EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN-EP-2022-018

1 Feb 2022

Revised version:

8 May 2022

Searches for lepton number violating $K^{+}\to\pi^{-}(\pi^{0})e^{+}e^{+}$
decays

The NA62 Collaboration

###### Abstract

Searches for lepton number violating $K^{+}\to\pi^{-}e^{+}e^{+}$ and
$K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ decays have been performed using the
complete dataset collected by the NA62 experiment at CERN in 2016–2018. Upper
limits of $5.3\times 10^{-11}$ and $8.5\times 10^{-10}$ are obtained on the
decay branching fractions at 90% confidence level. The former result improves
by a factor of four over the previous best limit, while the latter result
represents the first limit on the $K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ decay
rate.

Accepted for publication in Physics Letters B

The NA62 Collaboration 111Corresponding author: E. Goudzovski, email:
<EMAIL_ADDRESS>

Université Catholique de Louvain, Louvain-La-Neuve, Belgium
E. Cortina Gil, A. Kleimenova, E. Minucci 111Corresponding author: E.
Goudzovski, email: evgueni.goudzovski@cern.ch${}^{,}\,$222Deceased, S.
Padolski 333Present address: Brookhaven National Laboratory, Upton, NY 11973,
USA, P. Petrov, A. Shaikhiev 444Present address: School of Physics and
Astronomy, University of Birmingham, Birmingham, B15 2TT, UK, R. Volpe
555Present address: Faculty of Mathematics, Physics and Informatics, Comenius
University, 842 48, Bratislava, Slovakia

TRIUMF, Vancouver, British Columbia, Canada
T. Numao, Y. Petrov, B. Velghe, V. W. S. Wong

University of British Columbia, Vancouver, British Columbia, Canada
D. Bryman 666Also at TRIUMF, Vancouver, British Columbia, V6T 2A3, Canada, J.
Fu

Charles University, Prague, Czech Republic
T. Husek 777Present address: Department of Astronomy and Theoretical Physics,
Lund University, Lund, SE 223-62, Sweden, J. Jerhot 888Present address:
Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium, K. Kampf,
M. Zamkovsky 888Present address: Université Catholique de Louvain, B-1348
Louvain-La-Neuve, Belgium

Institut für Physik and PRISMA Cluster of Excellence, Universität Mainz,
Mainz, Germany
R. Aliberti 999Present address: Institut für Kernphysik and Helmholtz
Institute Mainz, Universität Mainz, Mainz, D-55099, Germany, G. Khoriauli
101010Present address: Universität Würzburg, D-97070 Würzburg, Germany, J.
Kunze, D. Lomidze 111111Present address: European XFEL GmbH, D-22761 Hamburg,
Germany, 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 121212Present address: University
of Glasgow, Glasgow, G12 8QQ, UK, F. Petrucci, H. Wahl 131313Present address:
Institut für Physik and PRISMA Cluster of Excellence, Universität Mainz,
D-55099 Mainz, Germany

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, G. Tinti

Dipartimento di Fisica “Ettore Pancini” e INFN, Sezione di Napoli, Napoli,
Italy
F. Ambrosino, T. Capussela, M. Corvino 111Corresponding author: E. Goudzovski,
email<EMAIL_ADDRESS>D. Di Filippo, R. Fiorenza 161616Present
address: Scuola Superiore Meridionale e INFN, Sezione di Napoli, I-80138
Napoli, Italy, 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 171717Present
address: University of Chinese Academy of Sciences, Beijing, 100049, China, C.
Santoni

INFN, Sezione di Perugia, Perugia, Italy
M. Barbanera, 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 131313Present address: Institut für Physik and
PRISMA Cluster of Excellence, Universität Mainz, D-55099 Mainz, Germany, N.
Doble 131313Present address: Institut für Physik and PRISMA Cluster of
Excellence, Universität Mainz, D-55099 Mainz, Germany, M. Giorgi, S. Giudici,
G. Lamanna, E. Lari, E. Pedreschi, M. Sozzi

INFN, Sezione di Pisa, Pisa, Italy
C. Cerri, R. Fantechi, L. Pontisso 191919Present address: INFN, Sezione di
Roma I, I-00185 Roma, Italy, 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, P. Cretaro, O. Frezza, E. Leonardi, A. Lonardo, M. Turisini, P.
Valente, P. Vicini

INFN, Sezione di Roma Tor Vergata, Roma, Italy
R. Ammendola, V. Bonaiuto 202020Also at Department of Industrial Engineering,
University of Roma Tor Vergata, I-00173 Roma, Italy, A. Fucci, A. Salamon, F.
Sargeni 212121Also 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 222222Also at Università degli Studi del Piemonte Orientale,
I-13100 Vercelli, Italy, B. Bloch-Devaux, M. Boretto 111Corresponding author:
E. Goudzovski, email<EMAIL_ADDRESS>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 232323Also at Universidad de Guanajuato,
36000 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 242424Also at L.N. Gumilyov Eurasian National
University, 010000 Nur-Sultan, Kazakhstan, D. Emelyanov, T. Enik, V. Falaleev
252525Also at Institute for Nuclear Research of the Russian Academy of
Sciences, 117312 Moscow, Russia, V. Kekelidze, A. Korotkova, L. Litov
151515Also at Faculty of Physics, University of Sofia, BG-1164 Sofia,
Bulgaria, D. Madigozhin, M. Misheva 262626Present 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 272727Also 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 of the Russian Federation State Research
Center “Kurchatov Institute”, 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 111Corresponding author: E. Goudzovski, email:
<EMAIL_ADDRESS>T. Blazek, V. Cerny, Z. Kucerova

CERN, European Organization for Nuclear Research, Geneva, Switzerland
J. Bernhard, A. Ceccucci, H. Danielsson, N. De Simone 282828Present address:
DESY, D-15738 Zeuthen, Germany, F. Duval, B. Döbrich, L. Federici, E.
Gamberini, L. Gatignon 292929Present address: University of Lancaster,
Lancaster, LA1 4YW, UK, R. Guida, F. Hahn 222Deceased, E. B. Holzer, B.
Jenninger, M. Koval 303030Present 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,
R. Marchevski 313131Present address: Dipartimento di Fisica e Astronomia
dell’Università e INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Italy,
K. Massri, M. Noy, V. Palladino 323232Present address: Physics Department,
Imperial College London, London, SW7 2BW, UK, M. Perrin-Terrin 333333Present
address: Aix Marseille University, CNRS/IN2P3, CPPM, F-13288, Marseille,
France${}^{,}\,$343434Also at Université Catholique de Louvain, B-1348
Louvain-La-Neuve, Belgium, J. Pinzino 353535Present address: INFN, Sezione di
Pisa, I-56100 Pisa, Italy, V. Ryjov, S. Schuchmann 131313Present 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 363636Present address: Department of Physics,
University of Warwick, Coventry, CV4 7AL, UK, V. Duk 373737Present address:
INFN, Sezione di Perugia, I-06100 Perugia, Italy, V. Fascianelli 383838Present
address: Center for theoretical neuroscience, Columbia University, New York,
NY 10027, USA, J. R. Fry, F. Gonnella, E. Goudzovski111Corresponding author:
E. Goudzovski, email<EMAIL_ADDRESS>J. Henshaw, L. Iacobuzio, C.
Lazzeroni, N. Lurkin 888Present address: Université Catholique de Louvain,
B-1348 Louvain-La-Neuve, Belgium, F. Newson, C. Parkinson, A. Romano, A. Sergi
393939Present address: Dipartimento di Fisica dell’Università e INFN, Sezione
di Genova, I-16146 Genova, Italy, A. Sturgess, J. Swallow 111Corresponding
author: E. Goudzovski, email<EMAIL_ADDRESS>A. Tomczak

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. Ruggiero 313131Present
address: Dipartimento di Fisica e Astronomia dell’Università e INFN, Sezione
di Firenze, I-50019 Sesto Fiorentino, Italy

University of Liverpool, Liverpool, United Kingdom
L. Fulton, D. Hutchcroft, E. Maurice 404040Present address: Laboratoire
Leprince Ringuet, F-91120 Palaiseau, France, B. Wrona

George Mason University, Fairfax, Virginia, USA
A. Conovaloff, P. Cooper, D. Coward 414141Also at SLAC National Accelerator
Laboratory, Stanford University, Menlo Park, CA 94025, USA, P. Rubin

11footnotetext: Present address: CERN, European Organization for Nuclear
Research, CH-1211 Geneva 23, Switzerland22footnotetext: Also at Laboratori
Nazionali di Frascati, I-00044 Frascati, Italy

## Introduction

In the Standard Model (SM), neutrinos are strictly massless due to the absence
of right-handed chiral states. The discovery of neutrino oscillations has
conclusively demonstrated the non-zero neutrino mass, making it possible, in
principle, to distinguish experimentally between the Dirac and Majorana
neutrino. In the minimal Type-I seesaw model [1], the neutrino is a Majorana
fermion with a mass term that violates lepton number by two units. Strong
evidence for the Majorana nature of the neutrino would be provided by
observation of lepton number violating (LNV) processes, such as charged kaon
decays [2]. The existing experimental limits on
$K^{+}\to\pi^{-}\ell^{+}_{1}\ell^{+}_{2}$ decays lead to stringent constraints
on active-sterile mixing angles between Majorana neutrinos. Below the kaon
mass, these constraints are competitive with those obtained from neutrinoless
double beta decays [3, 4, 5].

The NA62 experiment at CERN collected a large dataset of $K^{+}$ decays into
lepton pairs in 2016–2018, using dedicated trigger lines. Part of this dataset
has been analysed to establish upper limits on the rates of lepton number and
flavour violating decays $K^{+}\to\pi^{-}\ell^{+}\ell^{+}$ ($\ell=e,\mu$) [6],
and the full dataset has been analysed to obtain limits on
$K^{+}\to\pi^{\pm}\mu^{\mp}e^{+}$ and $\pi^{0}\to\mu^{-}e^{+}$ decays [7],
improving by up to an order of magnitude on earlier results. Searches for the
$K^{+}\to\pi^{-}e^{+}e^{+}$ and $K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ decays
based on the complete NA62 dataset collected in 2016–2018 are reported here,
representing the first search for the latter process.

##  1 Beam, detector and data sample

The layout of the NA62 beamline and detector [8] is shown schematically in
Fig. 1. An unseparated secondary beam of $\pi^{+}$ (70%), protons (23%) and
$K^{+}$ (6%) is created by directing 400 GeV/$c$ protons extracted from the
CERN SPS onto a beryllium target in spills of 3 s effective duration. The
central beam momentum is 75 GeV/$c$, with a momentum spread of 1% (rms).

Beam kaons are tagged with a time resolution of 70 ps by a differential
Cherenkov counter (KTAG), which uses nitrogen gas at 1.75 bar pressure
contained in a 5 m long vessel as radiator. Beam particle positions, momenta
and times (to better than 100 ps resolution) are measured by a silicon pixel
spectrometer consisting of three stations (GTK1,2,3) and four dipole magnets.
A muon scraper (SCR) is installed between GTK1 and GTK2. A 1.2 m thick steel
collimator (COL) with a $76\times 40$ mm2 central aperture and $1.7\times 1.8$
m2 outer dimensions is placed upstream of GTK3 to absorb hadrons from upstream
$K^{+}$ decays; a variable aperture collimator of $0.15\times 0.15$ m2 outer
dimensions was used up to early 2018. Inelastic interactions of beam particles
in GTK3 are detected by an array of scintillator hodoscopes (CHANTI). The beam
is delivered into a vacuum tank evacuated to a pressure of $10^{-6}$ mbar,
which contains a 75 m long fiducial volume (FV) starting 2.6 m downstream of
GTK3. The beam angular spread at the FV entrance is 0.11 mrad (rms) in both
horizontal and vertical planes. Downstream of the FV, undecayed beam particles
continue their path in vacuum.

SCR

COL

M

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

Momenta of charged particles produced in $K^{+}$ decays in the FV are measured
by a magnetic spectrometer (STRAW) located in the vacuum tank downstream of
the FV. The spectrometer consists of four tracking chambers made of straw
tubes, and a dipole magnet (M) located between the second and third chambers
that provides a horizontal momentum kick of 270 MeV/$c$. The momentum
resolution is $\sigma_{p}/p=(0.30\oplus 0.005p)\%$, with the momentum $p$
expressed in GeV/$c$.

A ring-imaging Cherenkov detector (RICH) consisting of a 17.5 m long vessel
filled with neon at atmospheric pressure (with a Cherenkov threshold of 12.5
GeV/$c$ for pions) provides particle identification, charged particle time
measurements with a typical resolution of 70 ps, and the trigger time. The
RICH optical system is optimised to collect light emitted by positively
charged particles, exploiting their deflection by the STRAW dipole magnet. Two
scintillator hodoscopes (CHOD), which include a matrix of tiles and two planes
of slabs arranged in four quadrants located downstream of the RICH, provide
trigger signals and time measurements with 200 ps precision.

A $27X_{0}$ thick quasi-homogeneous liquid krypton (LKr) electromagnetic
calorimeter is used for particle identification and photon detection. The
calorimeter has an active volume of 7 m3 segmented in the transverse direction
into 13248 projective cells of $2\times 2$ cm2 size, and provides an energy
resolution $\sigma_{E}/E=(4.8/\sqrt{E}\oplus 11/E\oplus 0.9)\%$, with $E$
expressed in GeV. To achieve hermetic acceptance for photons emitted in
$K^{+}$ decays in the FV at angles up to 50 mrad from the beam axis, the LKr
calorimeter is supplemented by annular lead glass detectors (LAV) installed in
12 positions inside and downstream of the vacuum tank, and two
lead/scintillator sampling calorimeters (IRC, SAC) located close to the beam
axis. An iron/scintillator sampling hadronic calorimeter formed of two modules
(MUV1,2) and a muon detector consisting of 148 scintillator tiles located
behind an 80 cm thick iron wall (MUV3) are used for particle identification.

The data sample analysed is obtained from $0.89\times 10^{6}$ SPS spills
recorded in 2016–2018, with the typical beam intensity increasing over time
from $1.3\times 10^{12}$ to $2.2\times 10^{12}$ protons per spill. The latter
value corresponds to a 500 MHz mean instantaneous beam particle rate at the FV
entrance, and a 3.7 MHz mean $K^{+}$ decay rate in the FV. Multi-track (MT)
and electron multi-track ($e$MT) trigger chains downscaled typically by
factors of 100 and 8, respectively, are used for the analysis. The low-level
(L0) trigger [9] for both chains is based on RICH signal multiplicity and
coincidence of signals in two opposite CHOD quadrants. The $e$MT chain
additionally involves a requirement of at least 20 GeV energy deposit in the
LKr calorimeter. The high-level software (L1) trigger used for both chains
involves beam $K^{+}$ identification by the KTAG and reconstruction of a
negatively charged STRAW track. For signal-like samples (which are
characterised by LKr energy deposit well above 20 GeV), the measured
inefficiencies of the CHOD (STRAW) trigger conditions are typically at the 1%
(5%) level, while those of the RICH, KTAG and LKr are ${\cal O}(10^{-3})$.

Monte Carlo (MC) simulations of particle interactions with the detector and
its response are performed with a software package based on the Geant4 toolkit
[10], and include pileup (i.e. coincidences of multiple events in time) and
the full trigger chain.

##  2 Event selection

The rates of the possible signal decays $K^{+}\to\pi^{-}(\pi^{0})e^{+}e^{+}$
are measured with respect to the rate of the normalisation decay
$K^{+}\to\pi^{+}e^{+}e^{-}$. This approach leads to significant cancellation
of the effects of detector inefficiencies, trigger inefficiencies and pileup.
The signal and normalisation processes are collected concurrently using the MT
and $e$MT trigger chains described above. The main selection criteria are
listed below.

  * •

Three-track vertices are reconstructed by backward extrapolation of STRAW
tracks into the FV, taking into account the measured residual magnetic field
in the vacuum tank, and selecting triplets of tracks consistent with
originating from the same point. Exactly one such vertex should be present in
the event. The total charge of the tracks forming the vertex should be $q=1$,
and the vertex longitudinal position ($z_{\rm vtx}$) should be within the FV.
The momentum of each track forming the vertex should be in the range 6–44
GeV/$c$, and its trajectory through the STRAW chambers, and its extrapolation
to the CHOD and LKr calorimeter, should be within the respective geometrical
acceptances. Each pair of tracks should be separated by at least 15 mm in each
STRAW chamber plane to suppress photon conversions, and by at least 200 mm in
the LKr front plane to reduce shower overlap effects.

  * •

Track times are initially defined using the CHOD information. The vertex CHOD
time is evaluated as the average of track CHOD times. The RICH signal pattern
within 3 ns of the vertex CHOD time is used to compute the likelihoods of mass
hypotheses for each track and evaluate track RICH times. Track and vertex time
estimates are then recomputed using the RICH information. Each track is
required to be within 2.5 ns of the trigger time.

  * •

To suppress backgrounds from $K^{+}\to\pi^{+}\pi^{0}_{D}$ and
$K^{+}\to\pi^{0}_{D}e^{+}\nu$ decays followed by the Dalitz decay
$\pi^{0}_{D}\to\gamma e^{+}e^{-}$, which are characterised by emission of soft
photons at large angles, no signals within 4 ns of the vertex time are allowed
in the LAV detectors located downstream of the reconstructed vertex position.
Since energetic photons emitted forward are already suppressed by the total
momentum condition (see below), no photon veto requirements are applied in the
LKr, IRC and SAC calorimeters.

  * •

The ratio of energy deposited in the LKr calorimeter to the momentum measured
by the spectrometer, $E/p$, identifies pion ($\pi^{\pm}$) and electron
($e^{\pm}$) candidates: $E/p<0.85$ for pions, and $0.9<E/p<1.1$ for electrons.
The vertex should consist of a $\pi^{\pm}$ candidate and two $e^{\pm}$
candidates.

The following condition is applied to select $K^{+}\to\pi^{+}e^{+}e^{-}$ and
$K^{+}\to\pi^{-}e^{+}e^{+}$ decays.

  * •

The total momentum of the three tracks, $p_{\pi ee}$, should satisfy $|p_{\pi
ee}-p_{\rm beam}|<2~{}{\rm GeV}/c$, where $p_{\rm beam}$ is the central beam
momentum. The total transverse momentum with respect to the beam axis should
be $p^{\pi ee}_{T}<30~{}{\rm MeV}/c$. The quantity $p_{\rm beam}$ and the beam
axis direction are monitored throughout the data taking period using fully
reconstructed $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decays.

The following conditions are applied to select $K^{+}\to\pi^{+}e^{+}e^{-}$
decays.

  * •

The reconstructed $e^{+}e^{-}$ mass should be $m_{ee}>140~{}{\rm MeV}/c^{2}$
to suppress backgrounds from the $K^{+}\to\pi^{+}\pi^{0}$ decay followed by
$\pi^{0}_{D}\to e^{+}e^{-}\gamma$, $\pi^{0}_{DD}\to e^{+}e^{-}e^{+}e^{-}$ and
$\pi^{0}\to e^{+}e^{-}$ decays. This leads to a 27% fractional loss in
acceptance.

  * •

The signal region of reconstructed $\pi^{+}e^{+}e^{-}$ mass, $m_{\pi ee}$, is
defined as 470–505 MeV/$c^{2}$, which accounts for the mass resolution of 1.7
MeV/$c^{2}$ and the radiative mass tail. The lower mass region, $m_{\pi
ee}<470~{}{\rm MeV}/c^{2}$, is used for validation of the background
estimates.

The following conditions are applied to select $K^{+}\to\pi^{-}e^{+}e^{+}$
decays.

  * •

RICH-based $e^{+}$ identification suppresses the otherwise dominant
backgrounds from $K^{+}\to\pi^{+}\pi^{0}_{D}$ and $K^{+}\to\pi^{+}e^{+}e^{-}$
decays with double $\pi^{+}\to e^{+}$ and $e^{-}\to\pi^{-}$ misidentification.
The identification condition is based on the likelihoods of $e^{+}$ and
$\pi^{+}$ mass hypotheses evaluated using the RICH signal pattern.
Additionally, the angles between track pairs in the RICH are required to
exceed 4 mrad to reduce overlaps of Cherenkov light-cones, which causes a
fractional reduction of 7% in the signal acceptance.

  * •

The signal region of reconstructed $\pi^{-}e^{+}e^{+}$ mass, $m_{\pi ee}$, is
defined as 488.6–498.8 MeV/$c^{2}$, corresponding to six times the mass
resolution of 1.7 MeV/$c^{2}$. The mass region 470–505 MeV/$c^{2}$, which
includes the signal region, is kept masked until the validation of the
background estimate. The lower and upper mass regions used for validation of
the background estimate are defined as $m_{\pi ee}<470~{}{\rm MeV}/c^{2}$ and
$505~{}{\rm MeV}/c^{2}<m_{\pi ee}<600~{}{\rm MeV}/c^{2}$, respectively.

The following conditions are applied to select
$K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ decays.

  * •

The $\pi^{0}$ is reconstructed by its prompt $\pi^{0}\to\gamma\gamma$ decay.
Exactly two photon candidates are required, defined as reconstructed LKr
energy deposit clusters with energy above 2 GeV, within 5 ns of the vertex
time, separated by at least 150 mm from each other and from each track impact
point in the nominal LKr transverse plane.

  * •

The longitudinal coordinate of the neutral vertex is defined assuming that the
two photons are emitted in a $\pi^{0}\to\gamma\gamma$ decay: $z_{\rm N}=z_{\rm
LKr}-D_{12}\sqrt{E_{1}E_{2}}/m_{\pi 0}$. Here $D_{12}$ is the distance between
the two clusters in the LKr transverse plane (with a $z$ coordinate $z_{\rm
LKr}$), $E_{1,2}$ are the photon candidate energies, and $m_{\pi 0}$ is the
nominal $\pi^{0}$ mass [11].

  * •

Consistency of the three-track and neutral vertices is required: $|z_{\rm
vtx}-z_{\rm N}|<8$ m. Vertex position resolutions evaluated with simulations
are $\delta z_{\rm vtx}=0.25$ m and $\delta z_{\rm N}=1.8$ m.

  * •

Photon momenta are computed using cluster energies and positions in the LKr
transverse plane, assuming emission at the three-track vertex. The $\pi^{0}$
momentum is then evaluated as the sum of photon momenta.

  * •

The total final state momentum, $p_{\pi\pi ee}$, should be consistent with the
central beam momentum: $|p_{\pi\pi ee}-p_{\rm beam}|<3~{}{\rm GeV}/c$. The
total transverse momentum with respect to the beam axis is required to be
$p^{\pi\pi ee}_{T}<30~{}{\rm MeV}/c$.

  * •

The signal region of reconstructed $\pi^{-}\pi^{0}e^{+}e^{+}$ mass, $m_{\pi\pi
ee}$, is defined as 484–504 MeV/$c^{2}$, and is kept masked until the
validation of the background estimates. The mass resolution is 1.9
MeV/$c^{2}$, and a loose signal region definition is adopted due to the lack
of background. The control region 400–600 MeV/$c^{2}$ (excluding the signal
region) is used for validation of the background estimate.

Figure 2: Left: $\pi^{\pm}$ misidentification probability $P_{\pi e}$ for the
LKr-based and RICH-based identification conditions, applied separately and
together, measured in momentum bins. Right: $e^{+}$ misidentification
probability $P_{e\pi}$ for the LKr-based $\pi^{+}$ identification condition,
and $e^{+}$ identification inefficiency $1-\varepsilon_{e}$ for the LKr-based
and RICH-based conditions, measured in momentum bins. Polynomial fits to the
measured values used for the positron identification model are shown.

##  3 Particle identification studies

Backgrounds to signal decays arise mainly from pion ($\pi^{\pm}$)
misidentification as electron ($e^{\pm}$) and vice versa. As discussed in
Section 2, LKr-based identification is used for $\pi^{\pm}$ and $e^{\pm}$, and
RICH-based identification is additionally employed for $e^{+}$ within the
$K^{+}\to\pi^{-}e^{+}e^{+}$ selection. The accuracy of Geant4-based particle
identification simulation is limited: the quantity $E/p$ is sensitive to
hadronic shower and cluster reconstruction simulation, while the RICH-based
algorithm depends critically on gas pressure, light yield and mirror alignment
calibrations. A dedicated data-driven model is used to simulate particle
identification for the MC samples: the measured identification probabilities
are applied as weights to MC events. This approach also improves the
statistical precision on the estimated backgrounds.

Pion misidentification probability ($P_{\pi e}$) and identification efficiency
($\varepsilon_{\pi}$) are measured as functions of momentum in the range 10–44
GeV/$c$ using a pure $\pi^{\pm}$ sample obtained by kinematic selection of
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decays. The lower bound of 10 GeV/$c$
represents the kinematic limit of $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decays. The
efficiency $\varepsilon_{\pi}$ varies from 98.2% to 98.7% as a function of
momentum. The $P_{\pi e}$ measurements are shown in Fig. 2 (left). The LKr-
based $e^{\pm}$ identification leads to $P_{\pi e}\approx 10^{-2}$ with a weak
momentum dependence, and $P_{\pi e}$ values are larger for $\pi^{+}$ than for
$\pi^{-}$ by about $5\times 10^{-4}$ in absolute terms which is attributed to
the larger $\pi^{+}$ charge exchange ($\pi^{+}n\to\pi^{0}p$) cross-section on
Krypton nuclei. The RICH-based $e^{+}$ identification provides an additional
$\pi^{+}$ rejection factor of up to $10^{3}$. The strongest suppression
corresponds to the $\pi^{+}$ momentum range for which the RICH is optimised,
while non-zero $P_{\pi e}$ values below the Cherenkov threshold are due to the
presence of additional in-time tracks in the fully reconstructed
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ events. The model takes into account the
dependence of the RICH-based $P_{\pi e}$ on the angle between the two
$\pi^{+}$ tracks in the RICH, caused by ring overlaps. The model also accounts
for the correlation between the measured track momentum $p$ and $E/p$.

Positron misidentification probability ($P_{e\pi}$) and identification
efficiency ($\varepsilon_{e}$) are measured in the full momentum range 6–44
GeV/$c$ used in the event selection using a positron sample obtained by
kinematic selection of $K^{+}\to\pi^{0}e^{+}\nu$ decays. The background
contamination from $K^{+}\to\pi^{+}\pi^{0}$ and $K^{+}\to\pi^{0}\mu^{+}\nu$
decays is estimated from simulations and subtracted: it varies from negligible
at low momentum to 0.6% at high momentum. The measurements are shown in Fig. 2
(right): both $P_{e\pi}$ and $\varepsilon_{e}$ exhibit a significant momentum
dependence. Small differences in the LKr calorimeter response for electrons
and positrons have negligible effect on this analysis, and the $e^{+}$
measurements are also used to model $e^{-}$ identification for the MC samples.

Figure 3: Reconstructed $m_{\pi ee}$ spectra of the data, simulated signal and
backgrounds obtained using the standard $K^{+}\to\pi^{+}e^{+}e^{-}$ selection
(left) and the control $K^{+}\to\pi^{+}e^{+}e^{-}$ selection involving RICH-
based $e^{+}$ identification (right). The signal $m_{\pi ee}$ region is
indicated with arrows.

Figure 4: Left: reconstructed $m_{\pi ee}$ spectra of the data and simulated
backgrounds obtained using the $K^{+}\to\pi^{-}e^{+}e^{+}$ selection. The
shaded vertical band indicates the $m_{\pi ee}$ region masked during the
analysis, including the signal region bounded by dashed lines. Right:
reconstructed $m_{\pi ee}$ spectra of the data and simulated backgrounds
obtained using the control $K^{+}\to\pi^{-}e^{+}e^{+}$ selection with a
missing momentum requirement. Inset: ratio of the data and simulated spectra,
and a fit to the ratio with a linear function.

##  4 Normalisation to the $K^{+}\to\pi^{+}e^{+}e^{-}$ decay

The $K^{+}\to\pi^{+}e^{+}e^{-}$ sample is used for normalisation of the
estimated backgrounds, and validation of the $\pi^{\pm}$ misidentification
modelling. In addition to the standard $K^{+}\to\pi^{+}e^{+}e^{-}$ selection
relying on LKr-based particle identification, a control selection with RICH-
based $e^{+}$ identification is considered for validation purposes. The
reconstructed $m_{\pi ee}$ spectra of the data, simulated signal and
backgrounds obtained with the two selections are displayed in Fig. 3.

The background in the signal $m_{\pi ee}$ region comes mainly from the
$K^{+}\to\pi^{+}\pi^{0}_{D}$ decay with double misidentification ($\pi^{+}\to
e^{+}$ and $e^{+}\to\pi^{+}$). Principal backgrounds in the lower $m_{\pi ee}$
region are due to the $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay with double
$\pi^{\pm}\to e^{\pm}$ misidentification, and the
$K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$ decay with $\pi^{-}\to e^{-}$
misidentification. The former source is reduced by a factor ${\cal O}(10^{3})$
by the RICH-based $e^{+}$ identification. To model the latter source, a
constant $P_{\pi e}$ value is assumed for LKr-based identification for pion
momenta below 10 GeV/$c$. Contributions involving pion decays in flight
$\pi^{\pm}\to e^{\pm}\nu$ are found to be negligible. The standard
$K^{+}\to\pi^{+}e^{+}e^{-}$ selection validates the modelling of backgrounds
due to $\pi^{\pm}\to e^{\pm}$ misidentification with the LKr-based condition
to 3% precision. The control selection validates the modelling of backgrounds
due to $\pi^{+}\to e^{+}$ misidentification with the RICH-based condition
within the statistical precision.

The number of $K^{+}$ decays in the FV is computed as

$N_{K}=\frac{(1-f)\cdot N_{\pi ee}}{{\cal B}_{\pi ee}\cdot A_{\pi
ee}}=(1.015\pm 0.010_{\rm stat}\pm 0.030_{\rm syst})\times 10^{12},$

where $N_{\pi ee}=11041$ is the number of data $K^{+}\to\pi^{+}e^{+}e^{-}$
candidates in the signal $m_{\pi ee}$ region (constituting the world’s largest
sample of these decays), ${\cal B}_{\pi ee}=(3.00\pm 0.09)\times 10^{-7}$ is
the $K^{+}\to\pi^{+}e^{+}e^{-}$ branching fraction [11], $A_{\pi ee}=(3.62\pm
0.02_{\rm syst})\%$ is the selection acceptance evaluated with simulations
including trigger inefficiency and random veto effects (with the uncertainty
estimated by stability checks with respect to variation of the selection
criteria), and $f=1.0\times 10^{-3}$ is the relative background contamination
evaluated with simulations. The systematic error on $N_{K}$ is dominated by
the external uncertainty on ${\cal B}_{\pi ee}$.

As a cross-check, the quantity $N_{K}$ is evaluated using the control
selection including RICH-based $e^{+}$ identification. In this case, a 9%
fractional reduction of the acceptance $A_{\pi ee}$ is observed (mainly due to
the track angular separation requirement in the RICH), $f$ is negligible,
$N_{\pi ee}$ becomes 9922, and the resulting $N_{K}$ value changes by less
than 1%.

##  5 Search for the $K^{+}\to\pi^{-}e^{+}e^{+}$ decay

The reconstructed $m_{\pi ee}$ spectra of the data, simulated signal and
backgrounds obtained using the $K^{+}\to\pi^{-}e^{+}e^{+}$ selection are
displayed in Fig. 4 (left). Identification of two positrons with both LKr-
based and RICH-based conditions leads to a stronger reduction of the
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ and $K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$
backgrounds than in the $K^{+}\to\pi^{+}e^{+}e^{-}$ case. This makes the
$K^{+}\to\pi^{0}_{D}e^{+}\nu$ decay the largest background source in the
lower, masked and upper $m_{\pi ee}$ regions. The $K^{+}\to e^{+}\nu
e^{+}e^{-}$ decay represents another background source in the masked region.

The $K^{+}\to\pi^{0}_{D}e^{+}\nu$ and $K^{+}\to e^{+}\nu e^{+}e^{-}$
backgrounds enter by $e^{-}\to\pi^{-}$ misidentification with the LKr-based
condition, and their description relies on the $P_{e\pi}$ measurement and
modelling (Fig. 2, right). To validate the model, a control
$K^{+}\to\pi^{-}e^{+}e^{+}$ selection is used, obtained from the standard
selection by replacing the $p_{\pi ee}$ condition with a missing momentum
requirement $p_{\rm beam}-p_{\pi ee}>10~{}{\rm GeV}/c$ and removing the
$p^{\pi ee}_{T}$ condition (thus removing the possible
$K^{+}\to\pi^{-}e^{+}e^{+}$ signal contribution). The data and simulated
$m_{\pi ee}$ spectra obtained using the control selection, and the ratio of
these spectra, are shown in Fig. 4 (right). The sample is dominated by
$K^{+}\to\pi^{0}_{D}e^{+}\nu$ decays. The variation of the data/MC ratio over
the $m_{\pi ee}$ range validates the background description to a 20% relative
precision. The limited accuracy is attributed to the MC description of the LAV
detector response for soft photons from $K^{+}\to\pi^{0}_{D}e^{+}\nu$ decays.
A 20% relative systematic uncertainty is assigned to the estimated backgrounds
to the signal decay $K^{+}\to\pi^{-}e^{+}e^{+}$ arising from $e^{-}\to\pi^{-}$
misidentification.

The background due to pileup is found to be negligible by studying the
sidebands of the track time distributions. Backgrounds from
$K^{+}\to\pi^{+}\pi^{0}_{DD}$ decays and multiple photon conversions are
studied using four control selections involving vertices with total charge
$q\neq 1$ ($\pi^{-}e^{-}e^{-}$, $\pi^{-}e^{+}e^{-}$, $\pi^{+}e^{-}e^{-}$ and
$\pi^{+}e^{+}e^{+}$), without any $m_{\pi ee}$ selection criteria. The first
two selections involving a $\pi^{-}$ are similar to the signal selection, and
yield no events from the data sample. The last two selections allow the
backgrounds from $K^{+}\to\pi^{+}\pi^{0}_{DD}$ and multiple conversions to
enter without misidentification, enhancing the background by at least a factor
of $10^{4}$. These selections yield 676 and 326 data events (mainly with
$m_{ee}<140$ MeV/$c^{2}$), which is consistent with the
$K^{+}\to\pi^{+}\pi^{0}_{DD}$ contribution expected from simulations.
Considering the above enhancement factor, the background to the
$K^{+}\to\pi^{-}e^{+}e^{+}$ signal is clearly negligible.

The estimated background contributions in the lower, upper, masked and signal
$m_{\pi ee}$ regions are listed in Table 1. The numbers of data and expected
background events in the lower and upper regions are compared before opening
the masked region, and found to be in agreement within statistical
fluctuations. The background in the signal region is estimated to be

$N_{B}=0.43\pm 0.09,$

where the uncertainty is dominated by the systematic contribution due to the
accuracy of $P_{e\pi}$ modelling. Background suppression in the signal region
relies on the LKr-based $\pi^{-}$ identification and LAV photon veto
conditions. After the masked region is opened, one data event is observed in
this region but outside the signal region (in agreement with the expected
background), and no events are observed in the signal region.

Table 1: Numbers of estimated background events and observed data events obtained using the $K^{+}\to\pi^{-}e^{+}e^{+}$ selection in the lower, upper, masked and signal $m_{\pi ee}$ regions. The uncertainties are dominated by the 20% systematic error on the estimated $K^{+}\to\pi^{0}_{D}e^{+}\nu$ and $K^{+}\to e^{+}\nu e^{+}e^{-}$ backgrounds (fully correlated between the two contributions). MC statistical errors and external uncertainties due to the background branching fractions and kinematic distributions [11] are negligible. The masked and signal regions are opened for the data sample only after the validation of the background estimates. Mode | Lower region | Upper region | Masked region | Signal region
---|---|---|---|---
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ | 0.9 | – | – | –
$K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$ | 3.3 | – | – | –
$K^{+}\to\pi^{+}\pi^{0}_{D}$ | – | 0.02 | 0.01 | –
$K^{+}\to\pi^{0}_{D}e^{+}\nu$ | $3.7\pm 0.7$ | $1.20\pm 0.24$ | $1.23\pm 0.25$ | $0.29\pm 0.06$
$K^{+}\to e^{+}\nu e^{+}e^{-}$ | $0.7\pm 0.1$ | $0.76\pm 0.15$ | $0.47\pm 0.09$ | $0.14\pm 0.03$
Total | $8.6\pm 0.9$ | $1.98\pm 0.39$ | $1.71\pm 0.34$ | $0.43\pm 0.09$
Data | 8 | 1 | 1 | 0

The signal acceptance evaluated with simulations assuming a uniform phase
space distribution is $A_{\pi ee}^{\rm LNV}=4.32\%$, which is larger than
$A_{\pi ee}$ due to the absence of the $m_{\pi ee}>140$ MeV/$c^{2}$ condition
in the $K^{+}\to\pi^{-}e^{+}e^{+}$ selection. The single event sensitivity
defined as the signal branching fraction corresponding to the observation of
one signal event is found to be

${\cal B}_{\rm SES}=\frac{1}{N_{K}\cdot A_{\pi ee}^{\rm LNV}}=(2.28\pm
0.07)\times 10^{-11}.$

An upper limit on the signal branching fraction is evaluated using the
quantity ${\cal B}_{\rm SES}$ and the numbers of expected background events
and observed data events using the CLS method [12]:

${\cal B}(K^{+}\to\pi^{-}e^{+}e^{+})<5.3\times 10^{-11}~{}~{}{\rm
at}~{}90\%~{}{\rm CL}.$

##  6 Search for the $K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ decay

The reconstructed $m_{\pi\pi ee}$ spectra of the data, simulated signal and
backgrounds obtained using the $K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ selection
are shown in Fig. 5. Background sources studied with simulations include
$K^{+}\to\pi^{+}\pi^{0}\pi^{0}_{D}$, $K^{+}\to\pi^{+}\pi^{0}_{D}\gamma$ and
$K^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$ decays with double particle
misidentification ($\pi^{+}\to e^{+}$ and $e^{-}\to\pi^{-}$), and the
$K^{+}\to\pi^{0}_{D}e^{+}\nu\gamma$ decay with single misidentification
($e^{-}\to\pi^{-}$). The two background sources with a radiative photon (which
gives rise to a photon candidate in $\pi^{0}\to\gamma\gamma$ reconstruction)
are dominated by the inner bremsstrahlung components. To minimise MC
statistical errors, $K^{+}\to\pi^{+}\pi^{0}_{D}\gamma$ components with the
radiative photon energy in the $K^{+}$ rest frame, $E_{\gamma}$, below and
above 10 MeV are simulated separately; the two contributions are found to be
of similar size. The $K^{+}\to\pi^{0}_{D}e^{+}\nu\gamma$ contribution entering
via single misidentification ($e^{-}\to\pi^{-}$) is evaluated for
$E_{\gamma}>10$ MeV only due to computational limitations; all simulated
events passing the selection are found to have $E_{\gamma}>25$ MeV.

Figure 5: Reconstructed $m_{\pi\pi ee}$ spectra of the data and simulated backgrounds obtained using the $K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ selection. The $K^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$ contribution is the smallest, and is not shown. The shaded vertical band indicates the signal $m_{\pi\pi ee}$ region masked during the analysis. Table 2: Numbers of estimated background events and observed data events obtained using the $K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ selection in the control and signal $m_{\pi\pi ee}$ regions. The signal region is unmasked for the data sample only after the validation of the background estimate. Mode | Control region | Signal region
---|---|---
$K^{+}\to\pi^{+}\pi^{0}\pi^{0}_{D}$ | $0.16\pm 0.01$ | 0.019
$K^{+}\to\pi^{+}\pi^{0}_{D}\gamma$ | $0.06\pm 0.01$ | 0.004
$K^{+}\to\pi^{0}_{D}e^{+}\nu\gamma$ | $0.05\pm 0.02$ | –
$K^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$ | 0.01 | 0.001
Pileup | $0.20\pm 0.20$ | $0.020\pm 0.020$
Total | $0.48\pm 0.20$ | $0.044\pm 0.020$
Data | 1 | 0

Background due to pileup is potentially larger than in the
$K^{+}\to\pi^{-}e^{+}e^{+}$ case due to the absence of topological constraints
on the photon candidates, as opposed to the three-track vertex condition for
the tracks. This background is studied with the data, using two control
selections with inverted photon timing conditions. In the first (second) of
these selections, one photon is (both photons are) required to be in the
sidebands of the photon time distribution, separated by 5–30 ns from the
vertex time. The selection with one out-of-time photon yields one data event
in the control mass region at $m_{\pi\pi ee}=573$ MeV/$c^{2}$ and no events in
the signal region, while the selection with both photons out of time yields no
data events. Background contamination due to pileup is estimated considering
that sidebands are five times wider than the signal time window, and assuming
that the background $m_{\pi\pi ee}$ distribution is uniform over the control
and signal regions.

The estimated background contributions in the control and signal regions are
summarised in Table 2. One data event is observed in the control region, in
agreement with the background expectation. The total expected background in
the signal region is found to be

$N_{B}=0.044\pm 0.020,$

where the uncertainty is dominated by the statistical error in the pileup
background estimate. After unmasking, no data events are found in the signal
region.

The signal acceptance evaluated with simulation assuming a uniform phase space
distribution is $A_{\pi\pi ee}^{\rm LNV}=(0.271\pm 0.003_{\rm syst})\%$. The
uncertainty is due to the $\pi^{0}\to\gamma\gamma$ decay reconstruction (which
does not cancel between signal and normalisation), and is evaluated from a
study of $K^{+}\to\pi^{0}e^{+}\nu$ decays in which variations to LKr
calorimeter simulated response and calibration are included. The acceptance is
suppressed with respect to the decays without $\pi^{0}$ emission ($A_{\pi\pi
ee}^{\rm LNV}/A_{\pi ee}^{\rm LNV}\approx 0.06$) by the geometric acceptance
for the photons from $\pi^{0}\to\gamma\gamma$ decay. Note that the acceptance
for the SM decay $K^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$ achieved with a selection
similar to the signal one, i.e. based on identification of each track, is
${\cal O}(10^{-5})$ due to the predominantly soft $e^{+}e^{-}$ pairs.
Therefore the $K^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$ decay is not used for
normalisation.

The single event sensitivity is evaluated as

${\cal B}_{\rm SES}=\frac{1}{N_{K}\cdot{\cal B}_{\gamma\gamma}\cdot A_{\pi\pi
ee}^{\rm LNV}}=(3.68\pm 0.12)\times 10^{-10},$

where ${\cal B}_{\gamma\gamma}=(98.823\pm 0.034)\%$ is the
$\pi^{0}\to\gamma\gamma$ branching fraction [11]. An upper limit on the signal
branching fraction is evaluated using the CLS method:

${\cal B}(K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+})<8.5\times 10^{-10}~{}~{}{\rm
at}~{}90\%~{}{\rm CL}.$

For comparison, the branching fraction of the corresponding SM decay is ${\cal
B}(K^{+}\to\pi^{+}\pi^{0}e^{+}e^{-})=(4.24\pm 0.14)\times 10^{-6}$, as
measured by the NA48/2 experiment [13].

## Summary

Searches for lepton number violating $K^{+}\to\pi^{-}e^{+}e^{+}$ and
$K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ decays have been performed using the
complete dataset collected by the NA62 experiment at CERN in 2016–2018. Upper
limits of $5.3\times 10^{-11}$ and $8.5\times 10^{-10}$ were obtained on the
decay branching fractions at 90% confidence level, assuming uniform phase
space distributions. The former result improves by a factor of four over the
previous best limit [6], while the latter result represents the first limit on
the $K^{+}\to\pi^{-}\pi^{0}e^{+}e^{+}$ decay rate. The sensitivity of both
searches is limited by the size of the dataset and not by backgrounds.
Similarly to most other limits for LNV decay rates, these results depend on
phase space density assumptions.

## Acknowledgements

It is a pleasure to express our appreciation to the staff of the CERN
laboratory and the technical staff of the participating laboratories and
universities for their efforts in the operation of the experiment and data
processing.

The cost of the experiment and its auxiliary systems was supported by the
funding agencies of the Collaboration Institutes. We are particularly indebted
to: F.R.S.-FNRS (Fonds de la Recherche Scientifique - FNRS), under Grants No.
4.4512.10, 1.B.258.20, Belgium; CECI (Consortium des Equipements de Calcul
Intensif), funded by the Fonds de la Recherche Scientifique de Belgique
(F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region, Belgium;
NSERC (Natural Sciences and Engineering Research Council), funding
SAPPJ-2018-0017, Canada; MEYS (Ministry of Education, Youth and Sports)
funding LM 2018104, Czech Republic; BMBF (Bundesministerium für Bildung und
Forschung) contracts 05H12UM5, 05H15UMCNA and 05H18UMCNA, Germany; INFN
(Istituto Nazionale di Fisica Nucleare), Italy; MIUR (Ministero
dell’Istruzione, dell’Università e della Ricerca), Italy; CONACyT (Consejo
Nacional de Ciencia y Tecnología), Mexico; IFA (Institute of Atomic Physics)
Romanian CERN-RO No. 1/16.03.2016 and Nucleus Programme PN 19 06 01 04,
Romania; INR-RAS (Institute for Nuclear Research of the Russian Academy of
Sciences), Moscow, Russia; JINR (Joint Institute for Nuclear Research), Dubna,
Russia; NRC (National Research Center) “Kurchatov Institute” and MESRF
(Ministry of Education and Science of the Russian Federation), Russia; MESRS
(Ministry of Education, Science, Research and Sport), Slovakia; CERN (European
Organization for Nuclear Research), Switzerland; STFC (Science and Technology
Facilities Council), United Kingdom; NSF (National Science Foundation) Award
Numbers 1506088 and 1806430, U.S.A.; ERC (European Research Council)
“UniversaLepto” advanced grant 268062, “KaonLepton” starting grant 336581,
Europe.

Individuals have received support from: Charles University Research Center
(UNCE/SCI/013), Czech Republic; Ministero dell’Istruzione, dell’Università e
della Ricerca (MIUR “Futuro in ricerca 2012” grant RBFR12JF2Z, Project GAP),
Italy; Russian Science Foundation (RSF 19-72-10096), Russia; the Royal Society
(grants UF100308, UF0758946), United Kingdom; STFC (Rutherford fellowships
ST/J00412X/1, ST/M005798/1), United Kingdom; ERC (grants 268062, 336581 and
starting grant 802836 “AxScale”); EU Horizon 2020 (Marie Skłodowska-Curie
grants 701386, 754496, 842407, 893101, 101023808).

## References

  * [1] R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.
  * [2] L. Littenberg and R. Shrock, Phys. Lett. B491 (2000) 285.
  * [3] A. Atre, V. Barger and T. Han, Phys. Rev. D71 (2005) 113014.
  * [4] A. Atre, T. Han, S. Pascoli and B. Zhang, JHEP 05 (2009) 030.
  * [5] A. Abada et al., JHEP 02 (2018) 169.
  * [6] E. Cortina Gil et al., Phys. Lett. B797 (2019) 134794.
  * [7] E. Cortina Gil et al., Phys. Rev. Lett. 127 (2021) 131802.
  * [8] E. Cortina Gil et al., JINST 12 (2017) P05025.
  * [9] R. Ammendola et al., Nucl. Instrum. Methods A929 (2019) 1.
  * [10] J. Allison et al., Nucl. Instrum. Methods A835 (2016) 186.
  * [11] P.A. Zyla et al., Prog. Theor. Exp. Phys. 2020 083C01 (2020).
  * [12] A.L. Read, J. Phys. G28 (2002) 2693.
  * [13] J.R. Batley et al., Phys. Lett. B788 (2019) 552.