EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN-EP-2020-132

July 14, 2020

An investigation of the very rare $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay
The NA62 Collaboration

###### Abstract

The NA62 experiment reports an investigation of the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ mode from a sample of $K^{+}$ decays
collected in 2017 at the CERN SPS. The experiment has achieved a single event
sensitivity of $(0.389\pm 0.024)\times 10^{-10}$, corresponding to 2.2 events
assuming the Standard Model branching ratio of $(8.4\pm 1.0)\times 10^{-11}$.
Two signal candidates are observed with an expected background of 1.5 events.
Combined with the result of a similar analysis conducted by NA62 on a smaller
data set recorded in 2016, the collaboration now reports an upper limit of
$1.78\times 10^{-10}$ for the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ branching
ratio at 90% CL. This, together with the corresponding 68% CL measurement of
$(0.48^{+0.72}_{-0.48})\times 10^{-10}$, are currently the most precise
results worldwide, and are able to constrain some New Physics models that
predict large enhancements still allowed by previous measurements.

Accepted for publication in JHEP

The NA62 Collaboration 111Corresponding authors: G. Ruggiero, R. Marchevski,
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Université Catholique de Louvain, Louvain-La-Neuve, Belgium
E. Cortina Gil, A. Kleimenova, E. Minucci 111Corresponding authors: G.
Ruggiero, R. Marchevski,
<EMAIL_ADDRESS>radoslav.marchevski@cern.ch${}^{,}\,$222Deceased, S. Padolski 333Present
address: Brookhaven National Laboratory, Upton, NY 11973, USA, P. Petrov, A.
Shaikhiev 444Also at Institute for Nuclear Research of the Russian Academy of
Sciences, 117312 Moscow, Russia, R. Volpe 555Present address: Faculty of
Mathematics, Physics and Informatics, Comenius University, 842 48, Bratislava,
Slovakia

TRIUMF, Vancouver, British Columbia, Canada
T. Numao, B. Velghe

University of British Columbia, Vancouver, British Columbia, Canada
D. Bryman 666Also at TRIUMF, Vancouver, British Columbia, V6T 2A3, Canada, J.
Fu 777Present address: UCLA Physics and Biology in Medicine, Los Angeles, CA
90095, USA

Charles University, Prague, Czech Republic
T. Husek 888Present address: IFIC, Universitat de València - CSIC, E-46071
València, Spain, J. Jerhot 999Present address: Université Catholique de
Louvain, B-1348 Louvain-La-Neuve, Belgium, K. Kampf, M. Zamkovsky

Institut für Physik and PRISMA Cluster of excellence, Universität Mainz,
Mainz, Germany
R. Aliberti 101010Present address: Institut für Kernphysik and Helmholtz
Institute Mainz, Universität Mainz, Mainz, D-55099, Germany, G. Khoriauli
111111Present address: Universität Würzburg, D-97070 Würzburg, Germany, J.
Kunze, D. Lomidze 121212Present address: Universität Hamburg, D-20146 Hamburg,
Germany, R. Marchevski 111Corresponding authors: G. Ruggiero, R. Marchevski,
<EMAIL_ADDRESS>radoslav.marchevski@cern.ch,131313Present
address: CERN, European Organization for Nuclear Research, CH-1211 Geneva 23,
Switzerland, L. Peruzzo, M. Vormstein, R. Wanke

Dipartimento di Fisica e Scienze della Terra dell’Università e INFN, Sezione
di Ferrara, 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’Università 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 141414Also at Dipartimento di Fisica, Università di Modena e Reggio
Emilia, I-41125 Modena, Italy, F. Bucci

Laboratori Nazionali di Frascati, Frascati, Italy
A. Antonelli, G. Georgiev 151515Also at Faculty of Physics, University of
Sofia, BG-1164 Sofia, Bulgaria, V. Kozhuharov 151515Also at Faculty of
Physics, University of Sofia, BG-1164 Sofia, Bulgaria, 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 131313Present address: CERN, European
Organization for Nuclear Research, CH-1211 Geneva 23, Switzerland, D. Di
Filippo, P. Massarotti, M. Mirra, M. Napolitano, G. Saracino

Dipartimento di Fisica e Geologia dell’Università e INFN, Sezione di Perugia,
Perugia, Italy
G. Anzivino, F. Brizioli, E. Imbergamo, R. Lollini, R. Piandani 161616Present
address: Institut für Experimentelle Teilchenphysik (KIT), D-76131 Karlsruhe,
Germany, C. Santoni

INFN, Sezione di Perugia, Perugia, Italy
M. Barbanera 171717Present address: INFN, Sezione di Pisa, I-56100 Pisa,
Italy, P. Cenci, B. Checcucci, P. Lubrano, M. Lupi 181818Present address:
Institut am Fachbereich Informatik und Mathematik, Goethe Universität, D-60323
Frankfurt am Main, Germany, M. Pepe, M. Piccini

Dipartimento di Fisica dell’Università 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

Scuola Normale Superiore e INFN, Sezione di Pisa, Pisa, Italy
I. Mannelli

Dipartimento di Fisica, Sapienza Università 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 191919Also at Department of Industrial Engineering,
University of Roma Tor Vergata, I-00173 Roma, Italy, A. Fucci, A. Salamon, F.
Sargeni 202020Also at Department of Electronic Engineering, University of Roma
Tor Vergata, I-00173 Roma, Italy

Dipartimento di Fisica dell’Università e INFN, Sezione di Torino, Torino,
Italy
R. Arcidiacono 212121Also at Università degli Studi del Piemonte Orientale,
I-13100 Vercelli, Italy, B. Bloch-Devaux, M. Boretto 131313Present address:
CERN, European Organization for Nuclear Research, CH-1211 Geneva 23,
Switzerland, E. Menichetti, E. Migliore, D. Soldi

INFN, Sezione di Torino, Torino, Italy
C. Biino, A. Filippi, F. Marchetto

Instituto de Física, Universidad Autónoma de San Luis Potosí, San Luis Potosí,
Mexico
J. Engelfried, N. Estrada-Tristan 222222Also at Universidad de Guanajuato,
Guanajuato, Mexico

Horia Hulubei National Institute of Physics for R&D in Physics and Nuclear
Engineering, 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 151515Also at Faculty of Physics, University of Sofia,
BG-1164 Sofia, Bulgaria, D. Madigozhin, M. Misheva 232323Present address:
Institute of Nuclear Research and Nuclear Energy of Bulgarian Academy of
Science (INRNE-BAS), BG-1784 Sofia, Bulgaria, N. Molokanova, S. Movchan, I.
Polenkevich, Yu. Potrebenikov, S. Shkarovskiy, A. Zinchenko 222Deceased

Institute for Nuclear Research of the Russian Academy of Sciences, Moscow,
Russia
S. Fedotov, E. Gushchin, A. Khotyantsev, Y. Kudenko 242424Also at National
Research Nuclear University (MEPhI), 115409 Moscow and Moscow Institute of
Physics and Technology, 141701 Moscow region, Moscow, Russia, 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 222Deceased, V.
Semenov 222Deceased, V. Sugonyaev, O. Yushchenko

Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava, Slovakia
L. Bician 131313Present address: CERN, European Organization for Nuclear
Research, CH-1211 Geneva 23, Switzerland, T. Blazek, V. Cerny, Z. Kucerova

CERN, European Organization for Nuclear Research, Geneva, Switzerland
J. Bernhard, A. Ceccucci, H. Danielsson, N. De Simone 252525Present address:
DESY, D-15738 Zeuthen, Germany, F. Duval, B. Döbrich, L. Federici, E.
Gamberini, L. Gatignon, R. Guida, F. Hahn 222Deceased, E. B. Holzer, B.
Jenninger, M. Koval 262626Present address: Charles University, 116 36 Prague
1, Czech Republic, P. Laycock 333Present address: Brookhaven National
Laboratory, Upton, NY 11973, USA, G. Lehmann Miotto, P. Lichard, A. Mapelli,
K. Massri, M. Noy, V. Palladino 272727Present address: Physics Department,
Imperial College London, London, SW7 2BW, UK, M. Perrin-Terrin 282828Present
address: Aix Marseille University, CNRS/IN2P3, CPPM, F-13288, Marseille,
France${}^{,}\,$292929Also at Université Catholique de Louvain, B-1348
Louvain-La-Neuve, Belgium, J. Pinzino 303030Present address: Department of
Physics, University of Toronto, Toronto, Ontario, M5S 1A7,
Canada${}^{,}\,$313131Also at INFN, Sezione di Pisa, I-56100 Pisa, Italy, V.
Ryjov, S. Schuchmann 323232Present address: Institut für Physik and PRISMA
Cluster of excellence, Universität Mainz, D-55099 Mainz, Germany, S. Venditti

University of Birmingham, Birmingham, United Kingdom
T. Bache, M. B. Brunetti 333333Present address: Department of Physics,
University of Warwick, Coventry, CV4 7AL, UK, V. Duk 343434Present address:
INFN, Sezione di Perugia, I-06100 Perugia, Italy, V. Fascianelli 353535Present
address: Dipartimento di Psicologia, Università di Roma La Sapienza, I-00185
Roma, Italy, J. R. Fry, F. Gonnella, E. Goudzovski, L. Iacobuzio, C.
Lazzeroni, N. Lurkin 999Present address: Université Catholique de Louvain,
B-1348 Louvain-La-Neuve, Belgium, F. Newson, C. Parkinson 999Present address:
Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium, A. Romano,
A. Sergi, A. Sturgess, J. Swallow

University 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. Ruggiero111Corresponding
authors: G. Ruggiero, R. Marchevski,
<EMAIL_ADDRESS><EMAIL_ADDRESS>

University of Liverpool, Liverpool, United Kingdom
L. Fulton, D. Hutchcroft, E. Maurice 363636Present address: Laboratoire
Leprince Ringuet, F-91120 Palaiseau, France, B. Wrona

George Mason University, Fairfax, Virginia, USA
A. Conovaloff, P. Cooper, D. Coward 373737Also at SLAC National Accelerator
Laboratory, Stanford University, Menlo Park, CA 94025, USA, P. Rubin

11footnotetext: Present address: Laboratori Nazionali di Frascati, I-00044
Frascati, Italy22footnotetext: Also at CERN, European Organization for Nuclear
Research, CH-1211 Geneva 23, Switzerland

###### Contents

  1. 1 Introduction
  2. 2 Principles of the experiment and analysis method
  3. 3 Experimental setup and data taking
  4. 4 Data reconstruction and calibration
  5. 5 Selection of signal and normalization decays
    1. 5.1 Downstream charged particle
    2. 5.2 Parent beam particle
    3. 5.3 Kaon decay
    4. 5.4 Kinematic regions
    5. 5.5 Pion identification
    6. 5.6 Signal selection
    7. 5.7 Normalization selection
  6. 6 Single Event Sensitivity determination
    1. 6.1 Number of $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays
    2. 6.2 Signal and normalization efficiencies
      1. 6.2.1 Monte Carlo efficiencies
      2. 6.2.2 $K^{+}\rightarrow\mu^{+}\nu$ branching ratio measurement
      3. 6.2.3 Random veto efficiency
    3. 6.3 Trigger efficiency
      1. 6.3.1 PNN L0 trigger efficiency
      2. 6.3.2 PNN L1 trigger efficiency
      3. 6.3.3 Trigger efficiency and $SES$
    4. 6.4 $SES$ result
  7. 7 Expected background
    1. 7.1 $K^{+}$ decay background
      1. 7.1.1 $K^{+}\to\pi^{+}\pi^{0}$ decay
      2. 7.1.2 $K^{+}\to\mu^{+}\nu$ decay
      3. 7.1.3 $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay
      4. 7.1.4 $K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$ decay
      5. 7.1.5 Other $K^{+}$ decays
    2. 7.2 Upstream background
      1. 7.2.1 Background sources
      2. 7.2.2 Upstream background evaluation
    3. 7.3 Summary
  8. 8 Results
  9. 9 Conclusions

##  1 Introduction

The $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay is a flavour-changing Neutral-
Current process that proceeds through electroweak box and penguin diagrams in
the Standard Model (SM), allowing an exploration of its flavour structure
thanks to unique theoretical cleanliness. A quadratic GIM mechanism and the
transition of the top quark to the down quark make this process extremely
rare. The SM prediction for the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$
branching ratio (BR) can be written as [1]:

$\text{BR}(K^{+}\to\pi^{+}\nu\bar{\nu})=\kappa_{+}(1+\Delta_{\rm
EM})\left[\left(\frac{Im\lambda_{t}}{\lambda^{5}}X(x_{t})\right)^{2}+\left(\frac{Re\lambda_{c}}{\lambda}P_{c}(X)+\frac{Re\lambda_{t}}{\lambda^{5}}X(x_{t})\right)^{2}\right],$
(1)

where $\Delta_{EM}=-0.003$ accounts for the electromagnetic radiative
corrections; $x_{t}=m_{t}^{2}/M_{W}^{2}$; $\lambda=|V_{us}|$ and
$\lambda_{i}=V^{*}_{is}V_{id}$ $(i=c,t)$ 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

$\kappa_{+}=(5.173\pm 0.025)\times
10^{-11}\left[\frac{\lambda}{0.225}\right]^{8}$ (2)

parameterizes hadronic matrix elements. It is worth noting that
$\text{BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ depends on the sum of the square of
the imaginary part of the top loop, which is CP-violating, and the square 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 the angle $\gamma$, as follows :

$\text{BR}(K^{+}\to\pi^{+}\nu\bar{\nu})=(8.39\pm 0.30)\times
10^{-11}\Big{[}\frac{|V_{cb}|}{40.7\times
10^{-3}}\Big{]}^{2.8}\Big{[}\frac{\gamma}{73.2^{\circ}}\Big{]}^{0.74},$ (3)

where the numerical uncertainty is due to theoretical uncertainties in the NLO
(NNLO) QCD corrections to the top (charm) quark contribution [2, 3] and NLO
electroweak corrections [4]. The intrinsic theoretical accuracy is at the
level of 3.6%. Uncertainties in the hadronic matrix element largely cancel
when it is evaluated from the precisely-measured branching ratio of the
$K^{+}\rightarrow\pi^{0}e^{+}\nu$ decay, including isospin-breaking and non-
perturbative effects calculated in detail [4, 5, 6]. Using tree-level elements
of the CKM matrix as external inputs [7], averaged over exclusive and
inclusive determinations, namely $|V_{cb}|=(40.7\pm 1.4)\times 10^{-3}$ and
$\gamma=(73.2^{+6.3}_{-7.0})$ degrees, the SM prediction of the branching
ratio is $(8.4\pm 1.0)\times 10^{-11}$ [1]. The current precision of the CKM
parameters dominates the BR uncertainty.

The $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay is sensitive to currently
proposed SM extensions and probes higher mass scales than other rare meson
decays. This arises because of the absence of tree-level contributions and the
quadratic GIM suppression at loop level in the SM, which together lead to a
very small BR. Moreover, the absence of long-distance contributions enables
the accurate BR calculation. The largest deviations from SM predictions are
expected in models with new sources of flavour violation, where constraints
from B physics are weaker [8, 9]. Models with currents of defined chirality
produce specific correlation patterns between the branching ratios of
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ decay modes, which are constrained by
the value of the CP-violating parameter $\varepsilon_{K}$ [10, 11]. Present
experimental constraints limit the range of variation within supersymmetric
models [12, 13, 14]. The $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay is also
sensitive to some aspects of lepton flavour non-universality [15] and can
constrain leptoquark models [16, 17] that aim to explain the measured CP-
violating ratio $\varepsilon^{\prime}/\varepsilon$ [7].

The E787 and E949 experiments at the Brookhaven National Laboratory (BNL)
studied the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay using a kaon decay-at-
rest technique, reaching an overall single event sensitivity of about
$0.8\times 10^{-10}$ and measuring the BR to be $(17.3^{+11.5}_{-10.5})\times
10^{-11}$ [18, 19]. The NA62 experiment at the CERN SPS will measure more
precisely the BR of the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay using a
decay-in-flight technique and data recorded from 2016 to 2018. The first NA62
result was based on the analysis of the data collected in 2016 and proved the
feasibility of the technique to study the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay [20]. In the following sections,
NA62 reports the investigation of the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$
decay, based on data recorded in 2017, corresponding to about 30% of the total
data set collected in 2016–18.

##  2 Principles of the experiment and analysis method

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 $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay presents two main
challenges: the extremely low value of the SM signal branching ratio of order
$10^{-10}$ and the open kinematics of the final state, as neutrinos remain
undetected. These challenges require both the production of a sufficient
number of $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decays, as can be achieved by
exploiting the high-intensity 75 GeV$/c$ secondary $K^{+}$ beam produced by
the CERN SPS; and the reduction of the contribution of the dominant $K^{+}$
decay modes by at least eleven orders of magnitude to bring the background to
a level lower than the signal.

The signature of the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay is a single
$\pi^{+}$ and missing energy. The squared missing mass,
$m^{2}_{\mathrm{miss}}=(P_{K}-P_{\pi^{+}})^{2}$, where $P_{K}$ and
$P_{\pi^{+}}$ indicate the 4-momenta of the $K^{+}$ and $\pi^{+}$, describes
the kinematics of the one-track final state. In particular, the presence of
two neutrinos makes the signal broadly distributed over the
$m^{2}_{\mathrm{miss}}$ range, as illustrated in Figure 1. The dominant
$K^{+}$ decay modes $K^{+}\rightarrow\mu^{+}\nu$,
$K^{+}\rightarrow\pi^{+}\pi^{0}$ and
$K^{+}\rightarrow\pi^{+}\pi^{+(0)}\pi^{-(0)}$ have different
$m^{2}_{\mathrm{miss}}$ distributions; it is therefore possible to define
regions, either side of the $K^{+}\rightarrow\pi^{+}\pi^{0}$ peak,
qualitatively indicated in Figure 1, where the search for the signal is
performed, also called signal regions.

Figure 1: Expected theoretical distributions of the $m^{2}_{\mathrm{miss}}$
variable relevant to the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ measurement,
before applying acceptance and resolution effects. The $m^{2}_{\mathrm{miss}}$
is computed under the hypothesis that the charged particle in the final state
is a $\pi^{+}$. The $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ signal (red line) is
multiplied by $10^{10}$ for visibility. The hatched areas include the signal
regions.

The $K^{+}\rightarrow\mu^{+}\nu$, $K^{+}\rightarrow\pi^{+}\pi^{0}$ and
$K^{+}\rightarrow\pi^{+}\pi^{+(0)}\pi^{-(0)}$ decays enter the signal regions
through radiative and/or resolution tails of the reconstructed
$m^{2}_{\mathrm{miss}}$. The signal selection, based on kinematics only,
relies on the accurate measurement of the $m^{2}_{\mathrm{miss}}$ quantity,
i.e. of the $K^{+}$ and $\pi^{+}$ momenta and directions. In contrast,
$K^{+}\rightarrow\pi^{0}\ell^{+}\nu$ or rarer decays, like
$K^{+}\rightarrow\pi^{+}\pi^{-}\ell^{+}\nu$, span over the signal regions
because of the presence of undetected neutrinos; however, these background
decay modes include a lepton in the final state and exhibit extra activity in
the form of photons or charged particles. A particle identification system
must therefore separate $\pi^{+}$ from $\mu^{+}$ and $e^{+}$. Photons and
additional charged particles in final state must be vetoed as efficiently as
possible.

The above conditions translate into the following experimental requirements:

  * •

the detection of incident $K^{+}$ and outgoing $\pi^{+}$ signals with 100 ps
time resolution to mitigate the impact of the pile-up effect due to the high
particle rates;

  * •

a low-mass $K^{+}$ and $\pi^{+}$ tracking system, which reconstructs precisely
the kinematics to suppress $K^{+}\rightarrow\pi^{+}\pi^{0}$ and
$K^{+}\rightarrow\mu^{+}\nu$ 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 decays with positrons and muons by seven to eight orders of
magnitude;

  * •

a set of electromagnetic calorimeters, to detect photons and reduce the number
of $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays by eight orders of magnitude; and

  * •

an experimental design which guarantees the geometric acceptance for
negatively charged particles in at least two detectors.

The decay-in-flight configuration has two main advantages:

  * •

the selection of $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decays with a $\pi^{+}$
momentum lower than 35 GeV/$c$ to facilitate the background rejection by
ensuring at least 40 GeV of missing energy, and to exploit the capability of
the RICH for $\pi^{+}/\mu^{+}$ separation; and

  * •

the achievement of sufficient $\pi^{0}$ suppression by using photon detection
coverage up to 50 mrad with respect to the $K^{+}$ direction, and by
efficiently detecting photons of energy above 1 GeV.

The experimental layout and the data-taking conditions are reviewed in section
3. The reconstruction algorithms are described in section 4. After the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection (section 5), the analysis
proceeds through the evaluation of the single event sensitivity, defined as
the branching ratio equivalent to the observation of one SM signal event
(section 6). The number of signal decays is normalized to the number of
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays, whose branching ratio is accurately
known [7]. This allows the precise determination of the single event
sensitivity without relying on the absolute measurement of the total number of
$K^{+}$ decays. The final step of the analysis is the evaluation of the
expected background in the signal regions (section 7). To avoid biasing the
selection of $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ events, the analysis
follows a “blind” procedure, with signal regions kept masked until completion
of all the analysis steps. Finally, the result is presented in section 8.

##  3 Experimental setup and data taking

The NA62 beam line and detector are sketched in Figure 2. A detailed
description of them can be found in [21]. The beam line defines the Z-axis of
the experiment’s right-handed laboratory coordinate system. The origin is the
kaon production target, and beam particles travel in the positive Z-direction.
The Y-axis is vertical (positive up), and the X-axis is horizontal (positive
left).

The kaon production target is a 40 cm long beryllium rod. A 400 GeV proton
beam extracted from the CERN Super Proton Synchrotron (SPS) impinges on the
target in spills of three seconds effective duration. Typical intensities
during data taking range from $1.7$ to $1.9\times 10^{12}$ protons per pulse
($ppp$). The resulting secondary hadron beam of positively charged particles
consists of 70% $\pi^{+}$, 23% protons, and 6% $K^{+}$, with a nominal
momentum of 75 GeV/$c$ (1% rms momentum bite).

Beam particles are characterized by a differential Cherenkov counter (KTAG)
and a three-station silicon pixel matrix (Gigatracker, GTK, with pixel size of
$300\times 300~{}\mu{\rm m}^{2}$). The KTAG uses N2 gas at 1.75 bar pressure
(contained in a 5 m long vessel) and is read out by photomultiplier 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 beam achromat), forming a spectrometer that measures beam particle
momentum, direction, and time with resolutions of 0.15 GeV/$c$, 16 $\mu$rad,
and 100 ps, respectively.

The last GTK station (GTK3) is immediately preceded by a 1 m thick, variable
aperture steel collimator (final collimator). Its inner aperture is typically
set at 66 mm $\times$ 33 mm, and its outer dimensions are 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 define a volume in which 13% of the kaons decay. The beam has a
rectangular transverse profile of 52 $\times$ 24 mm2 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 measured by a magnetic spectrometer (STRAW), a ring-imaging
Cherenkov counter (RICH), and two scintillator hodoscopes (CHOD and
NA48-CHOD). The STRAW, consisting of two pairs of straw chambers on either
side of a dipole magnet, measures momentum-vectors with a resolution,
$\sigma_{p}/p$, between 0.3% and 0.4%. The RICH, filled with neon at
atmospheric pressure, tags the decay particles with a timing precision of
better than 100 ps and provides particle identification. The CHOD, a matrix of
tiles read out by SiPMs, and the NA48-CHOD, comprising two orthogonal planes
of scintillating slabs reused from the NA48 experiment, are used for
triggering and timing, providing a time measurement with 200 ps resolution.

Other sub-detectors suppress decays into photons or into multiple charged
particles (electrons, pions or muons) or provide complementary particle
identification. Six stations of plastic scintillator bars (CHANTI) detect,
with 1 ns time resolution, extra activity, including inelastic interactions in
GTK3. Twelve stations of ring-shaped electromagnetic calorimeters (LAV1 to
LAV12), made of lead-glass blocks, surround the vacuum tank and downstream
sub-detectors to achieve hermetic acceptance for photons emitted by $K^{+}$
decays in the decay volume at polar angles between 10 and 50 mrad. A 27
radiation-length thick, quasi-homogeneous liquid krypton electromagnetic
calorimeter (LKr) detects photons from $K^{+}$ decays emitted at angles
between 1 and 10 mrad. The LKr also complements the RICH for particle
identification. Its energy resolution in NA62 conditions is
$\sigma_{E}/E=1.4\%$ for energy deposits of 25 GeV. Its spatial and time
resolutions are 1 mm and between 0.5 and 1 ns, respectively, depending on the
amount and type of energy released. Two hadronic iron/scintillator-strip
sampling calorimeters (MUV1,2) and an array of scintillator tiles located
behind 80 cm of iron (MUV3) supplement the pion/muon identification 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 the Z-axis, and a similar detector (SAC) placed on the Z-axis at
the downstream end of the apparatus, ensure the detection of photons down to
zero degrees in the forward direction. Additional counters (MUV0, HASC)
installed at optimized locations provide nearly hermetic coverage for charged
particles produced in multi-track kaon decays.

All detectors are read out with TDCs, except for LKr and MUV1, 2, which are
read out with 14-bit FADCs. The IRC and SAC are read out with both. All TDCs
are mounted on custom-made (TEL62) boards, except for GTK and STRAW, which
each have specialized TDC boards. TEL62 boards both read out data and provide
trigger information. A dedicated processor interprets calorimeter signals for
triggering. A dedicated board (L0TP) combines logical signals (primitives)
from the RICH, CHOD, NA48-CHOD, LKr, LAV, and MUV3 into a low-level trigger
(L0) whose decision is dispatched to sub-detectors for data readout [22]. 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 [21].

Figure 2: Schematic top view of the NA62 beam line and detector. Dipole
magnets are displayed as boxes with superimposed crosses. The label “COL”
denotes the collimator named “final collimator” in the text. The label “CHOD”
refers both to the CHOD and NA48-CHOD detectors. Also shown is the trajectory
of a beam particle in vacuum which crosses all the detector apertures, thus
avoiding interactions with material. A dipole magnet between MUV3 and SAC
deflects the beam particles out of the SAC acceptance.

The data come from $3\times 10^{5}$ SPS spills accumulated during a four-month
data-taking period in 2017, recorded at an average beam intensity of 450 MHz.
The instantaneous beam intensity is measured 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 $\pm$10% throughout the data-taking
period, while the instantaneous beam intensity showed fluctuations up to a
factor of two around the average value.

The data have been collected using a trigger specifically setup for the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ measurement, called PNN trigger,
concurrently with a minimum-bias trigger. The PNN trigger is defined as
follows. The L0 trigger requires a signal in the RICH to tag a charged
particle. The time of this signal, called trigger time, is used as a reference
to define a coincidence within 6.3 ns of: a signal in one to four CHOD tiles;
no signals in opposite CHOD quadrants to suppress
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays; no signals in MUV3 to reject
$K^{+}\rightarrow\mu^{+}\nu$ decays; less than 30 GeV energy deposited in LKr
and no more than one cluster to reject $K^{+}\rightarrow\pi^{+}\pi^{0}$
decays. The L1 trigger requires: a kaon identified in KTAG; signals within 10
ns of the trigger time in at most two blocks of each LAV station; at least one
STRAW track corresponding to a particle with momentum below 50 GeV$/c$ and
forming a vertex with the nominal beam axis upstream of the first STRAW
chamber. Events collected by the PNN trigger are referred to as PNN events or
data. The minimum-bias trigger is based on NA48-CHOD information downscaled by
a factor of 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 to estimate 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 [23] to describe detector geometry and response.
The $K^{+}$ decays are generated in the kaon rest frame using the appropriate
matrix elements and form factors. The simulation also includes a description
of the collimators and dipole and quadrupole magnets in the beam line,
necessary to accurately simulate the beam shape. Certain aspects of the
simulation are tuned using input from data, namely signal formation and
readout detector inefficiencies. Accidental activity is added to the KTAG
Cherenkov counter and to the GTK beam tracker assuming 450 MHz beam intensity,
and using a library of pileup beam particles built from data. No accidental
activity is simulated in the detectors downstream of the last station of the
beam tracker. Simulated data are subjected to the same reconstruction and
calibration procedures as real data.

##  4 Data reconstruction and calibration

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 $K^{+}$ KTAG candidate 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 trigger time and corrected for pulse-amplitude slewing. Signals from
the three GTK stations grouped within 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 in more
than one adjacent pixel in the same station if hitting the edge of a pixel or
because of $\delta$-rays. In this case, pulses in neighbouring pixels form a
cluster that is used to reconstruct the track. Fully reconstructed
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays in the STRAW spectrometer are
used to align the GTK stations transversely to a precision of better than 100
$\mu$m and to tune the GTK momentum scale.

The STRAW reconstruction relies on the trigger time as a reference to
determine the drift time. A track is defined by space-points in the chambers
describing a path compatible with magnetic bending. A Kalman-filter fit
provides the track parameters. The $\chi^{2}$ fit value and the number of
space-points characterize the track quality. Straight tracks collected with
the magnet off serve to align the straw tubes to 30 $\mu$m accuracy. The
average value of the $K^{+}$ mass reconstructed for
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays provides fine tuning of the
momentum scale to a part per thousand precision.

Two algorithms reconstruct RICH ring candidates, both grouping signals from
photomultipliers (PM) in time around the trigger time. The first one, called
track-seeded ring, makes use of a STRAW track as a seed to build a RICH ring
and compute a likelihood for several mass hypotheses ($e^{+}$, $\mu^{+}$,
$\pi^{+}$ and $K^{+}$). The second one, called single ring, fits the signals
to a ring assuming that they are produced by a single particle, with the fit
$\chi^{2}$ characterizing the quality of this hypothesis. Positrons are used
to calibrate the RICH response and align the twenty RICH mirrors to a
precision of $30$ $\mu$rad [24].

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 the passage of a charged
particle form NA48-CHOD candidates. Each slab is time-aligned to the trigger
time. Time offsets depending on the intersection position account for the
effect of light propagation along a slab.

Groups 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. Cluster energies, positions, and times are
reconstructed taking into account energy calibration, non-linearity, energy
sharing for nearby clusters and noisy cells. The final calibration is
performed using positrons from $K^{+}\rightarrow\pi^{0}e^{+}\nu$ decays. An
additional reconstruction algorithm is applied to maximise the reconstruction
efficiency. This is achieved by defining candidates as sets of cells with 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 in fewer 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 using weighting factors extracted
from dedicated simulations and tested on samples of $\pi^{+}$ and $\mu^{+}$.

Candidates in MUV3 are defined by time coincidences of the response of the two
PMs reading the same tile. The time of a candidate is defined by the later of
the two PM signals, to avoid the effect of the time spread due to the early
Cherenkov light produced by particles traversing the PM window.

CHANTI candidates are defined by signals clustered in time and belonging
either to adjacent parallel bars or to intersecting orthogonal bars.

Two threshold settings discriminate the CHANTI, LAV, IRC and SAC TDC signals
[21]. Thus up to four time measurements are associated with each signal,
corresponding to the leading and trailing edge times of the high and low
thresholds. The relation between the amplitude of the IRC and SAC pulses
provided by the FADC readout, and the energy release is calibrated for each
channel after baseline subtraction using a sample of
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays.

Signal times measured by GTK, KTAG, CHOD, RICH and LKr are further aligned to
the trigger time for each spill, resulting in a better than 10 ps stability
through the whole data sample.

##  5 Selection of signal and normalization decays

The selection of both $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ signal and
$K^{+}\rightarrow\pi^{+}\pi^{0}$ normalization decays requires the
identification of the downstream charged particle as a $\pi^{+}$ and the
parent beam particle as a $K^{+}$. Further specific criteria are applied to
separate signal and normalization events.

###  5.1 Downstream charged particle

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, in the electromagnetic calorimeter LKr, and in the RICH
counter.

The downstream track must include space-points reconstructed in all four
chambers of the STRAW spectrometer, satisfy suitable quality criteria, and be
consistent with a positively charged particle. The extrapolation of this track
to any downstream detector defines the expected position of the charged
particle’s impact point on that detector. These positions must lie within the
geometric acceptance of the corresponding downstream detectors and outside the
acceptance of the large and small angle calorimeters LAV and IRC. The impact
points of the charged particles are used to match the downstream tracks with
signals in the hodoscopes and the electromagnetic calorimeter.

Two discriminant variables are built using the difference of time and spatial
coordinates between each hodoscope candidate and the track. The NA48-CHOD
candidate with the lowest discriminant value and the CHOD candidate closest in
space to the particle impact point are matched to the track. The latter
candidate must be within $\pm 5$ ns of the assigned NA48-CHOD candidate. Cuts
on maximum allowed values of the discriminant variables are also implemented
to avoid fake or accidental signals in the hodoscopes.

A LKr cluster is matched to a charged particle if its distance from the
particle impact point is smaller than 100 mm. The energy released by the track
in the calorimeter is defined as the energy of the associated cluster. The
time of the associated cluster is the time of the most energetic cell of the
cluster. A 2 ns time coincidence is required between the cluster and the
NA48-CHOD candidate associated with the track.

The association between the track and a single ring of the RICH counter
exploits the relationship between the slope of the track and the position of
the ring center. A track-seeded ring is also considered for particle
identification purposes (section 5.5). Both types of RICH rings must be in
time within $\pm 3$ ns of the NA48-CHOD candidate associated to the track. The
time of the downstream charged particle is defined as the time of the
associated RICH single ring.

Track-matching with a CHOD, NA48-CHOD, RICH and LKr candidate is mandatory.

###  5.2 Parent beam particle

The parent $K^{+}$ of a selected downstream charged particle is defined by:
the $K^{+}$ candidate in KTAG closest in time and within $\pm 2$ ns of the
downstream particle; a beam track in GTK associated in time with the KTAG
candidate and in space with the downstream track in the STRAW.

The association between GTK, KTAG and STRAW candidates relies on a likelihood
discriminant built from two variables: the time difference between the KTAG
candidate and the beam track ($\Delta$T(KTAG-GTK)); and the closest distance
of approach of the beam track to the downstream charged particle (CDA)
computed taking into account bending of the particle trajectory in the stray
magnetic field in the vacuum tank. The templates of the $\Delta$T(KTAG-GTK)
and CDA distributions of the parent $K^{+}$ are derived from a sample of
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ selected on data. In this case, the
clean three-pion final state signature tags the $K^{+}$ track in the GTK and
one of the positively charged pions is chosen to be the downstream charged
particle. The resulting distributions are shown in Figure 3, together with the
corresponding distributions for events 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 size and
divergence of the beam, and the emission angle of the $\pi^{+}$. The beam
track with the largest discriminant value is, by construction, the parent
$K^{+}$; its momentum and direction must be consistent with the nominal beam
properties.

Figure 3: Distributions of $\Delta$T(KTAG-GTK) (left) and CDA (right) for
events with beam $K^{+}$ (shaded histogram) and accidental beam particle
(empty histogram), as obtained from fully reconstructed
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays in the data. The red curves
superimposed on the histograms describe the functions used to model the time
and CDA distributions of the beam $K^{+}$.

Because of the high particle rate in the beam tracker, several beam particles
may overlap with the $K^{+}$ within $\pm 1$ ns; they are referred to as pileup
(or accidental) particles and the corresponding GTK track is called a pileup
(or accidental) track. A wrong association occurs when 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 beam tracker and a pileup track is associated to the downstream charged
particle. A sharp cut on the minimum allowed value of the likelihood
discriminant reduces the probabilities of wrong and accidental association.
Events are also rejected if more than 5 pileup tracks are reconstructed or if
the likelihood discriminant values of different beam tracks matching the same
downstream charged particle are similar. Finally, a cut is applied on a
discriminant computed using the time difference between the beam track and the
downstream charged particle, instead of $\Delta$T(KTAG-GTK).

The $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays allow the performance of
the beam-track matching to be monitored. The probabilities of wrong and
accidental association depend on the instantaneous beam intensity and are
about 1.3% and 3.5% on average, respectively. The latter includes also the
probability that a pileup track time is within $\pm$1 ns of the KTAG time.
Both probabilities depend on the type of process under study.

###  5.3 Kaon decay

A downstream charged particle and its parent $K^{+}$ define the kaon decay.
The mid-point between the 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 result of overlapping accidental charged particles in the downstream
detectors. In particular, this occurs in the STRAW spectrometer which makes
use of a large 200 ns readout window. If two downstream charged particles are
reconstructed and both match a parent $K^{+}$, the one closer to the trigger
time is accepted. The same trigger time requirement is applied independently
to each detector signal matched with beam and downstream tracks. Further
conditions are applied to suppress $K^{+}$ decay like
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$: no more than two tracks reconstructed
in the STRAW are allowed in total; if there are two tracks, both must be
positively charged and should not form a vertex with a $Z$-position between
GTK3 and the first STRAW station.

Figure 4: Left: distribution of the longitudinal position of the reconstructed
decay vertex. The FV is defined between 105 and 165 m (vertical red lines).
Right: reconstructed $m^{2}_{\mathrm{miss}}$ as a function of the decay
particle momentum for minimum-bias events selected without applying $\pi^{+}$
identification and photon rejection, assuming the $K^{+}$ and $\pi^{+}$ mass
for the parent and decay particle, respectively. Signal regions 1 and 2
(hatched areas), as well as $3\pi$, $\pi^{+}\pi^{0}$, and $\mu^{+}\nu$
background regions (solid thick contours) are shown. The control regions are
located between the signal and background regions.

Figure 4 (left) displays the distribution of the longitudinal position
($Z_{vertex}$) of the reconstructed 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 is due
to nuclear interactions of beam particles grazing the edges of the final
collimator or passing through the last station of the beam tracker located at
103 m. Charged particles created by decays upstream of the final collimator or
by nuclear interactions can reach the detectors downstream and create fake
$K^{+}$ decays. To mitigate this effect, the decay vertex is required to lie
within a fiducial volume (FV) defined as 105 m to 165 m from the target. The
coordinates of this vertex must also be consistent with the beam envelope.
Wrong or accidental associations or mis-reconstruction of the $Z_{vertex}$ can
shift the origin of these events within the FV, imitating a $K^{+}$ decay.

Cuts on the direction of the decay particles as a function of $Z_{vertex}$ are
applied to reduce the number of events reconstructed within the FV, but which
actually originated upstream (section 7.2). These cuts are also useful against
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays with only one $\pi^{+}$
reconstructed. The CHANTI detector further protects the FV against nuclear
interactions by vetoing events with CHANTI signals within 3 ns of the decay
particle candidate. Extra pulses in at least two GTK stations in time with the
$K^{+}$ candidate may indicate that the $K^{+}$ has decayed before entering
the decay region. In this case the event is rejected if at least one pileup
track is reconstructed in the beam tracker in addition to the $K^{+}$
candidate. Finally, events are also discarded if the decay particle track
points back to the active area of GTK3.

###  5.4 Kinematic regions

Figure 4 (right) shows the $m^{2}_{\mathrm{miss}}$ distribution as a function
of the decay particle momentum for $K^{+}$ decays selected as above from
minimum-bias data. Here, the $m^{2}_{\mathrm{miss}}$ quantity is computed
using the three-momenta measured by the beam tracker and the STRAW
spectrometer, assuming $K^{+}$ and $\pi^{+}$ masses. Events from
$K^{+}\rightarrow\pi^{+}\pi^{0}$ and $K^{+}\rightarrow\mu^{+}\nu$ decays
accumulate at $m^{2}_{\mathrm{miss}}=m^{2}_{\pi^{0}}$ and
$m^{2}_{\mathrm{miss}}<0$, respectively. Events above
$m^{2}_{\mathrm{miss}}=4m^{2}_{\pi^{+}}(4m^{2}_{\pi^{0}})$ are mostly
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}(\pi^{+}\pi^{0}\pi^{0})$ decays. The
shape of the region at low momentum arises from the $Z_{vertex}$ cuts.

The $m^{2}_{\mathrm{miss}}$ resolution varies with $m^{2}_{\mathrm{miss}}$ and
is about $10^{-3}\,\text{GeV}^{2}/c^{4}$ at the
$K^{+}\rightarrow\pi^{+}\pi^{0}$ peak. This sets the definition of the
boundaries of signal region 1 and 2:

Region 1: $0<m^{2}_{\mathrm{miss}}<0.01\,\text{GeV}^{2}/c^{4}$;

Region 2: $0.026<m^{2}_{\mathrm{miss}}<0.068\,\text{GeV}^{2}/c^{4}$.
Additional momentum-dependent constraints supplement this definition by
selecting $m^{2}_{\mathrm{miss}}$ values computed using either the decay
particle momentum measured by the RICH under the $\pi^{+}$ mass hypothesis
instead of the STRAW momentum, or the nominal beam momentum and direction
instead of those measured by the GTK tracker. These requirements are intended
to reduce the probability of wrong reconstruction of the
$m^{2}_{\mathrm{miss}}$ quantity due to a mis-measurement of the momenta of
the decay particle or $K^{+}$ candidate.

The momentum of the decay particle in the range $15-35$ GeV$/c$ complements
the definition of the signal regions. The $\pi^{+}$ Cherenkov threshold of the
RICH sets the lower boundary at 15 GeV$/c$. The $K^{+}\rightarrow\mu^{+}\nu$
kinematics and the requirement of a large missing energy drive the choice of
the 35 GeV$/c$ upper boundary. The two signal regions are kept masked (blind)
until the completion of the analysis.

In addition to the signal regions, three exclusive background regions are
defined:

The $\mu\nu$ region: $-0.05<m^{2}_{\mathrm{miss}}<m^{2}_{\mu-kin}+3\sigma$,
where $m^{2}_{\mu-kin}$ is the $m^{2}_{\mathrm{miss}}$ of the
$K^{+}\rightarrow\mu^{+}\nu$ decays under the $\pi^{+}$ mass hypothesis and
$\sigma$ its resolution;

The $\pi^{+}\pi^{0}$ region:
$0.015<m^{2}_{\mathrm{miss}}<0.021\,\text{GeV}^{2}/c^{4}$;

The $3\pi$ region: $0.072<m^{2}_{\mathrm{miss}}<0.150\,\text{GeV}^{2}/c^{4}$.
Once photons, muons and positrons are rejected (sections 5.5 and 5.6),
simulations show that solely $K^{+}\rightarrow\mu^{+}\nu$,
$K^{+}\rightarrow\pi^{+}\pi^{0}$ and $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$
decays populate these regions, respectively.

Regions of the $m^{2}_{\mathrm{miss}}$ distribution between signal and
background regions, referred to as control regions, are masked until
backgrounds are estimated and then used to validate the estimates. Two regions
around the $\pi^{+}\pi^{0}$ peak, for the $\pi^{+}\pi^{0}$ background, and one
region each for the $K^{+}\rightarrow\mu^{+}\nu$ and
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ backgrounds, are identified. Both
background and control regions are restricted to the $15-35$ GeV/$c$ $\pi^{+}$
momentum range for consistency with the definition of the signal regions.

###  5.5 Pion identification

The PNN trigger (section 3) discards kaons decaying to muons by vetoing events
with a signal in the MUV3 detector. A similar requirement applied offline
reinforces the trigger condition, recovering possible online veto
inefficiencies and makes the $\pi^{+}$ identification in minimum-bias and PNN
data identical. Muons may fail to be detected by MUV3 because of inefficiency
or catastrophic interaction in the calorimeter, or if they decay upstream.

Pions can be distinguished from muons and positrons using information from the
LKr calorimeter and, should any be present, from the MUV1 and MUV2 hadronic
calorimeters. A multivariate classifier resulting from a Boosted Decision Tree
algorithm, BDT, combines 13 variables characterizing the calorimetric energy
depositions. A first group of variables consists of the ratios between the
calorimetric energy deposited and the particle momentum measured in the STRAW.
The energy in the LKr is used alone, and in combination with the hadronic
energies. A second group of variables describes the longitudinal and
transverse development of the calorimetric showers. The energy sharing between
LKr, MUV1 and MUV2 provides information about the longitudinal shape of the
energy deposition, and the shape of the clusters characterizes the transverse
size of the shower. Finally, the BDT makes use of the distance between the
particle impact point and the reconstructed cluster position. The BDT training
is performed using samples of $\mu^{+}$, $\pi^{+}$ and $e^{+}$ selected from
minimum-bias data recorded in 2016 and not used in the present analysis. The
BDT returns the probability for a particle to be a $\pi^{+}$, a $\mu^{+}$, or
a positron. Pion identification requires the $\pi^{+}$ probability to be
larger than a minimum value that depends on the particle momentum and is
optimised with data.

Samples of $K^{+}\rightarrow\pi^{+}\pi^{0}$ and $K^{+}\rightarrow\mu^{+}\nu$
decays selected from minimum-bias data are used to monitor the performance of
the $\pi^{+}$ identification efficiency and resulting $\mu^{+}$
misidentification probability, shown in Figure 5 (left).

Figure 5: Performance of the $\pi^{+}$ identification using calorimeters
(left) and RICH (right) measured on data. Performance is quantified in terms
of $\pi^{+}$ and $\mu^{+}$ efficiency, defined as the fraction of pions and
muons passing the pion identification criteria, respectively. These criteria
include the corresponding RICH and calorimeter reconstruction efficiency. On
each plot, the $\pi^{+}$ efficiency scale is shown on the left (black)
vertical axis, the $\mu^{+}$ efficiency (misidentification) scale is shown on
the right (blue) vertical axis.

Finally, the RICH separates $\pi^{+}$, $\mu^{+}$ and $e^{+}$ independently of
the calorimeter responses. The reconstructed mass and the likelihood of the
particle must be consistent with the $\pi^{+}$ hypothesis. Figure 5 (right)
shows the performance of the $\pi^{+}/\mu^{+}$ separation using the RICH as a
function of the particle momentum, evaluated using data.

The $\pi^{+}$ identification is required for both signal
$K^{+}\to\pi^{+}\nu\bar{\nu}$ and normalization $K^{+}\to\pi^{+}\pi^{0}$
selections.

###  5.6 Signal selection

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 physics process producing the downstream $\pi^{+}$.

Photon rejection discriminates against partially reconstructed
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays. An extra in-time photon in the LKr
calorimeter is defined as a cluster located at least 100 mm away from the
$\pi^{+}$ impact point and within a cluster energy-dependent time coincidence
with the $\pi^{+}$ time that ranges from $\pm 5$ ns below 1 GeV to $\pm$50 ns
above 15 GeV. Pileup clusters can overlap in space with the photon to be
rejected, spoiling the time of such a photon by as much as several tens of ns.
The choice of a broad timing window at high energy keeps the detection
inefficiency below $10^{-5}$.

An extra in-time photon in the LAV detector is defined as any signal in a LAV
station within $\pm 3$ ns of the $\pi^{+}$ time. Appropriate combinations of
the TDC leading and trailing edges of the high and low threshold channels
define a LAV signal [21]. A similar method identifies photons in the small
angle calorimeters IRC and SAC, using a time-window of $\pm 7$ ns around the
$\pi^{+}$ time. In addition to the signals from the TDC readout, photon
rejection in IRC and SAC exploits the FADC readout; here, a photon signal is
defined as an energy deposit larger than 1 GeV in a $\pm 7$ ns time window.

Multiplicity rejection discriminates against tracks produced by photons
interacting in the material before reaching the calorimeters, and against
tracks from $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays partially
reconstructed in the STRAW. The first category of charged particles is
expected to leave signals in the detectors downstream of the STRAW. The
rejection criteria exploit the time and 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 are also
included. The second category of charged particles is characterized by the
presence of track segments, defined as pairs of signals in the first-second or
third-fourth STRAW stations and consistent with a particle coming from the FV.

Figure 6:  Rejection inefficiency of the $\pi^{0}$ from
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays as a function of the $\pi^{+}$
momentum. The quoted uncertainties are statistical only.

Figure 7:  Distribution of the $m^{2}_{\mathrm{miss}}$ of events selected from
minimum-bias data for normalization. Data and MC simulation are superimposed.
The bottom insert shows the data/MC ratio. The error bars correspond to the
statistical uncertainty of the ratio, the yellow band is the systematic
uncertainty due to the imperfect simulation of the detector response.

The reduction of reconstructed $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays
quantifies the performance of the photon and multiplicity rejection. The
number of PNN events in the $\pi^{+}\pi^{0}$ region remaining after rejection
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 $\pi^{0}$ produced in
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays. This inefficiency depends on the
$\pi^{+}$ momentum and is about $1.3\times 10^{-8}$ on average, as shown in
Figure 7. The measured $\pi^{0}$ rejection can be explained in terms of
single-photon detection inefficiencies in the LKr, LAV, IRC and SAC
calorimeters which are measured from a sample of minimum-bias
$K^{+}\to\pi^{+}\pi^{0}$ data using a tag-and-probe method. The estimated
$\pi^{0}$ efficiency stems from the measured single-photon detection
efficiencies convoluted with simulated $K^{+}\to\pi^{+}\pi^{0}$ decays and is
in agreement with the measured $\pi^{0}$ efficiency within the statistical
uncertainty. The rise at low $\pi^{+}$ momentum is a consequence of lower
detection efficiency for photons travelling close to the beam axis and
interacting with the beam pipe.

In addition to the photon and multiplicity rejection, the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection enforces specific requirements
against particles entering the FV from upstream. The $\pi^{+}$ track is
extrapolated back to the $Z$-position of the final collimator and the $X,Y$
transverse coordinates are required to be outside of a box with $|X|<100$ mm
and $|Y|<500$ mm. This cut removes a region with weaker shielding against
particles coming from upstream and corresponds to the central aperture of the
last dipole magnet of the beam line (section 7.2). This condition is referred
to as the box cut in the following sections.

Finally, signal selection requires the $m^{2}_{\mathrm{miss}}$ value to be
within the signal regions defined in section 5.4. The set of criteria
described in this section is called PNN selection in the following.

###  5.7 Normalization selection

The $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays used for normalization are
selected from minimum-bias data, as defined in sections 5.3 and 5.5, and their
$m^{2}_{\mathrm{miss}}$ value must be in the 0.01–0.026 GeV${}^{2}/c^{4}$
range. Figure 7 shows the $m^{2}_{\mathrm{miss}}$ spectrum of these events
before the $m^{2}_{\mathrm{miss}}$ cut, together with the simulated
distribution. The shape of the $K^{+}\rightarrow\pi^{+}\pi^{0}$ peak depends
on the resolution of the STRAW spectrometer, on multiple scattering in the
tracker material, on the rate of pileup tracks, and on the calibration of the
beam and STRAW trackers. The uncertainty in the simulation of these effects
affects the data/MC agreement in the peak region only, and is taken into
account in the evaluation of the $SES$ (section 6). The overall background
under the peak is at the one part per thousand level and stems from
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays with $\pi^{0}\rightarrow
e^{+}e^{-}\gamma$.

##  6 Single Event Sensitivity determination

Denoting $N_{K^{+}}$ the number of kaon decays occurring in the FV, the
single-event sensitivity ($SES$) of the present data sample to
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ can be written as

$SES=\frac{1}{N_{K^{+}}\cdot\epsilon_{\pi\nu\nu}\cdot\epsilon_{trig}^{\scriptscriptstyle
PNN}}=\frac{\text{BR}(K^{+}\rightarrow\pi^{+}\pi^{0})}{D\cdot
N_{\pi\pi}}\frac{\epsilon_{\pi\pi}\cdot\epsilon_{trig}^{\scriptscriptstyle
MB}}{\epsilon_{\pi\nu\nu}\cdot\epsilon_{trig}^{\scriptscriptstyle
PNN}}~{}\cdot$ (4)

Here $N_{\pi\pi}$ is the number of $K^{+}\rightarrow\pi^{+}\pi^{0}$ events
reconstructed in the FV from minimum-bias data (section 5.7), also called
normalization events; $D$ is the reduction, or down-scaling, factor applied
online to reduce the minimum-bias contribution to the total trigger rate;
$\epsilon_{\pi\nu\nu}$ and $\epsilon_{\pi\pi}$ are the efficiencies to
identify a $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and a
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decay in the FV, also called signal and
normalization efficiencies, respectively; $\epsilon_{trig}^{\scriptscriptstyle
PNN}$ and $\epsilon_{trig}^{\scriptscriptstyle MB}$, are the trigger
efficiencies that account for the data loss after the event selection due to
the PNN and minimum-bias triggers. The efficiencies and $N_{\pi\pi}$ depend on
the $\pi^{+}$ momentum, $p_{\pi}$, and on the instantaneous beam intensity,
$I$. The $SES$ is consequently computed in bins of $p_{\pi}$ and $I$: the
momentum range 15–35 GeV/$c$ is subdivided into four bins of 5 GeV/$c$ width
and the instantaneous beam intensity into five bins of approximately the same
statistics of $K^{+}\rightarrow\pi^{+}\pi^{0}$ normalization events.

###  6.1 Number of $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays

The number of events satisfying the conditions described in section 5.7 is
$N_{\pi\pi}=68\times 10^{6}$.

The $\pi^{0}$ mainly decays to $\gamma\gamma$, but in about 1% of cases it
decays to $\gamma e^{+}e^{-}$, called a Dalitz decay ($\pi^{0}_{D}$). The
relative impact of $\pi^{0}_{D}$ decays on the $SES$ is estimated to be less
than 0.3% and is assigned as systematic uncertainty. In the following sections
$K^{+}\rightarrow\pi^{+}\pi^{0}$ refers only to
$\pi^{0}\rightarrow\gamma\gamma$ decays.

###  6.2 Signal and normalization efficiencies

The efficiencies $\epsilon_{\pi\nu\nu}$ and $\epsilon_{\pi\pi}$ quantify the
effects of reconstruction and selection (section 5) on the counting of signal
and normalization channels. Event losses can be grouped into 6 classes:

  1. 1.

geometric and kinematic acceptances;

  2. 2.

reconstruction of the $K^{+}$ and of the downstream charged particle;

  3. 3.

matching the $K^{+}$ with the downstream charged particle;

  4. 4.

$\pi^{+}$ identification by the RICH and calorimeters;

  5. 5.

decay region definition; and

  6. 6.

selection criteria unique to the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ mode.

The impact of these effects on $\epsilon_{\pi\nu\nu}$ and $\epsilon_{\pi\pi}$
depends on the kinematics of the decay, detector resolutions and efficiencies,
and the accidental presence of unassociated particles in an event.

The kinematics of the decays are studied with simulations, while detector
performance is studied with data and either reproduced by simulation or
factored out from $\epsilon_{\pi\nu\nu}$ and $\epsilon_{\pi\pi}$.

Accidental particles have a twofold effect. They affect detector response and
therefore the reconstruction of $K^{+}$ decays and kinematic resolution. They
also randomly satisfy conditions in the GTK, CHANTI, STRAW, MUV3,
calorimeters, CHOD, and NA48-CHOD that lead to an event being rejected,
referred to here as a random veto. The first effect is modelled with
simulation. The second effect, which is independent of decay mode topology, is
measured directly with data as a function of the instantaneous beam intensity
and factored out of $\epsilon_{\pi\nu\nu}$ and $\epsilon_{\pi\pi}$.

As a consequence, signal and normalization efficiencies may take the form:

$\epsilon_{\pi\nu\nu}=\epsilon^{MC}_{\pi\nu\nu}\cdot\epsilon^{Random}_{\pi\nu\nu}~{}~{}~{}~{}~{}~{}~{}~{}\epsilon_{\pi\pi}=\epsilon^{MC}_{\pi\pi}\cdot\epsilon^{Random}_{\pi\pi}.$
(5)

The Monte Carlo efficiency, $\epsilon^{MC}_{decay}$, quantifies the effects of
the factors listed above, except for random losses, and the random efficiency,
$\epsilon^{Random}_{decay}$, quantifies the fraction of events randomly lost
because of the accidental presence of at least one veto condition.

The $SES$ depends only on the ratio of the $\epsilon_{\pi\nu\nu}$ and
$\epsilon_{\pi\pi}$ efficiencies. As both signal and normalization channels
contain a $\pi^{+}$ in the final state, the ratio effectively cancels
significant components of the two efficiencies, decreasing the dependence of
the $SES$ on their magnitude and reducing significantly their contribution to
the SES uncertainty.

####  6.2.1 Monte Carlo efficiencies

The Monte Carlo efficiency, $\epsilon^{MC}_{decay}$, is the ratio of the
number of simulated events passing signal or normalization selection to the
corresponding number of generated events in the FV.

Figure 8 shows the values of $\epsilon^{MC}_{\pi\pi}$ and
$\epsilon^{MC}_{\pi\nu\nu}$ in bins of $\pi^{+}$ momentum. The sums over all
bins are $0.087\pm 0.009$ and $0.030\pm 0.003$, respectively. The
uncertainties are systematic, due mainly to the accuracy of $\pi^{+}$
identification and $K/\pi$ track matching in the simulation.

Table 1 shows estimates of the contributions to $\epsilon^{MC}_{\pi\nu\nu}$
and $\epsilon^{MC}_{\pi\pi}$ of the components listed in section 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
$\epsilon^{MC}_{\pi\nu\nu}$ and $\epsilon^{MC}_{\pi\pi}$ is attributable to
differences in acceptance, particle reconstruction, and cuts specific to the
signal channel. The accuracy with which these factors are simulated is the
primary source of uncertainty in the $SES$.

Figure 8: Left: $K^{+}\rightarrow\pi^{+}\pi^{0}$ MC efficiency in independent bins of $\pi^{+}$ momentum. Right: $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ MC efficiency in independent bins of $\pi^{+}$ momentum. The efficiencies in regions 1 and 2 are shown separately and summed (full symbols). The width of the coloured bands represents the uncertainties in the measured values. Table 1: Monte Carlo efficiencies for normalization and signal decay modes. The uncertainties in the total efficiencies are systematic and reflect the accuracy of the simulation. Source | $K^{+}\rightarrow\pi^{+}\pi^{0}$ | $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$
---|---|---
Acceptance | $0.27$ | $0.16$
Particle reconstruction | $0.64$ | $0.70$
$K^{+}$ matching | $0.84$ | $0.84$
$\pi^{+}$ identification | $0.72$ | $0.72$
Decay region selection | $0.83$ | $0.81$
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection | $-$ | $0.55$
Total | $0.087\pm 0.009$ | $0.030\pm 0.003$

The next paragraphs focus on the contributions from each component of the
Monte Carlo efficiencies listed in Table 1.

Acceptance
Events fail to be selected because of detector geometry as well as
restrictions on the $\pi^{+}$ momentum and $m^{2}_{\mathrm{miss}}$ ranges. The
effects of these three factors are different for signal and normalization
selection efficiencies and are therefore a potential source of SES
uncertainty.

The impact of the limited accuracy of the simulated $m^{2}_{\mathrm{miss}}$
distribution has been quantified by recalculating the $SES$ with
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays in a smaller $m^{2}_{\mathrm{miss}}$
region, (0.015,0.021) $\text{GeV}^{2}/c^{4}$, where data and MC marginally
agree (section 5.7). The corresponding variation of the SES is approximately
1% and assigned as systematic uncertainty due to the simulation of the
$m^{2}_{\mathrm{miss}}$.

Detector illumination and the momentum spectrum contribute to a lesser extent
to the difference between signal and normalization acceptances. The accuracy
of the simulation with respect to these is ascertained by measuring the
branching ratio of $K^{+}\rightarrow\mu^{+}\nu$ normalized to
$K^{+}\rightarrow\pi^{+}\pi^{0}$. A systematic uncertainty is assigned after
comparing the result of this measurement to the accepted value (section
6.2.2).

Particle reconstruction
The particle reconstruction efficiency is the product of the KTAG and GTK
efficiencies for reconstructing the parent $K^{+}$, and the STRAW, RICH, CHOD,
NA48-CHOD and LKr efficiencies for reconstructing the daughter $\pi^{+}$. The
RICH, CHOD, NA48-CHOD and LKr efficiencies include detector signal association
with a STRAW track.

The effect of local inefficiencies due to detector readout or to accidental
activity cancels at first order in the ratio of efficiencies, as signal and
normalization decays are recorded simultaneously. Nonetheless, these effects
are measured with data and added to the simulation. Table 2 details the impact
of the various subdetectors on the reconstruction efficiency. The numbers are
averages over $\pi^{+}$ momentum between 15 and 35 GeV$/c$ and instantaneous
beam intensity.

Table 2: Average detector efficiencies over $\pi^{+}$ 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 identification efficiencies from $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and $K^{+}\rightarrow\pi^{+}\pi^{0}$, respectively. All other detector efficiencies are equal for signal and normalization. Source | Efficiency
---|---
KTAG | $0.97\pm 0.01$
GTK detector | $0.92\pm 0.02$
GTK reconstruction | $0.94\pm 0.02$
STRAW | $0.95\pm 0.01$
RICH |  $0.95\pm 0.03$ $(0.87\pm 0.03)$
CHOD | $>0.99$
NA48-CHOD | $>0.99$
LKr | $0.95\pm 0.03$

KTAG and GTK efficiencies refer to $K^{+}$ detection and are equal for signal
and normalization. Both efficiencies are measured with data, using
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays. KTAG inefficiencies come
mainly from the readout. GTK inefficiencies arise from geometric acceptance
and identified readout malfunctioning (5%) and from the detector (3%) [25].
The GTK reconstruction efficiency is due to the conditions applied to identify
a track of good quality.

The efficiency to reconstruct a $\pi^{+}$ track with the STRAW is measured
with $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays in the data. In the $15-35$
GeV/$c$ momentum range, the efficiency depends only on the instantaneous beam
intensity, which is directly related to accidental activity in the detectors.

The RICH efficiency for reconstructing a $\pi^{+}$ with momentum between 15
and 35 GeV$/c$ is measured with data using $K^{+}\rightarrow\pi^{+}\pi^{0}$
decays. It is directly related to the statistics of Cherenkov photons and
depends only on the $\pi^{+}$ momentum. Simulation reproduces this efficiency
with a relative accuracy of about 3%. The simulation indicates that this
efficiency is about 7% higher for $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decays
than for $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays. This difference is
attributable to extra hits created when photons from $\pi^{0}$ decay in
$K^{+}\rightarrow\pi^{+}\pi^{0}$ events convert in RICH material and spoil the
charged pion ring shape. Therefore, the RICH reconstruction efficiency does
not cancel in the ratio of Equation (4). A sample of
$K^{+}\rightarrow\mu^{+}\nu$ decays in the data is used to test the accuracy
of RICH particle reconstruction in the simulation. The resulting ratio of data
to MC agrees with that of $K^{+}\rightarrow\pi^{+}\pi^{0}$ to within 1.5%.
This value is assigned as a relative systematic uncertainty 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
efficient. An overall 0.99 efficiency is assigned to account for small losses
in the association of detector signals with STRAW tracks that define
downstream charged particles.

The LKr calorimeter detects signals from minimum ionizing particles with an
efficiency greater than 99%, as measured with data. In the case of $\pi^{+}$
inelastic hadronic interactions, an additional inefficiency may arise in
associating LKr clusters with STRAW tracks.

$\boldsymbol{K^{+}}$ matching
The efficiency for matching a $K^{+}$ with a downstream charged particle is
0.84 and depends on the GTK efficiency and on time and CDA resolutions. The
simulation reproduces the matching performance measured with data to within 5%
relative accuracy, once accidental pileup in the GTK and GTK efficiency are
simulated. This measurement of the accuracy is taken as a systematic
uncertainty in the magnitudes of both $\epsilon^{MC}_{\pi\nu\nu}$ and
$\epsilon^{MC}_{\pi\pi}$. However, the effect of $K^{+}$ matching is equal for
signal and normalization, and therefore no corresponding uncertainty is
assigned to the $SES$. As a cross check, the $SES$ is found to be nearly
insensitive to the simulated level of GTK inefficiency.

$\boldsymbol{\pi^{+}}$ identification
Not every $\pi^{+}$ is identified due to the intrinsic efficiencies of the
RICH and calorimeters and to $\pi^{+}$ decays in flight.

The RICH efficiency for identifying undecayed $\pi^{+}$s from
$K^{+}\rightarrow\pi^{+}\pi^{0}$ events is measured with data and found to be
about 0.95. Simulation reproduces this number with 3% accuracy and indicates
that $\pi^{+}$s from $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decays are
identified with a comparable efficiency.

Simulation reproduces the measured efficiency for the RICH to reconstruct and
identify a $\pi^{+}$ with an accuracy of about 6%. This value is assigned as a
relative uncertainty to $\epsilon^{MC}_{\pi\nu\nu}$ and
$\epsilon^{MC}_{\pi\pi}$. However, no additional uncertainty is assigned to
the $SES$ beyond that from the RICH reconstruction efficiency, because the
RICH identification algorithm treats signal and normalization modes the same.

The average efficiency of $\pi^{+}$ identification with the calorimeters is
about 0.80, as measured with data. Simulation reproduces this result with 2%
accuracy. This degree of accuracy is propagated as a relative uncertainty to
$\epsilon^{MC}_{\pi\nu\nu}$ and $\epsilon^{MC}_{\pi\pi}$. Simulation also
shows that the efficiencies to 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 $\pi^{+}$ identification efficiencies reported in Table 1 include an
additional factor of 0.95 to account for the probability of $\pi^{+}$ decay.

Decay region
In addition to the definition of the 105–165 m FV, the decay region is shaped
by the cuts on the $\pi^{+}$ direction as a function of $Z_{vertex}$ as
discussed in section 5.3. These selection criteria reject a slightly different
number of signal and normalization events. The simulation accounts for the
corresponding effect in the $SES$ together with the kinematic and geometric
acceptances, as the various contributions are correlated.

Signal $\boldsymbol{K^{+}\rightarrow\pi^{+}\nu\bar{\nu}}$ selection
Photon and multiplicity rejection and the box cut are applied only to signal
events. These selection criteria, therefore, directly impact the measurement
of the $SES$.

In the absence of random activity, the box cut alone rejects about 40% of
signal events. The GTK, CHANTI, STRAW, and MUV3 veto conditions do not affect
$\epsilon^{MC}_{\pi\nu\nu}$ or $\epsilon^{MC}_{\pi\pi}$. On the other hand,
because charged pions may interact in RICH material, vetoing photons and extra
charged particles can inadvertently reject
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ events. The accuracy with which the
simulation models this effect is studied by selecting from data
$K^{+}\rightarrow\pi^{+}\pi^{0}$ events in which both photons from the
$\pi^{0}$ decay are detected in LAV stations. The loss of events because of
$\pi^{+}$ interactions is measured on these data and compared with simulation,
leading to about 6% discrepancy. The efficiency $\epsilon^{MC}_{\pi\nu\nu}$ is
corrected for half of this difference. An uncertainty of 100% is assigned to
this correction factor, resulting in about 3% relative uncertainty in the
$SES$.

####  6.2.2 $K^{+}\rightarrow\mu^{+}\nu$ branching ratio measurement

The measurement of the branching ratio of the $K^{+}\rightarrow\mu^{+}\nu$
decay provides a test of the accuracy of the MC simulation of the kinematic
and geometric acceptances.

The measurement follows a procedure similar to that adopted for the $SES$. The
decay $K^{+}\rightarrow\pi^{+}\pi^{0}$ is used for normalization and the
branching ratio can be expressed as:

$\text{BR}(K^{+}\rightarrow\mu^{+}\nu)=\text{BR}(K^{+}\rightarrow\pi^{+}\pi^{0})~{}\frac{N_{\mu
2}}{\hat{N}_{\pi\pi}}~{}\frac{\hat{\epsilon}_{\pi\pi}}{\epsilon_{\mu
2}}~{}\cdot$ (6)

Here $N_{\mu 2}$ and $\hat{N}_{\pi\pi}$ are the number of selected
$K^{+}\rightarrow\mu^{+}\nu$ and $K^{+}\rightarrow\pi^{+}\pi^{0}$ events,
$\epsilon_{\mu 2}$ and $\hat{\epsilon}_{\pi\pi}$ are the efficiencies for
selecting them.

Event selection for both modes differs slightly from the procedures described
in sections 5.1 and 5.2. Both modes require a RICH ring associated with the
STRAW track, but $\pi^{+}$ identification for the
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decay relies on the electromagnetic and
hadronic calorimeters only. MUV3 provides positive identification of the
$\mu^{+}$ from $K^{+}\rightarrow\mu^{+}\nu$ decay.

The kinematic range $0.01<m^{2}_{\mathrm{miss}}<0.026\,\text{GeV}^{2}/c^{4}$
defines $K^{+}\rightarrow\pi^{+}\pi^{0}$ events. The requirement that
$|m^{2}_{\mathrm{miss}}(\mu)|<0.01\,\text{GeV}^{2}/c^{4}$ defines
$K^{+}\rightarrow\mu^{+}\nu$ decays. Here, $m^{2}_{\mathrm{miss}}(\mu)$ is the
squared missing mass computed assuming the particle associated with the STRAW
track to be a muon. The background in both selected modes is of the order of
$10^{-3}$. Estimations of $\hat{\epsilon}_{\pi\pi}$ and $\epsilon_{\mu 2}$
rely on Monte Carlo simulations, as for the $SES$. Their magnitudes are about
0.09 and 0.10.

The procedure described in section 6.2 is adopted to quantify the efficiency
bias introduced by the simulation of the RICH reconstruction. The
corresponding correction factor applied to
$\text{BR}(K^{+}\rightarrow\mu^{+}\nu)$ is $+0.005\pm 0.005$.

The $\pi^{+}$ identification efficiency affects only the
$K^{+}\rightarrow\pi^{+}\pi^{0}$ 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
$\hat{\epsilon}_{\pi\pi}$. Assuming that the uncertainty in this correction is
100%, this amounts to correcting $\text{BR}(K^{+}\rightarrow\mu^{+}\nu)$ by
$-0.008\pm 0.008$.

The $K^{+}\rightarrow\mu^{+}\nu$ branching ratio is measured to be

$\text{BR}(K^{+}\rightarrow\mu^{+}\nu)=0.62\pm 0.01,$ (7)

in agreement within 2.5% with the PDG value [7]. The 0.01 uncertainty is
systematic, mainly attributable to the corrections described above. The
statistical uncertainty is negligible. The result is stable within
uncertainties when signal and normalization selection cuts are varied. It is
also stable throughout the 2017 data taking, as shown in Figure 9.

Figure 9:  Measured branching ratio of the $K^{+}\rightarrow\mu^{+}\nu$ decay
mode in different time periods of the 2017 data taking. The overall quoted
uncertainty is mostly systematic.

This result relies on simulation to account for the different acceptances of
the $K^{+}\rightarrow\mu^{+}\nu$ and $K^{+}\rightarrow\pi^{+}\pi^{0}$ decay
modes, as in the case of the $SES$ measurement. The comparison between the
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
$SES$.

####  6.2.3 Random veto efficiency

In both $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and
$K^{+}\rightarrow\pi^{+}\pi^{0}$ event selections, the GTK, CHANTI, STRAW, and
MUV3 are also used to veto backgrounds. Data are used to estimate the fraction
of kaon decays rejected due to accidental activity in these detectors.
Measurements on samples of $K^{+}\rightarrow\pi^{+}\pi^{0}$ and
$K^{+}\rightarrow\mu^{+}\nu$ 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% of signal and normalization
decays. Because the average beam intensity of selected normalization and
signal-like events is comparable, the effects of the GTK, CHANTI, STRAW and
MUV3 vetoes cancel in the ratio of Equation (4).

The criteria, collectively termed photon and multiplicity rejection (section
5.6), employed to veto $K^{+}$ decays with photons or more than one charged
particle in the final state also reject signal events if accidental particles
overlap the $\pi^{+}$ in time.

Figure 10:  Random veto efficiency $\epsilon_{RV}$ in bins of instantaneous
beam intensity after photon and multiplicity rejection, after photon
rejection, after LKr veto only, after LAV veto only, and after IRC and SAC
veto only. The error bars on the photon and multiplicity rejection points
indicates the total uncertainty. Lines are drawn as guides for the eye.

The fraction of signal events passing photon and multiplicity rejection is
denoted $\epsilon_{RV}$ and called the random veto efficiency. A sample of
$K^{+}\rightarrow\mu^{+}\nu$ decays selected from minimum-bias data is used to
estimate $\epsilon_{RV}$. The selection closely follows that of
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ , including GTK-, CHANTI-, and STRAW-
based veto criteria, except that: $|(P_{K}-P_{\mu})^{2}|<0.006$
GeV${}^{2}/\text{c}^{4}$ replaces the missing-mass squared regions; the
calorimeters and the MUV3 are used for $\mu^{+}$ identification; and no box
cut or the photon and multiplicity rejection criteria are applied. Simulation
shows that the background to $K^{+}\rightarrow\mu^{+}\nu$ is less than a part
per thousand.

Table 3: Contributions to the uncertainty of the random veto efficiency measurement. The total uncertainty is the sum in quadrature of the contributions. Source | Uncertainty in $\epsilon_{RV}$
---|---
$\mu^{+}$ interaction correction | $\pm 0.011$
$\mu^{+}$ identification | $\pm 0.008$
Momentum dependence | $\pm 0.003$
Statistical uncertainty | $<0.001$
Total | $\pm 0.014$

The random veto efficiency is computed as the ratio between the number of
$K^{+}\rightarrow\mu^{+}\nu$ events remaining before and after photon and
multiplicity rejection. Figure 10 displays $\epsilon_{RV}$ as a function of
the instantaneous beam intensity. This result is corrected for the probability
of event loss induced by $\mu^{+}$ interactions, such as $\delta$-ray
production in the RICH material, as estimated by simulation. This correction
increases $\epsilon_{RV}$ by about 1%. An uncertainty of 100% is assigned to
this correction, leading to a 1% systematic uncertainty in $\epsilon_{RV}$.
The stability of $\epsilon_{RV}$ is tested against cuts on
$(P_{K}-P_{\mu})^{2}$ and $\mu^{+}$ identification. The maximum observed
relative variation is $2.4\%$ due to the cut on the calorimetric BDT
probability. Half this variation is used to correct the measured
$\epsilon_{RV}$ and half is assigned as a systematic uncertainty. A residual
dependence on the $\mu^{+}$ momentum is observed and added to the total
systematic uncertainty. The final average random veto efficiency is $0.638\pm
0.014$. Table 3 summarizes the different contributions to the uncertainty.

Because the random veto affects only the signal, the uncertainty in
$\epsilon_{RV}$ contributes linearly to the uncertainty in the $SES$.

###  6.3 Trigger efficiency

Normalization events are selected from minimum-bias data, and signal events
are selected from PNN data. Problems in the hardware and trigger definitions
in conflict with offline cuts may cause the trigger to reject good
normalization and signal events. Because mimimum-bias and PNN triggers differ,
their efficiencies, denoted $\epsilon_{trig}^{\scriptscriptstyle MB}$ and
$\epsilon_{trig}^{\scriptscriptstyle PNN}$ in Equation (4), do not cancel in
the ratio, which therefore must be precisely evaluated. The L0 and L1 trigger
algorithms which identify signal candidates employ different sets of
detectors, so their efficiencies can be studied separately.

####  6.3.1 PNN L0 trigger efficiency

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
$K^{+}\rightarrow\pi^{+}\pi^{0}$ events selected from minimum-bias data using
PNN-like criteria allows the measurement of the L0NoCalo efficiency. The
contributions from the RICH and CHOD are also estimated with
$K^{+}\rightarrow\mu^{+}\nu$ decays. The measured L0NoCalo efficiency is about
0.980 at the mean intensity of 450 MHz and varies almost linearly as a
function of the instantaneous beam intensity, decreasing by about 1% at twice
the mean intensity. The main source of inefficiency comes from the MUV3 veto
criteria, because the veto timing window is larger online than offline due to
online time resolution. The uncertainty in the measured value is at the level
of 0.5%, is mostly systematic and reflects the deviation of the efficiency
from linearity.

The L0Calo efficiency is measured with a sample of
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays in which the two photons are detected
in LAV stations. Events of this type result in a $\pi^{+}$ with momentum
greater than 45 GeV$/c$ in the LKr. The L0Calo efficiency, defined as the
fraction of events passing the L0Calo conditions, is measured as a function of
the energy, $E_{LKr}$, that the $\pi^{+}$ deposits in the LKr. The dependence
on $E_{LKr}$ is converted into a dependence on the $\pi^{+}$ momentum,
$p_{\pi^{+}}$, in the $15-35$ GeV$/$c range, with a conversion factor
extracted from the $E_{LKr}/p_{\pi^{+}}$ distribution of a sample of
$\pi^{+}$s selected from $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays. The
L0Calo efficiency depends on the $\pi^{+}$ 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 the energy
resolution of the LKr, is the main source of inefficiency. The uncertainty in
the L0Calo trigger efficiency comes from the statistics of the
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ sample used to map $E_{LKr}$ into
$p_{\pi^{+}}$.

The overall L0 trigger efficiency is the product of the L0NoCalo and L0Calo
efficiencies as a function of $\pi^{+}$ and intensity. The measured value
decreases with both increasing $\pi^{+}$ momentum and intensity, ranging from
0.95 to 0.9.

####  6.3.2 PNN L1 trigger efficiency

The effects of independent KTAG, LAV, and STRAW requirements in the L1 trigger
efficiency are uncorrelated, such that the overall efficiency is the product
of the individual efficiencies. Samples of $K^{+}\rightarrow\mu^{+}\nu$
selected from minimum-bias data and of $K^{+}\rightarrow\pi^{+}\pi^{0}$
selected from data triggered by the PNN L0 conditions and recorded
irrespective of the L1 trigger decision were used to measure these
efficiencies. The L1 trigger algorithms were emulated offline, including the
effects of resolution.

After applying PNN selection criteria, the KTAG L1 requirements do not
introduce additional loss of signal. On the other hand, the LAV requirements
introduce intensity-dependent losses of events which pass signal offline
selection criteria, because the online LAV time resolution requires a larger
veto timing window than that used offline. The L1 LAV efficiency in the first
part of the 2017 data-taking period ranges from 0.965 to 0.955, depending on
the intensity. This efficiency is about 1% higher and exhibits less intensity
dependence in the second part of 2017 data taking as a consequence of an
optimization of the L1 LAV algorithm. The spread of the efficiency among data-
taking periods is used to set a systematic uncertainty for 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 intensity. A $\pm$0.2% uncertainty is assigned to this value to account for
an observed $\pi^{+}$ momentum dependence.

####  6.3.3 Trigger efficiency and $SES$

Figure 11:  Measured PNN trigger efficiency as a function of instantaneous
beam intensity in four bins of $\pi^{+}$ momentum. The shaded band corresponds
to the total uncertainty.

The effect of the trigger efficiency on the $SES$ is determined using Equation
(4) with the following assumptions: the total PNN trigger efficiency is the
product of the L0 and L1 efficiencies; the efficiency of the L0TP is the same
for the PNN and the minimum bias trigger and therefore cancels in Equation
(4); and the minimum-bias trigger is 100% efficient.

The following test is performed to check the accuracy of these assumptions.
The PNN selection, except $\pi^{+}$ identification with the RICH, is applied
to minimum-bias data, leading to $N^{MB}=701\pm 26$ events in the $\mu\nu$
region of the $m^{2}_{\mathrm{miss}}$ distribution. The expected number of PNN
data in this region passing the same selection can be written, under the above
assumptions, as:

$N^{PNN}(\text{expected})\,=\,D\cdot
N^{MB}\cdot\epsilon^{PNN}_{L0}\cdot\epsilon^{PNN}_{L1}.$ (8)

Here $D$ is the minimum-bias reduction factor and $\epsilon^{PNN}_{L0}$
($\epsilon^{PNN}_{L1}$) are the L0 (L1) PNN trigger efficiencies for
$K^{+}\rightarrow\mu^{+}\nu$ decays in which the muon resembles a pion in the
calorimeters and does not hit MUV3. Considering that
$K^{+}\rightarrow\mu^{+}\nu$ decays are fully efficient under the L0Calo
condition (section 6.3.1), the measured values of $\epsilon^{PNN}_{L0}$ and
$\epsilon^{PNN}_{L1}$ lead to $N^{PNN}(\text{expected})=(263\pm 10)\times
10^{3}$. The number of PNN data observed in the $\mu\nu$ region of the
$m^{2}_{\mathrm{miss}}$ distribution after removing the RICH identification
from the PNN selection is $N^{PNN}(\text{observed})=(255.6\pm 0.6)\times
10^{3}$, in agreement within $\pm\,3.8\%$ with $N^{PNN}(\text{expected})$.
This value is assigned as systematic uncertainty to the measured PNN trigger
efficiency (noted Global in Table 4).

The PNN trigger efficiency relevant to the measurement of $SES$ is shown in
Figure 11 as a function of instantaneous beam intensity and $\pi^{+}$
momentum. The overall average trigger efficiency is $0.87\pm 0.03$. Table 4
summarizes the various contribution to the uncertainty in the trigger
efficiency.

Table 4: Contributions to the uncertainty of the trigger efficiency. When quoted, the range corresponds to the efficiency dependence on the instantaneous beam intensity and the $\pi^{+}$ momentum. Source | Trigger efficiency uncertainty
---|---
L0NoCalo |  $\pm 0.002$ to $\pm 0.004$
L0Calo |  $\pm 0.003$ to $\pm 0.004$
L1 LAV |  $\pm 0.004$ to $\pm 0.014$
L1 Straw | $\pm 0.002$
Global | $\pm 0.035$

###  6.4 $SES$ result

The single event sensitivity and the total number of expected Standard Model
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decays are:

$\displaystyle SES=(0.389\pm 0.024_{syst})\times 10^{-10},$ (9) $\displaystyle
N^{exp}_{\pi\nu\nu}(SM)=2.16\pm 0.13_{syst}\pm 0.26_{ext}.$ (10)

The statistical uncertainty is negligible. Table 5 details the various
contributions to the $SES$, averaged over instantaneous beam intensity and
$\pi^{+}$ momentum.

Table 5: Contributions to $SES$, averaged over instantaneous beam intensity and $\pi^{+}$ momentum. Contribution | value
---|---
$N_{\pi\pi}$ | $68\times 10^{6}$
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ Monte Carlo efficiency, $\epsilon_{\pi\nu\nu}^{MC}$ | $0.030\pm 0.003$
$K^{+}\rightarrow\pi^{+}\pi^{0}$ Monte Carlo efficiency, $\epsilon_{\pi^{+}\pi^{0}}^{MC}$ | $0.087\pm 0.009$
Random veto efficiency (photon and multiplicity rejection) | $0.638\pm 0.014$
Trigger efficiency | $0.87\pm 0.03$

This list of contributions is for reference only, as the measured value of the
$SES$ comes from Equation (4) in bins of instantaneous beam intensity and
$\pi^{+}$ momentum.

The above $SES$ corresponds to about $1.5\times 10^{12}$ effective $K^{+}$
decays in the FV, defined as $(D\cdot
N_{\pi\pi})/(\epsilon_{\pi^{+}\pi^{0}}^{MC}\cdot\text{BR}(K^{+}\rightarrow\pi^{+}\pi^{0}))$.
This quantity is proportional to the actual number of $K^{+}$ decays in the
FV, although not strictly equal because $\epsilon_{\pi^{+}\pi^{0}}^{MC}$ does
not include the elements which factor out and cancel in Equation (4).

Table 6 lists the different sources of $SES$ uncertainty, including
contributions to the uncertainty of the Monte Carlo and trigger efficiency
ratios, as discussed in sections 6.2 and 6.3, respectively. The external error
on $N^{exp}_{\pi\nu\nu}$ stems from the uncertainty in the theoretical
prediction of $\text{BR}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})$. Figure 12
shows $N^{exp}_{\pi\nu\nu}$ in bins of $\pi^{+}$ momentum and instantaneous
beam intensity.

Table 6: Sources contributing to the uncertainty in the $SES$ measurement. “Normalization background” refers to the impact of $K^{+}\rightarrow\pi^{+}\pi^{0}_{D}$ decays on the normalization sample. The total uncertainty is the sum in quadrature of the four contributions listed in the first column. Source | Uncertainty in $SES$ ($\times 10^{10}$)
---|---
Monte Carlo efficiency ratio | $\pm 0.017$
|  $\pi^{+}$ interactions | $\pm 0.012$
| RICH reconstruction | $\pm 0.006$
|  $m^{2}_{\mathrm{miss}}$ Selection | $\pm 0.004$
| Acceptance simulation | $\pm 0.010$
Trigger efficiency | $\pm 0.015$
| L0 Efficiency | $\pm 0.002$
| L1 Efficiency | $\pm 0.003$
| Global | $\pm 0.015$
Random Veto efficiency | $\pm 0.008$
Normalization Background | $<0.001$
Total | $\pm 0.024$

Figure 12: Number of expected Standard Model
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ events in bins of $\pi^{+}$ momentum
(left) and average beam intensity (right). The average beam intensity per bin,
obtained from the $K^{+}\rightarrow\pi^{+}\pi^{0}$ sample used for
normalization, is plotted at the barycentre of the bin.

##  7 Expected background

The background to $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decays can be divided
into two classes. The $K^{+}$ decay background is due to kaon decays in the FV
other than $K^{+}\to\pi^{+}\nu\bar{\nu}$, while the upstream background is due
to $\pi^{+}$ particles produced either by beam particle interactions or by
kaon decays upstream of the FV. To mimic a signal, a background event should
have a $K^{+}$ reconstructed upstream and matched to a $\pi^{+}$ downstream,
and $m^{2}_{\mathrm{miss}}$ reconstructed in the signal region. Furthermore,
either the extra particles produced in association with the $\pi^{+}$ should
escape detection, or a lepton in the final state should be mis-identified as a
$\pi^{+}$.

###  7.1 $K^{+}$ decay background

The background from $K^{+}$ decays in the FV is primarily due to the
$K^{+}\to\pi^{+}\pi^{0}$, $K^{+}\to\mu^{+}\nu$,
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ and $K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$
decays.

The first three processes are constrained kinematically, and enter the signal
regions via $m^{2}_{\mathrm{miss}}$ mis-reconstruction due to large-angle
Coulomb scattering, elastic hadronic interactions in GTK and STRAW material,
incorrect $K/\pi$ association, pattern recognition errors, or position mis-
measurement in the spectrometers. In addition to $m^{2}_{\mathrm{miss}}$ mis-
reconstruction, at least one of the following should occur: photons from a
$K^{+}\to\pi^{+}\pi^{0}$ decay are not detected by electromagnetic
calorimeters; the muon from a $K^{+}\to\mu^{+}\nu$ decay is mis-identified as
$\pi^{+}$ by the RICH counter, hadronic calorimeters and MUV3; a
$\pi^{+}\pi^{-}$ pair from a $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decay is
undetected by the STRAW and the other downstream detectors.

The background from the three kinematically-constrained decays is evaluated
with data. Denoting by $N_{\rm decay}$ the number of events in the
corresponding background region of $m^{2}_{\mathrm{miss}}$ in the PNN data
sample passing the PNN selection, and by $f_{\rm kin}$ the probability that
$m^{2}_{\mathrm{miss}}$ is reconstructed in the signal region, the expected
number of background events from each decay is given by

$N^{\rm exp}_{\rm decay}=N_{\rm decay}\cdot f_{\rm kin}\cdot$ (11)

The value of $N_{\rm decay}$ is obtained directly from the PNN data, while the
probability $f_{\rm kin}$ is measured with minimum-bias data. This technique
does not require knowledge of photon and charged particle rejection
inefficiencies or of the $\pi^{+}$ mis-identification probability.
Nevertheless, the precision of the method relies on three assumptions whose
reasonableness is tested with both data and simulations:

  1. 1.

$f_{\rm kin}$ represents the probability that an event of a given decay mode
enters the signal region;

  2. 2.

$f_{\rm kin}$ and $N_{\rm decay}$ are uncorrelated; and

  3. 3.

$N_{\rm decay}$ accounts only for events of the corresponding decay mode.

Backgrounds from the $K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$ decay, as well as from
the rare decay $K^{+}\to\pi^{+}\gamma\gamma$ and the semileptonic decays
$K^{+}\to\pi^{0}\ell^{+}\nu$ ($\ell=e,\mu$), are evaluated with simulations.

In the following subsections the $K^{+}\to\pi^{+}\pi^{0}$ and the
$K^{+}\to\mu^{+}\nu$ backgrounds are shown in bins of $\pi^{+}$ momentum up to
40 GeV$/c$, albeit only the 15–35 GeV/$c$ momentum range is used to evaluate
the corresponding backgrounds in the signal regions.

####  7.1.1 $K^{+}\to\pi^{+}\pi^{0}$ decay

Figure 13:  Distribution of the PNN events in the ($\pi^{+}$ momentum,
$m^{2}_{\mathrm{miss}}$) plane after the PNN selection in the $\pi^{+}\pi^{0}$
(red box) and control $m^{2}_{\mathrm{miss}}$ regions (blue boxes). Both
control regions are used only for validation of the background estimation. The
shaded grey area represents the distribution of the simulated SM
$K^{+}\to\pi^{+}\nu\bar{\nu}$ events (arbitrarily normalized).

After the PNN selection, $N_{\pi\pi}=264$ events from the PNN sample remain in
the $\pi^{+}\pi^{0}$ region. The distribution of these events in the
($\pi^{+}$ momentum, $m^{2}_{\mathrm{miss}}$) plane is shown in Figure 13,
after the unblinding of the $K^{+}\to\pi^{+}\pi^{0}$ control regions. The
$\pi^{+}$ momentum lies in the $15-20$ GeV/$c$ range for 60% of these events,
due to the degradation of the $\pi^{0}$ detection efficiency for photons
emitted at small angles (section 5.6). The measurement of $f_{\rm kin}$ is
based on a $K^{+}\to\pi^{+}\pi^{0}$ sample selected from minimum-bias data.
The selection involves the $K^{+}\to\pi^{+}$ decay definition described in
sections 5.1, 5.2 and 5.5. The conditions of section 5.3 are applied as well,
however the decay region is defined as $115<Z_{\rm vertex}<165$ m. Specific
selection criteria are employed to tag the $\pi^{0}$ by reconstructing two
photons from the $\pi^{0}\to\gamma\gamma$ decay in the LKr calorimeter
independently of the $\pi^{+}$ and $K^{+}$ tracks. The quantity $Z_{\rm
vertex}$ is evaluated from the coordinates of the two photon energy clusters
in LKr by assuming that they originate from a $\pi^{0}$ decay on the nominal
beam axis. The vertex is required to be within the decay region, and its
position is used to reconstruct the photon and the $\pi^{0}$ momenta.
Consequently, the expected $\pi^{+}$ trajectory is reconstructed and is
required to be in the geometric acceptance of the detectors. The reconstructed
squared missing mass $(P_{K}-P_{0})^{2}$, where $P_{K}$ and $P_{0}$ are the
four-momenta of the nominal $K^{+}$ and the reconstructed $\pi^{0}$, peaks at
the squared $\pi^{+}$ mass for $K^{+}\to\pi^{+}\pi^{0}$ decays. A cut on this
quantity is applied to select an almost background-free
$K^{+}\to\pi^{+}\pi^{0}$ sample without biasing the $m^{2}_{\mathrm{miss}}$
reconstruction.

Figure 14 (top left) displays the $m^{2}_{\mathrm{miss}}$ spectrum of the
$K^{+}\to\pi^{+}\pi^{0}$ minimum-bias sample used for $f_{\rm kin}$
measurement: $f_{\rm kin}$ is evaluated for each of the signal regions 1 and 2
as the ratio of the numbers of events in the signal region and in the
$K^{+}\to\pi^{+}\pi^{0}$ region. The simulation reproduces the tails within
the statistical uncertainties, and the background is negligible. The measured
values of $f_{\rm kin}$ in bins of $\pi^{+}$ momentum are shown in Figure 14
(bottom left). Incorrect $K/\pi$ association due to the pileup in the GTK
accounts for 50% of the contribution to $f_{\rm kin}$ in region 1, and 30% in
region 2.

Figure 14:  Top left: reconstructed $m^{2}_{\mathrm{miss}}$ distribution of
the $K^{+}\rightarrow\pi^{+}\pi^{0}$ minimum-bias data events selected by
tagging the $\pi^{0}$ (full symbols, see text for details) integrated over the
$15-35\,\text{GeV}/c$ momentum range. Simulated samples of
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays and backgrounds (normalized to the
data in the $K^{+}\to\pi^{+}\pi^{0}$ region) are superimposed. Signal regions
1 and 2 are shown. The $K^{+}\rightarrow\pi^{+}\pi^{0}$ region, defined by the
condition $0.015<m^{2}_{\mathrm{miss}}<0.021\,\text{GeV}^{2}/c^{4}$, and the
control regions, comprised between the signal and
$K^{+}\rightarrow\pi^{+}\pi^{0}$ regions, are not shown. Top right: same as
top-left, but the simulated $K^{+}\rightarrow\pi^{+}\pi^{0}$ sample is
selected without applying the $\pi^{0}$ tagging; simulated backgrounds are not
shown. Bottom left : the probability $f_{\rm kin}$, defined in the text,
measured using the $K^{+}\to\pi^{+}\pi^{0}$ minimum-bias sample in bins of
$\pi^{+}$ momentum, separately for signal region 1 and 2 and combined, with
the statistical uncertainties. Bottom right: expected and observed numbers of
background events in the $K^{+}\to\pi^{+}\pi^{0}(\gamma)$ decay control
regions in $\pi^{+}$ momentum bins. The errors are statistical for the
observed numbers of events, and dominated by systematics for the expected
numbers of events.

The total expected background in the signal regions, evaluated by applying
Equation (11) in each $\pi^{+}$ momentum bin, is $N_{\pi\pi}^{\rm exp}=0.27\pm
0.026_{\rm stat}\pm 0.014_{\rm syst}$. The statistical uncertainty is mainly
due to $N_{\pi\pi}$. The systematic uncertainty accounts for a possible bias
to the shape of the $m^{2}_{\mathrm{miss}}$ spectrum induced by the $\pi^{0}$
tagging used to measure $f_{\rm kin}$. It is evaluated by comparing the
simulated shape of the $m^{2}_{\mathrm{miss}}$ spectrum in the minimum-bias
sample with that of $K^{+}\to\pi^{+}\pi^{0}$ decays used for normalization
(Figure 14, top right). The 5% difference between the numbers of events in
region 1 in the two samples is taken as a systematic uncertainty.

Radiative decays in the simulated $K^{+}\to\pi^{+}\pi^{0}$ sample are modeled
according to [26]. Simulation studies show that decays with radiative photons
energetic enough to shift the reconstructed $m^{2}_{\mathrm{miss}}$ value to
the signal regions are absent in the minimum-bias sample, due to the $\pi^{0}$
tagging suppression (Figure 14 top right). However the presence of an
additional photon in the final state improves the photon veto, compensating
for the weaker kinematic suppression. The contribution of the radiative
component to the $K^{+}\to\pi^{+}\pi^{0}$ decays is computed by applying the
measured single photon detection efficiency (section 5.6) to the simulated
$K^{+}\to\pi^{+}\pi^{0}$ decays entering signal region 2 because of the
presence of radiated photons. It is concluded that the presence of an
additional photon improves the rejection of $K^{+}\to\pi^{+}\pi^{0}$ in region
2 by a factor of almost 30 with respect to the case of the photons from the
$\pi^{0}$ decay only. This leads to an increase of the expected
$K^{+}\to\pi^{+}\pi^{0}$ background of 0.02 events. A systematic uncertainty
of 100% is conservatively considered for this value, mainly due to the
accuracy of the simulation and the modelling of the single photon detection
efficiency.

Table 7: Expected numbers of $K^{+}\to\pi^{+}\pi^{0}$ events in the signal regions, and expected and observed numbers of events in the control regions. “Control region 1” corresponds to $0.010<m^{2}_{\mathrm{miss}}<0.015$ GeV${}^{2}/c^{4}$, “Control region 2” to $0.021<m^{2}_{\mathrm{miss}}<0.026$ GeV${}^{2}/c^{4}$. Expected events in both signal and control regions 2 are corrected for the contribution from the radiative component of the decay. The uncertainties are the sums in quadrature of the statistical and systematic ones. Radiative decays are kinematically forbidden in both signal region 1 and control region 1. Region | Expected $K^{+}\rightarrow\pi^{+}\pi^{0}$ | Observed
---|---|---
Signal region 1 | $0.11\pm 0.01$ | masked
Signal region 2 | $0.18\pm 0.04$ | masked
Control region 1 | $2.6\pm 0.3$ | 2
Control region 2 | $5.2\pm 0.6$ | 5

The numbers of expected $K^{+}\to\pi^{+}\pi^{0}$ events in the signal regions
are presented in Table 7. The overall background expected in the signal
regions, including the effect of the radiative decays, is

$N^{\rm exp}_{\pi\pi}=0.29\pm 0.03_{\rm stat}\pm 0.03_{\rm syst}.$ (12)

To validate this result, the numbers of expected and observed events are
compared in the two $\pi^{+}\pi^{0}$ control regions. The probability $f_{\rm
kin}$ for the control regions is measured to be about 25 times higher than for
the corresponding signal regions, and the expected background scales
accordingly. The contribution from radiative decays in control regions is
found negligible. Table 7 and in Figure 14 (bottom right) present the numbers
of expected and observed events in the control regions, found to be in good
agreement. The uncertainties in the expected background in the control regions
are mostly systematic due to the modelling of the $m^{2}_{\mathrm{miss}}$
spectrum.

####  7.1.2 $K^{+}\to\mu^{+}\nu$ decay

After the PNN selection, $N_{\mu\nu}=479$ events from the PNN sample remain in
the $\mu\nu$ region. The numbers of events in bins of reconstructed $\pi^{+}$
momentum are presented in Table 8, and the distribution of these events in the
($\pi^{+}$ momentum, $m^{2}_{\mathrm{miss}}$) plane is shown in Figure 15 (top
left). The momentum dependence is a consequence of the $K^{+}\to\mu^{+}\nu$
kinematics when the $\pi^{+}$ mass is used to reconstruct
$m^{2}_{\mathrm{miss}}$ and of the better performance of the RICH in rejecting
$\mu^{+}$ at low momentum. The background to $N_{\mu\nu}$ is negligible.

Figure 15:  Top left: distributions of the PNN-triggered events in the ($\pi^{+}$ momentum, $m^{2}_{\mathrm{miss}}$) plane after the PNN selection in the $\mu\nu$ (red contour) and control $m^{2}_{\mathrm{miss}}$ (blue contour) regions. The control region is used only for validation of the background estimation. Top right: reconstructed $m^{2}_{\mathrm{miss}}$ distribution of the $K^{+}\to\mu^{+}\nu$ minimum-bias data events (full symbols, see text for details) integrated over the 15–35 GeV/$c$ momentum range. The distribution of simulated $K^{+}\to\mu^{+}\nu$ decays is superimposed. Signal regions 1 and 2 are shown. Bottom left: the probability $f_{\rm kin}$ measured using the $K^{+}\to\mu^{+}\nu$ minimum-bias sample in bins of reconstructed $\pi^{+}$ momentum, separately for signal region 1 and 2 and combined (black symbols), and the corresponding statistical uncertainties. Bottom right: expected and observed numbers of background events from $K^{+}\to\mu^{+}\nu$ decays in the $\mu\nu$ control region in $\pi^{+}$ momentum bins. The errors are statistically dominated. Table 8: Number of PNN-triggered events that pass the PNN selection and are reconstructed in the $\mu\nu$ region, observed in the data in bins of $\pi^{+}$ momentum. Momentum bins (GeV/$c$) | $15-20$ | $20-25$ | $25-30$ | $30-35$
---|---|---|---|---
Observed events | $1$ | $48$ | $143$ | $287$

Two methods are exploited to estimate the $K^{+}\to\mu^{+}\nu$ background.

In the first one, the measurement of $f_{\rm kin}$ is based on a
$K^{+}\to\mu^{+}\nu$ sample selected from minimum-bias data, as described in
sections 5.1 and 5.2. Additionally, the calorimetric BDT probability must be
consistent with the identification of a $\mu^{+}$, while events are discarded
if the STRAW track is identified as $\pi^{+}$ or $e^{+}$ in the calorimeters.
The decay region is defined as $115<Z_{\rm vertex}<165$ m. The rejection of
photons and extra charged particles is the same as in the PNN selection
(section 5.6). The box cut and the kinematic requirements on
$m^{2}_{\mathrm{miss}}$ are not applied. Figure 15 (top right) displays the
$m^{2}_{\mathrm{miss}}$ spectrum of the $K^{+}\to\mu^{+}\nu$ minimum-bias
sample used for $f_{\rm kin}$ measurement: $f_{\rm kin}$ is evaluated for each
of the signal regions 1 and 2 as the ratio of the numbers of events in the
signal and $\mu\nu$ regions. The signal region definition does not include the
cuts on $m^{2}_{\mathrm{miss}}$ computed using the momentum evaluated from the
RICH information. Simulation reproduces the shape of the
$m^{2}_{\mathrm{miss}}$ spectrum within the statistical uncertainties, and the
background is negligible. The measured $f_{\rm kin}$ values in bins of
reconstructed $\pi^{+}$ momentum are shown in Figure 15 (bottom left). At
large momentum, $f_{\rm kin}$ increases because the $m^{2}_{\mathrm{miss}}$ of
$K^{+}\rightarrow\mu^{+}\nu$ events computed assuming the $\pi^{+}$ mass
approaches signal region 1. Simulations show that the contribution to $f_{\rm
kin}$ due to incorrect $K/\pi$ association is sub-dominant with respect to
material effects. The total expected background in the signal region,
evaluated with this method by applying Equation (11) in each $\pi^{+}$
momentum bin, is $N_{\mu\nu}^{\rm exp}=0.14\pm 0.007_{\rm stat}\pm 0.007_{\rm
syst}$. The statistical uncertainty is due to $N_{\mu\nu}$. The systematic
uncertainty comes from the stability of the result with the variation of the
BDT probability cut, and accounts for a possible bias on $f_{\rm kin}$ due to
the $\mu^{+}$ identification criteria applied in the selection of the
$K^{+}\to\mu^{+}\nu$ minimum-bias sample. This result relies on the assumption
that the $\pi^{+}$ identification with the RICH and the shape of the
$m^{2}_{\mathrm{miss}}$ spectrum are uncorrelated. This assumption, in
principle, is violated because $K^{+}\to\mu^{+}\nu$ events may enter the
signal region due to track mis-reconstruction, which also affects particle
identification with the RICH. In addition, the background estimation procedure
does not include the cut on $m^{2}_{\mathrm{miss}}$ computed using the RICH to
measure $f_{\rm kin}$, which can bias the result.

To investigate the accuracy of these approximations, a second method is
employed to evaluate the $K^{+}\to\mu^{+}\nu$ background. In this case the
strategy is similar to that previously discussed, but the $\pi^{+}$
identification by the RICH is removed from the PNN selection used to derive
$N_{\mu\nu}$ and added to the selection of the $K^{+}\to\mu^{+}\nu$ minimum-
bias sample used to measure $f_{\rm kin}$. In addition, the determination of
$f_{\rm kin}$ includes the cuts on $m^{2}_{\mathrm{miss}}$ computed using the
$\pi^{+}$ momentum measured from the RICH in the $\pi^{+}$ hypothesis. The
$\mu^{+}$ rejection by the RICH suppresses $f_{\rm kin}$ by two orders of
magnitude with respect to that of Figure 15 (bottom left), keeping a similar
dependence on the $\pi^{+}$ momentum. This translates into a statistical
uncertainty of 20% on $f_{\rm kin}$. The expected background in the signal
region is evaluated to be $N_{\mu\nu}^{\rm exp}=0.08\pm 0.02_{\rm stat}$. This
method is free from the bias of the first method, as it relies only on the
assumption that $\mu^{+}$ rejection with the RICH and the calorimeters are
uncorrelated. Simulations show that this assumption is valid as long as the
$\mu^{+}$ does not decay upstream of the RICH.

Table 9: Expected numbers of $K^{+}\to\mu^{+}\nu$ events in signal regions, and numbers of expected and observed events in the control region. The uncertainties are the sums in quadrature of the statistical and systematic ones. Region | Expected $K^{+}\rightarrow\mu^{+}\nu$ | Expected $K^{+}\rightarrow\mu^{+}\nu$, $\mu^{+}\rightarrow e^{+}\nu\bar{\nu}$ | Observed
---|---|---|---
Signal region 1 | $0.11\pm 0.04$ | $0.04\pm 0.02$ | masked
Signal region 2 | $<0.005$ | $<0.005$ | masked
Control region | $10.5\pm 1.0$ | $0.5\pm 0.2$ | 12

The average of the estimates of $N_{\mu\nu}^{\rm exp}$ from the two methods is
used, and a systematic uncertainty equal to half of the difference ($\pm
0.03$) is assigned to account for a possible bias due to the correlation
between particle identification and shape of the $m^{2}_{\mathrm{miss}}$
spectrum. This background estimate includes also the contribution from the
radiative component of the $K^{+}\to\mu^{+}\nu$ decays, as radiative decays
enter the $K^{+}\to\mu^{+}\nu$ minimum-bias sample used to evaluate $f_{\rm
kin}$.

The background from muon decays in flight $\mu^{+}\to e^{+}\nu\bar{\nu}$ is
not included in the above estimate (as $f_{\rm kin}$ is measured requiring
muon identification), and is determined separately using simulation. The
rejection of this background depends on the muon decay position within the
detector setup. Decays in the FV affect the kinematics, however, positrons are
efficiently rejected by particle identification. Decays within the STRAW
spectrometer impact both kinematics and particle identification in the RICH,
while decays downstream of the STRAW affect particle identification only.
Simulations show that only $\mu^{+}$ decays between the third and fourth STRAW
chambers are relevant, leading to a worsening of both $K^{+}\to\mu^{+}\nu$
kinematics and particle identification. The background is found to contribute
to region 1 only, and is computed to be $0.04\pm 0.02$. The uncertainty quoted
includes statistical and systematic contributions of similar magnitudes. The
latter is evaluated by checks performed on data to validate the simulation of
the positron rejection.

The numbers of expected $K^{+}\to\mu^{+}\nu$ events in the signal regions are
presented in Table 9. The overall background expected is

$N^{\rm exp}_{\mu\nu}=0.15\pm 0.02_{\rm stat}\pm 0.04_{\rm syst},$ (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 $\mu\nu$ control region. The expected number of events is
evaluated with a technique similar to that described above. This comparison is
presented in Table 9 and Figure 15 (bottom right), showing good agreement
between expected and observed numbers of events.

####  7.1.3 $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay

After the PNN selection, $N_{3\pi}=161$ events from the PNN sample remain in
the $3\pi$ region. The distribution of these events in the ($\pi^{+}$
momentum, $m^{2}_{\mathrm{miss}}$) plane is shown in Figure 16 (left); the
$\pi^{+}$ momentum is constrained kinematically to the region below 25
GeV/$c$.

Figure 16:  Left: distribution of the PNN-triggered events in the ($\pi^{+}$
momentum, $m^{2}_{\mathrm{miss}}$) plane after the PNN selection in the $3\pi$
(red box) and control $m^{2}_{\mathrm{miss}}$ regions (blue box). No event is
found in the control region. The shaded grey area represents the distribution
of the simulated SM $K^{+}\to\pi^{+}\nu\bar{\nu}$ events (arbitrarily
normalized). Right: reconstructed $m^{2}_{\mathrm{miss}}$ distribution for the
unpaired $\pi^{+}$ from $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decays, obtained using
the $2\pi$ tagging method, selected from minimum-bias data and simulated
samples. Signal regions 1 and 2 are shown.

The measurement of $f_{\rm kin}$ is based on a $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$
sample selected from minimum-bias data. For this purpose, a $\pi^{+}\pi^{-}$
pair is used to tag the $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay without biasing
the reconstruction of the unpaired $\pi^{+}$. The $\pi^{+}$ to be paired to
the $\pi^{-}$ is chosen randomly event by event. The presence of a two-track
vertex in the FV is required, and the quantity
$(P_{\pi^{+}}+P_{\pi^{-}}-P_{K^{+}})^{2}$, where $P_{\pi^{\pm}}$ are the
reconstructed 4-momenta of the $\pi^{\pm}$ and $P_{K^{+}}$ is the nominal kaon
4-momentum, must be consistent with the squared $\pi^{+}$ mass. The selection
proceeds with respect to the unpaired $\pi^{+}$ as described in sections 5.1
and 5.2. Photon veto conditions are applied to the LAV, IRC and SAC only. The
box cut, the kinematic cuts on $m^{2}_{\mathrm{miss}}$ and the multiplicity
rejection are not applied.

The reconstructed $m^{2}_{\mathrm{miss}}$ spectra for the tagged unpaired
$\pi^{+}$ for the minimum-bias data and simulated samples are shown in Figure
16 (right). It is found that $f_{\rm kin}=(1.6^{+2.0}_{-1.0})\times 10^{-5}$,
where the uncertainties are statistical. Data and simulations are consistent
within the uncertainties.

The kinematics of the tagged $\pi^{+}$ differs from that of the $\pi^{+}$
remaining after the PNN selection, potentially biasing the $f_{\rm kin}$
measurement. In particular, the $m^{2}_{\mathrm{miss}}$ spectrum of the tagged
sample does not match the one of the residual events in the $3\pi$ region. The
impact on the $f_{\rm kin}$ measurement is evaluated with simulations, varying
the selection criteria. The full PNN selection cannot be applied to the
simulated samples due to statistical limitations. Modified PNN selections used
for the tests include those without the tagging, and with requirements of at
least one and exactly one $\pi^{+}$ reconstructed in the geometric acceptance.
The latter selection is the most PNN-like, and leads to a shape of the
$m^{2}_{\mathrm{miss}}$ spectrum in the $3\pi$ region matching that of the
data events passing the full PNN selection. The values of $f_{\rm kin}$
obtained from simulations with the modified selections are in agreement within
the uncertainties quoted above.

A possible bias comes from the dependence of $f_{\rm kin}$ on the $Z$-position
of the decay vertex, as the tagging affects the shape of the $Z_{\rm vertex}$
spectrum. To quantify this effect, $f_{\rm kin}$ is evaluated in bins of
$Z_{\rm vertex}$ for data and simulated samples. The variation of $f_{\rm
kin}$ across $Z_{\rm vertex}$ bins in simulated samples is conservatively
considered as a systematic uncertainty. The final result is $f_{\rm kin}=(5\pm
5)\times 10^{-5}$.

The background computed using Equation (11) is

$N_{3\pi}=0.008\pm 0.008.$ (14)

To validate this result, the numbers of expected and observed events are
compared in the (unmasked) $3\pi$ control region. The expected number of
events in the control region is sensitive to the shape of the
$m^{2}_{\mathrm{miss}}$ spectrum close to the kinematic threshold of the
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay. Simulation studies lead to a
conservative upper limit of $1.5\times 10^{-3}$ on $f_{\rm kin}$ in the
control region, corresponding to less than 0.24 expected background events.
This is consistent with the observation of zero events in the control region.

####  7.1.4 $K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$ decay

The $K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$ decay (denoted $K_{e4}$ below) is
characterized by large $m^{2}_{\mathrm{miss}}$ and therefore contributes to
region 2 only. This background is suppressed by the ${\cal O}(10^{-5})$
branching ratio [7], the kinematic definition of the signal region, and the
multiplicity rejection. The reconstructed $m^{2}_{\mathrm{miss}}$ value
depends on the kinematics of the undetected charged particles, which impacts
the multiplicity rejection. Because of this correlation, the $K_{e4}$
background estimation relies on simulation.

The efficiency of the PNN selection evaluated with a sample of $2\times
10^{9}$ simulated $K_{e4}$ decays using the same normalization procedure as
for the $SES$ computation is $\varepsilon_{Ke4}=(4\pm 2_{\rm stat})\times
10^{-9}$. This leads to an estimated background of $N_{Ke4}=(0.12\pm 0.05_{\rm
stat})$ events. To validate this estimate, four modified event selections
leading to samples enriched with $K_{e4}$ decays are used:

  1. 1.

the PNN selection, with inverted multiplicity conditions in the STRAW;

  2. 2.

the PNN selection applied to the $\pi^{-}$ with RICH identification criteria
not used, and inverted multiplicity conditions in the STRAW;

  3. 3.

similar to 2, with the standard STRAW multiplicity conditions used; and

  4. 4.

similar to 3, with RICH identification criteria used.

The selection efficiency for $K_{e4}$ decays ranges from $1.1\times 10^{-7}$
(selection 4) to $3.7\times 10^{-6}$ (selection 2). The corresponding data
events entering region 2 are solely $K_{e4}$. The reconstructed
$m^{2}_{\mathrm{miss}}$ distributions obtained within selection 2 for PNN data
and simulated $K_{e4}$ sample show agreement within the statistical
uncertainties (Figure 17 (left)). The expected and observed numbers of events
in region 2 within each of the four selections are summarized in Figure 17
(right). In particular, $3\pm 1_{\rm stat}$ events are expected and $6\pm
2_{\rm stat}$ events are observed within selection 4 which has the lowest
acceptance. This difference is conservatively considered as a systematic
uncertainty in $N_{Ke4}^{\rm exp}$, despite the agreement within the
statistical uncertainties, leading to the expected background from $K_{e4}$
decays

$N_{Ke4}=0.12\pm 0.05_{\rm stat}\pm 0.06_{\rm syst}.$ (15)

Figure 17: Left: reconstructed $m^{2}_{\mathrm{miss}}$ distribution of the
events of sample 2 selected in signal region 2 for data and $K_{e4}$
simulation. Right: expected number of $K_{e4}$ decays and observed number of
data events in region 2 for each of the four samples used to validate the
$K_{e4}$ simulation. The different samples are defined in the text.

####  7.1.5 Other $K^{+}$ decays

Semileptonic decays: the branching ratios of the semileptonic decays
$K^{+}\to\pi^{0}e^{+}\nu$ and $K^{+}\to\pi^{0}\mu^{+}\nu$ are 5.1% and 3.4%,
respectively [7]. The presence of the neutrino in the final state prevents
kinematic discrimination of these decays from the signal, and the background
is suppressed by exploiting the presence of a $\pi^{0}$ and a lepton in the
final state. The background estimation relies on simulation, with a
factorization approach used to overcome unavoidable statistical limitations.
Particle identification in the RICH and calorimeters are treated as
independent, and the corresponding efficiencies are factored out with respect
to the efficiency of the rest of the selection.

The measured muon misidentification probability as a pion in the RICH detector
depends on the particle momentum (section 5.5). On average such a probability
is about $2\times 10^{-3}$ for $K^{+}\to\pi^{0}\mu^{+}\nu$ decays passing the
PNN selection; this result is used to validate the simulations. Positron
misidentification probability as a pion in the RICH detector in the 15–35
GeV/$c$ momentum range is evaluated with a simulated sample to be about
$10^{-6}$. Calorimetric muon and positron misidentification probabilities as a
pion in this momentum range evaluated with simulations are about $10^{-5}$ and
$10^{-3}$, respectively.

The simulation accounts for the joint effect of $\pi^{0}$ rejection, and the
geometric and kinematic acceptances. Simulations show that the former is about
$10^{-5}$ (substantially weaker than for $K^{+}\to\pi^{+}\pi^{0}$ decays due
to the different photon kinematics), while the latter is about 10%. The
$K^{+}\to\pi^{+}\pi^{0}$ decay is used for normalization. This leads to a
systematic uncertainty in excess of 10%, mostly because the particle
identification efficiencies do not cancel in the ratio with that of
$K^{+}\to\pi^{+}\pi^{0}$. Including the measured random veto and trigger
efficiencies, the expected background is found to be less than 0.001 events
for both decay modes, and is therefore considered negligible.

$K^{+}\to\pi^{+}\gamma\gamma$: the branching ratio of this decay, occurring at
the loop level, is $1.0\times 10^{-6}$ [7]. The corresponding background is
evaluated with simulations. The decay dynamics favours values of the di-photon
invariant mass above the di-pion threshold, corresponding to
$m^{2}_{\mathrm{miss}}$ values in the $3\pi$ region. This procedure leads to
an overall efficiency of the PNN selection without photon rejection at the 1%
level. The rejection of events in the signal region benefits from the
correlation between $m^{2}_{\mathrm{miss}}$ and the photon energy, leading to
a photon rejection of order $10^{7}$. The $K^{+}\to\pi^{+}\pi^{0}$ decay is
used for normalization. Including the measured random veto and trigger
efficiencies, the background is estimated to be $N_{\pi\gamma\gamma}=0.005\pm
0.005$, where the conservative uncertainty accounts for the accuracy of the
photon rejection simulation.

###  7.2 Upstream background

####  7.2.1 Background sources

Upstream events are defined as interactions or decays of beam particles
upstream of the FV. An upstream event can mimic a
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay if:

  * •

a $\pi^{+}$ is produced and reaches the downstream detectors;

  * •

no additional particles associated to the $\pi^{+}$ are detected downstream;
and

  * •

a $K^{+}$ candidate is reconstructed and matched to the $\pi^{+}$.

Based on these conditions, upstream events can be classified as follows:

  1. 1.

Accidental upstream events: events in which the $\pi^{+}$ does not originate
from the reconstructed $K^{+}$ candidate. In this case the $K^{+}$ candidate
is a pileup GTK track associated accidentally with the $\pi^{+}$ and tagged as
a kaon by the KTAG.

The mechanisms giving rise to accidental upstream events are the following:

     * a)

the $\pi^{+}$ comes from a $K^{+}$ decaying in the region upstream of GTK3;
the KTAG signal produced by the parent $K^{+}$ is associated with a pileup
beam $\pi^{+}$ or proton track, which is reconstructed as a kaon in the GTK;
additional particles produced in the decay are absorbed by material in the
beam line;

     * b)

similar to a), but the matching GTK track belongs to another pileup $K^{+}$
identified correctly by the KTAG;

     * c)

similar to a), but the $\pi^{+}$ originates from an inelastic interaction of a
beam $K^{+}$ upstream of GTK3;

     * d)

similar to b), but the $\pi^{+}$ originates from an inelastic interaction of a
beam $\pi^{+}$ or proton upstream of GTK3.

  2. 2.

In-time upstream events: events in which the $\pi^{+}$ is a primary or a
secondary product of an inelastic interaction of a beam $K^{+}$ in GTK3. In
this case, additional particles produced in the interaction must escape
detection, as no beam line elements can absorb the particles.

Two processes may lead to in-time upstream events:

     * a)

the interacting $K^{+}$ produces a prompt $\pi^{+}$ that reaches the
downstream detectors;

     * b)

the interacting $K^{+}$ produces a relatively long-lived particle ($K_{S}$,
$K_{L}$, $K^{+}$ or $\Lambda$) that decays to a $\pi^{+}$ in the FV.

Figure 18: Reconstructed $m^{2}_{\mathrm{miss}}$ distributions of PNN data
sample and simulated samples obtained from the upstream event selection
described in the text.

Figure 19: Left: extrapolation of $\pi^{+}$ tracks in the upstream data sample
described in the text to the $(X,Y)$ plane at the $Z$-position of the final
collimator. The blue lines correspond to the last 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 $\pi^{+}$s
shown on the left plot.

The evidence for the above classification comes from studies based on data and
simulated samples. The PNN selection is modified as follows to provide an
almost pure sample of upstream data events: the matching conditions for the
$K^{+}$ candidate and the $\pi^{+}$ track are not applied; no constraints are
applied to the reconstructed $Z_{\rm vertex}$; the box cut is not applied; and
${\rm CDA}>4$ mm is required. The last condition ensures that the $K/\pi$
matching of the PNN selection is not satisfied, therefore the signal