Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

Snowmass 2021 White Paper:

Charged Lepton Flavour Violation in Heavy Particle Decays

W. Altmannshofer

Department of Physics, University of California Santa Cruz, and Santa Cruz
Institute for Particle Physics, 1156 High St., Santa Cruz, CA 95064, USA

C. Caillol

CERN, EP Department, 1 Esplanade des Particules, CH-1217 Meyrin, Switzerland

M. Dam111Corresponding author<EMAIL_ADDRESS>S. Xella

Niels Bohr Institute, Copenhagen University, Copenhagen, Denmark

Y. Zhang

School of Physics, Southeast University, Nanjing 211189, China

> ABSTRACT
>
>
>
>
> Charged lepton flavor violation is an unambiguous signature for New Physics.
> Here we present a summary of the theoretical and experimental status of the
> search for charged lepton flavor violation in heavy particle decays, in
> particular in the decays of the Z and Higgs bosons, and of the top quark.
> Decays of beyond-Standard-Model particles such as a Z′ or an additional
> scalar particle are also discussed. Finally the prospects for such searches
> at proposed future electron-positron colliders are reviewed.

##  1 Introduction

In the Standard Model (SM), with massless neutrinos, the three lepton-flavour
quantum numbers are individually conserved. The discovery of neutrino
oscillations demonstrate that neutrinos have mass and that lepton flavour can
be violated in the neutral-lepton sector. Via loop diagrams, neutrino
oscillations induce also lepton flavour violation (LFV) among charged leptons.
These processes are however suppressed to an unobservable level by the tiny
values of the neutrino masses. Any experimental observation of charged lepton
flavour violation (CLFV) would therefore be an unambiguous signal for physics
beyond the Standard Model (BSM) not related to the neutrino masses.

Searches for CLFV processes can be divided into low- and high-energy
categories. At low energy, the most sensitive probes are provided by the
decays of $\mu$ and $\tau$ leptons and of K mesons, and by the
$\mu\to\mbox{e}$ conversion in nuclei. At high energy, CLFV processes can be
searched for both inclusively, in processes such as
$\mbox{pp}\to\mu\text{e}+X$, and in the decays of heavy particles. In this
article we focus on the latter option and discuss the status of, and the
future prospects for, the search for CLFV in the decays of the heaviest SM
particles, the Z boson, the Higgs boson, and the top quark. Also a few BSM
processes will be considered, in particular processes involving a possible Z′
boson or an additional scalar beyond the 125-GeV particle.

The paper is organised as follows. Section 2 starts out with a theoretical
overview including the analysis of the sensitivity of CLFV decay processes to
New Physics as described within the Standard Model Effective Field Theory
framework. The discussion of CLFV processes involving BSM particles – the Z′
and additional scalars – will point also to how such phenomena could possibly
explain the $g-2$ anomaly and how future collider experiments will be
sensitive to the relevant regions of parameter space. Current results from the
LHC and prospects from the 20-fold larger statistics from the HL-LHC are
discusses in Section 3. Finally, Section 4 reviews the long-term prospects at
future electron-positron colliders, where the clean experimental environments
combined with very high luminosities, in particular at the proposed circular
colliders, CEPC and FCC-ee, promise very sensitive CLFV tests.

##  2 Theoretical Framework

###  2.1 Z Decays

As is the case with all LFV phenomena, SM predictions for the LFV Z decays,
$\text{Z}\to\tau\mu$, $\text{Z}\to\tau e$, and $\text{Z}\to\mu e$, are
suppressed by neutrino masses and thus completely beyond experimental reach.
Taking into account neutrino masses, SM predictions for the branching
fractions of such decays are in the ballpark of $10^{-50}$ to $10^{-40}$.
Hence, any experimental evidence for LFV Z decays would be a clear sign of
physics beyond the SM.

Direct searches for LFV Z decays into $\mu$e, $\tau$e, and $\tau\mu$ final
states were performed at LEP with branching fraction limits of about $2\times
10^{-6}$ for the $\mu$e mode, and $10^{-5}$ for the $\tau$e and $\tau\mu$
modes [1, 2]. These limits have been recently improved by measurements from
ATLAS [3, 4, 5], as detailed in Sect. 3.1. Improvements were by about a factor
of seven for the e$\mu$ mode and by factors of about two for the e$\tau$ and
$\mu\tau$ modes. The new limits are

$\mathcal{B}(\text{Z}\to\mu\text{e})<2.62\times
10^{-7},\quad\mathcal{B}(\text{Z}\to\tau\text{e})<5.0\times
10^{-6},\quad\mathcal{B}(\text{Z}\to\tau\mu)<6.5\times 10^{-6}.$ (1)

The presence of new physics could increase the branching fraction expectations
significantly bringing observations within experimental reach at the HL-LHC or
future colliders. Among the many BSM scenarios that can give
$\text{Z}\to\ell\ell^{\prime}$ decays are models with gauged flavor symmetries
[6], supersymmetric models [7], models with heavy sterile neutrinos [8, 9,
10], and leptoquark models [11].

New physics that leads to the LFV Z decays can be model independently
parameterized in the context of the Standard Model Effectiv Field Theory
(SMEFT) [12] which systematically extends the SM by higher dimensional
interactions

$\mathcal{L}_{\text{SMEFT}}=\mathcal{L}_{\text{SM}}+\sum_{n>4}\sum_{i_{n}}\frac{C_{i_{n}}}{\Lambda^{n-4}}Q_{i_{n}}~{}.$
(2)

Here, $n$ denotes the mass dimension of the operators $Q_{i_{n}}$, $C_{i_{n}}$
are so-called Wilson coefficients, and $\Lambda$ is the new physics scale.
Such a parameterization is valid as long as the new physics is heavy compared
to the electro-weak scale.

The leading interactions that can give $\text{Z}\to\ell\ell^{\prime}$ decays
at tree level are of dimension 6 and consist of dipole operators

$Q_{eW}=(\overline{\ell}_{i}\sigma^{\mu\nu}P_{R}\ell_{j})\tau^{I}\phi
W^{I}_{\mu\nu}~{},\quad
Q_{eB}=(\overline{\ell}_{i}\sigma^{\mu\nu}P_{R}\ell_{j})\phi B_{\mu\nu}~{},$
(3)

and Higgs current operators

$\displaystyle Q_{\phi\ell}^{(1)}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\overset{\leftrightarrow}{D_{\mu}}\phi)(\overline{\ell}_{i}\gamma^{\mu}P_{L}\ell_{j})~{},\quad
Q_{\phi\ell}^{(3)}=(\phi^{\dagger}i\tau^{I}\overset{\leftrightarrow}{D_{\mu}}\phi)(\overline{\ell}_{i}\tau^{I}\gamma^{\mu}P_{L}\ell_{j})~{},$
$\displaystyle\quad Q_{\phi e}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\overset{\leftrightarrow}{D_{\mu}}\phi)(\overline{\ell}_{i}\gamma^{\mu}P_{R}\ell_{j})~{}.$
(4)

In the above operators, $i,j=1,2,3$ label the lepton flavor, $\phi$ is the SM
Higgs doublet, and $W_{\mu\nu}$ and $B_{\mu\nu}$ are the field strength
tensors of the $SU(2)_{L}$ and $U(1)_{Y}$ gauge bosons. Assuming generic
Wilson coefficients $C\sim 1$, one finds the following order of magnitude
estimate for the branching ratios of flavor changing $Z$ decays

$\mathcal{B}(\text{Z}\to\ell\ell^{\prime})\sim\left(\frac{v}{\Lambda}\right)^{4}~{},$
(5)

where $v=246$ GeV is the vacuum expectation value of the Higgs. The existing
bounds thus already probe new physics at few TeV.

Figure 1: New physics sensitivity of the $\text{Z}\to\mu e$ decay compared to
several low energy lepton flavor violating processes. One SMEFT operator is
switched on at a time. Shown is the reach in the new physics scale setting the
Wilson coefficient to 1. Dark bars correspond to the current status, the light
bars show expected future sensitivities at a Tera-Z, MEG-II, Mu3e, and
Mu2e/COMET (from [13]). Figure 2: New physics sensitivity of the
$\text{Z}\to\tau\mbox{e}$ and $\text{Z}\to\tau\mu$ decays compared to several
low energy lepton flavor violating processes. One SMEFT operator is switched
on at a time. Shown is the reach in the new physics scale setting the Wilson
coefficient to 1. Dark bars correspond to the current status, the light bars
show expected future sensitivities at a Tera-Z and Belle II (from [13]).

It is important to note that in the presence of the effective operators listed
above, one not only expects LFV Z decays, but at the same time also various
LFV phenomena at low energies, including radiative, leptonic, and semi-
leptonic decays of muons and taus, e.g. $\mu\to\mbox{e}\gamma$,
$\tau\to\mu\gamma$, $\tau\to 3$e, $\tau\to\mu\pi^{0}$, etc. and
$\mu\to\mbox{e}$ conversion in nuclei. Barring accidental cancellations of the
new physics in the low energy processes, LFV muon decays
$\mu\to\mbox{e}\gamma$, $\mu\to\mbox{3e}$, and $\mu\to\mbox{e}$ conversion in
nuclei outperform $\text{Z}\to\mu\mbox{e}$ decays in terms of new physics
sensitivity. This is illustrated in Fig. 1 that shows the maximal new physics
scale that can be probed with current and future experimental sensitivities.

A different picture emerges for LFV processes involving taus. In this case,
the $\text{Z}\to\tau\mbox{e}$ and $\text{Z}\to\tau\mu$ decays have
sensitivities to new physics that are comparable to tau decays, see Fig. 2.
The Z decays and the $\tau$ decays are thus highly complementary in probing
new physics [13, 14].

###  2.2 Higgs Decays

SM predictions for LFV Higgs decays, $h\to\tau\mu$, $h\to\tau\mbox{e}$, and
$h\to\mu\mbox{e}$, are likewise suppressed by neutrino masses, and completely
beyond experimental reach. Any observation of such processes would be a clear
signature for new physics.

To provide a phenomenological parameterization of possible BSM effects in
Higgs decays, it is useful to introduce the following flavor violating Higgs
couplings in the fermion mass eigenstate basis

$\mathcal{L}_{\text{pheno}}\supset-\sum_{i\neq
j=1,2,3}\left[\left(\kappa_{\ell_{i}\ell_{j}}+i\tilde{\kappa}_{\ell_{i}\ell_{j}}\right)h\overline{\ell}_{i}P_{R}\ell_{j}~{}+~{}\text{h.c.}\right]~{},$
(6)

where $i,j=1,2,3$ label the three generations of leptons. The coefficients
$\kappa$ and $\tilde{\kappa}$ are the real and imaginary parts of the
couplings. In terms of these phenomenological couplings, branching ratios of
flavor violating Higgs decays are given by

$\mathcal{B}(h\to\ell_{i}\ell_{j})=\frac{m_{h}}{8\pi\Gamma_{h}}\left(\kappa_{\ell_{i}\ell_{j}}^{2}+\tilde{\kappa}_{\ell_{i}\ell_{j}}^{2}+\kappa_{\ell_{j}\ell_{i}}^{2}+\tilde{\kappa}_{\ell_{j}\ell_{i}}^{2}\right)~{},$
(7)

where $\Gamma_{h}$ is the total Higgs width.

There are indirect constraints on LFV Higgs couplings. The strongest ones are
from radiative muon and tau decays $\mu\to\mbox{e}\gamma$, $\tau\to\mu\gamma$,
and $\tau\to\mbox{e}\gamma$. Using existing bounds on the muon and tau decays,
one finds (adapted from [15], see also [16])

$\mathcal{B}(h\to\tau\mu)_{\text{ind.}}\lesssim
0.32~{},\qquad\mathcal{B}(h\to\tau\mbox{e})_{\text{ind.}}\lesssim
0.24~{},\qquad\mathcal{B}(h\to\mu\mbox{e})_{\text{ind.}}\lesssim 1.6\times
10^{-8}~{}.$ (8)

Therefore, observing $h\to\mu\mbox{e}$ in the foreseeable future would not
only imply new physics, but at the same time also requires a non-trivial
mechanism to suppress $\mu\to\mbox{e}\gamma$ below existing constraints. In
the following, we thus focus on LFV Higgs decays involving taus which could be
sizable.

Direct searches for the decays $h\to\tau\mu,\tau e$ are carried out at the LHC
[17, 18]. Existing limits are

$\displaystyle\mathcal{B}(h\to\tau\mu)_{\text{ATLAS}}\lesssim 2.8\times
10^{-3}~{},\qquad\mathcal{B}(h\to\tau\mu)_{\text{CMS}}\lesssim 1.5\times
10^{-3}~{},$ (9)
$\displaystyle\mathcal{B}(h\to\tau\mbox{e})_{\text{ATLAS}}\lesssim 4.7\times
10^{-3}~{},\ \qquad\mathcal{B}(h\to\tau\mbox{e})_{\text{CMS}}\lesssim
2.2\times 10^{-3}~{}.$ (10)

Sensitivities will improve considerably at the high luminosity LHC and at
future colliders (see Sects. 3 and 4).

The plots in Fig. 3 show current constraints (red shaded regions) and expected
sensitivities (orange and yellow dash-dotted lines) in the plane of LFV Higgs
couplings. The predicted values for $\mathcal{B}(h\to\tau\mu)$ and
$\mathcal{B}(h\to\tau e)$ are shown by the solid blue lines. Theoretically
motivated targets for the Higgs couplings are indicated by the dotted green
lines.

Figure 3: Current constraints (red shaded regions) and expected sensitivities
(orange and yellow dash-dotted lines) in the plane of lepton flavor violating
Higgs couplings. The solid blue lines show the predicted values for
$\mathcal{B}(h\to\tau\mu)$ and $\mathcal{B}(h\to\tau e)$. The dotted green
lines indicate theoretically motivated targets for the Higgs couplings.

Assuming that the new physics is sufficiently heavy compared to the electro-
weak scale $v=246$ GeV, the LFV Higgs couplings are expected to scale as
$\kappa_{\ell_{i}\ell_{j}},\tilde{\kappa}_{\ell_{i}\ell_{j}}\propto\frac{v^{2}}{\Lambda^{2}}$
where $\Lambda$ is the scale of new physics. In the context of SMEFT [12], the
following dimension six operators lead to the flavor violating couplings of
the Higgs

$\mathcal{L}_{\text{SMEFT}}\supset\sum_{i,j=1,2,3}\frac{C_{ij}}{\Lambda^{2}}\left[\overline{\ell}_{i}P_{R}\ell_{j}\phi~{}+~{}\text{h.c.}\right]\phi^{\dagger}\phi\quad\Rightarrow\quad\kappa_{\ell_{i}\ell_{j}}+i\tilde{\kappa}_{\ell_{i}\ell_{j}}=\frac{C_{ij}}{\sqrt{2}}\frac{v^{2}}{\Lambda^{2}}~{},$
(11)

where $\phi$ is the SM Higgs doublet. For generic Wilson coefficients
$C_{ij}\sim 1$, sensitivty to branching ratios of $\mathcal{O}(10^{-4})$
probes new physics scales of $\Lambda\sim 10$ TeV.

Many new physics models can lead to LFV Higgs couplings. They arise in models
with heavy vector-like fermions or lepto-quarks, or in models with a new
source of electro-weak symmetry breaking (e.g. a second Higgs doublet or a
technifermion condensate). The simplest models with vector-like leptons or
lepto-quarks are thightly constrained by LFV tau decays [19, 20]. Branching
ratios of $h\to\tau\mu$ and $h\to\tau e$ at an observable level cannot be
accommodated in these models. EFT arguments suggest this is a generic feature
in models without new sources of electro-weak symmetry breaking [21] (see
however the exceptions discussed in [22, 23]).

If a new physics model contains a new source of electro-weak symmetry
breaking, branching ratios of LFV Higgs decays can be sizable. Models with
extended Higgs sectors have been extensively studied in this respect [24, 25,
26, 27, 28, 29, 30, 21, 31, 32, 33, 34, 35]. It has been found that the
$h\to\tau\mu$ and $h\to\tau\mbox{e}$ branching ratios can saturate the current
direct bounds in many of such models.

###  2.3 Top Decays

In contrast to the LFV decays of the Z and Higgs, LFV decays of the top quark
are necessarily 3-body decays, $t\to q\ell\ell^{\prime}$, $q=u,c$. As these
3-body decays have to compete with a leading unsuppressed 2-body decay mode,
$t\to bW$, their generic new physics reach is slightly lower. However, the top
decays probe qualitatively different types of new physics and are thus highly
complementary to the Z and Higgs decays discussed above.

As is the case for the Z and Higgs decays, new physics that gives rise to the
LFV top decays can be systematically and model independently parameterized by
effective dimension 6 interactions of the SMEFT. The relevant operators are
[12, 36]

$\displaystyle Q_{\ell
q}^{(1)}=(\overline{\ell}_{i}\gamma^{\mu}P_{L}\ell_{j})(\overline{q}\gamma_{\mu}P_{L}t)~{}$
, $\displaystyle~{}~{}~{}Q_{\ell
q}^{(3)}=(\overline{\ell}_{i}\gamma^{\mu}\tau^{I}P_{L}\ell_{j})(\overline{q}\gamma_{\mu}\tau^{I}P_{L}t)~{},$
(12) $\displaystyle Q_{\ell
u}=(\overline{\ell}_{i}\gamma^{\mu}P_{L}\ell_{j})(\overline{q}\gamma_{\mu}P_{R}t)~{}$
,
$\displaystyle~{}~{}~{}Q_{qe}=(\overline{\ell}_{i}\gamma^{\mu}P_{R}\ell_{j})(\overline{q}\gamma_{\mu}P_{L}t)~{},$
(13) $\displaystyle
Q_{eu}=(\overline{\ell}_{i}\gamma^{\mu}P_{R}\ell_{j})(\overline{q}\gamma_{\mu}P_{R}t)~{}$
, (14) $\displaystyle Q_{\ell
equ}^{(1)}=(\overline{\ell}_{i}P_{R}\ell_{j})(\overline{q}P_{R}t)~{}$ ,
$\displaystyle~{}~{}~{}Q_{\ell
equ}^{(3)}=(\overline{\ell}_{i}\sigma_{\mu\nu}P_{R}\ell_{j})(\overline{q}\sigma^{\mu\nu}P_{R}t)~{},$
(15)

with $q=u,c$.

Assuming that the Wilson coefficients of these operators are of
$\mathcal{O}(1)$, the generic estimate for the top decay branching ratios is

$\mathcal{B}(t\to
q\ell\ell^{\prime})\sim\frac{1}{16\pi^{2}}\left(\frac{v}{\Lambda}\right)^{4}~{},$
(16)

with the Higgs vev $v$ and the scale of new physics $\Lambda$.

More than $10^{8}$ top quarks were produced at Run 2 of the LHC, opening up
the possibility to search for rare LFV top quark decays with high sensitivity.
A recent CMS analysis [37] finds bounds on the $t\to u\mu e$ branching ratio
of approximately $10^{-7}$ and on the $t\to c\mu e$ branching ratio of
approximately $10^{-6}$. The analysis, discussed in more detail in Sect. 3.4,
includes both searches for top decays as well as searches for single top
production in association with $\mu e$. Due to differences in the experimental
acceptances, the constraints differ by an $\mathcal{O}(1)$ amount for the
vector, scalar, and tensor operators in (12)–(15). The bounds obtained by CMS
already probe new physics above 1 TeV.

Some of the operators that give the $t\to q\ell\ell^{\prime}$ decays can also
be probed by different processes. In particular, the operators containing
left-handed charm or up quarks can lead to sizable rates of LFV kaon or $B$
meson decays like $K_{L}\to\mu e$, $B\to\pi\ell\ell^{\prime}$, or $B\to
K\ell\ell^{\prime}$ [36]. However, for operators with right-handed up and
charm quarks, the indirect constraints are very weak and the LFV top decays
provide the most sensitive probes.

###  2.4 Z′ Boson

A heavy or light Z′ boson exists in a large variety of new physics models, and
the corresponding phenomenologies are very rich; see e.g. Refs. [6, 38, 39].
The LFV couplings of the Z′ boson can originate from the mixing of SM fermions
with heavy exotic fermions [40]. The muon $g-2$ anomaly is an important
motivation for LFV couplings of a Z′ boson [41]. Without loss of generality,
the effective LFV couplings of the Z′ can be written as,

$\displaystyle{\cal L}\supset
Z_{\mu}^{\prime}\left(g^{\prime}_{L}\overline{\ell}_{\alpha\,L}\gamma^{\mu}\ell_{\beta\,L}+g^{\prime}_{R}\overline{\ell}_{\alpha\,R}\gamma^{\mu}\ell_{\beta\,R}\right)~{}+~{}{\rm
H.c.}\quad(\alpha\neq\beta)\,.$ (17)

Note that the couplings of the Z′ to left- and right-handed charged leptons
can be different. Direct searches of a heavy
Z${}^{\prime}\to\text{e}\mu,\,\text{e}\tau,\,\mu\tau$ have been performed at
the LHC [42]. Assuming the LFV couplings of the Z′ are the same as the
couplings of the Z boson to charged leptons, Z′ masses have been excluded up
to 5.0 TeV in the e$\mu$ channel, 4.3 TeV in the e$\tau$ channel, and 4.1 TeV
in the $\mu\tau$ channel [42].

Figure 4: Prospect of $m_{\mathrm{Z^{\prime}}}$ and $g^{\prime}_{R}$ at HL-LHC
(dashed orange line) and FCC-ee (dashed blue line) with
$g^{\prime}_{L}=g^{\prime}_{R}/10$. The green band correspond to the muon
$g-2$ anomaly at the $2\sigma$ CL. Also shown are the limits from the muon
$g-2$ discrepancy at the $5\sigma$ CL (gray), $\tau\to\mu\nu\overline{\nu}$
(red), LHC W data (black) and the LEP Z-pole data (purple). Figure remade
based on Ref. [43].

The LFV coupling $g^{\prime}_{L,\,R}$ of the light Z′ boson can be directly
measured at the high-energy hadron and lepton colliders via the process

$\displaystyle
pp,\,\text{e}^{+}\text{e}^{-}\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}\text{Z}^{\prime}\,.$
(18)

Take the explicit example of $\alpha=\mu$ and $\beta=\tau$, with the decay
Z${}^{\prime}\to\mu^{\pm}\tau^{\mp}$, the LFV coupling $g^{\prime}_{L,\,R}$
will generate same-sign dilepton pairs
$\mu^{\pm}\mu^{\pm}\tau^{\mp}\tau^{\mp}$. With
$g^{\prime}_{L}=g^{\prime}_{R}/10$, the prospects of Z′ mass
$m_{\mathrm{Z^{\prime}}}$ and $g^{\prime}_{R}$ at the HL-LHC 14 TeV with an
integrated luminosity of 3 ab-1 and the FCC-ee 91 GeV with 150 ab-1 are
presented respectively as the dashed orange and blue curves in Fig. 4 [43]. At
HL-LHC, the Z′ boson can be probed up to the TeV scale; at the future lepton
collider FCC-ee, the coupling $g^{\prime}_{R}$ can be probed down to
$10^{-3}$.

For low Z′ masses, $m_{\mathrm{Z^{\prime}}}\lesssim m_{\tau}-m_{\mu}$, the Z′
boson can be produced from the two-body tau decay
$\tau\to\mu\text{Z}^{\prime}$ via the gauge couplings in Eq. (17). Searches
for two-body decay $\tau\to\mu+$inv. have been performed by ARGUS, and the
null results exclude coupling $g^{\prime}_{R}\gtrsim 10^{-6}$ [44]. For higher
masses, $m_{\mathrm{Z^{\prime}}}\gtrsim m_{\tau}-m_{\mu}$, the coupling
$g^{\prime}_{R}$ is constrained by lepton flavor universality results in tau
decays, high-precision LHC W data, LEP Z-pole data, and the muon $g-2$
anomaly. These limits are shown, at $5\sigma$ CL, as the shaded regions in
Fig. 4. With $\tau$ running in the loop, LFV couplings of a Z′ boson can
explain the muon $g-2$ discrepancy, which is in some sense similar to the
neutral scalar case described below [41, 43]. The corresponding $2\sigma$
regions for the muon $g-2$ anomaly is shown as the green band in Fig. 4. As
seen in Fig. 4, when all these constraints are taken into consideration, a Z′
with mass $m_{\mathrm{Z^{\prime}}}\gtrsim 2$ GeV can provide a viable
interpretation of the muon $g-2$ anomaly. Furthermore, both the future high-
energy hadron and lepton colliders can probe large regions of the Z′
interpretation of muon $g-2$ anomaly.

With LFV couplings to electron and muon, the Z′ boson could contribute to
$\mu-\text{e}$ scattering [45, 46], e.g. in the MUonE experiment, which is
proposed to determine the contribution of hadronic vacuum polarization to muon
$g-2$ [47, 48]. However, the LFV couplings of Z′ boson is tightly constrained
by muonium-antimuonium oscillation and electron $g-2$. These limits have
precluded the sensitivities of the MUonE experiment to the lepton flavor
violating couplings of Z′ boson [45].

###  2.5 BSM Neutral Scalar

In a large variety of BSM scenarios, there exist BSM neutral scalars $H$,
which couple to the SM charged leptons in a flavor-violating way at the tree-
or 1-loop level [49, 50, 51, 52]. In the perspective of effective theory, the
couplings of $H$ to the SM charged leptons can be written in the following
way, without loss of generality,

${\cal L}_{Y}=h_{\alpha\beta}\overline{\ell}_{\alpha}H\ell_{\beta}~{}+~{}{\rm
H.c.}\,,$ (19)

where $h_{\alpha\beta}$ is the Yukawa coupling. For simplicity, we have
assumed $H$ to be a real field and CP-even. As a result, the matrix
$h_{\alpha\beta}$ is symmetric.

The couplings $h_{\alpha\beta}$ of the neutral scalar $H$ can induce very rich
LFV signals at high-energy colliders. In light of their clean experimental
conditions, future e+e- colliders are the primary facilities to search for
such smoking-gun signals. Given a single LFV coupling $h_{\alpha\beta}$ with
$\alpha\neq\beta$, if kinematically allowed, the scalar $H$ can be on-shell
produced at high-energy e+e- colliders via the process

$\displaystyle\mbox{e${}^{+}$e${}^{-}$}\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}H\quad(\alpha\neq\beta)\,,$
(20)

which is similar to the process (18) for the Z′. For simplicity, all other
Yukawa couplings in the matrix $h_{\alpha\beta}$ are assumed to be vanishing.
As a result, a single coupling $h_{\alpha\beta}$ cannot induce LFV decays,
such as $\ell_{\beta}\to\ell_{\alpha}\gamma$ and $\ell_{\beta}\to
3\ell_{\alpha}$, which depend on the combinations of $h_{\alpha\beta}$ with
other Yukawa couplings, e.g. $h_{\alpha\alpha}h_{\alpha\beta}$. It turns out
that only a few precise LFV measurements can be used to set limits on the
single coupling $h_{\alpha\beta}$. The current stringent limits on the LFV
couplings $h_{\mathrm{e\mu}}$, $h_{\mathrm{e\tau}}$ and $h_{\mu\tau}$ are
shown as the shaded regions in Fig. 5, including those from the anomalous
magnetic moment of electron [53] and muon [54, 55], muonium-antimuonium
oscillation [56], and the LEP e+e${}^{-}\to\ell^{+}\ell^{-}$ data [57]. The
prospects of the LFV couplings $h_{\mathrm{e\mu}}$, $h_{\mathrm{e\tau}}$ and
$h_{\mu\tau}$ at future lepton colliders are shown respectively in the left,
middle and right panels of Fig. 5, where the red and blue lines are,
respectively, for the CEPC 240 GeV with an integrated luminosity of 5 ab-1 and
the ILC 1 TeV with 1 ab-1 [50]. The long-dashed, short dashed and solid lines
correspond, respectively, to branching fractions of
$H\to\text{e}^{\pm}\mu^{\mp}$ to be 1%, 10% and 100%. It should be noted that,
with the coupling $h_{\mu\tau}$, the neutral scalar $H$ provide a simple
explanation for the muon $g-2$ anomaly via the $H$–$\tau$ loop. The
corresponding $1\sigma$ and $2\sigma$ ranges of $\Delta a_{\mu}$ are presented
in the right panel of Fig. 5 as the green and yellow bands, respectively. It
is promising that the explanation of muon $g-2$ discrepancy can be directly
tested at future lepton colliders.

Figure 5: Prospects of LFV couplings $h_{\alpha\beta}$ from the process (20)
at CEPC 240 GeV with 5 ab-1 (red) and ILC 1 TeV with 1 ab-1 (blue). A
branching fraction of 1% (long-dashed), 10% (short-dashed) or 100% (solid)
from $H$ decay is assumed to be visible. The shaded regions are excluded by
the corresponding limits. In the left and right panels, the green and yellow
bands cover the $1\sigma$ and $2\sigma$ ranges of $\Delta a_{\mu}$. See text
for more details. Figure from Ref. [50].

Figure 6: Prospects of $|h^{\dagger}_{\mathrm{ee}}h_{\mathrm{e}\tau}|$ (left),
$|h^{\dagger}_{\mathrm{ee}}h_{\mu\tau}|$ (middle) and
$|h^{\dagger}_{\mathrm{e}\mu}h_{\mathrm{e}\tau}|$ (right) from searches of
$\mathrm{e^{+}e^{-}}\to\mathrm{e}^{\pm}\tau^{\mp},\,\mu^{\pm}\tau^{\mp}$ at
CEPC 240 GeV with 5 ab-1 (red) and ILC 1 TeV with 1 ab-1 (blue). The shaded
regions are excluded by the corresponding limits. See text for more details.
Figure from Ref. [50].

If two Yukawa couplings are nonzero and at least one of them is LFV, for
instance the combinations $h_{\mathrm{ee}}h_{\mathrm{e\mu}}$,
$h_{\mathrm{ee}}h_{\mathrm{e\tau}}$, $h_{\mathrm{ee}}h_{\mu\tau}$ and
$h_{\mathrm{e\mu}}h_{e\tau}$, the scalar $H$ will induce the LFV process

$\displaystyle\mathrm{e^{+}e^{-}}\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}\;\;(\alpha\neq\beta)$
(21)

at high-energy lepton colliders. The prospects of
$h_{\mathrm{ee}}h_{\mathrm{e\mu}}$ at future lepton colliders have been
precluded by the stringent limit from $\mu\to\mbox{eee}$ [58]. In the $\tau$
sector, the limits are relatively much weaker. The coupling combination
$h_{\mathrm{ee}}h_{\mathrm{e\tau}}$ of $H$ can induce the process
$\mathrm{e^{+}e^{-}}\to\mathrm{e}^{\pm}\tau^{\mp}$, while the two combinations
$h_{\mathrm{ee}}h_{\mu\tau}$ and $h_{\mathrm{e\mu}}h_{\mathrm{e\tau}}$ will
both induce the process $\mathrm{e^{+}e^{-}}\to\mu^{\pm}\tau^{\mp}$
(respectively, $s$\- and $t$-channel). The prospects of these couplings at
CEPC 240 GeV and ILC 1 TeV are shown in Fig. 6 as the red and blue lines,
respectively, requiring at least ten signal events. All the shaded regions are
excluded by the limits from the rare LFV tau decays, electron $g-2$ and the
LEP $\mathrm{e^{+}e^{-}}\to\ell^{+}\ell^{-}$ data. The dips in the left and
middle panels of Fig. 6 are due to the resonant production of $H$ at CEPC and
ILC with $m_{H}\simeq\sqrt{s}$. In the limit of $m_{H}\gg\sqrt{s}$, the high-
energy lepton colliders probe the effective four-fermion interaction
$(\mathrm{\overline{e}e)}(\overline{\ell}_{\alpha}\ell_{\beta})/\Lambda^{2}$
with the cut-off scale $\Lambda\simeq m_{H}/\sqrt{|h^{\dagger}h|}$ [59, 60,
61, 62, 63].

##  3 LHC and prospects for HL-LHC

During LHC Run 2, in the period 2015–2018, the ATLAS and CMS experiments both
collected nearly 140 fb-1 of 13 TeV proton–proton collision data. These large
data samples, corresponding to the production, per experiment, of about
$8\times 10^{6}$ Higgs bosons, $10^{8}$ $t\overline{t}$ pairs, and $8\times
10^{9}$ Z bosons, form the basis for very sensitive searches for LFV processes
with e$\mu$, e$\tau$, or $\mu\tau$ final states. With the High-Luminosity
programme (HL-LHC), scheduled for 2029–2038, the integrated luminosity is
expected to increase significantly reaching eventually a total of 3 ab-1 per
experiment.

Decays to e$\mu$ final states have a clear experimental signature, and the
invariant mass is readily derived form the lepton momenta. For the e$\tau$ and
$\mu\tau$ channels, the $\tau$ either decays hadronically, $\tau_{h}$, or
leptonically, $\tau_{\ell}$, and the presence of either one or two neutrinos
in the final state makes the reconstruction of the dilepton invariant mass
challenging. Its reconstruction is based on the collinear approximation, where
the direction of neutrino momentum is approximated by the direction of the
other, visible decay products. The component of the event
$\vec{p}_{\mathrm{T}}$ (transverse missing momentum) in that direction is then
used as an estimate of the neutrino momentum. At the relevant high mass
scales, where the $\tau$-leptons are typically highly boosted, the _collinear
mass_ calculated this way is a good estimate.

###  3.1 Z decays

The decay $\text{Z}\rightarrow\text{e}\mu$ has been searched for by ATLAS in
their full Run-2 dataset [5]. The clean signature of a final state with an
e$\mu$ pair whose invariant mass equals the Z mass, allowed to distinguish
clearly signal from the steeply falling background of
$\text{Z}\rightarrow\tau\tau$ events, with the two $\tau$s decaying as
$\tau\to\mathrm{e}\overline{\nu}\nu$ and $\tau\to\mu\overline{\nu}\nu$, and
the flat combinatorial background from multijet events, where the two final
state leptons were either genuine or fake. Also $\text{Z}\rightarrow\mu\mu$
events where one muon emitted bremsstrahlung, and was consequently mis-
identified as an electron, was a non-negligible background. To reduce the
backgrounds, a multivariate boosted decision tree (BDT) was trained to
distinguish between signal and background events, and a threshold on the BDT
output was selected to optimize the search significance. The overall search
efficiency, including acceptance, of the search was about 10%. The final
invariant mass distribution is shown in Fig. 7.

The measurement was dominated by the statistical uncertainty being about twice
as large as the total systematic uncertainty. The dominant contribution to the
systematic uncertainty was due to the statistical uncertainty of the simulated
event samples used to form histograms of the $\text{Z}\rightarrow\tau\tau$ and
$\text{Z}\rightarrow\mu\mu$ backgrounds, which were applied independently to
each bin, allowing the bin content to vary. The expected upper limit including
this contribution was 9.5% higher than it would have been without this
contribution. A 95% CL branching fraction limit was established at
$\mathcal{B}(\mbox{Z}\to\mbox{e}\mu)<2.62\times 10^{-7}$.

ATLAS has published results on searches for the LFV decays $\mathrm{Z\to
e\tau}$ and $\mathrm{Z\to\mu\tau}$, using their full Run-2 dataset, and
considering both hadronic [3] and leptonic [4] $\tau$-lepton decay modes. In
all cases, the two final state leptons were required to be of opposite charge
and, in order to reduce backgrounds from $t\overline{t}$ events, no
$b$-flavoured jets were allowed. In the case of leptonic tau decays, the two
final state leptons were required to be of different flavour, _i.e._ a e$\mu$
pair.

In the case of hadronic tau decays, the main backgrounds were the irreducible
component $\text{Z}\rightarrow\tau\tau\rightarrow\ell+\mbox{\emph{hads}}+3\nu$
and reducible backgrounds from W+jets, $t\overline{t}$ and
$\text{Z}\rightarrow\ell\ell$ production, where either a jet or a light lepton
($\ell=\mathrm{e},\mu$) was misidentified as a hadronic tau decay. In the case
of leptonic tau decays, there was an additional reducible background
contribution from di-bosons events.

Neural network (NN) classifiers were used to identify the three major
background components: $\text{Z}\rightarrow\tau\tau$, W+jets, and
$\text{Z}\rightarrow\ell\ell$, for hadronic tau decays, and
$\text{Z}\rightarrow\tau\tau$, $t\overline{t}$ and di-boson production, for
leptonic tau decays. The NN classifiers were trained and validated with the MC
simulations. Components of the momenta of the light lepton and the $\tau_{h}$
visible momenta, their collinear mass and missing transverse momentum were
typically used as the NN input variables. In order to obtain a single
discriminating quantity, the combined NN classifier was built of individual NN
outputs trained each against one of the three main backgrounds. An example of
the combined NN classifier is shown in Fig. 7.

Figure 7: ATLAS LFV Z decay search. Left: Observed and best-fit predicted
distributions in the signal region for the
$\text{Z}\rightarrow\ell\tau_{\mathrm{had}}$ search. Distributions of the
combined NN output are shown for the $\text{e}\tau_{\mathrm{had}}$ channel,
where taus decay into three charged hadrons. Right: Distribution of the
invariant mass of $\text{Z}\rightarrow\text{e}\mu$ candidates, for data
(points) and expected backgrounds (lines) after the background-only likelihood
fit.

To minimise systematic uncertainties, the simulation of Z-boson production was
improved with a correction derived from $\text{Z}\rightarrow ee,\,\mu\mu$
events. The simulated Z boson $p_{\mathrm{T}}$ spectra were reweighed to match
the unfolded measured distribution. Furthermore, control regions, derived by
inverting some of the signal-region selection cuts, were exploited in a
combined fit to the NN score in the signal region, to improve uncertainties
related to hadronic tau decays, and derive remaining necessary normalisation
factors for the Z and top backgrounds. In the final state with hadronic tau
decays, jets faking $\tau_{h}$ signatures were still significant at high NN
score, where the sensitivity to LFV decays was maximal, and their
normalisation was derived from the combined fit.

In the scenario where the tau leptons are unpolarized, 95% CL branching
fraction limits were established at
$\mathcal{B}(\mbox{Z}\to\mbox{e}\tau)<5.0\times 10^{-6}$ and
$\mathcal{B}(\mbox{Z}\to\mu\tau)<6.5\times 10^{-6}$. For the establishment of
the search sensitivity, the statistical uncertainty contribution was found to
be dominant. The systematic uncertainties on the tau lepton energy and
identification in the hadronic tau final state and on the transverse missing
energy in the leptonic tau final state are unlikely to be further reducible.

Since the ATLAS LFV Z searches have backgrounds, the limits can be expected to
improve by a factor $1/\sqrt{\mathcal{L}}$, where $\mathcal{L}$ is the
collected luminosity, implying ultimately a factor of five improvement down to
sensitivities to branching fractions somewhat below $10^{-6}$ for
$\text{Z}\rightarrow\ell\tau$ and below $10^{-7}$ for
$\text{Z}\rightarrow\text{e}\mu$. The increase in the ATLAS tracking
detector’s acceptance in the HL-LHC phase, as well as a possible combination
with a similar analysis from the CMS experiment, could potentially improve the
sensitivities further.

###  3.2 Z′ and other resonances searches

Searches for heavy resonant (and non-resonant) production of e$\mu$, e$\tau$,
and $\mu\tau$ final states in 13 TeV LHC data have been reported by CMS from
their full Run 2 data set (137.1 fb-1) [42] and by ATLAS from a partial
dataset (36.1 fb-1) [64]. No evidence for physics beyond the standard model in
the dilepton invariant mass spectra was found in neither of the analyses,
allowing limits to be set on various BSM models, including models with heavy
Z′ gauge bosons with LVF transitions.

Processes leading to LFV final states have a clear experimental signature and
a low background from SM processes. Unlike for flavour-diagonal decays, there
is no irreducible background from the Drell–Yan process (dilepton production
in hadron–hadron collisions), and the dominant backgrounds are from
$\mathrm{t\overline{t}}$ and diboson (WW, WZ, ZZ) production, and from
multijet and W+jets processes, where jets are misidentified as leptons.

For both analysis it was assumed that the Z′ boson has couplings to quarks
similar to those of the SM Z boson, and that it decays also to LFV e$\mu$,
e$\tau$, and $\mu\tau$ final states. Whereas CMS assumed the flavour-diagonal
lepton couplings to be SM-like, but added a 10% branching fraction to each of
the LFV modes, ATLAS assumed that only LFV leptonic decays were allowed, and
that only one of these was active at the time with a couplings strength
similar to the leptonic couplings of the SM Z boson. From their larger data
sample, CMS was able to set the most stringent limits, excluding a Z′ up to
5.0 TeV in the e$\mu$ channel, 4.3 TeV in the e$\tau$ channel, and 4.1 TeV in
the $\mu\tau$ channel.

###  3.3 Higgs decays

Searches for LFV decays of the Higgs boson have been performed by ATLAS, CMS,
and LHCb. While some of these analyses focus on the 125-GeV particle, other
explore more generally LFV decays of a lighter or heavier scalar particle.

Searches for the decay of the 125-GeV Higgs boson to e$\tau$ or $\mu\tau$ can
be performed in different final states depending on the $\tau$ decay mode.
Final states with hadronically decaying taus, denoted $\tau_{h}$, benefit from
a larger branching fraction but suffer from larger backgrounds from quark- and
gluon-jets.

ATLAS and CMS have both explored the e$\mu$, e$\tau_{h}$, and $\mu\tau_{h}$
final states [17, 18]. Events were selected online with trigger paths based on
the presence of an electron or a muon with sufficient transverse momentum,
typically 30 GeV. The dominant backgrounds consisted of $\mbox{Z}\to\tau\tau$
events, $\mbox{W}(\to\ell\nu)+\mbox{jets}$ events with a jet misidentified as
a lepton, QCD multijet events with two misidentified jets, and $t\overline{t}$
and diboson productions. To increase the sensitivity, events were divided into
several categories with different signal-to-background ratios. While the
choice of the analysis categories depended on the experiment, both
collaborations isolated events with two jets with a large invariant mass,
characteristic of the vector-boson-fusion production mode.

In a CMS analysis, based on the full Run-2 dataset, results were extracted
from a maximum likelihood fit of the distribution of a boosted decision tree
(BDT) output in the different categories of each final state. The BDT was
trained against a mixture of the dominant backgrounds and included variables
such as the transverse momentum of the leptons, the collinear mass, the
visible mass of the dilepton system, the transverse mass between the missing
transverse momentum and either of the leptons, the pseudorapidity difference
between the leptons, the azimuthal-angle difference between the leptons, and
the azimuthal-angle difference between the missing transverse momentum and
either of the leptons. No excess over the expected background was oberved, and
95% CL limits on the Yukawa couplings were established at
$\sqrt{|Y_{\mu\tau}|^{2}+|Y_{\tau\mu}|^{2}}<1.11\times 10^{-3}$ and
$\sqrt{|Y_{e\tau}|^{2}+|Y_{\tau e}|^{2}}<1.35\times 10^{-3}$, as shown in Fig.
8. The limits on the branching fractions were
$\mathcal{B}(H\to\mu\tau)<0.15\%$ and $\mathcal{B}(H\to e\tau)<0.22\%$.

Figure 8: CMS $H\to\ell\tau$ search: Expected (red line) and observed (black
solid line) 95% CL upper limits on $|Y_{\mu\tau}|$ vs. $|Y_{\tau\mu}|$. The
green and yellow bands indicate the range containing 68% and 95% of all
observed limit variations from the expected limit. The shaded regions are
constraints obtained from null searches for $\tau\to 3\mu$ or $\tau\to 3e$
(dark blue) and $\tau\to\mu\gamma$ or $\tau\to e\gamma$ (purple). The blue
diagonal line is the theoretical naturalness limit
$|Y_{ij}Y_{ji}|=m_{i}m_{j}/v^{2}$ (from Ref.[18]).

ATLAS has currently analyzed 36 fb-1 of 13 TeV data collected in 2016 [17],
and has observed no excess over the background expectation either. A BDT was
used to separate the signal from the backgrounds in the different signal
regions. The collinear mass as well as the dilepton mass reconstructed with
the MMC algorithm were used as input variables, among other variables. The
backgrounds from top-quark and Z-boson productions were constrained from
dedicated control regions in data. The 95% CL branching fraction limits were
$\mathcal{B}(H\to\mu\tau)<0.28\%$ and $\mathcal{B}(H\to\mbox{e}\tau)<0.47\%$.

Significantly more stringent limits are expected to be set at the HL-LHC,
thanks to the increased integrated luminosity, as well as improvements to the
object reconstruction (in particular for hadronically decaying taus) and
analysis strategy (e.g. using deep neural network or using more production
information). Theory uncertainties for the signal are also expected to be
reduced [65]. Taking these points into account, a limit in the range
0.01–0.05% could potentially be reached on the $\mathcal{B}(H\to\mu\tau)$
branching fraction.

LHCb has also explored the possibility of LFV decays of a scalar particle to
$\mu\tau$ final states, scanning over a large mass range of 45–195 GeV, using
2 fb-1 of 8 TeV data [66]. The analysis targets scalar particles produced in
the forward region via the gluon-gluon fusion process. No excess over the
background estimate was observed. An upper limit on the production cross-
section multiplied by the branching fraction was set and ranges from 22 pb for
a boson mass of 45 GeV to 4 pb for a mass of 195 GeV at 95% CL, as shown in
Fig. 9. While the branching fraction limit at a mass of 125 GeV
($\mathcal{B}(H\to\mu\tau)<26\%$) is weaker than those of ATLAS and CMS, LHCb
provides the most sensitive results so far for scalar masses below 125 GeV.

Figure 9: LHCb search for LFV scalar particle decays, $H\to\mu\tau$: Cross-
section times branching fraction 95% CL limits as a function of $m_{H}$. The
observed limits from individual channels are also shown (from Ref.[66]).

CMS has scanned the higher mass spectrum from 200 to 900 GeV in a search for
decays of a heavy scalar to e$\tau$ or $\mu\tau$ final states, using 36 fb-1
of 13 TeV data [67]. The analysis strategy was close to that focussed on the
125-GeV particle, except that more emphasis was put on the precise
reconstruction of the dilepton mass spectrum, and fewer model-dependent
assumptions on the production of the scalar particle were made. The upper
limits on the production cross section multiplied by the branching fraction
range from 51.9 to 1.6 fb for the $\mu\tau$ mode and from 94.1 to 2.3 fb for
the $e\tau$ mode, at 95% CL.

Figure 10: CMS search for LFV decays of a heavy scalar particle,
$H\to\ell\tau$: Observed and expected 95% CL upper limits on the production
cross section multiplied by the branching fraction for the $\mu\tau$ (left)
and $e\tau$ (right) decay channels (From Ref.[67]).

Searches for Higgs boson decays to e$\mu$ can take advantage of the excellent
mass resolution of the lepton pair, and look for a narrow excess of events
over a smooth background. The ATLAS experiment has established the most
stringent limit on such a decay, $\mathcal{B}(H\to e\mu)<6.1\times 10^{-5}$,
from their full Run-2 dataset [68]. In the analysis, events were classified in
various categories to improve the analysis sensitivity, on the basis of the
following variables: the subleading lepton transverse momentum, the dilepton
transverse momentum, the lepton pseudorapidity, and the dijet pseudorapidity
and invariant mass if there were at least two jets. The results were extracted
from an unbinned parametric maximum likelihood fit, where the narrow signal
resonance was parameterized by the sum of a Crystal Ball function and a
Gaussian function, and a Bernstein polynomial of degree 2 was used to describe
the background mass spectrum. The result is significantly better than the
limit of $3.5\times 10^{-4}$ set by the CMS Collaboration using their Run-1
dataset [69].

Figure 11: Dilepton invariant mass for the $e\mu$ channel compared with the
background-only model. The signal expectation with $\mathcal{B}(H\to
e\mu)=0.05\%$ is shown in red. The bottom panel shows the difference between
data and the background-only fit (from Ref.[68]).

###  3.4 Top quark processes

A search for CLFV processes in top-quark reactions has been recently published
by CMS [37]. The analysis, based on 138 fb-1 of 13 TeV LHC proton-proton
collision data, includes both production and decay modes of the top quark,
_i.e._ $\mbox{q}\to\mbox{e}\mu\mbox{t}$ and $\mbox{t}\to\mbox{e}\mu\mbox{q}$,
respectively, with q referring to an up or charm quark. The result of the
search was interpreted in terms of limits on vector, scalar, and tensor four-
fermion interactions.

Whereas the production-mode process involves single-top events, the decay-mode
process is dominated by t$\overline{\mbox{t}}$ events. In both cases, signal
events contain an oppositely charged e$\mu$ pair together with multiple jets,
one of which stems from the hadronization of a bottom quark from a
$\mbox{t}\to\mbox{bW}$ decay. An event sample was selected by requiring one
electron, one muon of opposite charge, and at least one b-tagged jet. The
dominant source of background was from t$\overline{\mbox{t}}$ events. Hence,
the sample was divided into a signal region containing events with exactly one
b-tagged jet and a t$\overline{\mbox{t}}$ control region containing events
with at least two b-tagged jets. A boosted decision tree (BDT) combining five
discriminating variables was used to distinguish the signal from the
background. The distributions of the input variables, of which the
$p_{\mathrm{T}}$ of the leading lepton was the most powerful, were found to be
well described by the simulated background model. The BDT was trained on
simulated samples of CLFV signal events, on the one hand, against background
events, on the other. Events from the production channel
($\mbox{q}\to\mbox{e}\mu\mbox{t}$) were found to dominate over those from the
decay channel ($\mbox{t}\to\mbox{e}\mu\mbox{q}$) in the discrimination. Two
factors contributed to this observation: Firstly, the production channel had a
higher event yield than the decay channel. Secondly, the production channel
was better separated from the background via a harder final-state-particle
$p_{\mathrm{T}}$ distribution. The data was found to be consistent to
expectation of the SM in the absence of signal, and upper limits were set on
the strength of individual vector-, scalar-, and tensor-like four-fermion
effective operators. These were then converted into 95% CL limits on branching
fractions of the top quark $\mathcal{B}(\mbox{t}\to\mbox{e}\mu\mbox{q})$, q=u
(c), $<0.13\times 10^{-6}$ ($1.31\times 10^{-6}$), $0.07\times 10^{-6}$
($0.89\times 10^{-6}$), and $0.25\times 10^{-6}$ ($2.59\times 10^{-6}$) for
vector, scalar, and tensor CLFV interactions, respectively. The search limits
on $\mathcal{B}(\mbox{t}\to\mbox{e}\mu\mbox{c})$ and
$\mathcal{B}(\mbox{t}\to\mbox{e}\mu\mbox{u})$ are indeed correlated, as
illustrated in Fig. 12.

Figure 12: CMS CLFV top decay search: Observed 95% CL exclusion limits on
$\mathcal{B}(\mbox{t}\to\mbox{e}\mu\mbox{c})$ as a function of
$\mathcal{B}(\mbox{t}\to\mbox{e}\mu\mbox{u})$ for the vector-, scalar-, and
tensor-like CLFV interactions. The hatched bands indicate the regions
containing 68% of the distribution of limits expected under the background-
only hypothesis (from Ref. [37]).

It is interesting to observe that the CMS limits are about one order of
magnitude lower than that predicted in Ref. [36] from a similarly sized data
sample. The main reason for this difference shall be probably be traced to
CMS’s use of the production channel, which was not considered in the earlier
work.

##  4 Future e+e- colliders

Electron-positron colliders are largely complementary to proton-proton
colliers in the search for rare phenomena in heavy particle decays. Whereas
proton-proton colliders provide enormous event samples, electron-positron
colliders generally deliver smaller samples but in cleaner experimental
conditions. A further advantage arises from the constrained kinematics in e+e-
collisions, where the initial state is precisely known. As an illustration of
the complementarity between the two types of colliders, the recent ATLAS
limits on CLFV Z decays, were derived from event samples with two-to-three
orders of magnitude more Z decays than the previous limits from LEP, which
they succeeded by small factors.

Several future electron-positron colliders have been proposed. Until ten years
ago, it was generally believed that any future high-energy e+e- collider would
be linear, and there were two proposals on the market: The International
Linear Collider (ILC) [70, 71] and the Compact Linear Collider (CLIC) [72].
With the discovery in 2012 that the Higgs boson is light, circular colliders
had a strong revival with two proposed projects: The FCC-ee [73] at CERN and
the largely similar CEPC [74] in China. For the search for rare decays of
heavy SM particles, circular colliders have a marked advantage in their
superior luminosities at the relevant collisions energies: the Z pole
($\sqrt{s}=91.2$ GeV), the cross-section maximum for HZ production
($\sqrt{s}=240$ GeV), and the threshold for $\mathrm{t\overline{t}}$
production ($\sqrt{s}\gtrsim 350$ GeV). As an illustration, where a GigaZ
phase is discussed for the ILC as a possible upgrade option beyond the
15-years Higgs programme, a TeraZ phase is already included in the FCC-ee
baseline programme.

The FCC-ee is the first stage of the integrated Future Circular Colliders
(FCC) project to be based on a novel research infrastructure hosted in a
$\sim$100-km tunnel in the neighbourhood of CERN. The FCC-ee programme [75]
includes four major phases with precision measurements of the four heaviest
particles of the Standard Model: _i_) the Z boson, with $5\times 10^{12}$ Z
decays collected around the Z pole (4 years), _ii_) the W boson, with $10^{8}$
WW pairs collected close to threshold (2 years), _iii_) the Higgs boson, with
$1.2\times 10^{6}$ e+e${}^{-}\to$ HZ events produced at the cross-section
maximum (3 years), and _iv_) the top quark, with $10^{6}$
$\mathrm{t\overline{t}}$ pairs produced at and slightly above threshold (5
years). It is interesting to note that, with its extreme Z-pole luminosity,
exceeding $10^{36}\,\text{cm}^{-2}\,\text{s}^{-1}$, FCC-ee will produce a Z
sample larger by about two orders of magnitude than that of HL-LHC, and even
comparable to what is expected from FCC-hh.

FCC-ee will feature two (possible four) interaction points each equipped with
a powerful, state-of-the art detector system. Detector concepts being studied
feature a solenoidal magnetic field, a small pitch, thin layers vertex
detector providing an excellent impact parameter resolution for lifetime
measurements, a highly transparent tracking system providing a superior
momentum resolution, a finely segmented calorimeter system with excellent
energy resolution, and a highly efficient muon system. At least one of the
detector systems will be equipped with efficient particle-identification
capabilities allowing $\pi/\text{K}$ separation over a wide momentum range.
Important for the following discussion is a typical momentum resolution of
$\sigma_{p}/p\simeq 1.5\times 10^{-3}$ at $p=45.6$ GeV.

###  4.1 Z Decays

The prospects of searches for CLFV Z decays at the FCC-ee were first discussed
in Ref. [76], from which much of the following information has been extracted.

#### $\boldsymbol{\text{Z}\rightarrow\mu\mathrm{e}}$

From OPAL at LEP, a background-free search for $\text{Z}\to\mu$e decays
resulted in a limit of $1.7\times 10^{-6}$ (95% CL). Even with a much improved
detector performance, backgrounds will eventually show up in the
$\mathcal{O}(10^{5})$ larger FCC-ee samples. As at the LHC, an irreducible
background is expected from
$\text{Z}\to\tau\tau\to\mu\text{e}\overline{\nu}\nu\overline{\nu}\nu$, where
the two charged leptons are both emitted close to the end-point of their
momentum spectra. With the superior momentum resolution of FCC-ee detectors,
this background is tiny corresponding to a Z-boson branching fraction of only
$\mathcal{O}(10^{-11})$. Here, FCC-ee has an enormous advantage over LHC. Not
only is the momentum resolution better, but, more importantly, the Z-mass
constraint is immensely more powerful. Whereas in proton-proton collisions,
the mass constraint is limited by the 2.5 GeV natural width of the Z boson, at
FCC-ee the mass is precisely known from the collision energy with a dispersion
of only 85 MeV from the beam energy spread.

A potentially more serious background arises from so-called catastrophic
bremsstrahlung of muons in the material of the electromagnetic calorimeter
(ECAL), by which a muon radiates off a significant fraction of its energy,
with the subsequent risk of being mis-identified as an electron. In Ref. [76]
it is argued that this background can be controlled down to a level
corresponding to a Z-boson branching fraction of $\mathcal{O}(10^{-8})$ via a
careful design of the ECAL including a good energy resolution and longitudinal
segmentation, where the latter allows for the suppression of late starting
showers. Important for controlling this background is the ability to precisely
determine the mis-identification rate from the data themselves. For this, an
independent means of e/$\mu$ separation, such at that provided by a powerful
d$E$/d$x$ measurement, will be essential, and it is encouraging to notice,
that a 3–4 standard deviation e/$\mu$ separation at $p=45.6$ GeV may be
possible via the use of the cluster counting technique in a Helium based drift
chamber [77].

In conclusion, a sensitivity for the $\text{Z}\rightarrow\mu\text{e}$ mode at
the $10^{-8}$ level should be safely within reach at FCC-ee. An independent
method for e/$\mu$ separation, as that provided by a powerful d$E/$d$x$
measurement, could improve this sensitivity by one to two orders of magnitude
potentially all the way down to the $10^{-10}$ level.

####  $\boldsymbol{\text{Z}\rightarrow\tau\mathrm{e}}$ and
$\boldsymbol{\text{Z}\rightarrow\tau\mu}$

In e+e- collisions, the pursuit for $\text{Z}\rightarrow\tau\text{e}\
(\tau\mu)$ decays amounts to a search for events with a _clear tau decay_ in
one hemisphere recoiling against a _beam-momentum electron (muon)_ in the
other. This procedure was used at LEP, where sensitivities were at the
$10^{-5}$ level.

To illuminate the analysis procedure, first consider the term _clear tau
decay_. Here, the point is to restrict the analysis to those decay modes,
where the probability is minimal of misidentifying a final-state lepton from
$\text{Z}\to\mu\mu$ or $\text{Z}\to\text{ee}$ as a $\tau$ decay. Immediately
this excludes the leptonic modes $\tau\to\mu\overline{\nu}\nu$ and
$\tau\to\text{e}\overline{\nu}\nu$ and possible also the hadronic mode
$\tau\to\pi\nu$, where the pion may be mis-identified as a lepton. To reach
very high purities, it may be ultimately necessary to restrict the analysis to
exclusive modes such as $\tau\to\rho\nu\to\pi\pi^{0}\nu$ and modes with three
(or more) charged particles. Secondly, consider the term _beam-momentum
electron (muon)_. The momentum distribution of the final-state charged lepton
is given by [78]

$\frac{1}{\Gamma}\frac{\mathrm{d}\Gamma}{\mathrm{d}x}=\frac{1}{3}\left[\left(5-9x^{2}+4x^{3}\right)+P_{\tau}\left(1-9x^{2}+8x^{3}\right)\right],$
(22)

where $x=p/p_{\mathrm{beam}}$, and $P_{\tau}$ is the longitudinal polarisation
of the $\tau$ leptons. The density of events close to the beam-momentum
endpoint depends on $P_{\tau}$, which has the value $P_{\tau}\simeq-0.15$, as
measured at LEP [79]. The separation of signal and background now depends on
the experimental precision by which a _beam-momentum particle_ can be defined.
This is illustrated in Figure 13, where a momentum spread of $1.8\times
10^{-3}$ has been assumed. This value arises as a combination of the
$0.9\times 10^{-3}$ spread of the collision energy and the $1.5\times 10^{-3}$
momentum resolution typical for a FCC-ee detector at $p=45.6$ GeV.

Figure 13: FCC-ee search for lepton flavour violating decays
$\text{Z}\rightarrow\tau\ell$, $\ell=\mathrm{e},\mu$. Momentum distribution
(upper part) of the final state lepton $\ell$ for the signal (red) and for the
background from $\text{Z}\rightarrow\tau\tau$, with
$\tau\rightarrow\ell\overline{\nu}\nu$ (blue). A momentum resolution of
$1.8\times 10^{-3}$ has been assumed, see text. For illustration, the signal
branching fraction is set here to $\mathcal{B}(\text{Z}\to\tau\ell)=10^{-7}$.

A simple estimate of the search sensitivity was established by assuming a
signal efficiency of 25% and by defining the signal box via the simple
requirement $x>1$. For the quoted momentum spread, this procedure indicated a
sensitivity to branching fractions down to $10^{-9}$. The sensitivity was
found to scale linearly in the momentum spread.

###  4.2 Higgs Boson Decays

The sensitivity of searches for CLFV Higgs decays, $\text{H}\to\text{e}\mu$,
e$\tau$, and $\mu\tau$, at future e+e- colliders has been estimated in Ref.
[80]. The studies were based on the dominant Higgs production process
$\mbox{e}^{+}\mbox{e}^{-}\to\mbox{ZH}$, with an assumed event multiplicity of
about one million at circular colliders (CEPC and FCC-ee) and half a million
at the ILC. Longitudinal beam polarisation was assumed at the ILC, leading to
a somewhat larger HZ production cross section, but no other discernible
effect. The dominant hadronic decay mode of the Z into quark pairs was used
for all three Higgs decay modes, and relatively loose requirements on the di-
jet invariant mass ensured a high reconstruction efficiency for the Z.

For the $\text{H}\to\text{e}\mu$ mode, the electron–muon invariant mass was
asked to be consistent, within a window, with the Higgs boson mass, leading to
a signal detection efficiency of 41%, and very few accepted background events.
The upper bound on the $\text{H}\to\text{e}\mu$ branching fraction was found
to be $\mathcal{B}(\text{H}\to\text{e}\mu)<1.2\times 10^{-5}$ ($2.1\times
10^{-5}$) at CEPC and FCC-ee (ILC).

For the two modes involving $\tau$s, $\text{H}\to\text{e}\tau$ and
$\text{H}\to\mu\tau$, the same event signature was used, _i.e._ two jets, one
electron, and one muon. With this choice, targeting the leptonic $\tau$ decay
mode $\tau\rightarrow\mu\overline{\nu}\nu$ for $\text{H}\to\text{e}\tau$
($\tau\rightarrow\text{e}\overline{\nu}\nu$ for $\text{H}\to\mu\tau$), the
$\tau$ was then reconstructed from the muon (electron) and the missing energy.
With a ceiling on the reconstructed $\tau$ mass of a few GeV, the e$\tau$
($\mu\tau$) invariant mass was then required to be consistent with the Higgs
boson mass. The overall signal efficiency was at the 5% level, nearly one
order of magnitude lower than for the $\text{H}\to\text{e}\mu$ mode, with more
than half of the difference caused by the $\sim$18% $\tau$ leptonic branching
fraction. The upper bound on the $\text{H}\to\text{e}\tau$ and
$\text{H}\to\mu\tau$ branching fractions were found to be rather similar, so
here we quote the mean of the two: $\mathcal{B}(\text{H}\to\ell\tau)<1.5\times
10^{-4}$ ($2.4\times 10^{-4}$) at CEPC and FCC-ee (ILC).

##  5 Summary

The first precision results on CLFV decays of heavy particles came from LEP
and were based on about $4\times 10^{6}$ observed Z decays per experiment.
Branching fraction limits were established at about $2\times 10^{-6}$ for
$\mbox{Z}\to\mbox{e}\mu$ and $10^{-5}$ for
$\mbox{Z}\to\mbox{e}\tau,\,\mu\tau$. Recently, ATLAS has improved these limits
by a factor of about seven for the $\mu$e mode and about two for the,
obviously more difficult, $\tau$e and $\tau\mu$ modes. No results have so far
been released by CMS. Sensitivities can be expected to improve by a factor
$1/\sqrt{\mathcal{L}}$, where $\mathcal{L}$ is the collected luminosity,
implying ultimately an improvement by an additional factor of about five from
HL-LHC. Sensitivities would then be at the $10^{-7}$ level for
$\mbox{Z}\to\mbox{e}\mu$ and at the $10^{-6}$ level for
$\mbox{Z}\to\mbox{e}\tau,\,\mu\tau$. With its phenomenal Z-pole luminosity
performance, FCC-ee will produce $5\times 10^{12}$ Z decays exceeding HL-LHC
by about two orders of magnitude. Sensitivities to CLFV Z decays will depend
strongly on detector performance. Of particular importance is the optimization
of detectors in terms of momentum resolution and PID performance.
Sensitivities could reach down to the $10^{-9}$ level. For the $\mu$e mode the
sensitivity will depend critically on whether sufficient instrumental
redundency is available to control precisely the fraction of muons appearing
as electrons due to hard bremsstrahlung.

Searches for heavy resonant (and non-resonant) production of e$\mu$, e$\tau$,
and $\mu\tau$ final states above the Z resonance have been reported by ATLAS
and CMS. From their full Run-2 dataset, CMS set the most stringent limits,
excluding a Z′ up to 5.0 TeV in the e$\mu$ channel, 4.3 TeV in the e$\tau$
channel, and 4.1 TeV in the $\mu\tau$ channel.

Searches for LFV decays of the Higgs boson have been reported by ATLAS and
CMS. The most stringent limits, extracted from the full Run-2 datasets, are
$\mathcal{B}(H\to\mu\tau)<0.15\%$ and $\mathcal{B}(H\to e\tau)<0.22\%$, from
CMS, and $\mathcal{B}(H\to e\mu)<6.1\times 10^{-5}$, from ATLAS. Accounting
for the 20 times larger HL-LHC datasets and possible improvements to the
analysis, the modes involving taus could potentially reach sensitivities at
the $1$–$5\times 10^{-4}$ level, whereas the e$\mu$ mode could reach the
$10^{-5}$ level. Similar sensitivities have been estimated for CEPC and FCC-ee
from their samples of about one million ZH events.

Complementary searches for the decay to $\mu\tau$ of a scalar particle has
allowed LHCb and CMS to set branching fraction limits in a wide mass range.

From the more than $10^{8}$ top quarks produced during Run 2, CMS has
established bounds on the $t\to u\mu$e and $t\to c\mu$e branching fractions at
the level of $10^{-7}$ and $10^{-6}$, respectively. Improvements of about a
factor five can be expected from HL-LHC. Future e+e- colliders will be limited
by the about 106 $t\overline{t}$ events they will produce and will not be able
to improve significantly on the HL-LHC results.

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