###### Abstract

Searches for lepton flavor violation in tau decays are unambiguous signatures
of new physics. The branching ratios of tau leptons at the level of
$10^{-10}$–$10^{-9}$ can be probed using 50 $ab^{-1}$ of electron-positron
annihilation data being collected by the Belle II experiment at the world’s
highest luminosity accelerator, the SuperKEKB, located at the High Energy
Accelerator Research Organization, KEK, in Tsukuba, Japan. Searches with such
expected sensitivity will either discover new physics or strongly constrain
several new physics models.

###### keywords:

lepton flavor violation; new physics in tau decays; B-Factory; Belle II
experiment

8 9 480 Academic Editors: Robert H. Bernstein and Bertrand Echenard 11 April
2022 18 August 2022 13 September 2022 https://doi.org/10.3390/universe8090480
Searches for Lepton Flavor Violation in Tau Decays at Belle II Searches for
Lepton Flavor Violation in Tau Decays at Belle II Swagato Banerjee  Swagato
Banerjee Banerjee, Sw.

##  1 Introduction

Lepton flavor conservation stands out in the Standard Model (SM) among all
other symmetries because it is not associated with any underlying conserved
current. Lepton flavor violation (LFV) in the charged sector is predicted by
many new physics (NP) models. The small but finite mass of the neutrinos in
the SM allow charged LFV in neutrino-less two-body decays, e.g.,
$\tau^{-}\to\ell_{i}^{-}\gamma$ decays (charge conjugate modes are implied
throughout the text, unless otherwise specified), where $\ell_{i}^{-}$ ($i$ =
1, 2) denotes light charged leptons ($e^{-}$, $\mu^{-}$). However, such decays
are suppressed by a factor of $({m_{\nu}^{2}}/{m_{W}^{2}})^{2}$ Lee and Shrock
(1977), which produces experimentally unreachable rates of the order of
$10^{-54}$. For neutrino-less three-body decays, e.g.,
$\tau^{-}\to\ell_{i}^{-}\ell_{j}^{+}\ell_{a}^{-}$ decays (where
$a~{}=~{}i~{}\mathrm{or}~{}j$, and $i$ may or may not be equal to $j$), two
conflicting predictions existed in the literature: one of the order of
$10^{-55}$ Petcov (1977) and another of the order of $10^{-14}$ Pham (1999).
Recently, the contributions due to finite neutrino masses for such decays were
re-scrutinized and found to be in the range of $[10^{-56},10^{-54}]$
Hernández-Tomé et al. (2019); Blackstone et al. (2020), thereby laying to rest
the claim that such decays could be of the order of $10^{-14}$ in the SM.
Thus, any observation of charged LFV is an unambiguous signature of NP.

A discovery of LFV in the charged sector may provide deeper insight into
several unsolved mysteries such as the origin of the dark sector, large baryon
number asymmetry, number of flavor generations and extra dimensions. Many NP
models, such as low-scale seesaw models Cvetic et al. (2002), supersymmetric
standard models Ellis et al. (2000, 2002); Dedes et al. (2002); Brignole and
Rossi (2003); Masiero et al. (2003); Hisano et al. (2009); Fukuyama et al.
(2003), little Higgs models Choudhury et al. (2007); Blanke et al. (2007),
leptoquark models Davidson et al. (1994), non-universal $Z^{\prime}$ models
Yue et al. (2002), and extended Higgs models Akeroyd et al. (2009); Harnik et
al. (2013); Celis et al. (2014); Omura et al. (2015); Goudelis et al. (2012),
predict LFV in $\tau$ decays at the level of $10^{-10}$–$10^{-8}$ which are
just below the current experimental bounds. Predictions for two-body and
three-body neutrino-less $\tau$ decays from some of these NP models Cvetic et
al. (2002); Ellis et al. (2000, 2002); Dedes et al. (2002); Brignole and Rossi
(2003); Masiero et al. (2003); Fukuyama et al. (2003); Yue et al. (2002) are
tabulated in Table 1.

Table 1: The branching fractions (${\cal{B}}$) for $\tau^{-}\to\ell^{-}\gamma$
and $\tau^{-}\to\ell^{-}\ell^{+}\ell^{-}$ decays in some NP models.
---
|  ${\cal{B}}(\tau^{-}\to\ell^{-}\gamma)$ |  ${\cal{B}}(\tau^{-}\to\ell^{-}\ell^{+}\ell^{-})$
SM + seesaw Cvetic et al. (2002) |  $10^{-9}$ |  $10^{-10}$
SUSY + Higgs Dedes et al. (2002); Brignole and Rossi (2003) |  $10^{-10}$ |  $10^{-8}$
SUSY + SO(10) Masiero et al. (2003); Fukuyama et al. (2003) |  $10^{-8}$ |  $10^{-10}$
Non-universal Z′ Yue et al. (2002) |  $10^{-9}$ |  $10^{-8}$

Since $\tau$ is the only lepton that can decay hadronically, many neutrino-
less final states are allowed via LFV processes in NP models within the
observable parameter space Black et al. (2002). Example of such processes
include final states containing a light lepton and a meson e.g.,
$\tau^{-}\to\ell^{-}M^{0}$ (where
$M^{0}=\pi^{0},K_{S}^{0},\eta,\rho^{0},\omega,K^{\star 0},\bar{K}^{\star
0},\eta^{\prime},a_{0},f_{0},\phi$), or a light lepton and two mesons:
$\tau^{-}\to\ell^{-}h^{+}h^{-},\;\ell^{+}h^{-}h^{-}$, $\ell^{-}h^{0}h^{0}$
(where $h^{\pm}=\pi^{\pm}/K^{\pm},h^{0}=\pi^{0}/K_{S}^{0}$).

Searches for all possible LFV processes in decays of $\tau$ are necessary
because there are strong correlations between the expected rates of the
different channels in various models. For example, in some supersymmetric
seesaw models Babu and Kolda (2002); Sher (2002), the relative rates of
${\cal{B}}(\tau^{\pm}\to\mu^{\pm}\gamma)$ :
${\cal{B}}(\tau^{\pm}\to\mu^{\pm}\mu^{+}\mu^{-})$ :
${\cal{B}}(\tau^{\pm}\to\mu^{\pm}\eta)$ are predicted to have specific ratios,
depending on the model parameters. In the unconstrained minimal supersymmetric
model, which includes various correlations between the $\tau$ and $\mu$ LFV
rates, the LFV branching fractions of the $\tau$ lepton in some decay channels
can be as high as $10^{-8}$ Brignole and Rossi (2004); Goto et al. (2008)),
while still respecting the strong experimental bounds on the LFV decays of the
$\mu$ lepton Baldini et al. (2016); Bellgardt et al. (1988). Thus, it is
critical to probe all possible LFV modes of the $\tau$ lepton, because any
excess in a single channel will not provide sufficient information to nail
down the underlying LFV mechanism or even to identify an underlying theory.

More exotic decay modes, such as $\tau^{-}\to\pi^{+}\ell^{-}\ell^{-}$ and
$K^{+}\ell^{-}\ell^{-}$, accompanied by a violation of the lepton number
(LNV), are predicted at the level of $10^{-10}$–$10^{-8}$ in several scenarios
beyond the SM Lopez Castro and Quintero (2012). Several of these decay modes
are expected to have branching ratios close to existing experimental limits in
NP models, e.g., heavy Dirac neutrinos Gonzalez-Garcia and Valle (1992);
Ilakovac (2000), supersymmetric processes Sher (2002); Arganda et al. (2008),
flavor-changing $Z^{\prime}$ exchanges with non-universal couplings Li et al.
(2006), etc., to name a few. Wrong-sign
$\tau^{-}\to\ell_{i}^{+}\ell_{j}^{-}\ell_{j}^{-}$ decays are very intriguing
because they are expected at rates only one order of magnitude below the
present bounds in some NP models, e.g., the Littlest Higgs model with T-parity
realizing an inverse seesaw Pacheco and Roig (2022).

Most models for baryogenesis, a hypothetical physical process based on
different descriptions of the interaction between the fundamental particles
that took place during the early universe producing the observed
matter–antimatter asymmetry, require baryon number violation (BNV), which in
charged lepton decays automatically implies LNV and LFV Aaij et al. (2013).
Angular momentum conservation requires the difference of net baryon number (B)
and lepton number (L) to be equal to either 0 or 2. Although the SM conserves
this difference, the symmetry group for the sum of baryon number and lepton
number can be associated with an anomalous current. A set of models predicts
baryogenesis that conserves B-L but includes instanton induced B+L violating
currents Fuentes-Martin et al. (2015). In a large class of models Hou et al.
(2005), BNV in $\tau$ decay modes containing baryons in the final state, for
example,
$\tau^{-}\to\pi^{-}\Lambda,\pi^{-}\bar{\Lambda},K^{-}\Lambda,K^{-}\bar{\Lambda},\bar{p}\gamma,\bar{p}\ell_{i}\bar{\ell}_{j}~{}\mathrm{and}~{}p\ell_{i}\ell_{j}$,
are predicted at observable rates in the large $\tau$ data set that the Belle
II detector will record over the coming years.

##  2 Belle II Experiment at SuperKEKB

The most restrictive limits on LFV in $\tau$ decays at the level of $10^{-8}$
have been obtained by the first generation of the B-Factory experiments, Belle
and BABAR, where a big data sample of $\tau$’ s was generated thanks to large
and similar values of the production cross-sections of $B-$ mesons and $\tau-$
pairs around the $\Upsilon(4S)$ resonance at the level of a nanobarn ($nb$)
Banerjee et al. (2008). Belle and BABAR experiments collected approximately
one attobarn-inverse ($ab^{-1}$) and half an $ab^{-1}$ of $e^{-}e^{+}$
annihilation data, respectively. The next generation of the B-Factory
experiment, Belle II, is expected to collect 50 $ab^{-1}$ of data over the
next decade Aggarwal et al. (2022). Such a huge data sample corresponding to
$10^{11}$ single $\tau$-decays would lower the limits on LFV in $\tau$ decays
by one or two orders of magnitude.

###  2.1 Luminosity Upgrade of SuperKEKB

The asymmetric beam energy $e^{-}e^{+}$ collider, SuperKEKB, is an upgrade of
the KEKB accelerator facility in Tsukuba, Japan, and has a circumference of
about 3 km. The main components of the SuperKEKB collider complex are a 7 GeV
electron ring known as the high-energy ring (HER), a 4 GeV positron ring known
as the low-energy ring (LER), and an injector linear accelerator with a 1.1
GeV positron damping ring Akai et al. (2018). The HER and the LER have four
straight sections named Tsukuba, Oho, Fuji, and Nikko, with the interaction
point in the straight section of Tsukuba, where the Belle II detector is
located.

The target integrated luminosity of 50 $ab^{-1}$ to be collected by the Belle
II experiment will be achieved by increasing the instantaneous luminosity by a
factor of 30. Two major upgrades account for this increase: a modest two-fold
increase in the beam currents, and a fifteen-fold reduction of the vertical
beta function at the interaction point $(\beta^{\star}_{y})$ from 5.9 mm to
0.4 mm, according to the “nano-beam” scheme described below.

Compared to KEKB, the asymmetry between the beam energies for the HER/LER
beams were reduced from 8.0/3.5 GeV to 7.0/4.0 GeV, which reduces the beam
loss due to Touschek scattering. This also improves the solid-angle acceptance
of the experiment, which helps to analyze events with large missing energy.
Additionally, the effects of synchrotron radiation as a result of higher
currents are mitigated. Since synchrotron radiation is proportional to the
product of beam current and the fourth power of beam energy, the HER at
SuperKEKB emits $(7/8)^{4}=59\%$ as much synchrotron radiation per unit of
beam current compared to KEKB Wu and Summers (2015). This facilitates the
SuperKEKB collider to operate at a beam current twice the value of the KEKB.

The very high luminosity environment of SuperKEKB required significant
upgrades of the injection beams with high current and low emittance. The
upgraded accelerator complex houses a new electron-injection gun and a new
target for positron production. A new damping ring was installed for injection
of the positron beam with low-emittance, as well as for improving simultaneous
top-up injections needed for the high luminosity upgrade. The upgrade also
features completely redesigned lattices for the LER and HER, replacement of
short dipoles with longer ones in the LER, a new Titanium Nitride coated beam
pipe with antechambers to suppress the electron-cloud effect, a modified RF
system, and a completely redesigned interaction region Suetsugu et al. (2016).

The design of the beam parameters at SuperKEKB Ohnishi et al. (2013) follows
the “nano-beam” and the “crab-waist” schemes, which were originally proposed
for the SuperB-Factory in Italy Bona et al. (2007). Accordingly, the
transverse sizes of the beam bunches in the horizontal plane ($\sigma_{x}$)
are squeezed to have very small values and made to collide at a larger
horizontal crossing angle ($2\phi_{x}$) $=$ $83~{}{\rm{mrad}}$ at Belle II,
instead of =22 mrad at Belle. Thus, the effective size of the overlap region
($\tilde{\sigma}_{z}$) is much shorter than what it would have been in the
case of a normal head-on collision, which is given by the longitudinal size of
the beam bunches in the horizontal plane ($\sigma_{z}$) Ohnishi (2021).

The vertical beta function at the interaction point (IP) is constrained due to
the hour-glass effect as:
$\beta^{\star}_{y}>\tilde{\sigma}_{z}=\frac{\sigma^{\star}_{x}}{\sin\phi_{x}}=\frac{\sigma_{z}}{\Phi}$,
where $\Phi$ is the large Piwinski angle. With $\sigma_{z}$ of the order of 6
mm, the $\beta^{\star}_{y}$ is thus squeezed down to about
$400~{}{\upmu}{\rm{m}}$, which is much shorter than the real bunch length. In
addition to increasing the luminosity, a reduction of the interaction region
of the colliding beams restricts the vertex position along the beam axis, thus
providing an additional benefit of more precise estimation of the primary
vertex, which helps in the reconstruction of the complete event topology
during physics analysis.

The instantaneous luminosity in an $e^{-}e^{+}$ collider is

${\cal{L}}=\frac{N_{e^{-}}N_{e^{+}}f}{4\pi\sigma^{\star}_{x}\sigma^{\star}_{y}}R_{\cal{L}},$

where $N_{e^{-}}$ is the number of electrons per bunch, $N_{e^{+}}$ is the
number of positrons per bunch, $f$ is the collision frequency of the bunch,
$\sigma^{\star}_{x}$ and $\sigma^{\star}_{y}$ are the transverse beam-profile
sizes at the IP, and $R_{\cal{L}}$ is the luminosity-reduction factor (of the
order of unity) due to the finite beam-crossing angle. In terms of beam
currents $I_{\pm}=N_{e^{\pm}}ef$, the luminosity becomes

${\cal{L}}=\frac{N_{e^{\mp}}I_{e^{\pm}}}{e}\frac{1}{4\pi}\frac{R_{\cal{L}}}{\sigma^{\star}_{x}\sigma^{\star}_{y}},$

where $e$ is the charge of the electron. Each beam affects the stability of
the other, which can be characterized by the beam–beam tune shift parameters
given by

$\xi_{(x,y)}^{\pm}=\frac{r_{e}}{2\pi\gamma_{\pm}}\frac{N_{e^{\mp}}\beta^{\star}_{(x,y)}}{\sigma^{\star}_{(x,y)}\big{(}\sigma^{\star}_{x}+\sigma^{\star}_{y}\big{)}}R_{{\xi}_{(x,y)}},$

where $r_{e}$ is the classical radius of the electron, $\gamma_{\pm}$ is the
relativistic gamma factor of $e^{-}(e^{+})$ beams, and $R_{{\xi}_{(x,y)}}$ is
the geometric reduction factor (also of the order of unity) due to the hour-
glass effect. Putting all these factors altogether, we arrive at the following
expression for instantaneous luminosity:

${\cal{L}}=\frac{\gamma_{\pm}}{2er_{e}}\Big{(}1+\frac{\sigma^{\star}_{y}}{\sigma^{\star}_{x}}\Big{)}\frac{I_{e^{\pm}}\xi_{y}^{\pm}}{\beta^{\star}_{y}}\Big{(}\frac{R_{\cal{L}}}{R_{{\xi}_{y}^{\pm}}}\Big{)},$

where the design parameters for beam–beam tune shifts are $\xi_{(x,y)}$ $=$
$(0.0012,~{}0.0807)$ for HER and $=$ $(0.0028,~{}0.0881)$ for LER Ohnishi et
al. (2013). The horizontal/vertical beam sizes at the IP are reduced from
$\sigma_{x}^{\star}/\sigma_{y}^{\star}$ = 170 $\upmu$m/940 nm for HER and =147
$\upmu$m/940 nm for LER at Belle to =10.7 $\upmu$m/62 nm for HER and =10.1
$\upmu$m/48 nm for LER at Belle II, respectively Akai et al. (2018). The beam
currents for Belle were 1.19 A and 1.64 A for HER and LER, respectively,
compared to the design values of 2.6 A and 3.6 A for HER and LER,
respectively, at Belle II Akai et al. (2018). This allows one to improve upon
the value of instantaneous luminosity from ${\cal{L}}=2.1\times
10^{34}{~{}\rm{{cm^{2}}{s^{-1}}}}$ at Belle to $6.5\times
10^{35}{~{}\rm{{cm^{2}}{s^{-1}}}}$ at Belle II Aggarwal et al. (2022).

###  2.2 Detector Upgrade of Belle II

From the IP outward, the main components of the Belle II detector are
vertexing and tracking detectors, particle identification systems, calorimeter
and muon chambers, as shown in Figure 1 Abe et al. (2010). The tracking
detectors consist of an inner Silicon PiXel Detector (PXD), a Silicon Vertex
Detector (SVD) and a Central Drift Chamber (CDC). Two dedicated particle
identification systems are the Time-Of-Propagation (TOP) detector in the
barrel region and the Aerogel Ring-Imaging CHerenkov detector (ARICH) in the
forward endcap. These are surrounded by an Electromagnetic CaLorimeter (ECL)
and a superconducting solenoid providing a homogeneous magnetic field of 1.5
T. A $K_{L}^{0}$ and Muon detector (KLM) is the largest and outermost part of
the Belle II detector.

Figure 1: Schematic view of the Belle II detector Abe et al. (2010). The self-
annotated figure is based on overview available from SuperKEKB/Belle II public
page:
%****␣main.tex␣Line␣575␣****https://www.belle2.org/project/super_kekb_and_belle_ii
(accessed on 11 April 2022). Picture copyright and credit: KEK.

Some upgrades of the Belle II detector Abe et al. (2010); Altmannshofer et al.
(2019) over the Belle detector Abashian et al. (2002) are:

  * •

Vertexing:
In Belle, the beam pipe was at 15 mm Abe et al. (2004), the innermost layer of
a 4-layer silicon vertex detector Kibayashi (2006) was at 20 mm and the
outermost layer of the vertex detector was at a radius of 88 mm. In Belle II,
the beam pipe is at 10 mm, the inner two layers of the PXD, consisting of
silicon pixels, are closer to the IP at 14 mm and 22 mm, respectively, and the
outermost layer of the four layers of the SVD, consisting of silicon strips,
goes to a larger radius of 140 mm. The PXD is based on the Depleted Field
Effect Transistor (DEPFET) technology, which allows for thin sensors with 50
$\upmu$m thickness. The readout of the new silicon strip detector is based on
the APV25 chip, which has a much shorter shaping time to accommodate for
higher background rates in Belle II than the VAITA chip-based readout used at
Belle. As a result of these upgrades, considerably better performance is
expected in Belle II than Belle. For example, the vertex resolution at Belle
II is improved by the excellent spatial resolution of the two innermost pixel
detector layers.

  * •

Tracking:
The large volume CDC at Belle II, with 56 layers organized in 9 super-layers,
has smaller drift cells than in Belle. CDC starts just outside the expanded
silicon strip detector, and extends to a larger radius of 1130 mm in Belle II
as compared to 880 mm in Belle. The measured spatial resolution of the CDC is
about 100 $\upmu$m, while the relative precision of the $dE/dx$ measurement
for particles with an incident angle of $90^{\circ}$ is around $12\%$. The
angular resolution achieved between tracks is $\sim$4.5 mrad. The efficiency
to reconstruct $K_{S}^{0}\to\pi^{-}\pi^{+}$ decays in Belle II is also
improved because the silicon strip detector occupies a larger volume.

  * •

Particle Identification:
Belle II has two completely new, more compact particle identification devices
of the Cherenkov imaging type: TOP in the barrel and ARICH in the endcap
regions. Both detectors are equipped with very fast read-out electronics,
leading to very good kaon versus pion separation in the kinematic limits of
the experiment. The two Cherenkov detectors are designed to differentiate
between $K$ and $\pi$ particles over the entire momentum range, and also
differentiates among $\pi$, $\mu$, and $e$ below 1 GeV/c.

  * •

Calorimetry:
The ECL is made of CsI(Tl) scintillation crystals of size 6 cm $\times$ 6 cm
each with high light output, a short radiation length, and good mechanical
properties, covering the range of 12∘ $<\theta<$ 155∘ in the polar angle,
e.g., 90% of solid angle coverage in the center-of-mass system. The ECL is
divided into two parts: the barrel and the endcap. While the barrel part
consists of 6624 crystals, the endcap part consists of 2112 crystals. The new
electronics of the ECL are of the wave-form-sampling type, which has
particular relevance in missing-energy studies by reducing the noise due to
pile up considerably. The ECL is able to detect neutral particles in a wide
energy range, from 20 MeV up to 4 GeV, with a high resolution of
$\sigma_{E}/E$ $=$ 4% at 100 MeV, and angular resolution of 13 mrad (3 mrad)
at low (high) energies. This gives a mass resolution for reconstructing
$\pi^{0}\to\gamma\gamma$ of about 4.5 MeV/c2 Bevan et al. (2014).

  * •

$K_{L}^{0}$ and Muon Detection:
The $K_{L}^{0}$ and muon detector (KLM) at Belle was based on glass-electrode
resistive plate chambers (RPC). Since larger backgrounds are expected in the
high luminosity environment at Belle II, the upgraded KLM system consists of
RPC only in some parts of the barrel. The two innermost layers in the barrel
and the entire endcap section of KLM at Belle II consist of layers of
scintillator strips with wavelength shifting fibers, read out by silicon
photomultiplier (SiPMs) as light sensors Balagura et al. (2006). Although the
high neutron background can cause damage to the SiPMs, the upgraded KLM has
been demonstrated to operate reliably during irradiation tests by
appropriately setting the discrimination thresholds.

###  2.3 Daq Upgrade of Belle II

The new data acquisition (DAQ) system Yamada et al. (2015) meets the
requirements of considerably higher event rates at Belle II. It consists of a
Level One (L1) Iwasaki et al. (2011) and High Level Trigger (HLT). The L1
trigger has a latency of 5 $\upmu$s and a maximum trigger output rate of 30
kHz, limited by the read-in rate of the DAQ. The HLT must suppress online
event rates to 10 kHz for offline storage using complete reconstruction with
all available information from the entire detector. To enable readout from
high-speed data transmission, a peripheral component interconnect express
based readout module (PCIe40) with high data throughput of up to 100
Gigabytes/s was adopted for the upgrade of the Belle II DAQ system Zhou et al.
(2021). The trigger system at Belle II achieves almost 100 $\mathrm{\%}$
trigger efficiency for $\Upsilon(4S)\rightarrow B\bar{B}$ events and nearly
high efficiency for other physics processes of interest, e.g., $\tau$-pair
events.

##  3 Search Strategies

###  3.1 Event Topology

B-Factories typically operate at center-of-mass energies around the
$\Upsilon(4S)$ resonance, e.g., 10.58 GeV. Tau-pair production via
$e^{-}e^{+}$ annihilation in this energy regime leads to cleanly separated
event topology associated with the decay of each $\tau$ lepton, and are well
simulated by state-of-the-art event generators: KK2F Jadach et al. (2000);
Ward et al. (2003); Arbuzov et al. (2021), Tauola Jadach et al. (1993);
Chrzaszcz et al. (2018) and Photos Barberio and Was (1994); Davidson et al.
(2016). Searches for LFV in $\tau$ decays in B factories exploit these event
characteristics, assuming that only one of the two $\tau$’s produced in the
$e^{-}e^{+}\to\tau^{-}\tau^{+}$ process could have decayed in this rare mode,
and the other $\tau$ decays via the allowed SM processes. By dividing the
event into a pair of hemispheres perpendicular to the thrust axis Brandt et
al. (1964); Farhi (1977) in the center-of-mass frame, $\tau$ decay products
can thus be identified as coming from the signal-side and the tag-side,
corresponding to decay via LFV and the SM decays of the $\tau$ lepton,
respectively.

###  3.2 Signal Characteristics

The characteristic feature of $\tau$ decays via LFV is that the final state
does not contain $\nu_{\tau}$. Thus, there is no missing momentum associated
with the signal-side, and the kinematics of the signal $\tau$ lepton can be
completely reconstructed from measurements of the final state particles.
Simulation studies for more than a hundred possible decays via LFV that can be
searched with such signal characteristics for each sign of the $\tau$ lepton,
are possible with recent updates of the Tauola event generator Chrzaszcz et
al. (2018); Antropov et al. (2019); Banerjee et al. (2021), which have been
seamlessly integrated into the software of the Belle II experiment.

A very interesting feature of $\tau$-pair production in $e^{-}e^{+}$
annihilation is that the energy of each $\tau$ lepton is known to be exactly
half of the center-of-mass (CM) energy of the collision, except for
corrections due to initial and final state radiations. Therefore, the
uncertainty of the energy of the $\tau$ lepton is independent of the
performance of the detector, and is known from the beam energy spread of
SuperKEKB to be approximately 5 MeV Altmannshofer et al. (2019).

As a first example of the signal mode, let us consider
$\tau^{-}\to\ell^{-}\gamma$ decays, which are predicted with rates just lower
than the current experimental bounds in the widest variety of NP models and
are hence regarded as “golden modes” in searches for LFV. The total energy in
the CM frame of the $\tau$ decay products in the signal-side is
$E_{\ell\gamma}^{\mathrm{CM}}=\sqrt{s}/2$, and the invariant mass of the
$\ell\gamma$ pair can be calculated as
$m_{\ell\gamma}=2\sqrt{p_{\ell}E_{\gamma}}\sin(\frac{\theta}{2})$, where
$p_{\ell}$ is the magnitude of the three-momentum of the lepton, $E_{\gamma}$
is the energy of the photon, and $\theta$ is the opening angle between them.
The invariant mass is ideal as a discriminating variable, because its
resolution is given by Fabre (1992):

$\frac{\Delta
m_{\ell\gamma}}{m_{\ell\gamma}}=\frac{1}{2}\sqrt{\Big{(}\frac{\Delta
p_{\ell}}{p_{\ell}}\Big{)}^{2}+\Big{(}\frac{\Delta
E_{\gamma}}{E_{\gamma}}\Big{)}^{2}+\Big{(}\frac{\Delta\theta}{\tan\frac{\theta}{2}}\Big{)}^{2}}$

which simultaneously combines all the available experimental precision on the
measured energy/momentum from the calorimeter/tracking systems with the
measured uncertainties on the position measurements of the observable final
state decays products. The resolution of this kinematic variable is further
improved by considering the beam-energy-constrained mass, $M_{\mathrm{bc}}$,
given as:

$M_{\mathrm{bc}}=\sqrt{(E_{\mathrm{beam}}^{\mathrm{CM}})^{2}-|\vec{p}_{\ell\gamma}^{\mathrm{~{}CM}}|^{2}},$

where $E_{\mathrm{beam}}^{\mathrm{CM}}=\sqrt{s}/2$ and
$\vec{p}_{\ell\gamma}^{\mathrm{~{}CM}}$ is the sum of the lepton and photon
momenta in the CM frame, because the resolution of
$E_{\mathrm{beam}}^{\mathrm{CM}}$ comes from the accelerator instead of the
detector.

The beam-energy-constrained $\tau$ mass, labeled somewhat differently as
$\mathrm{m}_{\mathrm{E}C}$ for the BABAR search Aubert et al. (2010), is
typically obtained from a kinematic fit that constrains the CM energy of the
$\tau$ to be $\sqrt{s}/2$. Its resolution was further improved in the BABAR
search by assigning the origin of the photon candidate to the point of the
closest approach of the signal lepton track to the $e^{-}e^{+}$ collision
axis. Figure 2 shows a comparison study using a simulated sample of signal
$\tau^{-}\to\mu^{-}\gamma$ events, where the resolution of the invariant mass
is crudely estimated to be 18.2 MeV/c2 from a single Gaussian fit to the peak
of the distribution, while that of $\mathrm{m}_{\mathrm{E}C}$ is 12.1 MeV/c2,
which improves further to 8.3 MeV/c2 after the vertex constraint.

Figure 2: Invariant mass (left), $\mathrm{m}_{\mathrm{E}C}$ without vertex
constraint (middle) and $\mathrm{m}_{\mathrm{E}C}$ with vertex constraint
(right) obtained from simulated samples during analysis strategy development
studies for $\tau^{-}\to\mu^{-}\gamma$ search at the BABAR experiment Aubert
et al. (2010).

The most distinguishing feature of signal events is obtained by considering
the characteristic mass of the decay products of the LFV $\tau$ decays along
with the normalized difference in their energy from half the center-of-mass
energy in $e^{-}e^{+}$ annihilation, so that the search can be uniformly
performed at energies other than the $\Upsilon(4S)$ peak, to take advantage of
the larger luminosity including all the recorded data:

$\Delta E/\sqrt{s}=(E_{\ell\gamma}^{\mathrm{CM}}-\sqrt{s}/2)/\sqrt{s}.$

The signal events are clustered around $M_{\mathrm{bc}}\sim m_{\tau}$ and
$\Delta E/\sqrt{s}\sim 0$ in the two-dimensional plots of $\Delta E$ vs.
$M_{\mathrm{bc}}$, as shown in Figures 3 and 4 for $\tau^{-}\to e^{-}\gamma$
(left) and $\tau^{-}\to\mu^{-}\gamma$ (right) searches at Belle Abdesselam et
al. (2021) and BABAR Aubert et al. (2010) experiments, respectively, where the
variable $m_{EC}$ refers to the beam-energy-constrained mass in the latter, as
mentioned earlier.

Figure 3: Two-dimensional distributions of $\Delta E/\sqrt{s}$ versus
$M_{\mathrm{bc}}$ for Belle searches Abdesselam et al. (2021) for $\tau^{-}\to
e^{-}\gamma$ (left) and $\tau^{-}\to\mu^{-}\gamma$ (right) decays. Black
points are data, blue squares are $\tau^{-}\to\ell^{-}\gamma$ signal MC
events, and magenta ellipses show the signal region ($\pm 2\sigma$ region).
This figure has been reprinted with permission from Ref. Abdesselam et al.
(2021).

Figure 4:  Two-dimensional distributions in the $\mathrm{m}_{\mathrm{E}C}$ vs.
$\Delta{\rm{E}}$ plane for the BABAR searches Aubert et al. (2010) for
$\tau^{-}\to e^{-}\gamma$ (left) and $\tau^{-}\to\mu^{-}\gamma$ (right)
decays. Data are shown as dots and contours containing 90% (50%) of signal MC
events are shown as light-yellow (dark-green) shaded regions, along with the
$2\sigma$ contours shown as black ellipses. This figure has been reprinted
with permission from Ref. Aubert et al. (2010).

All analyses are developed in a blind manner, e.g., optimizing the event
selection before looking at the data events inside the signal region, to avoid
experimental bias in the search for LFV $\tau$ decays. The search sensitivity
can be optimized to give the smallest expected upper limit in the background-
only hypothesis inside a $2\sigma$ ellipse, for example, amongst other
possible choices. A typical $2\sigma$ signal region is defined as the
following elliptical regions:

$\displaystyle\frac{(M_{\mathrm{bc}}-\mu_{M_{\mathrm{bc}}})^{2}}{(2\sigma_{M_{\mathrm{bc}}})^{2}}$
$\displaystyle+$ $\displaystyle\frac{(\Delta E/\sqrt{s}-\mu_{\Delta
E/\sqrt{s}})^{2}}{(2\sigma_{\Delta E/\sqrt{s}})^{2}}<1.0,$
$\displaystyle\sigma_{M_{\mathrm{bc}}}$ $\displaystyle=$ $\displaystyle
0.5(\sigma_{M_{\mathrm{bc}}}^{\mathrm{high}}+\sigma_{M_{\mathrm{bc}}}^{\mathrm{low}}),$
$\displaystyle\sigma_{\Delta E/\sqrt{s}}$ $\displaystyle=$ $\displaystyle
0.5(\sigma_{\Delta E/\sqrt{s}}^{\mathrm{high}}+\sigma_{\Delta
E/\sqrt{s}}^{\mathrm{low}}).$

Here, $\sigma^{\mathrm{high/low}}_{M_{\mathrm{bc}}}$ and
$\sigma^{\mathrm{high/low}}_{\Delta E/\sqrt{s}}$ are the widths on the
higher$/$lower side of the peak obtained by fitting the signal distribution to
an asymmetric Gaussian function Hayasaka et al. (2008).

For the Belle search Abdesselam et al. (2021), the resolutions are
$\sigma_{M_{\mathrm{bc}}}^{\mathrm{high/low}}=11.55\pm 0.27/10.59\pm 0.19$
MeV/c2 and $\sigma_{\Delta E/\sqrt{s}}^{\mathrm{high/low}}=(6.1\pm
0.7)/(4.4\pm 0.3)\times 10^{-3}$ for $\tau^{-}\to e^{-}\gamma$ events; and
$\sigma_{M_{\mathrm{bc}}}^{\mathrm{high/low}}=11.08\pm 0.08/7.46\pm 0.23$
MeV/c2 and $\sigma_{\Delta E/\sqrt{s}}^{\mathrm{high/low}}=(5.6\pm
0.4)/(4.2\pm 0.2)\times 10^{-3}$ for $\tau^{-}\to\mu^{-}\gamma$ events. The
mean values are $\langle{M_{\mathrm{bc}}}\rangle=1.79$ MeV/c2 and
$\langle{\Delta E/\sqrt{s}}\rangle=-1.0\times 10^{-3}$ for $\tau^{-}\to
e^{-}\gamma$ events, and $\langle{M_{\mathrm{bc}}}\rangle=1.78$ MeV/c2 and
$\langle{\Delta E/\sqrt{s}}\rangle=-0.6\times 10^{-3}$ for
$\tau^{-}\to\mu^{-}\gamma$ events. For the BABAR search Aubert et al. (2010),
the resolutions for $\mathrm{m}_{\mathrm{E}C}$ and $\Delta{\rm{E}}$ are 8.6
MeV/c2 and 42.1 $\mathrm{\,Me\kern-1.00006ptV}$ for $\tau^{-}\to e^{-}\gamma$,
and 8.3 MeV/c2 and 42.2 $\mathrm{\,Me\kern-1.00006ptV}$ for
$\tau^{-}\to\mu^{-}\gamma$, respectively, centered on 1777.3 MeV/c2 and
$-$21.4 $\mathrm{\,Me\kern-1.00006ptV}$ for $\tau^{-}\to e^{-}\gamma$, and
1777.4 MeV/c2 and $-$18.3 $\mathrm{\,Me\kern-1.00006ptV}$ for
$\tau^{-}\to\mu^{-}\gamma$. Shifts from zero for
$\langle\Delta{\rm{E}}\rangle$ are mostly due to initial and final state
radiations.

The mass and energy kinematic variables typically have a small correlation
arising from initial and final state radiation, as well as energy/momentum
scale calibration effects. For the BABAR search Aubert et al. (2010), the
correlation was estimated to be $-$8.5% and $-$8.4% for the $\tau^{-}\to
e^{-}\gamma$ and $\tau^{-}\to\mu^{-}\gamma$ decays, respectively, around the
core region. Without the beam-energy constraint, the correlation between the
invariant mass and energy variables are typically much higher.

LFV process in $\tau$ decays containing a resonance in the final state are
identified by the presence of a peak in the invariant mass of the daughter
particles in the simulation of the signal process. For example, distributions
of invariant mass of the $\pi^{+}\pi^{-}$, $K^{+}K^{-}$,
$\pi^{+}\pi^{-}\pi^{0}$, $K^{+}\pi^{-}$, $\pi^{+}K^{-}$ and $\pi^{+}\pi^{-}$
systems in the signal-side are studied and confirmed to contain the respective
resonances in searches for $\tau^{-}\to\mu^{-}\rho^{0}$,
$\tau^{-}\to\mu^{-}\phi$, $\tau^{-}\to\mu^{-}\omega$,
$\tau^{-}\to\mu^{-}K^{\ast 0}$, $\tau^{-}\to\mu^{-}\bar{K}^{\ast 0}$ and
$\tau^{-}\to\mu^{-}f_{0}(980)$ decays, respectively, performed by the Belle
experiment Miyazaki et al. (2011, 2009). The selected mass regions ensure that
the signal is unambiguously selected in the corresponding searches.

###  3.3 Background Suppression

Background events containing leptons from decays of heavy quarks are easily
suppressed by appropriate cuts on Fox-Wolfram moments Fox and Wolfram (1978),
and on the invariant mass of all decay products on the tag-side. The
characteristic difference between $\tau$-pairs events with LFV decays and
backgrounds consisting of generic $\tau$-pair, di-lepton, two-photon
production and $q\bar{q}$ processes (where $q=u,~{}d\mathrm{~{}or~{}}s$), in
the number of neutrinos in the signal-side and tag-side, as defined by the
event topology in Section 3.1, are shown in Table 2.

Table 2: Number of neutrinos in the event for signal and background processes. # of $\nu$’s |  LFV Decays |  Generic $\tau$-Pair |  Other Backgrounds
---|---|---|---
Signal-side |  0 |  1–2 |  0
Tag-side |  1–2 |  1–2 |  0

Since decay products of the $\tau$ decay via LFV in the signal-side do not
contain any neutrino, the direction of the $\tau$ lepton in the tag-side can
be precisely obtained in the center-of-mass frame by reversing the total
momentum of the signal-side. This allows for good kinematic reconstruction of
the missing mass in the tag-side, assuming that in the CM frame, the tag-side
$\tau$ momentum is opposite that of the signal-side $\tau$ momentum and that
its energy is constrained to be half the center-of-mass energy. Thus,
selection of events with small values of the square of the missing mass
($m_{\nu}^{2}$) in the tag-side play an important role in the suppression of
the background events Abdesselam et al. (2021).

Additional selection criteria are also used to suppress the backgrounds in the
different LFV decay modes, which are mostly accidental in nature, except in
$\tau^{-}\to e^{-}\gamma$ and $\tau^{-}\to\mu^{-}\gamma$ searches. The
dominant background in the searches arise from $\tau^{+}\tau^{-}$ events
decaying via the $\tau^{\pm}\rightarrow e^{\pm}\nu_{e}\nu_{\tau}$
($\tau^{\pm}\rightarrow\mu^{\pm}\nu_{\mu}\nu_{\tau}$) channel with a photon
coming from initial-state radiation or beam background. The $e^{+}e^{-}\gamma$
and $\mu^{+}\mu^{-}\gamma$ events are subdominant, and are estimated to
contribute to <5% of the total backgrounds in the Belle search Abdesselam et
al. (2021). Contributions from other sources of backgrounds, such as two-
photon and $q\bar{q}$ processes, are estimated to be quite small in the signal
region.

Furthermore, each component of the background processes has distinctive
features as visible in their respective two-dimensional distributions in the
$(\Delta M,\Delta{\rm{E}})$ plane, where $\Delta M$ denotes the difference
between the characteristic mass of the system of $\tau$-daughters and the
well-known mass of the $\tau$-lepton = $(1776.86\pm
0.12)\mathrm{\,Me\kern-1.00006ptV}$ et al. (2020), and $\Delta{\rm{E}}$, as
defined above. The shapes of the leading backgrounds in search of
$\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ decays as performed at the BABAR experiment
Lees et al. (2010) are shown in Figure 5, where the red box indicates the
rectangular boundaries of a generic region mostly populated by the signal
processes. The SM $\tau\tau$ background events are generally restricted to
small negative values of both $\Delta M$ and $\Delta{\rm{E}}$ variables,
because the reconstruction of signal event topology does not account for the
neutrinos present in SM $\tau$ decays. QED background events are mostly
dominated by di-lepton production as the main underlying hard process and
typically lie within a narrow horizontal band across the $\Delta M$ variable
centered around slightly positive values of $\Delta{\rm{E}}$, due to the
presence of a pair of extra charged particles in such events. The QCD
background events from various $q\bar{q}$ processes tend to populate the plane
uniformly across the $\Delta M$ variable and drop towards large values of the
$\Delta{\rm{E}}$ variable. The expected background rates inside the signal
region can be obtained by fitting the observed data in the $(\Delta
M,\Delta{\rm{E}})$ plane to a sum of probability density functions. Such data-
driven estimates, based on the shapes predicted by respective simulation
samples and validated by data-driven control regions, scale well with larger
data statistics. Thus, the background uncertainties can be controlled in a
statistical manner, which is very useful in rare searches with high luminosity
data sets.

Figure 5: Two-dimensional distributions for $\tau\tau$, QED, and $q\bar{q}$
background processes in the $(\Delta M,\Delta{\rm{E}})$ plane obtained from
simulated samples during analysis strategy development studies for
$\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ search at the BABAR experiment Lees et al.
(2010). Rectangular box-shaped regions mostly populated by the signal
processes are shown in red.

###  3.4 Upper Limit Estimation

No excess of events has ever been observed in searches for LFV in $\tau$
decays. The upper limit at 90% confidence level (CL) on signal branching
fraction (${\cal{B}}^{90}_{UL}$) is calculated as:

${\cal{B}}^{90}_{UL}=\frac{S^{90}_{UL}}{2N_{\tau\tau}\epsilon},$

where $N_{\tau\tau}$ is the number of $\tau$-pairs produced, $\epsilon$ is the
reconstruction efficiency of the signal decay mode, and $S^{90}_{UL}$ is the
90% CL upper limit on number of signal events. The factor of two enters into
the denominator because either one of the two $\tau$ leptons produced in the
event can decay into the rare signal channel coming from LFV.

####  3.4.1 Number of $\tau$-Pairs Produced

$N_{\tau\tau}$ is obtained by summing over the product of luminosity
$({\cal{L}})$ and the $\tau$-pairs production cross-section
$(\sigma_{\tau\tau})$ at each of the center-of-mass energies ($\sqrt{s}$)
where the search is conducted. The cross-section of $\tau$-pair production in
$e^{-}e^{+}$ annihilation follows a characteristic ${1/s}$ dependence, but
receives additional contribution from decays of the $\Upsilon(nS)$ resonances
for $n=1,~{}2{~{}\rm{and}~{}}3$, according to the known branching fractions et
al. (2020): ${\cal{B}}(\Upsilon(1S)\to\tau^{-}\tau^{+})=(2.6\pm 0.1)\%$,
${\cal{B}}(\Upsilon({{2}}S)\to\tau^{-}\tau^{+})=(2.0\pm 0.2)\%$ and
${\cal{B}}(\Upsilon({{3}}S)\to\tau^{-}\tau^{+})=(2.3\pm 0.3)\%$. Beyond the
open-beauty threshold, contributions from resonances with $n\geq 4$ are
negligible.

The Belle experiment collected data with center-of-mass energies around the
peak of $\Upsilon(nS)$ resonances corresponding to luminosities of
$5.7\mbox{\,fb}^{-1}$, $24.9\mbox{\,fb}^{-1}$ and $2.9\mbox{\,fb}^{-1}$ at
$n=1,~{}2{~{}\rm{and}~{}}3$, respectively, while the BABAR experiment
collected luminosities of $13.6\mbox{\,fb}^{-1}$ and $28.0\mbox{\,fb}^{-1}$ at
center-of-mass energies corresponding to the $\Upsilon(nS)$ peak at
$n=2{~{}\rm{and}~{}}3$, respectively, as reported in Table 3.2.1 in the
reference Bevan et al. (2014). The statistical errors on these measured
luminosities are much smaller than the systematic errors, which are estimated
to be 1.4% at the Belle experiment, and 0.7% (0.6%) at the BABAR experiment
for $n=2{~{}\rm{and}~{}}3$, respectively Bevan et al. (2014).

The numbers of $\Upsilon(nS)$ produced in the Belle experiment are $(102\pm
2)\times 10^{6}$, $(158\pm 4)$ $\times 10^{6}$ and $(11\pm 0.3)\times 10^{6}$
for $n=1,~{}2{~{}\rm{and}~{}}3$, respectively, while in the BABAR experiment,
the numbers are $(98.3\pm 0.9)\times 10^{6}$ and $(121.3\pm 1.2)\times 10^{6}$
for $n=2{~{}\rm{and}~{}}3$, respectively, as obtained from Table 3.2.2 in the
reference Bevan et al. (2014). By dividing the numbers of $\Upsilon(nS)$
resonances produced with its corresponding luminosity, the resonant production
cross-sections are estimated for $\Upsilon(1S)$ to be $(17.89\pm 0.43)~{}nb$
in the Belle experiment, for $\Upsilon(2S)$ to be $(6.35\pm 0.18)~{}nb$ and
$(7.23\pm 0.08)~{}nb$ in the Belle and BABAR experiments, and for
$\Upsilon(3S)$ to be $(3.79\pm 0.12)~{}nb$ and $(4.33\pm 0.05)~{}nb$ in the
Belle and BABAR experiments, respectively. For the purposes of averaging the
measured values, the observed differences at $n=2{~{}\rm{and}~{}}3$ resonances
are accounted for by calculating the PDG-style scale factors et al. (2020)
$S=\sqrt{\chi^{2}/(N-1)}$ equal to 4.38 and 4.24, respectively, with $N=2$.
Thus, the average values are estimated to be $(7.08\pm 0.33)~{}nb$ for
$\Upsilon(2S)$ and $(4.24\pm 0.20)~{}nb$ for $\Upsilon(3S)$.

The total $\sigma_{\tau\tau}$ is obtained by adding the contributions to
$\tau$-pairs production from the continuum Banerjee et al. (2008) and the
contributions from the decays of the $\Upsilon(nS)$ resonances for
$n=1,~{}2{~{}\rm{and}~{}}3$. The estimated values of total $\sigma_{\tau\tau}$
at the peak of $\Upsilon(nS)$ resonances and at
$60\mathrm{\,Me\kern-1.00006ptV}$ below the corresponding resonances (labelled
with a “—off”) are listed in Table 3.

Table 3: $\sigma_{\tau\tau}$ at different center-of-mass energies
corresponding to data-taking at the B-Factories.
---
$\sqrt{s}$ |  $\sigma_{\tau\tau}$ $(nb)$ |  $\sqrt{s}$ |  $\sigma_{\tau\tau}$ $(nb)$
$\Upsilon(1S)$—off |  $1.142\pm 0.004$ |  $\Upsilon(1S)$ |  $1.593\pm 0.021$
$\Upsilon(2S)$—off |  $1.026\pm 0.004$ |  $\Upsilon(2S)$ |  $1.157\pm 0.017$
$\Upsilon(3S)$—off |  $0.966\pm 0.003$ |  $\Upsilon(3S)$ |  $1.052\pm 0.014$
$\Upsilon(4S)$—off |  $0.928\pm 0.003$ |  $\Upsilon(4S)$ |  $0.919\pm 0.003$
$\Upsilon(5S)$—off |  $0.881\pm 0.003$ |  $\Upsilon(5S)$ |  $0.873\pm 0.003$

####  3.4.2 Efficiency of Signal Reconstruction

The signal reconstruction efficiency receives multiplicative reduction factors
corresponding to the application of trigger, acceptance, and event topology
requirements, particle identification criteria, background suppression and
choice of the signal region in the two-dimensional plane given by mass versus
the normalized difference of energy of the $\tau$ decay products and
$\sqrt{s}/2$. At Belle and BABAR, the signal efficiencies were estimated to
lie approximately between $2\%$ to $12\%$, depending on the different decay
channels. For example, the overall signal efficiency estimated for the search
for $\tau^{-}\to e^{-}\eta^{\prime}$ reconstructed via the
$\eta^{\prime}\to\rho(\to\pi^{-}\pi^{+})\gamma$ and
$\eta^{\prime}\to\pi^{-}\pi^{+}\eta(\to\gamma\gamma))$ decay modes is
$(0.294\times 1\times 4.76+0.445\times 0.3943\times 4.27)\%=2.1\%$ Miyazaki et
al. (2007), while the search for $\tau^{-}\to\mu^{+}e^{-}e^{-}$ decays have an
efficiency of 11.5% Hayasaka et al. (2010). In the Belle II experiment, an
increase in the signal efficiency can be expected due to higher trigger
efficiencies, improvements in the vertex reconstruction, charged track and
neutral meson reconstructions, and particle identification. Refinements in the
analysis techniques will produce a more accurate understanding of the physics
backgrounds and would thus contribute to an increase in the signal detection
efficiency, which directly translates into higher sensitivities in searches
for LFV.

####  3.4.3 Upper Limit on the Number of Signal Events

In the case of searches with very low counts, the search becomes a single-bin
counting experiment following a Poisson probability distribution, with the
mean count given by the expected number of background events ($b$) and
possibly some signal events ($s$). The likelihood function $({\cal{L}})$ is
thus described by:

${\cal{L}}(s,b)=\frac{e^{-(s+b)}(s+b)^{N}}{N!},$

where $N$ is the number of observed events. If the experimental resolution of
the discriminating variables allows multiple bins, the difference in shapes of
the discriminating variables between the signal and background distributions
can be exploited in an extended unbinned maximum likelihood fit using:

${\cal{L}}(s,b)=\frac{e^{-(s+b)}}{N!}\prod_{i=1}^{N}(s\cdot
PDF^{sig}_{i}+b\cdot PDF^{bkg}_{i}),$

where $i$ indicates the $i$-th event, $PDF^{sig}$ and $PDF^{bkg}$ are the
probability density functions (PDF) for signal and sum of all background
process, respectively.

$S^{90}_{UL}$ is obtained by considering
${\cal{L}}(s)=\int_{0}^{\infty}{\cal{L}}(s,b)db$, and integrating the
likelihood ${\cal{L}}(s)$ up to the value that includes 90% of the total
integral of the likelihood function, following a flat prior Bayesian
prescription Helene (1983). Alternatively, following a Frequentist
prescription Narsky (2000), a toy Monte Carlo approach is used to generate
numerous samples with sizes that follow a Poisson distribution about the mean
value being given by the number of observed events. Each sample is then fitted
to obtain the number of signal and background events using the same extended
unbinned maximum likelihood fit procedure as that applied to the data.
$S^{90}_{UL}$ is obtained by varying the true branching fraction of the signal
such that 90% of the samples yield a fitted number of signal events greater
than the number of signal events in the observed data sample. In the unified
approach for finding confidence levels Feldman and Cousins (1998), the order
of samples in the acceptance interval for a specific value of the number of
signal events follows an ordering principle based on likelihood ratios, where
the denominator is determined by the best fit value in each sample.

In order to have an unbiased estimate of the expected sensitivity, a blinding
procedure should be followed to predict the expected background rate inside
the signal region (SR), which does not depend on the observed data inside a
blinding region (BR), defined as a part of a broad fit range (FR), but hiding
data events inside the SR. For well-controlled modeling of the total
background PDF, the number of expected background events ($N^{bkg}_{SR}$)
inside the SR can then be estimated directly from the data $N^{data}_{FR-BR}$
outside the blinded region using the formula:

$N^{bkg}_{SR}=\frac{\int_{SR}PDF_{bkg}}{\int_{FR-BR}PDF_{bkg}}\times
N^{data}_{FR-BR}$

where $\int_{SR}PDF_{bkg}$ and $\int_{FR-BR}PDF_{bkg}$ are the integrals of
the background probability density functions over the signal region and the
non-blinded parts of the fit region.

###  3.5 Systematic Uncertainties

In terms of the number of $\tau$ decays being studied,
$t=2N_{\tau\tau}\epsilon$, the number of signal events are written as $(s=\mu
t)$, where $\mu$ is the branching fraction of the signal process and the
normalization factor ${t}$ includes uncertainities on luminosity, cross-
section and the signal efficiency. The upper limit ${\cal{B}}^{90}_{UL}$
including all systematic effects using the technique of Cousins and Highland
Cousins and Highland (1992), is calculated by propagating all the measured
uncertainties onto the number of signal events $(s)$ and background events
$(b)$.

Implementation of systematic effects in the POLE (POisson Limit Estimator)
program Conrad et al. (2003) is based on the following likelihood function,
which is a convolution of a Poisson distribution with two Gaussian resolution
functions corresponding to the signal normalization factor and background, as
described by the following formula:

${{\cal L}}((\mu,t,b)={(\mu t+b)^{N}e^{-(\mu t+b)}\over
N!}\frac{1}{2\pi\sigma_{t}\sigma_{b}}e^{-{1\over
2}\left({\hat{t}-t\over\sigma_{t}}\right)^{2}-{1\over
2}\left({\hat{b}-b\over\sigma_{b}}\right)^{2}},$

where $\hat{t}$ and $\hat{b}$ are the average estimates corresponding to
measured uncertainties of $\sigma_{t}$ and $\sigma_{b}$, respectively.

In searches for rare processes such as LFV in $\tau$ decays, often a very
small number of events are expected in the signal region. Sometimes the
sensitivity of the search cannot easily distinguish a very small number of
signal events from the background-only hypothesis, and inappropriately tends
to exclude an unusually small signal value. To overcome such difficulties, the
upper limits can be calculated using the $CL_{s}$ method Junk (1999); Read
(2002), where the $CL_{s}$ is defined as the ratio of confidence levels for
the signal-plus-background hypothesis normalized by the confidence level for
the background-only hypothesis. Asymptotic calculations of the likelihood
ratios used as the test statistic in such methods allow for a computationally
efficient estimate of the $CL_{s}$ intervals Cowan et al. (2011).

A Neyman construction Neyman (1937) of $CL_{s}$ upper limits including
systematic uncertainties is provided by the HistFactory implementation Moneta
et al. (2010); Cranmer et al. (2012), based on the likelihood function
$\mathcal{L}(\mu,\theta_{j})$ defined as:

$\mathcal{L}(\mu,\theta_{j})=\prod_{\textrm{channel~{}}}\prod_{\textrm{category~{}}}\left[\prod_{i}\text{Poisson}(N_{i}|\mu\cdot
t_{i}+b_{i})\prod_{j}\text{Lognormal}(\theta_{j}|0,1)\right],$

where $N_{i}$ is the number of events observed in the $i^{th}$ bin with signal
normalization factor and background predictions given by $t_{i}$ and $b_{i}$,
respectively, of a multicategorical search describing, for example, different
tag-side decay modes each with different sensitivity over possibly multiple
decay channels of the signal mode. The systematic uncertainties are
constrained by nuisance parameters $\theta_{j}$ corresponding to various scale
factors as determined from dedicated calibration constants of efficiency
measurements and are obtained from simulation studies or analysis of control
regions in the data.

The HistFactory allows for the calculation of upper limits in both Bayesian
and Frequentist interpretations Cranmer et al. (2022), with slightly different
treatments of the nuisance parameters. While in the former interpretation, the
nuisance parameters are eliminated by marginalizing the posterior density,
using, for example, Markov Chain Monte Carlo integration, in the latter
interpretation, the nuisance parameters are determined by profiling the
likelihood function based on auxiliary measurements, such as control regions,
side-bands, or dedicated calibration measurements. Some uncertainties arising
from theoretical calculations or ad hoc estimates are not statistical in
nature and thus are not associated with auxiliary measurements. However, log-
normal probability density functions of nuisance parameters are used to
constrain all the uncertainties, by convention.

Bayesian limits can also be calculated using the Bayesian Analysis Toolkit
Caldwell et al. (2009).

##  4 Current Status and Future Prospects

Summary of observed limits obtained by CLEO, BABAR, Belle, ATLAS, CMS, and
LHCb experiments Amhis et al. (2021) are shown in Table 4 and Figure 6, along
with projections for two illustrative scenarios of luminosity ${\cal{L}}$ = 5
$\mbox{\,ab}^{-1}$ and 50 $\mbox{\,ab}^{-1}$ at the Belle II experiment
Aggarwal et al. (2022). Projections are extrapolated from expected limits
obtained at the Belle experiment. The expected limits for
$\tau^{-}\to\ell^{-}\gamma$ decays are obtained from Ref. Abdesselam et al.
(2021). We assume the presence of irreducible backgrounds for
$\tau^{-}\to\ell^{-}\gamma$ decays, thus approximating the sensitivity to
upper bounds as proportional to $1/\sqrt{\cal{L}}$. Given the expected number
of background events in each channel from the previous searches at the Belle
experiment and the improvements listed in Section 3.4.2, the background
expectations corresponding to the integrated luminosity at Belle II for all
other modes are still of the order of unity or less. For such accidental
backgrounds, the sensitivity for upper bounds is proportional to
$1/{\cal{L}}$, as discussed in Section 8.2.1.3 of Ref. Raidal et al. (2008).
The projections for the corresponding upper limits at Belle II are estimated
using the Feldman and Cousins approach Feldman and Cousins (1998).

Table 4: Current status of observed (obs) and expected (exp) upper limits (UL)
et al. (2020); Aggarwal et al. (2022).

---
| Observed Limits | Expected Limits
$\tau^{-}\to$ |  Experiment |  Luminosity |  UL (obs) |  Experiment |  Luminosity |  UL (exp)
$e^{-}\gamma$ |  Belle Abdesselam et al. (2021) |  988 $\mbox{\,fb}^{-1}$ |  5.6$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  9.0$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2010) |  516 $\mbox{\,fb}^{-1}$ |  3.3$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Edwards et al. (1997) |  4.68 $\mbox{\,fb}^{-1}$ |  2.7$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}\gamma$ |  Belle Abdesselam et al. (2021) |  988 $\mbox{\,fb}^{-1}$ |  4.2$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  6.9$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2010) |  516 $\mbox{\,fb}^{-1}$ |  4.4$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Ahmed et al. (2000) |  13.8 $\mbox{\,fb}^{-1}$ |  1.1$~{}\times~{}10^{-6}$ |  |  |
$e^{-}\pi^{0}$ |  Belle Miyazaki et al. (2007) |  401 $\mbox{\,fb}^{-1}$ |  8.0$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  7.3$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2007) |  339 $\mbox{\,fb}^{-1}$ |  1.3$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bonvicini et al. (1997) |  4.68 $\mbox{\,fb}^{-1}$ |  3.7$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}\pi^{0}$ |  Belle Miyazaki et al. (2007) |  401 $\mbox{\,fb}^{-1}$ |  1.2$~{}\times~{}10^{-7}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  7.1$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2007) |  339 $\mbox{\,fb}^{-1}$ |  1.1$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bonvicini et al. (1997) |  4.68 $\mbox{\,fb}^{-1}$ |  4.0$~{}\times~{}10^{-6}$ |  |  |
$e^{-}K_{S}^{0}$ |  Belle Miyazaki et al. (2010) |  671 $\mbox{\,fb}^{-1}$ |  2.6$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.0$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  469 $\mbox{\,fb}^{-1}$ |  3.3$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Chen et al. (2002) |  13.9 $\mbox{\,fb}^{-1}$ |  9.1$~{}\times~{}10^{-7}$ |  |  |
$\mu^{-}K_{S}^{0}$ |  Belle Miyazaki et al. (2010) |  671 $\mbox{\,fb}^{-1}$ |  2.3$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.0$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  469 $\mbox{\,fb}^{-1}$ |  4.0$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Chen et al. (2002) |  13.9 $\mbox{\,fb}^{-1}$ |  9.5$~{}\times~{}10^{-7}$ |  |  |
$e^{-}\eta$ |  Belle Miyazaki et al. (2007) |  401 $\mbox{\,fb}^{-1}$ |  9.2$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.2$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2007) |  339 $\mbox{\,fb}^{-1}$ |  1.6$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bonvicini et al. (1997) |  4.68 $\mbox{\,fb}^{-1}$ |  8.2$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}\eta$ |  Belle Miyazaki et al. (2007) |  401 $\mbox{\,fb}^{-1}$ |  6.5$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  8.0$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2007) |  339 $\mbox{\,fb}^{-1}$ |  1.5$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bonvicini et al. (1997) |  4.68 $\mbox{\,fb}^{-1}$ |  9.6$~{}\times~{}10^{-6}$ |  |  |
$e^{-}\eta^{\prime}$ |  Belle Miyazaki et al. (2007) |  401 $\mbox{\,fb}^{-1}$ |  1.6$~{}\times~{}10^{-7}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.2$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2007) |  339 $\mbox{\,fb}^{-1}$ |  2.4$~{}\times~{}10^{-7}$ |  |  |
$\mu^{-}\eta^{\prime}$ |  Belle Miyazaki et al. (2007) |  401 $\mbox{\,fb}^{-1}$ |  1.3$~{}\times~{}10^{-7}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.2$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2007) |  339 $\mbox{\,fb}^{-1}$ |  1.4$~{}\times~{}10^{-7}$ |  |  |
$e^{-}f_{0}(980)$ |  Belle Miyazaki et al. (2009) |  671 $\mbox{\,fb}^{-1}$ |  6.8$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  9.5$~{}\times~{}10^{-10}$
$\mu^{-}f_{0}(980)$ |  Belle Miyazaki et al. (2009) |  671 $\mbox{\,fb}^{-1}$ |  6.4$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  9.1$~{}\times~{}10^{-10}$
$e^{-}\rho^{0}$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  1.8$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  3.8$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  451 $\mbox{\,fb}^{-1}$ |  4.6$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  2.0$~{}\times~{}10^{-6}$ |  |  |

Table 5: Cont.

---
| Observed Limits | Expected Limits
$\tau^{-}\to$ |  Experiment |  Luminosity |  UL (obs) |  Experiment |  Luminosity |  UL (exp)
$\mu^{-}\rho^{0}$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  1.2$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  5.5$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  451 $\mbox{\,fb}^{-1}$ |  2.6$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  6.3$~{}\times~{}10^{-6}$ |  |  |
$e^{-}\omega$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  4.8$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.0$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2008) |  384 $\mbox{\,fb}^{-1}$ |  1.1$~{}\times~{}10^{-7}$ |  |  |
$\mu^{-}\omega$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  4.7$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.4$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2008) |  384 $\mbox{\,fb}^{-1}$ |  1.0$~{}\times~{}10^{-7}$ |  |  |
$e^{-}K^{\ast 0}$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  3.2$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  6.7$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  451 $\mbox{\,fb}^{-1}$ |  5.9$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  5.1$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}K^{\ast 0}$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  7.2$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  9.3$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  451 $\mbox{\,fb}^{-1}$ |  1.7$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  7.5$~{}\times~{}10^{-6}$ |  |  |
$e^{-}\bar{K}^{\ast 0}$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  3.4$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  6.2$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  451 $\mbox{\,fb}^{-1}$ |  4.6$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  7.4$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}\bar{K}^{\ast 0}$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  7.0$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  8.5$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  451 $\mbox{\,fb}^{-1}$ |  7.3$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  7.5$~{}\times~{}10^{-6}$ |  |  |
$e^{-}\phi$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  3.1$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  7.4$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  451 $\mbox{\,fb}^{-1}$ |  3.1$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  6.9$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}\phi$ |  Belle Miyazaki et al. (2011) |  854 $\mbox{\,fb}^{-1}$ |  8.4$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  8.4$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2009) |  451 $\mbox{\,fb}^{-1}$ |  1.9$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  7.0$~{}\times~{}10^{-6}$ |  |  |
$e^{-}e^{+}e^{-}$ |  Belle Hayasaka et al. (2010) |  782 $\mbox{\,fb}^{-1}$ |  2.7$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.7$~{}\times~{}10^{-10}$
|  BABAR Lees et al. (2010) |  468 $\mbox{\,fb}^{-1}$ |  2.9$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  2.9$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}e^{+}e^{-}$ |  Belle Hayasaka et al. (2010) |  782 $\mbox{\,fb}^{-1}$ |  1.8$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  2.9$~{}\times~{}10^{-10}$
|  BABAR Lees et al. (2010) |  468 $\mbox{\,fb}^{-1}$ |  2.2$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  1.7$~{}\times~{}10^{-6}$ |  |  |
$e^{-}\mu^{+}\mu^{-}$ |  Belle Hayasaka et al. (2010) |  782 $\mbox{\,fb}^{-1}$ |  2.7$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.5$~{}\times~{}10^{-10}$
|  BABAR Lees et al. (2010) |  468 $\mbox{\,fb}^{-1}$ |  3.2$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  1.8$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}\mu^{+}\mu^{-}$ |  Belle Hayasaka et al. (2010) |  782 $\mbox{\,fb}^{-1}$ |  2.1$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  3.6$~{}\times~{}10^{-10}$
|  BABAR Lees et al. (2010) |  468 $\mbox{\,fb}^{-1}$ |  3.3$~{}\times~{}10^{-8}$ |  |  |
|  LHCb Aaij et al. (2015) |  3 $\mbox{\,fb}^{-1}$ |  4.6$~{}\times~{}10^{-8}$ |  |  |
|  CMS CMS Collaboration (2021) |  33 $\mbox{\,fb}^{-1}$ |  8.0$~{}\times~{}10^{-8}$ |  |  |
|  ATLAS ATLAS Collaboration (2016) |  20 $\mbox{\,fb}^{-1}$ |  3.8$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  1.9$~{}\times~{}10^{-6}$ |  |  |
$e^{+}\mu^{-}\mu^{-}$ |  Belle Hayasaka et al. (2010) |  782 $\mbox{\,fb}^{-1}$ |  1.7$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  2.6$~{}\times~{}10^{-10}$
|  BABAR Lees et al. (2010) |  468 $\mbox{\,fb}^{-1}$ |  2.6$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  1.5$~{}\times~{}10^{-6}$ |  |  |
$\mu^{+}e^{-}e^{-}$ |  Belle Hayasaka et al. (2010) |  782 $\mbox{\,fb}^{-1}$ |  1.5$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  2.3$~{}\times~{}10^{-10}$
|  BABAR Lees et al. (2010) |  468 $\mbox{\,fb}^{-1}$ |  1.8$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  1.5$~{}\times~{}10^{-6}$ |  |  |

Table 6: Cont.

---
| Observed Limits | Expected Limits
$\tau^{-}\to$ |  Experiment |  Luminosity |  UL (obs) |  Experiment |  Luminosity |  UL (exp)
$e^{-}\pi^{+}\pi^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  2.3$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  5.8$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  1.2$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  2.2$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}\pi^{+}\pi^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  2.1$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  5.6$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  2.9$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  8.2$~{}\times~{}10^{-6}$ |  |  |
$e^{-}\pi^{+}K^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  3.7$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  7.1$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  3.2$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  6.4$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}\pi^{+}K^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  8.6$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.2$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  2.6$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  7.5$~{}\times~{}10^{-6}$ |  |  |
$e^{-}K^{+}\pi^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  3.1$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  7.8$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  1.7$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  3.8$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}K^{+}\pi^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  4.5$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.2$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  3.2$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  7.4$~{}\times~{}10^{-6}$ |  |  |
$e^{-}K^{+}K^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  3.4$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  6.5$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  1.4$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  6.0$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}K^{+}K^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  4.4$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.1$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  2.5$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  1.5$~{}\times~{}10^{-5}$ |  |  |
$e^{-}K_{S}^{0}K_{S}^{0}$ |  Belle Miyazaki et al. (2010) |  671 $\mbox{\,fb}^{-1}$ |  7.1$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  9.7$~{}\times~{}10^{-10}$
|  CLEO Chen et al. (2002) |  13.9 $\mbox{\,fb}^{-1}$ |  2.2$~{}\times~{}10^{-6}$ |  |  |
$\mu^{-}K_{S}^{0}K_{S}^{0}$ |  Belle Miyazaki et al. (2010) |  671 $\mbox{\,fb}^{-1}$ |  8.0$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.1$~{}\times~{}10^{-9}$
|  CLEO Chen et al. (2002) |  13.9 $\mbox{\,fb}^{-1}$ |  3.4$~{}\times~{}10^{-6}$ |  |  |
$e^{+}\pi^{-}\pi^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  2.0$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.6$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  2.7$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  1.9$~{}\times~{}10^{-6}$ |  |  |
$\mu^{+}\pi^{-}\pi^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  3.9$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.5$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  7.0$~{}\times~{}10^{-8}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  3.4$~{}\times~{}10^{-6}$ |  |  |
$e^{+}\pi^{-}K^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  3.2$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  7.7$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  1.8$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  2.1$~{}\times~{}10^{-6}$ |  |  |
$\mu^{+}\pi^{-}K^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  4.8$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  1.2$~{}\times~{}10^{-9}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  2.2$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  7.0$~{}\times~{}10^{-6}$ |  |  |
$e^{+}K^{-}K^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  3.3$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  5.8$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  1.5$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  3.8$~{}\times~{}10^{-6}$ |  |  |
$\mu^{+}K^{-}K^{-}$ |  Belle Miyazaki et al. (2013) |  854 $\mbox{\,fb}^{-1}$ |  4.7$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  9.7$~{}\times~{}10^{-10}$
|  BABAR Aubert et al. (2005) |  221 $\mbox{\,fb}^{-1}$ |  4.8$~{}\times~{}10^{-7}$ |  |  |
|  CLEO Bliss et al. (1998) |  4.79 $\mbox{\,fb}^{-1}$ |  6.0$~{}\times~{}10^{-6}$ |  |  |

Table 7: Cont.

---
| Observed Limits | Expected Limits
$\tau^{-}\to$ |  Experiment |  Luminosity |  UL (obs) |  Experiment |  Luminosity |  UL (exp)
$\pi^{-}\bar{\Lambda}$ |  Belle Miyazaki et al. (2006) |  154 $\mbox{\,fb}^{-1}$ |  1.4$~{}\times~{}10^{-7}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  5.5$~{}\times~{}10^{-10}$
$\pi^{-}\Lambda$ |  Belle Miyazaki et al. (2006) |  154 $\mbox{\,fb}^{-1}$ |  7.2$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  5.4$~{}\times~{}10^{-10}$
$\bar{p}^{-}e^{+}e^{-}$ |  Belle Sahoo et al. (2020) |  921 $\mbox{\,fb}^{-1}$ |  3.0$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.0$~{}\times~{}10^{-10}$
$\bar{p}^{-}e^{+}\mu^{-}$ |  Belle Sahoo et al. (2020) |  921 $\mbox{\,fb}^{-1}$ |  2.0$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.4$~{}\times~{}10^{-10}$
$\bar{p}^{-}\mu^{+}e^{-}$ |  Belle Sahoo et al. (2020) |  921 $\mbox{\,fb}^{-1}$ |  1.8$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  4.4$~{}\times~{}10^{-10}$
$\bar{p}^{-}\mu^{+}\mu^{-}$ |  Belle Sahoo et al. (2020) |  921 $\mbox{\,fb}^{-1}$ |  1.8$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  7.4$~{}\times~{}10^{-10}$
|  LHCb Aaij et al. (2013) |  1 $\mbox{\,fb}^{-1}$ |  3.3$~{}\times~{}10^{-7}$ |  |  |
$p^{+}e^{-}e^{-}$ |  Belle Sahoo et al. (2020) |  921 $\mbox{\,fb}^{-1}$ |  3.0$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  3.6$~{}\times~{}10^{-10}$
$p^{+}\mu^{-}\mu^{-}$ |  Belle Sahoo et al. (2020) |  921 $\mbox{\,fb}^{-1}$ |  4.0$~{}\times~{}10^{-8}$ |  Belle II Aggarwal et al. (2022) |  50 $\mbox{\,ab}^{-1}$ |  8.3$~{}\times~{}10^{-10}$
|  LHCb Aaij et al. (2013) |  1 $\mbox{\,fb}^{-1}$ |  4.4$~{}\times~{}10^{-7}$ |  |  |

A beam polarization upgrade of the SuperKEKB $e^{-}e^{+}$ collider can enhance
the sensitivity to LFV in $\tau$ decays at the Belle II experiment to levels
beyond the ones listed in Table 4 and Figure 6. The proposed upgrade Banerjee
and Roney (2022) will result in ${\sim}70\%$ longitudinal polarization of the
high energy electron beam, which will influence the angular distribution of
the $\tau$ decay products in the SM $\tau$-pair backgrounds. The
characteristic $\tau$ polarization dependence of the helicity angles of the
$\tau$ decay products with beam polarization can then be used to further
suppress the background in, for example, $\tau^{-}\to\mu^{-}\gamma$ searches,
where one $\tau$ decays to a muon and a photon, while the other $\tau$ decays
to a pion and a neutrino, the decay channel most sensitive to the polarization
of the $\tau$ lepton. Similar background suppression can also be obtained with
the other decay modes, which vary in their sensitivity to the $\tau$
polarization. In general, the maximal discriminating power is obtained by
studying the polar angles in the center-of-mass frame times the charge of the
$\tau$ decay.

The “irreducible background” from $\tau^{-}\to\mu^{-}\nu\bar{\nu}\gamma$
decays are studied in Figure 7 Hitlin et al. (2008). While the distributions
of the backgrounds show marked differences in the case of beam polarization
with respect to the case of no beam polarization, the signal distribution
modeled by uniform phase–phase does not change with beam polarization. By
removing events where the distribution of the irreducible background shows a
rising trend near unity, the background can be reduced significantly,
corresponding to a small loss in signal efficiency. An optimization study has
demonstrated that this would result in approximately a 10% improvement in the
sensitivity to LFV. Similar analyses are expected to yield comparable gain in
sensitivities for other decay modes.

Figure 6:  Current status of observed upper limits at CLEO, BABAR, Belle,
ATLAS, CMS, and LHCb experiments Amhis et al. (2021) and projections of
expected upper limits at the Belle II experiment Aggarwal et al. (2022).
Figure 7:  Distribution of the cosine of the signal-side muon multiplied by
the muon charge for signal and background events with and without electron
beam polarization in the $\tau^{-}\to\mu^{-}\gamma$ search analysis Hitlin et
al. (2008).

It is worth noting that the uniform phase space model of the signal
distribution is chosen because the underlying theory behind LFV is not known.
Different spin-dependent operators are predicted to give significantly
different features in the Dalitz plane of final state momenta distributions
of, for example, $\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ decays Matsuzaki and Sanda
(2008); Dassinger et al. (2007). One of the most interesting aspects of having
the beam polarization is the possibility to distinguish between these
different new physics models to understand the helicity structure of the
couplings producing LFV in $\tau$ decays, once such decays are observed.

Belle II limits will probe predictions from several theoretical models, as
listed in Table 1. Some theoretical expectations from different new physics
models and improvements on experimental limits over the last few decades for
$\tau^{-}\to\mu^{-}\gamma$ Hayes et al. (1982); Albrecht et al. (1992); Bean
et al. (1993); Abreu et al. (1995); Edwards et al. (1997); Ahmed et al.
(2000); Abe et al. (2004); Aubert et al. (2005); Hayasaka et al. (2008);
Aubert et al. (2010); Abdesselam et al. (2021) and
$\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ Hayes et al. (1982); Bowcock et al. (1990);
Albrecht et al. (1992); Bartelt et al. (1994); Bliss et al. (1998); Yusa et
al. (2004); Aubert et al. (2004, 2007); Miyazaki et al. (2008); Lees et al.
(2010); Hayasaka et al. (2010); Aaij et al. (2013, 2015); ATLAS Collaboration
(2016); CMS Collaboration (2021) decays, along with future prospects at Belle
II Aggarwal et al. (2022), are shown in Figure 8.

Figure 8: Evolution of experimental bounds on $\tau^{-}\to\mu^{-}\gamma$ Hayes
et al. (1982); Albrecht et al. (1992); Bean et al. (1993); Abreu et al.
(1995); Edwards et al. (1997); Ahmed et al. (2000); Abe et al. (2004); Aubert
et al. (2005); Hayasaka et al. (2008); Aubert et al. (2010); Abdesselam et al.
(2021) (left) and $\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ Hayes et al. (1982);
Bowcock et al. (1990); Albrecht et al. (1992); Bartelt et al. (1994); Bliss et
al. (1998); Yusa et al. (2004); Aubert et al. (2004, 2007); Miyazaki et al.
(2008); Lees et al. (2010); Hayasaka et al. (2010); Aaij et al. (2013, 2015);
ATLAS Collaboration (2016); CMS Collaboration (2021) decays, future prospects
at Belle II Aggarwal et al. (2022) and some predictions from Table 1.

##  5 Conclusions

LFV in $\tau$ decays are unambiguous signatures of new physics, and are thus
of great experimental and theoretical interest. Many models from
supersymmetric scenarios to leptoquarks predict LFV in $\tau$ decays at
experimentally observable rates, which will be probed at Belle II. Searches
for LFV in $\tau$ decays can discover new physics at the multi-TeV scale by
identifying the underlying mechanism beyond the SM, or strongly constrain the
flavor structure of TeV-scale extensions beyond the SM Husek et al. (2021);
Cirigliano et al. (2021), as discussed in the context of different
experimental efforts in Ref. Banerjee et al. (2022). The first generation
B-Factory experiments, Belle and BABAR, saw an order of magnitude improvement
on the upper limit on LFV in $\tau$ decays from $10^{-6}$ level down to
$10^{-8}$ level. The Belle II experiment will continue to improve the
sensitivity in searches of LFV in $\tau$ decays over the next decade. The
projected sensitivity at Belle II for LFV in $\tau$ decays with
50$\mbox{\,ab}^{-1}$ of data is at the $10^{-10}$–$10^{-9}$ level, which
constitutes one or two orders of magnitude of improvement over the previous
experiments.

This project was supported by the U.S. Department of Energy under research
Grant No. DE-SC0022350.

Not applicable.

Not applicable.

Not applicable.

###### Acknowledgements.

The author acknowledges fruitful discussions with Kiyoshi Hayasaka, Kenji
Inami, John Micheal Roney, Armine Rostomyan, Michel Hernández Villanueva, and
many others. Sources of all figures and tables included for discussions have
been appropriately cited, and reproduced with permissions from the respective
collaborations. The author declares no conflict of interest. [custom]
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