertainty on R K .
The acceptance correction A(K µ2 )/A(K e2 ) in lepton momentum bins is presented in Fig. 5a. The corrections evaluated without internal (IB) and external (EB) bremsstrahlung radiation are also presented to illustrate the magnitudes of the corresponding effects. The correction is enhanced at low lepton momentum because the radial distributions of positrons from K e2 decays in the DCH planes are wider than those of muons from K µ2 decays, and low momentum leptons are not fully contained within the geometric acceptance due to the limited transverse sizes of the DCHs.
The track reconstruction inefficiency due to interactions in spectrometer material is included into the acceptance correction. Simulation of the positron track reconstruction inefficiency (which is ∼ 10 −3 in the analysis track momentum range) has been validated with a sample of K + → π + π 0 D decays. The muon track reconstruction inefficiency evaluated with MC simulation is ∼ 2 × 10 −4 . Systematic effects due to imperfect simulation of the reconstruction efficiency are negligible.
Apart from helium purity, the main sources of systematic uncertainty of the acceptance correction are the limited knowledge of beam profile and divergence, and the simulation of soft radiative photons. A separate uncertainty has been assigned to account for the finite precision of the DCH alignment.

Lepton identification efficiencies
The E/p ratio provides powerful particle identification criteria. The momentum-dependent positron identification window (E/p) min < E/p < 1.1 includes more than 99% of the K e2 events, while suppressing muons by a factor of 1/P µe ∼ 10 6 . The requirement E/p < 0.85 leads to a negligible inefficiency of the muon identification.
A pure sample of 4 × 10 7 positrons, selected kinematically from K + → π 0 e + ν (charged K e3 ) decays collected with the K e2 trigger concurrently with the main K ℓ2 data set, is used to calibrate the energy response of each LKr cell and to study f e with respect to local position and time stability. However, the momentum range of the positrons from charged K e3 decays is kinematically limited, preventing a sufficiently precise measurement of f e above 50 GeV/c. Therefore a dedicated data sample was recorded in a special 15 hour long run with a broad momentum band K 0 L beam. Electrons and positrons from the 4 × 10 6 collected K 0 L → π ± e ∓ ν (neutral K e3 ) decays allow the determination of f e in the whole analysis momentum range.
The measurements of f e have been performed in bins of lepton momentum; a finer binning is used inside the lowest of the 10 standard bins to improve the determination of local inefficiencies, which peak at low momentum. Separate measurements have been performed for several identified groups of LKr cells with higher local inefficiencies. Efficiency measurements with the charged and neutral K e3 decays agree to better than 0.1%. Fig. 5b shows the measurements of 1 − f e in momentum bins used to evaluate corrections to R K , obtained as the weighted mean of charged and neutral K e3 measurements for momenta up to 50 GeV/c, and as neutral kaon measurements for higher momenta. The inefficiency averaged over the K e2 sample is 1 − f e = (0.73 ± 0.05)%, where the uncertainty takes into account the statistical precision and the small differences between charged and neutral kaon results.

Trigger and readout efficiencies
The efficiency of the Q 1 trigger condition has been measured using K µ2 events triggered with a control LKr signal. The inefficiency integrated over the K µ2 sample is (1.4 ± 0.1)%. As a consequence of its geometric uniformity and the similarity of the K e2 and K µ2 distributions over the HOD plane, it nearly cancels between the K e2 and K µ2 samples, and the residual systematic bias is negligible. The inefficiency of the 1-track condition also largely cancels in the ratio R K , but is anyway negligible.
Thus the trigger efficiency correction ǫ(K µ2 )/ǫ(K e2 ) is determined by the efficiency ǫ(E LKr ) of the LKr energy deposit trigger signal E LKr > 10 GeV entering the K e2 trigger chain only. The inefficiency 1 − ǫ(E LKr ) is only significant in the lowest lepton momentum bin of (13, 20) GeV/c, which is close to the trigger energy threshold and is thus affected by the online energy resolution. A sample of events triggered with a control Q 1 signal passing all K e2 selection criteria except the M 2 miss (e) constraint, therefore dominated by K e3 events with two lost photons, has been used to measure 1 − ǫ(E LKr ) in the lowest momentum bin to be (0.41 ± 0.05 stat. )%. Corrected for the difference of positron distributions in the LKr plane between the K e2 sample and the control sample, it translates into 1 − ǫ(E LKr ) = (0.61 ± 0.20)% for the K e2 sample. The correction and its uncertainty are significant due to the presence of several locally inefficient regions. The resulting uncertainty on R K is negligible.
Energetic photons not reconstructed in the LKr may initiate showers by interacting in the DCHs or the beam pipe, causing the DCH hit multiplicities to exceed the limits allowed by the 1-track trigger condition. Among the backgrounds, only the K + → e + νγ (SD + ) receives a non-negligible correction due to the 1-track inefficiency. The inefficiency for K + → π 0 e + ν events with two lost photons has been measured to vary in the range from 0.1 to 0.3 depending on  Figure 6: Measurements of R K in lepton momentum bins with their uncorrelated statistical uncertainties and the partially correlated total uncertainties. The average R K and its total uncertainty are indicated by a band.

NA62
track momentum. The extrapolation of this result to K + → e + νγ (SD + ) with one lost photon relies on simulation. The corresponding uncertainty has been propagated into R K . The global LKr readout inefficiency, affecting the K e2 reconstruction only, has been measured using an independent readout system to be 1 − f LKr = (0.20 ± 0.03)%, stable in time. This measurement has been cross-checked, with limited precision, by a study of the LKr response in a sample of π 0 DD → 4e ± decays reconstructed from spectrometer information only.

Result and discussion
A χ 2 fit to the measurements of R K in the 10 lepton momentum bins has been performed, taking into account the bin-to-bin correlations between the systematic errors. To validate the assigned systematic uncertainties, extensive stability checks have been performed in bins of kinematic variables and by varying selection criteria and analysis procedures. The fit result is R K = (2.487 ± 0.011 stat. ± 0.007 syst. ) × 10 −5 = (2.487 ± 0.013) × 10 −5 , with χ 2 /ndf = 3.6/9. The individual measurements with their statistical and total uncertainties, and the combined result are displayed in Fig. 6. The uncertainties of the combined result are summarised in Table 2. This is the most precise R K measurement to date. It is consistent with the KLOE measurement [10] and the SM expectation R SM K = (2.477 ± 0.001) × 10 −5 , and can be used to constrain multi-Higgs [2] and fourth generation [5] new physics scenarios. The experimental accuracy is still an order of magnitude behind the SM accuracy, which motivates further precision measurements of R K .