# Searching for LFV Flavon decays at hadron colliders

M. A. Arroyo-Ureña  Facultad de Ciencias Físico-Matemáticas, Benemérita
Universidad Autónoma de Puebla, Puebla, México.  A. Bolaños  Departamento de
Ciencias e Ingenierías, Universidad Iberoamericana, Campus Puebla, Puebla,
México.  J. L. Díaz-Cruz  Facultad de Ciencias Físico-Matemáticas, Benemérita
Universidad Autónoma de Puebla, Puebla, México.  G. Hernández-Tomé
Departamento de Física, CINVESTAV IPN, Apartado Postal 14-740, 07000, México
D. F., México.  G. Tavares-Velasco  Facultad de Ciencias Físico-Matemáticas,
Benemérita Universidad Autónoma de Puebla, Puebla, México.

###### Abstract

The search for Flavons with a mass of $\mathcal{O}$(1) TeV at current and
future colliders might probe low-scale flavor models. We are interested in the
simplest model that invokes the Froggatt-Nielsen (FN) mechanism with an
Abelian flavor symmetry, which includes a Higgs doublet and a FN complex
singlet. Assuming a CP conserving scalar potential, there are a $CP$-even
$H_{F}$ and a $CP$-odd $A_{F}$ Flavons with lepton flavor violating (LFV)
couplings. The former can mix with the standard-model-like Higgs boson,
thereby inducing tree-level LFV Higgs interactions that may be at the reach of
the LHC. We study the constraints on the parameter space of the model from
low-energy LFV processes, which are then used to evaluate the Flavon decay
widths and the $gg\to\phi\to\tau\mu$ ($\phi=H_{F},\,A_{F}$) production cross
section at hadron colliders. After imposing several kinematic cuts to reduce
the SM main background, we find that for $m_{H_{F}}$ about 200-350 GeV, the
decay $H_{F}\to\tau\mu$ might be at the reach of the LHC for a luminosity in
the range 1-3 ab-1, however, a luminosity of the order of 10 ab-1 would be
required to detect the $A_{F}\to\tau\mu$ decay. On the other hand a future 100
TeV $pp$ collider could probe masses as high as $\mathcal{O}$(10) TeV if it
reaches an integrated luminosity of at least 20 ab-1. Therefore, the 100 TeV
Collider could work as a Flavon factory.

##  I Introduction

After the discovery of a Higgs-like particle with a mass $m_{h}=125-126$ GeV
Aad et al. (2012); Chatrchyan et al. (2012), the search for new physics (NP)
has become the one of the next goals of the LHC. Although current measurements
of the spin, parity, and couplings of the Higgs boson seem consistent with the
standard model (SM) Gunion et al. (2000), its light mass seems troublesome,
i.e. the hierarchy problem, and calls for new physics (NP). The SM has also
other open issues, such as the flavor problem, unification, etc. Pomarol
(2012); Martin (1997), which also encourages the study of NP models.

The couplings of the Higgs particle to a pair of massive gauge bosons or
fermions have strengths proportional to the masses of such particles. However,
the LHC has tested only a few of such Higgs couplings, namely the ones to the
gauge bosons and the heaviest fermions. Along these lines, many studies have
been devoted to analyze the pattern of Higgs couplings derived from LHC data,
for instance Espinosa et al. (2012); Giardino et al. (2014). However, non-
standard Higgs couplings, including the flavor violating (FV) ones, are
predicted in many models of physics beyond the SM Branco et al. (2012); Diaz-
Cruz et al. (2004, 2005); Diaz-Cruz (2003). In particular, the observation of
neutrino oscillations, which is associated with massive neutrinos, motivates
the occurrence of lepton flavor violation (LFV) in nature Ma (2009). Within
the SM, LFV processes vanish at any order of perturbation theory, which
motivates the study of SM extensions that predict sizeable LFV effects that
could be at the reach of detection. Apart from decays such as $l_{i}\to
l_{j}\gamma$ and $l_{i}\to l_{j}\bar{l}_{k}l_{k}$, particularly interesting is
the decay $h\to\tau\mu$, which was studied first in Refs. Pilaftsis (1992);
Diaz-Cruz and Toscano (2000), with subsequent analyses on the detectability of
the signal appearing soon after Han and Marfatia (2001); Assamagan et al.
(2003). This motivated a plethora of calculations in the framework of several
SM extensions, such as theories with massive neutrinos, supersymmetric
theories, etc. Arganda et al. (2005); Diaz-Cruz (2003); Brignole and Rossi
(2004); Diaz-Cruz et al. (2009); Chamorro-Solano et al. (2017, 2016); Lami and
Roig (2016). After the Higgs boson discovery, the decay $h\to\tau\mu$ offers a
great opportunity to search for NP at the LHC. Although a slight excess of the
$h\to\tau\mu$ branching ratio was reported at the LHC run I, with a
significance of 2.4 standard deviations Khachatryan et al. (2015a), a
subsequent study Collaboration (2016) ruled out such an excess and put the
limit $BR(h\to\bar{\mu}\tau)<1.2\times 10^{-2}$ with 95% C.L. In the model we
are interested in, LFV effects are induced at the tree-level in the scalar
sector, so it is thus worth assessing their phenomenology.

Another open issue in the SM is the flavor problem Isidori et al. (2010),
which has long been the focus of interest, with several proposals meant to
address it, such as textures, GUT-inspired relations, symmetries, radiative
generation, etc. In particular, a flavor symmetry approach can be supplemented
with the Froggatt-Nielsen (FN) mechanism, which assumes that above some scale
$M_{F}$, there is a symmetry (perhaps of Abelian type $U(1)_{F}$) that forbids
the appearance of Yukawa couplings; SM fermions are charged under this
symmetry. However, the Yukawa matrices can arise through non-renormalizable
operators. The Higgs spectrum of these models could include a light Flavon
$H_{F}$, which could mix with the Higgs bosons when the flavor scale is of the
order of the TeVs. Quite recently, the phenomenology of Higgs-Flavons at
particle colliders has been the focus of attention Bolaños et al. (2016);
Bauer et al. (2016a); Huitu et al. (2016); Berger et al. (2014); Diaz-Cruz and
Saldaña-Salazar (2016).

Although the states we are interested in arise from the mixing of Higgs bosons
and Flavons, we will still call them Flavons for short. Depending on the
particular model, there could be several potentially detectable Flavon decays,
which would be indistinguishable from the decays of a heavy Higgs boson,
therefore, to search for a distinctive signature, we will focus on the one
arising from the LFV decay $H_{F}\to\tau\mu$, with
$\tau\mu=\tau^{-}\mu^{+}+\tau^{+}\mu^{-}$. The minimal model that introduces
the FN mechanism with an Abelian flavor symmetry includes a scalar sector
consisting of a Higgs doublet and a FN complex singlet. From now on we will
refer to this model as the FN extension of the Standard Model (FNSM). Such a
model predicts a $CP$-even Flavon $H_{F}$ and a $CP$-odd one $A_{F}$. Also,
the couplings of the light SM-like Higgs boson would deviate from the SM ones,
such that two possible scenarios are possible: firstly, the mixing of the real
part of the doublet with the real component of the FN singlet could induce
sizable LFV Higgs couplings of the light physical Higgs boson, which might
affect the light Higgs phenomenology; secondly, the $CP$-even Flavon could be
very heavy, so its mixing with the light Higgs boson would be negligible and
unconstrained by LHC Higgs data, though in such a case the $CP$-odd state
$A_{F}$ could be the lighter one, thereby giving rise to a potentially
detectable LFV signal.

In this paper we are interested in studying the possible detection of both
$CP$-even and $CP$-odd Flavons at the LHC and a future 100 TeV $pp$ collider
via their LFV decays. It has been pointed out that a 100 TeV $pp$ collider
would allow for a detailed study of several topics of interest in particle
physics, such as Higgs physics and the electroweak symmetry breaking mechanism
Contino et al. (2017). It will also be useful to search for possible signals
of dark matter, SUSY theories, and other extension models Golling et al.
(2017).

We will start our analysis by considering the constraints on the parameter
space of the FNSM obtained from Higgs data at the LHC, low energy LFV, and the
muon magnetic dipole moment. A set of benchmarks will then be used to estimate
the Flavon decay modes, focusing on the LFV ones, as well as their production
cross sections by gluon fusion at the LHC and a future 100 TeV $pp$ collider.
We will then explore the possibility that the $CP$-even and $CP$-odd Flavons
could be detected via the $\tau\mu$ decay channel, for which we will make a
Monte Carlo analysis of the signal and the SM main background.

The organization of our work is as follows. In Sec. II we describe the
realization of the FN mechanism within the simplest model, namely that with
one Higgs doublet and one FN singlet. In particular, we present the Higgs
potential and Yukawa Lagrangian, from which the Flavon couplings can be
extracted. Section III is devoted to the constraints on the parameters space
of the model and the benchmarks of parameter values we will be using in our
analysis, whereas Sec. IV is focused on the analysis of the decay modes of
both a $CP$-even and a $CP$-odd Flavon as well as their production cross
sections via gluon fusion at the LHC and a future 100 TeV $pp$ collider. We
also present the Monte Carlo analysis of the $\mu\tau$ signal and its main
background. The conclusions and outlook are presented in Sec. V.

##  II The scalar sector of the minimal FNSM

The scalar sector of the FNSM includes the usual SM Higgs doublet

$\Phi=\left(\begin{array}[]{c}G^{+}\\\
\frac{1}{\sqrt{2}}\left(v+\phi^{0}+iG_{Z}\right)\\\ \end{array}\right),$ (1)

and a complex singlet

$S_{F}=\frac{1}{\sqrt{2}}(u+s+ip),$ (2)

where $v$ denotes the SM vacuum expectation value (VEV) and $u$ that of the FN
singlet, whereas $G^{+}$ and $G^{0}$ will be identified with the pseudo-
goldstone bosons that become the longitudinal modes of the $W$ and $Z$ gauge
bosons.

###  II.1 The Higgs potential

We turn now to discuss the minimal $CP$-conserving Higgs potential with a
softly-broken $U(1)$ global symmetry, which is given as follows

$\displaystyle V$
$\displaystyle=-\frac{1}{2}{m_{1}^{2}}\Phi^{\dagger}\Phi-\frac{m_{s_{1}}^{2}}{2}S^{*}_{F}S_{F}-\frac{m_{s_{2}}^{2}}{2}(S^{*2}_{F}+S^{2}_{F})$
$\displaystyle+\frac{1}{2}\lambda_{1}\left(\Phi^{\dagger}\Phi\right)^{2}+\lambda_{s}(S^{*}_{F}S_{F})^{2}+\lambda_{11}(\Phi^{\dagger}\Phi)(S^{*}_{F}S_{F}).$
(3)

We are therefore left with the following $U(1)$-symmetric terms
($m_{1}^{2},m_{s_{1}}^{2},\lambda_{1},\lambda_{s},\lambda_{11}$), and the
$U(1)$-soft-breaking term $m_{s_{2}}^{2}$. The latter is required to avoid a
massless Goldstone boson when $\langle S_{F}\rangle\neq 0$. An extensive
analysis of this potential was presented in reference Bonilla et al. (2016),
where the parameter space that allows a viable model was identified.

After imposing the minimization conditions on the potential, the following
relations are obtained:

$\displaystyle m_{1}^{2}$ $\displaystyle=$ $\displaystyle
v^{2}\lambda_{1}+u^{2}\lambda_{11},$ (4) $\displaystyle m_{s_{1}}^{2}$
$\displaystyle=$
$\displaystyle-2m_{s_{2}}^{2}+2u^{2}\lambda_{s}+v^{2}\lambda_{11}.$ (5)

###  II.2 The Scalar Mass Matrix

In a $CP$-invariant potential, the $CP$-even (real) and $CP$-odd (imaginary)
components of the mass matrix do not mix. In this case the mass matrix for the
real components in the basis $(\phi^{0},s)$ is given by:

$M^{2}_{S}=\left(\begin{array}[]{cc}\lambda_{1}v^{2}&\lambda_{11}uv\\\
\lambda_{11}uv&2\lambda_{s}u^{2}\end{array}\right),$ (6)

whereas the mass matrix for the imaginary components, in the basis
$(G_{Z},p)$, reads

$M^{2}_{P}=\left(\begin{array}[]{cc}0&0\\\
0&2m^{2}_{s_{2}}\end{array}\right).$ (7)

We notice that the mass scale for the $CP$-odd state arising from the FN
singlet $A_{F}=p$ is different from the VEV $u$, which is the $U(1)$-breaking
scale, and therefore it could be much lighter. As for the mixing of the real
components of the doublet $\Phi$ and the singlet $s$, the mass eigenstates are
obtained through the standard $2\times 2$ rotation:

$\displaystyle\phi^{0}$ $\displaystyle=$
$\displaystyle\,\,\,\,\cos\,\alpha\,h+\sin\,\alpha\,H_{F},$ (8) $\displaystyle
s$ $\displaystyle=$ $\displaystyle-\sin\,\alpha\,h+\cos\,\alpha\,H_{F}.$ (9)

In what follows we will identify the mass eigenstate $h$ as the SM-like Higgs
boson with $m_{h}=125$ GeV, while the mass eigenstates $H_{F}$ and $A_{F}$
will be assumed to be heavier. Although they arise from Flavon-Higgs mixing,
in the present work we will still refer to $H_{F}$ and $A_{F}$ as Flavons for
short. The properties of the $CP$-even Flavon will depend on the size of its
mixing with the lightest state. On the other hand, the $CP$-odd state, which
does not couple to gauge bosons, it does couple to the SM fermions, including
both diagonal and non-diagonal interactions.

Our analysis of Flavon decays requires the knowledge of cubic interactions,
such as the trilinear vertex $H_{F}hh$, which is given in the minimal model
by:

$g_{H_{F}hh}=\frac{1}{2}[\lambda_{11}(u\cos^{3}\alpha+v\sin^{3}\alpha)+2u\sin^{2}\alpha\cos\alpha(3\lambda_{s}-\lambda_{11})+v\sin\alpha\cos^{2}\alpha(3\lambda_{1}-2\lambda_{11})].$
(10)

###  II.3 Yukawa sector and LFV interactions

The FN Lagrangian of the model includes the terms that become the Yukawa
couplings once the $U(1)_{F}$ flavor symmetry is spontaneously broken. It is
given by:

$-{\cal{L}}_{Y}=\rho^{d}_{ij}\left(\frac{S_{F}}{\Lambda_{F}}\right)^{q^{d}_{ij}}\bar{Q}_{i}d_{j}\tilde{\Phi}+\rho^{u}_{ij}\left(\frac{S_{F}}{\Lambda_{F}}\right)^{q^{u}_{ij}}\bar{Q}_{i}u_{j}\Phi+\rho^{l}_{ij}\left(\frac{S_{F}}{\Lambda_{F}}\right)^{q^{l}_{ij}}\bar{L}_{i}l_{j}\Phi+{\rm
H.c.},$ (11)

where $q^{f}_{ij}$ $(f=u,d,l)$ denote the Abelian charges that reproduce the
observed fermion masses, for each fermion type. The Flavon field $S_{F}$ is
assumed to have flavor charge equal to -1, such that ${\cal{L}}_{Y}$ is
$U(1)_{F}$-invariant. Then, the Yukawa couplings arise after the spontaneous
breaking of the flavor symmetry, i.e.
$\lambda_{x}=(\frac{<S_{F}>}{\Lambda_{F}})^{n_{x}}$, where $<S_{F}>$ denotes
the Flavon VEV, whereas $\Lambda_{F}$ denotes a heavy mass scale, which
represents the mass of heavy fields that transmit such symmetry breaking to
the quarks and leptons. For a detailed discussion of the viable structures for
the Yukawa matrices for the FN multi-Higgs model see Ref. Bauer et al.
(2016a). Here we shall discuss the generic features that can be identified by
studying specific Abelian charges for the charged leptons. With this purpose
we consider the charge assignment used by one of us in Ref. Diaz-Cruz and
Saldaña-Salazar (2016), where the Yukawa matrix $Y^{l}$ is of the form:

$Y^{l}\sim\begin{pmatrix}\lambda^{6}&\lambda^{6}&\lambda^{6}\\\
\lambda^{6}&\lambda^{4}&\lambda^{4}\\\
\lambda^{6}&\lambda^{4}&\lambda^{2}\end{pmatrix},$ (12)

which can be justified with the following Abelian charges

  * •

$\alpha_{i}$= $U(1)_{F}$ charge of lepton doublet $L_{i}$,

  * •

$\beta_{j}$= $U(1)_{F}$ charge of lepton singlet $l_{j}$.

Then, $Y_{ij}^{l}\sim\lambda^{|\alpha_{i}-\beta_{j}|}$, which means that
choosing $Y_{23}^{l}\sim\lambda^{4}$ and $Y_{33}^{l}\sim\lambda^{2}$ implies
$|\alpha_{2}-\beta_{3}|=4$ and $|\alpha_{3}-\beta_{3}|=2$. Other examples of
Abelian charges are presented in Ref. Bauer et al. (2016a); Baldes et al.
(2016); Grossman and Nir (1995). We then focus on the 2-3 submatrix, which
dictates the $\phi\tau\mu$ interaction. The corresponding squared mass matrix
can be written as:

$M_{l}^{2}=v^{2}|Y_{33}^{l}|^{2}\begin{pmatrix}\lambda^{2}&\lambda^{2}\\\
\lambda^{2}&1\end{pmatrix},$ (13)

with $Y_{33}^{l}\sim\lambda^{2}=(m_{\tau}-m_{\mu})/v$. This mass matrix can be
diagonalized by a $2\times 2$ rotation with mixing angle
$\theta\sim(m_{\mu}/m_{\tau})$.

In the unitary gauge we set $G^{\pm}\rightarrow 0$, $G_{Z}\rightarrow 0$, thus
we can write the doublet $\Phi$ as follows:

$\Phi=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ v+\phi^{0}\end{pmatrix},$ (14)

whereas the powers of the Flavon field can be expanded as

$\displaystyle
S^{q^{f}_{ij}}=\left(\frac{u+s+ip}{\sqrt{2}}\right)^{q^{f}_{ij}}\simeq\left(\frac{u}{\sqrt{2}}\right)^{q^{f}_{ij}}\left[1+q^{f}_{ij}\left(\frac{s+ip}{u}\right)\right].$
(15)

Then, after substituting the Higgs boson and Flavon mass eigenstates, one gets
finally the following interaction Lagrangian for the Higgs-fermion couplings

$\displaystyle-{\cal{L}}_{Y}$ $\displaystyle=$
$\displaystyle\frac{1}{v}[\bar{U}\tilde{M}_{u}U+\bar{D}\tilde{M}_{d}D+\bar{L}\tilde{M}_{l}L](c_{\alpha}h+s_{\alpha}H_{F})$
(16)
$\displaystyle+\frac{v}{\sqrt{2}u}[\bar{U}_{i}\tilde{Z}^{u}_{ij}U_{j}+\bar{D}_{i}\tilde{Z}^{d}_{ij}D_{j}+\bar{L}_{i}\tilde{Z}^{l}_{ij}L_{j}](-s_{\alpha}h+c_{\alpha}H_{F}+iA_{F})+{\rm
H.c.},$

where we use the usual short-hand notation $s_{\alpha}\equiv\sin\alpha$ and
$c_{\alpha}\equiv\cos\alpha$. Here, $\tilde{M}^{f}$ is the diagonal mass
matrix, whereas the information about the size of FV Higgs-Flavon couplings is
contained in the $\tilde{Z}^{f}=U_{L}^{f}{Z}^{f}U_{L}^{f\dagger}$ matrices,
with $Z^{f}_{ij}$ given in the flavor basis as

$Z^{f}_{ij}=\rho^{f}_{ij}\left(\frac{u}{\sqrt{2}\Lambda_{F}}\right)^{q^{f}_{ij}}q^{f}_{ij},$
(17)

which remains non-diagonal once the fermion mass matrices are diagonalized,
thereby giving rise to FV scalar couplings. For the Yukawa matrix we are
considering, the $Z$ matrix is given as follows (for the 2-3 subsystem):

$Z^{l}=\begin{pmatrix}4Y_{22}^{l}&4Y_{23}^{l}\\\
4Y_{32}^{l}&2Y_{33}^{l}\end{pmatrix}.$ (18)

Thus we can use the approximate diagonalization of the mass matrix to express
$Y_{ij}^{l}$ and $Z^{l}$ in terms of the mass ratios. To leading order we have

$\widetilde{Z}^{l}\backsimeq\begin{pmatrix}4\frac{m_{\mu}}{v}\frac{\sqrt{m_{\mu}}}{m_{\tau}}&6\frac{m_{\mu}}{v}\\\
6\frac{m_{\mu}}{v}&2\frac{m_{\tau}}{v}\end{pmatrix},$ (19)

and similary for up-type quarks, whose Higgs couplings play a fundamental role
for collider phenomenology. With the assignment of Abelian charges chosen in
Diaz-Cruz and Saldaña-Salazar (2016), the $Z^{u}$ matrix for the 2-family case
is given by:

$Z^{u}=\begin{pmatrix}2Y^{u}_{22}&2Y^{u}_{23}\\\ 2Y^{u}_{32}&0\end{pmatrix},$
(20)

with $Y^{u}_{22}\sim Y^{u}_{23}\sim\lambda^{2}$. However, in the mass-
eigenstates basis, the rotated matrix takes the following form to leading
order:

$\widetilde{Z}^{u}\sim\begin{pmatrix}2Y^{u}_{22}&2Y^{u}_{23}\\\
2Y^{u}_{32}&4s_{u}Y^{u}_{23}\end{pmatrix},$ (21)

where $s_{u}=\sin\theta_{u}$, with $\theta_{u}$ the mixing angle. It turns out
that the $\phi t\bar{t}$ coupling vanishes in the weak basis, but it is non-
vanishing as long as there is mixing between the top quark and the mass
eigenstates basis. The diagonal and non-diagonal interactions of the $h$,
$H_{F}$, and $A_{F}$ scalar bosons with massive fermions are given by:

$\displaystyle g_{hf_{i}f_{j}}$
$\displaystyle=\frac{c_{\alpha}}{v}\tilde{M}^{f}_{ij}-s_{\alpha}r_{s}{\tilde{Z}^{f}}_{ij},$
$\displaystyle g_{H_{F}f_{i}f_{j}}$
$\displaystyle=\frac{s_{\alpha}}{v}\tilde{M}^{f}_{ij}+c_{\alpha}r_{s}{\tilde{Z}^{f}}_{ij},$
$\displaystyle g_{A_{F}f_{i}f_{j}}$
$\displaystyle=ir_{s}{\tilde{Z}^{f}}_{ij},$ (22)

where the Feynman rule for the $A_{F}f_{i}f_{j}$ vertex includes a
$\gamma^{5}$ Dirac matrix and $r_{s}=v/(\sqrt{2}u)$.

Besides the Yukawa couplings, we also need to specify the scalar-to-gauge-
boson couplings. They can be readily extracted from the kinetic terms of the
Higgs doublet and the singlet, which transforms trivially under the SM gauge
group. Thus after substituting Eq. (8) in the kinetic term, we obtain that the
$h$ and $H_{F}$ couplings to gauge boson pairs are SM-like, with the coupling
constants given by $g_{h_{i}VV}=\chi_{V}^{h_{i}}g_{h_{SM}VV}$ $(V=W,Z)$, for
$h_{i}=h,\,H_{F}$, with
$\chi^{h}_{V}\;(\chi^{H_{F}}_{V})=\cos\alpha\;(\sin\alpha)$. Thus the coupling
constants are

$\displaystyle g_{hZZ}$ $\displaystyle=\frac{gm_{Z}}{c_{W}}\cos\alpha$ (23)
$\displaystyle g_{hWW}$ $\displaystyle=gm_{W}\cos\alpha,$ (24) $\displaystyle
g_{H_{F}ZZ}$ $\displaystyle=\frac{gm_{Z}}{c_{W}}\sin\alpha,$ (25)
$\displaystyle g_{H_{F}WW}$ $\displaystyle=gm_{W}\sin\alpha.$ (26)

We are interested in the possible detection of $CP$-even and $CP$-odd Flavons
with masses of the TeV order at both the LHC and a future $pp$ collider with a
center-of-mass energy of $100$ TeV. Depending on the particular model, there
could be several potentially detectable decays of such Flavons, but some of
them would also arise from heavy Higgs bosons, for instance within multi-Higgs
doublet models. Thus, in order to search for a distinctive Flavon signature,
we shall focus on the one arising from the LFV decay $\phi\to\tau\mu$
($\phi=H_{F},\,A_{F}$). In order to determine the detectability of this decay,
we will present a Monte Carlo analysis of the signal and the most relevant SM
backgrounds.

##  III Constraints on the parameter space of the FNSM

In order to evaluate the Flavon decays and production modes at a hadron
collider we need to analyze the most up-to-date constraints on the model
parameters. For the mixing angle $\alpha$ we can use the data obtained by the
LHC collaborations on the Higgs boson properties, whereas the LFV couplings
can be constrained via the experimental data on the muon anomalous magnetic
dipole moment (AMDM) $a_{\mu}$, the LFV decays of the tau lepton
$\tau\to\bar{l_{i}}\bar{l_{j}}l_{j}$ and $\tau\to l_{i}\gamma$, as well as the
experimental constraint on the $h\to\mu\tau$ decay. All the necessary formulas
to perform our analysis below are presented in Appendix A.

###  III.1 Mixing angle $\alpha$

We shall use the universal Higgs fit of Ref. Giardino et al. (2014), which
presents constraints on the parameters $\epsilon_{X}$, defined as (small)
deviations of the Higgs couplings from the SM values, i.e.
$\chi^{h}_{X}=g_{hXX}/g_{hXX}^{SM}=1+\epsilon_{X}$. For the $W$ and $Z$ gauge
bosons, the corresponding constraints are $\epsilon_{W}=-0.15\pm 0.14$ and
$\epsilon_{Z}=-0.01\pm 0.13$. Regarding the fermion couplings, we notice that
this universal Higgs fit is valid for the $CP$-conserving case, which we are
considering here; therefore we can apply them to constrain the properties of
the $CP$-even SM-like Higgs boson. Furthermore, the constraints derived from
the gauge interactions provide the strongest constraints on the mixing angle
$\alpha$. Since the lightest scalar boson $h$ couples with the SM gauge bosons
with a strength that deviates from the SM couplings by the factor
$c_{\alpha}$, to satisfy the bound on $\epsilon_{Z}$ we need to have
$0.86<c_{\alpha}<1$. We will use a conservative approach and use the benchmark
$c_{\alpha}=0.95$ in our analysis below.

###  III.2 Diagonal $\tilde{Z}_{22}^{l}$ and $\tilde{Z}_{33}^{l}$ matrix
elements

In this work we will use the 2-family approximation, neglecting FV with the
leptons of the first fermion family. We will also assume that there is no
$CP$-violating phase, which means that we have three free parameters:
$\tilde{Z}_{22}$, $\tilde{Z}_{23}$, and $\tilde{Z}_{33}$. To constrain the
diagonal $\tilde{Z}_{33}$ matrix element, we refer again to the universal
Higgs fit of Ref. Giardino et al. (2014) and consider the constraint on the
deviation of the SM $h\bar{\tau}\tau$ coupling, namely, $\epsilon_{\tau}=0\pm
0.18$. We thus show in Fig. 1 the allowed area on the $u-\tilde{Z}_{33}^{l}$
plane for two values of $c_{\alpha}$. We observe that in order to agree with
the universal Higgs fit constraint when $c_{\alpha}=0.9$, $\tilde{Z}_{33}^{l}$
must be of the order of $10^{-3}$ for $u=0.5$ TeV and $10^{-2}$ for $u=2$ TeV,
but when $c_{\alpha}=0.95$ we must have values of the order $10^{-2}$ in the
complete $u$ interval. Since we are considering $c_{\alpha}=0.95$, we will use
$\tilde{Z}_{33}^{l}=10^{-2}$ as benchmark.

Figure 1: Allowed values of $\tilde{Z}_{33}^{l}$ (upper plots) and
$\tilde{Z}_{22}^{l}$ (lower plots) as functions of $u$ and for two values of
$c_{\alpha}$. To constraint $\tilde{Z}_{33}^{l}$ we use the universal Higgs
fit on $\epsilon_{\tau}$ Giardino et al. (2014), whereas for
$\tilde{Z}_{22}^{l}$ we use the ATLAS and CMS bound
$BR(h\to\bar{\mu}\mu)\lesssim 1.5\times 10^{-3}$ with 95% C. L. Khachatryan et
al. (2015b); Aad et al. (2014, 2016).

As far as $\tilde{Z}_{22}^{l}$ is concerned, the ATLAS and CMS Collaborations
searched for the Higgs boson decay to $h\to\bar{\mu}\mu$ using the LHC data
collected at $\sqrt{s}=7$ TeV and $\sqrt{s}=8$ TeV Khachatryan et al. (2015b);
Aad et al. (2014, 2016). The upper bound $BR(h\to\bar{\mu}\mu)\lesssim
1.5\times 10^{-3}$ with 95% C. L. was found for $m_{h}=125.6$ GeV, which is
about one order of magnitude above the SM value $BR(h\to\bar{\mu}\mu)\simeq
2.04\times 10^{-4}$. This bound would allow $\tilde{Z}_{22}^{l}$ values as
large as $10^{-2}$ as shown in Fig. 1. We will take instead a conservative
approach and assume the hierarchy $\tilde{Z}_{22}^{l}<\tilde{Z}_{33}^{l}$, so
we fix $\tilde{Z}_{22}^{l}=10^{-4}$, otherwise the SM $h\bar{\mu}\mu$ coupling
would be swamped by the new corrections of the FNSM.

###  III.3 Non-diagonal $\tilde{Z}_{23}^{l}$ matrix element

We will consider the current experimental bounds on the muon AMDM $a_{\mu}$
Patrignani et al. (2016), the tau decay $\tau\to\mu\gamma$ Patrignani et al.
(2016), and the Higgs boson decay $h\to\tau\mu$ Aad et al. (2015); Khachatryan
et al. (2015a) to constrain the $Z_{ij}^{l}$ matrix elements Arroyo-Ureña et
al. (2016). Notice that the current bound on the $\tau\to 3\mu$ decay width
gives very weak constraints, so we will omit such a process in our analysis.
Two scenarios arise when dealing with constraints on LFV couplings:

  * •

Scenario I: the FNSM is assumed to be responsible for the current discrepancy
between the theoretical and experimental values of $a_{\mu}$, which requires a
positive contribution from new physics. Along this line, the one-loop
contribution from a $CP$-even scalar boson is positive, whereas that of the
$CP$-odd scalar boson is negative. Therefore a suitable scenario would be that
with a relatively light $CP$-even Flavon and a heavy $CP$-odd Flavon, which
would suppress the negative contribution. When $m_{H_{F}}\simeq m_{A_{F}}$,
the Flavon contributions would largely cancel one another out for
$c_{\alpha}\simeq 1$ and the remaining (positive) contribution would arise
from the corrections to the SM Higgs boson couplings, unless there were large
extra positive contributions arising from the $CP$ odd Flavon at the two-loop
level: such contributions could arise as long as $A_{F}$ is very light and
there was an enhanced one-loop-induced $A_{F}\gamma\gamma$ coupling, as
discussed in Marciano et al. (2016). We do not expect such an enhancement in
the FNSM, so large values of the mixing matrix element $Z_{23}^{l}$ would be
required to solve the $a_{\mu}$ discrepancy.

  * •

Scenario II: the FNSM Flavons give a negligible contribution to $a_{\mu}$ and
fail to explain the discrepancy, though the model still can remain viable. It
could happen for instance that extra positive contributions to $a_{\mu}$ would
arise from the ultraviolet completion of the FNSM, thereby solving the puzzle.
Also, it is still possible that more precise determinations of the SM hadronic
contribution and the experimental measurement would settle the discrepancy in
the future without requiring any NP effects. In any case, by requiring that
the Flavon contribution to $a_{\mu}$ is negligible, it is enough to satisfy
the LHC constrain from $h\to\tau\mu$ in order to have a viable parameter space
in this scenario.

We will now assess the viability of both scenarios. To avoid large corrections
to the diagonal lepton scalar couplings, we take $\tilde{Z}_{22}\simeq
10^{-3}$ and $\tilde{Z}_{33}\simeq 10^{-2}$. We then show in Fig. 2 the area
allowed in the $u-\tilde{Z}_{23}$ plane by the experimental constraints on the
$\tau\to\mu\gamma$ and $h\to\tau\mu$ decays for the indicated values of the
Flavon masses and the mixing angle $c_{\alpha}$. The blue strip is the region
where $u$ and $\tilde{Z}_{23}$ must lie in order to alleviate the $a_{\mu}$
discrepancy. We observe that the most stringent limits on $\tilde{Z}_{23}$ are
obtained from the experimental bound on the $h\to\mu\tau$ decay, which
requires $\tilde{Z}_{23}$ to be of the order of $10^{-2}$ ($10^{-1}$) for
$u=0.5$ ($1$) TeV. However, in order to solve the $a_{\mu}$ discrepancy,
values of $\tilde{Z}_{23}$ as large as 1 would be required for $u=1$ TeV,
which is due to the fact that we need large (positive) contributions arising
from the $CP$-even Flavon and the new corrections to the SM Higgs boson
couplings. We note that the blue strips shifts downwards as the mass of the
$CP$-odd Flavon increases as in this case the (negative) contribution is
smaller. Therefore, scenario I discussed above is not favored by the current
experimental data and we assume that the scenario II is fulfilled, with the
Flavon contribution to $a_{\mu}$ being rather small and not responsible for
the discrepancy between experimental and theoretical values. We thus take as
benchmark $Z_{23}\simeq 0.1$, which is allowed by the experimental bound on
the $\tau\to\mu\gamma$ and $h\to\mu\tau$ decays for $u\simeq 1.5$ TeV.

Figure 2: Allowed area on the $u-\tilde{Z}_{23}^{l}$ plane by the current
experimental bounds on the $\tau\to\mu\gamma$ and $h\to\tau\mu$ decays for the
indicated values of the model parameters. The blue strip is the region where
$u$ and $\tilde{Z}^{l}_{23}$ must lie to satisfy the $a_{\mu}$ discrepancy.

###  III.4 $\tilde{Z}^{u}_{33}$ and $\tilde{Z}^{u}_{23}$ matrix elements

According to the universal Higgs fit Giardino et al. (2014), the allowed value
for the deviation of the SM $h\bar{t}t$ coupling is $\epsilon_{t}=-0.21\pm
0.22$. We will use again the two-family approximation and take the values
$\tilde{Z}^{u}_{33}=0.01$ and $\tilde{Z}^{u}_{23}=0.1$.

###  III.5 Summary of Benchmarks for the model parameters

In summary, in our study below we will use the following benchmarks:

  1. 1.

Mixing angle $\alpha$: As discussed above, to satisfy the fit on the 125 GeV
Higgs couplings measured at the LHC, the following constraint must be obeyed
$0.86<c_{\alpha}<1$. We will thus use the benchmark $c_{\alpha}=0.95$.

  2. 2.

FN singlet VEV $u$: It appears in the Flavon couplings and the LFV SM Higgs
coupling. We will consider the value $1.5$ TeV.

  3. 3.

$\tilde{Z}^{f}_{ij}$ matrix: It determines the strength of the LFV scalar
couplings. We will use the 2-family approximation and take
$\tilde{Z}^{l}_{22}=10^{-4}$, $\tilde{Z}^{l}_{33}=10^{-2}$, and
$\tilde{Z}^{l}_{23}=10^{-1}$, which are consistent with the constraint on the
LFV decay $h\to\mu\tau$.

  4. 4.

$H_{F}hh$ interaction: This vertex depends on a combination of parameters that
appear in the Higgs potential. However, these parameters could be traded by an
effective coupling $\lambda_{H_{F}hh}$, which can take values of the order of
$O(1)$. We will thus fix $\lambda_{H_{F}hh}\simeq 0.1u$.

A summary of the benchmarks we are going to consider in our analysis is
presented in Table 1.

Table 1: Benchmarks used for the analysis of the production and detection of the Flavons $H_{F}$ and $A_{F}$ at the LHC and a future 100 TeV $pp$ collider in the context of the FNSM. Parameter | Benchmark
---|---
$c_{\alpha}$ | 0.95
$u$ |  $1.5$ TeV
$\tilde{Z}^{l}_{22}$ | $10^{-4}$
$\tilde{Z}^{l}_{33}$ | $10^{-2}$
$\tilde{Z}^{l}_{23}$ | $10^{-1}$
$\lambda_{H_{F}hh}$ | $0.1u$

##  IV Search for LFV Flavon decays at hadron colliders

As stated above, the aim of this work is to analyze the detectability of the
LFV signal arising from the Flavon decays, as predicted by the FNSM, at the
LHC and a future 100 TeV $pp$ collider. Below we will present an analysis
concentrating on the main Flavon production mechanism, i.e. gluon fusion, as
well as the branching ratios of its dominant decay modes. Then, we will
present the Monte Carlo analysis of the $H_{F}\to\tau\mu$ and
$A_{F}\to\tau\mu$ decay signatures, including the study of the potential SM
background. We will present a conservative analysis meant to find out whether
it is possible to have evidence of our signal at the LHC and the future 100
TeV $pp$ collider.

###  IV.1 Production cross-sections of the $CP$-even and $CP$-odd Flavons

We now turn to analyze the main production mode of both $H_{F}$ and $A_{F}$
Flavons at hadronic colliders, namely, by gluon fusion. The High-Luminosity
Large Hadron Collider project aims to increase potential discoveries
contemplating a luminosity of up to $\mathcal{L}$=3 ab-1 about 2025. As far as
a 100 TeV $pp$ collider is concerned the future circular collider (FCC)
contemplates an integrated luminosity of until $\mathcal{O}$(10 ab-1). We
consider the integrated luminosities shown in tha table 2.

Table 2: Integrated luminosities considered in our analysis. Collider | Luminosity
---|---
HL-LHC | 0.3-3 ab-1
FCC | 3-20 ab-1

In Fig. 3 we show the $pp\to H_{F}X$ production cross section of a $CP$-even
Flavon as a function of its mass $m_{H_{F}}$ at the LHC and a future 100 TeV
$pp$ collider. We also show the event numbers on the right axis of each plot.
As far as the $CP$-odd Flavon is concerned, the respective production cross
section and event numbers are presented in Fig. 4.

Figure 3: Production cross-section $\sigma(pp\to H_{F}X)$ of a $CP$-even
Flavon at a hadronic collider as a function of its mass $m_{H_{F}}$ for
$\sqrt{s}=14\;\;(100)$ TeV. The event numbers obtained with an integrated
luminosity of $\mathcal{L}=3\;\;(10)\,\;{\rm ab}^{-1}$ are presented on the
right axis.

Figure 4: The same as in Fig. 3 but for a $CP$-odd Flavon .

The dominant contribution to gluon fusion arises from loops carrying the top
quark as shown in Fig 5. This explains the suppression of the production of
the $CP$-odd Flavon as compared to that of the $CP$-even one, as observed in
Figs. 3 and 4, which stems from the appearance of the coupling
$g_{A_{F}\bar{t}t}$ in the corresponding cross section. For instance, taking
into account the parameter values of Table 1, we have
$g^{2}_{A_{F}\bar{t}t}\sim 10^{-6}$, whereas for the $CP$-even Flavon
$g^{2}_{H_{F}t\bar{t}}\sim 10^{-2}$.

Figure 5: Feynman diagram for the dominant contribution to Flavon production
via gluon fusion at the leading order.

###  IV.2 Flavon decays

####  IV.2.1 Two-body decays

We now analyze the behavior of the branching ratios of the dominant decay
modes, including the FV ones, of both the $CP$-even and $CP$-odd Flavons.
Analytical expressions for the partial decay widths are presented in Appendix
B. It is worth mentioning that a crosscheck was done by comparing the
numerical results obtained via our own C language code implementing the
analytic expressions of Appendix B, and those computed with the aid of the
CalcHEP package Belyaev et al. (2013), for which we used an implementation of
the FNSM Feynman rules obtained with LanHEP Semenov (2016). In Fig. 6 we
present the relevant branching ratios of the decays modes of a $CP$-even
Flavon as functions of its mass for the benchmarks of Table 1.

(a) (b)

Figure 6: Branching ratios of the two-body decay modes of a $CP$-even Flavon
as a function of its mass for the parameter values of Table 1.

We observe in Fig. 6 that in the scenario under study and for $m_{H_{F}}$
ranging between $200$ and $1000$ GeV, the dominant $H_{F}$ decay mode would be
$H_{F}\to WW$, followed by $H_{F}\to ZZ$. Once the $H_{F}\to\bar{t}t$ channel
became open, its branching ratio would be about the same order of magnitude as
that of the $H_{F}\to ZZ$ decay. Other relevant decay modes would be
$H_{F}\to\bar{t}c$, $H_{F}\to\bar{b}b$, and $H_{F}\to\tau\mu$, whereas the
one-loop induced decays $H_{F}\to\gamma\gamma$ and $H_{F}\to\gamma Z$ would
have tiny branching ratios.

As far as the $CP$-odd Flavon $A_{F}$ is concerned, since it does not couple
to gauge bosons at the tree-level, its main decay modes are into fermion
pairs. The corresponding branching ratios are shown in Fig. 7.

Figure 7: The same as in Fig. 6 but for the $CP$-odd Flavon.

We can conclude that the decay modes $A_{F}\to\bar{f}f$ could have branching
ratios up to two orders of magnitude larger than the analogue branching ratios
of the $CP$-even Flavon . For the parameter values used here, the main decay
channels of the $CP$-odd Flavon would be $A_{F}\to\bar{t}c$,
$A_{F}\to\bar{b}b$, $A_{F}\to\tau\mu$, whereas the $A_{F}\to\bar{t}t$ decay
would be very suppressed due to the $A_{F}\bar{t}t$ coupling. As for the one-
loop decays $A_{F}\to\gamma\gamma$ and $A_{F}\to\gamma Z$, they have very
small branching ratios, of the order of $10^{-8}$. Along this line, in Fig. 7
we also have included the branching ratio for the one-loop induced decay
$A_{F}\to hZ$, which proceeds via the fermion loops shown in Fig. 8. We have
calculated the corresponding decay width, which is shown in Appendix B. It has
pointed out recently Bauer et al. (2016b) that this decay could have a
relevant branching ratio in models with a $CP$-odd scalar boson arising from a
complex singlet and it could be useful to look for evidences of $CP$
violation. We observe in Fig. 7 that this decay has a somewhat suppressed
branching ratio in the scenario of the FNSM we are interested in: $BR(A_{F}\to
hZ)\simeq 10^{-4}$ for $m_{A_{F}}$ in the interval between 200 GeV and 1000
GeV. This value is much larger than the branching ratios of the loop induced
decays $A_{F}\to\gamma Z$ and $A_{F}\to\gamma\gamma$ but it is about the same
size than $BR(A_{F}\to gg)$. A rough estimate shows that other loop induced
decays such as $A_{F}\to ZZ$ and $A_{F}\to WW$ have also suppressed branching
ratios. This stems from the fact that not only they are of higher order in the
coupling constants but are suppressed by the loop factor $1/(16\pi^{2})$,
which enters squared into the decay width. In general, the one-loop induced
branching ratios $BR(A_{F}\to VV)$ ($V=W,Z$) and $BR(A_{F}\to hZ)$ are
suppressed with respect to those of the tree-level induced decays
$A_{F}\to\bar{f}f$ by a factor of $(3g^{2}m_{t}/(16\pi^{2}m_{W}))^{2}\simeq
3.3\times 10^{-4}$, assuming that the main contribution arises from the top
quark. Although these one-loop induced decays may be interesting by
themselves, they have no impact on our study of the lepton flavor decays of
the $CP$-odd Flavon.

Figure 8: Fermion loop contribution to the $A_{F}\to hZ$ decay in the FNSM,
where it is absent at the tree-level. There is one additional triangle diagram
where $h$ and $Z$ are exchanged. We denote the four-momenta as follows
$A_{F}(p)\to h(p_{1})Z(p_{2})$ ($p$ is incoming whereas $p_{1}$ and $p_{2}$
are outgoing).

####  IV.2.2 Three-body decays $H_{F}\to\bar{f}_{i}f_{j}h$

At the tree-level the $CP$-even Flavon also couples with a fermion pair and a
SM Higgs boson, thus it is worth analyzing the behavior of the three-body
decay modes $H_{F}\to\bar{f}_{i}f_{j}h$. The corresponding decay width is
presented in Appendix B. In Fig. 9 we show the branching ratios for these
three-body decays. We observe that, for a relatively light Flavon with mass
around 300 GeV, the decay channel $H_{F}\to\bar{b}bh$ would have a branching
ratio as large as $10^{-1}$. On the other hand, other kinematically allowed
$H_{F}\to\bar{f}_{i}f_{j}h$ decays would reach branching ratios as high as
$10^{-2}$. For a heavier Flavon with $m_{H_{F}}>600$ GeV, the decay
$H_{F}\to\bar{t}th$ would become open and could be the dominant three-body
decay mode for $m_{H_{F}}\simeq 900$ GeV. Although these decay channels seem
worth a more detailed study, we will content ourselves with obtaining the
event numbers that could be achieved at the LHC and the FCC. We consider
values for the luminosities of the table 2. In the Fig. 10 we shown the number
of events for the processes $pp\to H_{F}\to q\bar{q}h$ with $q=b,\,t$.

Figure 9: Branching ratios for the three-body decay modes
$H_{F}\to\bar{f_{i}}f_{j}h$ as functions of $m_{H_{F}}$ for the parameters of
Table 1.

(a) (b)

Figure 10: Event number for the process $pp\to H_{F}\to q\bar{q}h$ ($q=b,\,t$)
as a function of $m_{H_{F}}$: (a) $\sqrt{s}=14\;\;$TeV for an integrated
luminosity from $\mathcal{L}$=0.3 ab-1 (lower limit) to 3 ab-1 (upper limit)
and (b) $\sqrt{s}=100\;\;$TeV for an integrated luminosity from
$\mathcal{L}$=3 ab-1 (lower limit) to 30 ab-1 (upper limit). The parameters of
Table 1 were used.

###  IV.3 Search for LFV Flavon decays at the LHC and a future 100 TeV $pp$
collider

We are interested in the possible detection of the $CP$-even and $CP$-odd
Flavons via their LFV decay into a $\tau\mu$ pair at the LHC and the FCC. We
thus show in Fig. 11 the event numbers for the processes $pp\to
H_{F}\to\tau\mu$ and $pp\to A_{F}\to\tau\mu$, for $\sqrt{s}=14\;\;(100)$ TeV
and an integrated luminosity for the values displayed in the Table 2. We also
use the same set of parameter values of Table 1. We note that for a $CP$-even
Flavon with a mass about $1$ TeV, there would be about $\mathcal{O}$(10)
$\mu\tau$ signal events at the LHC and $\mathcal{O}(10^{4})$ events at the
FCC. These event numbers would decrease by about two orders of magnitude for a
$CP$-odd Flavon.

Figure 11: $\sigma(pp\to H_{F}\to\tau\mu)$ (top plots) and $\sigma(pp\to
A_{F}\to\tau\mu)$ (bottom plots) cross sections as functions of the Flavon
mass for $\sqrt{s}=14$ TeV (left plot) and $100$ TeV (right plots). On the
right axes we show the event numbers considering an integrated luminosity
$\mathcal{L}$ of $3$ ab-1 and $10$ ab-1.

We will now analyze the signature of the LFV Flavon decays $H_{F}\to\tau\mu$
and $A_{F}\to\tau\mu$, with $\tau\mu=\tau^{-}\mu^{+}+\tau^{+}\mu^{-}$, and
their potential SM background. We are inspired in the analysis carried out by
the CMS collaboration in Refs. Khachatryan et al. (2015a), ATLAS collaboration
Aad et al. (2017) and the work of the authors of Primulando and Uttayarat
(2017). The ATLAS and CMS collaborations considered the two following tau
decay channels: electron decay $\tau_{e}\mu$ and the hadron decay
$\tau_{h}\mu$. For our analysis we will concentrate instead on the electron
decay. As far as our computation scheme is concerned, we first use the LanHEP
routines to obtain the FNSM Feynman rules for Madgraph Alwall et al. (2011).
In this way, the signal and background events can be generated by MadGraph5
and MadGraph5$\\_$aMC$@$NLO, respectively, interfaced with Pythia 6 Sjostrand
et al. (2006) and Delphes 3 de Favereau et al. (2014) for detector
simulations. The background events were generated at NLO in QCD and the signal
at LO, using the CT10 parton distribution functions Gao et al. (2014). The
signal and main background events are as follows:

  * •

Signal: the signal is $gg\to\phi\to\tau\mu\to e\nu_{e}\nu_{\tau}\mu$ with
$\phi=H_{F},\,A_{F}$. The electron channel must contain exactly two opposite-
charge leptons, one an electron and the other a muon. Then, we search for the
final state $e\mu$+miss energy. We consider the specific case for which an
integrated luminosity $\mathcal{L}$ of $1$ ab-1 for the LHC is considered and
in the interval $1-20$ ab-1 for the FCC.

  * •

Background: the main SM background arises from $Z$ production via the Drell-
Yan process, followed by the decay $Z\to\tau\tau$ as well as $W^{+}W^{-}$ and
$ZZ$ pair production and jets. In this work we will only consider the main
background to assess how our signal could be searched for.

####  IV.3.1 Analysis at the LHC

We start by analyzing the possible detection of the $CP$-even Flavon at the
LHC with $\sqrt{s}$=14 TeV. For illustrative purpose, we use the following set
of values for $m_{H_{F}}$: 200, 250, 300, and 350 GeV. We generated $10^{5}$
events for the signal and the SM main background. Afterwards, the kinematic
analysis was done via MadAnalysis-5 Conte et al. (2013). The cuts applied to
both the signal and background are shown in Table 3, where we also show the
event numbers of the signal (S) and background (B) after the kinematic cuts
are applied, along with the signal significance $S/\sqrt{S+B}$ for
$m_{H_{F}}=200$ GeV. The effect of the cuts on the signal and background event
numbers is best illustrated in Fig. 12, where we show how the efficiencies
$\epsilon_{\text{signal}}$ and $\epsilon_{\text{background}}$ evolve after
each cut is successively applied. One can observe that once the kinematic cuts
are applied, the resulting signal efficiency is about $0.31$, whereas that for
the background is around $0.02$. With a luminosity of $0.3$ ab-1, the signal
significance is about $2\sigma$. However, if we take into account an
integrated luminosity of $3$ ab-1 it increases up to $\sim$6.5$\sigma$. The
net effect is shown in Fig. 13, where the signal significance is plotted as a
function of the luminosity for the chosen values of $m_{H_{F}}$.

Table 3: Kinematic cuts applied to the $pp\to H_{F}\to\tau\mu\to e\mu+\text{miss energy}$ signal and the SM background at the LHC with a center-of-mass energy $\sqrt{s}=14$ TeV and an integrated luminosity of $\mathcal{L}=1$ ab-1. We also show the corresponding event numbers obtained after the kinematic cuts and the signal significance $S/\sqrt{S+B}$ for $m_{H_{F}}=200$ GeV. The tranverse mass is defined as $M_{T}^{\ell}=\sqrt{2P_{T}^{\ell}E_{T}^{\text{miss}}(1-\text{cos}\Delta\phi_{P_{T}^{\ell}-E_{T}^{\text{miss}}})}$ .  Cut number | Cuts | Signal (S) | Background (B) | $\frac{S}{\sqrt{S+B}}$
---|---|---|---|---
| Initial (no cuts) | 9190 | 29320240 | 1.7
1 |  $|\eta^{e}|<$2.3 | 6348 | 9644078 | 2.04
2 |  $|\eta^{\mu}|<$2.1 | 5185 | 7736476 | 1.86
3 | 0.1$<\Delta$R($e,\,\mu$) | 5185 | 7727929 | 1.87
4 | 60$<P_{T}^{\mu}$ | 4856 | 3602928 | 2.56
5 | 20$<P_{T}^{e}$ | 4562 | 2031748 | 3.20
6 | 20$<M_{\text{inv}}(e,\,\mu)<$170 | 4450 | 1781998 | 3.20
7 | 10$<MET<$100 | 3504 | 1158653 | 3.33
8 | 75$<M_{T}^{e}$ | 2942 | 973054 | 3.55
9 | 60$<M_{T}^{\mu}$ | 2833 | 585342 | 3.7
Figure 12: Evolution of the efficiencies of the $pp\to H_{F}\to\tau\mu$ signal
and the SM background after the kinematic cuts of Table 3 are successively
applied. We use $m_{H_{F}}=200$ GeV and the parameters of Table 1. Figure 13:
Signal significance $S/\sqrt{S+B}$ as a function of the integrated luminosity
for the LFV Flavon $\tau\mu$ decay at the LHC. The horizontal lines show the
discovery and evidence thresholds.

Therefore, it seems troublesome that a $CP$-even Flavon with a mass greater
than 300 GeV could be detected at the LHC via the $H_{F}\to\tau\mu$ decay
channel. In such a case, we can turn to the decay modes $H_{F}\to WW$ and
$H_{F}\to ZZ$, which are the dominant ones. These decay channels seem more
promising for the detection of a heavy Flavon. Along this line, to assess the
potentiality of the $WW$ and $ZZ$ decay channels for the Flavon detection, we
show in Fig. 14 the corresponding event numbers that could be produced at the
LHC and a 100 TeV $pp$ collider. We note that about $10^{5}$ $WW$ events would
be produced at the LHC for $m_{H_{F}}=1$ TeV, whereas the number of $ZZ$
events would be slightly smaller. This seems more promising for the signal
detection, though a more detailed analysis of the background would be required
to draw a definitive conclusion.

Figure 14:  $pp\to H_{F}\to WW$ and $pp\to H_{F}\to ZZ$ event numbers as
functions of the Flavon mass at the LHC (left plot) and the FCC (right plot).

As for the $CP$-odd Flavon, we notice that it has a smaller $pp\to
A_{F}\to\tau\mu$ production rate, therefore, in order to have evidence of the
$A_{F}\to\tau\mu$ decay, higher luminosities, of the order of $10\;\;{\rm
ab}^{-1}$, would be required. This seems inaccessible for the LHC, however, we
expect that the search for this decay could be possible at the FCC.

####  IV.3.2 Analysis at the future 100 TeV $pp$ collider

The building of a 100 TeV $pp$ collider is under consideration Arkani-Hamed et
al. (2016). The luminosity and center-of-mass energy are crucial factors to
allow detectable LFV signatures of the FNSM. Luminosity goals for a 100 TeV
$pp$ collider are discussed by the authors of Ref. Arkani-Hamed et al. (2016).
Again, we consider the values of the Table 2, although up to 30 ab-1 might be
reached. In our analysis we consider the conservative kinematic cuts in order
to give a general overview assess how our signal could be searched for. The
applied cuts to both the signal and background are shown in Table 4, whereas
in Fig. 15 we show the signal significance as a function of the luminosity and
the Flavon mass $m_{\phi}$, with $\phi=H_{F},\,A_{F}$. We observe that at a
100 TeV $pp$ collider with integrated luminosity of 10 ab-1, it would be
possible to probe $CP$-even Flavon masses in the multi-TeV range, up to about
4 TeV. We also notice that $CP$-odd Flavon masses could be searched until the
order $\sim$1 TeV.

Table 4: Kinematic cuts applied to the $pp\to H_{F}\to\tau\mu$ signal and the SM main background at a 100 TeV $pp$ collider with an integrated luminosity of $\mathcal{L}=10$ ab-1. We use the parameter values of Table 1 and consider a $CP$-even Flavon with a mass $m_{H_{F}}=1000$ GeV. We also show the corresponding event numbers obtained after the kinematic cuts, and the signal significance $S/\sqrt{S+B}$.  Cut number | Cuts | Signal (S) | Background (B) | $\frac{S}{\sqrt{S+B}}$
---|---|---|---|---
| Initial (no cuts) | 90720 | 364075164 | 4.75
1 |  $|\eta^{e}|<$2.3 | 80574 | 310571620 | 4.57
2 |  $|\eta^{\mu}|<$2.1 | 73587 | 262283714 | 4.54
3 | 150$<M_{T}^{e}$ | 45334 | 117527474 | 4.18
4 | 80$<M_{T}^{\mu}$ | 43765 | 65354838 | 5.41
5 | 100$<P_{T}^{e}$ | 34252 | 57819738 | 4.5
6 | 220$<P_{T}^{\mu}$ | 30204 | 40582493 | 4.74
7 | 100$<MET<$200 | 28136 | 32475178 | 4.94

Figure 15: Density plot for the signal significance as a function of the
luminosity and the flavon masses $m_{H_{F}}$ and $m_{A_{F}}$.

##  V Conclusions

In this work we have explored the possibility that the LFV $\tau\mu$ decay
channel of $CP$-even $H_{F}$ and $CP$-odd $A_{F}$ Higgs Flavons with a mass of
a few hundreds of GeVs could be at the reach of detection at the LHC and a
future 100 TeV $pp$ collider, which would serve as a possible probe of low-
scale flavor models. For the theoretical framework, we have considered the
simplest Froggatt-Nielsen model, with an Abelian flavor symmetry and a
$CP$-conserving Higgs sector that includes a Higgs doublet and a Froggatt-
Nielsen complex singlet. In this model the $CP$-even Flavon can mix with the
SM-like Higgs boson, thereby inducing tree-level LFV interactions mediated by
the latter. In this work we concentrate instead on the LFV couplings of both
the $CP$-even and $CP$-odd Higgs Flavons. After studying the constraints on
the parameter space of the model from low-energy LFV processes, we choose a
set of benchmarks and estimate the relevant decay modes and the production
cross section of the Flavons via gluon fusion at the LHC and a future 100 TeV
$pp$ collider. We then consider a set of kinematic cuts for both the signal
and the SM main background. It is found that the LHC has the potential to
discover the LFV decay $H_{F}\to\tau\mu$ for $m_{H_{F}}$ between 200 and 350
GeVs provided that luminosities, of the order of 1-3 ab-1, are achieved. In
such a case other decay channels would be more appropriate to search for the
signal of a Flavon at the LHC. As far as a future 100 TeV $pp$ collider is
concerned, it would be able to probe the LFV $\tau\mu$ decay channel for
Flavon masses as heavy as 10 TeVs, as long as an integrated luminosity of at
least 20 ab-1 was available, which has been deemed viable in the literature
regarding the possible construction of such a collider. Therefore, besides
other physics goals, a 100 TeV Collider might also work as a Flavon factory.

###### Acknowledgements.

We acknowledge support from Conacyt and SNI (México). Partial support from
VIEP-BUAP is also acknowledged.

##  Appendix A Flavon contributions to LFV $h$ and $\tau$ decays and the muon
anomaly

In this Appendix we present the analytical expressions necessary to obtain the
constraints on the LFV Flavons couplings shown in Fig. 2. Although these
results were meant for $CP$-even and $CP$-odd scalar bosons, they are also
valid for the Flavons.

In the FNSM, the LFV decay $h\to\tau\mu$ proceeds at the tree-level. The decay
width can be obtained from Eq. (33) of Appendix B in the $m_{h}\gg m_{\tau}\gg
m_{\mu}$ limit. The result is given by

$\Gamma(h\to\tau\mu)=\frac{g_{h\mu\tau}m_{h}}{8\pi}.$ (27)

The CMS collaboration reported a bound on the respective branching ratio:
$BR(h\to\bar{\mu}\tau)<1.2\times 10^{-2}$ Collaboration (2016).

As far as the $\tau\to\mu\gamma$ decay is concerned, it arises at the one-loop
level and receives contributions of the SM Higgs boson and the Flavons via the
Feynman diagram of Fig. 16(a). The respective decay width is

$\Gamma(\tau\to\mu\gamma)=\frac{\alpha
m_{\tau}^{5}}{64\pi^{4}}\left(|C_{S}|^{2}+|C_{P}|^{2}\right),$ (28)

where the $C_{S}$ and $C_{P}$ coefficients stand for the contribution of
$CP$-even and $CP$-odd scalar bosons, respectively, which in the limit of
$g_{\phi\tau\tau}\gg g_{\phi\mu\mu}\gg g_{\phi ee}$ $(\phi=h,H_{F},A_{F})$ and
$m_{\tau}\gg m_{\mu}\gg m_{e}$, can be approximated as Harnik et al. (2013)

$C_{S}=C_{P}\simeq\sum_{\phi=h,H_{F},A_{F}}\frac{g_{\phi\tau\tau}g_{\phi\mu\tau}}{12m_{\phi}^{2}}\left(3\ln\left(\frac{m_{\phi}^{2}}{m_{\tau}^{2}}\right)-4\right).$
(29)

Two-loop contributions can be relevant and the respective expressions are
reported in Harnik et al. (2013). The current experimental limit on the
branching ratio is $BR(\tau\to\mu\gamma)<4.4\times 10^{-8}$ Patrignani et al.
(2016).

Figure 16: Feynman diagrams for the LFV decays $\tau\to\mu\gamma$ and
$\tau\to\mu\bar{\mu}\mu$ with exchange of a scalar boson $\phi$. We omit both
the bubble diagrams for the decay $\tau\to\mu\gamma$, which only serve to
cancel the ultraviolet divergences, and the diagram where there is the
exchange of the final muons in diagram (b).

As for the $\tau\to\mu\bar{\mu}\mu$ decay, it receives contributions from the
exchange of a SM-Higgs boson and the Flavons as depicted in the Feynman
diagram of Fig. 16(b). The tree-level decay width can be approximated as

$\displaystyle\Gamma(\tau\to\mu\bar{\mu}\mu)$
$\displaystyle\simeq\frac{m_{\tau}^{5}}{256\pi^{3}}\Bigg{(}\frac{S_{h}^{2}}{m_{h}^{4}}+\frac{S_{H_{F}}^{2}}{m_{H_{F}}^{4}}+\frac{S_{H_{F}}^{2}}{m_{H_{F}}^{4}}+\frac{2S_{h}S_{H_{F}}}{m_{h}^{2}m_{H_{F}}^{2}}+\frac{2S_{A_{F}}}{3m_{A_{F}}^{2}}\left(\frac{S_{h}}{m_{h}^{2}}+\frac{S_{H_{F}}}{m_{H_{F}}}\right)\Bigg{)},$
(30)

where $S_{\phi}=g_{\phi\mu\mu}g_{\phi\mu\tau}$. It has been pointed in Ref.
Harnik et al. (2013), however, that the one-loop contribution is dominant. We
refrain from presenting the corresponding expression as this process, for
which the experimental limit on the respective branching ratio is
$BR(\tau\to\mu\bar{\mu}\mu)<2.1\times 10^{-8}$ Patrignani et al. (2016), gives
very weak constraints on the FNSM parameters.

Finally, the muon AMDM also receives contributions from the SM Higgs boson and
the Flavons, which are induced by a triangle diagram similar to the diagram of
Fig. 16(a) but with two external muons. The corresponding contribution can be
approximated for $m_{\phi}\gg m_{l}$ as Harnik et al. (2013)

$\delta
a_{\mu}\sim\frac{m_{\mu}}{16\pi^{2}}\sum_{\phi=h,H_{F},A_{F}}\sum_{l=\mu,\tau}\frac{m_{l}g_{\phi\mu
l}^{2}}{m_{\phi}^{2}}\left(2\ln\left(\frac{m_{\phi}^{2}}{m_{l}^{2}}\right)-3\right),$
(31)

where one must take into account the NP corrections to the $g_{h\mu\mu}$
coupling only. If the Flavons are too heavy, the dominant NP contribution
would arise from the SM Higgs boson.

The discrepancy between SM theoretical prediction and the experimental value
is Patrignani et al. (2016)

$\Delta a_{\mu}=a_{\mu}^{exp}-a_{\mu}^{SM}=(2.88\pm 0.63\pm 0.49)\times
10^{-9}.$ (32)

Thus, the requirement that this discrepancy is accounted for by Eq. (31) leads
to the bound $1.32\times 10^{-9}\leq\Delta a_{\mu}\leq 4.44\times 10^{-9}$
with $95\%$ C.L.

##  Appendix B Decay widths of $CP$-even and $CP$-odd scalar bosons

###  B.1 $CP$-even scalar boson decays

The most relevant decays of both $CP$-even and $CP$-odd scalar bosons $\phi$
have been long studied in the literature. We will present the relevant decay
widths for the sake of completeness as they are also valid for the Flavons. We
will assume that all the couplings are SM-like, other than the
$g_{HP_{1}P_{2}}$ couplings, which stand for the couplings of a $CP$-odd
scalar boson $H$ with the $P_{1}$ and $P_{2}$ particles. The tree-level two-
body widths are as follows.

The $\phi\to\bar{f_{i}}f_{j}$ decay width is given by

$\Gamma(\phi\to\bar{f}_{i}f_{j})=\frac{g_{\phi
f_{i}f_{j}}^{2}N_{c}m_{\phi}}{128\pi}\left(4-(\sqrt{\tau_{f_{i}}}+\sqrt{\tau_{f_{j}}})^{2}\right)^{\frac{3}{2}}\sqrt{4-(\sqrt{\tau_{f_{i}}}-\sqrt{\tau_{f_{j}}})^{2}},$
(33)

where $\tau_{i}=4m_{i}^{2}/m_{\phi}^{2}$ and $N_{c}$ is the color number. From
here we easily obtain the flavor conserving decay width.

The decays of a heavy $CP$-even scalar boson into pairs of real electroweak
gauge bosons can also be kinematically allowed. The corresponding decay width
is

$\Gamma(\phi\to VV)=\frac{g_{HVV}^{2}m_{\phi}^{3}}{64n_{V}\pi
m_{V}^{4}}\sqrt{1-\tau_{V}}\left(1-\tau_{V}+\frac{3}{4}\tau_{V}^{2}\right),$
(34)

with $n_{V}=1\;(2)$ for $V=W\;(Z)$.

Other relevant decays are those arising at the one-loop level, such as
$\phi\to\gamma\gamma$ and $\phi\to gg$. The two-photon decay width can be
written as

$\Gamma(\phi\to\gamma\gamma)=\frac{\alpha^{2}m_{\phi}^{3}}{1024\pi^{3}m_{W}^{2}}\left|\sum_{s}A_{s}^{\phi\gamma\gamma}\left(\tau_{s}\right)\right|^{2},$
(35)

with the subscript $s$ standing for the spin of the charged particle
circulating into the loop. The $A^{s}_{\phi\gamma\gamma}$ function is given by

$A_{s}^{\phi\gamma\gamma}(\tau_{s})=\left\\{\begin{array}[]{cr}\sum_{f}\frac{2m_{W}g_{\phi
ff}N_{c}Q_{f}^{2}}{m_{f}}\left[-2\tau_{s}\left(1+(1-\tau_{s})f\left(\tau_{s}\right)\right)\right]&s=\frac{1}{2},\\\
\\\ \frac{g_{\phi
WW}}{m_{W}}\left[2+3\tau_{W}+3\tau_{W}(2-\tau_{W})f\left(\tau_{W}\right)\right]&s=1,\\\
\\\ \frac{m_{W}g_{\phi
H^{-}H^{+}}}{m_{S}^{2}}\left[\tau_{\phi^{\pm}}\left(1-\tau_{\phi^{\pm}}f\left(\tau_{\phi^{\pm}}\right)\right)\right]&s=0,\end{array}\right.$
(36)

where

$f(x)=\left\\{\begin{array}[]{cr}\left[\arcsin\left(\frac{1}{\sqrt{x}}\right)\right]^{2}&x\geq
1,\\\
-\frac{1}{4}\left[\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)-i\pi\right]^{2}&x<1.\end{array}\right.$
(37)

On the other hand, the two-gluon decay can receive contributions of quarks
only and the respective decay width can be obtained from Eq. (35) by summing
over quarks only and making the replacements $\alpha^{2}\to 2\alpha^{2}_{S}$,
$N_{c}Q_{f}^{2}\to 1$.

Formulas for other decay channels such as $\phi\to Z\gamma$, which has a
considerably suppressed decay width, as well as radiative corrections for the
above decay widths can be found in the literature Djouadi (2008a, b).

###  B.2 $CP$-odd scalar boson decays

The decay of a $CP$-odd scalar boson $A$ into a pair of fermions of distinct
flavor is given by

$\Gamma(A\to\bar{f}_{i}f_{j})=\frac{g_{Af_{i}f_{j}}^{2}N_{c}m_{A}}{128\pi}\left(4-(\sqrt{\eta_{f_{i}}}-\sqrt{\eta_{f_{j}}})^{2}\right)^{\frac{3}{2}}\sqrt{4-(\sqrt{\eta_{i}}+\sqrt{\eta_{j}})^{2}},$
(38)

where we now use the definition $\eta_{i}=4m_{i}^{2}/m_{A}^{2}$. The FC decay
width follows easily.

There are no decays into electroweak gauge bosons at the tree-level. On the
other hand, the two-photon decay proceeds through charged fermion loops only
and the corresponding decay width can be obtained from (35) by making the
replacement $\phi\to A$, $\tau_{f}\to\eta_{f}$, and summing over fermions
only, with

$A_{1/2}^{A\gamma\gamma}(\eta_{f})=\sum_{f}\frac{2m_{W}g_{A\bar{f}f}Q_{f}^{2}N_{c}}{m_{f}}\left(-2\eta_{f}f(\eta_{f})\right),$
(39)

whereas the two-gluon decay width can be obtained by summing over quarks only
and making the additional replacements $\alpha^{2}\to 2\alpha^{2}_{S}$ and
$N_{c}Q_{f}^{2}\to 1$.

###  B.3 Three-body decay $H_{F}\to\bar{f}fh$

As far as three-body decays are concerned, the study of the $H\to\bar{f}fh$
decay channel could be interesting as it can also have a sizeable branching
ratio. This decay receives contribution from the four Feynman diagrams shown
in Fig. 17. After some algebra, we can write the decay width as follows

$\displaystyle\Gamma(H_{F}\to h\bar{f}f)$ $\displaystyle=$
$\displaystyle\frac{m_{H_{F}}}{256\pi^{3}}\int dx_{a}\int
dx_{b}|\overline{\cal M}|^{2},$ (40)

where the integration domain is given by

$2\sqrt{x_{t}}\leq x_{a}\leq 1-x_{h}-2\sqrt{x_{t}x_{h}},$ (41)
$x_{b}\gtreqqless\frac{2(1-x_{h}+2x_{t})+x_{a}\left(x_{a}+x_{h}-2x_{t}-3\right)\mp\sqrt{x_{a}^{2}-4x_{t}}\sqrt{\left(x_{a}+x_{h}-1\right)^{2}-4x_{h}x_{t}}}{2\left(1-x_{a}+x_{t},\right)}$
(42)

and the average square amplitude is

$\displaystyle|\overline{\cal M}|^{2}$
$\displaystyle=\frac{1}{2\left(x_{a}+x_{b}+x_{h}-2\right)^{2}}\left(x_{a}+x_{b}+x_{h}-4x_{t}-1\right)\left(\left(x_{a}+x_{b}+x_{h}-2\right)C_{a}+C_{b}\right)^{2}$
$\displaystyle+\frac{2}{\left(x_{a}-1\right)^{2}\left(x_{b}-1\right)^{2}}\Bigg{(}\left(x_{a}-1\right)\left(x_{b}-1\right)\left(x_{a}-x_{b}\right)^{2}-16\left(x_{a}+x_{b}-2\right)^{2}x_{t}^{2}$
$\displaystyle+4\left(x_{a}+x_{b}-2\right)\left(2-3x_{b}+x_{a}\left(4x_{b}-3\right)\right)x_{t}+x_{h}\left(4\left(x_{a}+x_{b}-2\right)^{2}x_{t}-\left(x_{a}-x_{b}\right)^{2}\right)\Bigg{)}C_{c}^{2}$
$\displaystyle-\frac{4\sqrt{x_{t}}}{\left(x_{a}-1\right)\left(x_{b}-1\right)}\left(x_{a}^{2}+2\left(3x_{b}+x_{h}-4x_{t}-3\right)x_{a}+x_{b}^{2}-4x_{h}+2x_{b}\left(x_{h}-4x_{t}-3\right)+16x_{t}+4\right)C_{a}C_{c}$
$\displaystyle-\frac{4\sqrt{x_{t}}}{\left(x_{a}-1\right)\left(x_{b}-1\right)\left(x_{a}+x_{b}+x_{h}-2\right)}\Bigg{(}x_{a}^{2}+2\left(3x_{b}+x_{h}-4x_{t}-3\right)x_{a}+x_{b}^{2}-4x_{h}$
$\displaystyle+2x_{b}\left(x_{h}-4x_{t}-3\right)+16x_{t}+4\Bigg{)}C_{b}C_{c}.$
(43)

with $x_{a}=(m_{a}/m_{H_{F}})^{2}$. Also $C_{a}=g_{H_{F}ffh}$,
$C_{b}=g_{H_{F}hh}g_{hff}/m_{h}^{2}$, and $C_{c}=g_{H_{F}ff}g_{hff}/m_{h}$ are
the coupling constants involved in the Feynman diagrams of Fig. 17.

Figure 17: Feynman diagrams inducing the $H\to\bar{f}fh$ decay in the FNSM.

The general expressions for the FV decay $H_{F}\to f_{i}f_{j}h$ are to
cumbersome unless one of the masses is neglected, so we refrain from
presenting it here. The calculation was done instead via the CalcHEP software.

###  B.4 The decay $A_{F}\to hZ$

Although the coupling of the $Z$ gauge boson to a pair of $CP$-even and
$CP$-odd scalar bosons $AhZ$ appears at the tree level in multiple Higgs
doublet models, in the FNSM the scalar singlet does not couple to SM gauge
bosons, so the $A_{F}\to hZ$ decay proceeds up to the one-loop level via
fermion exchange through the Feynman diagrams of Fig. 8. It has been pointed
out recently that such a decay Bauer et al. (2016b) can be relevant to find
evidences of $CP$ violation. We have calculated the corresponding amplitude
via the Passarino-Veltman reduction scheme. Using the nomenclature of Fig. 8
we obtain

$\mathcal{M}(A_{F}\to
hZ)=ig_{A_{F}hZ}(w_{h},w_{Z})(p+p_{1})^{\mu}\epsilon(p_{2})_{\mu},$ (44)

where $w_{a}=m_{a}^{2}/m_{A_{F}}^{2}$, whereas the one-loop induced coupling
$g_{A_{F}hZ}(w_{h},w_{Z})$ is given as

$g_{A_{F}hZ}(w_{h},w_{Z})=\frac{g}{4\pi^{2}c_{W}}\frac{\sum_{f}N_{c}^{f}g_{A}^{f}g_{A_{F}ff}g_{hff}g(w_{f},w_{h},w_{Z})}{(1-(w_{h}+w_{Z})^{2})(1-(w_{h}-w_{Z})^{2})},$
(45)

with $g_{A}^{f}=\frac{1}{2}$ ($-\frac{1}{2}$) for up (down) fermions. The
$g_{A_{F}hZ}(w_{h},w_{Z})$ is given in terms of Passarino-Veltman scalar
functions as follows

$\displaystyle g(w_{f},w_{h},w_{Z})$
$\displaystyle=2\left(w_{h}w_{Z}+w_{f}\left(\left(w_{h}-1\right)^{2}+w_{Z}^{2}-2\left(w_{h}+1\right)w_{Z}\right)\right)C_{A_{F}\,h\,Z\,f}$
$\displaystyle+w_{f}w_{h}\left(w_{h}-w_{Z}-1\right)\Delta_{h\,f\,Z}-w_{f}\left(w_{h}^{2}-\left(2w_{Z}+1\right)w_{h}+\left(w_{Z}-1\right)w_{Z}\right)\Delta_{A_{F}\,f\,Z},$
(46)

where the dimensionless functions $C_{A_{F}\,h\,Z\,f}$ and
$\Delta_{\chi\,f\,Z}$ ($\chi=A_{F}\,,h$) are given by

$C_{A_{F}\,h\,Z\,f}=m_{f}^{2}C_{0}(m_{A_{F}}^{2},m_{h}^{2},m_{Z}^{2},m_{f}^{2},m_{f}^{2},m_{f}^{2}),$
(47)

and

$\Delta_{\chi\,f\,Z}=B_{0}(m_{\chi}^{2},m_{f}^{2},m_{f}^{2})-B_{0}(m_{Z}^{2},m_{f}^{2},m_{f}^{2}),$
(48)

where $B_{0}$ and $C_{0}$ are two-point and three-point Passarino-Veltman
scalar functions written in the usual notation.

The $A_{F}\to hZ$ decay width can be written as

$\Gamma(A_{F}\to
hZ)=\frac{|g_{A_{F}hZ(w_{h},w_{Z})}|^{2}m_{A_{F}}}{256\pi\eta_{Z}}\left((4-(\sqrt{\eta_{h}}-\sqrt{\eta_{Z}})^{2})(4-(\sqrt{\eta_{h}}+\sqrt{\eta_{Z}})^{2})\right)^{\frac{3}{2}}.$
(49)

with $\eta_{a}=4m_{a}^{2}/m_{A_{F}}^{2}$. The above expressions are also valid
for the decay $A_{F}\to H_{F}Z$, with the replacement $w_{h}\to w_{H_{F}}$,
provided that it is kinematically allowed.

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