# Ultra Localized Optoelectronic Properties of Nanobubbles in 2D
Semiconductors

Sara Shabani  Department of Physics, Columbia University, New York, NY, USA
Thomas P. Darlington  Department of Mechanical Engineering, Columbia
University, New York, NY, USA  Colin Gordon  Department of Physics and
Astronomy, Stony Brook University, Stony Brook, NY, USA  Wenjing Wu
Department of Chemistry, Columbia University, New York, NY, USA  Emanuil Yanev
Department of Mechanical Engineering, Columbia University, New York, NY, USA
James Hone  Department of Mechanical Engineering, Columbia University, New
York, NY, USA  Xiaoyang Zhu  Department of Chemistry, Columbia University, New
York, NY, USA  Cyrus E. Dreyer<EMAIL_ADDRESS>Center for
Computational Quantum Physics, Flatiron Institute, New York, NY, USA
Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY,
USA  P. James Schuck<EMAIL_ADDRESS>Department of Mechanical
Engineering, Columbia University, New York, NY, USA  Abhay N. Pasupathy
<EMAIL_ADDRESS>Department of Physics, Columbia University, New York, NY,
USA

###### Abstract

The optical properties of transition metal dichalcogenides have previously
been modified at the nanoscale by using mechanical and electrical
nanostructuring. However, a clear experimental picture relating the local
electronic structure with emission properties in such structures has so far
been lacking. Here, we use a combination of scanning tunneling microscopy
(STM) and near-field photoluminescence (nano-PL) to probe the electronic and
optical properties of single nano-bubbles in bilayer heterostructures of WSe2
on MoSe2. We show from tunneling spectroscopy that there are electronic states
deeply localized in the gap at the edge of such bubbles, which are independent
of the presence of chemical defects in the layers. We also show a significant
change in the local bandgap on the bubble, with a continuous evolution to the
edge of the bubble over a length scale of $\sim$20 nm. Nano-PL measurements
observe a continuous redshift of the interlayer exciton on entering the
bubble, in agreement with the band to band transitions measured by STM. We use
self-consistent Schrödinger-Poisson (SP) simulations to capture the essence of
the experimental results, and find that strong doping in the bubble region is
a key ingredient to achieving the observed localized states, together with
mechanical strain.

##  1 Introduction

Two-dimensional (2d) transition metal dichalcogenides (TMDs) have been
investigated as a possible route toward producing well-controlled optical
emitters, critical components in quantum technologies and photonics 1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12. The bandgap of monolayer TMDs is uniquely
sensitive to doping13, 14, 15, 16, 17, dielectric environment18 and mechanical
strain19, 20, 8, 11, 21, thereby providing an ideal platform to create quantum
confinement by local mechanical and electrical fields. Indeed, previous
research has investigated moire patterns 22, 23, 24, 25, 26, wrinkles 4, 2, 8,
bubbles 1, 27 and lithographically fabricated dielectric arrays as effective
methods 28 to create localized optical emitters. In spite of these notable
achievements, a basic band diagram picture across such localized structures
remains elusive. For example, recent theoretical calculations29, 28, 30 have
predicted large bandgap changes at the edges of wrinkles and nanobubbles.
However, such predictions have not been experimentally verified, and it
remains unclear what the major contributors are to achieving localized optical
emitters in TMD materials.

In order to address these questions, high spatial resolution spectroscopic
measurements across local perturbations in TMDs are necessary. Our approach is
to use the local imaging and spectroscopic capabilities of scanning tunneling
microscopy(STM) to measure the local band diagram across nano-bubbles, and
directly correlate these measurements with near-field optical measurements
with 10 nm spatial resolution.

##  2 Results and Discussion

Our STM measurements are conducted on nearly aligned (61.7 ∘) WSe2/MoSe2
heterobilayers that were stacked on Graphite/hBN as a conducting electrode for
STM measurements (see Fig. 1a for a schematic). The relative orientation
between two TMD monolayers was determined by second harmonic generation (SHG)
(see supporting information figure S1 and accompanying discussion). The
details of sample fabrication are presented in methods. During the stacking
process, nanobubbles are commonly formed between layers. Figure 1b shows a
large scale atomic force microscopic (AFM ) image of such nanobubbles. The
nanobubbles range in size from 20 nm to 400 nm and in height from 2 nm to 50
nm. STM topographic measurements on one such nanobubble are shown in Fig. 1c.
Figure 1d shows height profile of the nanobubble boundary corresponding to the
dashed arrow in Fig. 1c. In order to obtain the STM topography, the tunneling
current remains constant at a particular voltage bias, through a feedback
loop; however, the tunneling current depends on the integrated local density
of states between the sample Fermi level and the bias voltage. This dependency
of current on electronic structure affects the topographic data. Thus, both
electronic structure and actual topographic height influence the apparent
topographic height in STM and the apparent heights are not directly
translatable into real vertical displacements. This consideration is not
relevant to AFM measurements; however, AFM measurements are dictated by tip-
sample force interactions which can strongly influence the apparent height in
AFM topographic scans. However, it is clearly evident from the line profile
across the boundary shown in Fig. 1d that the edge of the bubble features a
sharp step. Such a sharp step indicates that there is localized strain at the
edge of the bubble. Finally, a moiré pattern is clearly visible both on and
off the bubble. Its presence in all regions shows that the WSe2 and MoSe2
layers are in good contact throughout, and the bubble is under the
heterobilayer. As a result of the device fabrication process under ambient
conditions, compounds in air such as water molecules often are trapped between
layers. While the chemical identity of the material in the bubble is unknown,
our data below, which shows a highly doped region on the nanobubble, indicate
the presence of polarized (or ionized) molecules trapped in the nanobubble
during the fabrication. Such polarized molecules induce doping and electric
field to the layers above and modify the electronic structure.

To characterize the electronic properties in the vicinity of the nanobubble,
we perform STM $dI/dV$ spectroscopy measurements at various points on and off
the bubble (see Fig. 2a). Shown in Fig. 2b and c are a sequence of such
measurements taken across the edge of the bubble, color coded according to the
markers on Fig. 2a. Figure 2b shows the typical spectra well inside (blue) and
outside (red) the nanobubble area. It is seen that the conduction band (CB)
edge shows a substantial shift from the outer edge of the bubble into the
interior while VB is nearly unchanged. This indicates a substantial reduction
in the bandgap on the bubble as well a large electron doping.

Shown in Fig. 2c are a sequence of spectra taken in the transition region
across the bubble edge, with a focus on the CB edge where the most dramatic
changes are observed. It is seen that at the edge of the bubble (red curve),
electronic states exist that are deeply bound in the semiconductor gap. As we
transition from outside to inside the bubble (green curve), these states move
towards the CB edge, while at the same time the CB edge moves towards the
Fermi level. The entire evolution is shown as a heat map of $dI/dV$ in Fig. 2d
as a function of position along the arrow overlaid on Fig. 2a. The dashed
lines overlaid on Fig. 2d indicate the evolution of the localized states and
the conduction band edge upon moving from the outside to the inside the
bubble.

In order to confirm that the observations above are representative of the
entire bubble and not specific to a particular region, we performed
spectroscopic imaging experiments across the bubble interface at various
energies, a subset of which are shown in Fig. 2e-g. At the valence band edge
(Fig. 2e), the map does not distinguish between the interior and exterior of
the bubble, showing the uniformity of the valence band edge across the bubble.
Figure 2f and 2g, taken within the semiconducting gap, show the presence of
the localized states clearly, and their evolution towards the interior of the
bubble. At large positive energies (not shown), these localized states merge
into the conduction band, and a clear contrast is seen between the interior
and exterior of the bubble. These maps confirm the basic picture provided by
point spectra – the bubble region is characterized by localized states at its
edge and a shifted conduction band edge throughout. We present larger maps
across the entire bubble in the supporting information, to rule out moiré
effects on the observed phenomena. Finally, to rule out possible damage to the
bubble during spectroscopic maps, we measure the topographic heights before
and after spectroscopy is performed.

We next proceed to compare our measurements of the single particle
spectroscopic properties of individual bubbles with their optical emission. To
do this, we employed hyperspectral nanoscale photoluminescence measurements to
map the changes in exciton energy across the bubble. We have previously
applied this technique to the imaging of localized exciton states in
transition metal dichalcogenide monolayers 27, however the spatial resolution
was limited due to the finite radius of curvature of the probe. To push the
nano-optical resolution to scales on order the localized state, we prepared
hetero-nanobubbles on template stripped gold (TS-Au), which allows for optical
resolutions on order of the gap formed between the nano-optic probe and the
substrate 31. In addition, the quenching of photoluminence of the monolayer
and heterobilayer regions in contact with the substrate greatly reduces
background, improving nanobubble’s PL contrast 32.

Figure 3a shows an AFM image of a nanobubble WSe2/MoSe2 on TS-Au, that shows
localized interlayer exciton emission (Fig. 3b) when compared with the flat
heterostructure. The greater strength of the interlayer exciton emission on
the nanobubble edge can be attributed to two factors. First, there is an
enhancement from the strain of the nano-bubble itself, which shifts the
absorption of both constituent layers to the red, allowing for greater
absorption of the excitation photons. Second, recent work has shown that like
their intra-layer cousins, the interlayer excitons transition dipole is
primarily in the plane of the 2D layers33. As the nano-optical probe moves
across the edge of the nanobubble, more of the transition dipole is aligned
with the polarization of the nano-optical field, allowing for significantly
greater in- and out-coupling for optical fields. This alignment we believe is
critical for near-field observation of inter-layer exciton luminescence, which
has substantially weaker oscillator strength. We also expect that strain-
localized PL signal is enhanced by funneling effects, where excitons are
preferentially shuttled towards the lower energy states34, 35, 36

Figure 3c shows point spectra at various locations across the nanobubble
edges, identified by the colored markers in Fig. 3a and 3d. Black arrows show
visual identification of the localized interlayer exciton peak. Clear red
shifting of the emission is observed as one moves from the outer edge of the
bubble into the interior. To visualize this more clearly, in Fig. 3c we plot a
hyperspectral linescan along the vector defined by colored points in Fig. 3a.
The general trend is towards redder emission, shifting 200 meV over 15 nm into
the bubble, after which the emission quickly falls to zero in the nanobubble
interior. In Figs. 3e - 3g we show this same trend with spatial maps with
energy bins of 1.40, 1.30, and 1.25 eV, respectively. The size of the
localized interlayer exciton emission “ring” clearly shrinks for the redder
energy bins.

The shifting of the localized interlayer exciton emission energy reflects the
energic shift of the MoSe2 conduction band observed in the STS. Indeed,
comparing the STS and nano-PL hyperspectral linescans, the roll off of the
interlayer emission energy and the conduction band shows remarkable
similarities at the nanobubble, but the nano-PL shows a sudden drop-off in
emission intensity soon after the edge, which is not seen in STS. This drop-
off however is expected due to the silicon detector used for the nano-PL
spectroscopy (see methods for details), which rapidly lose quantum-efficiency
at energies lower than $\sim$1.3 eV. Similarly, we do not observe emission
from the deeply localized defect states seen in Fig. 2 given that this would
correspond to emission energies $\sim$0.7 eV, which is far below the cutoff
our detector. While quantitative comparison between the STM and nano-PL data
is not possible due to the sample differences, evaluation of the STS-derived
band gap and exciton PL energy provides information on the binding energy for
excitons associated with the conduction band near the bubble edge. Our results
are consistent with a conduction band interlayer exciton binding energy with
magnitude of a couple hundred meV within the bubble37.

To understand the contributions of doping and strain to the observed
spectroscopic features in experiment, we performed Schrödinger-Poisson (SP)
simulations using the approach and code of Ref. 38. The lengthscale associated
with the entire bubble region is too large to model directly, but we will show
that the pertinent experimental features can be captured with a significantly
simplified model. Our focus on the conduction band of the heterostructure,
which is derived from the MoSe2 layer. Thus, we take band parameters for MoSe2
from Ref. 39 and electromechanical parameters from Ref. 40. Since the
important physics is localized to the bubble edge, we approximate the bubble
region by a 2D inclusion in monolayer MoSe2 (large enough to isolate the two
interfaces required by periodic boundary conditions), with a downward shift of
the conduction band in the bubble region, as well as a background doping, that
both increase with strain 41. The strain is assumed to be slightly larger at
the edge of the bubble region. The DOS is calculated assuming a 2D parabolic
dispersion in the direction parallel to the inclusion above the conduction-
band minimum and below the valence band maximum.

Figure 4a show the band diagram obtained by self-consistent solution of the SP
equations, focusing on the CB. By construction, the Fermi level is at the
conduction band within the bubble region due to the background doping. The
dips in the electrostatic potential of the conduction-band at the edges of the
“bubble” have two origins. The first is the band bending induced by the
interface between the doped and undoped region. The second is that we assume a
higher strain at the interface, which increases the effect of this band
bending. We found that adding a piezoelectric polarization charge at the
interface (not shown) did not change the qualitative features, other than
introducing an asymmetry in the band-bending on the opposite sides of the
bubble region.

The red dot-dashed lines in Fig. 4a are the energies of the bound states in
the bubble region, and the green dotted lines are their squared wavefunctions
(shifted so that the zero-density level is at the energy for clarity). We can
see that the band bending at the interface region results in a bound state,
which can also be clearly seen in Fig. 4b as spikes in the electron density at
the edge of the bubble regions. In Fig. 4c we plot the DOS of the total
system, including the sum of 1D states in the bubble region, and the 2D DOS in
the bulk, at different x points near the interface of the bubble (see the
inset, denoted by curve color). Superimposed on the step-like increase from
the 2D DOS at the conduction and valence band edges in the bubble, we can see
a small enhancement in the DOS near the interface, where the localized state
resides. As we move toward the center of the bubble region, this enhancement
weakens, and ultimately disappears. We focused our model on the MoSe2 layer of
the bilayer structure since the MoSe2 is the main contributor to the
conduction band. The size of the bubble is much larger than the crystal cell.
If you were to zoom in at the atomic level the edge of the bubble can be
approximated by its tangent on the surface of the MoSe2 layer. Hence using the
deformation gradients as a baseline for the energy of the different strained
portions we can accurately model the band edge with the one- dimensional
model.

##  3 Conclusion

Our theoretical results, though the result of a simplified model, accurately
captures the essential features of the experimental observations. They
indicate that sharp lateral junctions in doping are an excellent way to
engineer localized states in 2D TMD semiconductors. Such localized states do
not depend on the presence of specific chemical defects or specialized strain
fields. Recently, several 2D materials interfaces have shown the ability to
realize large charge transfer at the interface 42, 43. Our results indicate
that such interfacial engineering together with nanostructuring can be
employed to create optical emitters with arbitrarily desired shapes at the
nanoscale. Our work provides a clear route to achieving this in the future.

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##  4 Data availability

The data used in this paper are available from the corresponding authors upon
reasonable request.

##  5 Acknowledgement

Single crystal synthesis and heterostructure fabrication for this work was
supported by the the NSF MRSEC program through Columbia in the Center for
Precision-Assembled Quantum Materials (PAQM), grant number DMR-2011738. STM
measurements were supported by the NSF GOALI program under grant DMR-1809122
and by the Air Force Office of ScientificResearch via grant FA9550-16-1-0601.
CED and CG acknowledge support from the National Science Foundation under
Grant No. DMR-1918455. The Flatiron Institute is a division of the Simons
Foundation. PJS, TD, and EY acknowledge support from the National Science
Foundation through award DMR-2004437.

##  6 Author Contributions

S.S. carried out the STM/STS measurements and analysis. T.D. conducted optical
measurements. T.D. and S.S. analyzed the nano-optical data. W.W. and E.Y
fabricated the device and performed SHG experiments. C.G. performed the
theoretical calculations. J.H., X.Z., C.E.D., P.J.S. and A.N.P. provided
advice. All authors contributed to writing the manuscript.

##  7 Competing interests

The authors declare no competing interests.

Figure 1: Nanobubbles in twisted WSe2/MoSe2 heterobilayers a Schematic of the
experimental setup. b 3.8 micron AFM image of a WSe2/MoSe2 heterobilayer on
gold substrate used in nano-PL measurements showing several nanobubbles c STM
topographic image ($V_{b}$ = -1.8 V, I = -50 pA), of a single nanobubble. The
continuous moiré pattern across the nanobubble shows that contact between the
two TMD layers is maintained throughout. d STM Height profile across the
dashed arrow in (c) showing an apparent step size of 1 angstrom at the
nanobubble edge highlighted in yellow. e Schematic band diagram of the
conduction band edge (CB) , the bound states, the valence band edge(VB), and
interlayer exciton energy as a function of position deduced from STM and nano-
PL measurements.

Figure 2:  STM spectroscopic properties of nanobubbles a STM topography
indicating locations where the point spectra shown in b,c were taken. b Point
spectra taken inside (blue) and outside (Red) the bubble show a pronounced
shift of the conduction band edge (CBE) inside the bubble while the valence
band is unaffected. c Evolution of $dI/dV$point spectra from the outside to
the inside of the nanobubble corresponding to the markers in a. At the edge of
the bubble, deeply localized states are seen within the semiconductor gap. d
Heat map of position dependent $dI/dV$spectra at the conduction band edge
across the bubble. The dashed lines are guides to the eye that show the
evolution of the bound states and the conduction band edge. e-g Spectroscopic
maps across the nanobubble edge at energies of -0.78 V (valence band edge),
-0.39 V, and -0.22 V (within the bandgap) The set points are $V_{b}$ = -1.8 V,
I = -200 pA. Bound states are clearly observed at the nanobubble edge at -0.39
V and -0.22 V.

Figure 3:  Hyperspectral nano-PL maps of localized interlayer exciton spectra
a Atomic force micrograph of MoSe2-WSe2 nanobubbles on a template-stripped Au
substrate. b Average nano-PL spectra on (blue) and off (red) the nanobubble
region. c Nano-PL point spectra across the bubble edge corresponding the
colored points in a. For points farther into the interior of the nanobubble,
the emissions energy redshifts. d Hyperspectral heat map along the vector
defined by points in (a). The redshifting of the emission is clearly seen,
with the spectral weight shifting from above 1.5 eV at the very edge of the
bubble (red point) to below 1.3 eV at 35 nm inside the bubble (blue point).
Past the position past 40 nm (orange point) the intensity shift below the low
energy detection cutoff of the EMCCD of 1.2 eV. e-g nano-PL maps (average with
a window sized 2.5 nm) binned different energy intervals, ordered by
decreasing emission energy. The approximate physical nanobubble edge is shown
with the green dashed line. The data clearly shows the redshifting of the
emission on going into the bubble.

Figure 4:  Schrödinger-Poisson (SP) simulations a Band diagram for two-
dimensional Schrödinger-Poisson (SP) model of the bubble. Blue solid lines are
the self-consistent band potentials, red dot-dashed lines are the energies of
some of the confined states, black dashed line is the Fermi level, and green
dotted lines are the confined wavefunctions. b Self-consistent electron
density versus $x$ position in (a). c Density of states versus energy
including the 1D states confined to the well, with 2D DOS outside of the
bubble, and in the bubble interior, superimposed.


# An arbitrary Lagrangian-Eulerian formulation for the numerical simulation of
flow patterns generated by the hydromedusa Aequorea victoria

Mehmet SAHIN and Kamran MOHSENI

Department of Aerospace Engineering Sciences,
University of Colorado, Boulder,
Colorado, 80309, USA

###### Abstract

A new geometrically conservative arbitrary Lagrangian-Eulerian (ALE)
formulation is presented for the moving boundary problems in the swirl-free
cylindrical coordinates. The governing equations are multiplied with the
radial distance and integrated over arbitrary moving Lagrangian-Eulerian
quadrilateral elements. Therefore, the continuity and the geometric
conservation equations take very simple form similar to those of the Cartesian
coordinates. The continuity equation is satisfied exactly within each element
and a special attention is given to satisfy the geometric conservation law
(GCL) at the discrete level. The equation of motion of a deforming body is
solved in addition to the Navier-Stokes equations in a fully-coupled form. The
mesh deformation is achieved by solving the linear elasticity equation at each
time level while avoiding remeshing in order to enhance numerical robustness.
The resulting algebraic linear systems are solved using an ILU(k)
preconditioned GMRES method provided by the PETSc library. The present ALE
method is validated for the steady and oscillatory flow around a sphere in a
cylindrical tube and applied to the investigation of the flow patterns around
a free-swimming hydromedusa Aequorea victoria (crystal jellyfish). The
calculations for the hydromedusa indicate the shed of the opposite signed
vortex rings very close to each other and the formation of large induced
velocities along the line of interaction while the ring vortices moving away
from the hydromedusa. In addition, the propulsion efficiency of the free-
swimming hydromedusa is computed and its value is compared with values from
the literature for several other species. The fluid dynamics video presented
here shows the time variation of the instantaneous three-dimensional vorticity
isosurfaces around a free-swimming hydromedusa Aequorea victoria.

##  1 Introduction

The present dynamics video presented here shows the time variation of the
instantaneous three-dimensional vorticity isosurfaces around a free-swimming
hydromedusa Aequorea victoria. To solve the flow pattern around the free-
swimming hydromedusa Aequorea victoria, a new geometrically conservative
arbitrary Lagrangian-Eulerian (ALE) formulation presented in [1] has been
used. The maximum bell radius of the medusa is approximately $2.3$ cm and the
period of one cycle T is being approximately equal to 1.16 seconds. To compute
the velocity of the medusa, the equation of motion is solved in addition to
the Navier Stokes equations in a fully coupled form. The calculations are
carried out on high resolution computational meshes: a coarse mesh M1 with
63,099 vertices and 62,610 quadrilateral elements and a fine mesh M2 with
205,714 vertices and 204,784 quadrilateral elements. The computed average
swimming velocity is computed to be $1.453$ cm/s and $1.462$ cm/s for the
meshes M1 and M2, respectively. Based on the average medusa velocity on mesh
M2 and the maximum bell diameter, the dimensionless parameters Reynolds and
Strouhal numbers are computed to be 672 and 8.47, respectively. The details of
the present work can be found in the papers listed in the references.

## References

  * [1] M. Sahin and K. Mohseni, An Arbitrary Lagrangian-Eulerian Formulation for the Numerical Simulation of Flow Patterns Generated by the Hydromedusa Aequorea Victoria. J. Comput. Phys., (2009), 228:4588-4605.
  * [2] M. Sahin, K. Mohseni and S. Colin, The Numerical Comparison of Flow Patterns and Propulsive Performances for the Hydromedusae Sarsia Tubulosa and Aequorea Victoria. J. Exp. Biol., (2009), 212:2656-2667.

[a]

[b]  Figure 1: The mesh convergence is given for the wake structure behind a
free-swimming hydromedusa Aequorea victoria on meshes M1 [a] and M2 [b].


# Searches for Lepton Flavour Violation and Lepton Number Violation in Hadron
Decays

P. Seyfert on behalf of the LHCb Collaboration  Physikalisches Institut,
Heidelberg University, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany

###### Abstract

In the Standard Model of particle physics, lepton flavour and lepton number
are conserved quantities although no fundamental symmetry demands their
conservation. I present recent results of searches for lepton flavour and
lepton number violating hadron decays measured at the $\PB$ factories and
LHCb.

In addition, the LHCb collaboration has recently performed a search for the
lepton flavour violating decay $\Ptauon\to\Pmuon\Pmuon\APmuon$. The obtained
upper exclusion limit, that has been presented in this talk for the first
time, is of the same order of magnitude as those observed at the $\PB$
factories. This is the first search for a lepton flavour violating $\Ptau$
decay at a hadron collider.

##  I Introduction

In the Standard Model both lepton number as well as lepton flavour are
conserved quantities SM1 ; SM2 . Since both can be broken in extensions of the
standard model, observation of either of them would be a clear sign for new
physics.

Results of the search for lepton number violation (LNV) and lepton flavour
violation (LFV) in decays of hadrons are presented. These comprise the $\PB$
decay modes $\PBplus\to h^{+}\ell^{+}\ell^{\prime-}$, $\PBplus\to
h^{-}\ell^{+}\ell^{\prime+}$, and the corresponding $\PD$ modes $\PDplus\to
h^{+}\ell^{+}\ell^{\prime-}$ and $\PDplus\to h^{-}\ell^{+}\ell^{\prime+}$. The
final state meson $h$ may hereby either be a stable meson (pions or kaons) or
in case of $\PB$ decays also a $\PD$ meson. Additionally to these modes, the
first limit on the branching fraction $\PBplus\to\PDzero\Ppiplus\Pmuon\Pmuon$
is presented as well as a new result on the search for LFV in the decay of
$\Ptau$ leptons at the LHC.

Throughout this document charge conjugate decays are implied.

###  I.1 Lepton Number Violation

Numerous models without lepton number conservation have been proposed, see
majorana  for an overview. Similar to the fundamental diagram in the
neutrinoless double beta decay, any neutrinoless hadron decay with two same
sign leptons in the final state probes the existence of Majorana neutrinos. Of
the two lowest order diagrams for LNV in meson decays, one can go through an
on-shell neutrino, while the other contains a virtual neutrino (Fig. 1). In
$\PBplus$ decays one of them is Cabbibo favoured depending on the final state.
Thus for Majorana neutrino masses in the accessible mass range (up to
$\unit{5140}{\MeVovercsq}$) the modes
$\PBplus\to\PDminus\Pleptonplus\Pleptonplus$ and
$\PBplus\to\PDstar^{-}\Pleptonplus\Pleptonplus$ are more sensitive and also
provide a mass measurement. Beyond the accessible mass range other final
states ($\Ppiminus\Pleptonplus\Pleptonplus$,
$\PDsminus\Pleptonplus\Pleptonplus$) are more sensitive.

(a)  (b)

Figure 1: Lowest order diagrams for LNV in meson decays, involving on-shell or
virtual Majorana neutrinos. Depending on the individual quark flavours either
of them can be Cabbibo favoured. Reproduced from lhcb-Blnv2 .

(a)  (b)

Figure 2: Constraints on charged lepton couplings $V_{\Pe 4}$ and $V_{\Pmu 4}$
to a fourth heavy Majorana neutrino from 2009 as a function of the mass
$m_{4}$majorana .

In the framework of LNV through a fourth neutrino $N$ with a large Majorana
mass, an observation of LNV not only provides information about the mass
$m_{4}$ of the fourth neutrino, but also on the $\PW N\ell$ coupling strength
$|V_{\Plepton 4}|$. A compilation of different exclusion limits is shown in
Fig. 2. For the coupling to the muon the strongest constraints come from kaon
physics.

Complementary to the modes with one meson in the final state, it has been
suggested in 1108.6009  to also consider
$\PBplus\to\PDzero\Ppiplus\Pmuon\Pmuon$ with the diagram shown in Fig. 3.
Until 2012 no limit on the branching fraction of this decay has been measured.

Figure 3: Tree level Feynman diagram for the decay
$\PBminus\to\PDzero\Ppiplus\Pmuon\Pmuon$.

###  I.2 Lepton Flavour Violation

In contrast to violation of lepton number, lepton flavour violation has been
observed experimentally in the neutrino sector. Through loop diagrams,
neutrino oscillation can also enter the charged sector as illustrated in Fig.
4 – the predicted rates however are immeasurable small, suppressed by powers
of $m^{2}_{\Pneutrino}/m^{2}_{\PW}$ lfvcalc .

Two examples how to introduce sizeable lepton flavour violation are multi
Higgs extensions by means of new scalar particles (see e. g. d44_1461 ) as in
the diagram in Fig. LABEL:fig:phis or by means of heavy neutrinos as
introduced in low scale seesaw models (e. g. D73_074011 ) which couple to
electrons and muons as shown in Fig. LABEL:fig:hadlfvbsmtwo. Other ways to
embed LFV in the standard model are given e. g. in leptoquark .

Figure 4: Feynman diagram for lepton flavour violating meson decays in the
Standard Model with neutrino oscillation.

(a)  (b)

Figure 5: Examples for introduction of lepton flavour violating in meson
decays

Particularly interesting about LFV in $\PB$ decays compared to $\PD$ decays is
that the $\PB$ mass is high enough to produce a $\APtauon,\Pmuon$ pair in the
final state. For new physics introduced in a Higgs coupling, this final state
is most sensitive due to the high masses, and thereby Higgs couplings of the
leptons involved.

(a)  (b)

Figure 6: Feynman diagrams for $\Ptauon\to\Pmuon\Pmuon\APmuon$ in different
models.

Similarly the decay $\Ptauon\to\Pmuon\Pmuon\APmuon$ is not entirely forbidden,
but neutrino oscillation at loop level alone cannot bring the branching
fraction to an observable level. As presented in renga  the strongest limits
on LFV in lepton decays come from the $\Pmuon\to\Pelectron\Pphoton$ mode, the
search for $\Ptauon\to\Pmuon\Pmuon\APmuon$ is particularly interesting because
some new physics models (e. g. Littlest Higgs littlesthiggs ), as in Fig. 6,
have enhanced lepton flavour violating couplings to heavy leptons (favouring
$\Ptau$ over $\Pmu$ decays) and do not involve photon couplings and therefore
enhance the three lepton final state over the $\Plepton\Pphoton$ final state
tau23mutheory . Moreover to identify the character of new physics, a search in
both $\Plepton\to\Plepton^{\prime}\Pphoton$ and
$\Plepton\to\Plepton^{\prime}\Plepton^{\prime}\Plepton^{\prime}$ must be
performed.

##  II Experimental Results

The study of rare decays naturally needs large event samples, which, for $\PB$
and $\PD$ mesons, is available at the $\PB$ factories and at the LHC. The most
stringent constraints on LFV and LNV in modes involving electrons come from
BaBar and Belle, while muonic final states are now best constrained by recent
LHCb measurements.

###  II.1 Limits on Lepton Number Violation

Decays of $\PBplus$, $\PDplus$, and $\PKplus$ mesons were used to search for
Majorana neutrinos of different masses. The mass difference of the decaying
meson and the final state lepton is the upper limit on the mass of the on-
shell neutrino which can be produced. Since the neutrino mass is the invariant
mass of the final state meson-lepton pair, the sum of their rest masses is the
lower limit on the accessible mass range.

The strongest limits on the lepton coupling $|V_{\Pe 4}|^{2}$ and $|V_{\Pmu
4}|^{2}$ to a fourth neutrino are in the low neutrino mass region between
$\unit{140}{\MeVovercsq}$ and $\unit{353}{\MeVovercsq}$ coming from searches
for the decays $\PKplus\to\APelectron\APelectron\Ppiminus$ and
$\PKplus\to\APmuon\APmuon\Ppiminus$ respectively. Couplings down to
$|V_{\Plepton 4}|^{2}\lesssim 10^{-8}$ are thereby ruled out in the most
sensitive range.

The currently most stringent limits on LNV in charm decays and thereby higher
neutrino masses have been obtained by the BaBar collaboration babar-charm
shown in Tab. LABEL:tab:babar-charm.

The extension of the search range to higher masses is only possible in $\PB$
decays, the enormous production cross section in hadron collisions makes the
LHC the optimal place for searches for LNV in $\PB$ decays. LHCb recently
provided new results on the on-shell modes $\PBplus\to\Ppiminus\APmuon\APmuon$
and $\PBplus\to\PDsminus\APmuon\APmuon$, as well as the virtual modes
$\PBplus\to\PDplus\APmuon\APmuon$ and $\PBplus\to\PD^{*-}\APmuon\APmuon$ lhcb-
Blnv2 .

Limits on the branching fraction are hereby set as a function of the neutrino
mass for the on-shell modes. For comparison, assuming a flat distribution of
the decay products in phase space, the observed branching fraction is shown in
Tab. LABEL:tab:BLNV along with the modes which are sensitive to virtual
Majorana neutrinos and previous measurements.

The first search for $\PBplus\to\PDzero\Ppiminus\APmuon\APmuon$ has been
performed by LHCb lhcb-Blnv2  and showed no excess over the background. Since
this channel involves an on-shell Majorana neutrino, the limit is given as a
function of the neutrino mass as well.

Table 1: Current limits on lepton number violating charm (a) and bottom (b)
meson decays.

channel | limit |  |
---|---|---|---
$\mathcal{B}(\PDplus\to\Ppiminus\APelectron\APelectron)$ | $<1.9\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDplus\to\Ppiminus\APmuon\APmuon)$ | $<2.0\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDplus\to\Ppiminus\APmuon\APelectron)$ | $<2.0\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDsplus\to\Ppiminus\APelectron\APelectron)$ | $<4.1\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDsplus\to\Ppiminus\APmuon\APmuon)$ | $<14\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDsplus\to\Ppiminus\APmuon\APelectron)$ | $<8.4\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDplus\to\PKminus\APelectron\APelectron)$ | $<0.9\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDplus\to\PKminus\APmuon\APmuon)$ | $<10\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDplus\to\PKminus\APmuon\APelectron)$ | $<1.9\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDsplus\to\PKminus\APelectron\APelectron)$ | $<5.2\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDsplus\to\PKminus\APmuon\APmuon)$ | $<13\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PDsplus\to\PKminus\APmuon\APelectron)$ | $<6.1\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PLambdac\to\APproton\APelectron\APelectron)$ | $<2.7\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PLambdac\to\APproton\APmuon\APmuon)$ | $<9.4\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
$\mathcal{B}(\PLambdac\to\APproton\APmuon\APelectron)$ | $<16\times 10^{-6}$ | @$90\,\%$ CL |  babar-charm  BaBar
(a)

channel | limit |  |  |
---|---|---|---|---
$\mathcal{B}(\PBminus\to\Ppi^{+}\Pelectron\Pelectron)$ | $<2.3\times 10^{-8}$ | @$90\,\%$ CL | babar-Blnv  | BaBar
$\mathcal{B}(\PBminus\to\PK^{+}\Pelectron\Pelectron)$ | $<3.0\times 10^{-8}$ | @$90\,\%$ CL | babar-Blnv  | BaBar
$\mathcal{B}(\PBminus\to\PK^{*+}\Pelectron\Pelectron)$ | $<2.8\times 10^{-6}$ | @$90\,\%$ CL | cleo-Blnv  | CLEO
$\mathcal{B}(\PBminus\to\Prho^{+}\Pelectron\Pelectron)$ | $<2.6\times 10^{-6}$ | @$90\,\%$ CL | cleo-Blnv  | CLEO
$\mathcal{B}(\PBminus\to\PD^{+}\Pelectron\Pelectron)$ | $<2.6\times 10^{-6}$ | @$90\,\%$ CL | belle-Blnv  | Belle
$\mathcal{B}(\PBminus\to\PD^{+}\Pelectron\Pmuon)$ | $<1.8\times 10^{-6}$ | @$90\,\%$ CL | belle-Blnv  | Belle
$\mathcal{B}(\PBminus\to\Ppi^{+}\Pmuon\Pmuon)$ | $<1.3\times 10^{-8}$ | @$95\,\%$ CL | lhcb-Blnv2  | LHCb
$\mathcal{B}(\PBminus\to\PK^{+}\Pmuon\Pmuon)$ | $<5.4\times 10^{-7}$ | @$95\,\%$ CL | lhcb-Blnv  | LHCb
$\mathcal{B}(\PBminus\to\PK^{*+}\Pmuon\Pmuon)$ | $<4.4\times 10^{-6}$ | @$90\,\%$ CL | cleo-Blnv  | CLEO
$\mathcal{B}(\PBminus\to\Prho^{+}\Pmuon\Pmuon)$ | $<5.0\times 10^{-6}$ | @$90\,\%$ CL | cleo-Blnv  | CLEO
$\mathcal{B}(\PBminus\to\PD^{+}\Pmuon\Pmuon)$ | $<6.9\times 10^{-7}$ | @$95\,\%$ CL | lhcb-Blnv2  | LHCb
$\mathcal{B}(\PBminus\to\PD^{*+}\Pmuon\Pmuon)$ | $<2.4\times 10^{-6}$ | @$95\,\%$ CL | lhcb-Blnv2  | LHCb
$\mathcal{B}(\PBminus\to\PDs^{+}\Pmuon\Pmuon)$ | $<5.8\times 10^{-7}$ | @$95\,\%$ CL | lhcb-Blnv2  | LHCb
$\mathcal{B}(\PBminus\to\PDzero\Ppiplus\Pmuon\Pmuon)$ | $<1.5\times 10^{-6}$ | @$95\,\%$ CL | lhcb-Blnv2  | LHCb
(b)

LHCb also provides the strongest limits on $|V_{\mu 4}|$ up to the $\PBplus$
mass considering these results come from $\PBplus\to\Ppiminus\APmuon\APmuon$,
shown in Fig. 7.

Figure 7: Limit on $|V_{\mu 4}|^{2}$ from $\PBminus\to\Ppi^{+}\Pmuon\Pmuon$
measured by LHCb lhcb-Blnv2 . Figure 8: Relative reconstruction efficiency as
a function of the Majorana neutrino lifetime. The branching fraction limits
from lhcb-Blnv2  have been computed for the assumption of infinitively short
lifetimes ($100\,\%$ relative efficiency). For longer lifetimes, the
reconstruction efficiency decreases and the observed limit has to be scaled
down. The efficiencies are given for (a) $\PBplus\to\Ppiminus\APmuon\APmuon$,
(b) $\PBplus\to\PDsminus\APmuon\APmuon$, and (c)
$\PBplus\to\PDzero\Ppiminus\APmuon\APmuon$.

A natural way to search for a lepton number violating decay is to search for
same sign leptons from a common vertex. Thereby the analysis’ implications on
an intermediate on-shell neutrino are only drawn correctly if the lifetime of
the neutrino is short enough not to degrade the reconstruction. To estimate
how the observed limits are to be understood in models with long lived heavy
neutrinos, LHCb also provides the relative reconstruction efficiency as a
function of the neutrino lifetime, shown in Fig. 8.

###  II.2 Limits on Lepton Flavour Violation

The tightest constraints on lepton flavour violating processes in charm decays
have been found by the BaBar collaboration, listed in Tab. LABEL:tab:babar-
charmlfv. For bottom decays, Tab. LABEL:tab:babar-bottomlfv shows the recent
results, involving $\Ptau$ leptons in the final state. Details are given in
giovanni . These results improved the limit on the energy scale at which LFV
can occur LFV_energyscale  significantly.

The implication for new physics is that the energy scale for LFV effective
operators is pushed up from $\unit{2.2}{}$ to $\unit{11}{}$ or from
$\unit{2.6}{}$ to $\unit{15}{}$ for the $\Pbottom\to\Pdown$ and the
$\Pbottom\to\Pstrange$ transition respectively LFV_energyscale .

Table 2: Limits on lepton flavour violating hadron decays at $90\,\%$
confidence level. All listed limits from the BaBar collaboration.

channel | limit |
---|---|---
$\mathcal{B}(\PDplus\to\Ppiplus\APmuon\Pelectron)$ | $<3.6\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDplus\to\Ppiplus\APelectron\Pmuon)$ | $<2.9\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDsplus\to\Ppiplus\APmuon\Pelectron)$ | $<20\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDsplus\to\Ppiplus\APelectron\Pmuon)$ | $<12\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDplus\to\PKplus\APmuon\Pelectron)$ | $<2.8\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDplus\to\PKplus\APelectron\Pmuon)$ | $<1.2\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDsplus\to\PKplus\APmuon\Pelectron)$ | $<9.7\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PDsplus\to\PKplus\APelectron\Pmuon)$ | $<14\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PLambdac\to\Pproton\APmuon\Pelectron)$ | $<19\times 10^{-6}$ | babar-charm
$\mathcal{B}(\PLambdac\to\Pproton\APelectron\Pmuon)$ | $<9.9\times 10^{-6}$ | babar-charm
(a)

channel | limit |
---|---|---
$\mathcal{B}(\PBplus\to\PKplus\Ptauon\APmuon)$ | $<4.5\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\PKplus\APtauon\Pmuon)$ | $<2.8\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\PKplus\Ptauon\APelectron)$ | $<4.3\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\PKplus\APtauon\Pelectron)$ | $<1.5\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\Ptauon\APmuon)$ | $<6.2\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\APtauon\Pmuon)$ | $<4.5\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\Ptauon\APelectron)$ | $<7.4\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\APtauon\Pelectron)$ | $<2.0\times 10^{-5}$ | babar-Blfv2
$\mathcal{B}(\PBplus\to\Ppiplus\Pmu^{\pm}\Pe^{\mp})$ | $<1.7\times 10^{-7}$ | babar-Blfv
(b)

The most recent result in this talk is the limit on LFV in
$\Ptauon\to\Pmuon\Pmuon\APmuon$ achieved by LHCb tau23muCONF . The hadron
collider environment introduces special experimental challenges compared to
the $\PB$ factories.

####  $\Ptau$ tag

At the $\PB$ factories, $\Ptau$ are produced in pairs. A clean event selection
therefore is to look at events with four tracks – three from the signal
candidate and one from a standard model one prong $\Ptau$ decay. At the LHC
the main source for $\Ptau$ is the leptonic $\PDsminus\to\Ptauon\APnut$ decay
tau23muCONF .

#### Normalisation

The $\Ptau$ tag automatically provides the number of produced $\Ptau$ which
enter the analysis. Since the main production mode for $\Ptau$ at the LHC does
not provide any further charged particles, the number of $\Ptau$ entering the
analysis is not directly accessible. A normalisation to allowed $\Ptau$ decays
is not possible since they are indistinguishable from more abundant $\PDplus$
decays with $\Ppizero$ in the decay chain.

#### Background

Having no production tag, background from events without $\Ptau$, such as
$\PB$ and $\PD$ cascade decays, is more severe in the LHCb analysis than for
the $\PB$ factories.

The main advantage of LHCb however is the huge production cross section for
$\Ptau$ from $\PDs$ decays. Considering the charm and bottom production cross
sections measured by LHCb LHCb-CONF-2010-013 ; Bprod  and the known
semileptonic branching fractions pdg , about $8\times 10^{10}$ $\Ptau$ leptons
have been produced at LHCb in 2011 compared to a total of $10^{9}$ $\Ptau$
pairs at the $\PB$ factories.

The analysis strategy of tau23muCONF  is similar to other rare decay searches
at LHCb. A loose cut based selection is applied to get a processable data
sample. All events passing this selection are classified in a three
dimensional likelihood space. The discriminating variables are the invariant
mass of the $\Ptauon\to\Pmuon\Pmuon\APmuon$ candidate, a multivariate
classifier $\mathcal{M}_{\text{3body}}$ for the three body decay properties
(geometry, displacement, track quality, isolation, and kinematics), and a
multivariate classifier for the particle identification
$\mathcal{M}_{\text{PID}}$ (combining information from muon stations, RICH
detectors, and the calorimeter signature). The latter classifiers use boosted
decision trees bdt  with adaptive boosting bdt_boosted  as implemented by TMVA
tmva .

(a)  (b)

Figure 9: Distribution of signal events in the two multivariate likelihoods
for signal (blue / solid) and background (red / dashed).

(a)  (b)

Figure 10: Invariant mass distribution for (a) simulated signal candidates and
(b) observed events in the two highest $\mathcal{M}_{\text{3body}}$ and
$\mathcal{M}_{\text{PID}}$ bins. The background fit
($\PDsplus\to\Peta\APmuon\Pnu$ in green / dotted; combinatorial in red /
dashed; combined in blue / solid) is shown in the range which is used for the
fit.

The signal efficiency of the multivariate classifiers as well as the invariant
mass resolution come from simulation and are calibrated on a control channel –
$\PDsplus\to\Ppiplus\Pphi(\APmuon\Pmuon)$ in the case for the three body
classifier and the invariant mass and $\PBplus\to\PJpsi(\APmuon\Pmuon)\PKplus$
for the particle identification.

The $\PDsplus\to\Pphi\Ppiplus$ calibration channel also serves as a
normalisation, since the branching fractions
$\mathcal{B}(\PDsplus\to\Pphi\Ppiplus)$ and
$\mathcal{B}(\PDsplus\to\APtauon\Pnut)$ are known – yielding the number of
$\Ptau$ which have been produced in $\PDs$ decays. To determine the fraction
of $\Ptau$ from $\PDs$ decays, $f(\PDs)$, the bottom and charm cross sections
measured by LHCb LHCb-CONF-2010-013 ; Bprod , as well as the branching
fractions of charm and bottom hadrons to $\Ptau$ are used. Hereby most of the
systematic uncertainties (e. g. luminosity measurement, reconstruction
efficiencies) cancel, i. e. $f(\PDs)$ is more accurately known than the
inclusive $\Ptau$ production cross section. Contributions from gauge bosons or
Drell-Yan processes have been evaluated to be negligible.

Using the above normalisation as well as the efficiencies for selection,
reconstruction, and trigger the branching fraction can be written as follows:

$\displaystyle\mathcal{B}(\Ptauon\to\Pmuon\Pmuon\APmuon)$
$\displaystyle=\frac{\mathcal{B}(\PDsplus\to\Pphi(\APmuon\Pmuon)\Ppiplus)}{\mathcal{B}(\PDsplus\to\APtauon\Pnut)}\times
f(\PDs)$
$\displaystyle\quad\times\frac{\varepsilon_{\text{norm}}}{\varepsilon_{\text{sig}}}\frac{N_{\Ptauon\to\Pmuon\Pmuon\APmuon}}{N_{\PDsplus\to\Pphi(\APmuon\Pmuon)\Ppiplus}}$

where $\varepsilon_{\text{norm}}$ is the total efficiency to trigger,
reconstruct and select the normalisation decay and $\varepsilon_{\text{sig}}$
is the total efficiency for the signal channel.

The dimuon decay of the $\Pphi$ is chosen to provide similar trigger and
particle identification properties compared to the signal being sought for.
The non resonant contribution from $\PDsplus\to\APmuon\Pmuon\Ppiplus$ decays
was found to be below $2\,\%$.

The three dimensional likelihood space is subdivided into 150 bins (five for
$\mathcal{M}_{\text{3body}}$ and $\mathcal{M}_{\text{PID}}$, and six for the
invariant mass) as shown in Fig. 9 and LABEL:fig:massbinning. The signal
efficiency for each bin is evaluated from the calibration channels and the
background in each bin is estimated from the sidebands in the invariant mass.

The background consists mainly of two components. Firstly combinatorial
background which is modelled by an exponential and secondly by
$\PDsplus\to\Peta(\APmuon\Pmuon\Pphoton)\APmuon\Pnum$ decays. This physical
background is not discriminated in the current analysis by either
$\mathcal{M}_{\text{3body}}$ or $\mathcal{M}_{\text{PID}}$ as it has the same
behaviour as the signal in all input quantities. Rejecting this decay will be
subject of future improvements. It is modelled by an exponential multiplied by
a second order polynomial for which all shape parameters have been fixed on
simulated events. The normalisation is left free in the final fit within one
standard deviation from the expected yield which is determined using the
normalisation channel and the branching fractions
$\mathcal{B}(\PDsplus\to\Peta\APmuon\Pnum),\mathcal{B}(\Peta\to\APmuon\Pmuon\Pphoton)$,
and $\mathcal{B}(\PDsplus\to\Pphi\Ppiplus)$. For illustration the invariant
mass distribution and the combined fit for the combination of the two highest
$\mathcal{M}_{\text{3body}}$ and two highest $\mathcal{M}_{\text{PID}}$ bins
is shown in Fig. LABEL:fig:unblinded.

For the final limit, all bins are combined using the CLs method CLs1 ; CLs2 .

Table 3: Limits on the branching fraction for $\Ptauon\to\Pmuon\Pmuon\APmuon$ obtained by different experiments. collaboration | limit |  |
---|---|---|---
Belle | $<2.1\times 10^{-8}$ | @$90\,\%$ CL | tau23muBELLE
BaBar | $<3.3\times 10^{-8}$ | @$90\,\%$ CL | tau23muBABAR
LHCb | $<6.3\times 10^{-8}$ | @$90\,\%$ CL | tau23muCONF

The observed limits at $90\,\%$ confidence level is $6.3\times 10^{-8}$, in
agreement with the expected limit for the absence of a signal ($8.2\times
10^{-8}$). Tab. 3 shows the comparison to the limits from BaBar and Belle.

##  III Conclusion

Hadron decays measured at the $\PB$ factories and at the LHC provide an
excellent and abundant probe to search for LNV and LFV. So far no signal has
been observed and only lower limits for the branching fractions are given. The
$\PB$ factories have achieved high sensitivity and ruled out branching
fractions to the level of $10^{-5}$. The huge production cross section of
$\PB$ mesons at the LHC furthermore enabled LHCb to improve the limits on LNV
in $\PB$ decays further. Finally, the first LFV $\Ptau$ decay search performed
at a hadron collider has been performed.

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