EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-306 LHCb-PAPER-2015-048 26 January 2016

Search for the lepton-flavour violating decay $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$

The LHCb collaboration†††Authors are listed at the end of this paper.

A search for the lepton-flavour violating decay $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ is made with a dataset corresponding to an integrated
luminosity of 3.0 $\mbox{\,fb}^{-1}$ of proton-proton collisions at centre-of-
mass energies of $7$ $\mathrm{\,Te\kern-1.00006ptV}$ and $8$
$\mathrm{\,Te\kern-1.00006ptV}$, collected by the LHCb experiment. Candidate
${{D}^{0}}$ mesons are selected using the decay
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ and the $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ branching fraction is measured using the decay mode
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ as a normalisation channel. No
significant excess of $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ candidates over the
expected background is seen, and a limit is set on the branching fraction,
${\cal B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})<1.3\times 10^{-8}$, at 90%
confidence level. This is an order of magnitude lower than the previous limit
and it further constrains the parameter space in some leptoquark models and in
supersymmetric models with R-parity violation.

Published in Phys. Lett. B754 (2016) 167

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

Searches for decays that are forbidden in the Standard Model (SM) probe
potential contributions from new processes and particles at mass scales beyond
the reach of direct searches. The decay $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ is
an example of a forbidden decay, in which lepton flavour is not
conserved.111The inclusion of charge-conjugate processes is implied. The
contributions to this process from neutrino oscillations would give a rate
that is well below the reach of any currently feasible experiment. However,
the decay is predicted to occur in several other models that extend the SM,
with rates varying by up to eight orders of magnitude.

In Ref. [1] three extensions to the SM are considered: in a minimal
supersymmetric (SUSY) SM with R-parity violation (RPV) the branching fraction
${\cal B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})$ could be as large as
$\cal{O}$$(10^{-6})$; in a theory with multiple Higgs doublets it would be
less than about $7\times 10^{-10}$; and in the SM extended with extra fermions
the branching fraction would be less than $\cal{O}$$(10^{-14})$. In Ref. [2]
an RPV SUSY model is considered in which limits on products of couplings are
obtained from the experimental upper limit on the branching fraction ${\cal
B}({{D}^{+}_{s}}\rightarrow{{K}^{+}}{e^{\pm}}{\mu^{\mp}})$; from these limits,
${\cal B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})$ could be as large as $3\times
10^{-8}$. A similar study of constraints on coupling constants in RPV SUSY
[3], obtained from limits on the branching fraction ${\cal
B}({{D}^{+}}\rightarrow{{\pi}^{+}}{e^{\pm}}{\mu^{\mp}})$, showed that ${\cal
B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})$ could reach $10^{-7}$. LHCb has
previously set limits [4] on branching fractions for the $B$ meson decays
$B^{0}\rightarrow e^{\pm}\mu^{\mp}$ and $B_{s}^{0}\rightarrow
e^{\pm}\mu^{\mp}$, using them to put lower limits on the masses of Pati-Salam
leptoquarks [5]. As is shown in Ref. [6], lepton-flavour violating charm
decays are relatively insensitive to the presence of such leptoquarks.
However, in a recent paper [7] it is shown that in other leptoquark scenarios
${\cal B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})$ could be as large as $4\times
10^{-8}$.

The first experimental limit on ${\cal B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})$
was from Mark II [8], and more recent results have come from E791 [9] and
BaBar [10]. The most stringent limit is from Belle [11], ${\cal
B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})<2.6\times{}10^{-7}$ at $90\%$ confidence
level (CL). An improved limit, below $\cal{O}$$(10^{-7})$, would provide
tighter constraints on coupling constants in RPV SUSY models [1, 2, 3], while
a limit below $4\times 10^{-8}$ would also constrain the parameter space in
some leptoquark models [7].

This Letter presents a search for the decay $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ using $pp$ collision data corresponding to integrated
luminosities of 1.0 $\mbox{\,fb}^{-1}$ at a centre-of-mass energy of $7$
$\mathrm{\,Te\kern-1.00006ptV}$ and 2.0 $\mbox{\,fb}^{-1}$ at $8$
$\mathrm{\,Te\kern-1.00006ptV}$, collected by the LHCb experiment in 2011 and
2012, respectively. In the analysis, signal candidates are selected using the
decay ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ and the measurements are
normalized using the well-measured channel
$D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$, which has the same topology as the
signal. A multivariate analysis based on a boosted decision tree algorithm
(BDT) is used to help separate signal and background. The mass spectrum in the
signal region, defined as $1815-1915$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, is not examined until all analysis
choices are finalized.

##  2 Detector and simulation

The LHCb detector [12, 13] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream of the
magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with a relative uncertainty that varies from $0.5\%$ at low
momentum to $1.0\%$ at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum
distance of a track to a primary vertex (PV), the impact parameter, is
measured with a resolution of $(15+29/\mbox{$p_{\rm T}$}){\,\upmu\rm m}$,
where $p_{\rm T}$ is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.

The online event selection is performed by a trigger [14], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage in which all charged particles with
$\mbox{$p_{\rm T}$}>500\,(300){\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ are
reconstructed for 2011 (2012) data. At the hardware trigger stage, events are
required to have a muon with high $p_{\rm T}$, or a hadron, photon or electron
with high transverse energy in the calorimeters. The software trigger requires
a two-, three- or four-track secondary vertex with a significant displacement
from the primary $pp$ interaction vertices. At least one charged particle must
have a transverse momentum $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be inconsistent with
originating from a PV. A multivariate algorithm [15] is used for the
identification of secondary vertices consistent with the decay of a $b$ or $c$
hadron.

In the simulation, $pp$ collisions are generated using Pythia [16,
*Sjostrand:2006za] with a specific LHCb configuration [18]. Decays of hadronic
particles are described by EvtGen [19], in which final-state radiation is
generated using Photos [20]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [21,
*Agostinelli:2002hh] as described in Ref. [23]. Samples of simulated events
are generated for the signal $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ channel, for
the normalization $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ channel and for
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$, which is an important background
channel.

Figure 1: Mass spectra from simulation for $D^{0}\rightarrow e^{\pm}\mu^{\mp}$
decays (solid line) and $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays
reconstructed as $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ (dashed line). Each
spectrum is normalized to unit area. The vertical line indicates the mass of
the ${D}^{0}$ meson.

##  3 Event selection and efficiencies

In the first stage of the offline event selection, the
${{D}^{*+}}\rightarrow{{D}^{0}}({e^{\pm}}{\mu^{\mp}}){{\pi}^{+}}$ and
${{D}^{*+}}\rightarrow{{D}^{0}}({{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$ candidates
that pass the trigger selection are required to have a vertex, formed from two
good-quality tracks associated with particles of opposite charge, that is well
separated from any PV, with the summed momentum vector of the two particles
pointing to a PV (the mean number of PVs per beam crossing is $1.6$). The
measured momentum of the electron candidates is corrected to account for loss
of momentum by bremsstrahlung in the detector, using the photon energy
deposition in the electromagnetic calorimeter [24]. Muon and electron
candidates, and pions and kaons from the
$D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ candidates, are required to have $p>4$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $p_{T}>0.75$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and to be positively identified by the
particle identification systems. The soft pion from the candidate
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ decay is required to have
$p_{T}>110$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and to be consistent with
coming from the PV. A kinematic fit is performed, with the two ${D}^{0}$ decay
tracks constrained to a secondary vertex and the soft pion and ${D}^{0}$
candidates constrained to come from the PV. This fit improves the resolution
on the mass difference between the reconstructed ${D}^{*+}$ and ${D}^{0}$
mesons, which is required to be in the range $135-155$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. About $2\%$ of events contain more
than one ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ candidate and in these
events one is chosen at random. After the above selections, $2114$ candidates
remain in the signal mass region for $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ and
$330\,359$ for $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ (the trigger accept rate
for the latter channel is scaled to retain only $1\%$ of candidates).

An important source of background in the sample of $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ candidates comes from
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays where one pion is
misidentified as an electron and the other as a muon. From simulations and
calibration samples in the data [13], the probability for a
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ event to be selected in the final
sample of candidate signal events is found to be $(1.0\pm 0.6)\times 10^{-8}$
in the $7$ $\mathrm{\,Te\kern-1.00006ptV}$ data and $(1.8\pm 0.4)\times
10^{-8}$ in the $8$ $\mathrm{\,Te\kern-1.00006ptV}$ data. Figure 1 shows a
comparison of the mass spectra, from simulation, for $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ decays and for $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
decays reconstructed as $D^{0}\rightarrow e^{\pm}\mu^{\mp}$, with each
spectrum normalized to unit area. The low-mass tail for genuine
$D^{0}\rightarrow e^{\pm}\mu^{\mp}$ decays is caused by bremsstrahlung from
the electrons; about $15\%$ of the signal lies below
$1810\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The misidentified
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays produce a peak at a mass about
$15$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ below the signal mass.
Misidentified $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays always have
reconstructed mass below the region selected for the analysis, because of the
large mass difference between kaons and electrons or muons; as a consequence,
there is no background from this source. Other sources of background include
the semileptonic decay modes
${{D}^{0}}\rightarrow{{\pi}^{-}}{e^{+}}{{\nu}_{e}}$ and
${{D}^{0}}\rightarrow{{\pi}^{-}}{\mu^{+}}{{\nu}_{\mu}}$, with the pion
misidentified as a muon or an electron, respectively. Since, as part of
bremsstrahlung recovery, the energy of unrelated photons may be incorrectly
added to the energy of the electron candidates, these semileptonic backgrounds
extend smoothly above the signal region and are treated as part of the
combinatorial background of $e^{\pm}\mu^{\mp}$ pairs where the two lepton
candidates have different sources.

Trigger, selection and particle identification efficiencies, and
misidentification probabilities, are obtained from a combination of simulation
and data. Control samples of well-identified electrons, muons, pions and kaons
in data are obtained from ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson
decays into pairs of electrons or muons and from
${{D}^{*+}}\rightarrow{{D}^{0}}({{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$ decays,
selected using different requirements from those used in the current analysis.
These control samples are binned in pseudorapidity and transverse momentum of
the tracks, and in the track multiplicity of the event. The hardware trigger
efficiency for signal is evaluated using data, while the efficiency for the
software trigger and offline selections is evaluated using simulation after
validation with the data control samples. Where efficiencies are taken from
the simulation, the samples are weighted to take into account differences
between simulation and data, particularly in the distribution of per-event
track multiplicities.

##  4 Multivariate classifier

A multivariate classifier based on a BDT [25] with a gradient boost [26] is
used to divide the selected sample into bins of different signal purity. The
following variables are used as inputs to the BDT: the smallest distance of
closest approach of the ${D}^{0}$ candidate to any PV; an isolation variable
that depends on how much additional charged particle momentum is in a region
of radius $R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}=1$ around the
${D}^{*+}$ candidate, where $\eta$ and $\phi$ are pseudorapidity and azimuthal
angle; $\chi^{2}$ of the kinematic fit; and $\chi^{2}_{\rm IP}$, the impact
parameter $\chi^{2}$ with respect to the associated PV, for each of the
${D}^{*+}$ and ${D}^{0}$ candidates, and for the two ${D}^{0}$ decay tracks.
The variable $\chi^{2}_{\rm IP}$ is defined as the difference in vertex fit
$\chi^{2}$ with and without the particle considered. None of the BDT input
variables contains particle identification information. It therefore performs
equally well for the signal and normalization channels (and for the
misidentified $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays).

The BDT is trained separately for the $7$ $\mathrm{\,Te\kern-1.00006ptV}$ and
$8$ $\mathrm{\,Te\kern-1.00006ptV}$ data samples, to exploit the dependence of
some input variables, for example the isolation variable, on the collision
energy. The background sample used for the training comprises selected
candidates with invariant mass within $300$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known ${D}^{0}$ mass, but
excluding the signal region, $1815-1915$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The training for signal is done
with the simulated $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ events. One half of
each sample is used for training the BDT, while the other half is used to test
for over-training. No evidence for over-training is seen. Following procedures
used in Refs. [27, 28], the BDT output value, which lies between $-1$ (most
background-like) and $1$ (most signal-like), is used to separate the data
sample into three sub-samples with ranges chosen to give optimum separation
between the background-only and signal-plus-background hypotheses.

##  5 Fits to mass spectra

In order to determine the number of signal decays, extended maximum likelihood
fits are made simultaneously to unbinned distributions of $m({{D}^{0}})$ and
$\Delta m=m({{D}^{*+}})-m({{D}^{0}})$ for the $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ candidates in each of the three BDT bins for the $7$
$\mathrm{\,Te\kern-1.00006ptV}$ and $8$ $\mathrm{\,Te\kern-1.00006ptV}$ data.
Hereinafter, $m({{D}^{0}})$ denotes the mass of the ${D}^{0}$ candidate for
both signal and normalization channels, and $\Delta m$ denotes the mass
difference between the ${D}^{*+}$ and ${D}^{0}$ candidates. In these fits,
from which the branching fraction is extracted directly, all systematic
uncertainties, as discussed in Sect. 6, are included as Gaussian constraints
on the appropriate parameters.

The $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ signal probability density functions
(PDF) in the three BDT bins are obtained from the simulation. The simulated
$D^{0}\rightarrow e^{\pm}\mu^{\mp}$ mass spectra are fitted using the sum of
two Crystal Ball functions [29] with a common peak value but different widths.
One of the Crystal Ball functions has a low-mass tail to account for energy
loss due to bremsstrahlung while the other is modified to have a high-mass
tail to accommodate events where a bremsstrahlung photon is incorrectly
assigned to an electron candidate. The per-event particle multiplicity affects
the amount of bremsstrahlung radiation recovered for the electron candidates,
and this differs between simulation and data. Therefore both the simulation
and the data are classified in three bins of the variable $N_{\rm SPD}$, the
number of hits in the scintillating pad detector, which is a measure of the
particle multiplicity. The parameters of the signal PDF are obtained as
averages of their values in the three bins of $N_{\rm SPD}$, weighted to
account for data-simulation differences. The PDF shapes for the peaking
background due to misidentified $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
decays (see Fig. 1) are obtained in the same way as for $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$, using the same functional form for the signal shapes, and
their yields are Gaussian-constrained in the fits. The combinatorial
background for the ${D}^{0}$ candidate mass is described by a second-order
polynomial.

The signal shapes in the $\Delta m$ distributions for the $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ and $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ channels are
each parametrised as a sum of three Gaussian functions; for $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ two of the Gaussians functions have the same mean, but the
one with the largest width is allowed to have a different mean, while the
three mean values are independent for the
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ shape. In each case all three
Gaussian functions have independent widths. The combinatorial background in
$\Delta m$ is fitted using an empirical function of the form

$f(\Delta m)=N\left[\left(1-\exp\left(-\frac{\Delta m-{(\Delta
m)}_{0}}{c}\right)\right)\times{}\left(\frac{\Delta m}{{(\Delta
m)}_{0}}\right)^{a}+b\left(\frac{\Delta m}{{(\Delta
m)}_{0}}-1\right)\right]~{},$ (1)

where $N$ is a normalization factor, ${(\Delta m)}_{0}$ is the threshold mass
difference, and $a$, $b$ and $c$ are free parameters. In the fits to the
$D^{0}\rightarrow e^{\pm}\mu^{\mp}$ candidates, the parameter $a$ is fixed to
zero. A fraction of the $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ and the
misidentified $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays is associated to
a random soft pion, and therefore peaks in $m({{D}^{0}})$ but not in $\Delta
m$. This fraction is Gaussian constrained to the value $23.7\pm 0.2\%$ found
in the fits to the $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ normalisation
channel, discussed below.

Figure 2 shows the fit results for the combined $7$
$\mathrm{\,Te\kern-1.00006ptV}$ and $8$ $\mathrm{\,Te\kern-1.00006ptV}$
dataset, separately for the three bins of BDT output. The peaks seen in the
$m({{D}^{0}})$ and $\Delta m$ distributions are due to misidentified
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays. No evidence is seen for any
$D^{0}\rightarrow e^{\pm}\mu^{\mp}$ signal. The fits return a total of $-7\pm
15$ signal decays.

Figure 2: Distributions of (left) $m({{D}^{0}})$ and (right) $\Delta m$ for
$D^{0}\rightarrow e^{\pm}\mu^{\mp}$ candidates reconstructed in the combined
$7$ $\mathrm{\,Te\kern-0.90005ptV}$ and $8$ $\mathrm{\,Te\kern-0.90005ptV}$
data, with fit functions overlaid. The rows correspond to the three bins of
BDT output, with the top row corresponding to the most background-like and the
bottom row to the most signal-like. The solid (blue) lines show the total fit
results, while the thick (grey) lines show the total $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ component, the thin (purple) lines show the total
misidentified $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and the dashed (grey)
lines indicate the combinatorial background.

For the normalisation channel $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$, for
which there are many candidates, binned fits are done separately to the $7$
$\mathrm{\,Te\kern-1.00006ptV}$ and $8$ $\mathrm{\,Te\kern-1.00006ptV}$
samples, using a sum of two Gaussian functions with a common mean to model the
${D}^{0}$ candidate mass distribution, and a sum of three Gaussian functions
for the $\Delta m$ distribution. In the latter case, two of the Gaussian
functions have the same mean, but the one with the largest width is allowed to
have a different mean. The function defined by Eq. (1) is used for the
background in the $\Delta m$ spectrum, with all parameters allowed to vary in
the fit. Figure 3 shows the results of the fit for the
$D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ normalization samples in the $8$
$\mathrm{\,Te\kern-1.00006ptV}$ data, for both the $m({{D}^{0}})$ and $\Delta
m$ distributions. Totals of $80\times 10^{3}$ and $182\times 10^{3}$
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
decays are observed in the $7$ $\mathrm{\,Te\kern-1.00006ptV}$ and $8$
$\mathrm{\,Te\kern-1.00006ptV}$ data, respectively.

Figure 3: Distributions of (left) $m({{D}^{0}})$ and (right) $\Delta m$ for
$K^{-}\pi^{+}$ candidates for the $8$ $\mathrm{\,Te\kern-0.90005ptV}$ data.
The dark (blue) line shows the overall fit, the lighter grey line shows the
signal, and the dot-dash line shows genuine ${D}^{0}$ events where the soft
pion does not come from a ${D}^{*+}$ decay. The combinatorial background is
too small to be visible.

##  6 Systematic uncertainties

The uncertainty on the fitted $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ signal rate
is dominated by statistical fluctuations of the combinatorial background.
Sources of systematic uncertainty that could affect the final result include
those on the yield of the normalization $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$
decay, uncertainties in the shapes of the PDFs used for $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ and $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$, and
uncertainties in the selection efficiencies and particle misidentification
probabilities. All of these uncertainties are included as Gaussian constraints
in the fits described in Sect. 5.

In the nominal fit to signal candidates, the parameters of the signal PDF,
obtained from the simulation, are Gaussian constrained according to their
uncertainties. To obtain these uncertainties, samples of
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{e^{+}}{e^{-}}$ are selected in both simulation and data,
and the ${e^{+}}{e^{-}}$ mass spectra are fitted using the same functional
form as used for $D^{0}\rightarrow e^{\pm}\mu^{\mp}$. The fractional
differences in the parameter values between the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{e^{+}}{e^{-}}$ fits
to the data and to the simulation are taken as the fractional systematic
uncertainties on the corresponding parameters of the PDF for the
$D^{0}\rightarrow e^{\pm}\mu^{\mp}$ candidate mass spectra.

For the fits to the fully simulated, misidentified
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ mass spectra, some selection
requirements are removed in order to have enough events to obtain reliable
fits. The efficiency of the selection requirements that are not applied varies
linearly by a relative $9.4\%$ with reconstructed mass across the fit region.
The PDF for the peak shape in the misidentified
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays is corrected for this
variation of efficiency, and the resulting contribution to the systematic
uncertainty on the yield is taken as $4.7\%$.

To allow for uncertainties in the fractions of $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ signal and misidentified
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays that are estimated in the
three bins of BDT output, a comparison is made between these fractions for
simulated $D^{0}\rightarrow e^{\pm}\mu^{\mp}$, simulated
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and well identified
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays in the data. Since the BDT
does not take into account particle identification, the largest differences
between these fractions in each bin, typically $2.5\%$, are taken as the
systematic uncertainties on the fractions in the data.

To account for differences between data and simulation in the per-event track
multiplicity, the reconstruction efficiencies and misidentification
probabilities for simulated events are evaluated in three bins of $N_{\rm
SPD}$. These are then weighted to match the multiplicity distribution in the
data. Half of the differences between the unweighted and the weighted
efficiencies and misidentification probabilities, typically $5\%$, are taken
as the systematic uncertainties on these quantities. Further uncertainties, of
$2.5\%$ for each of $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ and
$D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$, are included to account for limited
knowledge of the tracking efficiencies.

Using the calibration samples, particle identification and trigger
efficiencies are estimated in bins of pseudorapidity, transverse momentum and
event multiplicity. Overall efficiencies are determined by scaling the
simulation so that the distributions in these variables match the data. To
estimate systematic uncertainties from this procedure, different binning
schemes are used and the resulting changes in the efficiency values are
treated as systematic uncertainties. Overall systematic uncertainties are
$6\%$ on the $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ selection efficiency and
$30\%$ on the $D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ misidentification
probability.

To study systematic effects in the fit to the normalization channel, the order
of the background polynomial is increased, the number of bins changed, fixed
parameters are varied and the Gaussian mean values in the $\Delta m$ fits are
constrained to be equal. From these studies a contribution of $1\%$ is
assigned to the systematic uncertainty on the yield. Similar procedures as
described above for the signal channel are also used to evaluate the other
systematic uncertainties for the $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$
normalization channel. The resulting overall systematic uncertainty in the
measured number of $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays is $5\%$.

##  7 Results and conclusions

The measured branching fraction for the signal channel is given by

${\cal B}(D^{0}\rightarrow
e^{\pm}\mu^{\mp})={\frac{N_{e\mu}/\epsilon_{e\mu}}{{N_{K\pi}/\epsilon_{K\pi}}}}\times{\cal
B}(D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}),$ (2)

where $N_{e\mu}$ and $N_{K\pi}$ are the fitted numbers of $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ and $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays, the
corresponding $\epsilon$ are the overall efficiencies, and the branching
fraction for the normalization channel, ${{\cal
B}(D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}})}=(3.88\pm 0.05)\%$, is taken from
Ref. [30]. The efficiencies $\epsilon_{e\mu}=(4.4\pm 0.3)\times 10^{-4}$ and
$\epsilon_{K\pi}=(2.5\pm 0.1)\times 10^{-6}$, for the signal and normalization
channels, are the products of the reconstruction efficiencies for the final-
state particles, including the geometric detector acceptance, the selection
efficiencies, and the trigger efficiencies (including the $1\%$ scaling in the
trigger for the $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ channel).

No evidence is seen for a $D^{0}\rightarrow e^{\pm}\mu^{\mp}$ signal in the
overall mass spectrum, nor in any individual bin of BDT output, and the
measured branching fraction is ${\cal B}(D^{0}\rightarrow
e^{\pm}\mu^{\mp})=(-0.6\pm 1.2)\times 10^{-8}$, where the uncertainty accounts
for both statistical and systematic effects. An upper limit on the branching
fraction is obtained using the CL${}_{\text{S}}$ method [31], where the
p-value for the signal-plus-background hypothesis is compared to that for the
background-only hypothesis. The expected and observed CL${}_{\text{S}}$ values
as functions of the assumed branching fraction are shown in Fig. 4, where the
expected CL${}_{\text{S}}$ values are obtained using an Asimov dataset [32] as
described in Ref. [33], and are the median expected limits under the
assumption of no signal. Expected limits based on pseudoexperiments give
consistent results. There is excellent correspondence between the expected and
observed CL${}_{\text{S}}$ values, and an upper limit is set on the branching
fraction, ${\cal B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})$ $<1.3\times 10^{-8}$
at $90\%$ CL (and $<1.6\times 10^{-8}$ at $95\%$ CL). This limit will help to
further constrain products of couplings in supersymmetric models that
incorporate R-parity violation [1, 2, 3] and constrains the parameter space in
some leptoquark scenarios [7].

In summary, a search for the lepton-flavour violating decay $D^{0}\rightarrow
e^{\pm}\mu^{\mp}$ is performed on a data sample corresponding to an integrated
luminosity of 3.0$\mbox{\,fb}^{-1}$ collected in $pp$ collisions at centre-of-
mass energies of $7$ and $8$ TeV. The data are consistent with the background-
only hypothesis, and a limit is set on the branching fraction, ${\cal
B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})$ $<1.3\times 10^{-8}$ at $90\%$ CL,
which is an order of magnitude lower than the previous limit.

Figure 4: Distribution of CL${}_{\text{S}}$ as a function of ${\cal
B}(D^{0}\rightarrow e^{\pm}\mu^{\mp})$. The expected distribution is shown by
the dashed line, with the $\pm 1\sigma$ and $\pm 2\sigma$ regions shaded. The
observed distribution is shown by the solid line connecting the data points.
The horizontal line indicates the $90\%$ confidence level.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI (Russia),
CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC
(USA). We are indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia). Individual groups or members have received support from AvH
Foundation (Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC
(European Union), Conseil Général de Haute-Savoie, Labex ENIGMASS and OCEVU,
Région Auvergne (France), RFBR (Russia), GVA, XuntaGal and GENCAT (Spain), The
Royal Society and Royal Commission for the Exhibition of 1851 (United
Kingdom).

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Wraight52, S. Wright48, K. Wyllie39, Y. Xie62, Z. Xu40, Z. Yang3, J. Yu62, X.
Yuan35, O. Yushchenko36, M. Zangoli15, M. Zavertyaev11,c, L. Zhang3, Y.
Zhang3, A. Zhelezov12, A. Zhokhov32, L. Zhong3, V. Zhukov9, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
36Institute for High Energy Physics (IHEP), Protvino, Russia
37Universitat de Barcelona, Barcelona, Spain
38Universidad de Santiago de Compostela, Santiago de Compostela, Spain
39European Organization for Nuclear Research (CERN), Geneva, Switzerland
40Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
41Physik-Institut, Universität Zürich, Zürich, Switzerland
42Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
43Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
44NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
45Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
46University of Birmingham, Birmingham, United Kingdom
47H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
48Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
49Department of Physics, University of Warwick, Coventry, United Kingdom
50STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
51School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
52School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
53Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
54Imperial College London, London, United Kingdom
55School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
56Department of Physics, University of Oxford, Oxford, United Kingdom
57Massachusetts Institute of Technology, Cambridge, MA, United States
58University of Cincinnati, Cincinnati, OH, United States
59University of Maryland, College Park, MD, United States
60Syracuse University, Syracuse, NY, United States
61Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
62Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
63Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
64Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
65National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
66Yandex School of Data Analysis, Moscow, Russia, associated to 32
67Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
68Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 42

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
†Deceased