An investigation of the very rare K + → π + ν ν ¯ decay
EEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERN-EP-2020-132July 14, 2020
An investigation of the very rare K + → π + ν ¯ ν decay The NA62 Collaboration
Abstract
The NA62 experiment reports an investigation of the K + → π + ν ¯ ν mode from a sample of K + decays collected in 2017 at the CERN SPS. The experiment has achieved a single eventsensitivity of (0 . ± . × − , corresponding to 2.2 events assuming the StandardModel branching ratio of (8 . ± . × − . Two signal candidates are observed with anexpected background of 1.5 events. Combined with the result of a similar analysis conductedby NA62 on a smaller data set recorded in 2016, the collaboration now reports an upperlimit of 1 . × − for the K + → π + ν ¯ ν branching ratio at 90% CL. This, together with thecorresponding 68% CL measurement of (0 . +0 . − . ) × − , are currently the most preciseresults worldwide, and are able to constrain some New Physics models that predict largeenhancements still allowed by previous measurements. Submitted to JHEP a r X i v : . [ h e p - e x ] J u l he NA62 Collaboration ∗ Universit´e Catholique de Louvain, Louvain-La-Neuve, Belgium
E. Cortina Gil, A. Kleimenova, E. Minucci , , S. Padolski , P. Petrov, A. Shaikhiev ,R. Volpe TRIUMF, Vancouver, British Columbia, Canada
T. Numao, B. Velghe
University of British Columbia, Vancouver, British Columbia, Canada
D. Bryman , J. Fu Charles University, Prague, Czech Republic
T. Husek , J. Jerhot , K. Kampf, M. Zamkovsky Institut f¨ur Physik and PRISMA Cluster of excellence, Universit¨at Mainz, Mainz,Germany
R. Aliberti , G. Khoriauli , J. Kunze, D. Lomidze , R. Marchevski ∗ , , L. Peruzzo,M. Vormstein, R. Wanke Dipartimento di Fisica e Scienze della Terra dell’Universit`a e INFN, Sezione diFerrara, Ferrara, Italy
P. Dalpiaz, M. Fiorini, I. Neri, A. Norton, F. Petrucci, H. Wahl
INFN, Sezione di Ferrara, Ferrara, Italy
A. Cotta Ramusino, A. Gianoli
Dipartimento di Fisica e Astronomia dell’Universit`a e INFN, Sezione di Firenze,Sesto Fiorentino, Italy
E. Iacopini, G. Latino, M. Lenti, A. Parenti
INFN, Sezione di Firenze, Sesto Fiorentino, Italy
A. Bizzeti , F. Bucci Laboratori Nazionali di Frascati, Frascati, Italy
A. Antonelli, G. Georgiev , V. Kozhuharov , G. Lanfranchi, S. Martellotti, M. Moulson,T. Spadaro Dipartimento di Fisica “Ettore Pancini” e INFN, Sezione di Napoli, Napoli, Italy
F. Ambrosino, T. Capussela, M. Corvino, D. Di Filippo, P. Massarotti, M. Mirra,M. Napolitano, G. Saracino
Dipartimento di Fisica e Geologia dell’Universit`a e INFN, Sezione di Perugia,Perugia, Italy
G. Anzivino, F. Brizioli, E. Imbergamo, R. Lollini, R. Piandani , C. Santoni INFN, Sezione di Perugia, Perugia, Italy
M. Barbanera , P. Cenci, B. Checcucci, P. Lubrano, M. Lupi , M. Pepe, M. Piccini Dipartimento di Fisica dell’Universit`a e INFN, Sezione di Pisa, Pisa, Italy
F. Costantini, L. Di Lella, N. Doble, M. Giorgi, S. Giudici, G. Lamanna, E. Lari, E. Pedreschi,M. Sozzi
INFN, Sezione di Pisa, Pisa, Italy
C. Cerri, R. Fantechi, L. Pontisso, F. Spinella 2 cuola Normale Superiore e INFN, Sezione di Pisa, Pisa, Italy
I. Mannelli
Dipartimento di Fisica, Sapienza Universit`a di Roma e INFN, Sezione di Roma I,Roma, Italy
G. D’Agostini, M. Raggi
INFN, Sezione di Roma I, Roma, Italy
A. Biagioni, E. Leonardi, A. Lonardo, P. Valente, P. Vicini
INFN, Sezione di Roma Tor Vergata, Roma, Italy
R. Ammendola, V. Bonaiuto , A. Fucci, A. Salamon, F. Sargeni Dipartimento di Fisica dell’Universit`a e INFN, Sezione di Torino, Torino, Italy
R. Arcidiacono , B. Bloch-Devaux, M. Boretto , E. Menichetti, E. Migliore, D. Soldi INFN, Sezione di Torino, Torino, Italy
C. Biino, A. Filippi, F. Marchetto
Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı, San Luis Potos´ı,Mexico
J. Engelfried, N. Estrada-Tristan Horia Hulubei National Institute of Physics for R&D in Physics and NuclearEngineering, Bucharest-Magurele, Romania
A. M. Bragadireanu, S. A. Ghinescu, O. E. Hutanu
Joint Institute for Nuclear Research, Dubna, Russia
A. Baeva, D. Baigarashev, D. Emelyanov, T. Enik, V. Falaleev, V. Kekelidze, A. Korotkova,L. Litov , D. Madigozhin, M. Misheva , N. Molokanova, S. Movchan, I. Polenkevich,Yu. Potrebenikov, S. Shkarovskiy, A. Zinchenko † Institute for Nuclear Research of the Russian Academy of Sciences, Moscow,Russia
S. Fedotov, E. Gushchin, A. Khotyantsev, Y. Kudenko , V. Kurochka, M. Medvedeva,A. Mefodev Institute for High Energy Physics - State Research Center of Russian Federation,Protvino, Russia
S. Kholodenko, V. Kurshetsov, V. Obraztsov, A. Ostankov † , V. Semenov † , V. Sugonyaev,O. Yushchenko Faculty of Mathematics, Physics and Informatics, Comenius University,Bratislava, Slovakia
L. Bician , T. Blazek, V. Cerny, Z. Kucerova CERN, European Organization for Nuclear Research, Geneva, Switzerland
J. Bernhard, A. Ceccucci, H. Danielsson, N. De Simone , F. Duval, B. D¨obrich, L. Federici,E. Gamberini, L. Gatignon, R. Guida, F. Hahn † , E. B. Holzer, B. Jenninger, M. Koval ,P. Laycock , G. Lehmann Miotto, P. Lichard, A. Mapelli, K. Massri, M. Noy, V. Palladino ,M. Perrin-Terrin , , J. Pinzino , , V. Ryjov, S. Schuchmann , S. Venditti University of Birmingham, Birmingham, United Kingdom
T. Bache, M. B. Brunetti , V. Duk , V. Fascianelli , J. R. Fry, F. Gonnella,E. Goudzovski, L. Iacobuzio, C. Lazzeroni, N. Lurkin , F. Newson, C. Parkinson ,A. Romano, A. Sergi, A. Sturgess, J. Swallow 3 niversity of Bristol, Bristol, United Kingdom H. Heath, R. Page, S. Trilov
University of Glasgow, Glasgow, United Kingdom
B. Angelucci, D. Britton, C. Graham, D. Protopopescu
University of Lancaster, Lancaster, United Kingdom
J. Carmignani, J. B. Dainton, R. W. L. Jones, G. Ruggiero ∗ University of Liverpool, Liverpool, United Kingdom
L. Fulton, D. Hutchcroft, E. Maurice , B. Wrona George Mason University, Fairfax, Virginia, USA
A. Conovaloff, P. Cooper, D. Coward , P. Rubin ∗ Corresponding authors: G. Ruggiero, R. Marchevski,email:[email protected], [email protected] † Deceased Present address: Laboratori Nazionali di Frascati, I-00044 Frascati, Italy Also at CERN, European Organization for Nuclear Research, CH-1211 Geneva 23, Switzerland Present address: Brookhaven National Laboratory, Upton, NY 11973, USA Also at Institute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia Present address: Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48, Bratislava,Slovakia Also at TRIUMF, Vancouver, British Columbia, V6T 2A3, Canada Present address: UCLA Physics and Biology in Medicine, Los Angeles, CA 90095, USA Present address: IFIC, Universitat de Val`encia - CSIC, E-46071 Val`encia, Spain Present address: Universit´e Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium Present address: Institut f¨ur Kernphysik and Helmholtz Institute Mainz, Universit¨at Mainz, Mainz, D-55099,Germany Present address: Universit¨at W¨urzburg, D-97070 W¨urzburg, Germany Present address: Universit¨at Hamburg, D-20146 Hamburg, Germany Present address: CERN, European Organization for Nuclear Research, CH-1211 Geneva 23, Switzerland Also at Dipartimento di Fisica, Universit`a di Modena e Reggio Emilia, I-41125 Modena, Italy Also at Faculty of Physics, University of Sofia, BG-1164 Sofia, Bulgaria Present address: Institut f¨ur Experimentelle Teilchenphysik (KIT), D-76131 Karlsruhe, Germany Present address: INFN, Sezione di Pisa, I-56100 Pisa, Italy Present address: Institut am Fachbereich Informatik und Mathematik, Goethe Universit¨at, D-60323 Frankfurtam Main, Germany Also at Department of Industrial Engineering, University of Roma Tor Vergata, I-00173 Roma, Italy Also at Department of Electronic Engineering, University of Roma Tor Vergata, I-00173 Roma, Italy Also at Universit`a degli Studi del Piemonte Orientale, I-13100 Vercelli, Italy Also at Universidad de Guanajuato, Guanajuato, Mexico Present address: Institute of Nuclear Research and Nuclear Energy of Bulgarian Academy of Science (INRNE-BAS), BG-1784 Sofia, Bulgaria Also at National Research Nuclear University (MEPhI), 115409 Moscow and Moscow Institute of Physics andTechnology, 141701 Moscow region, Moscow, Russia Present address: DESY, D-15738 Zeuthen, Germany Present address: Charles University, 116 36 Prague 1, Czech Republic Present address: Physics Department, Imperial College London, London, SW7 2BW, UK Present address: Centre de Physique des Particules de Marseille, Universit´e Aix Marseille, CNRS/IN2P3,F-13288, Marseille, France Also at Universit´e Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium Present address: Department of Physics, University of Toronto, Toronto, Ontario, M5S 1A7, Canada Also at INFN, Sezione di Pisa, I-56100 Pisa, Italy Present address: Institut f¨ur Physik and PRISMA Cluster of excellence, Universit¨at Mainz, D-55099 Mainz,Germany Present address: Department of Physics, University of Warwick, Coventry, CV4 7AL, UK Present address: INFN, Sezione di Perugia, I-06100 Perugia, Italy Present address: Dipartimento di Psicologia, Universit`a di Roma La Sapienza, I-00185 Roma, Italy Present address: Laboratoire Leprince Ringuet, F-91120 Palaiseau, France Also at SLAC National Accelerator Laboratory, Stanford University, Menlo Park, CA 94025, USA ontents K + → π + π decays . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Signal and normalization efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . 206.2.1 Monte Carlo efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2.2 K + → µ + ν branching ratio measurement . . . . . . . . . . . . . . . . . 246.2.3 Random veto efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 Trigger efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3.1 PNN L0 trigger efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3.2 PNN L1 trigger efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.3.3 Trigger efficiency and SES . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4
SES result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 K + decay background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.1.1 K + → π + π ( γ ) decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.1.2 K + → µ + ν decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.1.3 K + → π + π + π − decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.1.4 K + → π + π − e + ν decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.1.5 Other K + decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.2 Upstream background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2.1 Background sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2.2 Upstream background evaluation . . . . . . . . . . . . . . . . . . . . . . . 437.3 Additional check and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Introduction
The K + → π + ν ¯ ν decay is a flavour-changing Neutral-Current process that proceeds throughelectroweak box and penguin diagrams in the Standard Model (SM), allowing an exploration ofits flavour structure thanks to unique theoretical cleanliness [1]. A quadratic GIM mechanismand the transition of the top quark to the down quark make this process extremely rare. TheSM prediction for the K + → π + ν ¯ ν branching ratio (BR) can be written as [2]:BR( K + → π + ν ¯ ν ) = κ + (1 + ∆ EM ) (cid:34)(cid:18) Imλ t λ X ( x t ) (cid:19) + (cid:18) Reλ c λ P c ( X ) + Reλ t λ X ( x t ) (cid:19) (cid:35) , (1)where ∆ EM = − .
003 accounts for the electromagnetic radiative corrections; x t = m t /M W ; λ = | V us | and λ i = V ∗ is V id ( i = c, t ) are combinations of Cabibbo-Kobayashi-Maskawa (CKM)matrix elements; X and P c ( X ) are the loop functions for the top and charm quark respectively;and κ + = (5 . ± . × − (cid:20) λ . (cid:21) (2)parameterizes hadronic matrix elements. It is worth noting that BR( K + → π + ν ¯ ν ) depends onthe sum of the square of the imaginary part of the top loop, which is CP-violating, and thesquare of the sum of the charm contribution and the real part of the top loop. Numerically,the branching ratio can be written as an explicit function of the CKM parameters, V cb and theangle γ , as follows :BR( K + → π + ν ¯ ν ) = (8 . ± . × − (cid:104) | V cb | . × − (cid:105) . (cid:104) γ . ◦ (cid:105) . , (3)where the numerical uncertainty is due to theoretical uncertainties in the NLO (NNLO) QCDcorrections to the top (charm) quark contribution [3, 4] and NLO electroweak corrections [5]. Theintrinsic theoretical accuracy is at the level of 3.6%. Uncertainties in the hadronic matrix elementlargely cancel when it is evaluated from the precisely-measured branching ratio of the K + → π e + ν decay, including isospin-breaking and non-perturbative effects calculated in detail [5, 6, 7].Using tree-level elements of the CKM matrix as external inputs [8], averaged over exclusive andinclusive determinations, namely | V cb | = (40 . ± . × − and γ = (73 . +6 . − . ) degrees, the SMprediction of the branching ratio is (8 . ± . × − [2]. The current precision of the CKMparameters dominates the BR uncertainty.The K + → π + ν ¯ ν decay is sensitive to currently proposed SM extensions and probes highermass scales than other rare meson decays. This arises because of the absence of tree-levelcontributions and the quadratic GIM suppression at loop level in the SM, which together leadto a very small BR. Moreover, the absence of long-distance contributions enables the accurateBR calculation. The largest deviations from SM predictions are expected in models with newsources of flavour violation, where constraints from B physics are weaker [9, 10]. Models withcurrents of defined chirality produce specific correlation patterns between the branching ratiosof K + → π + ν ¯ ν and K L → π ν ¯ ν decay modes, which are constrained by the value of the CP-violating parameter ε K [11, 12]. Present experimental constraints limit the range of variationwithin supersymmetric models [13, 14, 15, 16]. The K + → π + ν ¯ ν decay is also sensitive to someaspects of lepton flavour non-universality [17] and can constrain leptoquark models [18, 19] thataim to explain the measured CP-violating ratio ε (cid:48) /ε [8].The E787 and E949 experiments at the Brookhaven National Laboratory (BNL) studiedthe K + → π + ν ¯ ν decay using a kaon decay-at-rest technique, reaching an overall single event6ensitivity of about 0 . × − and measuring the BR to be (17 . +11 . − . ) × − [20, 21]. The NA62experiment at the CERN SPS will measure more precisely the BR of the K + → π + ν ¯ ν decayusing a decay-in-flight technique and data recorded from 2016 to 2018. The first NA62 resultwas based on the analysis of the data collected in 2016 and proved the feasibility of the techniqueto study the K + → π + ν ¯ ν decay [22]. In the following sections, NA62 reports the investigationof the K + → π + ν ¯ ν decay, based on data recorded in 2017, corresponding to about 30% of thetotal data set collected in 2016–18. The NA62 experiment is designed to reconstruct charged kaons and their daughter particles,when the kaons decay in flight inside a defined fiducial volume. The K + → π + ν ¯ ν decay presentstwo main challenges: the extremely low value of the SM signal branching ratio of order 10 − and the open kinematics of the final state, as neutrinos remain undetected. These challengesrequire both the production of a sufficient number of K + → π + ν ¯ ν decays, as can be achieved byexploiting the high-intensity 75 GeV /c secondary K + beam produced by the CERN SPS; andthe reduction of the contribution of the dominant K + decay modes by at least eleven orders ofmagnitude to bring the background to a level lower than the signal.The signature of the K + → π + ν ¯ ν decay is a single π + and missing energy. The squaredmissing mass, m = ( P K − P π + ) , where P K and P π + indicate the 4-momenta of the K + and π + , describes the kinematics of the one-track final state. In particular, the presence of twoneutrinos makes the signal broadly distributed over the m range, as illustrated in Figure 1.The dominant K + decay modes K + → µ + ν , K + → π + π and K + → π + π +(0) π − (0) havedifferent m distributions; it is therefore possible to define regions, either side of the K + → π + π peak, qualitatively indicated in Figure 1, where the search for the signal is performed, alsocalled signal regions .The K + → µ + ν , K + → π + π and K + → π + π +(0) π − (0) decays enter the signal regionsthrough radiative and/or resolution tails of the reconstructed m . The signal selection, basedon kinematics only, relies on the accurate measurement of the m quantity, i.e. of the K + and π + momenta and directions. In contrast, K + → π (cid:96) + ν or rarer decays, like K + → π + π − (cid:96) + ν ,span over the signal regions because of the presence of undetected neutrinos; however, thesebackground decay modes include a lepton in the final state and exhibit extra activity in theform of photons or charged particles. A particle identification system must therefore separate π + from µ + and e + . Photons and additional charged particles in final state must be vetoed asefficiently as possible.The above conditions translate into the following experimental requirements: • the detection of incident K + and outgoing π + signals with 100 ps time resolution to miti-gate the impact of the pile-up effect due to the high particle rates; • a low-mass K + and π + tracking system, which reconstructs precisely the kinematics tosuppress K + → π + π and K + → µ + ν backgrounds by at least three orders of magnitude,while keeping the background from hadronic interactions low; • a system of calorimeters and a Ring Imaging Cherenkov counter (RICH) to suppress decayswith positrons and muons by seven to eight orders of magnitude; • a set of electromagnetic calorimeters, to detect photons and reduce the number of K + → π + π decays by eight orders of magnitude; and • an experimental design which guarantees the geometric acceptance for negatively chargedparticles in at least two detectors. 7 .04 -
0 0.04 0.08 0.12] /c [GeV miss2 m - - - - - - -
10 1 ) m i ss / d m G ) ( d t o t G ( / ) g ( m n + mfi + K e n p + e fi + K m n p + mfi + K ) g ( p + pfi + K ) · ( nn + pfi + K - p + p + pfi + K p p + pfi + K e n + e - p + pfi + K Figure 1: Expected theoretical distributions of the m variable relevant to the K + → π + ν ¯ ν measurement, before applying acceptance and resolution effects. The m is computedunder the hypothesis that the charged particle in the final state is a π + . The K + → π + ν ¯ ν signal(red line) is multiplied by 10 for visibility. The hatched areas include the signal regions.The decay-in-flight configuration has two main advantages: • the selection of K + → π + ν ¯ ν decays with a π + momentum lower than 35 GeV/ c to facilitatethe background rejection by ensuring at least 40 GeV of missing energy, and to exploit thecapability of the RICH for π + /µ + separation; and • the achievement of sufficient π suppression by using photon detection coverage up to50 mrad with respect to the K + direction, and by efficiently detecting photons of energyabove 1 GeV.The experimental layout and the data-taking conditions are reviewed in section 3. The recon-struction algorithms are described in section 4. After the K + → π + ν ¯ ν selection (section 5), theanalysis proceeds through the evaluation of the single event sensitivity, defined as the branchingratio equivalent to the observation of one SM signal event (section 6). The number of signaldecays is normalized to the number of K + → π + π decays, whose branching ratio is accuratelyknown [8]. This allows the precise determination of the single event sensitivity without relyingon the absolute measurement of the total number of K + decays. The final step of the analysisis the evaluation of the expected background in the signal regions (section 7). To avoid biasingthe selection of K + → π + ν ¯ ν events, the analysis follows a “blind” procedure, with signal re-gions kept masked until completion of all the analysis steps. Finally, the result is presented insection 8. 8 Experimental setup and data taking
The NA62 beam line and detector are sketched in Figure 2. A detailed description of them canbe found in [23]. The beam line defines the Z-axis of the experiment’s right-handed laboratorycoordinate system. The origin is the kaon production target, and beam particles travel in thepositive Z-direction. The Y-axis is vertical (positive up), and the X-axis is horizontal (positiveleft).The kaon production target is a 40 cm long beryllium rod. A 400 GeV proton beam extractedfrom the CERN Super Proton Synchrotron (SPS) impinges on the target in spills of three secondseffective duration. Typical intensities during data taking range from 1 . . × protonsper pulse ( ppp ). The resulting secondary hadron beam of positively charged particles consists of70% π + , 23% protons, and 6% K + , with a nominal momentum of 75 GeV/ c (1% rms momentumbite).Beam particles are characterized by a differential Cherenkov counter (KTAG) and a three-station silicon pixel matrix (Gigatracker, GTK, with pixel size of 300 × µ m ). The KTAGuses N gas at 1.75 bar pressure (contained in a 5 m long vessel) and is read out by photo-multiplier tubes grouped in eight sectors. It tags incoming kaons with 70 ps time-resolution.The GTK stations are located before, between, and after two pairs of dipole magnets (a beamachromat), forming a spectrometer that measures beam particle momentum, direction, and timewith resolutions of 0.15 GeV/ c , 16 µ rad, and 100 ps, respectively.The last GTK station (GTK3) is immediately preceded by a 1 m thick, variable aperturesteel collimator. Its inner aperture is typically set at 66 mm ×
33 mm, and its outer dimensionsare about 15 cm. It serves as a partial shield against hadrons produced by upstream K + decays.GTK3 marks the beginning of a 117 m long vacuum tank. The first 80 m of the tank definea volume in which 13% of the kaons decay. The beam has a rectangular transverse profile of 52 ×
24 mm and a divergence of 0.11 mrad (rms) in each plane at the decay volume entrance.The time, momentum, and direction of charged daughters of kaon decays-in-flight are mea-sured by a magnetic spectrometer (STRAW), a ring-imaging Cherenkov counter (RICH), andtwo scintillator hodoscopes (CHOD and NA48-CHOD). The STRAW, consisting of two pairs ofstraw chambers on either side of a dipole magnet, measures momentum-vectors with a resolu-tion, σ p /p , between 0.3% and 0.4%. The RICH, filled with neon at atmospheric pressure, tagsthe decay particles with a timing precision of better than 100 ps and provides particle identifi-cation. The CHOD, a matrix of tiles read out by SiPMs, and the NA48-CHOD, comprising twoorthogonal planes of scintillating slabs reused from the NA48 experiment, are used for triggeringand timing, providing a time measurement with 200 ps resolution.Other sub-detectors suppress decays into photons or into multiple charged particles (elec-trons, pions or muons) or provide complementary particle identification. Six stations of plasticscintillator bars (CHANTI) detect, with 1 ns time resolution, extra activity, including inelasticinteractions in GTK3. Twelve stations of ring-shaped electromagnetic calorimeters (LAV1 toLAV12), made of lead-glass blocks, surround the vacuum tank and downstream sub-detectorsto achieve hermetic acceptance for photons emitted by K + decays in the decay volume at polarangles between 10 and 50 mrad. A 27 radiation-length thick, quasi-homogeneous liquid kryptonelectromagnetic calorimeter (LKr) detects photons from K + decays emitted at angles between1 and 10 mrad. The LKr also complements the RICH for particle identification. Its energy res-olution in NA62 conditions is σ E /E = 1 .
4% for energy deposits of 25 GeV. Its spatial and timeresolutions are 1 mm and between 0.5 and 1 ns, respectively, depending on the amount and typeof energy released. Two hadronic iron/scintillator-strip sampling calorimeters (MUV1,2) and anarray of scintillator tiles located behind 80 cm of iron (MUV3) supplement the pion/muon iden-tification system. MUV3 has a time resolution of 400 ps. A lead/scintillator shashlik calorimeter(IRC) located in front of the LKr, covering an annular region between 65 and 135 mm from the9 -1-212
100 150 200 250 Z [m]
GTK X [ m ] xx x HASC
MUV0
CHANTI LAVKTAGTarget MUV1,2STRAW IRCLKr
Vacuum
MUV3Iron
RICHRICH
DumpCHODSAC
Figure 2: Schematic top view of the NA62 beam line and detector. Dipole magnets are displayedas boxes with superimposed crosses. Also shown is the trajectory of a beam particle in vacuumwhich crosses all the detector apertures, thus avoiding interactions with material. A dipolemagnet between MUV3 and SAC deflects the beam particles out of the SAC acceptance.Z-axis, and a similar detector (SAC) placed on the Z-axis at the downstream end of the appara-tus, ensure the detection of photons down to zero degrees in the forward direction. Additionalcounters (MUV0, HASC) installed at optimized locations provide nearly hermetic coverage forcharged particles produced in multi-track kaon decays.All detectors are read out with TDCs, except for LKr and MUV1, 2, which are read outwith 14-bit FADCs. The IRC and SAC are read out with both. All TDCs are mounted oncustom-made (TEL62) boards, except for GTK and STRAW, which each have specialized TDCboards. TEL62 boards both read out data and provide trigger information. A dedicated pro-cessor interprets calorimeter signals for triggering. A dedicated board combines logical signals(primitives) from the RICH, CHOD, NA48-CHOD, LKr, LAV, and MUV3 into a low-level trig-ger (L0) whose decision is dispatched to sub-detectors for data readout [24]. A software trigger(L1) exploits reconstruction algorithms similar to those used offline with data from KTAG, LAV,and STRAW to further cull the data before storing it on disk [23].The data come from 3 × SPS spills accumulated during a four-month data-taking periodin 2017, recorded at an average beam intensity of 450 MHz. The instantaneous beam intensity ismeasured event-by-event using the number of signals recorded out-of-time in the GTK detector.The average beam intensity per spill was stable within ±
10% throughout the data-taking period,while the instantaneous beam intensity showed fluctuations up to a factor of two around theaverage value.The data have been collected using a trigger specifically setup for the K + → π + ν ¯ ν mea-surement, called PNN trigger , concurrently with a minimum-bias trigger. The PNN trigger isdefined as follows. The L0 trigger requires a signal in the RICH to tag a charged particle. Thetime of this signal, called trigger time, is used as a reference to define a coincidence within 6.3 nsof: a signal in one to four CHOD tiles; no signals in opposite CHOD quadrants to suppress K + → π + π + π − decays; no signals in MUV3 to reject K + → µ + ν decays; less than 30 GeVenergy deposited in LKr and no more than one cluster to reject K + → π + π decays. The L1trigger requires: a kaon identified in KTAG; signals within 10 ns of the trigger time in at mosttwo blocks of each LAV station; at least one STRAW track corresponding to a particle with10omentum below 50 GeV /c and forming a vertex with the nominal beam axis upstream of thefirst STRAW chamber. Events collected by the PNN trigger are referred to as PNN events ordata. The minimum-bias trigger is based on NA48-CHOD information downscaled by a factorof 400. The trigger time is the time of the NA48-CHOD signal. Data collected by the minimum-bias trigger are used at analysis level to determine the K + flux, to measure efficiencies, and toestimate backgrounds. These data are called minimum-bias events or data.Acceptances and backgrounds are evaluated using Monte Carlo (MC) simulation based on the GEANT4 toolkit [25] to describe detector geometry and response. The K + decays are generated inthe kaon rest frame using the appropriate matrix elements and form factors. The simulation alsoincludes a description of the collimators and dipole and quadrupole magnets in the beam line.Certain aspects of the simulation are tuned using input from data, namely signal formation andreadout detector inefficiencies. Accidental activity is added to the KTAG Cherenkov counterand to the GTK beam tracker assuming 450 MHz beam intensity, and using a library of pileupbeam particles built from data. No accidental activity is simulated in the detectors downstreamof the last station of the beam tracker. Simulated data are subjected to the same reconstructionand calibration procedures as real data. The channels of the Cherenkov beam counter KTAG are time-aligned with the trigger time,and signals are grouped within 2 ns wide windows to define KTAG candidates. A K + KTAGcandidate must have signals in at least five of eight sectors.The arrival time of the pulses measured in each of the GTK pixels is aligned to the triggertime and corrected for pulse-amplitude slewing. Signals from the three GTK stations groupedwithin 10 ns of the trigger time form a beam track. A track must have pulses in all three stations,therefore it is made of at least three hit pixels. Nevertheless, a particle can leave a signal inmore than one adjacent pixel in the same station if hitting the edge of a pixel or because of δ -rays. In this case, pulses in neighbouring pixels form a cluster that is used to reconstructthe track. Fully reconstructed K + → π + π + π − decays in the STRAW spectrometer are used toalign the GTK stations transversely to a precision of better than 100 µ m and to tune the GTKmomentum scale.The STRAW reconstruction relies on the trigger time as a reference to determine the drifttime. A track is defined by space-points in the chambers describing a path compatible withmagnetic bending. A Kalman-filter fit provides the track parameters. The χ fit value andthe number of space-points characterize the track quality. Straight tracks collected with themagnet off serve to align the straw tubes to 30 µ m accuracy. The average value of the K + massreconstructed for K + → π + π + π − decays provides fine tuning of the momentum scale to a partper thousand precision.Two algorithms reconstruct RICH ring candidates, both grouping signals from photomulti-pliers (PM) in time around the trigger time. The first one, called track-seeded ring , makes useof a STRAW track as a seed to build a RICH ring and compute a likelihood for several masshypotheses ( e + , µ + , π + and K + ). The second one, called single ring , fits the signals to a ringassuming that they are produced by a single particle, with the fit χ characterizing the quality ofthis hypothesis. Positrons are used to calibrate the RICH response and align the twenty RICHmirrors to a precision of 30 µ rad [26].The CHOD candidates are defined by the response of two silicon-photomultipliers (SiPM)reading out the same tile. Signals in crossing horizontal and vertical slabs compatible with thepassage of a charged particle form NA48-CHOD candidates. Each slab is time-aligned to thetrigger time. Time offsets depending on the intersection position account for the effect of lightpropagation along a slab. 11roups of LKr cells with deposited energy within 100 mm of a seed form LKr candidates(clusters). A seed is defined by a cell in which an energy of at least 250 MeV is released. Clusterenergies, positions, and times are reconstructed taking into account energy calibration, non-linearity, energy sharing for nearby clusters and noisy cells. The final calibration is performedusing positrons from K + → π e + ν decays. An additional reconstruction algorithm is applied tomaximise the reconstruction efficiency. This is achieved by defining candidates as sets of cellswith at least 40 MeV energy, closer than 100 mm and in time within 40 ns of each other.The reconstruction of MUV1(2) candidates relies on the track impact point. Signals infewer than 8 (6) nearby scintillator strips around the track are grouped to form a candidate.The energy of a candidate is defined as the sum of the energies in the strips, calibrated usingweighting factors extracted from dedicated simulations and tested on samples of π + and µ + .Candidates in MUV3 are defined by time coincidences of the response of the two PMs readingthe same tile. The time of a candidate is defined by the later of the two PM signals, to avoidthe effect of the time spread due to the early Cherenkov light produced by particles traversingthe PM window.CHANTI candidates are defined by signals clustered in time and belonging either to adjacentparallel bars or to intersecting orthogonal bars.Two threshold settings discriminate the CHANTI, LAV, IRC and SAC TDC signals [23].Thus up to four time measurements are associated with each signal, corresponding to the leadingand trailing edge times of the high and low thresholds. The relation between the amplitude ofthe IRC and SAC pulses provided by the FADC readout, and the energy release is calibratedfor each channel after baseline subtraction using a sample of K + → π + π decays.Signal times measured by GTK, KTAG, CHOD, RICH and LKr are further aligned to thetrigger time for each spill, resulting in a better than 10 ps stability through the whole datasample. The selection of both K + → π + ν ¯ ν signal and K + → π + π normalization decays requires theidentification of the downstream charged particle as a π + and the parent beam particle as a K + .Further specific criteria are applied to separate signal and normalization events. A downstream charged particle is defined as a track reconstructed in the STRAW spectrometer(downstream track) and matching signals in the two hodoscopes CHOD and NA48-CHOD, inthe electromagnetic calorimeter LKr, and in the RICH counter.The downstream track must include space-points reconstructed in all four chambers of theSTRAW spectrometer, satisfy suitable quality criteria, and be consistent with a positivelycharged particle. The extrapolation of this track to any downstream detector defines the ex-pected position of the charged particle’s impact point on that detector. These positions mustlie within the geometric acceptance of the corresponding downstream detectors and outside theacceptance of the large and small angle calorimeters LAV and IRC. The impact points of thecharged particles are used to match the downstream tracks with signals in the hodoscopes andthe electromagnetic calorimeter.Two discriminant variables are built using the difference of time and spatial coordinatesbetween each hodoscope candidate and the track. The NA48-CHOD candidate with the lowestdiscriminant value and the CHOD candidate closest in space to the particle impact point arematched to the track. The latter candidate must be within ± ± The parent K + of a selected downstream charged particle is defined by: the K + candidatein KTAG closest in time and within ± K + are derived from a sample of K + → π + π + π − selectedon data. In this case, the clean three-pion final state signature tags the K + track in the GTKand one of the positively charged pions is chosen to be the downstream charged particle. Theresulting distributions are shown in Figure 3, together with the corresponding distributions forevents including a random GTK track instead of the K + track. In contrast with the parent K + ,the shape of the CDA distribution in the presence of random beam tracks depends on the sizeand divergence of the beam, and the emission angle of the π + . The beam track with the largestdiscriminant value is, by construction, the parent K + ; its momentum and direction must beconsistent with the nominal beam properties.Because of the high particle rate in the beam tracker, several beam particles may overlapwith the K + within ± pileup (or accidental ) particles and thecorresponding GTK track is called a pileup (or accidental) track. A wrong association occurswhen a pileup track leads to a likelihood discriminant value larger than that of the actual K + track. An accidental association occurs when the K + track is not reconstructed in the beamtracker and a pileup track is associated to the downstream charged particle. A sharp cut on theminimum allowed value of the likelihood discriminant reduces the probabilities of wrong andaccidental association. Events are also rejected if more than 5 pileup tracks are reconstructedor if the likelihood discriminant values of different beam tracks matching the same downstreamcharged particle are similar. Finally, a cut is applied on a discriminant computed using the timedifference between the beam track and the downstream charged particle, instead of ∆T(KTAG-GTK).The K + → π + π + π − decays allow the performance of the beam-track matching to be moni-tored. The probabilities of wrong and accidental association depend on the instantaneous beamintensity and are about 1.3% and 3.5% on average, respectively. The latter includes also theprobability that a pileup track time is within ± .6 - - - T(KTAG-GTK) [ns] D E n t r i e s / ( . n s ) + Beam Kbeam particleAccidental
CDA [mm] E n t r i e s / ( . mm ) + Beam Kbeam particleAccidental
Figure 3: Distributions of ∆T(KTAG-GTK) (left) and CDA (right) for events with beam K + (shaded histogram) and accidental beam particle (empty histogram), as obtained from fully re-constructed K + → π + π + π − decays in the data. The red curves superimposed on the histogramsdescribe the functions used to model the time and CDA distributions of the beam K + .depend on the type of process under study. A downstream charged particle and its parent K + define the kaon decay. The mid-point betweenthe beam and downstream track at the closest distance of approach defines the position of the K + decay, called decay vertex .Several downstream charged particles may be reconstructed in the same event as a resultof overlapping accidental charged particles in the downstream detectors. In particular, thisoccurs in the STRAW spectrometer which makes use of a large 200 ns readout window. If twodownstream charged particles are reconstructed and both match a parent K + , the one closer tothe trigger time is accepted. The same trigger time requirement is applied independently to eachdetector signal matched with beam and downstream tracks. Further conditions are applied tosuppress K + decay like K + → π + π + π − : no more than two tracks reconstructed in the STRAWare allowed in total; if there are two tracks, both must be positively charged and should notform a vertex with a Z -position between GTK3 and the first STRAW station.Figure 4 (left) displays the distribution of the longitudinal position ( Z vertex ) of the recon-structed decay vertex of a K + decay. The events with Z vertex <
100 m mostly originate from K + decays upstream of the final collimator. The peaking structure starting at about 100 m isdue to nuclear interactions of beam particles grazing the edges of the final collimator or pass-ing through the last station of the beam tracker located at 103 m. Charged particles createdby decays upstream of the final collimator or by nuclear interactions can reach the detectorsdownstream and create fake K + decays. To mitigate this effect, the decay vertex is required tolie within a fiducial volume (FV) defined as 105 m to 165 m from the target. The coordinates ofthis vertex must also be consistent with the beam envelope. Wrong or accidental associationsor mis-reconstruction of the Z vertex can shift the origin of these events within the FV, imitating14 [m] vertex Z · E n t r i e s / ( . m ) Fiducial volume (FV) 5 10 15 20 25 30 35 40 45 50 55 momentum [GeV/c] + p - - ] / c [ G e V m i ss m p n + m p + p Region 1
Region 2
Control regionsControl regions
Figure 4:
Left : distribution of the longitudinal position of the reconstructed decay vertex. TheFV is defined between 105 and 165 m (vertical red lines).
Right : reconstructed m as afunction of the decay particle momentum for minimum-bias events selected without applying π + identification and photon rejection, assuming the K + and π + mass for the parent and decayparticle, respectively. Signal regions 1 and 2 (hatched areas), as well as 3 π , π + π , and µ + ν background regions (solid thick contours) are shown. The control regions are located betweenthe signal and background regions.a K + decay.Cuts on the direction of the decay particles as a function of Z vertex are applied to reducethe number of events reconstructed within the FV, but which actually originated upstream(section 7.2). These cuts are also useful against K + → π + π + π − decays with only one π + reconstructed. The CHANTI detector further protects the FV against nuclear interactions byvetoing events with CHANTI signals within 3 ns of the decay particle candidate. Extra pulsesin at least two GTK stations in time with the K + candidate may indicate that the K + hasdecayed before entering the decay region. In this case the event is rejected if at least one pileuptrack is reconstructed in the beam tracker in addition to the K + candidate. Finally, events arealso discarded if the decay particle track points back to the active area of GTK3. Figure 4 (right) shows the m distribution as a function of the decay particle momentumfor K + decays selected as above from minimum-bias data. Here, the m quantity is com-puted using the three-momenta measured by the beam tracker and the STRAW spectrometer,assuming K + and π + masses. Events from K + → π + π and K + → µ + ν decays accumulateat m = m π and m <
0, respectively. Events above m = 4 m π + (4 m π ) are mostly K + → π + π + π − ( π + π π ) decays. The shape of the region at low momentum arises from the Z vertex cuts.The m resolution varies with m and is about 10 − GeV /c at the K + → π + π peak.This sets the definition of the boundaries of signal region 1 and 2: Region 1 : 0 < m < .
01 GeV /c ; 15 egion 2 : 0 . < m < .
068 GeV /c .Additional momentum-dependent constraints supplement this definition by selecting m val-ues computed using either the decay particle momentum measured by the RICH under the π + mass hypothesis instead of the STRAW momentum, or the nominal beam momentum and direc-tion instead of those measured by the GTK tracker. These requirements are intended to reducethe probability of wrong reconstruction of the m quantity due to a mis-measurement of themomenta of the decay particle or K + candidate.The momentum of the decay particle in the range 15 −
35 GeV /c complements the definitionof the signal regions. The π + Cherenkov threshold of the RICH sets the lower boundary at15 GeV /c . The K + → µ + ν kinematics and the requirement of a large missing energy drive thechoice of the 35 GeV /c upper boundary. The two signal regions are kept masked (blind) untilthe completion of the analysis.In addition to the signal regions, three exclusive background regions are defined: The µν region : − . < m < m µ − kin +3 σ , where m µ − kin is the m of the K + → µ + ν decays under the π + mass hypothesis and σ its resolution; The π + π region : 0 . < m < .
021 GeV /c ; The 3 π region : 0 . < m < .
150 GeV /c .Once photons, muons and positrons are rejected (sections 5.5 and 5.6), simulations show thatsolely K + → µ + ν , K + → π + π and K + → π + π + π − decays populate these regions, respectively.Regions of the m distribution between signal and background regions, referred to as control regions , are masked until backgrounds are estimated and then used to validate theestimates. Two regions around the π + π peak, for the π + π background, and one region eachfor the K + → µ + ν and K + → π + π + π − backgrounds, are identified. Both background andcontrol regions are restricted to the 15 −
35 GeV/ c π + momentum range for consistency withthe definition of the signal regions. The PNN trigger (section 3) discards kaons decaying to muons by vetoing events with a signalin the MUV3 detector. A similar requirement applied offline reinforces the trigger condition,recovering possible online veto inefficiencies and makes the π + identification in minimum-biasand PNN data identical. Muons may fail to be detected by MUV3 because of inefficiency orcatastrophic interaction in the calorimeter, or if they decay upstream.Pions can be distinguished from muons and positrons using information from the LKrcalorimeter and, should any be present, from the MUV1 and MUV2 hadronic calorimeters.A multivariate classifier resulting from a Boosted Decision Tree algorithm, BDT, combines 13variables characterizing the calorimetric energy depositions. A first group of variables consistsof the ratios between the calorimetric energy deposited and the particle momentum measuredin the STRAW. The energy in the LKr is used alone, and in combination with the hadronicenergies. A second group of variables describes the longitudinal and transverse developmentof the calorimetric showers. The energy sharing between LKr, MUV1 and MUV2 provides in-formation about the longitudinal shape of the energy deposition, and the shape of the clusterscharacterizes the transverse size of the shower. Finally, the BDT makes use of the distancebetween the particle impact point and the reconstructed cluster position. The BDT training isperformed using samples of µ + , π + and e + selected from minimum-bias data recorded in 2016and not used in the present analysis. The BDT returns the probability for a particle to be a π + ,a µ + , or a positron. Pion identification requires the π + probability to be larger than a minimumvalue that depends on the particle momentum and is optimised with data.Samples of K + → π + π and K + → µ + ν decays selected from minimum-bias data are usedto monitor the performance of the π + identification efficiency and resulting µ + misidentification16 Particle momentum [GeV/c] e ff i c i e n c y + p
15 20 25 30 35 40
Particle momentum [GeV/c] e ff i c i e n c y + p - · e ff i c i e n c y + m efficiency data + p efficiency data + m
15 20 25 30 35 40
Particle momentum [GeV/c] e ff i c i e n c y + p
15 20 25 30 35 40
Particle momentum [GeV/c] e ff i c i e n c y + p - · e ff i c i e n c y + m efficiency data + p efficiency data + m Figure 5: Performance of the π + identification using calorimeters (left) and RICH (right) mea-sured on data. Performance is quantified in terms of π + and µ + efficiency, defined as the fractionof pions and muons passing the pion identification criteria, respectively. These criteria includethe corresponding RICH and calorimeter reconstruction efficiency. On each plot, the π + effi-ciency scale is shown on the left (black) vertical axis, the µ + efficiency (misidentification) scaleis shown on the right (blue) vertical axis.probability, shown in Figure 5 (left).Finally, the RICH separates π + , µ + and e + independently of the calorimeter responses. Thereconstructed mass and the likelihood of the particle must be consistent with the π + hypothesis.Figure 5 (right) shows the performance of the π + /µ + separation using the RICH as a functionof the particle momentum, evaluated using data.The π + identification is required for both signal K + → π + ν ¯ ν and normalization K + → π + π selections. Additional requirements are applied to PNN data to reject events with in-time photons or non-accidental additional charged particles in the final state that are compatible with a physicsprocess producing the downstream π + .Photon rejection discriminates against partially reconstructed K + → π + π decays. An extrain-time photon in the LKr calorimeter is defined as a cluster located at least 100 mm away fromthe π + impact point and within a cluster energy-dependent time coincidence with the π + timethat ranges from ± ±
50 ns above 15 GeV. Pileup clusters can overlap inspace with the photon to be rejected, spoiling the time of such a photon by as much as severaltens of ns. The choice of a broad timing window at high energy keeps the detection inefficiencybelow 10 − .An extra in-time photon in the LAV detector is defined as any signal in a LAV station within ± π + time. Appropriate combinations of the TDC leading and trailing edges of thehigh and low threshold channels define a LAV signal [23]. A similar method identifies photons inthe small angle calorimeters IRC and SAC, using a time-window of ± π + time.17 momentum [GeV/c] + p · re j ec t i o n i n e ff i c i e n c y p Figure 6: Rejection inefficiency of the π from K + → π + π decays as a function of the π + momentum. The quoted uncertainties arestatistical only. ] /c [GeV miss2 m ) / c E n t r i e s / ( . G e V Data p + pfi + K p p + pfi + K D0 p + pfi + K gg + pfi + K - - ] /c [GeV miss2 m D a t a / M C MC systematic uncertainty
Figure 7: Distribution of the m of eventsselected from minimum-bias data for normal-ization. Data and MC simulation are su-perimposed. The bottom insert shows thedata/MC ratio. The error bars correspondto the statistical uncertainty of the ratio, theyellow band is the systematic uncertainty dueto the imperfect simulation of the detector re-sponse.In addition to the signals from the TDC readout, photon rejection in IRC and SAC exploitsthe FADC readout; here, a photon signal is defined as an energy deposit larger than 1 GeV in a ± K + → π + π + π − decayspartially reconstructed in the STRAW. The first category of charged particles is expected toleave signals in the detectors downstream of the STRAW. The rejection criteria exploit the timeand spatial coincidence of isolated signals reconstructed in at least two of the CHOD, NA48-CHOD and LKr detectors. In-time signals in the peripheral detectors MUV0 and HASC arealso included. The second category of charged particles is characterized by the presence of tracksegments, defined as pairs of signals in the first-second or third-fourth STRAW stations andconsistent with a particle coming from the FV.The reduction of reconstructed K + → π + π decays quantifies the performance of the photonand multiplicity rejection. The number of PNN events in the π + π region remaining afterrejection is compared to the number of minimum-bias events in the same region before rejection.The ratio of these two numbers, corrected for the minimum-bias downscaling factor (section 6)and trigger efficiency (section 6.3), is the rejection inefficiency of the π produced in K + → π + π decays. This inefficiency depends on the π + momentum and is about 1 . × − on average,as shown in Figure 6. The measured π rejection can be explained in terms of single-photondetection inefficiencies in the LKr, LAV, IRC and SAC calorimeters which are measured froma sample of minimum-bias K + → π + π data using a tag-and-probe method. The estimated π K + → π + π decays and is in agreement with the measured π efficiency within the statisticaluncertainty. The rise at low π + momentum is a consequence of lower detection efficiency forphotons travelling close to the beam axis and interacting with the beam pipe.In addition to the photon and multiplicity rejection, the K + → π + ν ¯ ν selection enforces spe-cific requirements against particles entering the FV from upstream. The π + track is extrapolatedback to the Z -position of the final collimator and the X, Y transverse coordinates are requiredto be outside of a box with | X | <
100 mm and | Y | <
500 mm. This cut removes a regionwith weaker shielding against particles coming from upstream and corresponds to the centralaperture of the last dipole magnet of the beam line (section 7.2). This condition is referred toas the box cut in the following sections.Finally, signal selection requires the m value to be within the signal regions defined insection 5.4. The set of criteria described in this section is called PNN selection in the following.
The K + → π + π decays used for normalization are selected from minimum-bias data, as definedin sections 5.3 and 5.5, and their m value must be in the 0.01–0.026 GeV /c range. Figure 7shows the m spectrum of these events before the m cut, together with the simulateddistribution. The shape of the K + → π + π peak depends on the resolution of the STRAWspectrometer, on multiple scattering in the tracker material, on the rate of pileup tracks, and onthe calibration of the beam and STRAW trackers. The uncertainty in the simulation of theseeffects affects the data/MC agreement in the peak region only, and is taken into account in theevaluation of the SES (section 6). The overall background under the peak is at the one partper thousand level and stems from K + → π + π decays with π → e + e − γ . Denoting N K + the number of kaon decays occurring in the FV, the single-event sensitivity( SES ) of the present data sample to K + → π + ν ¯ ν can be written as SES = 1 N K + · (cid:15) πνν · (cid:15) PNN trig = BR( K + → π + π ) D · N ππ (cid:15) ππ · (cid:15) MB trig (cid:15) πνν · (cid:15) PNN trig · (4)Here N ππ is the number of K + → π + π events reconstructed in the FV from minimum-biasdata (section 5.7), also called normalization events; D is the reduction, or down-scaling, factorapplied online to reduce the minimum-bias contribution to the total trigger rate; (cid:15) πνν and (cid:15) ππ are the efficiencies to identify a K + → π + ν ¯ ν and a K + → π + π decay in the FV, also calledsignal and normalization efficiencies, respectively; (cid:15) PNN trig and (cid:15) MB trig , are the trigger efficiencies thataccount for the data loss after the event selection due to the PNN and minimum-bias triggers.The efficiencies and N ππ depend on the π + momentum, p π , and on the instantaneous beamintensity, I . The SES is consequently computed in bins of p π and I : the momentum range 15–35 GeV/ c is subdivided into four bins of 5 GeV/ c width and the instantaneous beam intensityinto five bins of approximately the same statistics of K + → π + π normalization events. K + → π + π decays The number of events satisfying the conditions described in section 5.7 is N ππ = 68 × .The π mainly decays to γγ , but in about 1% of cases it decays to γe + e − , called a Dalitzdecay ( π D ). The relative impact of π D decays on the SES is estimated to be less than 0.3%and is assigned as systematic uncertainty. In the following sections K + → π + π refers only to π → γγ decays. 19 .2 Signal and normalization efficiencies The efficiencies (cid:15) πνν and (cid:15) ππ quantify the effects of reconstruction and selection (section 5) onthe counting of signal and normalization channels. Event losses can be grouped into 6 classes:1. geometric and kinematic acceptances;2. reconstruction of the K + and of the downstream charged particle;3. matching the K + with the downstream charged particle;4. π + identification by the RICH and calorimeters;5. decay region definition; and6. selection criteria unique to the K + → π + ν ¯ ν mode.The impact of these effects on (cid:15) πνν and (cid:15) ππ depends on the kinematics of the decay, detectorresolutions and efficiencies, and the accidental presence of unassociated particles in an event.The kinematics of the decays are studied with simulations, while detector performance isstudied with data and either reproduced by simulation or factored out from (cid:15) πνν and (cid:15) ππ .Accidental particles have a twofold effect. They affect detector response and therefore thereconstruction of K + decays and kinematic resolution. They also randomly satisfy conditionsin the GTK, CHANTI, STRAW, MUV3, calorimeters, CHOD, and NA48-CHOD that lead toan event being rejected, referred to here as a random veto. The first effect is modelled withsimulation. The second effect, which is independent of decay mode topology, is measured directlywith data as a function of the instantaneous beam intensity and factored out of (cid:15) πνν and (cid:15) ππ .As a consequence, signal and normalization efficiencies may take the form: (cid:15) πνν = (cid:15) MCπνν · (cid:15) Randomπνν (cid:15) ππ = (cid:15) MCππ · (cid:15) Randomππ . (5)The Monte Carlo efficiency, (cid:15) MCdecay , quantifies the effects of the factors listed above, except forrandom losses, and the random efficiency, (cid:15)
Randomdecay , quantifies the fraction of events randomlylost because of the accidental presence of at least one veto condition.The
SES depends only on the ratio of the (cid:15) πνν and (cid:15) ππ efficiencies. As both signal andnormalization channels contain a π + in the final state, the ratio effectively cancels significantcomponents of the two efficiencies, decreasing the dependence of the SES on their magnitudeand reducing their contribution to its uncertainty to a negligible level.
The Monte Carlo efficiency, (cid:15)
MCdecay , is the ratio of the number of simulated events passing signalor normalization selection to the corresponding number of generated events in the FV.Figure 8 shows the values of (cid:15)
MCππ and (cid:15)
MCπνν in bins of π + momentum. The sums over all binsare 0 . ± .
009 and 0 . ± . π + identification and K/π track matching in the simulation.Table 1 shows estimates of the contributions to (cid:15)
MCπνν and (cid:15)
MCππ of the components listed in sec-tion 6.2. The values in the table are approximated, due to correlations among the components.A 10% relative uncertainty is assigned to each component and conservatively considered as 100%correlated. The difference between (cid:15)
MCπνν and (cid:15)
MCππ is attributable to differences in acceptance,particle reconstruction, and cuts specific to the signal channel. The accuracy with which thesefactors are simulated is the primary source of uncertainty in the
SES . The next paragraphsfocus on the contributions from each component of the Monte Carlo efficiencies listed in Table 1.20 momentum [GeV/c] + p M C E ff i c i e n c y ggfi p ), g ( p + pfi + K
15 20 25 30 35 momentum [GeV/c] + p M C E ff i c i e n c y Region 1+2Region 1Region 2 nn + pfi + K Figure 8:
Left : K + → π + π MC efficiency in independent bins of π + momentum. Right : K + → π + ν ¯ ν MC efficiency in independent bins of π + momentum. The efficiencies in regions1 and 2 are shown separately and summed (full symbols). The width of the coloured bandsrepresents the uncertainties in the measured values.Table 1: Monte Carlo efficiencies for normalization and signal decay modes. The uncertaintiesin the total efficiencies are systematic and reflect the accuracy of the simulation.Source K + → π + π K + → π + ν ¯ ν Acceptance 0 .
27 0 . .
64 0 . K + matching 0 .
84 0 . π + identification 0 .
72 0 . .
83 0 . K + → π + ν ¯ ν selection − . . ± .
009 0 . ± . Acceptance
Events fail to be selected because of detector geometry as well as restrictions on the π + momen-tum and m ranges. The effects of these three factors are different for signal and normalizationselection efficiencies and are therefore a potential source of SES uncertainty.The impact of the limited accuracy of the simulated m distribution has been quantified byrecalculating the SES with K + → π + π decays in a smaller m region, (0.015,0.021) GeV /c ,where data and MC marginally agree (section 5.7). The corresponding variation of the SES isapproximately 1% and assigned as systematic uncertainty due to the simulation of the m .Detector illumination and the momentum spectrum contribute to a lesser extent to the dif-21able 2: Average detector efficiencies over π + momentum and instantaneous beam intensity.The uncertainties are estimated by comparing data and simulation, and with systematic studies,such as checks of time stability. The values for the RICH efficiency refer to pion identificationefficiencies from K + → π + ν ¯ ν and K + → π + π , respectively. All other detector efficiencies areequal for signal and normalization.Source EfficiencyKTAG 0 . ± . . ± . . ± . . ± . . ± .
03 (0 . ± . > . > . . ± . K + → µ + ν normalized to K + → π + π . A systematic uncertainty is assigned after comparing the result of this measure-ment to the accepted value (section 6.2.2). Particle reconstruction
The particle reconstruction efficiency is the product of the KTAG and GTK efficiencies for recon-structing the parent K + , and the STRAW, RICH, CHOD, NA48-CHOD and LKr efficiencies forreconstructing the daughter π + . The RICH, CHOD, NA48-CHOD and LKr efficiencies includedetector signal association with a STRAW track.The effect of local inefficiencies due to detector readout or to accidental activity cancels atfirst order in the ratio of efficiencies, as signal and normalization decays are recorded simultane-ously. Nonetheless, these effects are measured with data and added to the simulation. Table 2details the impact of the various subdetectors on the reconstruction efficiency. The numbers areaverages over π + momentum between 15 and 35 GeV /c and instantaneous beam intensity.KTAG and GTK efficiencies refer to K + detection and are equal for signal and normalization.Both efficiencies are measured with data, using K + → π + π + π − decays. KTAG inefficienciescome mainly from the readout. GTK inefficiencies arise from geometric acceptance and identifiedreadout malfunctioning (5%) and from the detector (3%) [27]. The GTK reconstruction efficiencyis due to the conditions applied to identify a track of good quality.The efficiency to reconstruct a π + track with the STRAW is measured with K + → π + π decays in the data. In the 15 −
35 GeV/ c momentum range, the efficiency depends only on theinstantaneous beam intensity, which is directly related to accidental activity in the detectors.The RICH efficiency for reconstructing a π + with momentum between 15 and 35 GeV /c ismeasured with data using K + → π + π decays. It is directly related to the statistics of Cherenkovphotons and depends only on the π + momentum. Simulation reproduces this efficiency with arelative accuracy of about 3%. The simulation indicates that this efficiency is about 7% higherfor K + → π + ν ¯ ν decays than for K + → π + π decays. This difference is attributable to extrahits created when photons from π decay in K + → π + π events convert in RICH material and22poil the charged pion ring shape. Therefore, the RICH reconstruction efficiency does not cancelin the ratio of Equation (4). A sample of K + → µ + ν decays in the data is used to test theaccuracy of RICH particle reconstruction in the simulation. The resulting ratio of data to MCagrees with that of K + → π + π to within 1.5%. This value is assigned as a relative systematicuncertainty in the SES due to the simulation of the RICH reconstruction efficiency.Measurements with data show that the CHOD and NA48-CHOD detectors are highly ef-ficient. An overall 0.99 efficiency is assigned to account for small losses in the association ofdetector signals with STRAW tracks that define downstream charged particles.The LKr calorimeter detects signals from minimum ionizing particles with an efficiencygreater than 99%, as measured with data. In the case of π + inelastic hadronic interactions, anadditional inefficiency may arise in associating LKr clusters with STRAW tracks. K + matching The efficiency for matching a K + with a downstream charged particle is 0.84 and depends onthe GTK efficiency and on time and CDA resolutions. The simulation reproduces the matchingperformance measured with data to within 5% relative accuracy, once accidental pileup in theGTK and GTK efficiency are simulated. This measurement of the accuracy is taken as a system-atic uncertainty in the magnitudes of both (cid:15) MCπνν and (cid:15)
MCππ . However, the effect of K + matchingis equal for signal and normalization, and therefore no corresponding uncertainty is assigned tothe SES . As a cross check, the
SES is found to be nearly insensitive to the simulated level ofGTK inefficiency. π + identification Not every π + is identified due to the intrinsic efficiencies of the RICH and calorimeters and to π + decays in flight.The RICH efficiency for identifying undecayed π + s from K + → π + π events is measuredwith data and found to be about 0.95. Simulation reproduces this number with 3% accuracyand indicates that π + s from K + → π + ν ¯ ν decays are identified with a comparable efficiency.Simulation reproduces the measured efficiency for the RICH to reconstruct and identify a π + with an accuracy of about 6%. This value is assigned as a relative uncertainty to (cid:15) MCπνν and (cid:15)
MCππ . However, no additional uncertainty is assigned to the
SES beyond that from theRICH reconstruction efficiency, because the RICH identification algorithm treats signal andnormalization modes the same.The average efficiency of π + identification with the calorimeters is about 0.80, as measuredwith data. Simulation reproduces this result with 2% accuracy. This degree of accuracy ispropagated as a relative uncertainty to (cid:15) MCπνν and (cid:15)
MCππ . Simulation also shows that the efficienciesto identify charged pions with the calorimeters are the same for signal and normalization modes.Therefore, the accuracy of calorimeter simulations does not affect the
SES measurement.The π + identification efficiencies reported in Table 1 include an additional factor of 0.95 toaccount for the probability of π + decay. Decay region
In addition to the definition of the 105–165 m FV, the decay region is shaped by the cuts on the π + direction as a function of Z vertex as discussed in section 5.3. These selection criteria rejecta slightly different number of signal and normalization events. The simulation accounts for thecorresponding effect in the SES together with the kinematic and geometric acceptances, as thevarious contributions are correlated.
Signal K + → π + ν ¯ ν selection Photon and multiplicity rejection and the box cut are applied only to signal events. These23election criteria, therefore, directly impact the measurement of the
SES .In the absence of random activity, the GTK, CHANTI, STRAW, and MUV3 veto conditionsdo not affect (cid:15)
MCπνν or (cid:15) MCππ . On the other hand, because charged pions may interact in RICH ma-terial, vetoing photons and extra charged particles can inadvertently reject K + → π + ν ¯ ν events.The accuracy with which the simulation models this effect is studied by selecting from data K + → π + π events in which both photons from the π decay are detected in LAV stations.The loss of events because of π + interactions is measured on these data and compared withsimulation, leading to about 6% discrepancy. The efficiency (cid:15) MCπνν is corrected for half of thisdifference. An uncertainty of 100% is assigned to this correction factor, resulting in about 3%relative uncertainty in the
SES . K + → µ + ν branching ratio measurement The measurement of the branching ratio of the K + → µ + ν decay provides a test of the accuracyof the MC simulation of the kinematic and geometric acceptances.The measurement follows a procedure similar to that adopted for the SES . The decay K + → π + π is used for normalization and the branching ratio can be expressed as:BR( K + → µ + ν ) = BR( K + → π + π ) N µ ˆ N ππ ˆ (cid:15) ππ (cid:15) µ · (6)Here N µ and ˆ N ππ are the number of selected K + → µ + ν and K + → π + π events, (cid:15) µ and ˆ (cid:15) ππ are the efficiencies for selecting them.Event selection for both modes differs slightly from the procedures described in sections 5.1and 5.2. Both modes require a RICH ring associated with the STRAW track, but π + identifi-cation for the K + → π + π decay relies on the electromagnetic and hadronic calorimeters only.MUV3 provides positive identification of the µ + from K + → µ + ν decay.The kinematic range 0 . < m < .
026 GeV /c defines K + → π + π events. Therequirement that | m ( µ ) | < .
01 GeV /c defines K + → µ + ν decays. Here, m ( µ ) is thesquared missing mass computed assuming the particle associated with the STRAW track to bea muon. The background in both selected modes is of the order of 10 − . Estimations of ˆ (cid:15) ππ and (cid:15) µ rely on Monte Carlo simulations, as for the SES . Their magnitudes are about 0.09 and0.10.The procedure described in section 6.2 is adopted to quantify the efficiency bias introducedby the simulation of the RICH reconstruction. The corresponding correction factor applied toBR( K + → µ + ν ) is +0 . ± . π + identification efficiency affects only the K + → π + π mode. As stated in section 6.2,this efficiency can be measured with data, and the simulation reproduces the value within 2%accuracy. Half this discrepancy is applied as a correction to ˆ (cid:15) ππ . Assuming that the uncertaintyin this correction is 100%, this amounts to correcting BR( K + → µ + ν ) by − . ± . K + → µ + ν branching ratio is measured to beBR( K + → µ + ν ) = 0 . ± . , (7)in agreement within 2.5% with the PDG value [8]. The 0.01 uncertainty is systematic, mainlyattributable to the corrections described above. The statistical uncertainty is negligible. Theresult is stable within uncertainties when signal and normalization selection cuts are varied. Itis also stable throughout the 2017 data taking, as shown in Figure 9.This result relies on simulation to account for the different acceptances of the K + → µ + ν and K + → π + π decay modes, as in the case of the SES measurement. The comparison betweenthe measured and PDG branching ratios is used to set the level of accuracy in the simulations,leading to a relative uncertainty of 2 .
5% being propagated to
SES .24
600 7700 7800 7900 8000 8100 8200 8300
Time period ) n + mfi + B R ( K NA62 Measurements per periodNA62 MeasurementPDG Value
Figure 9: Measured branching ratio of the K + → µ + ν decay mode in different time periods ofthe 2017 data taking. The overall quoted uncertainty is mostly systematic. In both K + → π + ν ¯ ν and K + → π + π event selections, the GTK, CHANTI, STRAW, andMUV3 are also used to veto backgrounds. Data are used to estimate the fraction of kaon decaysrejected due to accidental activity in these detectors. Measurements on samples of K + → π + π and K + → µ + ν show that the fraction of events accepted by each of these detectors is about 0.9,0.97, 0.9, and 0.95. These veto requirements are uncorrelated and, in total, reject about 25% ofsignal and normalization decays. Because the average beam intensity of selected normalizationand signal-like events is comparable, the effects of the GTK, CHANTI, STRAW and MUV3vetoes cancel in the ratio of Equation (4).The criteria, collectively termed photon and multiplicity rejection (section 5.6), employed toveto K + decays with photons or more than one charged particle in the final state also rejectsignal events if accidental particles overlap the π + in time.The fraction of signal events passing photon and multiplicity rejection is denoted (cid:15) RV andcalled the random veto efficiency . A sample of K + → µ + ν decays selected from minimum-biasdata is used to estimate (cid:15) RV . The selection closely follows that of K + → π + ν ¯ ν , including GTK-,CHANTI-, and STRAW-based veto criteria, except that: | ( P K − P µ ) | < .
006 GeV / c replacesthe missing-mass squared regions; the calorimeters and the MUV3 are used for µ + identification;and no box cut or the photon and multiplicity rejection criteria are applied. Simulation showsthat the background to K + → µ + ν is less than a part per thousand.The random veto efficiency is computed as the ratio between the number of K + → µ + ν events remaining before and after photon and multiplicity rejection. Figure 10 displays (cid:15) RV as afunction of the instantaneous beam intensity. This result is corrected for the probability of eventloss induced by µ + interactions, such as δ -ray production in the RICH material, as estimatedby simulation. This correction increases (cid:15) RV by about 1%. An uncertainty of 100% is assignedto this correction, leading to a 1% systematic uncertainty in (cid:15) RV . The stability of (cid:15) RV is testedagainst cuts on ( P K − P µ ) and µ + identification. The maximum observed relative variation is2 .
4% due to the cut on the calorimetric BDT probability. Half this variation is used to correctthe measured (cid:15) RV and half is assigned as a systematic uncertainty. A residual dependence on25
100 200 300 400 500 600 700 800 900 1000
Intensity [MHz] R a nd o m v e t o e ff i c i e n c y Photon + multiplicity rejectionPhoton rejectionLKr vetoLAV vetoIRC+SAC veto
Figure 10: Random veto efficiency (cid:15) RV in bins of instantaneous beam intensity after photonand multiplicity rejection, after photon rejection, after LKr veto only, after LAV veto only, andafter IRC and SAC veto only. The error bars on the photon and multiplicity rejection pointsindicates the total uncertainty. Lines are drawn as guides for the eye.the µ + momentum is observed and added to the total systematic uncertainty. The final averagerandom veto efficiency is 0 . ± . (cid:15) RV contributes linearlyto the uncertainty in the SES . Normalization events are selected from minimum-bias data, and signal events are selected fromPNN data. Problems in the hardware and trigger definitions in conflict with offline cuts maycause the trigger to reject good normalization and signal events. Because mimimum-bias andPNN triggers differ, their efficiencies, denoted (cid:15) MB trig and (cid:15) PNN trig in Equation (4), do not cancel inthe ratio, which therefore must be precisely evaluated. The L0 and L1 trigger algorithms whichidentify signal candidates employ different sets of detectors, so their efficiencies can be studiedseparately.
The L0 efficiency stems from conditions in the RICH, CHOD, and MUV3, termed L0NoCalo,and veto conditions in the LKr, called L0Calo. A sample of K + → π + π events selected fromminimum-bias data using PNN-like criteria allows the measurement of the L0NoCalo efficiency.The contributions from the RICH and CHOD are also estimated with K + → µ + ν decays. Themeasured L0NoCalo efficiency is about 0.980 at the mean intensity of 450 MHz and varies almostlinearly as a function of the instantaneous beam intensity, decreasing by about 1% at twice themean intensity. The main source of inefficiency comes from the MUV3 veto criteria, because the26able 3: Contributions to the uncertainty of the random veto efficiency measurement. The totaluncertainty is the sum in quadrature of the contributions.Source Uncertainty in (cid:15) RV µ + interaction correction ± . µ + identification ± . ± . < . ± . K + → π + π decays in which the twophotons are detected in LAV stations. Events of this type result in a π + with momentum greaterthan 45 GeV /c in the LKr. The L0Calo efficiency, defined as the fraction of events passing theL0Calo conditions, is measured as a function of the energy, E LKr , that the π + deposits in theLKr. The dependence on E LKr is converted into a dependence on the π + momentum, p π + , inthe 15 −
35 GeV / c range, with a conversion factor extracted from the E LKr /p π + distribution ofa sample of π + s selected from K + → π + π + π − decays. The L0Calo efficiency depends on the π + momentum, and decreases from 0.965 to 0.910 between the first and the last momentum bin.The requirement that there be no more than 30 GeV detected in the LKr, convoluted with theenergy resolution of the LKr, is the main source of inefficiency. The uncertainty in the L0Calotrigger efficiency comes from the statistics of the K + → π + π + π − sample used to map E LKr into p π + .The overall L0 trigger efficiency is the product of the L0NoCalo and L0Calo efficiencies as afunction of π + and intensity. The measured value decreases with both increasing π + momentumand intensity, ranging from 0.95 to 0.9. The effects of independent KTAG, LAV, and STRAW requirements in the L1 trigger efficiencyare uncorrelated, such that the overall efficiency is the product of the individual efficiencies.Samples of K + → µ + ν selected from minimum-bias data and of K + → π + π selected from datatriggered by the PNN L0 conditions and recorded irrespective of the L1 trigger decision wereused to measure these efficiencies. The L1 trigger algorithms were emulated offline, includingthe effects of resolution.After applying PNN selection criteria, the KTAG L1 requirements do not introduce addi-tional loss of signal. On the other hand, the LAV requirements introduce intensity-dependentlosses of events which pass signal offline selection criteria, because the online LAV time reso-lution requires a larger veto timing window than that used offline. The L1 LAV efficiency inthe first part of the 2017 data-taking period ranges from 0.965 to 0.955, depending on the in-tensity. This efficiency is about 1% higher and exhibits less intensity dependence in the secondpart of 2017 data taking as a consequence of an optimization of the L1 LAV algorithm. Thespread of the efficiency among data-taking periods is used to set a systematic uncertainty for27 ntensity [MHz] momentum [20,25] GeV/c + p Intensity [MHz] P NN t r i gg er e ff i c i e n c y momentum [15,20] GeV/c + p Statistical uncertaintysystematic uncertaintyStatistical and
Intensity [MHz] momentum [30,35) GeV/c + p Intensity [MHz] + p Figure 11: Measured PNN trigger efficiency as a function of instantaneous beam intensity infour bins of π + momentum. The shaded band corresponds to the total uncertainty.this measurement, which amounts to about 0.4% (1.4%) at low (high) intensity.The efficiency of the L1 STRAW algorithm is greater than 0.99 and independent of inten-sity. A ± π + momentumdependence. SES
The effect of the trigger efficiency on the
SES is determined using Equation (4) with the followingassumptions: the total PNN trigger efficiency is the product of the L0 and L1 efficiencies; theefficiencies of the minimum-bias and PNN triggers due to hardware performance are equal andtherefore cancel in Equation (4); and the minimum-bias trigger is 100% efficient.The following test is performed to check the accuracy of these assumptions. The PNNselection, except π + identification with the RICH, is applied to minimum-bias data, leading to N MB = 701 ±
26 events in the µν region of the m distribution. The expected number of PNNdata in this region passing the same selection can be written, under the above assumptions, as: N P NN (expected) = D · N MB · (cid:15) P NNL · (cid:15) P NNL . (8)Here D is the minimum-bias reduction factor and (cid:15) P NNL ( (cid:15) P NNL ) are the L0 (L1) PNN trig-28able 4: Contributions to the uncertainty of the trigger efficiency. When quoted, the rangecorresponds to the efficiency dependence on the instantaneous beam intensity and the π + mo-mentum. Source Trigger efficiency uncertaintyL0NoCalo ± .
002 to ± . ± .
003 to ± . ± .
004 to ± . ± . ± . K + → µ + ν decays in which the muon resembles a pion in the calorime-ters and does not hit MUV3. Considering that K + → µ + ν decays are fully efficient un-der the L0Calo condition (section 6.3.1), the measured values of (cid:15) P NNL and (cid:15) P NNL lead to N P NN (expected) = (263 ± × . The number of PNN data observed in the µν regionof the m distribution after removing the RICH identification from the PNN selection is N P NN (observed) = (255 . ± . × , in agreement within ± .
8% with N P NN (expected).This value is assigned as systematic uncertainty to the measured PNN trigger efficiency (notedGlobal in Table 4).The PNN trigger efficiency relevant to the measurement of
SES is shown in Figure 11 asa function of instantaneous beam intensity and π + momentum. The overall average triggerefficiency is 0 . ± .
03. Table 4 summarizes the various contribution to the uncertainty in thetrigger efficiency.
SES result
The single event sensitivity and the total number of expected Standard Model K + → π + ν ¯ ν de-cays are: SES = (0 . ± . syst ) × − , (9) N expπνν ( SM ) = 2 . ± . syst ± . ext . (10)The statistical uncertainty is negligible. Table 5 details the various contributions to the SES ,averaged over instantaneous beam intensity and π + momentum. This list of contributionsis for reference only, as the measured value of the SES comes from Equation (4) in bins ofinstantaneous beam intensity and π + momentum.Table 6 lists the different sources of SES uncertainty, including contributions to the un-certainty of the Monte Carlo and trigger efficiency ratios, as discussed in sections 6.2 and 6.3,respectively. The external error on N expπνν stems from the uncertainty in the theoretical predictionof BR( K + → π + ν ¯ ν ). Figure 12 shows N expπνν in bins of π + momentum and instantaneous beamintensity. The background to K + → π + ν ¯ ν decays can be divided into two classes. The K + decay back-ground is due to kaon decays in the FV other than K + → π + ν ¯ ν , while the upstream background29able 5: Contributions to SES , averaged over instantaneous beam intensity and π + momentum.Contribution value N ππ × K + → π + ν ¯ ν Monte Carlo efficiency 0 . ± . K + → π + π Monte Carlo efficiency 0 . ± . . ± . . ± .
15 20 25 30 35 momentum [GeV/c] + p nn + pfi + E x p ec t e d S M K Expected eventsExternal error
Intensity [MHz] nn + pfi + E x p ec t e d S M K Expected eventsExternal error
Figure 12: Number of expected Standard Model K + → π + ν ¯ ν events in bins of π + momentum(left) and average beam intensity (right). The average beam intensity per bin, obtained fromthe K + → π + π sample used for normalization, is plotted at the barycentre of the bin.is due to π + particles produced either by beam particle interactions or by kaon decays upstreamof the FV. To mimic a signal, a background event should have a K + reconstructed upstream andmatched to a π + downstream, and m reconstructed in the signal region. Furthermore, eitherthe extra particles produced in association with the π + should escape detection, or a lepton inthe final state should be mis-identified as a π + . K + decay background The background from K + decays in the FV is primarily due to the K + → π + π ( γ ), K + → µ + ν ( γ ), K + → π + π + π − and K + → π + π − e + ν decays.The first three processes are constrained kinematically, and enter the signal regions via m mis-reconstruction due to large-angle Coulomb scattering, elastic hadronic interactions inGTK and STRAW material, incorrect K/π association, pattern recognition errors, or positionmis-measurement in the spectrometers. In addition to m mis-reconstruction, at least one30able 6: Sources contributing to the uncertainty in the SES measurement. “Normalizationbackground” refers to the impact of K + → π + π D decays on the normalization sample. Thetotal uncertainty is the sum in quadrature of the four contributions listed in the first column.Source Uncertainty in SES ( × )Monte Carlo efficiency ratio ± . π + interactions ± . ± . m Selection ± . ± . ± . ± . ± . ± . ± . < . ± . K + → π + π ( γ ) decay are not detected byelectromagnetic calorimeters; the muon from a K + → µ + ν ( γ ) decay is mis-identified as π + bythe RICH counter, hadronic calorimeters and MUV3; a π + π − pair from a K + → π + π + π − decayis undetected by the STRAW and the other downstream detectors.The background from the three kinematically-constrained decays is evaluated with data.Denoting by N decay the number of events in the corresponding background region of m inthe PNN data sample passing the PNN selection, and by f kin the probability that m isreconstructed in the signal region, the expected number of background events from each decayis given by N expdecay = N decay · f kin · (11)The value of N decay is obtained directly from the PNN data, while the probability f kin is measuredwith minimum-bias data. This technique does not require knowledge of photon and chargedparticle rejection inefficiencies or of the π + mis-identification probability. Nevertheless, theprecision of the method relies on three assumptions whose reasonableness is tested with bothdata and simulations:1. f kin represents the probability that an event of a given decay mode enters the signal region;2. f kin and N decay are uncorrelated; and3. N decay accounts only for events of the corresponding decay mode.Backgrounds from the K + → π + π − e + ν decay, as well as from the rare decay K + → π + γγ and the semileptonic decays K + → π (cid:96) + ν ( (cid:96) = e, µ ), are evaluated with simulations.In the following subsections the K + → π + π ( γ ) and the K + → µ + ν backgrounds are shownin bins of π + momentum up to 40 GeV /c , albeit only the 15–35 GeV/ c momentum range is usedto evaluate the corresponding backgrounds in the signal regions.31igure 13: Distribution of the PNN events in the ( π + momentum, m ) plane after the PNNselection in the π + π (red box) and control m regions (blue boxes). Both control regionsare used only for validation of the background estimation. The shaded grey area represents thedistribution of the simulated SM K + → π + ν ¯ ν events (arbitrarily normalized). K + → π + π ( γ ) decay After the PNN selection, N ππ = 264 events from the PNN sample remain in the π + π region.The distribution of these events in the ( π + momentum, m ) plane is shown in Figure 13. The π + momentum lies in the 15 −
20 GeV/ c range for 60% of these events, due to the degradation ofthe π detection efficiency for photons emitted at small angles (section 5.6). The measurementof f kin is based on a K + → π + π sample selected from minimum-bias data. The selectioninvolves the K + → π + decay definition described in sections 5.1, 5.2 and 5.5. The conditionsof section 5.3 are applied as well, however the decay region is defined as 115 < Z vertex <
165 m.Specific selection criteria are employed to tag the π by reconstructing two photons from the π → γγ decay in the LKr calorimeter independently of the π + and K + tracks. The quantity Z vertex is evaluated from the coordinates of the two photon energy clusters in LKr by assumingthat they originate from a π decay on the nominal beam axis. The vertex is required to bewithin the decay region, and its position is used to reconstruct the photon and the π momenta.Consequently, the expected π + trajectory is reconstructed and is required to be in the geometricacceptance of the detectors. The reconstructed squared missing mass ( P K − P ) , where P K and P are the four-momenta of the nominal K + and the reconstructed π , peaks at the squared π + mass for K + → π + π decays. A cut on this quantity is applied to select an almost background-free K + → π + π sample without biasing the m reconstruction.Figure 14 (top left) displays the m spectrum of the K + → π + π control sample used for f kin measurement: f kin is evaluated for each of the signal regions 1 and 2 as the ratio of thenumbers of events in the signal region and in the K + → π + π region. The simulation reproducesthe tails within the statistical uncertainties, and the background is negligible. The measuredvalues of f kin in bins of π + momentum are shown in Figure 14 (bottom left). Incorrect K/π association due to the pileup in the GTK accounts for 50% of the contribution to f kin in region32able 7: Expected numbers of K + → π + π ( γ ) events in the signal regions, and expected andobserved numbers of events in the control regions. The uncertainties are the sums in quadratureof the statistical and systematic ones. Radiative decays are kinematically forbidden in bothsignal region 1 and control region 1.Region Expected K + → π + π Expected K + → π + π γ ObservedSignal region 1 0 . ± . − maskedSignal region 2 0 . ± .
03 0 . ± .
02 maskedControl region 1 2 . ± . − . ± . . ± . π + momentum bin, is N exp ππ = 0 . ± . stat ± . syst . The statistical uncertainty ismainly due to N ππ . The systematic uncertainty accounts for a possible bias to the shape of the m spectrum induced by the π tagging used to measure f kin . It is evaluated by comparingthe simulated shape of the m spectrum in the control sample with that of K + → π + π decays used for normalization (Figure 14, top right). The 5% difference between the numbersof events in region 1 in the two samples is taken as a systematic uncertainty.Radiative decays K + → π + π γ are modelled according to [28]. Simulation studies showthat decays with radiative photons energetic enough to shift the reconstructed m value tothe signal regions are absent in the control sample, due to the π tagging suppression (Figure 14top right). However the presence of an additional photon in the final state improves the photonveto, compensating for the weaker kinematic suppression. The contribution of radiative decaysto the background is computed by applying the measured single photon detection efficiency(section 5.6) to the simulated K + → π + π γ decays. It is concluded that the presence of anadditional photon improves the K + → π + π γ rejection in region 2 by a factor of almost 30with respect to the case of the photons from the π decay only. This leads to an expectedbackground of 0.02 events. A systematic uncertainty of 100% is conservatively considered forthis value, mainly due to the modelling of the single photon detection efficiency.The numbers of expected K + → π + π ( γ ) events in the signal regions are presented inTable 7. The overall background expected in the signal regions is N exp ππ = 0 . ± . stat ± . syst . (12)To validate this result, the numbers of expected and observed events are compared in thetwo π + π control regions. The probability f kin for the control regions is measured to be about25 times higher than for the corresponding signal regions, and the expected background scalesaccordingly. Table 7 and in Figure 14 (bottom right) present the numbers of expected andobserved events in the control regions, found to be in good agreement. The uncertainties in theexpected background in the control regions are mostly systematic due to the modelling of the m spectrum. K + → µ + ν decay After the PNN selection, N µν = 479 events from the PNN sample remain in the µν region. Thenumbers of events in bins of reconstructed π + momentum are presented in Table 8, and the33 .02 - ] /c [GeV miss2 m ) / c E n t r i e s / ( . G e V Region 1 Region 2
Data p + pfi + MC K p p + pfi + MC K uncertainty p + pfi + MC K - ] /c [GeV miss2 m ) / c E n t r i e s / ( . G e V Region 1 Region 2
Data tagging) p (no p + pfi + MC K
15 20 25 30 35 40 momentum [GeV/c] + p · k i n f Data Region 1+2 uncertainty) s Data Region 1 (1 uncertainty) s Data Region 2 (1
15 20 25 30 35 40 momentum [GeV/c] + p E v e n t s / ( G e V / c ) expected p + pfi + KData
Figure 14:
Top left : reconstructed m distribution of the K + → π + π control data eventsselected by tagging the π (full symbols, see text for details) integrated over the 15 −
35 GeV /c momentum range. Simulated samples of K + → π + π decays and backgrounds (normalizedto the data in the K + → π + π region) are superimposed. Signal regions 1 and 2 are shown.The K + → π + π region, defined by the condition 0 . < m < .
021 GeV /c , and thecontrol regions, comprised between the signal and K + → π + π regions, are not shown. Topright : same as top-left, but the simulated K + → π + π sample is selected without applyingthe π tagging; simulated backgrounds are not shown. Bottom left : the probability f kin ,defined in the text, measured using the K + → π + π control sample in bins of π + momentum,separately for signal region 1 and 2 and combined, with the statistical uncertainties. Bottomright : expected and observed numbers of background events in the K + → π + π ( γ ) decaycontrol regions in π + momentum bins. The errors are statistical for the observed numbers ofevents, and dominated by systematics for the expected numbers of events.34able 8: Number of PNN-triggered events that pass the PNN selection and are reconstructedin the µν region, observed in the data in bins of π + momentum.Momentum bins (GeV/ c ) 15 −
20 20 −
25 25 −
30 30 − π + momentum, m ) plane is shown in Figure 15 (top left).The momentum dependence is a consequence of the K + → µ + ν kinematics when the π + massis used to reconstruct m and of the better performance of the RICH in rejecting µ + at lowmomentum. The background to N µν is negligible.The measurement of f kin is based on a K + → µ + ν sample selected from minimum-bias data.The selection proceeds as described in sections 5.1 and 5.2. Additionally, the calorimetric BDTprobability must be consistent with the identification of a µ + , while events are discarded if theSTRAW track is identified as π + or e + in the calorimeters. The decay region is defined as115 < Z vertex <
165 m. The rejection of photons and extra charged particles is the same as inthe PNN selection (section 5.6). The box cut and the kinematic requirements on m are notapplied. Figure 15 (top right) displays the m spectrum of the K + → µ + ν control sample usedfor f kin measurement: f kin is evaluated for each of the signal regions 1 and 2 as the ratio of thenumbers of events in the signal region and the µν region. The signal region definition does notinclude the cuts on m computed using the momentum evaluated from the RICH information.Simulation reproduces the m spectrum shape within the statistical uncertainties, and thebackground is negligible. The measured f kin values in bins of reconstructed π + momentum areshown in Figure 15 (bottom left). At large momentum, f kin increases because the m of K + → µ + ν events computed assuming the π + mass approaches signal region 1. Simulationsshow that the contribution to f kin due to incorrect K/π association is sub-dominant with respectto material effects.The total expected background in the signal region, evaluated by applying Equation (11) ineach π + momentum bin, is N exp µν = 0 . ± . stat ± . syst . The statistical uncertainty isdue to N µν . The systematic uncertainty is evaluated from the stability of the result with thevariation of the BDT probability cut, and accounts for a possible bias on f kin due to the µ + identification criteria applied in the selection of the K + → µ + ν control sample.The result relies on the assumption that the π + identification with the RICH and the shapeof the m spectrum are uncorrelated. This assumption, in principle, is violated because K + → µ + ν events may enter the signal region due to track mis-reconstruction, which alsoaffects particle identification with the RICH. In addition, the background estimation proceduredoes not include the cut on m computed using the RICH to measure f kin , which can biasthe result. To investigate the accuracy of these approximations, the K + → µ + ν backgroundevaluation procedure is modified as follows: the π + identification by the RICH is removed fromthe PNN selection used for the measurement of N µν ; it is added instead to the selection of the K + → µ + ν control sample used to measure f kin ; and the determination of f kin includes the cutson m computed using the π + momentum measured from the RICH in the π + hypothesis.With these modifications, it is found that 1% of the events in the µν region are K + → e + ν decays. The µ + rejection by the RICH suppresses f kin by two orders of magnitude with respectto that of Figure 15 (bottom left), keeping a similar dependence on the π + momentum. Thistranslates into a statistical uncertainty of 20% on f kin . The expected background in the signalregion is evaluated in these conditions to be N exp µν = 0 . ± . stat . This method is free from35 momentum [GeV/c] + p - - - - - ] / c [ G e V m i ss m Data - ] /c [GeV miss2 m ) / c E n t r i e s / ( . G e V Region 1 Region 2
Data n + mfi + MC KMC Uncertainty
15 20 25 30 35 40 momentum [GeV/c] + p · k i n f Data Region 1+2 uncertainty) s Data Region 1 (1 uncertainty) s Data Region 2 (1
15 20 25 30 35 40 momentum [GeV/c] + p E v e n t s / ( G e V / c ) expected n + mfi + KData
Figure 15:
Top left : distributions of the PNN-triggered events in the ( π + momentum, m )plane after the PNN selection in the µν (red contour) and control m (blue contour) re-gions. The control region is used only for validation of the background estimation. Top right :reconstructed m distribution of the K + → µ + ν ( γ ) control data events (full symbols, seetext for details) integrated over the 15–35 GeV/ c momentum range. The distribution of sim-ulated K + → µ + ν ( γ ) decays is superimposed. Signal regions 1 and 2 are shown. Bottomleft : the probability f kin measured using the K + → µ + ν ( γ ) control sample in bins of recon-structed π + momentum, separately for signal region 1 and 2 and combined (black symbols), andthe corresponding statistical uncertainties. Bottom right : expected and observed numbers ofbackground events from K + → µ + ν ( γ ) decays in the µν control region in π + momentum bins.The errors are statistically dominated. 36able 9: Expected numbers of K + → µ + ν ( γ ) events in signal regions, and numbers of expectedand observed events in the control region. The uncertainties are the sums in quadrature of thestatistical and systematic ones.Region Expected K + → µ + ν ( γ ) Expected K + → µ + ν , µ + → e + ν ¯ ν ObservedSignal region 1 0 . ± .
04 0 . ± .
02 maskedSignal region 2 < . < .
005 maskedControl region 10 . ± . . ± . µ + rejection with the RICHand the calorimeters are uncorrelated. Simulations show that this assumption is valid as longas the µ + does not decay upstream of the RICH.The average of the two estimates of N exp µν is used, and a systematic uncertainty equal to halfof the difference ( ± .
03) is assigned to account for a possible bias due to the correlation betweenparticle identification and shape of the m spectrum. This background estimate accounts alsofor radiative K + → µ + νγ decays, as they enter the control sample used to evaluate f kin .The background from muon decays in flight µ + → e + ν ¯ ν is not included in the above es-timate (as f kin is measured requiring muon identification), and is determined separately usingsimulation. The rejection of this background depends on the muon decay position within the de-tector setup. Decays in the FV affect the kinematics, however, positrons are efficiently rejectedby particle identification. Decays within the STRAW spectrometer impact both kinematics andparticle identification in the RICH, while decays downstream of the STRAW affect particle iden-tification only. Simulations show that only µ + decays between the third and fourth STRAWchambers are relevant, leading to a worsening of both K + → µ + ν kinematics and particleidentification. The background is found to contribute to region 1 only, and is computed to be0 . ± .
02. The uncertainty quoted includes statistical and systematic contributions of similarmagnitudes. The latter is evaluated by checks performed on data to validate the simulation ofthe positron rejection.The numbers of expected K + → µ + ν ( γ ) events in the signal regions are presented in Table 9.The overall background expected is N exp µν = 0 . ± . stat ± . syst , (13)with the contributions to the statistical and systematic uncertainties detailed above.To validate this result, the numbers of expected and observed events are compared in the µν control region. The expected number of events is evaluated with a technique similar to thatdescribed above. This comparison is presented in Table 9 and Figure 15 (bottom right), showinggood agreement between expected and observed numbers of events. K + → π + π + π − decay After the PNN selection, N π = 161 events from the PNN sample remain in the 3 π region. Thedistribution of these events in the ( π + momentum, m ) plane is shown in Figure 16 (left); the π + momentum is constrained kinematically to the region below 25 GeV/ c .The measurement of f kin is based on a K + → π + π + π − sample selected from minimum-biasdata. For this purpose, a π + π − pair is used to tag the K + → π + π + π − decay without biasingthe reconstruction of the unpaired π + . The π + to be paired to the π − is chosen randomly event37 momentum [GeV/c] + p ] / c [ G e V m i ss m Data nn + pfi + SM K
0. 0.02 0.04 0.06 0.08 0.10 0.12 ] /c [GeV miss2 m ) / c E n t r i e s / ( . G e V Region 1 Region 2 tagging) p Data (2 tagging) p (2 - p + p + pfi + MC KMC Uncertainty
Figure 16:
Left : distribution of the PNN-triggered events in the ( π + momentum, m ) planeafter the PNN selection in the 3 π (red box) and control m regions (blue box). No event isfound in the control region. The shaded grey area represents the distribution of the simulatedSM K + → π + ν ¯ ν events (arbitrarily normalized). Right : reconstructed m distribution forthe unpaired π + from K + → π + π + π − decays, obtained using the 2 π tagging method, selectedfrom control data and simulated samples. Signal regions 1 and 2 are shown.by event. The presence of a two-track vertex in the FV is required, and the quantity ( P π + + P π − − P K + ) , where P π ± are the reconstructed 4-momenta of the π ± and P K + is the nominalkaon 4-momentum, must be consistent with the squared π + mass. The selection proceeds withrespect to the unpaired π + as described in sections 5.1 and 5.2. Photon veto conditions areapplied to the LAV, IRC and SAC only. The box cut, the kinematic cuts on m and themultiplicity rejection are not applied.The reconstructed m spectra for the tagged unpaired π + for the control data and simu-lated samples are shown in Figure 16 (right). It is found that f kin = (1 . +2 . − . ) × − , where theuncertainties are statistical. Data and simulations are consistent within the uncertainties.The kinematics of the tagged π + differs from that of the π + remaining after the PNNselection, potentially biasing the f kin measurement. In particular, the m spectrum of thetagged sample does not match the one of the residual events in the 3 π region. The impact onthe f kin measurement is evaluated with simulations, varying the selection criteria. The full PNNselection cannot be applied to the simulated samples due to statistical limitations. ModifiedPNN selections used for the tests include those without the tagging, and with requirements of atleast one and exactly one π + reconstructed in the geometric acceptance. The latter selection isthe most PNN-like, and leads to a shape of the m spectrum in the 3 π region matching thatof the data events passing the full PNN selection. The values of f kin obtained from simulationswith the modified selections are in agreement within the uncertainties quoted above.A possible bias comes from the dependence of f kin on the Z -position of the decay vertex, asthe tagging affects the shape of the Z vertex spectrum. To quantify this effect, f kin is evaluatedin bins of Z vertex for data and simulated samples. The variation of f kin across Z vertex bins insimulated samples is conservatively considered as a systematic uncertainty. The final result is f kin = (5 ± × − . 38he background computed using Equation (11) is N π = 0 . ± . . (14)To validate this result, the numbers of expected and observed events are compared in the (un-masked) 3 π control region. The expected number of events in the control region is sensitiveto the shape of the m spectrum close to the kinematic threshold of the K + → π + π + π − decay. Simulation studies lead to a conservative upper limit of 1 . × − on f kin in the controlregion, corresponding to less than 0.24 expected background events. This is consistent with theobservation of zero events in the control region. K + → π + π − e + ν decay The K + → π + π − e + ν decay (denoted K e below) is characterized by large m and thereforecontributes to region 2 only. This background is suppressed by the O (10 − ) branching ratio [8],the kinematic definition of the signal region, and the multiplicity rejection. The reconstructed m value depends on the kinematics of the undetected charged particles, which impacts themultiplicity rejection. Because of this correlation, the K e background estimation relies onsimulation.The efficiency of the PNN selection evaluated with a sample of 2 × simulated K e decaysusing the same normalization procedure as for the SES computation is ε Ke = (4 ± stat ) × − .This leads to an estimated background of N Ke = (0 . ± . stat ) events. To validate thisestimate, four modified event selections leading to samples enriched with K e decays are used:1. the PNN selection, with inverted multiplicity conditions in the STRAW;2. the PNN selection applied to the π − with RICH identification criteria not used, andinverted multiplicity conditions in the STRAW;3. similar to 2, with the standard STRAW multiplicity conditions used; and4. similar to 3, with RICH identification criteria used.The selection efficiency for K e decays ranges from 1 . × − (selection 4) to 3 . × − (selection2). The corresponding data events entering region 2 are solely K e . The reconstructed m distributions obtained within selection 2 for PNN data and simulated K e sample show agree-ment within the statistical uncertainties (Figure 17 (left)). The expected and observed numbersof events in region 2 within each of the four selections are summarized in Figure 17 (right).In particular, 3 ± stat events are expected and 6 ± stat events are observed within selection 4which has the lowest acceptance. This difference is conservatively considered as a systematicuncertainty in N exp Ke , despite the agreement within the statistical uncertainties, leading to theexpected background from K e decays N Ke = 0 . ± . stat ± . syst . (15) K + decaysSemileptonic decays: the branching ratios of the semileptonic decays K + → π e + ν and K + → π µ + ν are 5.1% and 3.4%, respectively [8]. The presence of the neutrino in the finalstate prevents kinematic discrimination of these decays from the signal, and the background issuppressed by exploiting the presence of a π and a lepton in the final state. The backgroundestimation relies on simulation, with a factorization approach used to overcome unavoidable39 .03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 ] /c [GeV miss2 m ) / c E n t r i e s / ( . G e V n + e - p + pfi + MC KData 1 2 3 4
Sample N u m b er o f e v e n t s ExpectationData
Figure 17:
Left : reconstructed m distribution of the events of sample 2 selected in signalregion 2 for data and K e simulation. Right : expected number of K e decays and observednumber of data events in region 2 for each of the four samples used to validate the K e simulation.The different samples are defined in the text.statistical limitations. Particle identification in the RICH and calorimeters are treated as inde-pendent, and the corresponding efficiencies are factored out with respect to the efficiency of therest of the selection.The measured muon misidentification probability as a pion in the RICH detector depends onthe particle momentum (section 5.5). On average such a probability is about 2 × − for K + → π µ + ν decays passing the PNN selection; this result is used to validate the simulations. Positronmisidentification probability as a pion in the RICH detector in the 15–35 GeV/ c momentumrange is evaluated with a simulated sample to be about 10 − . Calorimetric muon and positronmisidentification probabilities as a pion in this momentum range evaluated with simulations areabout 10 − and 10 − , respectively.The simulation accounts for the joint effect of π rejection, and the geometric and kinematicacceptances. Simulations show that the former is about 10 − (substantially weaker than for K + → π + π decays due to the different photon kinematics), while the latter is about 10%.The K + → π + π decay is used for normalization. This leads to a systematic uncertainty inexcess of 10%, mostly because the particle identification efficiencies do not cancel in the ratiowith that of K + → π + π . Including the measured random veto and trigger efficiencies, theexpected background is found to be less than 0.001 events for both decay modes, and is thereforeconsidered negligible. K + → π + γγ : the branching ratio of this decay, occurring at the loop level, is 1 . × − [8]. Thecorresponding background is evaluated with simulations. The decay dynamics favours valuesof the di-photon invariant mass above the di-pion threshold, corresponding to m valuesin the 3 π region. This procedure leads to an overall efficiency of the PNN selection withoutphoton rejection at the 1% level. The rejection of events in the signal region benefits from thecorrelation between m and the photon energy, leading to a photon rejection of order 10 . The K + → π + π decay is used for normalization. Including the measured random veto and triggerefficiencies, the background is estimated to be N πγγ = 0 . ± . Upstream events are defined as interactions or decays of beam particles upstream of the FV. Anupstream event can mimic a K + → π + ν ¯ ν decay if: • a π + is produced and reaches the downstream detectors; • no additional particles associated to the π + are detected downstream; and • a K + candidate is reconstructed and matched to the π + .Based on these conditions, upstream events can be classified as follows:1. Accidental upstream events: events in which the π + does not originate from the recon-structed K + candidate. In this case the K + candidate is a pileup GTK track associatedaccidentally with the π + and tagged as a kaon by the KTAG.The mechanisms giving rise to accidental upstream events are the following:a) the π + comes from a K + decaying in the region upstream of GTK3; the KTAGsignal produced by the parent K + is associated with a pileup beam π + or protontrack, which is reconstructed as a kaon in the GTK; additional particles produced inthe decay are absorbed by material in the beam line;b) similar to a), but the matching GTK track belongs to another pileup K + identifiedcorrectly by the KTAG;c) similar to a), but the π + originates from an inelastic interaction of a beam K + upstream of GTK3;d) similar to b), but the π + originates from an inelastic interaction of a beam π + orproton upstream of GTK3.2. In-time upstream events: events in which the π + is a primary or a secondary productof an inelastic interaction of a beam K + in GTK3. In this case, additional particlesproduced in the interaction must escape detection, as no beam line elements can absorbthe particles.Two processes may lead to in-time upstream events:a) the interacting K + produces a prompt π + that reaches the downstream detectors;b) the interacting K + produces a relatively long-lived particle ( K S , K L , K + or Λ) thatdecays to a π + in the FV.The evidence for the above classification comes from studies based on data and simulatedsamples. The PNN selection is modified as follows to provide an almost pure sample of upstreamdata events: the matching conditions for the K + candidate and the π + track are not applied;no constraints are applied to the reconstructed Z vertex ; the box cut is not applied; and CDA > K/π matching of the PNN selectionis not satisfied, therefore the signal m regions can be explored in the PNN data samplewithout violating the blind analysis principle. The distribution of m for the selected dataand simulated events is shown in Figure 18: simulated upstream events explain the shape ofthe data. The sample is dominated by K + → π + π + π − and K + → π + π decays occurringdownstream of the first GTK station (GTK1).41 .1 - - ] /c [GeV miss2 m A r b i t r a r y un i t s DataMC accidental upstream (Type 1a,b)MC interactions (Types 1c,d + 2a,b)
Figure 18: Reconstructed m distributions of PNN data sample and simulated samples ob-tained from the upstream event selection described in the text.The X, Y coordinates of the pions selected in the data sample, obtained by extrapolating theirtracks to the (
X, Y ) plane of the final collimator, are shown in Figure 19 (left). In most cases,the pion passes through the beam hole in the collimator. The shape of the distribution outsideof the hole is determined by the material in the beam line: most of the pions outside the holeare contained in the aperture of the last dipole magnet of the beam line. Pions from upstreamin-time events, originating from GTK3, have an
X, Y distribution at the final collimator whichoverlaps with the
X, Y distribution of pions from accidental upstream events. The box cut usedin the PNN selection, | X | <
100 mm, | Y | <
500 mm (section 5.6) is defined to exclude the wholeaperture of the magnet.The time structure of the selected upstream events is shown in Figure 19 (right). Accidentalcoincidence between KTAG and GTK signals is necessary to reconstruct a K + candidate inevents with a K + decaying or interacting upstream of GTK3. On the other hand, the π + inthese events produces a RICH signal in time with the KTAG signal of the parent K + . Thereforeaccidental upstream events of types a) and c) populate the horizontal band in the timing plot.Accidental upstream events of types b) and d) require a pileup K + in the GTK. In this case oneof the two KTAG candidates and the GTK track are in time, while the π + signal in the RICHaccidentally coincides with the same KTAG candidate. As a consequence, accidental upstreamevents of types b) and d) form the vertical band in the timing plot. In-time upstream eventspopulate the central region of the plot. The distribution of data events in the central region isconsistent with that formed by the overlap of the horizontal and vertical bands, and indicatesthat in-time upstream events account for less than 10% of the sample, which is in agreementwith simulations.The PNN selection criteria mostly effective against accidental upstream background are: • the K/π association: a coincidence between the two independent particles can only occuraccidentally; • the Z vertex conditions defining the FV: a K + → π + decay vertex can only be reconstructed42 - - T(GTK - KTAG) [ns] D - - T ( K T A G - R I C H ) [ n s ] D Figure 19:
Left : extrapolation of π + tracks in the upstream data sample described in the textto the ( X, Y ) plane at the Z -position of the last collimator. The blue lines correspond to thelast dipole of the second achromat; the contour of the final collimator is shown with a red line. Right : time difference between KTAG and RICH versus GTK and KTAG for the π + s shownon the left plot.in the FV accidentally; • the box cut: the π + satisfies this condition only if mis-reconstructed or suffering large-anglescattering at STRAW1; • the rejection of events with extra hits in at least two GTK stations: the beam particleproducing the π + disappears along the beam line in the GTK region; simulations indicatethat π + produced upstream of GTK1 cannot reach the FV; mostly K + travelling outsidethe GTK acceptance can pass this condition, as the probability to lose a hit in a GTKstation is negligible.The first three criteria also suppress in-time upstream events along with the CHANTI vetoconditions.An accidental upstream event of type a) contributing to the background is sketched inFigure 20. The parent kaon decays ( K + → π + π ) downstream of GTK2. The photons from π → γγ decay are absorbed by the final collimator, while the π + propagates in the magnetic fieldthrough the collimator aperture. Finally, the π + direction is modified by large-angle scatteringat STRAW1. The evaluation of the upstream background in the PNN sample does not rely on Monte Carlosimulation, but follows a data-driven approach. A sample of PNN data enriched with upstreamevents, called the “upstream sample” below, is selected using modified PNN criteria: CDA > K/π matching conditions. The number of events from the PNNsample passing this selection is N data = 16. The background from K + decays in the FV in thissample is estimated to be 0.2 events by analysing background regions of m with the methodsdescribed in sections 7.1.1 and 7.1.2. 43igure 20: Sketch of an accidental upstream event of type a) in the horizontal plane (not to scale).The GTK stations GTK1, GTK2 and GTK3 are displayed together with the final collimator.The reconstructed π + fails the box cut because of the large-angle scattering at STRAW1.The upstream background is evaluated considering the probability P mistag that an upstreamevent satisfies the K/π matching criteria. This probability depends only on the shape of theCDA distribution and the time difference ∆T(GTK–KTAG) for the events in the horizontalband of Figure 19 (right), and ∆T(KTAG–RICH) for the events in the vertical band. The CDAdistribution model is established from simulations of accidental upstream events. This modelis validated using a data sample selected similarly to the upstream sample with the followingmodifications: GTK and CHANTI veto conditions are removed, the condition CDA > . < | T(KTAG–GTK) | < P mistag evaluated with simulations in bins of ∆ T is shown in Figure 21 (right). The number of upstreambackground events is estimated in each of the two bands shown in Figure 19 (right) as N upstream = f scale · (cid:88) i =1 N idata P i mistag , (16)where the sum runs over twelve 100 ps wide ∆T bins covering the ( − . , .
6) ns range; N i data isthe number of events found in the upstream sample in bin i , P i mistag is the corresponding mis-tagging probability shown in Figure 21 (right), and f scale = 1 .
06 accounts for upstream eventswith CDA ≤ N data definition. The last factor is obtained from a studyof the T(GTK–KTAG) sidebands, as data and simulations show that the CDA is independentof this quantity.The procedure described above is validated using seven different data samples selected mod-ifying the PNN criteria as follows:1. | X col | <
100 mm, | Y col | <
140 mm for the pion position in the final collimator plane,replacing the box cut;2. | X col | <
100 mm, | Y col | ≥
140 mm, replacing the box cut;44
10 20 30 40 50 60 70 80 90 100
CDA [mm] N o r m a li ze d e n t r i e s / ( mm ) decays in fiducial region + KUpstream events from dataUpstream background model
T(GTK - KTAG)| [ns] D | T | ) D ( | m i s t ag P Figure 21:
Left : CDA distribution of upstream events selected as described in the text. Thedistribution is compared with a model obtained from simulations. The CDA distribution of dataevents coming from K + decays in the FV is also shown. Right : probability for an upstreamevent to satisfy the
K/π matching conditions as a function of ∆T(GTK–KTAG) obtained usingthe CDA model shown in the left plot.3. m < − .
05 GeV /c , replacing the signal region mass definition;4. as 1), without GTK and CHANTI veto conditions;5. as 2), without GTK and CHANTI veto conditions;6. as 3), without GTK and CHANTI veto conditions;7. GTK and CHANTI veto conditions inverted.Simulations show that the contributions of the various types of upstream background differamong the samples. The numbers of expected and observed background events in each sampleare presented in Figure 22: they agree within one standard deviation in each sample.The number of expected upstream background events is found to be N upstream = 0 . ± . stat ± . syst . (17)The statistical uncertainty stems from N data . A systematic uncertainty of 12% is due to themodelling of the CDA distribution, and is derived from the comparison between data and simu-lations. An additional systematic uncertainty of 20% is assigned as half of the difference betweenthe expected and observed number of events in sample 6 (with statistics similar to the upstreamsample). This uncertainty accounts for the accuracy of the assumption that all the categoriesof upstream events have the same CDA distribution. As an additional check, the expected and observed numbers of events in the PNN sample arecompared in a control region defined by the same m range as the signal regions 1 and 2 but inthe 35–40 GeV /c π + momentum range. The expected number of background events is between0.4 and 0.8 at 90% CL, almost equally shared between K + decays in the FV and upstream45 Sample N u m b er o f e v e n t s ExpectationData
Figure 22: Number of expected and observed events in the seven different upstream backgroundvalidation samples.events. The corresponding expected number of SM K + → π + ν ¯ ν events is 0 . ± .
02. Oneevent is observed with a π + momentum of 38 GeV/c and m (cid:39) .
03 GeV /c , in agreementwith the expectation.The expected backgrounds are summarized in Table 10. After unmasking the signal regions, two candidate events are found, as shown in Figure 23.The second and third columns of Table 11 summarize the characteristics of these events.Figure 24 shows the m distribution of the events with momentum between 15 and35 GeV/ c passing the PNN selection, compared with that expected from SM K + → π + ν ¯ ν de-cays and from the various sources of background. In this plot the m distribution of the K + → π + π , K + → µ + ν and K + → π + π + π − decays come from the control samples selectedon minimum-bias data, and normalized to the number of events in the corresponding back-ground regions (sections 7.1.1, 7.1.2 and 7.1.3). The distribution of the m of the upstreambackground is extracted from an upstream-event-enriched data sample and is normalized to thenumber of upstream background events expected in the signal regions. The distributions of theother background sources are modelled using MC simulations and normalized to the expectednumber of events in the signal regions.The two candidate K + → π + ν ¯ ν events of this analysis complement the one found by NA62in the same signal region from the analysis of the 2016 data [22]. The characteristics of the 2016candidate are displayed in the fourth column of Table 11. Table 12 summarizes the numericalresults obtained in the K + → π + ν ¯ ν analysis of the 2017 and 2016 independent data samples.The statistical interpretation of the result is obtained from an event counting approachin the full range of the signal region. The level of the expected background does not allowa claim of signal observation nor a claim of inconsistency with the presence of SM K + → π + ν ¯ ν decays. Therefore both an upper limit and a measurement of the branching ratio of the K + → π + ν ¯ ν decay are presented.A fully frequentist hypothesis test, with a profile likelihood ratio as test statistic, is used46able 10: Expected numbers of SM K + → π + ν ¯ ν decays and of background events in the signalregions. Process Events expected K + → π + ν ¯ ν (SM) 2 . ± . syst ± . ext K + → π + π ( γ ) 0 . ± . stat ± . syst K + → µ + ν ( γ ) 0 . ± . stat ± . syst K + → π + π − e + ν . ± . stat ± . syst K + → π + π + π − . ± . syst K + → π + γγ . ± . syst K + → π (cid:96) + ν ( (cid:96) = µ, e ) < . . ± . stat ± . syst Total background 1 . ± . stat ± . syst to combine the results of the 2017 and 2016 analyses. The parameter of interest is the signalstrength µ defined as the branching ratio in units of the Standard Model one. The nuisanceparameters are the total expected number of background events in the signal regions ( B ) and thesingle event sensitivity ( SES ), obtained separately from the 2016 and 2017 datasets. Followingthe method described in [29] and according to [30], the number of background events is con-strained to follow a Poisson distribution with mean value (
B/δ B ) where δ B is the uncertaintyof B (Table 12). The mean ( B/δ B ) accounts for an equivalent number of events counted incontrol regions through the auxiliary measurements leading to B as described in section 7. Alog-normal distribution function is used to constrain the SES around the measured value.The likelihood functions of the results of the 2016 and 2017 analyses are multiplied to form asingle combined function, which is profiled with respect to the nuisance parameters. The upperlimit on the branching ratio of the K + → π + ν ¯ ν decay is obtained using a CL S method [31]for several values of the signal strength µ (Figure 25). The 90% CL expected upper limit isBR( K + → π + ν ¯ ν ) < . × − and the observed one is:BR( K + → π + ν ¯ ν ) < . × − . (18)This result translates to a Grossman-Nir limit [32] of the SM K L → π ν ¯ ν branching ratio equalto 7 . × − .The data also allow the setting of a 68% CL interval on the SM branching ratio of the K + → π + ν ¯ ν decay. Using the prescriptions of [33] and [34], the measured K + → π + ν ¯ ν branchingratio is: BR( K + → π + ν ¯ ν ) = (0 . +0 . − . ) × − . (19) An investigation of K + → π + ν ¯ ν has been performed using the data collected by the NA62experiment at CERN in 2017. The experiment has reached the best single event sensitivity sofar in this decay mode, corresponding to (0 . ± . syst ) × − . This translates into anexpectation of (2 . ± . syst ± . ext ) K + → π + ν ¯ ν events in the signal regions, assuming the47igure 23: Reconstructed m as a function of π + momentum for PNN events (full symbols)satisfying the PNN selection, except the m and π + momentum criteria. The grey areacorresponds to the expected distribution of SM K + → π + ν ¯ ν MC events (arbitrarily normalized).Red contours define the signal regions. The events observed in the signal regions are showntogether with the events found in the background and control regions.Standard Model BR of (8 . ± . × − . A further 1.5 background events are expected in thesame signal regions, mainly due to a single π + produced along the beam line upstream of the K + decay volume and accidentally matched to a beam kaon. Using a blind analysis procedure,two candidate events have been observed in the signal regions, consistent with expectation.These two candidates, together with the single candidate observed from the analysis of the2016 data, lead to the most stringent upper limit on the branching ratio BR( K + → π + ν ¯ ν ) < . × − at 90% CL and set the Grossman-Nir limit on BR( K L → π ν ¯ ν ) to 7 . × − . Thecorresponding 68% CL measurement of the K + → π + ν ¯ ν branching ratio is (0 . +0 . − . ) × − .This result constrains some New Physics models that predict large enhancements previouslyallowed by the measurements published by the E787 and E949 BNL experiments. The NA62experiment has collected and is now analysing almost twice as much data in 2018 as that reportedupon here, and further optimization of the analysis strategy is expected significantly to reducethe uncertainty in the measured BR of the K + → π + ν ¯ ν decay.48 .02 - ] /c [GeV miss2 m - -
10 110 ) / c E n t r i e s / ( . G e V Region 1 Region 2
Data nn + pfi + K ) g ( n + mfi + K + e fi + m , n + mfi + K p + pfi + K g p + pfi + K - p + p + pfi + K n + e - p + pfi + K n + e fi + KUpstream
Figure 24: Reconstructed m distribution of data events with π + momentum between 15 and35 GeV/ c , passing the PNN selection (full symbols). The expected background and SM signalevents contributions are superimposed as stacked histograms. The m distributions of the K + → π + π , K + → µ + ν and K + → π + π + π − decays and of the upstream events are extractedfrom data. The other contributions are obtained from simulations.Table 11: Observed events in the signal regions after PNN selection. Events 1 and 2 come fromthe analysis of the 2017 data presented here. Event 3 comes from the analysis of the 2016 data.Event 1 Event 2 Event 3Year 2017 2017 2016 m .
038 GeV /c .
064 GeV /c .
031 GeV /c π + momentum 26 . /c . /c . /cZ vertex
140 m 159 m 146 m∆T(KTAG - GTK) − .
171 ns 0 .
028 ns 0 .
006 ns∆T(RICH - KTAG) − .
082 ns 0 .
209 ns 0 .
040 ns(
X, Y ) at Final Collimator (228 . , .
1) mm (189 . , − .
7) mm ( − . , .
8) mm49able 12: Summary from the K + → π + ν ¯ ν analyses of the data recorded in 2017 and 2016.2017 2016Single Event Sensitivity SES (0 . ± . × − (3 . ± . × − Expected SM K + → π + ν ¯ ν decays 2 . ± . ± . ext . ± . ± . ext Expected background B ± δ B . ± .
30 0 . ± . ) n n + p fi + (K SM BR ) n n + p fi + BR(K = m - -
10 1 S C L Observed CLsExpected CLs - Median s – Expected CLs s – Expected CLs
Figure 25: CL S p -values as a function of the branching ratio of the K + → π + ν ¯ ν decay expressedin units of the Standard Model value. The red (blue) line corresponds to the 90% (95%) CL.50 cknowledgements It is a pleasure to express our appreciation to the staff of the CERN laboratory and the technicalstaff of the participating laboratories and universities for their efforts in the operation of theexperiment and data processing.The cost of the experiment and its auxiliary systems was supported by the funding agenciesof the Collaboration Institutes. We are particularly indebted to: F.R.S.-FNRS (Fonds de laRecherche Scientifique - FNRS), Belgium; BMES (Ministry of Education, Youth and Science),Bulgaria; NSERC (Natural Sciences and Engineering Research Council), funding SAPPJ-2018-0017 Canada; NRC (National Research Council) contribution to TRIUMF, Canada; MEYS(Ministry of Education, Youth and Sports), Czech Republic; BMBF (Bundesministerium f¨urBildung und Forschung) contracts 05H12UM5, 05H15UMCNA and 05H18UMCNA, Germany;INFN (Istituto Nazionale di Fisica Nucleare), Italy; MIUR (Ministero dell’Istruzione, dell’Universit`ae della Ricerca), Italy; CONACyT (Consejo Nacional de Ciencia y Tecnolog´ıa), Mexico; IFA (In-stitute of Atomic Physics) Romanian CERN-RO No.1/16.03.2016 and Nucleus Programme PN19 06 01 04, Romania; INR-RAS (Institute for Nuclear Research of the Russian Academy ofSciences), Moscow, Russia; JINR (Joint Institute for Nuclear Research), Dubna, Russia; NRC(National Research Center) “Kurchatov Institute” and MESRF (Ministry of Education and Sci-ence of the Russian Federation), Russia; MESRS (Ministry of Education, Science, Research andSport), Slovakia; CERN (European Organization for Nuclear Research), Switzerland; STFC(Science and Technology Facilities Council), United Kingdom; NSF (National Science Founda-tion) Award Numbers 1506088 and 1806430, U.S.A.; ERC (European Research Council) “Uni-versaLepto” advanced grant 268062, “KaonLepton” starting grant 336581, Europe.Individuals have received support from: Charles University Research Center (UNCE/SCI/013),Czech Republic; Ministry of Education, Universities and Research (MIUR “Futuro in ricerca2012” grant RBFR12JF2Z, Project GAP), Italy; Russian Foundation for Basic Research (RFBRgrants 18-32-00072, 18-32-00245), Russia; Russian Science Foundation (RSF 19-72-10096), Rus-sia; the Royal Society (grants UF100308, UF0758946), United Kingdom; STFC (Rutherford fel-lowships ST/J00412X/1, ST/M005798/1), United Kingdom; ERC (grants 268062, 336581 andstarting grant 802836 “AxScale”); EU Horizon 2020 (Marie Sk(cid:32)lodowska-Curie grants 701386,842407, 893101).
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