Search for a feebly interacting particle X in the decay K+ → π+X

A search for the K+ → π+X decay, where X is a long-lived feebly interacting particle, is performed through an interpretation of the K+ → π+νν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\pi}^{+}\nu \overline{\nu} $$\end{document} analysis of data collected in 2017 by the NA62 experiment at CERN. Two ranges of X masses, 0–110 MeV/c2 and 154–260 MeV/c2, and lifetimes above 100 ps are considered. The limits set on the branching ratio, BR(K+ → π+X), are competitive with previously reported searches in the first mass range, and improve on current limits in the second mass range by more than an order of magnitude.

The NA62 experiment, beamline and detector are described in detail in [17] and a schematic of the detector is shown in figure 1. A right-handed coordinate system, (x, y, z), is defined with the target at the origin and the beam travelling towards positive z, the y axis is vertical (positive up) and the x-axis is horizontal (positive left). A 400 GeV/c proton beam extracted from the CERN Super Proton Synchrotron (SPS) impinges on a beryllium target creating a 75 GeV/c secondary hadron beam with a 1% rms momentum spread and a composition of 70% pions, 23% protons and 6% kaons. Kaons (K + ) are positively tagged with 70 ps timing precision by the KTAG detector, a differential Cherenkov counter filled with nitrogen gas. The momentum and position of the K + are measured by the Giga-Tracker (GTK), a spectrometer formed of three silicon pixel tracker stations and a set of four dipole magnets. GTK measurements have momentum, direction and time resolutions of 0.15 GeV/c, 16 µrad and 100 ps, respectively. After traversing the GTK magnets, a magnetized scraper used to sweep away muons, and a bending magnet (B), the beam at the FV entrance has a rectangular profile of 52 × 24 mm 2 and a divergence of 0.11 mrad. The experiment is designed to study K + decays occurring in the 60 m fiducial volume (FV) starting 2.6 m downstream of GTK3 and housed inside a 117 m long vacuum tank, containing a magnetic spectrometer, and ending at the ring imaging Cherenkov counter (RICH). Momentum and position measurements for charged particles produced in K + decays in the FV are provided by the magnetic spectrometer composed of four STRAW tracking stations, two on either side of a dipole magnet (M). This spectrometer provides JHEP03(2021)058 a momentum measurement with resolution σ p /p of 0.3-0.4%. The RICH is filled with neon gas at atmospheric pressure and provides particle identification for charged particles, and a time measurement with a precision better than 100 ps. Two adjacent scintillator hodoscopes (CHOD and NA48-CHOD), provide time measurements for charged particles with a 200 ps resolution.
A system of veto detectors is key to the experiment. Interactions of beam particles in GTK3 are detected by the charged particle anti-counter (CHANTI), formed of six stations of scintillator bar counters. Downstream, a photon veto system is used to reject the K + → π + π 0 background. This analysis selects π + particles with momenta in the range 15-35 GeV/c. This means that a π 0 from the K + → π + π 0 background has momentum of at least 40 GeV/c and the subsequent π 0 → γγ decay, BR = 98.8%, produces two energetic photons which can be detected with high efficiency. There are twelve large angle veto (LAV) stations positioned to ensure hermetic coverage for photon emission angles of 8.5-50 mrad. The liquid krypton calorimeter (LKr) provides coverage for 1-8.5 mrad. The small angle photon veto (SAV) covers angles below 1 mrad using two sampling calorimeters of shashlyk design (IRC and SAC).
Downstream of the LKr are two hadronic sampling calorimeters (MUV1 and MUV2). Together with the LKr, these provide particle identification information through the pattern of energy deposition. Electrons/positrons produce electromagnetic showers that are well-contained in the LKr, which has a depth of 27 radiation lengths. Pions may pass through the LKr without losing all of their energy and can produce a hadronic shower in MUV1 and MUV2. In contrast, muons are minimum ionising particles in the calorimetric system. The MUV3 detector is positioned downstream of a 0.8 m iron absorber and consists of a plane of scintillator tiles. It provides measurements of muons with 400 ps time resolution.
A two-level trigger system is employed with a hardware level 0 (L0) selection followed by a level 1 (L1) decision made by software algorithms. The primary trigger stream of the experiment is dedicated to collection of K + → π + νν events and uses information from the CHOD, RICH, LKr, MUV3 at L0 [18] and KTAG, LAV, STRAW at L1 [16]. The NA48-CHOD also provides a 99% efficient minimum bias trigger, used for collection of K + → π + π 0 events that are used for normalisation. The data sample collected in 2017 for the study of the K + → π + νν decay is used for this analysis.

Signal selection
The observable for the K + → π + X search is the reconstructed squared missing mass where P K and P π are the K + and π + 4-momenta, derived from the measured 3-momenta of the GTK and STRAW tracks under the K + and π + mass hypotheses, respectively. The event selection is identical to that used for the K + → π + νν measurement [16] and is summarised below.

JHEP03(2021)058
Candidate events must have fewer than three reconstructed STRAW tracks with no negatively charged tracks. Only one track can fulfil additional criteria to become a π + candidate but, for example, an additional out-of-time halo muon track may exist. The time assigned to the π + candidate is calculated using the mean times measured in the STRAW, NA48-CHOD and RICH weighted by their respective measured resolutions. A π + candidate track must have momentum in the range 15-35 GeV/c and be within the sensitive regions of the downstream detectors (RICH, CHODs, LKr and MUV1,2,3) with geometrically and time-coincident associated signals recorded in the CHODs, LKr and RICH.
The candidate track must be consistent with the π + hypothesis for the RICH reconstructed mass and likelihood. The candidate must also satisfy a multivariate classifier based on calorimetric information. On average, for 15-35 GeV/c tracks, the two methods achieve π + identification efficiencies of 82% and 78%, with probabilities of misidentification of µ + as π + of 2.3 × 10 −3 and 6.3 × 10 −6 , respectively. A MUV3 veto condition rejects events with signals geometrically associated with the track within a time window of 7 ns. No signals are allowed in any LAV station (or SAV) within 3 (7) ns of the π + time. No LKr clusters are allowed beyond a distance of 100 mm from the π + impact point within cluster-energy dependent time windows of 10 to 100 ns. The STRAW, CHODs and LKr are used to veto events with additional activity, including tracks produced by photon interactions upstream of the calorimeters and partially reconstructed multi-track decays. Overall rejection of π 0 → γγ decays is achieved with an inefficiency of 1.3 × 10 −8 .
A K + is tagged upstream by the KTAG if Cherenkov photons are detected within 2 ns of the π + track time in at least five out of its total of eight sectors. A GTK track is associated with the K + if its time is within 0.6 ns of the KTAG time and the closest distance of approach (CDA) to the π + track is less than 4 mm. The K + /π + matching is based on time coincidence and spatial information and has an efficiency of 75%. The average probability for wrong (accidental) association with pileup GTK tracks is 1.3% (3.5%) when the K + track is (is not) correctly reconstructed.
Upstream backgrounds arise from a combination of early K + decays (upstream of the FV), beam particle interactions in the GTK stations, additional GTK tracks, and largeangle π + scattering in the first STRAW station. To minimise such backgrounds, the vertex formed between the selected K + and π + tracks must be inside the FV with no additional activity in the CHANTI within 3 ns of the π + candidate time. Additionally, a 'box cut' is applied requiring that the projection of the π + candidate track back to the final collimator (COL) is outside the area defined by |x| < 100 mm and |y| < 500 mm.
The m 2 miss observable is used to discriminate between a peaking two-body K + → π + X signal and backgrounds. Two signal regions are defined, called region 1 and region 2, to minimise large backgrounds from K + → π + π 0 , K + → µ + ν µ and K + → π + π + π − decays. The reconstructed m 2 miss for region 1 must be between 0 and 0.01 GeV 2 /c 4 and that for region 2 between 0.026 and 0.068 GeV 2 /c 4 . Additional momentum-dependent constraints supplement the definition of the signal regions using alternative squared missing mass variables, constructed either by replacing the GTK measurement of the beam 3-momentum with the average beam momentum and direction, or the STRAW 3-momentum measurement with one measured by the RICH under the π + mass hypothesis. These requirements reject events with incorrect reconstruction of m 2 miss due to momenta mismeasurements and improve background rejection, but decrease acceptance at the boundaries of the signal regions.

Signal and background models
Geant4-based [19] Monte Carlo simulations of K + → π + X decays are performed with the assumption that X is stable, for X masses covering the search range at 1.4 MeV/c 2 intervals. This value corresponds to intervals of the squared missing mass that are always smaller than its resolution. These simulations include decay kinematics, interactions in material, and the responses of the detectors. In this study, a scan is performed searching for K + → π + X signals with X mass, m X , in the ranges 0-110 MeV/c 2 and 154-260 MeV/c 2 . These m X ranges extend beyond the K + → π + νν signal regions because of the resolution of the reconstructed m 2 miss observable. The resolution of m 2 miss , σ m 2 miss , as a function of simulated m X is shown in figure 2 (left). The reconstructed m 2 miss resolution for a control sample of selected K + → π + π 0 events is found to be 4% better in simulations than in data. The resolution derived from simulations is therefore corrected by increasing it by 4% and a systematic uncertainty of 10% is assigned to the m 2 miss resolution. The acceptance for the selection described in section 3, as obtained using simulations, is displayed in figure 2 (centre). The single event sensitivity, BR SES , defined as the branching ratio corresponding to the observation of one signal event, is calculated by following the procedure adopted for the K + → π + νν analysis using the K + → π + π 0 decay for normalisation [16]; the resulting values are shown in figure 2 (right). The uncertainty of BR SES is 10% and is mainly systematic. The largest contributions to this uncertainty are associated with the trigger efficiency, signal and normalisation reconstruction and selection efficiencies [16], and differences between K + → π + νν and K + → π + X kinematics.
The sensitivity for low X masses is limited by the K + → π + νν signal region definition m 2 miss > 0, which is necessary to suppress the background from K + → µ + ν µ decays. This effect reduces the acceptance by half for m X = 0, and equivalently at each signal region boundary (figure 2 centre).
The acceptance for X with finite lifetime, τ X and m X = 0, is computed under the following assumptions: X decays only to visible SM particles; decays upstream of MUV3 are detected with 100% efficiency. The efficiency is 99.9%, and the uncertainty in this quantity  is included in the systematic uncertainty. The acceptance for a set of τ X values is calculated by weighting simulated events by the probability that X does not decay upstream of MUV3. The acceptance increases as a function of lifetime reaching a plateau for τ X > 10 ns. For m X < 20 MeV/c 2 , losses of acceptance at lower lifetimes are compensated by the increase in the Lorentz factor.
The background contributions for the K + → π + X search are the same as for the K + → π + νν analysis with the addition of the K + → π + νν decay itself, which becomes the dominant background. The SM description of the K + → π + νν decay is assumed. The total expected background and the reconstructed m 2 miss distributions for each component are obtained from auxiliary measurements, as described in [16]. The resulting numbers of background events in the signal regions are summarised in table 1. The contributions from kaon decays other than K + → π + νν are grouped in the row other K + decays, and their distribution in m 2 miss is known with good accuracy. For the upstream background, an additional systematic uncertainty of 30% is included, to account for the uncertainty in the estimation of its distribution in m 2 miss resulting from the limited size of the control sample used for the auxiliary measurements. The total background is described, as a function of the reconstructed m 2 miss , by fitting polynomial functions to the expectations in signal regions 1 and 2, as shown in figure 3.

Statistical analysis
The search procedure involves a fully frequentist hypothesis-test using a shape analysis with observable m 2 miss and an unbinned profile likelihood ratio test statistic. Each X mass JHEP03(2021)058 Observed events 0 2 Table 1. Summary of the predicted numbers of background events in the signal regions and the observed events. The statistical uncertainty for SM K + → π + νν is negligible and the external uncertainty arises from the uncertainty of the SM K + → π + νν branching ratio.
hypothesis is treated independently. The parameter of interest, BR(K + → π + X), is related to the expected number of signal events, n S , by BR(K + → π + X) = n S × BR SES .
The likelihood function has the form: where n is the observed number of events, n tot = n B +n S and n B is the expected number of background events; f B (m 2 miss ) is a polynomial function of m 2 miss normalised to unity which describes the total background in the signal region relevant for a certain mass hypothesis m X ; and f S (m 2 miss |µ X , σ X ) is the Gaussian function, normalised to unity, with parameters µ X and σ X obtained from a fit to the distribution of the reconstructed simulated events. Index j runs over the n observed events and their reconstructed m 2 miss are denoted m 2 miss , j . The N nuis nuisance parameters considered, p i nuis , are n B , BR SES , µ X , σ X , and are estimated by auxiliary measurements. These estimations, p i meas =n B ,BR SES ,μ X ,σ X , are treated as global observables [20]. The constraint terms, C i (p i meas |p i nuis ), are the probability density functions describing the distribution of each nuisance parameter. The constraint term for n B is a Poisson distribution with mean value (n B /σ B ) 2 wheren B and σ B are the central value and uncertainty of the background expectation [21]. The constraint term for BR SES is a log-normal function with parameters corresponding to a relative uncertainty of 10%. A Gaussian constraint term is used for µ X , with relative uncertainty depending on the mass hypothesis m X . A log-normal constraint term is used for σ X , with the mean corresponding to the estimated value after the 4% correction (described in section 4), and relative uncertainty of 10%. The normalised polynomial functions, describing the background distribution in m 2 miss , are considered to be known exactly. For each mass hypothesis the fully frequentist test is performed according to the CLs method [22] to exclude the presence of a signal with 90% confidence level (CL) for the observed data. A cross-check was performed, using single bin counting experiments in windows of width equal to four times σ m 2 miss around each mass hypothesis, with a hybrid frequentist treatment using a log-likelihood ratio test statistic. A comparable expected sensitivity was obtained.

Results and discussion
Two candidate K + → π + X events are observed [16] at reconstructed m miss values of 196 and 252 MeV/c 2 . Upper limits are established on BR(K + → π + X) at 90% CL for each X mass hypothesis: expected and observed upper limits, assuming stable or invisibly decaying X, are displayed in figure 4 (left). The observed upper limits are compared to the previous results from the E787/E949 experiments [23] in figure 4 (right), as a function of m X and for different values of τ X , assuming X decays to visible SM particles. The strongest limits of 5 × 10 −11 are obtained at large X masses (160-250 MeV/c 2 ) and long X lifetimes (> 5 ns). Under the assumption of stable or invisibly decaying X these upper limits improve by a factor of O(10) in signal region 2, and are competitive in region 1. For unstable X, assuming decays only to visible SM particles, the same pattern holds in general. However, in region 1 the limits obtained improve across an increasingly large range of mass hypotheses as the assumed lifetime becomes shorter. Despite differences in experimental set-up between E787/E949 (stopped K + decay-at-rest) and NA62 (highly boosted K + decay-in-flight), the two results exhibit similar dependence on τ X . This is because the ratios of the Lorentz factor for the X particle to the decay length are similar in the two experiments.
In a Higgs portal model with a dark sector scalar mixing with the Higgs boson, X production and decay are driven by the mixing parameter sin 2 θ (model BC4 [1,24]). This gives rise to K + → π + X decays with branching ratio proportional to sin 2 θ. The constraints derived on sin 2 θ from this search, alongside results from other studies, are shown in figure 5.
In a scenario where X is an ALP with couplings proportional to SM Yukawa couplings (model BC10 [1,14]) the K + → π + X decay occurs with a branching ratio proportional to the square of the coupling constant g Y . The constraints on g Y derived from this and other searches are shown in figure 6.  Figure 5. Excluded regions of the parameter space (m S , sin 2 θ) for a dark scalar, S, of the BC4 model [1] decaying only (left) to visible SM particles as in the BC4 model and (right) invisibly. The exclusion bound from the present search for the decay K + → π + S is labelled as "K + → π + + inv." and is shaded in red. In the π 0 mass region the independent NA62 search for π 0 → invisible decays [25] provides constraints, shown in purple. Other bounds, shown in grey, are derived from the experiments E949 [23], CHARM [24], NA48/2 [26], LHCb [27,28] and Belle [29].
If X decays only to invisible particles, such as dark matter, bounds on the coupling parameter (sin 2 θ or g Y for the scalar and ALP models, respectively) are directly derived from its relationship with the branching ratio, with results shown in the right-hand panels of figures 5 and 6. If X decays only to visible SM particles, τ X is inversely proportional to the coupling parameters [14,24], limiting the reach of this analysis for large coupling because of lower acceptance for shorter lifetimes. The X → e + e − decays dominate the visible decay width up to the di-muon threshold beyond which an additional channel opens and τ X decreases, limiting the sensitivity of this search. The model-dependent relationship between the lifetime and coupling therefore determines the shape of the exclusion regions shown in the left-hand panels of figures 5 and 6.

Conclusions
A search for the K + → π + X decay, where X is a long-lived feebly interacting particle, is performed through an interpretation of the K + → π + νν analysis of data collected in 2017 by the NA62 experiment at CERN. Two candidate K + → π + X events are observed, in agreement with the expected background. Upper limits on BR(K + → π + X) are established at 90% CL, with the strongest limits of 5 × 10 −11 at large X masses (160-250 MeV/c 2 ) and long X lifetimes (> 5 ns), improving on current results by up to a factor of O(10). An interpretation of these results to constrain BSM models is presented in scenarios where X is a dark scalar mixing with the Higgs boson or is an ALP with couplings to fermions.  Figure 6. Excluded regions of the parameter space (m a , g Y ) for an ALP, a, of the BC10 model [1] decaying only (left) to visible particles and (right) invisibly. The exclusion bound from the present search for the decay K + → π + a is labelled as "K + → π + + inv." and is shaded in red. In the π 0 mass region the independent NA62 search for π 0 → invisible decays [25] provides constraints, shown in purple. Other bounds, shown in grey, are derived from the experiments E949 [23], K µ2 [30], CLEO [31], CHARM [32], KTeV [33], LHCb [27,28] and from Big Bang nucleosynthesis (BBN) [1].