aainstitutetext: Laboratory of High Energy and Computational Physics, NICPB,
Rävala pst. 10, 10143 Tallinn, Estonia

# Production and decays of 146 GeV flavons into $e\mu$ final state at the LHC

Niko Koivunen a Martti Raidal<EMAIL_ADDRESS>
###### Abstract

The CMS experiment at CERN has reported a possible signal for a resonance at
146 GeV decaying into the $e\mu$ final state which, presently, is the only
experimental hint for lepton flavour violation in any low- and high-energy
experiment. The Froggatt-Nielsen mechanism naturally predicts the existence of
new scalars, the flavons, with flavour off-diagonal couplings. We study this
framework in the context of the CMS result and find that the minimal, purely
leptophilic model is too restricted to match the claimed signal. Thereafter we
show how models with additional flavon couplings to quarks can explain the
claimed signal while satisfying all the existing constraints on lepton flavour
violation.

##  1 Introduction

The CMS Collaboration has recently reported a possible signal for a new
resonance at $146$ GeV decaying into charged lepton flavour violating (LFV)
$e\mu$ final state111The $e\mu$ final state includes both the $e^{+}\mu^{-}$
and $e^{-}\mu^{+}$ final states. CMS:2023pqk ; CMS:Moriond . This is the first
time when a hint for such a signal is reported, with no similar result from
the ATLAS Collaboration CMS:Moriond , however. The claimed signal is based on
$\sqrt{s}=13$ TeV CMS data, with integrated luminosity of 138 $\rm{fb}^{-1}$.
The global (local) significance of the claimed signal is $2.8\sigma$
($3.8\sigma$) over the expected background. At the same time, the CMS does not
find any excess in the corresponding Higgs boson decay channel $h\to e\mu$
CMS:Moriond .

The standard model (SM) of particle physics does not contain any new resonance
nor have any sources of LFV. Therefore, if confirmed, this signal should be
interpreted as a sign for new physics beyond the SM. The searches for LFV
decays of $\mu$ and $\tau$ leptons, and for $\mu\leftrightarrow e$ conversion
have given null result so far Workman:2022ynf , consistently with the SM.
Similarly, no LFV decays of the SM Higgs boson have been observed. The LFV
decays $h\to e\mu$ have been constrained by the CMS and ATLAS Collaborations
as $\rm{BR}(h\to e\mu)<4.4\times 10^{-5}$ CMS:2023pqk  and $\rm{BR}(h\to
e\mu)<6.2\times 10^{-5}$ ATLAS:2019old , respectively, while the CMS searches
for decays $h\to e\tau$ and $h\to\mu\tau$ place upper bounds on the
corresponding branching ratios as $\rm{BR}(h\to e\tau)<2.2\times 10^{-4}$ and
$\rm{BR}(h\to\mu\tau)<1.4\times 10^{-4}$ CMS:2021rsq . Consequently, if the
claimed CMS hint for such a new physics will be confirmed, this would be the
first signal of LFV in any low- and high-energy experiment.

Assuming that the CMS hint for new resonance with LFV couplings actually
corresponds to reality, as we do in this work, it is extremely challenging to
reconcile the claimed signal with the stringent bounds on LFV. Indeed, in this
case the scale of new physics alone does not suppress any LFV process.
Therefore, to explain simultaneously the presence of the claimed CMS signal
and the absence of $\mu\to e\gamma$ and $\mu\leftrightarrow e$ conversion in
nuclei, one must involve some additional mechanism to suppress the latter
processes. For example, one could try to identify the new resonance with a
spin-1 particle, like a $Z^{\prime}$ with generation non-universal $U(1)$
couplings to charged leptons. This approach seems difficult, considering that
there is no other degrees of freedom to cancel the $Z^{\prime}$ contribution
to $\mu\to e\gamma$, for example. However, in the case of complex scalars
there exists a generic built-in cancellation between the scalar and pseudo-
scalar components to loop-induced LFV processes. Any new physics scenario
introducing LFV couplings at the scale ${\cal O}(100)$ GeV scale must involve
such a mechanism to comply with observations. Therefore, we choose to work
with scalars to address the new CMS hint for new physics.

The Froggatt-Nielsen (FN) mechanism Froggatt:1978nt  is one of the most well-
known methods of explaining the observed hierarchy in masses of charged
fermions. An integral part of the mechanism is the existence of _flavons_ ,
scalars with flavour violating interactions to fermions, whose vacuum
expectation values (VEVs) generate the observed charged fermion masses and
mixing via higher-order operators. The physics motivation for the Froggatt-
Nielsen mechanism goes beyond collider physics. Nevertheless, it is
interesting to ask whether the existence of low-scale flavons is compatible
with the present particle physics phenomenology.

In this work we study the possibility that the claimed CMS signal represents
the very first experimental hint for the Froggatt-Nielsen flavon. First we
shall concentrate on the _leptophilic_ flavon which only generates the flavour
structure of the lepton sector and, therefore, does not couple to quarks. This
allows us to avoid quite stringent additional constraints from the flavour
violation in the quark sector, that would force the flavon VEV to higher
scales and suppress the effects of the $e\mu$-coupling. In this set-up the SM
Higgs boson and the flavon mix due to portal coupling in the scalar potential,
producing two mass eigenstates, $H_{1}$, identified as the $125$ GeV Higgs
boson and $H_{2}$, identified as the $146$ GeV particle. In the leptophilic
Froggatt-Nielsen framework the state $H_{2}$, with the dominant flavon
component, obtains couplings to gauge bosons and quarks through the mixing
with the SM Higgs, thus allowing it to be produced at the LHC through the same
processes as the SM $125$ GeV Higgs boson. Importantly, it is well established
Huitu:2016pwk  that this scenario does not have any problem to satisfy all the
LFV constraints as the pseudo-scalar contribution naturally cancels the scalar
contribution to $\mu\to e\gamma$.

We find that, for the maximally allowed Higgs boson-flavon mixing, the
maximally allowed production cross-section for the $146$ GeV resonance is an
order of magnitude smaller than the one claimed by the CMS experiment. To
increase the production cross section, we introduce additional flavour-
diagonal couplings of the flavon to quarks. This, however, re-introduces the
problem of LFV because, now, the tree level contribution to
$\mu\leftrightarrow e$ conversion is no longer suppressed by the mixing angle.
The problem can be solved if the flavon couplings to quarks introduce
additional cancellation between the tree level amplitudes to
$\mu\leftrightarrow e$ conversion. We demonstrate that this can be arranged
simultaneously for the experiments based on $Ti$ and $Au$ nuclei.

We conclude that viable Froggatt-Nielsen scenarios with low-scale flavons can
be constructed to address the hint for new $146$ GeV resonance. However, these
scenarios require non-trivial model building and some cancellation between the
model parameters. More experimental data is needed to clarify the status of
the claimed CMS hint for LFV.

##  2 The leptophilic Froggatt-Nielsen model

The claimed CMS result CMS:2023pqk ; CMS:Moriond  hints for a new resonance at
the electroweak scale. The Froggatt-Nielsen mechanism should preferably be
taking place at least at that scale, in order not to suffer from heavily
suppressed flavon couplings. This does not seem to be possible if we apply the
Froggatt-Nielsen mechanism to both quark and lepton sectors simultaneously.
This is due to the tree-level flavon mediated neutral meson mixing, such as
the $K^{0}$-$\bar{K}^{0}$ mixing. The flavon VEV would then have to be close
to TeV scale to avoid those constraints Bauer:2016rxs , which is an order of
magnitude higher than the scale indicated by the CMS experiment. We,
therefore, apply the Froggatt-Nielsen mechanism to charged leptons only,
allowing the flavon VEV to be at the electroweak scale, as demonstrated in
Huitu:2016pwk 222Each fermion sector could, in principle, have their own
flavon field, with different VEVs..

The Froggatt-Nielsen mechanism has been previously studied in the context of
flavour violation in the Higgs boson decays Dery:2013rta ; Dery:2014kxa ;
Huitu:2016pwk  and in the perspective of the LHC phenomenology Tsumura:2009yf
; Bauer:2016rxs .

###  2.1 The Froggatt-Nielsen mechanism and LFV

The Froggatt-Nielsen framework extends the SM with a flavour symmetry which in
the simplest case is global or local $U(1)$ or a $Z_{N}$ symmetry. We will
take the flavour symmetry to be global $U(1)$. The framework introduces, as
new fields, heavy fermion messengers and a complex scalar flavon. The SM
fermions and the new particles are also charged under the flavour symmetry.
The purpose of the flavour symmetry is to forbid the SM Yukawa couplings, with
the possible exception for the top quark. The Yukawa couplings are, instead,
generated from effective operators. The heavy fermion messengers connect the
SM fermions to Higgs boson at tree-level. Once the heavy fermion messengers
are integrated out, one is left with an effective operator, which in the case
of charged leptons is

$\mathcal{L}\supset
c_{ij}\left(\frac{\Phi}{\Lambda}\right)^{n_{ij}}\bar{L}_{L,i}He_{R,j}+h.c.,$
(1)

where $c_{ij}$ are dimensionless order-one couplings, $\Lambda$ is the mass
scale of the integrated out messenger fermions, $L_{L,i}$ are the $SU(2)_{L}$
lepton doublets and $e_{R,i}$ are the $SU(2)_{L}$ charged lepton singlets, and
$H$ is the SM Higgs doublet. $\Phi$ stands for a complex scalar flavon that is
decomposed into real and imaginary parts as $\Phi=1/\sqrt{2}(\phi+iA)$. The
conservation of flavour charge fixes the power $n_{ij}$ as

$n_{ij}=-\frac{1}{q_{\phi}}(q_{\bar{L},i}+q_{R,j}+q_{H}).$ (2)

The power $n_{ij}$ is the number of fermion messengers that were integrated
out from the diagram that generated the corresponding effective term. The
$n_{ij}$ is therefore a positive integer.

We assume that the flavon couples only to leptons, so that the Froggatt-
Nielsen mechanism generates only the charged lepton flavour structure. The
operator (1) gives rise to the SM Yukawa couplings, as the flavon acquires a
non-zero VEV. Expanding the operator (1) around the vacuum yields

$\displaystyle\mathcal{L}\supset
c_{ij}\left(\frac{\Phi+\frac{v_{\phi}}{\sqrt{2}}}{\Lambda}\right)^{n_{ij}}\bar{L}_{L,i}(H+\langle
H\rangle)e_{R,j}+h.c.$ (3) $\displaystyle=$ $\displaystyle
c_{ij}\left(\frac{v_{\phi}}{\sqrt{2}\Lambda}\right)^{n_{ij}}\bar{e}_{L,i}e_{R,j}\frac{1}{\sqrt{2}}(h+v)+n_{ij}c_{ij}\left(\frac{v_{\phi}}{\sqrt{2}\Lambda}\right)^{n_{ij}}\frac{v}{v_{\phi}}\bar{e}_{L,i}e_{R,j}~{}\Phi+h.c.+\cdots,$

where we have kept only the renormalizable terms. The first term in the second
line gives the SM Yukawa coupling

$Y_{ij}\equiv c_{ij}\left(\frac{v_{\phi}}{\sqrt{2}\Lambda}\right)^{n_{ij}}.$
(4)

The charged lepton mass hierarchy is explained by assuming $\epsilon\equiv
v_{\phi}/(\sqrt{2}\Lambda)<1$ and by assigning larger Froggatt-Nielsen charges
to the lighter leptons compared to the heavier ones. The flavour charge
assignment determines the hierarchy of the Yukawa couplings. This is in
contrast to the Standard Model where the hierarchy is obtained by tuning the
couplings themselves.

The flavon also has Yukawa-like interaction to leptons as can be seen from the
second line of Eq. (3),

$\widetilde{\kappa}_{ij}\equiv\frac{v}{v_{\phi}}n_{ij}Y_{ij}.$ (5)

This coupling is not proportional to the Yukawa matrix. It is not diagonalized
simultaneously with the charged lepton mass matrix and, therefore, the flavon
couplings are flavour violating.

The physical couplings are obtained by diagonalizing the Higgs Yukawa coupling
matrix

$Y_{\rm diag}=U_{L}YU_{R}^{\dagger}.$ (6)

The equation (3) then becomes

$\mathcal{L}=\frac{1}{\sqrt{2}}\bar{e}_{L}Y_{\rm
diag}e_{R}(h+v)+\bar{e}_{L}\kappa e_{R}~{}\Phi+h.c.,$ (7)

where

$\kappa_{ij}=\frac{v}{v_{\phi}}U_{L}(n\cdot
Y)U_{R}^{\dagger},\quad\textrm{with}\quad(n\cdot Y)_{ij}=n_{ij}Y_{ij}.$ (8)

The flavon coupling $\kappa$ can be written in a form

$\kappa_{ij}=\frac{v}{v_{\phi}}\sum_{k}\left[Y_{j}q_{\bar{L},k}(U_{L})_{ik}(U_{L}^{\dagger})_{kj}+Y_{i}q_{\bar{R},k}(U_{R})_{ik}(U_{R}^{\dagger})_{kj}\right],$
(9)

where $Y_{i}$ is the lepton SM Yukawa coupling. From this expression one can
deduce the maximal values for the flavour violating couplings. These are
obtained when either the left-handed or the right-handed rotation matrix has
”maximal” mixing, that is two elements $\sim 1/\sqrt{2}$ on the same row or
column333This can happen if all the left- or the right-handed flavour charges
are similar. If all the left-handed charges are identical, each mass matrix
entry in a column has the same order of magnitude, and if all the right-handed
charges are identical, each entry in each row has same order of magnitude.
Even though identical charges for a handedness produces large entries in
mixing matrix, the effect is lost due to unitarity. One has to break the
degeneracy slightly, in order to preserve the contribution of large mixing in
(9).. The maximum flavour violating flavon coupling to electron and muon
therefore is

$|\kappa^{\rm max}_{e\mu~{}\rm{or}~{}\mu
e}|\sim\frac{v}{v_{\phi}}\frac{Y_{\mu}}{2}\approx 3.0\times
10^{-4}\frac{v}{v_{\phi}}.$ (10)

This coupling is boosted if the flavon VEV is small compared to that of the SM
Higgs. We will use this maximally large $e\mu$ coupling to estimate the
maximum cross section obtainable in the leptophilic case in addition numerical
benchmark. We set $\epsilon=0.1$ and use in both cases the flavour charges
presented in Table 1.

Particle | $e_{L}^{c}$ | $e_{R}$ | $\mu_{L}^{c}$ | $\mu_{R}$ | $\tau_{L}^{c}$ | $\tau_{R}$ | $H$ | $\phi$
---|---|---|---|---|---|---|---|---
Charge | 6 | 1 | 4 | 0 | 2 | 0 | 0 | -1
Table 1: The $U(1)$ flavour charges used in the numerical and analytical
estimates.

We have chosen right-handed charges so that they are almost the same and
broken the degeneracy in the first generation. This will produce large mixing
in the right-handed rotation matrix. The left-handed charges are different and
produce the required hierarchy in the eigenvalues.

In the analytical estimate for the largest possible $e\mu$-cross section we
assume that only off-diagonal coupling of the flavon is $\kappa_{\mu e}$ and
that it is given by (10). This will maximize the brancing ratio into $e\mu$.
The diagonal flavon couplings in this case are approximately
$\widetilde{\kappa}_{ee}\approx 6Y_{e},\,\widetilde{\kappa}_{\mu\mu}\approx
4Y_{\mu}$ and $\widetilde{\kappa}_{\tau\tau}\approx 2Y_{\tau}$. We will use
these values in our analytical estimate.

We also provide the following numerical benchmark that produces the correct
masses for electron, muon and tau:

$Y=\left(\begin{array}[]{ccc}4.1\epsilon^{7}&0.63\epsilon^{6}&4.3\epsilon^{6}\\\
6.3\epsilon^{5}&-5.5927\epsilon^{4}&6.0\epsilon^{4}\\\
-5.7\epsilon^{3}&-0.2\epsilon^{2}&0.8219\epsilon^{2}\end{array}\right),\quad\widetilde{\kappa}\approx\left(\begin{array}[]{ccc}1.9\times
10^{-5}&-2.7\times 10^{-6}&1.0\times 10^{-5}\\\ -2.1\times 10^{-4}&2.6\times
10^{-3}&-9.1\times 10^{-4}\\\ 3.1\times 10^{-3}&3.5\times 10^{-3}&2.4\times
10^{-2}\end{array}\right).$ (11)

The diagonalization matrices are given by

$U_{L}\approx\left(\begin{array}[]{ccc}1&-1.4\times 10^{-3}&-2.3\times
10^{-4}\\\ -1.4\times 10^{-3}&1&5.5\times 10^{-2}\\\ 3.0\times
10^{-4}&5.5\times 10^{-2}&1\end{array}\right)\quad\rm{and}\quad
U_{R}\approx\left(\begin{array}[]{ccc}0.55&0.64&0.54\\\ -0.62&0.74&-0.25\\\
-0.56&-0.20&0.81\end{array}\right).$ (12)

Note that the element $\widetilde{\kappa}_{\mu e}$ is $\sim 2/3$ of the
maximum coupling and allows to produce large $\rm{BR}(H_{2}\to e\mu)$. The
flavon VEV should be $\gtrsim 100$ GeV, in order to have large enough
messenger scale, $\Lambda\gtrsim 1$ TeV.

###  2.2 Phenomenology at hadron colliders

The scalar potential we work with is of the Higgs-portal type,

$V=-\mu_{h}^{2}(H^{\dagger}H)-\mu_{\phi}^{2}(\Phi^{\ast}\Phi)+\lambda_{h}(H^{\dagger}H)^{2}+\lambda_{h}(\Phi^{\ast}\Phi)^{2}+\lambda_{h\phi}(H^{\dagger}H)(\Phi^{\ast}\Phi)+{\mu^{\prime}_{\phi}}^{2}(\Phi^{2}+\Phi^{\ast
2}).$ (13)

The last term in the potential explicitly breaks the $U(1)$ flavour symmetry
and gives a mass, $m_{A}^{2}={\mu^{\prime}_{\phi}}^{2}$, to the pseudo-scalar
$A$, that would otherwise be a massless Goldstone boson.

The real part of the flavon mixes with the Higgs boson, as both the SM Higgs
and the flavon acquire non-zero VEVs,

$\left(\begin{array}[]{c}h\\\
\phi\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{c}H_{1}\\\
H_{2}\end{array}\right).$ (14)

We identify $H_{1}$ as the $125$ GeV Higgs boson and $H_{2}$ as the flavon
like scalar. We interpret $H_{2}$ as the particle with mass $146$ GeV that is
responsible for the CMS signal. To constrain the mixing angle we use the Higgs
signal strength value $\mu=1.02\pm^{0.07}_{0.06}$, obtained for $137$
$\rm{fb}^{-1}$ of $\sqrt{s}=13$ TeV LHC data CMS:2020gsy . This is compatible
with the constraint $|\sin\theta|\lesssim 0.3$.

Taking into account the mixing with the Higgs boson, the lepton couplings
become

$\mathcal{L}=\frac{1}{\sqrt{2}}\bar{e}_{L}\Big{(}\cos\theta Y_{\rm
diag}-\sin\theta\kappa\Big{)}e_{R}H_{1}+\frac{1}{\sqrt{2}}\bar{e}_{L}\Big{(}\sin\theta
Y_{\rm
diag}+\cos\theta\kappa\Big{)}e_{R}H_{2}+\frac{i}{\sqrt{2}}\bar{e}_{L}\kappa
e_{R}A+\text{h.c.}.$ (15)

The decay rate of $H_{2}$ to $e\mu$-final state at tree-level is

$\displaystyle\Gamma(H_{2}\to
e\mu)=\frac{m_{H_{2}}}{16\pi}\cos^{2}\theta\left(|\kappa_{e\mu}|^{2}+|\kappa_{\mu
e}|^{2}\right).$ (16)

The mixing with flavon introduces flavor violating couplings to the $125$ GeV
Higgs boson and also changes the decay rates to flavor conserving final
states,

$\displaystyle\Gamma(H_{1}\to
e_{i}e_{i})=\frac{m_{H_{1}}}{16\pi}\left|\cos\theta Y_{\rm
diag}^{i}-\sin\theta\kappa_{ii}\right|^{2},$ (17)
$\displaystyle\Gamma(H_{1}\to
e_{i}e_{j})=\frac{m_{H_{1}}}{16\pi}\sin^{2}\theta\left(|\kappa_{ij}|^{2}+|\kappa_{ji}|^{2}\right).$
(18)

These will place significant constraints on the Higgs-flavon mixing angle and
on the flavon VEV. We find that for the numerical benchmark we use the
measurements of $\rm{BR}(h\to\mu\mu)$ CMS:2020xwi  and $\rm{BR}(h\to\tau\tau)$
CMS:2017zyp  are more constraining than the searches for the LFV decays $h\to
e\mu,e\tau,\mu\tau$ CMS:2021rsq ; CMS:2023pqk . The excluded parameter space
in $(v_{\phi},\sin\theta)$ plane is presented in Fig. 1. The $h\to\mu\mu$
measurement imposes the more stringent bound compared to the one of
$h\to\tau\tau$. This is due to the flavon coupling dependence on the flavour
charges: the muon coupling is enhanced by larger flavour charges compared to
the one of tau lepton.

Figure 1: _Left panel:_ The parameter space excluded by searches for the LFV
decays of Higgs boson. The dark gray area is excluded by the $h\to e\mu$
searches CMS:2023pqk , the medium gray area is excluded by the $h\to e\tau$
searches CMS:2021rsq  and the light gray area by the $h\to\mu\tau$ searches
CMS:2021rsq . _Right panel:_ The parameter regions excluded by the
measurements of $\rm{BR}(h\to\mu\mu)$ (blue area) and $\rm{BR}(h\to\tau\tau)$
(red area).

While the flavon $\phi$ does not couple to quarks, the flavon-like mass
eigenstate $H_{2}$ obtains coupling to quarks through mixing with the SM Higgs
boson. The couplings of $H_{2}$ to quarks and gauge bosons are proportional to
the SM couplings but scaled with $\sin\theta$. Therefore, $H_{2}$ can be
produced at the LHC analogously to the Higgs boson, mainly in gluon-gluon
fusion and vector boson fusion, which were the production channels considered
in the CMS analysis. The production cross section of $H_{2}$ is suppressed by
$\sin^{2}\theta$ compared to a hypothetical SM Higgs-like scalar of a mass
$m=146$ GeV,

$\sigma(pp\to H_{2})=\sin^{2}\theta~{}\sigma^{\rm SM}_{pp\to
h}(m_{h}=146\rm{GeV}),$ (19)

where we assume that the gluon-gluon and vector boson fusions are the only
production channels. In the narrow width approximation the production cross
section for the $e\mu$ final state through $H_{2}$ decay is given by

$\sigma(pp\to H_{2}\to e\mu)=\sigma(pp\to H_{2})\times\rm{BR}(H_{2}\to e\mu),$
(20)

where the branching ratio is

$\rm{BR}(H_{2}\to e\mu)=\frac{\Gamma(H_{2}\to e\mu)}{\Gamma_{\rm
tot}(H_{2})},$ (21)

with

$\Gamma_{\rm tot}(H_{2})=\sin^{2}\theta\big{(}\Gamma^{\rm
SM}_{h\to\rm{all}}-\sum_{i}\Gamma^{\rm SM}_{h\to
e_{i}e_{i}}\big{)}\Big{|}_{m_{h}=146\rm{GeV}}+\sum_{i}\Gamma(H_{2}\to
e_{i}e_{i})+\sum_{i\neq j}\Gamma(H_{2}\to e_{i}e_{j}).$ (22)

Here we have assumed that, $m_{H_{2}}<2m_{A}$, to prevent $H_{2}$ decays into
imaginary part of the flavon and reducing the relevant branching ratio (21).

Figure 2: _Left panel:_ Cross-section $\sigma(pp\to H_{2}\to e\mu)$ as a
function of flavon VEV at 13 TeV LHC for different Higgs-flavon mixing angles.
_Right panel:_ $H_{2}\to e\mu$ branching ratio as a function of flavon VEV. In
both panels solid lines correspond to numerical benchmark and the dashed lines
correspond to analytical estimate with maximal $e\mu$ coupling. In both panels
the red area is excluded by $\rm{BR}(h\to\tau\tau)$ and $\rm{BR}(h\to\mu\mu)$
measurements for $\sin\theta=0.1$. For $\sin\theta=0.2$ red and green areas
are excluded, and for $\sin\theta=0.3$ all colored areas are excluded.

We compute the $e\mu$ production cross section for the numerical benchmark
assuming the maximally allowed $e\mu$ coupling. The $e\mu$ production cross
section and branching ratios are presented in Fig. 2. The gluon-gluon fusion
is the dominant $H_{2}$ production channel, similar to the SM Higgs
production. The production of $H_{2}$ is increased with increasing the mixing
angle, whereas the $\rm{BR}(H_{2}\to e\mu)$ is decreased when $H_{2}$ decays
into $WW$ and $b\bar{b}$ final states become more significant.
$\rm{BR}(H_{2}\to e\mu)$ decreases as the $v_{\phi}$ grows, as the leptonic
couplings become smaller and the other decay channels start to dominate. For
relatively low values of $v_{\phi}$, below $400$ GeV, the $\tau\bar{\tau}$
final state is the dominant decay channel. This decay channel is not
suppressed by $\sin\theta$, due to direct flavon coupling to leptons, and,
therefore, the total $e\mu$ cross section grows with the mixing angle.

The combination of large mixing angle and small flavon VEV is excluded by the
LHC data, as can be seen in Fig. 1. This means that the 146 GeV particle
production cross section can maximally be $\sim 0.04$ fb, over two orders of
magnitude smaller than the cross section reported by the CMS experiment,
$5.77$ fb. Both scenarios are in addition constrained by the searches for LFV
decays of charged leptons and for $\mu\leftrightarrow e$ conversion in nuclei.
These LFV processes depend on the imaginary part of the flavon, unlike the
collider processes. This allows for a freedom in the parameters space to avoid
those LFV constraints. These constraints are discussed in more detail in
Section 3.1. The parameter space compatible with these scenarios is presented
in Fig. 4 and will be discussed shortly.

##  3 Addition of diagonal quark couplings

The scenario based on leptophilic version of the Froggatt-Nielsen mechanism is
not able to produce large enough $pp\to H_{2}\to e\mu$ cross section, as found
in the previous section. In the leptophilic case the flavon-like state,
$H_{2}$, couples to quarks and gauge bosons only due to the mixing with SM
Higgs, which suppresses its production. The $\sqrt{s}=13$ TeV LHC production
cross section of the $146$ GeV scalar with the SM couplings is $\sim 38$ pb,
of which the dominant part is coming from the gluon-gluon fusion. In the
numerical benchmark of the previous section the branching ratio for $H_{2}\to
e\mu$ is maximally $7.3\times 10^{-5}$. With this value, the production cross
section of $H_{2}$ would have to be $\sim 80$ pb, in order to reach the
reported total cross section $5.77$ fb. The flavon would need to have
significant quark couplings in order to reach the required production cross
section for $H_{2}$. We will now consider adding direct quark couplings to the
flavon, thus departing from the leptophilic case in order to study the
Froggatt-Nielsen scenario more broadly. We consider the addition of flavour
diagonal couplings of flavon to quarks,

$\mathcal{L}_{\rm
extra}=\sum_{i=u,d,s,c,b,t}\frac{c_{i}Y_{i}}{\sqrt{2}}\bar{q}_{L,i}q_{R,i}\Phi+\text{h.c.},$
(23)

where $Y_{i}$ are the SM Yukawa couplings and $c_{i}$ are free parameters.

We remain agnostic about the origin of these couplings. They might originate
from the Froggatt-Nielsen messenger sector in some fashion, but cannot
originate from the usual Froggatt-Nielsen mechanism, analogously to (1)
without flavour violating couplings accompanying them. Also, the Froggatt-
Nielsen mechanism would greatly limit the relative magnitude of these diagonal
couplings444The application of Froggatt-Nielsen mechanism to quarks with the
same flavon is out of the question due to quark flavour constraints, as
already stated above. The other possible option would be to apply the
Froggatt-Nielsen mechanims to quarks by adding a second flavon field for them.
In this case the leptonic flavon could have low VEV, as in the previous
section, but the quarky flavon would have to have its VEV at TeV scale to
avoid quark flavour constraints. The two flavons would them mix. Even if one
assumes that the two flavons mix strongly, it does not help to boost the
$H_{2}$ production, as the quarky flavon VEV would be too suppressed by its
large VEV.. Here we simply assume that the charged leptons acquire their
masses from the Froggatt-Nielsen mechanism, due to flavon $\phi$, and the
quarks obtain their masses in some other fashion, perhaps via other flavon or
flavons. One way to justify the different treatment of quarks and leptons is
the apparent disparity in the PMNS and CKM matrices, the former exhibiting
order one elements and the latter hierarchical elements. Also the lepton
sector shows more drastic differences in masses, neutrinos being at least six
orders of magnitude lighter than electron. Nevertheless, we assume that the
mass generation of charged leptons and quarks are linked to each other in some
fashion and, hence, the couplings in (23) arise.

Now that we have introduced direct couplings to quarks for the flavon, we set
the mixing angle with the SM Higgs boson to zero as it is no longer required
for $H_{2}$ production at the LHC. From now on the mass eigenstate $H_{2}$ is
the pure flavon. As there is no mixing, the properties of the Higgs boson stay
those of the SM and we are free from constraints presented in Fig. 1. One
might consider the production of $H_{2}$ directly from the light quarks,
instead of gluon-gluon fusion through the top-loop. Relatively large couplings
to up and down, $c_{u}Y_{u}\sim c_{d}Y_{d}\sim 5\times 10^{-2}$, would yield
the desired $H_{2}$ production cross section $\sim 80$ pb. The large up and
down couplings will also boost those $H_{2}$ decay channels that suppress the
$e\mu$ branching ratio, effectively killing this signature. The light quarks
cannot, therefore, be used for producing the flavon.

The $H_{2}$ production cross section needs to be increased at least by the
factor of $2$, compared to $146$ GeV SM Higgs-like scalar, in order to reach
the cross section indicated by the CMS experiment. This can be accomplished
with flavon coupling to top quark of the magnitude $\sim\sqrt{2}$. This
coupling alone would ensure sufficient production cross section for $H_{2}$.
The flavon cannot decay directly into top-antitop pairs, thus avoiding the
suppression of $H_{2}\to e\mu$ branching ratio. The large top coupling,
however, makes the $H_{2}\to gg$ decay rate significant for large values of
$v_{\phi}$. Couplings other than to the top quark are not required for
efficient production of $H_{2}$. However, they are required to avoid flavour
constraints that become more restrictive due to the additional quark-flavon
couplings.

###  3.1 Constraints from the LFV searches

$\mu$$e,\mu,\tau$$e$$H_{2},A$$\gamma$$\mu$$e$$e$$H_{2},A$$\gamma$$t$$t$$\gamma$$N$$N$$\mu$$e$$H_{2}$

Figure 3: _Left and middle panels:_ The $1$\- and $2$-loop contributions to
$\mu\to e\gamma$ in the case of additional quark couplings. _Right panel:_ The
tree-level contribution to $\mu\leftrightarrow e$ conversion in nuclei in the
case of additional quark couplings.

The constraints from LFV decays of leptons are the most stringent in the
$(e,\mu)$ sector, unlike in the case of SM Higgs for which they are the
weakest (as seen in Fig. 1). The searches for LFV decays of muon $\mu\to
e\gamma$ MEG:2016leq , $\mu\to eee$ SINDRUM:1987nra  and various tau decay
channels impose stringent constraints on the LFV couplings. In addition, the
$\mu\leftrightarrow e$ conversion in various nuclei, such as gold
SINDRUMII:2006dvw  and titanium SINDRUMII:1993gxf , impose relevant
constraints. In our analysis of LFV decays and $\mu\leftrightarrow$ conversion
we follow references Huitu:2016pwk ; Harnik:2012pb ; Kitano:2002mt . We find
that the $\mu\to e\gamma$ and $\mu\leftrightarrow e$ conversion in gold impose
the most stringent constraints on the scenarios we study.

In this scenario, $\mu\to e\gamma$ acquires significant and comparable
contributions both at 1-loop and 2-loop level. In the leptophilic model with
Higgs-flavon mixing the contributing diagrams along with the associated
formulae are presented in Ref. Huitu:2016pwk . Here we shall concentrate on
the case with additional flavon-quark couplings. The relevant diagrams in the
case of additional quark couplings are presented in the left and in the middle
panels of Fig. 3. The 2-loop Barr-Zee diagram Barr:1990vd  contains large top
coupling that compensates for the additional loop suppression. Both $1$\- and
$2$-loop diagrams are mediated by real and imaginary parts of the flavon.
Their contributions come with opposite signs, yielding cancellations in
certain regions of the parameter space, allowing to bypass the stringent
$\mu\to e\gamma$ constraint, despite of large flavour violating coupling and
relatively low mediator masses.

Figure 4: _Left panel_ : Regions of the $(v_{\phi},\,m_{A})$ parameter space
excluded by searches for $\mu\to e\gamma$ in the leptophilic model for
different values of the mixing angle. _Right panel_ : The LFV constraints in
the case of additional quark-flavon couplings for $c_{t}=1.9$ and three
different values of the charm coupling, $c_{c}=-1.9$, $0$ and $1$. The allowed
white wedge shaped area continues until the corner of the orange area, but is
too fine to be visible by eye.

The $\mu\to e\gamma$ decay rate is given by

$\Gamma(\mu\to e\gamma)=\frac{m_{\mu}^{3}}{4\pi}(|A_{L}|^{2}+|A_{R}|^{2}),$
(24)

where

$A_{L,R}=A_{L,R}^{1-\rm{loop}}+A_{L,R}^{2-\rm{loop}}.$ (25)

In the case of additional quark couplings the $1$\- and $2$-loop contributions
are

$\displaystyle A_{L}^{1-\rm{loop}}=$
$\displaystyle\sum_{i=e,\mu,\tau}\left(-\frac{ie}{64\pi}\right)\kappa_{ie}^{\ast}\int_{0}^{1}\frac{\kappa_{i\mu}y(y-x)m_{\mu}+\kappa_{\mu
i}^{\ast}(y-1)m_{i}}{y(y-x)m_{\mu}^{2}+(1-y)m_{i}^{2}+m_{H_{2}}^{2}y},$ (26)
$\displaystyle+$
$\displaystyle\sum_{i=e,\mu,\tau}\left(\frac{ie}{64\pi}\right)\kappa_{ie}^{\ast}\int_{0}^{1}\frac{-\kappa_{i\mu}y(y-x)m_{\mu}+\kappa_{\mu
i}^{\ast}(y-1)m_{i}}{y(y-x)m_{\mu}^{2}+(1-y)m_{i}^{2}+m_{A}^{2}y},$

and

$\displaystyle A_{L}^{2-\rm{loop}}=-i\frac{e\alpha
G_{F}v}{12\pi^{3}}\kappa_{\mu
e}^{\ast}c_{t}\Big{[}f(z_{t\phi})-g(z_{tA})\Big{]},$ (27)

with

$\displaystyle
f(z)=\frac{z}{2}\int_{0}^{1}\frac{1-2x(1-x)}{x(1-x)-z}\log\left[\frac{x(1-x)}{z}\right],$
(28) $\displaystyle
g(z)=\frac{z}{2}\int_{0}^{1}\frac{1}{x(1-x)-z}\log\left[\frac{x(1-x)}{z}\right].$

The arguments of these functions are defined by
$z_{t\phi}=m_{t}^{2}/m_{H_{2}}^{2}$ and $z_{tA}=m_{t}^{2}/m_{A}^{2}$. The
$A_{R}$-terms are obtained from $A_{L}$ by replacing $\kappa_{ij}$ with
$\kappa_{ji}^{\ast}$. Note the sign differences between $H_{2}$ and $A$
contributions in Eqs. (26) and (27). As the pseudo-scalar $A$ does not enter
the relevant collider processes, its mass is an independent parameter. This
fact allows for the cancellations between different amplitudes and, thus,
suppression of $\mu\to e\gamma$ decays.

The $\mu\leftrightarrow e$-conversion rate is given by

$\displaystyle\Gamma(\mu\leftrightarrow e)$
$\displaystyle=\left|\frac{iD}{2m_{\mu}}A_{L}+\widetilde{g}^{(p)}_{LS}S^{(p)}+\widetilde{g}^{(n)}_{LS}S^{(n)}+\widetilde{g}^{(p)}_{LV}V^{(p)}\right|^{2}$
(29)
$\displaystyle+\left|\frac{iD}{2m_{\mu}}A_{R}+\widetilde{g}^{(p)}_{RS}S^{(p)}+\widetilde{g}^{(n)}_{RS}S^{(n)}+\widetilde{g}^{(p)}_{RV}V^{(p)}\right|^{2}.$

The $\mu\leftrightarrow e$ coversion receives contributions generated by
1-loop and 2-loop diagrams in Fig. 3 and the coefficients $A_{L}$ and $A_{R}$
are given in Eqs. (25), (26) and (27). The $\mu\leftrightarrow e$ conversion
receives possibly dominant tree-level contribution presented by the right
diagram in Fig. 3. The tree-level process does not receive contribution from
pseudo-scalar $A$, as its contribution vanishes for coherent scattering
Kitano:2002mt . Finally, there is sub-leading vector contribution with
expression given in Ref. Harnik:2012pb .

The contributions from tree-level scalar interactions (Fig. 3) with the proton
and the neutron are:

$\displaystyle\widetilde{g}^{(p)}_{LS}=-\frac{\sqrt{2}}{m_{H_{2}}^{2}}\frac{m_{p}}{v}\kappa_{e\mu}\sum_{i}c_{i}f^{(i,p)},\quad\widetilde{g}^{(n)}_{LS}=-\frac{\sqrt{2}}{m_{H_{2}}^{2}}\frac{m_{n}}{v}\kappa_{e\mu}\sum_{i}c_{i}f^{(i,n)}.$
(30)

The summation is over all the quarks: $i=u,d,s,c,b,t$, and $m_{p}$ and $m_{n}$
are the proton and neutron masses respectively. The $\widetilde{g}^{(p)}_{RS}$
and $\widetilde{g}^{(n)}_{RS}$ are obtained from $\widetilde{g}^{(p)}_{LS}$
and $\widetilde{g}^{(n)}_{LS}$ by replacing $\kappa_{e\mu}$ with $\kappa_{\mu
e}^{\ast}$. The overlap integrals for gold555The overlap integrals for other
nuclei can be found in Kitano:2002mt . are $D=0.189$, $S^{(p)}=0.0614$,
$S^{(n)}=0.0918$ and $V^{(p)}=0.0974$, in units of $m_{\mu}^{5/2}$. The
nucleon matrix elements for light quarks are

$f^{(u,p)}=f^{(d,n)}=0.024,\quad f^{(d,p)}=f^{(u,n)}=0.033,\quad
f^{(s,p)}=f^{(s,n)}=0.25,$ (31)

and

$f^{(c,p)}=f^{(c,n)}=f^{(b,p)}=f^{(b,n)}=f^{(t,p)}=f^{(t,n)}=0.051,$ (32)

for the heavy quarks Harnik:2012pb .

In the leptophilic model the quark coupling to flavon is suppressed by
$\sin\theta$ and, therefore, the tree-level contribution to
$\mu\leftrightarrow e$-conversion is not competitive with the dipole
contribution $A_{L,R}$. In the leptophilic case the $\mu\to e\gamma$ provides
the most stringent constraint on the model parameters. The allowed parameter
space in leptophilic model is presented in the left panel of Fig. 4. The
effect of cancellation of scalar $H_{2}$ and pseudo-scalar $A$ is clearly
visible: large flavon couplings (small $v_{\phi}$) are allowed for $m_{A}\sim
m_{H_{2}}$, where the cancellation is most effective.

For the case of additional quark couplings the situation is more involved. The
process $\mu\to e\gamma$ enjoys the cancellation between real and imaginary
parts of flavon, just like in the leptophilic case, and the $\mu\to e\gamma$
bound can be avoided at small $v_{\phi}$, even for the large flavon-top
coupling. The $\mu\leftrightarrow e$ conversion, however, excludes the small
flavon VEVs in this case. This can be alleviated by adding flavon coupling to
quark(s), other than top, with an opposite sign. This can cancel the large top
contribution to $\mu\leftrightarrow e$ conversion. In the light of Eqs. (30)
and (32), the cancellation between two tree-level contributions of heavy
quarks takes place when the coupling parameters $c_{i}$ have the same absolute
value but come with opposite signs. The cancellation of top contribution is
also possible with other quarks. For the flavour constraints the choice
between $b$ and $c$ quark is not relevant, but for later collider results it
is. The bottom SM Yukawa coupling is larger than that of charm. The addition
of bottom will, therefore, dilute the $H_{2}\to e\mu$ branching ratio more. We
will, therefore, choose charm to cancel the top contribution.

We study the flavon production through gluon-gluon fusion at the LHC and set
the top coupling to be $c_{t}=1.9$. The results are shown in the right panel
of Fig. 4: for the opposite contributions of top and charm the tree-level
contributions cancel, leaving $\mu\to e\gamma$ as the dominant process. In
this case low values of $v_{\phi}\sim 100$ GeV are allowed. With top coupling
only, the $\mu\leftrightarrow e$ conversion constrains the low flavon VEVs,
exluding VEVs below $\sim 260$ GeV. For charm coupling equal to the SM Higgs
charm coupling, the flavon VEVs below $\sim 410$ GeV are excluded.

###  3.2 Collider results

Figure 5: _Left panel:_ Cross-section $\sigma(pp\to H_{2}\to e\mu)$ as a
function of the flavon VEV at the $\sqrt{s}=13$ TeV LHC for different quark
couplings as indicated in the figure. The gray shaded region is excluded by
$\mu\leftrightarrow e$ conversion for all values of $m_{A}$. The horizontal
line corresponds to $5.77$ fb. _Right panel:_ The relevant branching ratios
for the same case. The solid lines correspond to $c_{t}=-c_{c}=1.9$, dashed
line to $1.7$ and dot dashed line to $1.5$.

We consider here the flavon production through gluon-gluon fusion with large
coupling to top quark. We also have flavon coupling to charm, in order to
avoid the flavour bounds, but it offers negligible contribution to flavon
production. As there is no mixing with the Higgs boson, the relevant $H_{2}$
production channels are the dominant gluon-gluon fusion and the sub-dominant
$tt$-fusion. The resulting $e\mu$ production cross section is presented in
Fig. 5. One can see that large flavon coupling to top, $c_{t}Y_{t}\gtrsim
1.7$, can reproduce the reported CMS cross section for the $e\mu$ final state
for $v_{\phi}\gtrsim 100$ GeV.

Other possible collider constraints on the $146$ GeV flavon are avoided
because there is no mixing with the SM Higgs boson and, therefore, $H_{2}$
does not couple directly to gauge bosons. Therefore the searches for $ZZ$
final state from the second Higgs boson at low mass CMS:2018amk  do not impose
constraints. Most other second scalar searches by the CMS and ATLAS impose
limits on masses above $146$ GeV. The ATLAS searches for $WW$ and $ZZ$ final
states place a limit on masses above $300$ GeV ATLAS:2022eap , whereas the CMS
search for $WW$ final state set a limit above $200$ GeV CMS:2019bnu . Finally,
the $\gamma\gamma$ final state searches restrict masses above $200$ GeV
(ATLAS) ATLAS:2021uiz  and $500$ GeV (CMS) CMS:2018dqv . As a result, the
Froggatt-Nielsen scenario with additional quark couplings is, therefore, not
constrained by other collider searches, even though the required top coupling
is large.

##  4 Conclusions

In this work we studied the phenomenology of low-energy Froggatt-Nielsen
mechanism with the aim to address the recent CMS hint for a new $146$ GeV
resonance with LFV couplings. We first studied the purely leptophilic model
which is known to be consistent with all stringent constraints on LFV. We
found that the leptophilic Froggatt-Nielsen model cannot reach the production
cross section $\sigma=5.77$ fb which is indicated by the CMS result. The
simplest realizations of the Froggatt-Nielsen mechanism are, thus,
incompatible with the claimed CMS signal.

We modified the purely leptophilic model by adding diagonal flavon couplings
to quarks. A large flavon coupling to top quark allows for sufficient
production of flavon and the experimentally hinted cross-section can be
obtained. Additionally, couplings to lighter quarks are also required in order
to avoid the bounds arising from tree level mediated $\mu\leftrightarrow e$
conversion. The latter can be avoided when the lighter quark couplings to
flavon have different sign compared to the top coupling.

We conclude that non-trivial model building and some cancellation between the
model parameters are needed to identify the CMS hint for a new resonance with
the low-scale Froggatt-Nielsen flavon. More experimental data is needed to
clarify the origin and properties of the claimed CMS signal.

Acknowledgements. This work was supported by the Estonian Research Council
grants PRG803 and PRG1677.

Note added. The CMS hint for the 146 GeV resonance as a new scalar particle
has also been studied in Ref. Primulando:2023ugc  in the context of two Higgs
doublet models.

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