###### Contents

  1. 1 Introduction
  2. 2 The Effective Lagrangian
    1. 2.1 Flavour Contexts
  3. 3 Bounds from EDMs and Collider Higgs Boson Data
    1. 3.1 Bounds from EDMs
    2. 3.2 Bounds From Colliders
      1. 3.2.1 NP In Only One Yukawa Coupling At A Time
      2. 3.2.2 NP In All The Yukawa Couplings
  4. 4 Bounds from Flavour Changing Processes
  5. 5 Flavour Changing Tree-Level Higgs and Top Decays
  6. 6 Conclusions

##  1 Introduction

So far, no signal of beyond the Standard Model (SM) particles has been found
in direct searches. Therefore, indirect searches of new physics (NP) effects
in high precision experiments may play similarly important role as the
precision electroweak (EW) measurements played in constraining the top quark
and Higgs boson masses. A useful tool for interpreting the precision
measurements is the effective field theory (EFT) approach, where higher
dimension operators are added to the SM Lagrangian. In this paper we
investigate in the framework of the Standard Model Effective Field Theory
(SMEFT) [1, 2] approach 111 An alternative approach that will not be
considered here is the so-called Higgs Effective Field Theory [3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16] that treats the physical Higgs field and the
SM Goldstone bosons as independent degrees of freedom. the discovery potential
for NP effects in the Higgs boson Yukawa couplings.

There are several experimental sources of information about potential
contributions of beyond the SM (BSM) physics to the Higgs boson couplings to
fermions. One is the Higgs boson production and decays at colliders. So far,
only flavour conserving decays have been observed. An increase in the
precision of such measurements is one of the main goals of the future collider
experimental programmes. One may even hope that in the future Higgs factories
theoretically interesting bounds on flavour violating decays can be obtained,
or they can be discovered. Another source are the low energy observables:
magnetic (MDM) and electric dipole moments (EDM) (sensitive to the diagonal
Yukawa couplings) and a large variety of flavour changing neutral current
(FCNC) processes, including radiative decays and the CP symmetry violating
ones. If BSM physics gives rise to flavour violating Higgs couplings, those
processes are sensitive to them.

In the SMEFT approach, at the level of dim 6 operators, there is only one
operator contributing to the Yukawa couplings of the structure
${(\overline{f}}Hf)HH^{\dagger}$ , with three $3\times 3$ matrices in the
flavour space of the Wilson coefficients. Each measured observable puts bounds
on some combination of the Wilson coefficients expressed in the fermion mass
eigenstate basis.

It is a common procedure to use the bounds on the Wilson coefficients of
higher dimension operators added to the SM Lagrangian for estimating
sensitivity of various observables to NP scales. For instance, very recently,
the European Particle Physics Strategy committee has provided a qualitative
illustration of the sensitivity to the NP scales in the present and future
experiments based on the contributions of dim 6 operators, one at a time and
with $\mathcal{O}(1)$ couplings in the Wilson coefficients, to various
observables, see Fig. 5.1 in Ref. [17]. It has been shown in Ref. [17] that,
putting aside several (important) caveats of that analysis, in principle the
flavour precision-low energy observables can be sensitive to much higher NP
scales than the collider observables.

Focusing on NP in the Yukawa couplings, we investigate the sensitivity to NP
scale in both mentioned above sets of experimental data, that is collider and
flavour data, under the assumption that the Wilson coefficients have some
flavour structure: either are consistent with the Minimal Flavour Violation
(MFV) hypothesis [18, 19, 20, 21, 22] or respect a flavour symmetry, often
invoked to explain the structure of fermion masses and mixing. In UV complete
models extending the SM, once the flavour symmetry is imposed, it should also
manifest itself in the structure of the Wilson coefficients of the operators
contributing to the Yukawa couplings in the SMEFT framework. This fact has a
strong impact on the lower bounds on the NP scales derived from the
experimental data since the couplings in the Wilson coefficients are
controlled by the symmetry. With the Wilson coefficients respecting a flavour
symmetry, we investigate the complementarity of different processes in
establishing the most stringent bounds, that is of their sensitivity to NP
scales. The emerging picture is different from the one obtained with
uncontrolled Wilson coefficients, providing valuable indications on where to
look for NP that is related to the Higgs sector.

In the first part of the paper we systematically reanalyse the bounds on the
diagonal Higgs couplings following from the electron EDM and from the Higgs
boson production and decays at the LHC. In the second part we use the bounds
on the off-diagonal couplings following from the FCNC processes obtained in
Ref. [23], to investigate their complementarity with the collider bounds, once
a flavour symmetry is assumed. In conclusions we summarise the emerging
picture for the lower bounds on the NP scales, much more optimistic from the
point of view of direct searches for new particles than what follows when no
flavour symmetry is invoked.

Within the same framework, it has recently been shown [24] that the bounds
from the electron EDM on CP violation in the electron Yukawa coupling can be
translated into bounds on CP violation in tau Yukawa coupling that are two
orders of magnitude stronger than if no flavour symmetry is considered, with
strong implications for the electroweak baryogenesis. The idea of linking the
diagonal to off-diagonal Higgs-quark couplings with the purpose of
constraining the diagonal light generation Yukawas by the tree-level
contribution of the off-diagonal ones to the $K$, $B$ and $D$ meson
oscillations has been proposed in Ref. [25]. In that paper, the misalignment
between the quark mass and the Yukawa matrices is introduced by adding dim 6
operators with anarchical flavour structure to the SM Yukawa coupling. The
diagonal and off-diagonal couplings are related by arbitrary coefficients that
are randomly generated within a specific interval in a numerical analysis.
Moreover, in the recent paper Ref. [26], the implications of the muon $g-2$
results on the lepton mass matrix have been studied considering effective
$d=6$ dipole operators.

The structure of the paper can be seen in the table of Contents.

##  2 The Effective Lagrangian

Generic NP that affects the Yukawa couplings can be encoded at low energies
into effective operators. The relevant part of the Lagrangian can be written
within the SMEFT approach as

$\begin{split}\mathscr{L}=&-\overline{Q_{L}^{\prime}}\tilde{H}Y^{\prime}_{u}u^{\prime}_{R}-\overline{Q_{L}^{\prime}}HY^{\prime}_{d}d^{\prime}_{R}-\overline{L_{L}^{\prime}}HY^{\prime}_{e}e^{\prime}_{R}+\\\
&-\left(\overline{Q_{L}^{\prime}}\tilde{H}C^{\prime}_{u}u^{\prime}_{R}+\overline{Q_{L}^{\prime}}HC^{\prime}_{d}d^{\prime}_{R}\right)\dfrac{H^{\dagger}H}{\Lambda_{q}^{2}}-\overline{L_{L}^{\prime}}HC^{\prime}_{e}e^{\prime}_{R}\dfrac{H^{\dagger}H}{\Lambda_{\ell}^{2}}+\text{h.c.}\,,\end{split}$
(2.1)

where the prime denotes fields and quantities in the flavour basis.
$Q_{L}^{\prime}$ and $L_{L}^{\prime}$ are respectively the quark and lepton EW
doublets, $u^{\prime}_{R}$, $d^{\prime}_{R}$ and $e^{\prime}_{R}$ stand for
the EW singlets and $H$ is the Higgs EW doublet. The scales $\Lambda_{q,\ell}$
are the effective scales of NP, in general different for the quark and lepton
sectors. Finally $Y^{\prime}_{f}$ and $C^{\prime}_{f}$ are $3\times 3$ complex
matrices in the flavour space. Notice that flavour indices are omitted.

After the electroweak symmetry breaking (EWSB), the Lagrangian in Eq. (2.1)
can be rewritten as

$\begin{split}\mathscr{L}=&-\left[\overline{u^{\prime}_{L}}\left(Y^{\prime}_{u}+\dfrac{v^{2}}{2\Lambda_{q}^{2}}C^{\prime}_{u}\right)u^{\prime}_{R}+\overline{d^{\prime}_{L}}\left(Y^{\prime}_{d}+\dfrac{v^{2}}{2\Lambda_{q}^{2}}C^{\prime}_{d}\right)d^{\prime}_{R}+\overline{e^{\prime}_{L}}\left(Y^{\prime}_{e}+\dfrac{v^{2}}{2\Lambda_{\ell}^{2}}C_{e}^{\prime}\right)e^{\prime}_{R}\right]\dfrac{v}{\sqrt{2}}+\\\
&-\left[\overline{u^{\prime}_{L}}\left(Y^{\prime}_{u}+\dfrac{3v^{2}}{2\Lambda_{q}^{2}}C^{\prime}_{u}\right)u^{\prime}_{R}+\overline{d^{\prime}_{L}}\left(Y^{\prime}_{d}+\dfrac{3v^{2}}{2\Lambda_{q}^{2}}C^{\prime}_{d}\right)d^{\prime}_{R}+\overline{e^{\prime}_{L}}\left(Y^{\prime}_{e}+\dfrac{3v^{2}}{2\Lambda_{\ell}^{2}}C^{\prime}_{e}\right)e^{\prime}_{R}\right]\dfrac{h}{\sqrt{2}}+\\\
&+\text{h.c.}+\ldots\,,\end{split}$ (2.2)

where $v=246\ \text{GeV}$ is the EW symmetry breaking vacuum expectation value
(VEV) of the Higgs field. The first line gives the fermion mass terms, and the
second line describes the fermion interactions with the physical Higgs field
$h$. Dots stand for terms with more than one Higgs field that will not be
discussed in this paper.

The effective Yukawa matrices are diagonalised by bi-unitary rotations
involving the left- and right-handed fields:

$Y^{\prime}_{f}+\dfrac{v^{2}}{2\Lambda^{2}}C^{\prime}_{f}=V_{f}Y_{f}U_{f}^{\dagger}\,,$
(2.3)

where $V_{f}$ and $U_{f}$ are unitary $3\times 3$ matrices. Here and in the
following, $f=u,d,e$ refer to the up and down quark sectors, and to the lepton
sector, respectively. The matrices $V_{f}$ enter into the definitions of the
CKM and PMNS matrices. In this expression and the following ones where no
index is explicitly indicated, $\Lambda$ should be identified with
$\Lambda_{q}$ when $f=u,d$, and with $\Lambda_{\ell}$ when $f=e$. The matrices
$Y_{f}$ are diagonal matrices whose entries are defined in terms of fermion
masses:

$\begin{gathered}Y_{u}=\frac{\sqrt{2}}{v}M_{u}=\frac{\sqrt{2}}{v}\left(m_{u},m_{c},m_{t}\right)\,,\\\
Y_{d}=\frac{\sqrt{2}}{v}M_{d}=\frac{\sqrt{2}}{v}\left(m_{d},m_{s},m_{b}\right)\,,\\\
Y_{e}=\frac{\sqrt{2}}{v}M_{e}=\frac{\sqrt{2}}{v}\left(m_{e},m_{\mu},m_{\tau}\right)\,,\end{gathered}$
(2.4)

whose experimental determination [27] is

$\displaystyle m_{u}$ $\displaystyle=2.16^{+0.49}_{-0.26}\ \text{MeV}\qquad$
$\displaystyle m_{c}=1.27\pm 0.02\ \text{GeV}\qquad$ $\displaystyle
m_{t}=172.76\pm 0.30\ \text{GeV}$ $\displaystyle m_{d}$
$\displaystyle=4.67^{+0.48}_{-0.17}\ \text{MeV}\qquad$ $\displaystyle
m_{s}=93^{+11}_{-5}\ \text{MeV}\qquad$ $\displaystyle
m_{b}=4.18^{+0.03}_{-0.02}\ \text{GeV}$ (2.5) $\displaystyle m_{e}$
$\displaystyle=0.511\ \text{MeV}\qquad$ $\displaystyle m_{\mu}=105.66\
\text{MeV}\qquad$ $\displaystyle m_{\tau}=1776.86\pm 0.12\ \text{MeV}\,,$

where the errors on the electron (muon) mass is of $6\times 10^{-7}\%$
($2\times 10^{-6}\%$).

Finally, the Lagrangian in the mass eigenstate basis reads

$\begin{split}\mathscr{L}=&-\overline{u_{L}}Y_{u}u_{R}\dfrac{v}{\sqrt{2}}-\overline{d_{L}}Y_{d}d_{R}\dfrac{v}{\sqrt{2}}-\overline{e_{L}}Y_{e}e_{R}\dfrac{v}{\sqrt{2}}+\\\
&-\left[\overline{u_{L}}\left(Y_{u}+\dfrac{v^{2}}{\Lambda_{q}^{2}}C_{u}\right)u_{R}+\overline{d_{L}}\left(Y_{d}+\dfrac{v^{2}}{\Lambda_{q}^{2}}C_{d}\right)d_{R}+\overline{e_{L}}\left(Y_{e}+\dfrac{v^{2}}{\Lambda_{\ell}^{2}}C_{e}\right)e_{R}\right]\dfrac{h}{\sqrt{2}}+\\\
&+\text{h.c.}+\ldots\,,\end{split}$ (2.6)

where

$C_{f}=V_{f}^{\dagger}C^{\prime}_{f}U_{f}$ (2.7)

are the Wilson coefficients of the fermion-Higgs operators in this basis. It
will be useful in the following to introduce the following notation for the
effective fermion-Higgs couplings:

$\hat{Y}_{f}\equiv Y_{f}+\dfrac{v^{2}}{\Lambda^{2}}C_{f}\,.$ (2.8)

The effective Lagrangian below the EWSB scale, in the mass eigenstate basis,
describing flavour conserving Higgs-fermion couplings, that is usually
introduced in the literature is [28]

$\mathscr{L}_{\text{eff}}=-\dfrac{y_{\psi}}{\sqrt{2}}\left(\kappa_{\psi}\overline{\psi}\psi+i\tilde{\kappa}_{\psi}\overline{\psi}\gamma_{5}\psi\right)h\,,$
(2.9)

where $\psi$ refers to a specific quark or charged lepton,
$y_{\psi}=\sqrt{2}m_{\psi}/v$, $m_{\psi}$ is the $\psi$ mass. The
$\kappa_{\psi}$ and $\tilde{\kappa}_{\psi}$ are real numbers parametrising the
CP conserving and CP violating parts of these couplings. In the SM limit,
$\kappa_{\psi}=1$ and $\tilde{\kappa}_{\psi}=0$. Matching this effective
Lagrangian with the one in Eq. (2.6), we conclude that

$Y_{f}K_{f}=Y_{f}+\dfrac{v^{2}}{\Lambda^{2}}{\tt
diag}(\texttt{Re}C_{f})\,,\qquad
Y_{f}{\tilde{K}_{f}}=\dfrac{v^{2}}{\Lambda^{2}}{\tt
diag}(\texttt{Im}C_{f})\,,$ (2.10)

where $K_{f}$ is the diagonal matrix containing the $\kappa_{\psi}$
parameters. For instance, for the charged leptons: $K_{e}\equiv{\tt
diag}(\kappa_{e},\,\kappa_{\mu},\,\kappa_{\tau})$, and similarly for the
quarks.

We also introduce a parameter that will be useful in discussing the
constraints on the diagonal couplings of the Higgs with fermions: this
parameter measures the deviation from the SM predicted value of the Yukawa
couplings and can be defined as

$r^{2}_{\psi}\equiv\dfrac{v^{2}|\hat{y}_{\psi\psi}|^{2}}{2m^{2}_{\psi}}=\kappa_{\psi}^{2}+\tilde{\kappa}_{\psi}^{2}\,,$
(2.11)

where $\hat{y}_{\psi\psi}$ are the entries of the matrices $\hat{Y}_{f}$
defined in Eq. (2.8). On the right-hand side of the previous expression, we
express the parameter $r_{\psi}$ in terms of the $\kappa_{\psi}$ and
$\tilde{\kappa}_{\psi}$.

Once a flavour symmetry is in action to describe fermion masses and mixings,
fermions transform under such a symmetry determining the flavour structure of
both $Y^{\prime}_{f}$ and $C^{\prime}_{f}$. Although these two matrices may be
independent of each other, they own in general the same flavour structure: it
is this fact that allows to link the different entries of $\hat{Y}_{f}$. We
will consider in the next subsection two benchmark scenarios that are
representative of what occurs in general when flavour symmetries act at the
Lagrangian level.

###  2.1 Flavour Contexts

#### Minimal Flavour Violation

The first framework we will consider is the so-called Minimal Flavour
Violation (MFV) [18, 19, 20, 21, 22]: according to it, any source of flavour
and CP violation within the SM and beyond is encoded into the Yukawa couplings
of the SM. The modern description of the MFV consists in considering fermions
transforming non-trivially under a product of $U(3)$ symmetries, one for each
fermion Weyl spinors. The Yukawa terms are made invariant under this symmetry
by promoting the Yukawa couplings to spurion fields transforming as bi-
triplets of the flavour symmetry. Fermion masses and mixings are then
described (but not explained as pointed out in Refs. [29, 30, 31, 32]) once
these spurions acquire a proper background value.

Any higher dimensional operator encoding flavour changing effects needs
insertions of the Yukawa spurions in order to be invariant under the flavour
symmetry. Once the spurions acquire their background values, the Wilson
coefficients of these operators are suppressed by factors proportional to
fermions masses and mixings. It follows that present experimental bounds from
different experiments, both low-energy flavour factories and high-energy
particle colliders, allow NP contributions with a cut-off of a few TeV [19,
20, 21, 22, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45].

Here we apply the MFV ansatz to the dim 6 operators contributing to the Yukawa
couplings. Implementing the MFV construction in the Lagrangian in Eq. (2.1)
leads to

$C^{\prime}_{f}=c^{\prime}_{f}Y^{\prime}_{f}$ (2.12)

where $c^{\prime}_{f}$ are flavour blind complex coefficients. It follows that
fermion Yukawas and fermion-Higgs couplings are aligned: both are diagonal in
the fermion mass eigenstate basis, with the eigenvalues of the latter rescaled
by the factor $(1+v^{2}c^{\prime}/\Lambda^{2})$. Therefore, in the MFV
framework, the Higgs boson does not acquire flavour violating couplings to
fermions and, moreover, the $\kappa_{\psi}$ and $\tilde{\kappa}_{\psi}$ are
flavour independent in each sector:

$\begin{gathered}\kappa_{t}=\kappa_{c}=\kappa_{u}\simeq
1+\dfrac{v^{2}}{\Lambda_{q}^{2}}\texttt{Re}\,c^{\prime}_{u}\,,\qquad\qquad\tilde{\kappa}_{t}=\tilde{\kappa}_{c}=\tilde{\kappa}_{u}\simeq\dfrac{v^{2}}{\Lambda_{q}^{2}}\texttt{Im}\,c^{\prime}_{u}\,,\end{gathered}$
(2.13)

and similarly for the down-quarks and leptons. A useful relation is the one
connecting $\kappa^{2}_{\psi}$ and $\tilde{\kappa}^{2}_{\psi}$,

$(\kappa_{\psi}-1)^{2}+\tilde{\kappa}_{\psi}^{2}\simeq|c^{\prime}_{f}|^{2}\frac{v^{4}}{\Lambda^{4}},$
(2.14)

for any fermion $\psi$ in the sector $f$. Finally, the explicit expression for
$r_{\psi}$ turns out to be

$r^{2}_{t}=r^{2}_{c}=r^{2}_{u}\simeq
1+\dfrac{v^{4}}{\Lambda_{q}^{4}}|c^{\prime}_{u}|^{2}+2\dfrac{v^{2}}{\Lambda_{q}^{2}}\texttt{Re}\,c^{\prime}_{u}\,,$
(2.15)

and similarly for the down-quarks and the charged leptons.

#### Froggatt-Nielsen

The second benchmark framework we consider is the Froggatt-Nielsen (FN) one
[46], that is the simplest flavour model one can think of in terms of spectrum
and dimension of the flavour symmetry. Indeed, the latter is a global Abelian
$U(1)$ factor and the spectrum is the SM one augmented by an additional scalar
field $\phi$, singlet under the SM gauge symmetries and transforming non-
trivially only under the $U(1)$. Fermions may transform under this flavour
symmetry and the Yukawa terms are made invariant under it by the insertion of
powers of the scalar field, originating non-renormalisable operators.

Taking for concreteness, but without any loss of generality, the
$U(1)$-charges of the left-handed (LH) adjoint fields and of the right-handed
(RH) fields as positive integers and the $U(1)$-charge for the scalar $\phi$
as $n_{\phi}=-1$, the Yukawa Lagrangian then reads

$\begin{split}\mathscr{L}_{\text{FN}}=&-y^{\prime}_{u,ij}\,\overline{Q^{\prime}_{Li}}\tilde{H}u^{\prime}_{Rj}\left(\dfrac{\phi}{\Lambda_{F}}\right)^{(n_{Q_{i}}+n_{u_{j}})}-y^{\prime}_{d,ij}\,\overline{Q^{\prime}_{Li}}Hd^{\prime}_{Rj}\left(\dfrac{\phi}{\Lambda_{F}}\right)^{(n_{Q_{i}}+n_{d_{j}})}+\\\
&-y^{\prime}_{e,ij}\,\overline{L^{\prime}_{Li}}He^{\prime}_{Rj}\left(\dfrac{\phi}{\Lambda_{F}}\right)^{(n_{L_{i}}+n_{e_{j}})}+\\\
&-\Bigg{[}c^{\prime}_{u,ij}\,\overline{L^{\prime}_{Li}}\tilde{H}u^{\prime}_{Rj}\left(\dfrac{\phi}{\Lambda_{F}}\right)^{(n_{Q_{i}}+n_{u_{j}})}+c^{\prime}_{d,ij}\,\overline{L^{\prime}_{Li}}Hd^{\prime}_{Rj}\left(\dfrac{\phi}{\Lambda_{F}}\right)^{(n_{Q_{i}}+n_{d_{j}})}\Bigg{]}\dfrac{H^{\dagger}H}{\Lambda_{q}^{2}}+\\\
&-c^{\prime}_{e,ij}\,\overline{L^{\prime}_{Li}}He^{\prime}_{Rj}\left(\dfrac{\phi}{\Lambda_{F}}\right)^{(n_{L_{i}}+n_{e_{j}})}\dfrac{H^{\dagger}H}{\Lambda_{\ell}^{2}}+\text{h.c.}\,,\end{split}$
(2.16)

where $i$ and $j$ are flavour indices, and $y^{\prime}_{f,ij}$ and
$c^{\prime}_{f,ij}$ are free complex parameter with moduli of order
$\mathcal{O}(1)$. The cut-off scale $\Lambda_{F}$ is different from the scales
$\Lambda_{q,\ell}$ that suppresses the $d=6$ operators in Eq. (2.1): these
scales indeed correspond to two different sectors of the theory and may have
nothing in common.

Once the scalar field $\phi$ acquires VEV, the different operators get
suppressed by powers of the ratio

$\epsilon\equiv\langle\phi\rangle/\Lambda_{F}\,.$ (2.17)

Typically $n_{Q_{1}}>n_{Q_{2}}>n_{Q_{3}}$, $n_{u_{1}}>n_{u_{2}}>n_{u_{3}}$ and
similarly for the remaining fermion Weyl spinors 222For a discussion of
various constraints on the charge assignment in the FN models see, for
instance, Refs. [47, 48, 49, 50, 51] and references therein.: this guarantees
the correct charged fermion mass hierarchies and small mixings. While the
latter is the general expectation for the CKM matrix, it is not a very strong
requirement in the lepton sector, but follows from the prejudice that the
large mixings of the PMNS matrix arise mainly from the neutrino sector. The
condition imposed on the charges guarantees a simple parametrisation of the
mixing angles. According to the definition in Eq. (2.3), for the up sector we
get:

$\displaystyle Y_{u}={\tt
diag}(y_{u}\,\epsilon^{n_{Q_{1}}+n_{u_{1}}},\,y_{c}\,\epsilon^{n_{Q_{2}}+n_{u_{2}}},\,y_{t}\,\epsilon^{n_{Q_{3}}+n_{u_{3}}})$
$\displaystyle
V_{u,ij}=\delta_{ij}+(1-\delta_{ij})\frac{(j-i)}{|j-i|}\frac{y_{u,ij}}{y_{u,jj}}\epsilon^{|{n_{Q_{i}}-n_{Q_{j}}}|}$
(2.18) $\displaystyle
U_{u,ij}=\delta_{ij}+(1-\delta_{ij})\frac{(j-i)}{|j-i|}\frac{y_{u,ji}}{y_{u,jj}}\epsilon^{|{n_{u_{i}}-n_{u_{j}}}|}\,,$

and similarly for the down-quark and charged lepton sectors with the
corresponding substitutions. $y_{u,ij}$ and $y_{i}$ are complex and real
parameters of $\mathcal{O}(1)$, respectively.

In the mass eigenstate basis, the Higgs boson does have flavour changing
couplings with fermions and the diagonal ones have the same suppressions as
the Yukawa couplings. The matrices $C_{f}$ given by Eq. (2.7), using the
expressions for $C^{\prime}_{f}$, $V_{f}$ and $U_{f}$ above, read:

$\displaystyle
C_{u,ij}\approx\mathcal{O}(1)\epsilon^{n_{Q_{i}}+n_{u_{j}}}e^{i\theta_{u,ij}}\,,\quad
C_{d,ij}\approx\mathcal{O}(1)\epsilon^{n_{Q_{i}}+n_{d_{j}}}e^{i\theta_{d,ij}}\,,\quad
C_{e,ij}\approx\mathcal{O}(1)\epsilon^{n_{L_{i}}+n_{e_{j}}}e^{i\theta_{e,ij}}\,,$
$\displaystyle C_{f,ii}\approx\mathcal{O}(Y_{f,i})e^{i\theta_{f,ii}}\,,$
(2.19)

where $\theta_{f,ij}$ are effective phases resulting from the product in Eq.
(2.7). The resulting expressions for the $\kappa_{\psi}$ and
$\tilde{\kappa}_{\psi}$ then read

$\displaystyle K_{f}=$ $\displaystyle 1+\dfrac{v^{2}}{\Lambda^{2}}{\tt
diag}\left(\mathcal{O}(1)\cos\theta_{f,11},\,\mathcal{O}(1)\cos\theta_{f,22},\,\mathcal{O}(1)\cos\theta_{f,33}\right)\,,$
(2.20) $\displaystyle\tilde{K}_{f}=$
$\displaystyle\dfrac{v^{2}}{\Lambda^{2}}{\tt
diag}\left(\mathcal{O}(1)\sin\theta_{f,11},\,\mathcal{O}(1)\sin\theta_{f,22},\,\mathcal{O}(1)\sin\theta_{f,33}\right)\,,$

where $\mathcal{O}(1)$ factors reflect the presence of the free coefficients
associated to each effective operator and typical of the FN construction.
Notice the difference with the MFV scenario, where the angle $\theta$ is
flavour independent. Instead, similarly to Eq. (2.14) in the MFV case, we have

$(\kappa_{\psi}-1)^{2}+\tilde{\kappa}_{\psi}^{2}=\mathcal{O}(1)^{2}\frac{v^{4}}{\Lambda^{4}}\,.$
(2.21)

where $\mathcal{O}(1)$’s are due to free flavour dependent parameters.
Finally, for $r_{\psi}$ the Eq. (2.15) gets replaced by

$r_{\psi}^{2}\simeq
1+\mathcal{O}(1)^{2}\dfrac{v^{4}}{\Lambda^{4}}+2\mathcal{O}(1)\dfrac{v^{2}}{\Lambda^{2}}\cos\theta_{f,\psi}\,,$
(2.22)

for each fermion $\psi$.

To fully determine the size of the Yukawa couplings it is necessary to
consider a concrete set of FN charges and a value for the parameter
$\epsilon$. Tab. 1 contains possible settings that agree with data once fixing
$\epsilon=0.23$. For the lepton sector, three distinct models have been
considered, Anarchy ($A$), $\mu-\tau$-Anarchy ($A_{\mu\tau}$) and Hierarchy
($H$) as studied in Refs. [49, 50, 51]: the first realisation correspond to
the anarchical ansatz for the neutrino Yukawa matrix [52, 53, 54]; the second
introduces some degree of hierarchy in the entries associated to the first
interacting neutrino; while the last model envisage a hierarchical structure
within all the three interacting neutrinos. With the present sensitivity on
neutrino oscillation data, the three models are compatible with the
measurements, although the hierarchical models have a higher evidence with
respect to the anarchical one in a bayesian analysis [51]. Interestingly, in
the numerical analysis that follows, only the LH field charges are relevant:
it is indeed possible to express the dependence on the RH field charges in
terms of fermion masses and the LH fermion charges.

Quarks | $Q^{\prime}_{L}$ | $u^{\prime}_{R}$ | $d^{\prime}_{R}$
---|---|---|---
| $(2,1,0)$ | $(5,2,0)$ | $(5,4,2)$

Leptons | $L^{\prime}_{L}$ | $e^{\prime}_{R}$
---|---|---
Anarchy ($A$) | $(0,0,0)$ | $(10,5,3)$
$\mu\tau$-Anarchy ($A_{\mu\tau}$) | $(1,0,0)$ | $(9,5,3)$
Hierarchy ($H$) | $(2,1,0)$ | $(8,4,3)$

Table 1: FN charge assignments for realistic scenarios with $\epsilon=0.23$.

##  3 Bounds from EDMs and Collider Higgs Boson Data

This section is dedicated to the discussion of the constraints on the diagonal
fermion-Higgs couplings. There are two type of constraints that can be taken
into consideration: bounds from EDMs and bounds from colliders.

###  3.1 Bounds from EDMs

With some BSM CP violation effects encoded in non-vanishing parameters
$\tilde{\kappa}_{\psi}$, one obtains contributions to the electron EDM from
the Barr-Zee diagrams with the fermions $\psi$ running in the loop. Most
recent studies such as Ref. [55], assuming only one third generation fermion
running in the loop at a time, give

$|\tilde{\kappa}_{t}|\lesssim 0.0011\,,\qquad|\tilde{\kappa}_{b}|\lesssim
0.25\,,\qquad|\tilde{\kappa}_{\tau}|\lesssim 0.3\,,$ (3.1)

while for the charm quark and the muon

$|\tilde{\kappa}_{c}|\lesssim 0.37\,,\qquad|\tilde{\kappa}_{\mu}|\lesssim
31\,.$ (3.2)

On the other hand, assuming that only the $e\overline{e}h$ Yukawa coupling
received contributions from NP, the Barr-Zee contribution to the electron EDM
gives very strong bound on $\tilde{\kappa}_{e}$,

$|\tilde{\kappa}_{e}|\lesssim 0.0017\,.$ (3.3)

As reported in Ref. [24], once the presence of an underlying flavour symmetry
is assumed, the bound in Eq. (3.3) can be translated into one on
$|\tilde{\kappa}_{\tau}|$:

$|\tilde{\kappa}_{\tau}|\lesssim\alpha\,0.0017\,,$ (3.4)

with $\alpha=1$ in the MFV case and $\alpha=\mathcal{O}(1)$ in the FN one.
Although this result has been obtained assuming NP contributions only to the
charged lepton sector, it approximatively holds also when such contributions
to all the Yukawa couplings are allowed (under the hypothesis of the absence
of 1:1000 cancellation among the electron and top contributions). This result
is two order of magnitudes stronger than the one listed in Eq. (3.1).

Before moving on to the new section and analysing bounds from colliders, let
us discuss the impact of this result on the original parameter space. When
$\tilde{\kappa}_{\psi}$ is strongly constrained, there are two possible
explanations: either the corresponding phase $\theta_{f,ii}$ is extremely
small by accident or due to a symmetry, or the cut-off scale $\Lambda$ is
large. In this second case,

$\begin{cases}|\tilde{\kappa}_{t}|\lesssim
0.0011&\Longrightarrow\Lambda_{q}\gtrsim 7.4\ \text{TeV}\\\\[5.69054pt]
|\tilde{\kappa}_{e}|,|\tilde{\kappa}_{\tau}|\lesssim
0.0017&\Longrightarrow\Lambda_{\ell}\gtrsim 6.0\ \text{TeV}\,,\end{cases}$
(3.5)

taking the $\mathcal{O}(1)$ coefficients equal to $1$.

###  3.2 Bounds From Colliders

The LHC signal strength of the Higgs production by a mechanism $P$ with the
Higgs decaying into a final state $F$ , normalised to the SM prediction, is
defined as

$\mu_{P}^{F}=\frac{\sigma_{P}}{\sigma^{\text{SM}}_{P}}\,\frac{\Gamma\left(h\rightarrow
F\right)}{\Gamma^{\text{SM}}\left(h\rightarrow
F\right)}\,\left(\frac{\Gamma_{h,\text{tot}}}{\Gamma^{\text{SM}}_{h,\text{tot}}}\right)^{-1}\,,$
(3.6)

where $\sigma_{P}$ is the production cross section by the mechanism $P$,
$\Gamma\left(h\rightarrow F\right)$ is the decay rate into the final state $F$
and $\Gamma_{h,\text{tot}}$ is the total decay rate of the Higgs. The “SM”
prefix indicates that the quantity is the predicted one within the SM. The
parameter controlling the deviation from the SM expectation in the Higgs
signal strength is $r_{\psi}^{2}$ introduced in Eq. (2.11).

The most recent combination of different Higgs signal strengths has been
released by the ATLAS collaboration in Ref. [56], using data at $\sqrt{s}=13\
\text{TeV}$ with $80\text{ fb}^{-1}$ of integrated luminosity. The results
that will be used in this analysis are reported in Tab. 2.

ATLAS $\sqrt{s}=13\ \text{TeV}$ $24.5-79.8\text{ fb}^{-1}$
---
Production mech. $P$ | Final state $F$ |  $\sigma\times BR$ normalised to SM
ggF | $\gamma\gamma$ | $0.96\pm 0.14$
$ZZ^{*}$ | $1.04^{+0.16}_{-0.15}$
$WW^{*}$ | $1.08\pm 0.19$
$\tau\overline{\tau}$ | $0.96^{+0.59}_{-0.52}$
VBF | $\gamma\gamma$ | $1.39^{+0.40}_{-0.35}$
$ZZ^{*}$ | $2.68^{+0.98}_{-0.83}$
$WW^{*}$ | $0.59^{+0.36}_{-0.35}$
$\tau\overline{\tau}$ | $1.16^{+0.58}_{-0.53}$
$b\overline{b}$ | $3.01^{+1.67}_{-1.61}$
VH | $\gamma\gamma$ | $1.09^{+0.58}_{-0.54}$
$ZZ^{*}$ | $0.68^{+1.20}_{-0.78}$
$b\overline{b}$ | $1.19^{+0.27}_{-0.25}$
$t\overline{t}H+tH$ | $\gamma\gamma$ | $1.10^{+0.41}_{-0.35}$
$VV^{*}$ | $1.50^{+0.59}_{-0.57}$
$\tau\overline{\tau}$ | $1.38^{+1.13}_{-0.96}$
$b\overline{b}$ | $0.79^{+0.60}_{-0.59}$
Table 2: Higgs signal strength from ATLAS collaboration in Ref. [56]. ggF
stands for gluon fusion, VBF for vector boson fusion, VH for vector boson-
Higgs boson associated production, $t\overline{t}H+tH$ for associated Higgs
boson production with $t\overline{t}$ or single $t$.

In addition, ATLAS collaboration reported in Ref. [57] the observations of
$h\to\mu\overline{\mu}$ in collisions at $\sqrt{s}=13\ \text{TeV}$ with
$139\text{ fb}^{-1}$ of integrated luminosity. The only numerical data that
will be used in our analysis is the one with associated jet emission and VBF,
reported in Tab. 3 (more details follow).

ATLAS $h\to\mu\overline{\mu}$ $\sqrt{s}=13\ \text{TeV}$ $139\text{ fb}^{-1}$
---
Production mech. $P$ | Category |  $\sigma\times{\rm BR}$ normalised to SM
ggF |  $0$-jet | $-0.4\pm 1.6$
$1$-jet | $2.4\pm 1.2$
$2$-jet | $-0.6\pm 1.2$
VBF |  | $1.8\pm 1.0$
VH+$t\overline{t}H$ |  | $5.0\pm 3.5$
Table 3: The Higgs signal strength parameter for $h\to\mu\overline{\mu}$ from
ATLAS collaboration in Ref. [57]. The production mechanisms are the ggF, the
VBF, and the combination of VH and $t\overline{t}H$; in the first case, the
categories considered are with $0$, $1$ and $2$ gluon associated emission.

In Ref. [58], the CMS collaboration reported the most stringent bound on $h\to
c\overline{c}$, analysing data on vector boson-Higgs (VH) associated
production with the Higgs decaying into two charm quarks. The best fit value,
$\mu_{cc}=37\pm 20$, is however affected by very large uncertainties, much
larger than the ones on the previous reported data, and for this reason will
not be used in our analysis.

Let us report the last determination of the total decay width of the Higgs
boson presented by the CMS collaboration in Ref. [59]:

$\Gamma^{\text{CMS}}_{h,\text{tot}}=3.2^{+2.8}_{-2.2}\ \text{MeV}\,,$ (3.7)

while the SM prediction is $\Gamma^{\text{SM}}_{h,\text{tot}}=4.1\ \text{MeV}$
[60].

We can now proceed with the determination of the constraints on the parameters
of our effective Lagrangian considering the above data. First we will discuss
the case in which only one Yukawa coupling deviates from the corresponding SM
prediction at a time, leaving for the end of the section the discussion of NP
effects in all the Yukawas couplings simultanously, and the impact of the
action of a flavour symmetry.

####  3.2.1 NP In Only One Yukawa Coupling At A Time

Let us first consider the case in which NP affects only one Yukawa coupling of
the fermion $\psi$ at a time, focusing on any fermion but the top quark, which
will be discussed separately later on. In this case, the Higgs boson
production cross section by any mechanism is approximatively the one predicted
in the SM :

$\dfrac{\sigma_{P}}{\sigma^{\text{SM}}_{P}}=1\,,\qquad\text{for any $P$}\,.$
(3.8)

On the other hand, the partial decay width does deviates from the SM
prediction, accordingly to

$\dfrac{\Gamma\left(h\rightarrow
F\right)}{\Gamma^{\text{SM}}\left(h\rightarrow
F\right)}=\begin{cases}1\,,\quad&\text{for }F\neq\psi\overline{\psi}\\\
r_{\psi}^{2}\,,\quad&\text{for }F=\psi\overline{\psi}\,.\end{cases}$ (3.9)

Finally, the total decay width of the Higgs boson is given by

$\dfrac{\Gamma_{h,\text{tot}}}{\Gamma^{\text{SM}}_{h,\text{tot}}}=1+{\rm
BR}^{\text{SM}}_{\psi\psi}\left(r_{\psi}^{2}-1\right)\,,$ (3.10)

where ${\rm BR}^{\text{SM}}_{\psi\psi}$ stand for the Higgs branching ratios
into $\psi\overline{\psi}$ in the SM, whose numerical values are reported in
Tab. 4.

${\rm BR}^{\rm SM}_{bb}$ | $\text{BR}^{\rm SM}_{gg}$ | $\text{BR}^{\rm SM}_{\tau\tau}$ | $\text{BR}^{\rm SM}_{cc}$ | $\text{BR}^{\rm SM}_{\gamma\gamma}$ | $\text{BR}^{\rm SM}_{\mu\mu}$
---|---|---|---|---|---
$0.58$ | $0.082$ | $0.062$ | $0.029$ | $0.002$ | $0.0002$
Table 4: SM Branching Ratios for the Higgs boson decaying into
$\psi\overline{\psi}$, taking $m_{h}=125.10\ \text{GeV}$ [60].

All in all, the Higgs signal strength assuming that the top-Higgs coupling is
strictly the SM one, while allowing NP contributions to other fermion-Higgs
couplings, is given by the following expression:

$\mu_{P}^{F}=\begin{cases}\dfrac{1}{1+\text{BR}^{\rm
SM}_{\psi\psi}\left(r_{\psi}^{2}-1\right)}\,,&\qquad\text{for
}F\neq\psi\overline{\psi}\\\ \dfrac{r_{\psi}^{2}}{1+\text{BR}^{\rm
SM}_{\psi\psi}\left(r_{\psi}^{2}-1\right)}\,,&\qquad\text{for
}F=\psi\overline{\psi}\,\,,\end{cases}$ (3.11)

The case in which NP enters the top Yukawa coupling differs from the previous
one because the top-loops provide the largest contributions to some of the
production mechanisms. Indeed, the explicit expression of the normalised
production cross section is given by

$\dfrac{\sigma_{P}}{\sigma^{\text{SM}}_{P}}=\begin{cases}1\,,\qquad&\text{for
}P=\text{VBF},\text{VH}\\\ r_{t}^{2}\,,\qquad&\text{for
}P=\text{ggF},\text{ttH}+\text{tH}\,.\end{cases}$ (3.12)

The partial decay width also deviates from the SM prediction, in particular
the channel with $\gamma\gamma$ in the final state is affected:

$\frac{\Gamma\left(h\rightarrow F\right)}{\Gamma^{\text{SM}}\left(h\rightarrow
F\right)}=\begin{cases}1\,,\qquad&\text{for }F\neq\gamma\gamma\\\
r_{t}^{2}\,,\qquad&\text{for }F=\gamma\gamma\,.\end{cases}$ (3.13)

The Higgs total decay width is modified due to the NP contribution to the
$t\overline{t}$-Higgs coupling present in the loop-induced process with gluons
and photons in the final state:

$\dfrac{\Gamma_{h,\text{tot}}}{\Gamma^{\text{SM}}_{h,\text{tot}}}=1+\left(\text{BR}^{\rm
SM}_{gg}+\text{BR}^{\rm SM}_{\gamma\gamma}\right)\left(r_{t}^{2}-1\right)\,.$
(3.14)

All in all, the Higgs signal strength when NP modifies only the
$t\overline{t}$-Higgs coupling is then given by

$\mu_{P}^{F}=\begin{cases}\dfrac{1}{1+\left(\text{BR}^{\rm
SM}_{gg}+\text{BR}^{\rm
SM}_{\gamma\gamma}\right)\left(r_{t}^{2}-1\right)}\,,\qquad&\text{for
}P=\text{VBF},\text{VH}\text{ and }F\neq\gamma\gamma\\\
\dfrac{r_{t}^{2}}{1+\left(\text{BR}^{\rm SM}_{gg}+\text{BR}^{\rm
SM}_{\gamma\gamma}\right)\left(r_{t}^{2}-1\right)}\,,\qquad&\text{for
}P=\text{VBF},\text{VH}\text{ and }F=\gamma\gamma\\\
\dfrac{r_{t}^{2}}{1+\left(\text{BR}^{\rm SM}_{gg}+\text{BR}^{\rm
SM}_{\gamma\gamma}\right)\left(r_{t}^{2}-1\right)}\,,\qquad&\text{for
}P=\text{ggF},\text{ttH}+\text{tH}\text{ and }F\neq\gamma\gamma\\\
\dfrac{r_{t}^{4}}{1+\left(\text{BR}^{\rm SM}_{gg}+\text{BR}^{\rm
SM}_{\gamma\gamma}\right)\left(r_{t}^{2}-1\right)}\,,\qquad&\text{for
}P=\text{ggF},\text{ttH}+\text{tH}\text{ and }F=\gamma\gamma\,.\end{cases}$
(3.15)

We performed $\chi^{2}$ fits on the data listed before, obtaining constraints
on the $r_{\psi}^{2}$ parameters. The results of the fit are shown at $95\%$
C.L.. Each value has been obtained switching on NP contributions only to one
Yukawa coupling at a time. The results are shown in Tab. 5. In the second
column we show the bounds obtained from the data presented in Ref. [56] (see
Tab. 2). The third column contains the bounds extracted from Ref. [61] (see
Tab. 3), using only the data from ggF and VBF. Indeed, the inclusion of the
associated production data, given in the third line of Tab. 3, would require a
different expression for the cross section,
$\sigma_{P}/\sigma^{\text{SM}}_{P}=(\sigma^{\text{SM}}_{\text{VH}}+r_{t}^{2}\sigma^{\text{SM}}_{\text{ttH}})/(\sigma^{\text{SM}}_{\text{VH}}+\sigma^{\text{SM}}_{\text{ttH}})$,
and it turns out that the inclusion of this data does not change the result of
the $\chi^{2}$ at the approximation considered. For this reason and without
any impact on the results, we decided to adopt a simplified approach not
including this data. Finally, in the fourth column the constraints arising
from the total decay width of the Higgs boson are shown, using its
experimental determination by CMS in Ref. [59] and reported in Eq. (3.7).

Modified Coupling |  $\bm{\mu_{F}}$ Ref. [56] |  $\bm{\mu_{\mu\mu}}$ Ref. [57] |  $\bm{\Gamma^{\text{{CMS}}}_{h,\text{{tot.}}}}$ Ref. [59]
---|---|---|---
$t\overline{t}h$ | $0.91\lesssim r_{t}^{2}\lesssim 1.13$ | $r_{t}^{2}\lesssim 2.26$ | $r_{t}^{2}\lesssim 13$
$c\overline{c}h$ | $r_{c}^{2}\lesssim 4.07$ | $-$ | $r_{c}^{2}\lesssim 35$
$b\overline{b}h$ | $0.73\lesssim r_{b}^{2}\lesssim 1.20$ | $0.05\lesssim r_{b}^{2}$ | $r_{b}^{2}\lesssim 2.68$
$\tau\overline{\tau}h$ | $0.47\lesssim r_{\tau}^{2}\lesssim 1.60$ | $-$ | $r_{\tau}^{2}\lesssim 17$
$\mu\overline{\mu}h$ | $r_{\mu}^{2}\lesssim 460$ | $r_{\mu}^{2}\lesssim 2.22$ | $r_{\mu}^{2}\lesssim 4600$
Table 5: $95\%$ C.L. limits on $r_{\psi}^{2}$ after $\chi^{2}$ fits to data on
the Higgs signal strength from Ref. [56] in the second column, from Ref. [57]
in the third column, and using the total Higgs decay width from Ref. [59] in
the fourth column, when NP modifies only a single fermion-Higgs coupling at a
time. “$-$” refers to the fact that no bound can be obtained at $95\%$ C.L..

The best constraints on the third generation fermion couplings as well as the
strongest bound on $c\overline{c}h$ are obtained in the second column of Tab.
5 . Instead, the best constraint on $\mu\overline{\mu}h$ comes from the third
column. The bound on $r_{t}^{2}$ from the $\mu\overline{\mu}h$ data is close
to the one obtained from combining the data on different channels given in
Tab. 2: this is expected as the top enters at the loop-level in both cases.
The data on the total Higgs decay width does not provide bounds stronger than
those from the signal strengths.

Fig. 1 shows the results of Tab. 5 expressed in terms of $\kappa_{\psi}$ and
$\tilde{\kappa}_{\psi}$. The results in the second column of Tab. 5 from the
data reported in Ref. [56] are shown in red and those in the third column from
Ref. [57] are in blue: continuous lines refer to the upper bounds, while
dashed to the lower ones. The values corresponding to the Higgs total decay
width, in the fourth column of the table, are in brown. In green we shown the
bounds from EDMs discussed in the previous sub-section and listed in Eq.
(3.1). The star indicates the SM case with
$\kappa_{\psi}-1=\tilde{\kappa}_{\psi}=0$. If a line is not present in the
plot, it means that the corresponding bound is too weak and would correspond
to lines out of scale.

It is interesting to note that the EDM bounds can have very strong impact. In
the top case, the constraint on $\tilde{\kappa}_{t}$ is so strong that the
$\kappa_{t}$ can only be very close to $1$ and $-1$. For the bottom and $\tau$
cases, $\kappa_{b,\tau}$ can span a slightly larger range of values around $1$
and $-1$. For charm, the absence of a lower bound on $r_{c}^{2}$ implies that
$\kappa_{c}$ can take any value between $2$ and $-2$. Finally, for the $\mu$
plot, there is a lower (rather weak) bound on $r_{\mu}^{2}$, but the EDM one
is too weak to further reduce the parameter space and therefore $\kappa_{\mu}$
and $\tilde{\kappa}_{\mu}$ can take almost any value between $1.5$ and $-1.5$.

Figure 1: Bounds in the
$\left((\kappa_{\psi}-1),\tilde{\kappa}_{\psi}\right)$-plane, with NP only in
the single $\psi\overline{\psi}h$ coupling. These limits are derived from
$\chi^{2}$ fits to the signal strengths reported in Ref. [56] (red lines)
and/or reported in Ref. [57] (blue lines), from the Higgs total decay width of
Ref. [59] (brown line), and from electron EDM constraints from the ACME II
collaboration [62] (green lines). Continuous (dashed) lines refer to the upper
(lower) bound in Tab. 5. The SM case
($\kappa_{\psi}-1=\tilde{\kappa}_{\psi}=0$) is represented with a star.

####  3.2.2 NP In All The Yukawa Couplings

We can now consider the case in which all the Yukawa couplings receive
contributions from NP. In this case, the cross sections for the Higgs
production are:

$\frac{\sigma_{P}}{\sigma^{\text{SM}}_{P}}=\begin{cases}1\,,\qquad&\text{for
}P=\text{VBF},\text{VH}\\\ r_{t}^{2}\,,\qquad&\text{for
}P=\text{ggF},\text{ttH}+\text{tH}\,.\end{cases}$ (3.16)

The Higgs partial widths are given by the following expressions:

$\frac{\Gamma\left(h\rightarrow F\right)}{\Gamma^{\text{SM}}\left(h\rightarrow
F\right)}=\begin{cases}1,\qquad&\text{for }F=VV^{\ast}\,,\\\
r_{t}^{2},\qquad&\text{for }F=\gamma\gamma\,,\\\ r_{b}^{2},\qquad&\text{for
}F=b\overline{b}\,,\\\ r_{\tau}^{2},\qquad&\text{for
}F=\tau\overline{\tau}\,,\\\ r_{\mu}^{2},\qquad&\text{for
}F=\mu\overline{\mu}\,.\end{cases}$ (3.17)

Finally, the Higgs total width reads

$\frac{\Gamma_{h,\text{tot.}}}{\Gamma^{\text{SM}}_{h,\text{tot.}}}=1+\text{BR}^{\rm
SM}_{bb}\left(r_{b}^{2}-1\right)+\left(\text{BR}^{\rm SM}_{gg}+\text{BR}^{\rm
SM}_{\gamma\gamma}+\text{BR}^{\rm
SM}_{cc}\right)\left(r_{t}^{2}-1\right)+\text{BR}^{\rm
SM}_{\tau\tau}\left(r_{\tau}^{2}-1\right)\,.$ (3.18)

The signal strengths for the different processes are obtained combining
properly the previous three effects. Unfortunately, the parameter space is so
large that a $\chi^{2}$ would not be conclusive. However, assuming a flavour
symmetry reduces the freedom in the parameters as already discussed in Eqs.
(2.15) and (2.22). We will present the results explicitly for the MFV case;
the FN one is very similar as the only difference is the presence of the
$\mathcal{O}(1)$ free coefficients.

Figure 2: Bounds in the $r_{\psi}^{2}$-$r_{\psi^{\prime 2}}$ planes, with NP
contributions to all the Yukawa couplings. The MFV case is shown and the FN
one is very similar. The plot on top-left is the main result of the $\chi^{2}$
fit, and the following three ones are the projections in two-dimensions. The
last plot shows the $\kappa_{\tau}-\tilde{\kappa}_{\tau}$ parameter space: in
orange the result of the global $\chi^{2}$ fit and in green the EDM bound as
reported in Eq. (3.4). The SM case ($r^{2}_{\psi}=1$ or
$\kappa_{\tau}-1=\tilde{\kappa}_{\tau}=0$) is represented with a star.

In Fig. 2 we plot the final result of the $\chi^{2}$ fit on the full parameter
space, where $r_{u}^{2}=r_{c}^{2}=r_{t}^{2}$, $r_{d}^{2}=r_{s}^{2}=r_{b}^{2}$
and $r_{e}^{2}=r_{\mu}^{2}=r_{\tau}^{2}$. Moreover, we show the projections in
two dimensions that allow to extract the bounds on the different parameters.
The data used for the $\chi^{2}$ fit are those from Refs. [56] and [57]: as
for the previous analysis whose results are shown in Tab. 5, the only data
that have not been used are the ones from the last category of events in Tab.
3 denoted as VH$+t\overline{t}$H. Moreover, when analysing the data from Ref.
[56], we implemented the correlations provided in the original paper; on the
other side, we assumed that data from Refs. [56] and [57] are uncorrelated. In
the last plot of Fig. 2, we show the $\kappa_{\tau}-\tilde{\kappa}_{\tau}$
parameter space: while the ones for top and bottom are very similar to the
ones in Fig. 1, for the $\tau$ there is a clear change represented by the EDM
bound in green. For the MFV case, and very similarly for the FN one, the
induced EDM bound is very strong and comparable to the one on the top, as
discussed in Eq. (3.4).

Bounds from Ref. [56] | Bounds from Refs. [56]+[57]
---|---
$0.73\lesssim r_{t,c,u}^{2}\lesssim 1.36$ | $0.73\lesssim r_{t,c,u}^{2}\lesssim 1.36$
$0.47\lesssim r_{b,s,d}^{2}\lesssim 1.83$ | $0.46\lesssim r_{b,s,d}^{2}\lesssim 1.81$
$0.26\lesssim r_{\tau,\mu,e}^{2}\lesssim 1.97$ | $0.33\lesssim r_{\tau,\mu,e}^{2}\lesssim 1.86$
Table 6: $95\%$ C.L. limits on $r_{\psi}^{2}$ after $\chi^{2}$ fits obtained
on the data in Ref. [56], and Refs. [56]+[57], when NP contributes to all the
Yukawa couplings. These results holds for the MFV case, while the FN one is
likely to be very similar.

In tab. 6, we report in the second column the bounds for $r_{t,c,u}^{2}$,
$r_{b,s,d}^{2}$ and $r_{\tau,\mu,e}^{2}$ from Fig. 2. In the first column, we
show the results if only data from Ref. [56] are used: the main difference is
on $r_{\tau,\mu,e}^{2}$, as expected.

Comparing these results with those in Tab. 5, we can see that the bounds are
slightly weaker. On the other side, the $r_{\psi}^{2}$ of the lighter
generations are also constrained: for the $c\overline{c}h$ and
$\mu\overline{\mu}h$ couplings, we can see that the inferred bounds are
stronger than those we could put in the absence of any symmetry; moreover, we
can find constraints on $u\overline{u}h$, $d\overline{d}h$, $s\overline{s}h$
and $e\overline{e}h$.

The idea of constraining theoretically the diagonal light generation Yukawas
has been proposed in Ref. [25]. In that paper, they are linked to off-diagonal
Higgs-quark couplings contributing at the tree level to the $K$, $B$ and $D$
meson oscillations. The misalignment between the quark mass and the Yukawa
matrices is introduces by adding dim 6 operators with anarchical flavour
structure to the SM Yukawa coupling. The diagonal and off-diagonal couplings
are related by arbitrary coefficients that are randomly generated within a
specific numerical interval in a numerical analysis. The bounds of Ref. [25]
parametrised in terms of $r_{d}$ read:

$0.16\lesssim r_{d}^{2}\lesssim 2.89\,.$ (3.19)

Comparing this result to the one in Tab. 6 we can see that, once a flavour
symmetry is acting in the SM Lagrangian, present colliders bounds on
$\hat{Y}_{d}$ are already stronger than those from flavour observables.

Before moving to discuss NP in the Higgs boson Yukawa couplings from the
perspective of flavour changing processes, we summarise the results of this
section in terms of the lower bounds of the cut-off scales $\Lambda_{q,\ell}$
emerging from our analyses based on the assumption of an underlying flavour
symmetry. For the sake of a transparent presentation, one can distinguish two
cases: large CP violating phases of the effective Yukawa couplings and
approximately CP conserving scenario. The bounds on the cut-off scales
$\Lambda_{q,\ell}$ following from the electron EDM and the collider Higgs
boson production and flavour conserving decays are then summarised in Tab. 7.
These values strictly holds for the MFV case and also, up to $\mathcal{O}(1)$
coefficients, in the FN case.

| Conditions on $\theta_{f,ii}$ | Bound
---|---|---
EDM | $\sin\theta_{u,33}=1$ | $\Lambda_{q}\gtrsim 7.4\ \text{TeV}$
$\sin\theta_{e,11}=1=\sin\theta_{e,33}$ | $\Lambda_{\ell}\gtrsim 6.0\ \text{TeV}$
Collider (diag couplings) | $\sin\theta_{u,33}=0$ | $\Lambda_{q}\gtrsim 0.60\ \text{TeV}$
$\sin\theta_{\tau,33}=0=\sin\theta_{\tau,11}$ | $\Lambda_{\ell}\gtrsim 0.41\ \text{TeV}$
Table 7: Summary of the bounds on $\Lambda_{q,\ell}$ due to EDMs and collider
Higgs boson production data. In the second column, the corresponding
conditions on the phases $\theta_{f,ii}$ are also reported.

The results in the first line are those given in Eq.(3.5). The values of
$\Lambda_{q,\ell}$ explain the experimental bounds on
$\tilde{\kappa}_{t,e,\tau}$, with the corresponding $\sin\theta=1$ in Eq.
(2.20). The results in the second line follow from Eqs. (2.11) and (2.22): the
strong experimental EDM bounds on the $\tilde{\kappa}_{t,e,\tau}$ imply
$r_{\psi}\approx\kappa_{\psi}$ for $\psi=t,\,e,\,\tau$. Moreover, we have
assumed that in Eq. (2.20), $\sin\theta=0$ for $\psi=t,\,e,\,\tau$. It follows
from our fits that the role of flavour symmetry in obtaining those bounds
appears only at the quantitative rather than qualitative level. Without an
underlying flavour symmetry the bounds become flavour dependent but in the
same bulk part. The only significant difference is for the $\tau$: the EDM
bound on $\Lambda_{\ell}$ would be 13 times weaker without an underlying
flavour symmetry and assuming CP violating NP only in the
$\tau\overline{\tau}h$ coupling. On the contrary, an underlying flavour
symmetry has strong implications for the discussion in the next section.

##  4 Bounds from Flavour Changing Processes

In this section we use the upper bounds on the effective flavour non-diagonal
Yukawa couplings derived from tree-level Higgs exchange contributions to
various $\Delta F=2$ processes and radiative decays of quarks and leptons and
calculate the lower bounds on the NP scales, under the assumption that the
corresponding Wilson coefficients are controlled by a FN symmetry. The
observables considered in this section may receive NP contributions different
from those we have considered so far and that can also be described at low-
energy through four fermion interactions. For this reason, the results
obtained in the following should be considered as a conservative estimation.

We use previous results presented in Ref. [23], where the EFT approach has
been adopted, with the physical Higgs integrated out. In that article,
observables in both quark and lepton sectors have been taken into account. In
Tab. 8, we list the combination of the effective Yukawa couplings
$\hat{y}_{\psi\psi^{\prime}}$, defined as the entries of $\hat{Y}$ in Eq.
(2.8), for the quark sector: in the second column we report the bound on the
absolute value of the given combination, and in the fourth column on the
imaginary part of that combination. In the present analysis we always asume
that the EDM constraints on $\tilde{\kappa}_{t}$ and $\tilde{\kappa}_{e,\tau}$
are satisfied due to small enough phases of the corresponding Higgs couplings,
that is $\sin\theta=0$ for $t,\,e,\,\tau$. Notice that the notation used in
Ref. [23] and in these tables is different: the $c_{\psi\psi^{\prime}}$
coefficients of Ref. [23] are here the effective Yukawas
$\hat{y}_{\psi\psi^{\prime}}$ and the relation that links the two quantities
is very simple, $\hat{y}_{\psi\psi^{\prime}}=\sqrt{2}c_{\psi\psi^{\prime}}$.

Eff. Coupl. | Bound on $|y_{\text{eff.}}|$ [23] |  $\Lambda_{q}$ [ TeV] | Bound on $|{\texttt{Im}}(y_{\text{eff.}})|$ [23] |  $\Lambda_{q}$ [ TeV]
---|---|---|---|---
$\hat{y}_{sd}\hat{y}_{ds}^{*}$ | $2.2\times 10^{-10}$ | $0.7$ | $8.2\times 10^{-13}$ | $3$
$\hat{y}_{cu}\hat{y}_{uc}^{*}$ | $1.8\times 10^{-9}$ | $0.7$ | $3.4\times 10^{-10}$ | $1$
$\hat{y}_{bd}\hat{y}_{db}^{*}$ | $1.8\times 10^{-8}$ | $0.6$ | $5.4\times 10^{-9}$ | $0.8$
$\hat{y}_{bs}\hat{y}_{sb}^{*}$ | $4.0\times 10^{-7}$ | $0.6$ | $4.0\times 10^{-7}$ | $0.6$
$\hat{y}_{ds}^{2}$ | $4.4\times 10^{-10}$ | $0.6$ | $1.6\times 10^{-12}$ | $2$
$\hat{y}_{sd}^{2}$ | $4.4\times 10^{-10}$ | $0.6$ | $1.6\times 10^{-12}$ | $2$
$\hat{y}_{uc}^{2}$ | $2.8\times 10^{-9}$ | $1$ | $5.0\times 10^{-10}$ | $2$
$\hat{y}_{cu}^{2}$ | $2.8\times 10^{-9}$ | $0.2$ | $5.0\times 10^{-10}$ | $0.4$
$\hat{y}_{db}^{2}$ | $2.0\times 10^{-8}$ | $0.4$ | $6.0\times 10^{-9}$ | $0.5$
$\hat{y}_{bd}^{2}$ | $2.0\times 10^{-8}$ | $1$ | $6.0\times 10^{-9}$ | $1$
$\hat{y}_{sb}^{2}$ | $4.4\times 10^{-7}$ | $0.3$ | $4.4\times 10^{-7}$ | $0.3$
$\hat{y}_{bs}^{2}$ | $4.4\times 10^{-7}$ | $1$ | $4.4\times 10^{-7}$ | $1$
Table 8: FN lower bounds on the NP cut-off scale $\Lambda_{q}$ in TeV
following from the off-diagonal effective Yukawa couplings in the quark
sector. The bounds in the first half of the table are independent of the FN
charges, and the others are obtained taking $\epsilon=0.23$ and with the
charge assignments given in Tab. 1.

Those bounds, taken at the face value, suggest a strong power of flavour
observables in searching for NP in the Yukawa couplings (see Fig. 5.1 in Ref.
[17]). Indeed, assuming the structure of the effective couplings to be
$\mathcal{O}(1)v^{4}/\Lambda_{q}^{4}$, those observables would be sensitive to
the scales as high as $\Lambda_{q}\approx(60-300)\ \text{TeV}$, with the lower
value given by the bounds on the absolute value of the effective Yukawas
$\hat{y}_{sd}\hat{y}_{ds}^{*}$ and the upper one by its imaginary part. The
picture is, however, strikingly different if the Wilson coefficients of the
dim 6 operators contributing to the Yukawa couplings respect an underlying
flavour symmetry. This clearly emerges from the third and fifth columns of
Tab. 8, where we show the corresponding lower bounds on the NP scales when the
Wilson coefficients respect the FN symmetry. The bounds given in the third
column follow from the limits on the absolute values of the effective
coefficients and are independent of the phases $\theta_{f,ij}$. The bounds in
the fifth column follow from the limits on the imaginary parts, once we assume
the corresponding $\sin\theta_{f,ij}=1$. The latter case corresponds to a
hybrid scenario, where $\sin\theta=0$ for $t,e,\tau$, so that the bounds from
the EDM do not apply, and the non-diagonal couplings have maximal CP violating
phases. Of course, if the NP is (approximately) CP conserving, there apply the
bounds from the third column.

The obtained values for the lower bounds on NP scales are subject to three
main sources of uncertainty: the first one is due to the $\mathcal{O}(1)$
coefficients typical of the FN context; the second is associated to the
experimental errors on the lightest quark masses; the third one is linked to
the running between the scale $\Lambda_{F}$ down to the scale at which the
different observables are computed. Therefore, they should be considered as
approximate, qualitative results.

The bounds in the upper part of the table are independent from the FN charge
assignment, because for the corresponding $\hat{y}$ combinations it is
possible to re-express the specific FN charge dependence in terms of quark
masses. This does not occurs for the bounds in the lower part of the table for
which the charges in Tab. 1 have been taken. As can be seen, for most of the
combinations of effective Yukawas, the values of the scale $\Lambda_{q}$
following from the experimental limits on their absolute values are very
similar to the collider value $\Lambda_{q}=0.6$ TeV. The exceptions are
$\hat{y}^{2}_{uc}$, $\hat{y}^{2}_{bs}$, $\hat{y}^{2}_{bd}$ which give

$\Lambda_{q}\gtrsim 1\ \text{TeV}\,,$ (4.1)

still very close to the collider limits. This is the effect of the underlying
flavour symmetry.

Concerning the imaginary parts of the effective Yukawa combinations and
assuming maximal phases $\sin\theta=1$ (but still $\sin\theta_{u,33}=0$), the
obtained bounds on several of $\Lambda_{q}$’s are in a couple of TeV range.
The stronger one reads

$\Lambda_{q}\gtrsim 3\
\text{TeV}\qquad\qquad({\texttt{Im}}(\hat{y}_{ds}\hat{y}^{*}_{ds})\text{ from
$K^{0}$ meson oscillation}).$ (4.2)

These values are different quantitatively from the one extract from colliders
data, but are still orders of magnitude below the values of $\Lambda_{q}$
obtained when Wilson coefficients are not controlled by a flavour symmetry.

In Tab. 9, we discuss leptons: the last three columns correspond to the three
leptonic FN models. The limits in the upper part of the table are independent
from the FN charges and the bounds are in common to all the models: also in
this case, it is possible to express the FN charge dependence on the
corresponding effective Yukawa combinations in terms of lepton masses. This is
not the case for the bounds in the lower part of the table.

Eff. Coupl. | Bound from [23] |  $\Lambda_{\ell}$ [ TeV]
---|---|---
|  | $A$ | $A_{\mu\tau}$ | $H$
$|\hat{y}_{e\tau}\hat{y}_{\tau e}|$ | $2.2\times 10^{-2}$ | $8.\times 10^{-3}$ | $=$ | $=$
$|{\texttt{Re}}(\hat{y}_{e\tau}\hat{y}_{\tau e})|$ |  $1.4\times 10^{-4}$[27] | $3\times 10^{-2}$ | $=$ | $=$
$|{\texttt{Im}}(\hat{y}_{e\tau}\hat{y}_{\tau e})|$ |  $1.1\times 10^{-10}$ [27] | $1$ | $=$ | $=$
$|\hat{y}_{e\mu}\hat{y}_{\mu e}|$ | $3.6\times 10^{-1}$ | $2\times 10^{-3}$ | $=$ | $=$
$|{\texttt{Re}}(\hat{y}_{e\mu}\hat{y}_{\mu e})|$ |  $1.4\times 10^{-3}$[27] | $8\times 10^{-3}$ | $=$ | $=$
$|{\texttt{Im}}(\hat{y}_{e\mu}\hat{y}_{\mu e})|$ |  $1.1\times 10^{-9}$[27] | $0.3$ | $=$ | $=$
$|\hat{y}_{\mu\tau}\hat{y}_{\tau\mu}|$ | $4.0$ | $9\times 10^{-3}$ | $=$ | $=$
$|{\texttt{Re}}(\hat{y}_{\mu\tau}\hat{y}_{\tau\mu})|$ |  $2.5\times 10^{-3}$[27] | $5\times 10^{-2}$ | $=$ | $=$
$|{\texttt{Im}}(\hat{y}_{\mu\tau}\hat{y}_{\tau\mu})|$ | $1.2$ | $1\times 10^{-2}$ | $=$ | $=$
$|\hat{y}_{e\tau}\hat{y}_{\tau\mu}|$ |  $6\times 10^{-8}$ [27] | $0.8$ | $0.5$ | $0.5$
$|\hat{y}_{\tau e}\hat{y}_{\mu\tau}|$ |  $6\times 10^{-8}$ [27] | $0.2$ | $0.3$ | $0.3$
$|\hat{y}_{\mu\tau}|^{2}$ | $1.0\times 10^{-3}$ | $0.1$ | $0.1$ | $7\times 10^{-2}$
$|\hat{y}_{\tau\mu}|$ | $1.0\times 10^{-3}$ | $3\times 10^{-2}$ | $3\times 10^{-2}$ | $7\times 10^{-2}$
$|\hat{y}_{e\tau}|^{2}$ | $6.0\times 10^{-4}$ | $0.2$ | $8\times 10^{-2}$ | $4\times 10^{-2}$
$|\hat{y}_{\tau e}|^{2}$ | $6.0\times 10^{-4}$ | $3\times 10^{-3}$ | $6\times 10^{-3}$ | $1\times 10^{-2}$
$|\hat{y}_{\mu e}|^{2}$ |  $3.5\times 10^{-12}$[27] | $0.3$ | $0.6$ | $0.6$
$|\hat{y}_{e\mu}|^{2}$ |  $3.5\times 10^{-12}$ [27] | $4$ | $2$ | $2$
$|\hat{y}_{\mu e}\hat{y}_{e\tau}^{*}|$ | $1.8\times 10^{-4}$ | $3\times 10^{-2}$ | $3\times 10^{-2}$ | $2\times 10^{-2}$
$|\hat{y}_{\mu e}\hat{y}_{\tau e}|$ | $1.8\times 10^{-4}$ | $4\times 10^{-3}$ | $8\times 10^{-3}$ | $1\times 10^{-2}$
$|\hat{y}_{e\mu}^{*}\hat{y}_{e\tau}^{*}|$ | $1.8\times 10^{-4}$ | $0.1$ | $5\times 10^{-2}$ | $4\times 10^{-2}$
$|\hat{y}_{e\mu}^{*}\hat{y}_{\tau e}|$ | $1.8\times 10^{-4}$ | $1\times 10^{-2}$ | $1\times 10^{-2}$ | $2\times 10^{-2}$
$|\hat{y}_{e\mu}\hat{y}_{\mu\tau}^{*}|$ | $2.0\times 10^{-4}$ | $0.1$ | $7\times 10^{-2}$ | $5\times 10^{-2}$
$|\hat{y}_{e\mu}\hat{y}_{\tau\mu}|$ | $2.0\times 10^{-4}$ | $5\times 10^{-2}$ | $4\times 10^{-2}$ | $5\times 10^{-2}$
$|\hat{y}_{\mu e}^{*}\hat{y}_{\mu\tau}^{*}|$ | $2.0\times 10^{-4}$ | $3\times 10^{-2}$ | $4\times 10^{-2}$ | $3\times 10^{-2}$
$|\hat{y}_{\mu e}^{*}\hat{y}_{\tau\mu}|$ | $2.0\times 10^{-4}$ | $1\times 10^{-2}$ | $2\times 10^{-2}$ | $3\times 10^{-2}$
Table 9: FN lower bounds on the NP cut-off scale $\Lambda_{\ell}$ in TeV
following from the off-diagonal effective Yukawa couplings in the leptonic
sector. Most of the bounds are taken from Ref. [23], except those that show
explicitly a different reference: in these cases, the experimental bound has
improved as compared to the one reported in Ref. [23]. The bounds in the first
half of the table are independent of the FN charges, while the others are
obtained taking $\epsilon=0.23$ and with the charge assignments given in Tab.
1. “=” indicates that the bounds in the $A_{\mu\nu}$ and $H$ models are the
same as those in the $A$ model.

For most of the combinations of effective Yukawas for leptons, the lower
bounds on the scale $\Lambda_{\ell}$ are (much) lower than
$\Lambda_{\ell}=0.41\ \text{TeV}$ obtained from the collider data. The main
exception is $|\hat{y}_{e\mu}|^{2}$ that is linked to the muon radiative
decay. Barring large cancellations between free $\mathcal{O}(1)$ coefficients,
we obtain

$\begin{cases}\Lambda_{\ell}=4\ \text{TeV}&\qquad\text{for the $A$ Model}\\\
\Lambda_{\ell}=2\ \text{TeV}&\qquad\text{for the $A_{\mu\nu}$ and $H$
Models}\\\ \end{cases}\qquad(|\hat{y}^{2}_{e\mu}|\text{ from $\mu\to e\gamma$
decay})\,,$ (4.3)

as the lower bound on the NP scale (independently of the phases of the
individual Yukawa couplings). Similarly to the quark case, these values for
$\Lambda_{\ell}$ should be contrasted with the value $\Lambda_{\ell}=300\
\text{TeV}$ for the NP scale if no flavour symmetry is invoked.

In Fig. 3, we give the graphical representation of the experimental bounds and
the FN predictions for quarks and leptons with $\Lambda_{q}=1\ \text{TeV}$ and
$\Lambda_{\ell}=4\ \text{TeV}$. These two values are the lower bounds on the
scale $\Lambda_{q}$ and $\Lambda_{\ell}$ consistent with all the considered
experimental constraints when the Wilson coefficients respect the FN
symmetries, for the CP conserving case, that is $\sin\theta=0$ for any
effective phase.

Figure 3: Graphical representation of the experimental bounds and FN
predictions on the effective Yukawas for the CP conserving case. In the first
plot the results in the quark sector: in grey the bounds on the absolute value
of the combination of the effective Yukawas from Tab. 8, and in red the
prediction of the FN model assuming $\Lambda_{q}=1\ \text{TeV}$. The second
plot shows the results in the lepton sector when the combination of the
effective Yukawas do not depend on the FN charge assignment: in the dark
(light) grey the bounds on the absolute value (real part) from Tab. 9, while
the FN predictions are in red assuming $\Lambda_{\ell}=4\ \text{TeV}$. The
last plot still refers to the lepton sector and to the effective Yukawas that
do depend on the FN charge assignment: in grey there are the bounds on the
absolute value from Tab. 9, in red the Anarchical model, in green the
$A_{\mu\nu}$ model and in blue the hierarchical model for $\Lambda_{\ell}=4\
\text{TeV}$.

The corresponding bound on the $r_{\psi}^{2}$ parameters reads

$\displaystyle 0.88\lesssim r^{2}_{q}$ $\displaystyle\lesssim
1.12\qquad\text{for any quark}$ (4.4) $\displaystyle 0.99\lesssim
r^{2}_{\ell}$ $\displaystyle\lesssim 1.01\qquad\text{for any lepton}\,,$

where the lower (upper) bounds are obtained for the $\cos\theta_{\psi}=-1$
($\cos\theta_{\psi}=+1$). The values corresponding to $\cos\theta_{\psi}=0$
are $r_{q}^{2}=1.004$ and $r_{\ell}^{2}=1.00001$

The opposite case is the one with large phases of the effective third
generation Yukawas and then the strongest bounds arise from the electron EDM
data: in Fig. 4, we give the graphical representation of the experimental
bounds and the FN predictions for quarks and leptons with $\Lambda_{q}=7.4\
\text{TeV}$ and $\Lambda_{\ell}=6.0\ \text{TeV}$. As can be seen, once the
electron EDMs bounds are satisfied, the FN predictions for the other CPV
observables are well below the present experimental limits.

Figure 4: Graphical representation of the experimental bounds and FN
predictions on the effective Yukawas for the case with large CP phases. In the
first plot the results in the quark sector: in grey the bounds on the
imaginary part of the combination of the effective Yukawas from Tab. 8, and in
red the prediction of the FN model assuming $\Lambda_{q}=7.4\ \text{TeV}$. The
second plot shows the results in the lepton sector: in grey the bounds on the
Imaginary part from Tab. 9, while the FN predictions are in red assuming
$\Lambda_{\ell}=6.0\ \text{TeV}$.

The corresponding bound on the $r_{\psi}^{2}$ parameters reads

$\displaystyle 0.998\lesssim r^{2}_{q}$ $\displaystyle\lesssim
1.002\qquad\text{for any quark}$ (4.5) $\displaystyle 0.997\lesssim
r^{2}_{\ell}$ $\displaystyle\lesssim 1.003\qquad\text{for any lepton}\,,$

where the lower (upper) bounds are obtained for the $\cos\theta_{\psi}=-1$
($\cos\theta_{\psi}=+1$).

We can conclude that if the NP is approximately CP conserving then the
$q\overline{q}h$ couplings may still deviate from the SM values by a few
percent and the future collider data on the Higgs production and decay are
very much welcome. The $\ell\overline{\ell}h$ couplings are much more strongly
constrained even in the CP conserving case. An interesting chance for testing
the $\mu\overline{e}h$ couplings is provided by the $\mu\to e$ conversion in
nuclei. The present experimental bound is ${\rm BR}(\mu\to eN)<7\times
10^{-13}$. Using the scale of $\Lambda_{\ell}=4\ \text{TeV}$ and FN symmetry,
the bound we get is ${\rm BR}(\mu\to eN)<3.8\times 10^{-16}$. Future
experiments claim to reach a sensitivity of $10^{-16}$ [63], which is of the
same order of magnitude.

In the case of large CP violating phases in the Higgs couplings with $t$, $e$
and $\tau$, all the $h$ couplings are constrained to be very close to their SM
values and only higher precision selected flavour data may show the deviations
from the SM, as can be seen in Fig. 4.

##  5 Flavour Changing Tree-Level Higgs and Top Decays

This section is devoted to the analysis of processes that occur at the tree
level due to flavour changing Yukawa couplings: in particular we will focus on
flavour changing Higgs decays into leptons or quarks, and on flavour changing
top decays. Differently from the observables considered in the previous
section, these decays are safe from NP contributions different from the ones
in Eq. (2.6). Consistently with the results in the previous section, we will
adopt $\Lambda_{q}=1\ \text{TeV}$ and $\Lambda_{\ell}=4\ \text{TeV}$.

The first observables we consider are flavour violating Higgs decays into
fermions, but the top. When the final state particles are leptons, the decay
rates are quite strongly constrained experimentally. The explicit expression
for the decay width is given by [64, 65]

$\begin{split}\Gamma\left(h\to
f_{i}f_{j}\right)=&\Gamma\left(h\to\overline{f}_{i}f_{j}\right)+\Gamma\left(h\to
f_{i}\overline{f}_{j}\right)=\\\
=&\frac{N_{C}}{16\pi}m_{h}\sqrt{\left[1-\left(x_{i}+x_{j}\right)^{2}\right]\left[1-\left(x_{i}-x_{j}\right)^{2}\right]}\times\\\
&\times\Bigg{[}(1-x_{i}^{2}+x_{j}^{2})\left(\left|\hat{Y}_{f,ij}\right|^{2}+\left|\hat{Y}_{f,ji}\right|^{2}\right)-4\,x_{i}x_{j}\,{\texttt{Re}}\left(\hat{Y}_{f,ij}\hat{Y}_{f,ji}\right)\Bigg{]}\,,\end{split}$
(5.1)

for $i\neq j$, $N_{C}=3$ for quarks and $N_{C}=1$ for leptons, $x_{i}\equiv
m_{f_{i}}/m_{h}$, and where $\hat{Y}_{f}$ are defined in Eq. (2.8). From Eq.
(2.19), we get

$\begin{split}&\left|C_{f,ij}\right|^{2}\simeq\mathcal{O}(1)\,Y^{2}_{f,j}\,\epsilon^{2(n_{Q_{i}}-n_{Q_{j}})}\qquad\text{for
$f=u,d$}\,,\\\
&\left|C_{e,ij}\right|^{2}\simeq\mathcal{O}(1)\,Y^{2}_{e,j}\,\epsilon^{2(n_{L_{i}}-n_{L_{j}})}\,,\\\
&{\texttt{Re}}\left[C_{f,ij}C_{f,ji}\right]\simeq\mathcal{O}(1)\,Y_{f,i}Y_{f,j}\,\cos\left(\theta_{f,ij}+\theta_{f,ji}\right)\,,\end{split}$
(5.2)

and an explicit computation of the two terms in the last line of Eq. (5.1)
reveals that the first term is always dominating for any set
$\\{f_{i},f_{j}\\}$. Neglecting the fermion masses with respect to the Higgs
one, Eq. (5.1) reduces to

$\displaystyle\Gamma\left(h\to f_{i}f_{j}\right)$
$\displaystyle\simeq\frac{3}{8\,\pi}m_{h}\,\dfrac{v^{2}}{\Lambda^{4}}\left[\mathcal{O}(1)\,M^{2}_{f,j}\,\epsilon^{2(n_{Q_{i}}-n_{Q_{j}})}+\mathcal{O}(1)\,M^{2}_{f,i}\,\epsilon^{-2(n_{Q_{i}}-n_{Q_{j}})}\right]\quad\text{for
$f=u,d$}\,,$ $\displaystyle\Gamma\left(h\to e_{i}e_{j}\right)$
$\displaystyle\simeq\frac{1}{8\,\pi}m_{h}\,\dfrac{v^{2}}{\Lambda^{4}}\left[\mathcal{O}(1)\,M^{2}_{e,j}\,\epsilon^{2(n_{L_{i}}-n_{L_{j}})}+\mathcal{O}(1)\,M^{2}_{e,i}\,\epsilon^{-2(n_{L_{i}}-n_{L_{j}})}\right]\,.$
(5.3)

The branching ratios for these decays can be obtained dividing this last
expression by the Higgs total decay width deduced from Eq. (3.18). The results
can be read in Tab. 10 for decays in both quarks and leptons. As we can see,
the predicted values for all the decays are much below the present
experimental limit.

Notice that for the decays into quarks, the proper observables involve jets or
mesons in the final state. It is however beyond the scope of this paper to
enter into further details.

BR | Experimental Bound 95%C.L. | FN Prediction
---|---|---
$h\to uc$ | $-$ | $6\times 10^{-8}$
$h\to ds$ | $-$ | $6\times 10^{-10}$
$h\to db$ | $-$ | $4\times 10^{-8}$
$h\to sb$ | $-$ | $8\times 10^{-7}$
BR | Experimental Bound 95%C.L. | $A$ | $A_{\mu\tau}$ | $H$
$h\to e\mu$ | $6.1\times 10^{-5}\text{\leavevmode\nobreak\ \cite[cite]{[\@@bibref{}{ATLAS:2019old}{}{}]}}$ | $3\times 10^{-9}$ | $10^{-10}$ | $1\times 10^{-10}$
$h\to e\tau$ | $4.7\times 10^{-3}\text{\leavevmode\nobreak\ \cite[cite]{[\@@bibref{}{ATLAS:2019pmk}{}{}]}}$ | $8\times 10^{-7}$ | $4\times 10^{-8}$ | $2\times 10^{-9}$
$h\to\mu\tau$ | $2.5\times 10^{-3}\text{\leavevmode\nobreak\ \cite[cite]{[\@@bibref{}{CMS:2017con}{}{}]}}$ | $8\times 10^{-7}$ | $8\times 10^{-7}$ | $9\times 10^{-8}$
Table 10: Branching ratios for the flavour changing Higgs decays. In the
second column we report the experimental upper bounds when available, while in
the other columns the FN predictions, specifying the leptonic FN realisation.
The cut-off scales are fixed to $\Lambda_{q}=1\ \text{TeV}$ and
$\Lambda_{\ell}=4\ \text{TeV}$ and all the $\mathcal{O}(1)$ parameters are
taken equal to $1$.

The last observable we will consider in this section is flavour changing top
decays, $t\to hc$ and $t\to hu$. The explicit expression for the decay width
is given by [69]

$\Gamma\left(t\to
hu_{i}\right)=\frac{1}{32\pi}m_{h}\left|\hat{Y}_{u,i3}\right|^{2}\left[\left(1+x_{i}\right)^{2}-x_{h}^{2}\right]\sqrt{1-(x_{h}+x_{i})^{2}}\sqrt{1-(x_{h}-x_{i})^{2}}\,,$
(5.4)

where $x_{i}\equiv m_{u_{i}}/m_{t}$ and $x_{h}\equiv m_{h}/m_{t}$. Considering
the definition of $\hat{Y}_{u}$ and the expression for $C_{u,ij}$ in Eq.
(5.2), we get

$\left|\hat{Y}_{u,i3}\right|^{2}\simeq\mathcal{O}(1)\dfrac{2m_{t}^{2}v^{2}}{\Lambda_{q}^{4}}\epsilon^{2(n_{Q_{i}}-n_{Q_{3}})}$
(5.5)

and the generic expression, neglecting $x_{i}$ with respect to $x_{h}$,
reduces to

$\Gamma\left(t\to
hu_{i}\right)\simeq\frac{\mathcal{O}(1)}{16\pi}m_{h}\dfrac{m_{t}^{2}v^{2}}{\Lambda_{q}^{4}}\epsilon^{2(n_{Q_{i}}-n_{Q_{3}})}\left(1-x_{h}^{2}\right)^{2}\,.$
(5.6)

The numerical results for the corresponding branching ratios, considering the
total decay width $\Gamma_{t}=1.42^{+19}_{-15}\ \text{GeV}$ [27], can be read
in Tab. 11: the FN predictions are below the present experimental limits of at
least two orders of magnitude.

BR | Experimental Bound 95%C.L. | FN Prediction
---|---|---
$t\to hu$ | $4.5\times 10^{-3}\text{\leavevmode\nobreak\ \cite[cite]{[\@@bibref{}{ATLAS:2015ncl}{}{}]}}$ | $2\times 10^{-6}$
$t\to hc$ |  $4.6\times 10^{-3}\text{\leavevmode\nobreak\ \cite[cite]{[\@@bibref{}{ATLAS:2015ncl}{}{}]}}$, $5.6\times 10^{-3}\text{\leavevmode\nobreak\ \cite[cite]{[\@@bibref{}{CMS:2014qxa}{}{}]}}$ | $4\times 10^{-5}$
Table 11: Branching ratios for the flavour changing top decays. In the second
column we report the experimental upper bounds, while in the third column the
FN predictions. The cut-off scale is fixed to $\Lambda_{q}=1\ \text{TeV}$ and
all the $\mathcal{O}(1)$ parameters are taken equal to $1$.

##  6 Conclusions

In contrast to the electroweak part of the Standard Model, beautifully
understood in terms of the underlying gauge symmetries, its flavour sector is
very mysterious. It relies on purely phenomenological description encoded in
the fermion masses and mixings, which originate from the Yukawa coupling
matrices introduced at the level of the $SU(2)_{L}\times U(1)_{Y}$ invariant
renormalisable Lagrangian, with the SM fermion spectrum. The latter are
however beyond any SM theoretical control. Better theoretical understanding of
the flavour physics is one of the main BSM challenges, parallel to the
naturalness problem. It may well be that the scales of BSM physics that can
shed some light on the flavour problem are very different from the BSM physics
linked to the Brout-Englert-Higgs mechanism.

In this paper we have determined the lower bounds on the scale of new physics
possibly contributing to the $f\overline{f}h$ effective couplings following
from the variety of processes dependent on those couplings. We have worked in
the framework of SMEFT and our main assumption is that the Wilson coefficients
of the relevant dim 6 operators respect certain flavour structure: either the
Minimal Flavour Violation ansatz or a flavour symmetry, often invoked to
explain the observed pattern of fermion masses and mixings. Thus, our results
determine the sensitivity of different observables to the flavour BSM physics
under the mentioned above assumption.

There are several sources of experimental information on the effective Yukawa
couplings. The absolute values of the third generation couplings (and some of
the second one) are directly measured in the Higgs boson production and decays
at the LHC experiments. Their imaginary parts are bounded by the limits on
EDMs. The non-diagonal couplings are bounded by a vast number of FCNC
processes and radiative decays. The latter bounds are conservative as those
processes may be subject also to other new physics contributions at the same
level of precision calculations but we take them on equal footing with the
collider bounds.

The emerging picture of the bounds on new physics scales is discussed in
detail in Sect. 4, while the predictions for flavour changing Higgs and top
decays in Sect. 5. Here are the main points:

  * -

First, we disregard the bounds on the imaginary parts of the Yukawas (assuming
that they are controlled by approximately CP conserving BSM theory). With the
present precision, the FCNC data give stronger bounds on the scale of NP than
the collider data (obviously this does not apply to the MFV ansatz, where the
off-diagonal couplings are absent and only collider data are relevant). We
have then $\Lambda_{q}=1(0.6)$ TeV and $\Lambda_{\ell}=4(0.4)$ TeV for
FCNC(collider) data.

  * -

The low FCNC bound for quarks is consistent with a few percent deviations from
the SM in the Yukawa couplings and high precision collider data may show such
deviations. For leptons, the FCNC bound is stronger. Its violation by the
collider data would mean the violation of the flavour symmetry or the presence
of additional NP affecting the flavour observables that leads to
cancellations.

  * -

Flavour changing Higgs and top decays are predicted to be at least two orders
of magnitude below the present experimental limits. In this case, a violation
of those bounds would univocally mean the violation of the flavour symmetry.

  * -

A promising observable is the $\mu\to e$ conversion in nuclei. Future
experiments claim to reach a sensitivity of $10^{-16}$ [63], which is of the
same order of magnitude as our predicted upper bound.

  * -

In the extreme case, if the phases in $\tilde{\kappa}_{t,e,\tau}$ are maximal,
then $\Lambda_{q}=7.4\ \text{TeV}$ and $\Lambda_{\ell}=6\ \text{TeV}$ from the
upper limit on the electron EDM and only selected CP violating observables may
show some deviations from the SM predictions, provided the precision of the
data improves by an order of magnitude (see Fig. 4).

## Acknowledgements

J.A.G., A.d.G. and L.M. acknowledge partial financial support by the Spanish
MINECO through the Centro de excelencia Severo Ochoa Program under grant
SEV-2016-0597, by the Spanish “Agencia Estatal de Investigacíon”(AEI) and the
EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project
PID2019-108892RB-I00/AEI/ 10.13039/ 501100011033, by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 860881-HIDDeN. L.M. acknowledges partial financial
support by the Spanish MINECO through the “Ramón y Cajal” programme
(RYC-2015-17173). A.d.G. acknowledges financial support by the ETH Zürich and
the Institut für Theoretische Physik. The research of S.P. has received
funding from the Norwegian Financial Mechanism for years 2014-2021, grant nr
2019/34/H/ST2/00707. S.P thanks Anna Lipniacka (University of Bergen) for
discussions on the Higgs boson LHC data.

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