EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN-EP-2018-314

November 15, 2018

Revised version

December 21, 2018

First search for $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ using the decay-in-
flight technique
The NA62 Collaboration

###### Abstract

The NA62 experiment at the CERN SPS reports the first search for
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ using the decay-in-flight technique,
based on a sample of $1.21\times 10^{11}$ $K^{+}$ decays collected in 2016.
The single event sensitivity is $3.15\times 10^{-10}$, corresponding to 0.267
Standard Model events. One signal candidate is observed while the expected
background is 0.152 events. This leads to an upper limit of $14\times
10^{-10}$ on the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ branching ratio at 95%
CL.

This paper is dedicated to the memory of our colleagues S. Balev and F. Hahn.

Accepted for publication in Physics Letters B

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

Université Catholique de Louvain, Louvain-La-Neuve, Belgium
E. Cortina Gil, E. Minucci, S. Padolski 111Corresponding authors: G. Ruggiero,
R. Marchevski,
<EMAIL_ADDRESS><EMAIL_ADDRESS>P. Petrov, B.
Velghe 222Deceased

Faculty of Physics, University of Sofia, Sofia, Bulgaria
G. Georgiev 333Also at Laboratori Nazionali di Frascati, I-00044 Frascati,
Italy, V. Kozhuharov 333Also at Laboratori Nazionali di Frascati, I-00044
Frascati, Italy, L. Litov

TRIUMF, Vancouver, British Columbia, Canada
T. Numao

University of British Columbia, Vancouver, British Columbia, Canada
D. Bryman, J. Fu 444Present address: UCLA Physics and Biology in Medicine, Los
Angeles, CA 90095, USA

Charles University, Prague, Czech Republic
T. Husek 555Present address: IFIC, Universitat de València - CSIC, E-46071
València, Spain, K. Kampf, M. Zamkovsky

Institut für Physik and PRISMA Cluster of excellence, Universität Mainz,
Mainz, Germany
R. Aliberti, G. Khoriauli 666Present address: Universität Würzburg, D-97070
Würzburg, Germany, J. Kunze, D. Lomidze 777Present address: Universität
Hamburg, D-20146 Hamburg, Germany, R. Marchevski 111Corresponding authors: G.
Ruggiero, R. Marchevski,
<EMAIL_ADDRESS>radoslav.marchevski@cern.ch,888Present
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, E. Gamberini 888Present address: CERN, European
Organization for Nuclear Research, CH-1211 Geneva 23, Switzerland, 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

INFN, Sezione di Firenze, Sesto Fiorentino, Italy
A. Bizzeti 999Also at Dipartimento di Fisica, Università di Modena e Reggio
Emilia, I-41125 Modena, Italy, F. Bucci, R. Volpe 101010Present address:
Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium

Laboratori Nazionali di Frascati, Frascati, Italy
A. Antonelli, G. Lamanna 111111Present address: Dipartimento di Fisica
dell’Università e INFN, Sezione di Pisa, I-56100 Pisa, Italy, G. Lanfranchi,
G. Mannocchi, S. Martellotti, M. Moulson, M. Raggi 121212Present address:
Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy,
T. Spadaro

Dipartimento di Fisica “Ettore Pancini” e INFN, Sezione di Napoli, Napoli,
Italy
F. Ambrosino, T. Capussela, M. Corvino, D. Di Filippo, P. Massarotti, M.
Mirra, M. Napolitano, G. Saracino

Dipartimento di Fisica e Geologia dell’Università e INFN, Sezione di Perugia,
Perugia, Italy
G. Anzivino, F. Brizioli, E. Imbergamo, R. Lollini, C. Santoni

INFN, Sezione di Perugia, Perugia, Italy
M. Barbanera 131313Present address: INFN, Sezione di Pisa, I-56100 Pisa,
Italy, P. Cenci, B. Checcucci, V. Duk 141414Present address: School of Physics
and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK, P. Lubrano,
M. Lupi 888Present address: CERN, European Organization for Nuclear Research,
CH-1211 Geneva 23, Switzerland, 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, E. Pedreschi, M.
Sozzi

INFN, Sezione di Pisa, Pisa, Italy
C. Cerri, R. Fantechi, R. Piandani 151515Present address: Dipartimento di
Fisica e Geologia dell’Università e INFN, Sezione di Perugia, I-06100 Perugia,
Italy, J. Pinzino 888Present address: CERN, European Organization for Nuclear
Research, CH-1211 Geneva 23, Switzerland, 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

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 161616Also at Department of Industrial Engineering,
University of Roma Tor Vergata, I-00173 Roma, Italy, L. Federici 888Present
address: CERN, European Organization for Nuclear Research, CH-1211 Geneva 23,
Switzerland, A. Fucci, A. Salamon, F. Sargeni 171717Also 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 181818Also at Università degli Studi del Piemonte Orientale,
I-13100 Vercelli, Italy, B. Bloch-Devaux, M. Boretto 888Present 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 191919Also at Universidad de Guanajuato,
Guanajuato, Mexico

Horia Hulubei national Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
A. M. Bragadireanu, S. A. Ghinescu, O. E. Hutanu

Joint Institute for Nuclear Research, Dubna, Russia
T. Enik, V. Falaleev, V. Kekelidze, A. Korotkova, D. Madigozhin, M. Misheva
202020Present 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, A. Kleimenova 101010Present address:
Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium, Y. Kudenko
212121Also 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, A. Shaikhiev 101010Present
address: Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium

Institute for High Energy Physics - State Research Center of Russian
Federation, Protvino, Russia
S. Kholodenko, V. Kurshetsov, V. Obraztsov, A. Ostankov, V. Semenov, V.
Sugonyaev, O. Yushchenko

Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava, Slovakia
L. Bician, T. Blazek, V. Cerny, M. Koval 888Present address: CERN, European
Organization for Nuclear Research, CH-1211 Geneva 23, Switzerland, Z. Kucerova

CERN, European Organization for Nuclear Research, Geneva, Switzerland
A. Ceccucci, H. Danielsson, N. De Simone 222222Present address: DESY, D-15738
Zeuthen, Germany, F. Duval, B. Döbrich, L. Gatignon, R. Guida, F. Hahn
222Deceased, B. Jenninger, P. Laycock 111Corresponding authors: G. Ruggiero,
R. Marchevski,
<EMAIL_ADDRESS><EMAIL_ADDRESS>G. Lehmann
Miotto, P. Lichard, A. Mapelli, K. Massri, M. Noy, V. Palladino 232323Present
address: Physics Department, Imperial College London, London, SW7 2BW, UK, M.
Perrin-Terrin 242424Present address: Centre de Physique des Particules de
Marseille, Université Aix Marseille, CNRS/IN2P3, F-13288, Marseille,
France${}^{,}\,$252525Also at Université Catholique de Louvain, B-1348
Louvain-La-Neuve, Belgium, V. Ryjov, S. Venditti

University of Birmingham, Birmingham, United Kingdom
M. B. Brunetti, V. Fascianelli 262626Present address: Dipartimento di
Psicologia, Università di Roma La Sapienza, I-00185 Roma, Italy, F. Gonnella,
E. Goudzovski, L. Iacobuzio, C. Lazzeroni, N. Lurkin, F. Newson, C. Parkinson,
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 Liverpool, Liverpool, United Kingdom
J. B. Dainton 272727Present address: Physics Department, University of
Lancaster, Lancaster, LA1 4YW, UK, J. R. Fry 141414Present address: School of
Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK, L.
Fulton, D. Hutchcroft, E. Maurice 282828Present address: Laboratoire Leprince
Ringuet, F-91120 Palaiseau, France, G. Ruggiero 111Corresponding authors: G.
Ruggiero, R. Marchevski,
<EMAIL_ADDRESS>radoslav.marchevski@cern.ch,272727Present
address: Physics Department, University of Lancaster, Lancaster, LA1 4YW, UK,
B. Wrona

George Mason University, Fairfax, Virginia, USA
A. Conovaloff, P. Cooper, D. Coward 292929Also at SLAC National Accelerator
Laboratory, Stanford University, Menlo Park, CA 94025, USA, P. Rubin

11footnotetext: Present address: Brookhaven National Laboratory, Upton, NY
11973, USA22footnotetext: Present address: TRIUMF, Vancouver, British
Columbia, V6T 2A3, Canada

##  1 Introduction

The flavour-changing neutral current decay
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ proceeds at the lowest order in the
Standard Model (SM) through electroweak box and penguin diagrams largely
dominated by t-quark exchange. The quadratic GIM mechanism and the small value
of the CKM element $|V_{\mathrm{td}}|$ describing the transition of a top into
a down quark make this process extremely rare. Using tree-level elements of
the CKM matrix as external inputs, the SM predicts [1] the branching ratio to
be ${\rm BR}=(8.4\pm 1.0)\times 10^{-11}$, where the uncertainty is dominated
by the current precision on the CKM parameters. The intrinsic theoretical
accuracy is at the 2% level, as the computation includes NLO (NNLO) QCD
corrections to the top (charm) quark contribution [2, 3] and NLO electroweak
corrections [4]. Moreover, the hadronic matrix element largely cancels when
normalised to the precisely measured BR of the
$K^{+}\rightarrow\pi^{0}e^{+}\nu$ decay, with isospin breaking and non-
perturbative effects calculated in detail [4, 5].

The $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay is sensitive to physics beyond
the SM. The largest deviations from the SM are expected in models with new
sources of flavour violation, where constraints from $B$ physics are weaker
[6, 7]. The experimental value of the CP-violation parameter $\varepsilon_{K}$
limits the expected $\text{BR}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})$ range
within models with currents of defined chirality, producing specific
correlation patterns between the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ decay rates [8]. Present experimental
constraints limit the range of variation within supersymmetric models [9, 10,
11]. The $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay can also be sensitive to
effects of lepton flavour non-universality [12] and can constrain leptoquark
models [13] aiming to explain the measured value of the CP-violation parameter
$\varepsilon^{\prime}/\varepsilon$ [14].

The E787 and E949 experiments at BNL [15, 16] studied the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay using a decay-at-rest technique
and obtained $\text{BR}=(17.3^{+11.5}_{-10.5})\times 10^{-11}$. The NA62
experiment at the CERN SPS aims to measure
$\text{BR}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})$ more precisely with a novel
decay-in-flight technique. This letter reports a result from the analysis of
data collected by NA62 in 2016, corresponding to 2% of the full statistics
accumulated during the 2016–2018 data-taking period.

##  2 Beam line and detector

The choice of the decay-in-flight technique is motivated by the possibility of
obtaining an integrated flux of $\cal O$$(10^{13})$ kaon decays over a few
years of data-taking with a signal acceptance of a few percent, leading to the
collection of $\cal O$(100) SM events in the $K^{+}\to\pi^{+}\nu\bar{\nu}$
channel. This technique utilises a high energy beam to boost the kaon decay
products to high energy where their detection and identification can be
efficient. The boost folds the decay products into a small angular region
close to the beam axis allowing for a classic fixed target geometry
experiment. The detector should measure the incoming kaon and the outgoing
pion while reducing background to the level of $10^{-11}$ of accepted kaon
decays in the high rate environment necessary to achieve the required
sensitivity. To this end, the detector consists of a collection of sub-
detectors, each designed to perform one main task, and with complementary
performance.

The NA62 beam line and detector layout is described in detail in [17] and
shown in Fig. 1 together with the scale and reference system: the beam line
defines the Z axis with its origin at the kaon production target and beam
particles travelling in the positive direction, the Y axis points vertically
up, and the X axis is horizontal and directed to form a right-handed
coordinate system.

Figure 1: Schematic top view of the NA62 beam line and detector. Dipole
magnets are displayed as boxes with superimposed crosses. The trajectory is
shown of an un-decayed beam particle in vacuum, crossing the detector
apertures which avoid interactions with material. A dipole magnet between MUV3
and SAC deflects the charged particles of the beam out of the SAC acceptance.

A secondary hadron beam of positive charge containing 70% $\pi^{+}$, 23%
protons and 6% $K^{+}$, with a nominal momentum of 75 GeV/$c$ and 1% rms
momentum bite, is derived from 400 GeV/$c$ protons extracted from the SPS in
spills of 3 s effective duration and interacting with a 40 cm long beryllium
target. Typical intensities for the present measurement range from $1.0$ to
$1.3\times 10^{12}$ protons per pulse ($ppp$).

The time, momentum and direction of the charged components of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay are measured by the following detectors. A
differential Cherenkov counter (KTAG, filled with N2 at 1.75 bar pressure and
read out by PMs grouped in eight sectors) tags incoming kaons with a 70 ps
time resolution. Three silicon pixel stations (GTK) located before, between
and after two pairs of dipole magnets (beam achromat), form a spectrometer to
measure momentum, direction and time of beam particles with 0.15 GeV/$c$, 16
$\mu$rad and 100 ps resolutions. A magnetic spectrometer (STRAW, comprising
two pairs of straw chambers on either side of a dipole magnet) measures the
momentum-vector of the outgoing particle with a momentum resolution
$\sigma_{p}/p$ in the 0.3–0.4% range. A ring-imaging Cherenkov counter (RICH,
filled with neon at atmospheric pressure) tags the decay particle with a
precision better than 100 ps and provides particle identification.

Two scintillator hodoscopes (CHOD, a matrix of tiles read out by SiPMs and
NA48-CHOD, composed of two orthogonal planes of slabs, reused from the NA48
experiment) are used for triggering and timing purposes, providing a 99%
efficient trigger and a time measurement with 200 ps resolution for charged
particles.

The third GTK station (GTK3) is immediately preceded by a final collimator to
partly block particles produced in upstream decays, and marks the beginning of
a 117 m-long vacuum tank. The first 80 m of the tank define a fiducial volume
(FV) 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 FV entrance. The typical beam particle rate is 300 MHz.

Other sub-detectors are used as vetoes to suppress decays into photons or
multiple charged particles (electrons, pions or muons) or as complementary
particle-identifiers. Six stations of plastic scintillator bars (CHANTI)
detect with 99% efficiency and 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 the detector to achieve hermetic acceptance for photons emitted in
$K^{+}$ decays in the FV 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 and complements the RICH for particle identification.
The LKr energy, spatial and time resolutions in NA62 conditions are
$\sigma_{E}/E=1.4\%$ at an energy deposit of 25 GeV, 1 mm and between 0.5 and
1 ns, respectively, depending on the amount and type of energy release. Two
hadronic iron/scintillator-strip sampling calorimeters (MUV1,2) and an array
of scintillator tiles located behind 80 cm of iron (MUV3, with 400 ps time
resolution) supplement the pion/muon identification system. 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 hermetic coverage for charged particles produced in multi-track kaon
decays.

Most detectors are read out with TDCs, except LKr and MUV1,2 read out with
14-bit FADCs, and IRC, SAC read out with both. With the exception of GTK and
STRAW, read out by specific boards, the TDC modules are mounted on custom-made
boards (TEL62), which both produce trigger information and perform data read
out. Calorimeters use a dedicated processor for triggering [18].

A low-level trigger (L0) exploits the logical signals (primitives) produced by
the RICH, CHOD, NA48-CHOD, LKr, LAV and MUV3. A dedicated board combines the
primitives to build and dispatch trigger decisions to the detectors for data
readout [19]. A software trigger (L1) processes data from KTAG, LAV and STRAW
to produce higher level information exploiting reconstruction algorithms
similar to those used offline (Sec. 3), but adapted to the online environment.

The data sample is obtained from about $5\times 10^{4}$ SPS spills recorded
during one month of data-taking in 2016. The main trigger chain (called PNN)
is defined as follows. The L0 trigger requires a signal in RICH to tag a
charged particle in coincidence within 10 ns with: a signal in at least one
CHOD tile; no signals in opposite CHOD quadrants to reduce
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays; no signals in MUV3 and LAV12
to reduce $K^{+}\rightarrow\mu^{+}\nu$ and $K^{+}\rightarrow\pi^{+}\pi^{0}$
decays; and LKr energy deposit below 20 GeV to suppress
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays. The L1 trigger requires: a kaon
identified in KTAG and signals in at most two blocks of each LAV station,
within 10 ns of the L0 trigger RICH time; at least one STRAW track
corresponding to a particle with momentum below 50 GeV/$c$ which forms a
vertex with the nominal beam axis upstream of the first STRAW chamber. The
analysis also uses data taken with a minimum-bias L0 trigger based on
NA48-CHOD information downscaled by a factor of 400 (“control triggers”) to
measure efficiencies and estimate backgrounds. Events collected with the PNN
(control) trigger are referred to as “PNN-triggered events” (“control events”)
in the following.

##  3 Reconstruction and calibration

The KTAG channels are time-aligned, and signals are grouped within 2 ns wide
windows to define candidates. The KTAG candidate time is used as a reference
to adjust the time response of the other sub-detectors. Signals from the GTK
stations grouped within 2 ns of a KTAG candidate form a beam track. Fully
reconstructed $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays in the STRAW
spectrometer are used to align the GTK stations transversally to a precision
better than 100 $\mu$m and tune the GTK momentum scale.

The STRAW reconstruction relies on the NA48-CHOD time as a reference to
determine the drift time. Space-points in the chambers describing a path
compatible with the magnetic bending define a track, and its parameters are
obtained using a Kalman-filter fit. The $\chi^{2}$ 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 permille precision.

Two algorithms reconstruct RICH candidates, both grouping signals from PMs in
time around the L0 trigger. The first one 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 (“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 [20].

The CHOD candidates are defined by the response of the two SiPMs reading out
the same tile. Signals in crossing horizontal and vertical slabs compatible
with the passage of a charged particle form NA48-CHOD candidates; 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 maximize the photon veto
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 are grouped to form a
candidate, the energy of which 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 induced by the
Cherenkov light produced by particles traversing the PM window.

The 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. 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 determined for each channel after baseline
subtraction using a sample of $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays.

Signal times measured by GTK, KTAG, CHOD and RICH are further aligned on a
spill basis to the L0 RICH time, resulting in 20 ps stability through the
whole data sample. Early checks on reconstructed data help identifying spills
with hardware malfunctioning, which are excluded from the analysis.

Samples of $K^{+}$ decays are produced using a Geant4-based [21] Monte Carlo
(MC) simulation of the setup and used in the analysis to validate the detector
response, compute acceptances and estimate backgrounds. The simulation
includes the modeling of the time development of the signals, the response of
the front-end electronic and the effects of miscalibration as derived from
data. Accidental activity is added in GTK and KTAG assuming 300 MHz beam
intensity, and using a pileup beam particle library. No accidental activity is
simulated in the detectors downstream of the last GTK station. Simulated data
are subjected to the same reconstruction and calibration steps as described
above.

##  4 Event selection

Figure 2: Left: true $m^{2}_{\mathrm{miss}}$ distribution of the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay and the main $K^{+}$ decays,
computed under $\pi^{+}$ mass hypothesis for the charged particle in the final
state. Signal (red) is multiplied by $10^{10}$ for visibility and the dashed
areas show the signal search regions. Right: reconstructed
$m^{2}_{\mathrm{miss}}$ as a function of $\pi^{+}$ momentum for control events
selected without applying $\pi^{+}$ identification and photon rejection.
Signal regions 1 and 2, as well as the $3\pi$, $\pi^{+}\pi^{0}$ and
$\mu^{+}\nu$ background regions are also shown. The control regions located
between the signal and background regions are indicated by dashed lines.

The $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ signature consists of a $K^{+}$ with
4-momentum $p_{K}$ in the initial state, and a $\pi^{+}$ with 4-momentum
$p_{\pi}$ with missing energy in the final state. The squared missing mass
$m^{2}_{\mathrm{miss}}\equiv\left(p_{K}-p_{\pi}\right)^{2}$ is used to
discriminate kinematically the main $K^{+}$ decay modes from the signal. The
signal is searched for in two $m^{2}_{\mathrm{miss}}$ regions on each side of
the $K^{+}\rightarrow\pi^{+}\pi^{0}$ peak (Fig. 2, left). Selection criteria
based on $m^{2}_{\mathrm{miss}}$ alone are not sufficient to reduce the
backgrounds to the desired level, and additional suppression by $\pi^{+}$
identification and photon rejection is required. The principal selection
criteria are listed below.

Up to two positively-charged STRAW tracks are allowed in an event, provided
there are no negatively-charged tracks and these tracks do not form a vertex
in the FV. To be accepted, a track must be within the RICH, CHOD, LKr, and
MUV1,2,3 sensitive regions, and must be associated in position with candidates
in the CHOD, NA48-CHOD, LKr and RICH. Association in the RICH relies on the
consistency between the track direction downstream of the spectrometer magnet
and the position of the ring centre. Timing constraints are applied to the LKr
and RICH candidates with respect to the NA48-CHOD candidate time. Track times
are measured with 100 ps resolution by combining the sub-detector signals
associated with each track.

A KTAG candidate with Cherenkov photons detected in at least five out of eight
sectors tags a $K^{+}$. The parent $K^{+}$ is defined as that closest in time
to the $\pi^{+}$ candidate within 2 ns. The identification of the GTK track of
the parent $K^{+}$ relies on its time coincidence with the KTAG and RICH, and
its geometric compatibility with the STRAW track quantified by the closest
distance of approach (CDA). Probability density functions of the time and CDA
distributions for both $K^{+}$ and accidental beam particles obtained from
reconstructed $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays are used to
define a discriminant. The GTK track within 0.6 ns of the KTAG time with the
highest discriminant value is associated with the parent $K^{+}$. An
additional requirement is applied on the discriminant leading to a $K/\pi$
association efficiency value of 75%, as measured with a
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ data sample. The corresponding
fraction of incorrect $K/\pi$ associations is below 1% when the $K^{+}$ is
correctly reconstructed. If the $K^{+}$ is not reconstructed, the probability
of associating a pile-up beam track to the $\pi^{+}$ is 3.5%.

The $K^{+}$ and $\pi^{+}$ tracks define the decay vertex. Charged pions
originating from $K^{+}$ decays upstream of the final collimator or from
interactions of beam particles in the GTK stations can mimic a
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay if an accidental beam particle in
GTK matches the $\pi^{+}$, leading to incorrect vertex reconstruction. In
addition to $K/\pi$ association, further conditions suppress these events:
$Z_{\mathrm{vertex}}>Z_{0}$, where $Z_{\mathrm{vertex}}$ is the vertex
longitudinal coordinate and $Z_{0}$ lies in the range 110–115 m depending on
the $\pi^{+}$ direction; the $\pi^{+}$ extrapolated to the final collimator Z
plane must lie outside a $200\times 1000$ mm2 area centred around the Z axis
(“box cut”); no activity in CHANTI is allowed within 3 ns of the $\pi^{+}$;
and fewer than five GTK tracks may be reconstructed within 2 ns of the KTAG
time. To reduce background from $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays
and optimize the $\pi^{0}$ rejection, $Z_{\mathrm{vertex}}$ is required to be
upstream of 160–165 m, depending on the track slope.

If two STRAW tracks in the same event satisfy the above conditions, the one
closer in time to the trigger and the associated $K^{+}$ is considered. The
CHOD, KTAG, RICH and LKr candidates and the GTK track matched to the $\pi^{+}$
are required to be the closest to the trigger time. The analysis is restricted
to the pion momentum ($P_{\pi^{+}}$) range of $(15,35)$ GeV/$c$. This
condition ensures the presence of at least 40 GeV energy in addition to the
$\pi^{+}$, which improves the rejection of backgrounds such as
$K^{+}\rightarrow\pi^{+}\pi^{0}$. It also enhances the kinematic separation
between signal and $K^{+}\rightarrow\mu^{+}\nu$ decay, and makes the optimal
use of the RICH for $\pi^{+}/\mu^{+}$ separation. The $m^{2}_{\mathrm{miss}}$
is reconstructed from the $K^{+}$ and $\pi^{+}$ 4-momenta measured by the GTK
and the STRAW. The $m^{2}_{\mathrm{miss}}$ resolution at the
$K^{+}\rightarrow\pi^{+}\pi^{0}$ peak is about $10^{-3}$ GeV2/$c^{4}$, which
determines the choice of two $m^{2}_{\mathrm{miss}}$ signal regions defined as
$(0,0.01)$ GeV2/$c^{4}$ (“Region 1”) and $(0.026,0.068)$ GeV2/$c^{4}$ (“Region
2”). Three background regions ($\pi^{+}\pi^{0}$, $\mu^{+}\nu$ and $3\pi$) and
suitable control regions are also chosen (Fig. 2, right). Signal and control
regions are kept masked for PNN-triggered events until the completion of the
analysis. The $m^{2}_{\mathrm{miss}}$ is also computed either assuming the
average $K^{+}$ beam momentum and direction, or using $p_{\pi}$ measured by
the RICH single-ring algorithm assuming the $\pi^{+}$ mass. Conditions imposed
on the $m^{2}_{\mathrm{miss}}$ computed in these alternative ways refine the
definition of signal regions, reducing the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ acceptance by a further relative 7% and
providing additional background suppression in case of mis-reconstruction in
STRAW or GTK.

A multivariate classifier based on a Boosted Decision Tree algorithm [22]
combines 13 variables describing the energy associated with the $\pi^{+}$, the
shape of the clusters and energy sharing between the calorimeters. Pion
identification with the RICH exploits the ratio of likelihoods under the
$\pi^{+}$ and $\mu^{+}$ hypotheses. Additional constraints are applied on the
particle mass calculated from the ring radius computed by the single-ring RICH
algorithm and the momentum measured by the STRAW. The $\pi^{+}$ identification
efficiency measured in the $(15,35)$ GeV/$c$ momentum range is 78% (82%) with
calorimeters (RICH), and the corresponding probability of $\mu^{+}$ mis-
identification as $\pi^{+}$ is $0.6\times 10^{-5}$ ($2.1\times 10^{-3}$). MC
simulations reproduce these results with 10–20% accuracy. A requirement of no
particle detected in MUV3 within 7 ns of the $\pi^{+}$ reinforces the MUV3
trigger condition.

Photon rejection suppresses $K^{+}\rightarrow\pi^{+}\pi^{0}$ decays which can
mimic the signal if the two-body kinematics is not well reconstructed. The
main requirements are: no energy deposited in any LAV station (IRC and SAC)
within 3 (7) ns of the $\pi^{+}$ time; no clusters in the LKr beyond 100 mm
from the $\pi^{+}$ impact point location within time windows ranging from $\pm
5$ ns for cluster energies below 5 GeV to $\pm 50$ ns above 15 GeV.
Multiplicity rejection criteria against photons interacting in the detector
material upstream of the LKr include: no in-time activity in the CHOD and
NA48-CHOD unrelated to the $\pi^{+}$ but in spatial coincidence with an energy
deposit of at least 40 MeV in the LKr; no additional segments reconstructed in
the STRAW compatible with the decay vertex; no in-time signals in HASC and
MUV0; fewer than four extra signals in the NA48-CHOD in time with the
$\pi^{+}$. The resulting $\pi^{0}\rightarrow\gamma\gamma$ rejection
inefficiency is measured to be $2.5\times 10^{-8}$ by counting selected
control (PNN-triggered) events in the $\pi^{+}\pi^{0}$ region before (after)
photon and multiplicity rejection. Alternatively the single photon detection
efficiencies of LAV, LKr, IRC and SAC are measured using
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays with one reconstructed photon used to
tag the other photon, and convolved with a $K^{+}\rightarrow\pi^{+}\pi^{0}$
simulation; this leads to a similar $\pi^{0}$ rejection estimate. The
corresponding signal loss is 34%, with 10% (24%) due to $\pi^{+}$ interactions
(accidentals). Photon and multiplicity rejection is also effective against
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ and
$K^{+}\rightarrow\pi^{+}\pi^{-}e^{+}\nu$ backgrounds.

##  5 Single event sensitivity

The single event sensitivity is defined as ${\rm
SES}=1/(N_{K}\cdot\varepsilon_{\pi\nu\nu})$, where $N_{K}$ is the number of
$K^{+}$ decays in the FV and $\varepsilon_{\pi\nu\nu}$ is the signal
efficiency. The former quantity is computed as $N_{K}=(N_{\pi\pi}\cdot
D)/(A_{\pi\pi}\cdot\text{BR}_{\pi\pi})$, where $N_{\pi\pi}$ is the number of
$K^{+}\rightarrow\pi^{+}\pi^{0}$ decays selected from control events using the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ criteria except photon and multiplicity
rejection, and requiring $m^{2}_{\mathrm{miss}}$ to be in the $\pi^{+}\pi^{0}$
region; $A_{\pi\pi}=(9.9\pm 0.3)$% is the acceptance of this selection
estimated from MC simulation, where the error is due to the accuracy in the
simulation of $m^{2}_{\mathrm{miss}}$ resolution; $\text{BR}_{\pi\pi}$ is the
$K^{+}\rightarrow\pi^{+}\pi^{0}$ branching ratio [14] and $D=400$ is the
downscaling factor of the control trigger. This leads to $N_{K}=(1.21\pm
0.04_{syst})\times 10^{11}$, where the uncertainty is dominated by the error
on $A_{\pi\pi}$.

The signal efficiency is evaluated in four 5 GeV/$c$ wide $P_{\pi^{+}}$ bins
as
$\varepsilon_{\pi\nu\nu}=A_{\pi\nu\nu}\cdot\varepsilon_{trig}\cdot\varepsilon_{RV}$.
Here $A_{\pi\nu\nu}$ is the selection acceptance for the signal;
$\varepsilon_{trig}$ is the PNN trigger efficiency; $1-\varepsilon_{RV}$ is
the fraction of signal events discarded by the photon and multiplicity
rejection as a consequence of accidental activity in the detectors (Random
Veto). Other effects inducing signal loss are either included in
$A_{\pi\nu\nu}$ or cancel in the ratio to $A_{\pi\pi}$ when computing the SES.
An example is the signal loss due to random veto induced by accidental counts
in the MUV3 detector.

Figure 3: Left: Total acceptance $A_{\pi\nu\nu}$ in bins of $\pi^{+}$
momentum, and in Regions 1, 2 separately with their respective uncertainties.
Right: signal efficiency $\varepsilon_{RV}$ in bins of instantaneous beam
intensity after photon and multiplicity rejection with total uncertainty,
after photon rejection, after IRC and SAC veto only, after LAV veto only,
after LKr veto only. Lines are for eye guidance only. The instantaneous
intensity (estimated from out-of-time activity in GTK) can vary up to a factor
of two within a spill with respect to the average intensity.

The $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay is simulated using form
factors derived from the $K^{+}\rightarrow\pi^{0}e^{+}\nu$ decay. The
selection acceptance $A_{\pi\nu\nu}=(4.0\pm 0.1)$% is evaluated using MC
simulation and includes the particle identification efficiency (Fig. 3, left).
The main sources of acceptance losses are: detector geometry, $\pi^{+}$
momentum range, $m^{2}_{\mathrm{miss}}$ regions, particle identification and
$K^{+}/\pi^{+}$ association in the FV. The uncertainty on $A_{\pi\nu\nu}$ is
due to the accuracy of the simulation of the signal losses resulting from
photon and multiplicity rejection induced by $\pi^{+}$ interactions with the
detector material.

The quantity $\varepsilon_{trig}$ is the product of L0 and L1 trigger
efficiencies. The efficiency of each L0 component is measured with control
events. These efficiencies, with the exception of the LKr condition, are
evaluated with the $K^{+}\rightarrow\pi^{+}\pi^{0}$ sample used for the
$N_{K}$ computation, while the LKr efficiency is measured using a
$K^{+}\rightarrow\pi^{+}\pi^{0}$ sample selected with both photons in
LAV1–LAV11. The resulting L0 efficiency ranges from 0.93 to 0.85 depending on
$P_{\pi^{+}}$, where the main losses come from the LKr and MUV3 conditions,
with a systematic uncertainty of 0.02. The L1 trigger efficiency is measured
to be $0.97\pm 0.01$ using a $K^{+}\rightarrow\pi^{+}\pi^{0}$ sample passing
the PNN L0 condition. The quoted uncertainty reflects the stability during the
data taking.

The factor $\varepsilon_{RV}$ is measured as the fraction of
$K^{+}\rightarrow\mu^{+}\nu$ decays in the data surviving the photon and
multiplicity rejection. The selection used for this measurement is similar to
the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection, apart from particle
identification (replaced by positive $\mu^{+}$ identification in MUV3 and
calorimeters) and the $m^{2}_{\mathrm{miss}}$ range requirement. The result is
$\varepsilon_{RV}=0.76\pm 0.04$; this quantity is independent of $P_{\pi^{+}}$
and depends on beam intensity as shown in Fig. 3, right. The quoted value
includes a correction of $+0.02$ based on simulation to account for activity
in CHOD and LAV induced by $\delta$-rays produced by muons in the RICH
mirrors. The uncertainty on $\varepsilon_{RV}$ is evaluated as the difference
between the loss of acceptance as calculated by simulation and the measured
$\varepsilon_{RV}$ extrapolated to zero intensity.

The SES and the corresponding number of SM
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decays expected in the signal regions
are:

$\displaystyle\mathrm{SES}$ $\displaystyle=$ $\displaystyle(3.15\pm
0.01_{stat}\pm 0.24_{syst})\times 10^{-10},$ (1) $\displaystyle
N^{exp}_{\pi\nu\nu}\mathrm{(SM)}$ $\displaystyle=$ $\displaystyle 0.267\pm
0.001_{stat}\pm 0.020_{syst}\pm 0.032_{ext}.$ (2)

Systematic uncertainties include those on $N_{K}$, $A_{\pi\nu\nu}$,
$\varepsilon_{trig}$ and $\varepsilon_{RV}$. An additional uncertainty is
assigned to beam pileup effects; it is evaluated by comparing the acceptances
computed including or not the pileup simulation. The external error on
$N_{exp}^{\pi\nu\nu}$(SM) comes from the uncertainty of the SM prediction.

##  6 Expected background

Background from $K^{+}$ decays in the FV is mainly due to
$K^{+}\rightarrow\pi^{+}\pi^{0}(\gamma)$,
$K^{+}\rightarrow\mu^{+}\nu(\gamma)$, $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$
and $K^{+}\rightarrow\pi^{+}\pi^{-}e^{+}\nu$ decays. The first three processes
may enter the signal regions via $m^{2}_{\mathrm{miss}}$ mis-reconstruction.
The estimate of the corresponding backgrounds relies on the assumption that
$\pi^{0}$ rejection for $K^{+}\rightarrow\pi^{+}\pi^{0}$, particle
identification for $K^{+}\rightarrow\mu^{+}\nu$ and multiplicity rejection for
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ are independent of the
$m^{2}_{\mathrm{miss}}$ criteria defining the signal regions. Possible
violations of this assumption, such as the impact of the radiative component
of $K^{+}\rightarrow\pi^{+}\pi^{0}(\gamma)$ decay, are investigated separately
in a dedicated study. In this framework, the number of expected background
events in each signal region (1 and 2) from these processes is computed as
$N_{bkg}\cdot f^{kin}$. Here $N_{bkg}$ is the number of PNN-triggered events
remaining in the corresponding background $m^{2}_{\mathrm{miss}}$ region after
the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection; $f^{kin}$ (“tails”) is
the proportion of background events entering the signal region through tails
of $m^{2}_{\mathrm{miss}}$ which are modelled separately. The above procedure
is applied in four bins of $P_{\pi^{+}}$ for $K^{+}\rightarrow\pi^{+}\pi^{0}$
and $K^{+}\rightarrow\mu^{+}\nu$ backgrounds. Background in the control
regions is evaluated similarly.

Figure 4: Left: reconstructed $m^{2}_{\mathrm{miss}}$ distribution of the
$K^{+}\rightarrow\pi^{+}\pi^{0}$ control events selected from data by tagging
the $\pi^{0}$ (dots, see text for details). Two
$K^{+}\rightarrow\pi^{+}\pi^{0}$ MC samples are superimposed: one selected as
in data (red line), the other selected as
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ (blue line, referred to as MC
$K^{+}\rightarrow\pi^{+}\pi^{0}(\gamma)$ in the legend). Signal regions 1 and
2 are also shown. The MC distributions are normalised to the data in the
$\pi^{+}\pi^{0}$ region. Right: expected
$K^{+}\rightarrow\pi^{+}\pi^{0}(\gamma)$ background in bins of $\pi^{+}$
momentum compared to the expected number of SM
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ events.

$K^{+}\rightarrow\pi^{+}\pi^{0}(\gamma)$ background: forty-two events remain
in the $\pi^{+}\pi^{0}$ region after the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$
selection. The $m^{2}_{\mathrm{miss}}$ tails in Regions 1 and 2 are at the
$10^{-3}$ level and do not depend on $P_{\pi^{+}}$. They are evaluated from
$K^{+}\rightarrow\pi^{+}\pi^{0}$ control events selected with the same
criteria as for $N_{K}$ computation, except for the $m^{2}_{\mathrm{miss}}$
condition. The two photons from the $\pi^{0}\to\gamma\gamma$ decay are
required to be in the LKr acceptance, and $K^{+}\rightarrow\pi^{+}\pi^{0}$
decays are reconstructed independently of the measured $\pi^{+}$ and $K^{+}$
tracks by imposing $\pi^{0}$ mass, nominal beam momentum and missing mass
constraints. Simulation reproduces the tails to an accuracy better than 20%.
The kinematic constraints do not affect the reconstruction tails in Region 1,
but suppress the contribution from $K^{+}\rightarrow\pi^{+}\pi^{0}\gamma$
decays in Region 2 (Fig. 4, left). The estimate of
$K^{+}\rightarrow\pi^{+}\pi^{0}\gamma$ (inner bremsstrahlung) background is
based on simulation [23] and single photon detection efficiencies measured
from data. A systematic uncertainty is assigned to account for the precision
of the above measurement. The background dependence on $P_{\pi^{+}}$ is shown
in Fig. 4, right and is due to photon detection inefficiency at small angles.
After unblinding the control regions between the $\pi^{+}\pi^{0}$ and the two
signal regions (Fig. 2, right), one event is observed, while $1.46\pm
0.16_{stat}\pm 0.06_{syst}$ events are expected.

Figure 5: Left: reconstructed $m^{2}_{\mathrm{miss}}$ distribution of the
$K^{+}\rightarrow\mu^{+}\nu(\gamma)$ control events selected by assigning the
$\pi^{+}$ mass to the $\mu^{+}$ for data (dots), and for two MC
$K^{+}\rightarrow\mu^{+}\nu(\gamma)$ samples superimposed: one selected as in
data (red line), the other selected as $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$
without particle identification (blue line, referred as MC
$K^{+}\rightarrow\mu^{+}\nu(\gamma)$ in the legend). Signal regions 1 and 2
are also shown. Right: expected $K^{+}\rightarrow\mu^{+}\nu(\gamma)$
background in bins of $\pi^{+}$ momentum compared to the expected number of SM
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ events.

$K^{+}\rightarrow\mu^{+}\nu(\gamma)$ background: forty-five events remain in
the $\mu^{+}\nu$ region after the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$
selection. A $K^{+}\rightarrow\mu^{+}\nu$ sample selected as for the
$\varepsilon_{RV}$ measurement but without the $m^{2}_{\mathrm{miss}}$
condition is used to evaluate the reconstruction tails (Fig. 5, left). Tails
in Region 1 range from $5\times 10^{-5}$ in the first $P_{\pi^{+}}$ bin to
$5\times 10^{-4}$ in the last bin; tails in Region 2 are $2\times 10^{-5}$ in
all $P_{\pi^{+}}$ bins. Simulations reproduce these results to an accuracy of
30–40% in Region 1 and the control region, and better than 10% in Region 2.
The radiative contribution is included in the measured tails. The calorimetric
conditions used to identify $\mu^{+}$ in the control sample lead to
underestimation of the tails in Region 2 by up to 50%, limited by the current
data statistics, and a corresponding systematic uncertainty is assigned to the
tail measurement in this region. Mis-measurement of $P_{\pi^{+}}$ in the STRAW
may introduce a correlation between the reconstructed $m^{2}_{\mathrm{miss}}$
and $\pi^{+}$ identification in the RICH. The effect of this correlation is
estimated by comparing the RICH performance measured with data
$K^{+}\rightarrow\mu^{+}\nu$ events in $\mu^{+}\nu$ and signal regions. This
leads to a 30% uncertainty on the background estimate. The background
dependence on $P_{\pi^{+}}$ (Fig. 5, right) is driven by the increase of both
the muon mis-identification probability and the tails with $P_{\pi^{+}}$.
After unblinding the control region, between Region 1 and the $\mu^{+}\nu$
region (Fig. 2, right), two events are observed while $1.02\pm 0.16_{stat}\pm
0.31_{syst}$ events are expected.

$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ background: it populates mainly Region
2 due to the large $m^{2}_{\mathrm{miss}}$. Twenty events remain in the $3\pi$
region after the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection. A
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ sample used to evaluate the
$m^{2}_{\mathrm{miss}}$ tails is selected from control events using a
$\pi^{+}\pi^{-}$ pair to tag the decay. Simulations reproduce the
$m^{2}_{\mathrm{miss}}$ distribution of this sample over four orders of
magnitude. The tails are conservatively estimated to be $10^{-4}$, with a
systematic uncertainty of 100% assigned to account for the bias induced by the
above selection, as demonstrated by simulations. Multiplicity rejection and
kinematic cuts are effective against $K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$
decays, and the expected background is almost negligible.

$K^{+}\rightarrow\pi^{+}\pi^{-}e^{+}\nu$ background: it is characterized by
large $m^{2}_{\mathrm{miss}}$ and therefore enters Region 2. It is suppressed
by its $\cal O$$(10^{-5})$ branching ratio, multiplicity rejection, particle
identification and kinematics. The approach adopted to estimate the other
major backgrounds cannot be used in this case because the number of particles
in the detector acceptance and thus the multiplicity rejection are correlated
with the kinematics. The background is therefore estimated from simulation.
Out of 6$\times 10^{8}$ simulated decays, two events pass the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection. Modifications to the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection, such as requiring a $\pi^{-}$
instead of $\pi^{+}$, or inverting specific multiplicity rejection conditions,
define exclusive samples with $\cal O$(10) data events passing these new
selections. The agreement between the observed and expected numbers of events
in these samples validates the simulation.

Other backgrounds from $K^{+}$ decays: the contributions from
$K^{+}\rightarrow\mu^{+}\pi^{0}\nu$, $K^{+}\rightarrow\pi^{0}e^{+}\nu$and
$K^{+}\rightarrow\pi^{+}\gamma\gamma$ decays are found to be negligible
considering particle identification and photon rejection performance applied
to simulated samples. Upper limits of $\cal O$$(10^{-3})$ events are obtained
for these contributions.

Figure 6: Left: transverse position at the FV entrance of pions from a data
sample enriched with upstream events. Blue lines correspond to the contour of
the last dipole of the second achromat; red lines show the contour of the
final collimator; the black line indicates the acceptance region covered by
CHANTI. Right: time difference between RICH and KTAG versus that of GTK and
KTAG for the same sample of pions.

Upstream backgrounds: they are due to pions originating upstream of the FV and
are classified as follows.

  * •

$\pi^{+}$ from $K^{+}$ decays between GTK stations 2 and 3, matched to an
accidental beam particle;

  * •

$\pi^{+}$ from interactions of a beam $\pi^{+}$ at GTK stations 2 and 3,
matched to an accidental $K^{+}$;

  * •

$\pi^{+}$ from interactions of a $K^{+}$ with material in the beam line,
produced either promptly or as a decay product of a neutral kaon.

The interpretation of the upstream events in terms of the above mechanisms is
supported by a detailed analysis of a sample selected as
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ apart from the conditions defining the
FV and the $K/\pi$ association, which are inverted to enrich the sample with
upstream events. The $\pi^{+}$ position distribution at the GTK3 Z plane (Fig.
6, left) indicates that they originate from upstream decays or interactions in
the GTK stations and justifies the $200\times 1000$ mm2 box cut (Sec. 4). The
distribution of RICH–KTAG vs GTK–KTAG time difference suggests the accidental
origin of these events (Fig. 6, right). Installation of an additional
shielding insert in 2017 and a new collimator of larger transverse size in
2018 have substantially reduced the backgrounds at $|Y|>100$ mm (Fig. 6,
left).

Upstream background estimation is performed with data using a bifurcation
technique [16]. The $K/\pi$ matching ($cut1$) and the box cut ($cut2$) are the
two selection criteria to be inverted. The combination of $cut1$ and $cut2$
defines four samples, denoted as $A(cut1\cdot cut2)$ corresponding to signal,
$B(cut1\cdot\overline{cut2})$, $C(\overline{cut1}\cdot cut2)$ and
$D(\overline{cut1}\cdot\overline{cut2})$. The number of expected events in
sample $A$ is $N_{A}=N_{B}\cdot N_{C}/N_{D}$ assuming $cut1$ and $cut2$ are
uncorrelated. This assumption is validated with data by applying the same
procedure to different samples obtained modifying the selection conditions.
The accuracy of the results is limited by the size of the bifurcation samples.

The background estimates in the signal regions are summarized in Table 1.
Errors are added in quadrature to obtain the uncertainty on the total expected
background.

Table 1: Summary of the background estimates summed over the two signal regions.  Process | Expected events
---|---
$K^{+}\rightarrow\pi^{+}\pi^{0}(\gamma)$ | $0.064\pm 0.007_{stat}\pm 0.006_{syst}$
$K^{+}\rightarrow\mu^{+}\nu(\gamma)$ | $0.020\pm 0.003_{stat}\pm 0.006_{syst}$
$K^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ | $0.002\pm 0.001_{stat}\pm 0.002_{syst}$
$K^{+}\rightarrow\pi^{+}\pi^{-}e^{+}\nu$ | $0.013^{+0.017}_{-0.012}|_{stat}\pm 0.009_{syst}$
$K^{+}\rightarrow\pi^{0}\mu^{+}\nu$, $K^{+}\rightarrow\pi^{0}e^{+}\nu$ | $<0.001$
$K^{+}\rightarrow\pi^{+}\gamma\gamma$ | $<0.002$
Upstream background | $0.050^{+0.090}_{-0.030}|_{stat}$
Total background | ${{0.152^{+0.092}_{-0.033}|_{stat}\pm 0.013_{syst}}}$

##  7 Result and conclusion

After unblinding the signal regions, one event is found in Region 2, as shown
in Fig. 7, left. The STRAW track momentum is 15.3 GeV/$c$. The RICH response
is consistent with a $\pi^{+}$ hypothesis (Fig. 7, right). The track deposits
50% (20%) of its energy in the LKr (MUV1), and the $p$-value of the muon
hypothesis based on calorimetric identification is 0.2%. The candidates in the
RICH, CHOD and LKr associated to the $\pi^{+}$, as well as the KTAG and GTK
candidates associated to the $K^{+}$, are mutually consistent in time within
one standard deviation of the corresponding resolutions. The KTAG candidate is
identified by signals in seven of the eight sectors, and the kinematics of the
$K^{+}$ measured with the GTK is consistent with the nominal beam properties.
The decay vertex properties are $Z_{\mathrm{vertex}}=146$ m and
$\mathrm{CDA}=1.7$ mm. The $\pi^{+}$ transverse position extrapolated at the
FV entrance is $(x,y)=(-373,30)$ mm.

The hybrid Bayesian-frequentist approach [24] and the CLs method [25] are used
for the statistical interpretation of the result. A counting experiment
analysis is performed considering an expected signal of 0.267 events and an
expected background of $0.152\pm 0.090$ events, where a symmetric uncertainty
is considered. The corresponding expected upper limit is
$\text{BR}_{exp}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})<10\times 10^{-10}$ at
95% CL. Considering the observation of one event, the $p$-value of the signal
and background hypothesis is 15% and the corresponding observed upper limit is

$\text{BR}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})<14\times 10^{-10}~{}{\rm
at}~{}95\%~{}\text{CL}.$ (3)

The result, based on 2% of the total NA62 exposure in 2016–2018, demonstrates
the validity of the decay-in-flight technique in terms of background rejection
and in view of the measurement in progress using the full data sample.

Figure 7: Left: reconstructed $m^{2}_{\mathrm{miss}}$ as a function of
$\pi^{+}$ momentum for PNN-triggered events (markers) satisfying the
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ selection, except the
$m^{2}_{\mathrm{miss}}$ and $\pi^{+}$ momentum criteria. The grey area
corresponds to the expected distribution of
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ MC events. Red contours define the
signal regions. The event observed in Region 2 is shown together with the
events found in the control regions. Right: Signal in RICH (open circles)
detected in the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ candidate event with
rings superimposed as built under different particle mass hypotheses.

## 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 of its auxiliary systems were supported by the
funding agencies of the Collaboration Institutes. We are particularly indebted
to: F.R.S.-FNRS (Fonds de la Recherche Scientifique - FNRS), Belgium; BMES
(Ministry of Education, Youth and Science), Bulgaria; NSERC (Natural Sciences
and Engineering Research Council), Canada; NRC (National Research Council)
contribution to TRIUMF, Canada; MEYS (Ministry of Education, Youth and
Sports), Czech Republic; BMBF (Bundesministerium für Bildung und Forschung)
contracts 05H12UM5 and 05H15UMCNA, 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), 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 Number 1506088, U.S.A.; ERC (European
Research Council) “UniversaLepto” advanced grant 268062, “KaonLepton” starting
grant 336581, Europe.

Individuals have received support from: Charles University (project GA UK
number 404716), Czech Republic; Ministry of Education, Universities and
Research (MIUR “Futuro in ricerca 2012” grant RBFR12JF2Z, Project GAP), Italy;
Russian Foundation for Basic Research (RFBR grants 18-32-00072, 18-32-00245),
Russia; the Royal Society (grants UF100308, UF0758946), United Kingdom; STFC
(Rutherford fellowships ST/J00412X/1, ST/M005798/1), United Kingdom; ERC
(grants 268062, 336581).

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# Corrections to Estimated Accretion Disk Size due to Color Correction, Disk
Truncation and Disk Wind

Andrzej A. Zdziarski Nicolaus Copernicus Astronomical Center, Polish Academy
of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland<EMAIL_ADDRESS>Bei You
School of Physics and Technology, Wuhan University, Wuhan 430072, China
Astronomical Center, Wuhan University, Wuhan 430072, China

###### Abstract

We consider corrections to the disk sizes measured at a given frequency with
accretion models due to the color correction, a disk truncation at an inner
radius larger than the innermost stable circular orbit, and disk winds. We
apply our results to the estimates of the disk sizes based on microlensing. We
find these three effects combined can explain the long-standing problem of the
disk sizes from microlensing being larger than those estimated using the
standard disk model (i.e., that without accounting for the above effects). In
particular, an increase of the color correction with the increasing
temperature can lead to a strong increase of the half-light radius even if
this correction is close to unity at the temperature corresponding to an
observed frequency. Our proposed formalism for calculating the half-light
radius also resolves the long-standing issue of discrepancies between the disk
size estimates based on the accretion rate and on the observed flux.

##  1 Introduction

About a hundred of distant quasars have so far been observed to have multiple
images due to gravitational lensing by a galaxy in the line of sight. A famous
example is the Einstein cross source, G2237+0305, which has four
gravitationally lensed images (Schneider et al., 1988). Then, stars in the
lensing galaxies cause microlensing (Paczyński, 1986) of each of the image,
which effect allows us to determine the size of the source of light, in
particular, an accretion disk surrounding the quasar black hole, see
Wambsganss (2006) for a review.

A number of papers have reported that the sizes estimated by O/IR monitoring
observations are larger by a factor of a few than those predicted by the
standard accretion disk theory (Shakura & Sunyaev, 1973) given the observed
flux (e.g., Pooley et al. 2007; Dai et al. 2010; Morgan et al. 2010; Chartas
et al. 2016). Equivalently, the optical flux measured from the source is
significantly lower than that from a standard geometrically-thin disk emitting
at the wavelength of the observation. We consider here ways to explain this
discrepancy.

In this work, we consider a color correction to the disk blackbody emission of
$f_{\rm col}\geq 1$. At a given observed flux and frequency, the size of the
emitting region is $\propto f_{\rm col}^{2}$, and $f_{\rm col}\sim 2$ would be
sufficient. The study of Hubeny et al. (2001) indicates that $f_{\rm
col}\approx 1$ in the optical wavelengths. However, that result assumes that
there is no dissipation in disk surface layers. If this assumption is relaxed
and a moderate dissipation in the surface layer is allowed, the above
discrepancy can be resolved. Indeed, there have been a number of papers
considering accretion disks supported by magnetic pressure, which are
substantially hotter than the standard ones (Begelman & Pringle, 2007;
Salvesen et al., 2016), see also Begelman & Silk (2017), Dexter & Begelman
(2019), Mishra et al. (2020). The effect of scattering atmospheres on the disk
size estimates was also considered by Hall et al. (2018). Furthermore, even if
the color correction is about a unity at the measured frequencies, its
increase at higher frequencies can lead to a substantial increase of the half-
light radius (defined by Mortonson et al. 2005; Pooley et al. 2007), for which
a half of the radiation at a $\nu$ is emitted interior of it, and a half,
exterior to it.

A further effect is the possible truncation of the disk at some inner radius.
This is likely, given the X-ray emission also observed from those quasars. If
the disk is truncated, the half-light radius will be increased further. Yet
another effect is a possible decrease of the accretion rate with the
decreasing distance to the black hole due to wind mass loss (e.g., You et al.
2016b, which can, in principle, reconcile the theoretical predictions with the
observations (Li et al., 2019; Sun et al., 2019). In this Letter, we study the
effects of all three of these phenomena. Furthermore, we take into accoutn
that the disk may be covered by an X-ray emitting corona, in which a fraction
of the gravitational energy is dissipated as well as it upscatters a fraction
of the disk photons to higher energies (Svensson & Zdziarski, 1994; Gierliński
et al., 1999; Done & Kubota, 2006; You et al., 2016a). That effect is included
phenomenologically by a multiplicative coefficient.

##  2 Disc size

Let us consider a geometrically-thin, optically-thick, accretion disk (Shakura
& Sunyaev, 1973). We assume the disk locally, at each radius, $R$, emits a
blackbody spectrum diluted by a color correction, $f_{\rm col}\geq 1$, which,
in general, can be a function of the effective temperature, $T_{\rm eff}$, and
the color temperature is then $T=f_{\rm col}\left[T_{\rm eff}(R)\right]T_{\rm
eff}$, see, e.g., Hubeny et al. (2001), Chiang (2002). We take into account a
possible existence of a corona covering the disk, via the coefficient $f_{\rm
c}\leq 1$, equal to 1 in the absence of a corona. Then the observed flux at an
observed frequency, $\nu$, is given by,

$\displaystyle\nu F_{\nu}$ $\displaystyle={4\pi h\nu^{4}(1+z)^{4}f_{\rm c}\cos
i\over D_{L}^{2}c^{2}}\times$ $\displaystyle\int^{R_{\rm out}}_{R_{\rm
in}}{R{\rm d}R\over f_{\rm col}\left[T_{\rm
eff}(R)\right]^{4}\left\\{\exp\left[h\nu(1+z)/kT(R)\right]-1\right\\}},$ (1)

where $z$ is the redshift, $D_{L}$ is the luminosity distance, $i$ is the
inclination at which the disk is observed, $R_{\rm out}$ is the disk outer
radius, and $R_{\rm in}$ is the inner radius down to which the above optically
disk extends. This radius may correspond to the innermost stable orbit (ISCO),
but it may be substantially larger if the disk is truncated. In fact, the
presence of X-ray emission from quasars requires that a part of the available
gravitational energy is converted into heating of a relativistic gas, capable
of X-ray emission. The energy conservation requires that the same part is
subtracted from the disk dissipation. This can be achieved by either some of
the dissipation taking place at the corona (as given by a value of $f_{\rm
c}$), at the expense of the underlying disk (Svensson & Zdziarski, 1994; Done
& Kubota, 2006), or by truncation of the optically thick disk at $R_{\rm in}$,
below which it is replaced by a hot flow (e.g., Narayan & Yi 1994; Zdziarski &
Gierliński 2004; Done, Gierliński, & Kubota 2007; You, Cao, & Yuan 2016a), or
by a combination of both. Hereafter, we include $f_{\rm c}$ in equations
below, but set $f_{\rm c}=1$ in numerical estimates. We also note that
$D_{L}=D_{\rm A}(1+z)^{2}$, where $D_{\rm A}$ is the angular diameter
distance, which allows for a minor simplification of the above expression.

In the Newtonian limit and assuming the zero-torque inner boundary condition,
the effective temperature is111We note the presence of a redistribution of the
gravitationally-released energy is responsible for the appearance of the
factor of 3 in equation (2) regardless of the boundary condition, cf.
equations (4.30) and (5.14–5.15) of Frank et al. (2002). The locally released
gravitational energy would give only the factor of 1 in that equation. In the
case of zero-stress inner boundary condition, the factor of 3 also corresponds
to $\int_{R_{\rm in}}^{\infty}R^{-2}{\rm d}R=1/R_{\rm in}$ vs. $\int_{R_{\rm
in}}^{\infty}R^{-2}[1-(R_{\rm in}/R)]^{1/2}{\rm d}R=(1/(3R_{\rm in})$,
independent of the value of $R_{\rm in}$. We note that this factor of 3 should
appear in the denominator of equation (12) of Zdziarski et al. (2021).

$T_{\rm eff}(R)=\left[3f_{\rm c}GM\dot{M}(R)\over 8\pi\sigma
R^{3}\right]^{1/4}\left[1-\left(R_{\rm b}\over R\right)^{1/2}\right]^{1/4},$
(2)

where $M$ is the black-hole mass, $\dot{M}(R)$ is the disk mass flow rate at
$R$, $R_{\rm b}$ is here the radius at which the zero-torque boundary
condition is imposed, and $\sigma$ is the Stefan-Boltzmann constant. We note
that in order for the accretion to proceed, the angular momentum of the
accreted matter has to be transferred outside, which implies the presence of a
boundary term. However, the dissipation may continue in a hot flow below a
disk truncation radius, $R_{\rm in}$, which can be $\geq R_{\rm b}$. Given
theoretical uncertainties, we assume $R_{\rm in}=R_{\rm b}$ in the examples
given below. Note that equations (1) and (2) are the basic formulae
determining the disk size.

We are concerned here with determining a characteristic disk radius from which
the bulk of the radiation at a frequency $\nu$ is emitted. For that purpose,
we define the radius, $R_{\nu}$, at which the local color temperature
corresponds to the source-frame frequency, $h\nu(1+z)=kT_{\rm
eff}(R_{\nu})f_{\rm col}\left[T_{\rm eff}(R_{\nu})\right]$ (hereafter $\nu$
denotes a frequency in the observer frame). If we are concerned with emission
at $R_{\nu}\gg R_{\rm in}$, the second factor in equation (2) can be
neglected, in which case $T_{\rm eff}\propto R^{-3/4}$. The expression for
$R_{\nu}$ in terms of $M$ and $\dot{M}$ can be then obtained

$R_{\nu,T}\approx\left[kf_{\rm col}(R_{\nu,1})\over
h\nu(1+z)\right]^{4/3}\left[3f_{\rm c}GM\dot{M}(R_{\nu,1})\over
8\pi\sigma\right]^{1/3}.$ (3)

We note that this expression, often called the ’theory size’, depends on
$\dot{M}(R_{\nu})$, which is usually not directly known.

If $\dot{M}$ and the color correction are constant, we have
$kT(x)=h\nu(1+z)x^{-3/4}$, where $x\equiv R/R_{\nu}$. We can then rewrite
equation (1) as

$\displaystyle\nu F_{\nu}\approx{4\pi h\nu^{4}(1+z)^{4}R_{\nu}^{2}I_{1}f_{\rm
c}\cos i\over{f_{\rm col}^{4}D_{L}^{2}c^{2}}},$ (4) $\displaystyle
I_{1}\equiv\int^{\infty}_{0}\\!\\!\\!{x\,{\rm
d}x\over\exp(x^{3/4})-1}=\frac{4}{3}\Gamma\left(\frac{8}{3}\right)\zeta\left(\frac{8}{3}\right)\approx
2.5762,$ (5)

where $\Gamma$ and $\zeta$ are the Euler Gamma function and the Riemann zeta
function, respectively. This yields

$R_{\nu,F}\approx{f_{\rm col}^{2}D_{L}c(\nu F_{\nu})^{1/2}\over
2\nu^{2}(1+z)^{2}(\pi I_{1}hf_{\rm c}\cos i)^{1/2}},$ (6)

which is often called the ’flux size’. Note that in the considered limit,
$R_{\nu,F}$ is independent of the black-hole mass, and depends only on
$F_{\nu}$ and $i$. It is also $\propto f_{\rm col}^{2}f_{\rm c}^{-1/2}$. We
also note that the ratio $R_{\nu,F}/R_{\nu,T}\propto f_{\rm col}^{2/3}f_{\rm
c}^{-5/6}$; thus it is larger than unity for $f_{\rm col}>1$, $f_{\rm c}<1$,
which can explain values of $R_{\nu,T}$ being larger than $R_{\nu,F}$ at the
assumed $f_{\rm col}=f_{\rm c}=1$ in microlensed quasars (e.g., Morgan et al.
2010). If we equate $R_{\nu,T}=R_{\nu,F}$, we obtain the expression for the
color-corrected disk blackbody spectrum in the intermediate region,
$F_{\nu}\propto\nu^{1/3}$, between the low and high energy cutoffs due to the
disk outer and inner radii, respectively,

$\nu F_{\nu}=\frac{\pi^{1/3}I_{1}k^{8/3}\cos i\,(3f_{\rm
c}GM\dot{M}/\sigma)^{2/3}}{c^{2}D_{L}^{2}h^{5/3}}\left[\frac{(1+z)\nu}{f_{\rm
col}}\right]^{4/3}.$ (7)

A discrepancy between the values of $R_{\nu,T}$ and $R_{\nu,F}$ may indicate
that either the local emission is not given by the disk blackbody spectrum,
equation (7), or $\dot{M}$ and/or $f_{\rm col}$ are not properly estimated.

We also define the half-light radius (Mortonson et al., 2005; Pooley et al.,
2007), at which a half of the emission at $\nu$ is emitted interior of
$R_{1/2,\nu}$. This is a nearly model-independent quantity, and thus preferred
in comparison of models with observations (Mortonson et al., 2005). For
constant $\dot{M}$ and $f_{\rm col}$, it corresponds to the solution of

$\int^{x_{1/2}}_{0}\\!\\!\\!\\!\\!{x\,{\rm
d}x\over\exp(x^{3/4})-1}=\int^{\infty}_{x_{1/2}}\\!\\!{x\,{\rm
d}x\over\exp(x^{3/4})-1},\,\,x_{1/2}\approx 2.4356,$ (8)

i.e., $R_{1/2,\nu}={x_{1/2}}R_{\nu,F}$. In the considered limit,
$R_{1/2,\nu^{\prime}}=(\nu^{\prime}/\nu)^{-4/3}R_{1/2,\nu}$.

On the other hand, an accurate determination of the disk radius requires using
directly equations (1) and (2) rather than the above approximations, and
taking into account relevant physical effect. In particular, the emitting
region can be close to the disk inner boundary, in which case the boundary
term needs to be taken into account. Also, the color correction is, in
general, temperature dependent. Furthermore, winds can be launched from the
accretion disk accretion disks via a variety of mechanisms: line-driven,
thermal and magnetic. Details of those mechanisms remain still uncertain. In
their studies of the effect of winds on the measured size of disks, Li et al.
(2019) and Sun et al. (2019) assumed the decrease with the decreasing radius
of the mass flow rate due to the wind for a non-rotating BH to be
$\dot{M}(R)=\dot{M}_{\rm in}{(R/R_{\rm in})}^{s}$, where $R_{\rm in}=6R_{\rm
g}$, $R_{\rm g}=GM/c^{2}$, $\dot{M}_{\rm in}$ is the accretion rate onto the
BH, and $s>0$ is a parameter. Li et al. (2019) fitted this model to the sizes
obtained from microlensing, and found that the value of $s$ as high as
$\approx$1.3 was required. If the above functional dependence applies up to
the outer truncation radius, which in AGNs is typically
$\sim\\!10^{3}$–$10^{4}R_{\rm g}$ (e.g., Goodman 2003, and fig. 6 in You et
al. 2012), the mass flow rate supplied to the disk is $\sim\\!800$–$1.5\times
10^{4}$ higher than that accreting onto the BH, which we consider highly
unlikely. In the case of sources emitting at a substantial fraction of the
Eddington luminosity, e.g., $\sim$1/4–1 (typical for microlensing sources,
Kollmeier et al. 2006), such an increase of $\dot{M}$ with the increasing
radius would cause the accretion rate to become progressively more super-
Eddington with the increasing radius, while there have no indications of such
rates. Also, the calculations of the emitted spectrum in such case assuming
the thin disk become not self-consistent. In the case of accreting X-ray
binaries, Coriat et al. (2012) found that estimating the average mass transfer
rate from the donor based on the observed X-ray luminosity assuming the
standard accretion efficiency, i.e., neglecting the effect of winds in
reducing the BH accretion rate, gives a very good agreement with the
prediction of the disk instability model of the X-ray transients (Dubus et
al., 2001). Therefore, we believe it is highly unlikely that the presence of
disk winds results in a radial dependence of the accretion rate as strong as
the above power-law. Therefore, we consider alternative descriptions of the
effect of winds on $\dot{M}(R)$. A number of them were listed by You et al.
(2016b). Among those, we have chosen the most physically-motivated one,
following from the jet/disk outflow model of Blandford & Payne (1982),

$\dot{M}(R)=\dot{M}_{\rm in}\left[1+\left(\frac{\dot{M}_{\rm
out}}{\dot{M}_{\rm in}}-1\right)\frac{\ln(R/R_{\rm in})}{\ln(R_{\rm
out}/R_{\rm in})}\right],$ (9)

where hereafter we adopt $R_{\rm out}=10^{3}R_{\rm g}$, and $\dot{M}_{\rm
out}/\dot{M}_{\rm in}\geq 1,\,\dot{M}_{\rm in},\,R_{\rm in}$ are free
parameters. This prescription can give a relatively moderate increase of
$\dot{M}$ with the increasing radius. For the case of no wind effect on
$\dot{M}$ (which is then constant), we set $\dot{M}_{\rm out}/\dot{M}_{\rm
in}=1$.

Therefore, we write equations (1–2) as

$\displaystyle\nu F_{\nu}={4\pi h\nu^{4}(1+z)^{4}(GM)^{2}f_{\rm c}\cos i\over
D_{L}^{2}c^{6}}\times$ (10) $\displaystyle\quad\int^{r_{\rm out}}_{r_{\rm
in}}\\!\\!{r{\rm d}r\over f_{\rm col}\left[T_{\rm
eff}(r)\right]^{4}\left\\{\exp\left[h\nu(1+z)/kT(r)\right]-1\right\\}},$
$\displaystyle T_{\rm eff}(r)=\left[3f_{\rm c}c^{6}\dot{M}(r)\over 8\pi\sigma
G^{2}M^{2}r^{3}\right]^{1/4}\left[1-\left(r_{\rm b}\over
r\right)^{1/2}\right]^{1/4},$ (11) $\displaystyle T(r)=f_{\rm col}\left[T_{\rm
eff}(r)\right]T_{\rm eff}(r),$ (12)

where we used another radial variable, $r\equiv R/R_{\rm g}$. For the color
correction, we use a formula of Chiang (2002), who fitted the numerical
results given in fig. 13 of of Hubeny et al. (2001),

$f_{\rm col}(T_{\rm eff})=f_{\infty}-{(f_{\infty}-1)\left[1+\exp(-\nu_{\rm
b}/\Delta\nu)\right]\over 1+\exp\left[(\nu_{\rm p}-\nu_{\rm
b})/\Delta\nu\right]},$ (13)

where $\nu_{\rm p}=2.82kT_{\rm eff}/h$, $f_{\infty}=2.3$, $\nu_{\rm
b}=\Delta\nu=5\times 10^{15}$ Hz (in the source frame). This dependence is
shown in Fig. 1. We note it roughly agrees with the general trend shown in
fig. 1 of Hubeny et al. (2001), while a constant value of $f_{\rm col}$
disagrees with the results of Hubeny et al. (2001). On the other hand, an
$f_{\rm col}$ substantially greater than 1 is ruled out for emission at
frequencies $\sim 10^{15}$ Hz in the source frame in the framework of the
standard disk model, since it would predict a strong decrease of the flux, see
equation (10), not seen in the results of Hubeny et al. (2001).

Figure 1: The dependence of $f_{\rm col}$ on the disk temperature from the
formula of Chiang (2002) approximating results of Hubeny et al. (2001).

Figure 2: The disk blackbody spectra shown at the observer frame for $r_{\rm
in}=6$ for the parameters of SDSS 0924+0219 (Morgan et al., 2010). The red
dashed and blue dotted curves correspond to $f_{\rm col}=1$ and $f_{\rm col}$
of Chiang (2002). The black solid curve gives the spectrum corresponding to
$r_{\rm in}=r_{\rm b}=20$, $f_{\rm col}$ of Chiang (2002), and $\dot{M}(R)$
given by equation (9) with $\dot{M}_{\rm out}/\dot{M}_{\rm in}=5$. The
normalization is set by the observed flux at the $I$-band mid frequency
($3.7\times 10^{14}$ Hz).

Figure 3: The radial emissivity profiles, ${\rm d}F_{\nu}{\rm d}\ln R$, at the
source-frame wavelength of 2500 Å for $R_{\rm in}=0$, $f_{\rm c}=1$ at $f_{\rm
col}=1$, dashed red curve, compared to $f_{\rm col}$ of Chiang (2002), magenta
dot-dashed curve. The blue dotted curve shows the latter case case modified
for $R_{\rm in}=R_{\rm b}=20$, while the solid black curve shows this case
with adding variability of $\dot{M}(R)$, with $\dot{M}_{\rm out}/\dot{M}_{\rm
in}=5$. We see that including the variable $\dot{M}(R)$ has a relatively small
effect on the profile. The parameters correspond to those of SDSS 0924+0219
(Morgan et al., 2010), and the overall normalization is arbitrary.

Since the flux can be measured at a different frequency, $\nu_{1}$, than that,
$\nu_{2}$, of the size measurement (in particular from microlensing), we first
solve equations (9–12) at $\nu_{1}$ for $\dot{M}_{\rm in}$ for given $r_{\rm
in}$ and $r_{\rm out}$, by setting $F_{\nu_{1}}$ equal to that observed. Then,
the half-light radius at $\nu_{2}$ is given by $r_{1/2}$, which splits the
integral in equation (10) in two halves (analogously to equation 8). The
formalism based on equations (9–12) yields only one disk size, and thus there
is no more an ambiguity between the ’theory size’ and the ’disk size’.

As an example illustrating the above results, we use the parameters of the
quasar SDSS 0924+0219 ($z=1.524$) as given by Morgan et al. (2010)222We note
that Morgan et al. (2010) gave their estimates of the radii from microlensing,
$R_{\rm S}$ in their notation, as $R_{\nu}$, not $R_{1/2,\nu}$., namely
$M=1.1\times 10^{8}{\rm M}_{\sun}$, $D_{L}\approx 11.1$ Gpc, $\nu_{1}\approx
3.7\times 10^{14}$ Hz (the $I$ band), $F_{\nu_{1}}\approx 7.7\times 10^{-29}$
erg cm-2 s-1 Hz-1, $\nu_{2}\approx 1.2\times 10^{15}/(1+z)$ Hz
[$\lambda_{2}=2500(1+z)$ Å]. The half-light radius at $\nu_{2}$ from
microlensing following from the estimate by Morgan et al. (2010) is
$\approx\\!150R_{\rm g}$.

Fig. 2 shows the effect of using the above color correction on the integrated
spectra. We see the UV emission is greatly enhanced. We also show a case with
a disk truncation and variable $\dot{M}(R)$. At low frequencies, we also see
the effect of the outer disk radius (taken as $10^{3}R_{\rm g}$), which
hardens the spectral slope. Thus, there is not any extended range with
$F_{\nu}\propto\nu^{1/3}$, assumed in the approximations of equations (3) and
(6).

Fig. 3 shows the radial emissivity profiles, ${\rm d}F_{\nu}{\rm d}\ln R=R{\rm
d}F_{\nu}{\rm d}R$ for $R_{\rm in}=R_{\rm b}=0$ (corresponding to the neglect
of the inner boundary condition), $f_{\rm c}=1$ and $f_{\rm col}=1$ compared
to the same case except for $f_{\rm col}$ of Chiang (2002). We see that the
effect of using the color correction is to substantially enhance the relative
contributions of outer disk regions, in spite of the integrated spectra being
relatively similar, Fig. 2. For $f_{\rm col}=1$, $R_{\nu_{2},F}\approx
27R_{\rm g}$, $R_{1/2,\nu_{2}}\approx 65R_{\rm g}$, and $\dot{M}_{\rm
in}\approx 1.9\dot{M}_{\rm Edd}$, where $\dot{M}_{\rm Edd}\equiv L_{\rm
Edd}/c^{2}$ and $L_{\rm Edd}$ is the Eddington luminosity for pure hydrogen
(note that we do not include an efficiency in the defition of $\dot{M}_{\rm
Edd}$). On the other hand, $R_{1/2,\nu_{2}}\approx 101R_{\rm g}$, and
$\dot{M}_{\rm in}\approx 2.8\dot{M}_{\rm Edd}$ in the case of $f_{\rm col}$ of
Chiang (2002). Thus, the effect of the color correction is quite strong. As
found, e.g., by Mosquera et al. (2013), the sizes of the X-ray sources in
quasars estimated by microlensing are $\sim 20R_{\rm g}$. Thus, it is
reasonable to assume $R_{\rm in}=20R_{\rm g}$ as a disk truncation radius.
With the disk truncation and $R_{\rm b}=R_{\rm in}=20$, we find a further
increase of the half-light radius, $R_{1/2,\nu_{2}}\approx 123R_{\rm g}$, and
$\dot{M}_{\rm in}\approx 4.5\dot{M}_{\rm Edd}$. Then, taking into account a
variability of $\dot{M}$ with $\dot{M}_{\rm out}/\dot{M}_{\rm in}=5$ due to
disk winds increases $R_{1/2,\nu_{2}}$ to $\approx 141R_{\rm g}$
($\dot{M}_{\rm in}\approx 1.3\dot{M}_{\rm Edd}$). It can be increased to
$\approx 150R_{\rm g}$ only for $\dot{M}_{\rm out}/\dot{M}_{\rm in}=20$
($\dot{M}_{\rm in}\approx 0.37\dot{M}_{\rm Edd}$).

On the other hand, Dai et al. (2010) made a comparison of the disk sizes at a
given frequency, both $R_{\nu}$ and $R_{1/2,\nu}$ between the standard disk
model (Shakura & Sunyaev, 1973) and the model of Hubeny et al. (2001), either
relativistic and non-relativistic, and either assuming the local blackbody and
with the NLTE disk atmosphere. They found that the model of Hubeny et al.
(2001) gives generally smaller radii at an assumed value of $\dot{M}$ than
equation (3). However, it appears that Dai et al. (2010) defined the $R_{\nu}$
by $h\nu=kT_{\rm eff}$, without including the color correction. In general,
the blackbody emission is the most efficient radiation process possible (in
the absence of coherent processes), and any departure from the full
thermodynamical equilibrium has to lead to an increase of the emitting area.

##  3 Conclusions

We have studied the effects of including a color correction factor (based on
the results of Hubeny et al. 2001), a disk truncation at an inner radius
$>$ISCO, and of a decrease of $\dot{M}$ with the decreasing disk radius due to
winds on the half-light radii of standard accretion disks. In the case of the
color correction, we have found it is important to estimate the size using
half-light radii rather than the radii based on monochromatic approximations,
equations (3) and (6). The cause of this is that while $f_{\rm col}\approx 1$
at $R_{\nu}$, it increases fast at lower radii, which, in turn, moves the
emissivity radial profile to larger radii, which can be seen by comparing the
dashed and dot-dashed curves in Fig. 2. In the considered example, it
increases the half-light radius from $65R_{\rm g}$ to about $100R_{\rm g}$.

Then, including the effect of an inner truncation radius of the disk being
above the ISCO (compatible with the presence of X-ray emission) leads to a
further increase. In the considered example, a truncation at a radius of
$20R_{\rm g}$ together with the inclusion of the color correction increases
the half-radius twice, to about $120R_{\rm g}$. Thus, these effects are
important, and should be included in interpreting the microlensing results.

Furthermore, we have considered the effect of disk winds, which can cause the
local accretion rate to decrease with the decreasing radius. However, we have
found that this assumption has a relatively minor effect on the half-light
radius. Assuming a large $\dot{M}_{\rm out}/\dot{M}_{\rm in}=5$, we find the
half-light radius to increase to about $140R_{\rm g}$. This is similar to the
half-light radius from microlensing, of $\approx\\!150R_{\rm g}$ in the
considered example of SDSS 0924+0219 (Morgan et al., 2010).

The formalism used in this paper gives also the mass accretion rate of a given
model, removing the ambiguity between the ’theory’ and ’flux’ radii. For the
model with the disk truncation and a constant accretion rate, the implied
value is $\dot{M}_{\rm in}\approx 4.5\dot{M}_{\rm Edd}$. For an assumed
accretion efficiency of 0.1, this corresponds to about a half of the Eddington
luminosity, in agreement with the estimates of the Eddington ratios of quasars
of Kollmeier et al. (2006).

Finally, we note that the above results were derived assuming the standard
optically-thick accretion disk, i.e., one in which the energy dissipation
occurs close to the disk midplane. As we mentioned in Section 1, accretion
disks supported by magnetic pressure are substantially hotter than the
standard ones, and thus have substantially larger characteristic sizes at a
given emitted frequency (e.g., Begelman & Pringle 2007; Begelman & Silk 2017),
which provides an alternative solution to the problem of the accretion disk
sizes.

## Acknowledgments

We thank Eric Agol, Christopher Kochanek, Christopher Morgan, and Feng Yuan
for valuable discussions. We acknowledge support from the Polish National
Science Center under the grant 2019/35/B/ST9/03944, and from the Natural
Science Foundation of China (U1838103, U1931203 and 11903024). Our work also
benefitted from discussions during Team Meetings in the International Space
Science Institute (Bern).

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