# Composite Flavon-Higgs Models

Yi Chung<EMAIL_ADDRESS>Center for Quantum Mathematics and Physics
(QMAP), Department of Physics,
University of California, Davis, CA 95616, U.S.A.

###### Abstract

We consider a composite Higgs model based on the $SU(6)/Sp(6)$ coset, where an
$U(1)$ subgroup of $Sp(6)$ is identified as the flavor symmetry. A complex
scalar field $s$, which is a pseudo-Nambu-Goldstone boson of the broken
symmetry, carries a flavor charge and plays the role of a flavon field. The
$U(1)_{F}$ flavor symmetry is then broken by a VEV of the flavon field, which
leads to a small parameter and generates the mass hierarchy between the top
and bottom quarks. A light flavon below the TeV scale can be naturally
introduced, which provides a fully testable model for the origin of flavor
hierarchy. A light flavon also leads to substantial flavor changing neutral
currents, which are strongly constrained by the flavor experiments. The direct
search of additional scalar bosons can also be conducted in HL-LHC and future
hadron colliders.

##  I Introduction

The Standard Model (SM) of particle physics successfully describes all known
elementary particles and their interactions. However, there are still a few
puzzles that have yet to be understood, including two mysterious hierarchies.
One is the well-known hierarchy problem. With the discovery of light Higgs
bosons in 2012 [1, 2], the last missing piece of the SM seemed to be filled,
but SM does not address the UV-sensitive nature of scalar bosons. The Higgs
mass-squared receives quadratically divergent radiative corrections from the
interactions with SM fields, which require an extremely sensitive cancellation
to have a $125$ GeV Higgs boson. The other puzzle is related to the large
hierarchies in the masses and mixings of the SM fermions. Even within the
quark sector, the masses of quarks span over six orders of magnitude. The
mixing angles also show a hierarchical structure. The problem is known as the
flavor puzzle [3], which represents the mysterious structure of SM Yukawa
couplings.

One such appealing solution to the hierarchy problem is the composite Higgs
model (CHM), where the Higgs doublet is the pseudo-Nambu-Goldstone bosons
(pNGB) of a spontaneously broken global symmetry of the underlying strong
dynamics [4, 5]. Through the analogy of the chiral symmetry breaking in
quantum chromodynamics (QCD), which naturally introduces light scalar fields,
i.e., pions, we can construct models with light Higgs bosons. In a CHM, an
approximate global symmetry $G$ is spontaneously broken by some strong
dynamics down to a subgroup $H$ with a symmetry breaking scale $f$. The heavy
resonances of the strong dynamics are expected to be around the compositeness
scale $\sim 4\pi f$. The pNGBs of the symmetry breaking, on the other hand,
can naturally be light with masses $<f$ as they are protected by the shift
symmetry.

For the flavor puzzle, the hierarchy in the masses and mixings of the SM
fermions can be achieved by assuming an abelian $U(1)_{F}$ flavor symmetry
[6], where different SM fermions carry different charges. The low-energy
effective Yukawa coupling terms require the insertion of additional scalar
fields as

$\mathcal{L}_{\text{Yukawa}}=y_{ij}\left(\frac{s}{\Lambda_{F}}\right)^{a_{ij}}\bar{q}_{L,i}{H}q_{R,j},$
(1)

where $y_{ij}$ is a $\mathcal{O}(1)$ coupling, the complex scalar field $s$ is
called flavon field, and ${\Lambda_{F}}$ is the scale of flavor dynamics.
After the flavon field acquires a VEV, it will lead to a small parameter
$\epsilon=\langle s\rangle/\Lambda_{F}$ and result in the hierarchy of SM
Yukawa couplings. It is known as the Froggatt-Nielsen (FN) mechanism. Despite
the success in explaining the flavor structure, the scale of flavor dynamics
is not predicted and can be arbitrarily high. Also, the flavon as a scalar
boson receives large radiative corrections from the interactions with SM
fields and is expected to be well beyond the collider search.

In this paper, we explore models that can address these two problems at once
and provide predictive experimental signatures which can be probed by
colliders. We choose the specific CHMs with the unbroken subgroup large enough
to include the $U(1)_{F}$ symmetry. That is, the flavor symmetry arises as
part of the accidental global symmetry of the strong dynamics. Under this
construction, the Higgs doublet and the flavon are pNGBs of the spontaneously
broken global symmetries. In this case, the hierarchy problem is relieved, and
a light flavon is naturally introduced, which provides a testable theory for
the origin of flavor hierarchy.

Efforts to generate flavons as pNGBs have been implemented in the little
flavon model [7, 8], which is aimed at realizing collective symmetry breaking
on the flavon field. Versions combined with Higgs doublet were also studied
[9, 10], but the large symmetry group makes them uncompelling. They also
failed to treat the generation of Yukawa coupling carefully. Other attempts
aiming at generating the Higgs and flavon from a common source have been
studied recently [11], inspired by axiflavon models [12, 13]. However, the
scalar flavon in the model is not the pNGB mode but the heavy unstable radial
mode, which is hard to be detected, and the FN fields are elementary vector-
like fermions added by hands. There are also other efforts to relate the
flavor breaking scale to the electroweak scale but within the framework of
2HDM [14, 15].

For a concrete model, we consider a composite Higgs model based on the
$SU(6)/Sp(6)$ coset, where the unbroken $Sp(6)$ is large enough to include
both the SM gauge group and the global flavor symmetry group $SU(2)_{W}\times
U(1)_{Y}\times U(1)_{F}$. The flavons as well as two Higgs doublets are the
pNGBs of the coset. We then show how a suppressed Yukawa coupling can be
generated through partial compositeness with specific flavor charge
assignments. We discuss different scenarios to realize the Froggatt-Nielsen
mechanism and generate the top-bottom mass hierarchy. The experimental
constraints of different cases will also be discussed.

##  II The $SU(6)/Sp(6)$ CHM

The $SU(6)/Sp(6)$ coset is one of the earliest cosets employed in little Higgs
models [16] where the collective symmetry breaking for the quartic term was
realized. Recently, it was considered for dark matter study [17] and natural
Higgs potential [18]. It was pointed out in [18] that there is a $U(1)$
Peccei-Quinn like subgroup [19], which protects the theory from dangerous
tadpole terms and flavor changing neutral currents. In this paper, this
subgroup is identified as $U(1)_{F}$ flavor symmetry to realize the Froggatt-
Nielsen mechanism. For our purpose, we will focus on the fermion sector and
Yukawa couplings in the main text. The gauge sector and the pNGB potential are
discussed in the appendix. A more comprehensive discussion on these topics can
also be found in [18].

###  II.1 Basics of $SU(6)/Sp(6)$

The $SU(6)/Sp(6)$ non-linear sigma model can be parametrized by a sigma field
$\Sigma^{ij}$, which transforms as an anti-symmetric tensor representation
$\mathbf{15}$ of $SU(6)$, where $i,j=1,\ldots 6$ are $SU(6)$ indices. The
transformation under $SU(6)$ can be expressed as $\Sigma\to g\,\Sigma\,g^{T}$
with $g\in SU(6)$ or as
$\Sigma^{ij}\to{g^{i}}_{k}{g^{j}}_{\ell}\Sigma^{k\ell}$ with indices
explicitly written out. The scalar field $\Sigma$ has an anti-symmetric VEV
$\langle\Sigma\rangle=\Sigma_{0}^{\alpha\beta}$ (with $\alpha$, $\beta$
representing $Sp(6)$ indices), where

$\Sigma_{0}=\begin{pmatrix}0&-\mathbb{I}_{3\times 3}\\\ \mathbb{I}_{3\times
3}&0\\\ \end{pmatrix}.$ (2)

The $\Sigma$ VEV breaks $SU(6)$ down to $Sp(6)$, producing 14 Nambu-Goldstone
bosons.

The 35 $SU(6)$ generators can be divided into unbroken ones and broken ones
with each type satisfying

$\begin{cases}\text{unbroken
generators}&~{}T_{a}:T_{a}\Sigma_{0}+\Sigma_{0}T_{a}^{T}=0~{},\\\ \text{broken
generators}&X_{a}:X_{a}\Sigma_{0}-\Sigma_{0}X_{a}^{T}=0~{}.\end{cases}$ (3)

The Nambu-Goldstone fields can be written as a matrix with the broken
generators:

$\xi(x)={\xi^{i}}_{\alpha}(x)\equiv e^{\frac{i\pi_{a}(x)X_{a}}{2f}}.$ (4)

Under $SU(6)$, the $\xi$ field transforms as $\xi\to g\,\xi\,h^{\dagger}$
where $g\in SU(6)$ and $h\in Sp(6)$, so $\xi$ carries one $SU(6)$ index and
one $Sp(6)$ index. The relation between $\xi$ and $\Sigma$ field is given by

$\Sigma(x)=\Sigma^{ij}(x)\equiv\xi\,\Sigma_{0}\,\xi^{T}=e^{\frac{i\pi_{a}(x)X_{a}}{f}}\Sigma_{0}~{}.$
(5)

The complex conjugation raises or lowers the indices. The fundamental
representation of $Sp(6)$ is (pseudo-)real and the $Sp(6)$ index can be raised
or lowered by $\Sigma_{0}^{\alpha\beta}$ or $\Sigma_{0,\alpha\beta}$.

The broken generators and the corresponding fields in the matrix can be
organized as follows ($\epsilon=i\sigma^{2}$):

$\pi_{a}X_{a}=\begin{pmatrix}\frac{\phi_{a}}{\sqrt{2}}\sigma^{a}-\frac{\eta}{\sqrt{6}}\bf{1}&~{}H_{2}&~{}\epsilon
s&~{}H_{1}\\\ H_{2}^{\dagger}&~{}\frac{2\eta}{\sqrt{6}}&~{}-H_{1}^{T}&~{}0\\\
\epsilon^{T}s^{*}&~{}-H_{1}^{*}&~{}\frac{\phi_{a}}{\sqrt{2}}\sigma^{a*}-\frac{\eta}{\sqrt{6}}\bf{1}&~{}H_{2}^{*}\\\
H_{1}^{\dagger}&~{}0&~{}H_{2}^{T}&~{}\frac{2\eta}{\sqrt{6}}\\\ \end{pmatrix}.$
(6)

In this matrix, there are 14 independent fields. They are (under $SU(2)_{W}$):
a real triplet $\phi_{a}$, a real singlet $\eta$, a complex singlet $s$ (as
the flavon field), and two Higgs (complex) doublets $H_{1}$ and $H_{2}$. We
effectively end up with a two-Higgs-doublet model (2HDM). The observed Higgs
boson will correspond to a mixture of $h_{1}$ and $h_{2}$ inside two Higgs
doublets
$H_{1}=H_{1/2}\supset\frac{1}{\sqrt{2}}\bigl{(}\begin{smallmatrix}0\\\
h_{1}\end{smallmatrix}\bigr{)}$ and
$H_{2}=H_{-1/2}\supset\frac{1}{\sqrt{2}}\bigl{(}\begin{smallmatrix}h_{2}\\\
0\end{smallmatrix}\bigr{)}$. Using the $\xi$ and $\Sigma$ matrices, we can
construct the low energy effective Lagrangian for the flavon field, the Higgs
fields, and all the other pNGBs.

###  II.2 Unbroken subgroups of $Sp(6)$

To realize the FN mechanism, we need a global symmetry with scalars and
fermions charged under it. Within the $Sp(6)$ symmetry, there are several
unbroken $U(1)$ symmetries. The symmetries with generators

$\displaystyle\frac{1}{2}$ $\displaystyle\begin{pmatrix}\sigma^{a}&0&0&0\\\
0&0&0&0\\\ 0&0&-\sigma^{a*}&0\\\ 0&0&0&0\\\ \end{pmatrix}\text{ and
}\frac{1}{2}\begin{pmatrix}0_{2\times 2}&0&0&0\\\ 0&1&0&0\\\ 0&0&0_{2\times
2}&0\\\ 0&0&0&-1\\\ \end{pmatrix}+X\mathbf{I}$

are identified as the SM gauge group $SU(2)_{W}$ and $U(1)_{Y}$, which are
discussed in appendix A.

Besides the SM gauge group, there is one more $U(1)_{F}$ global symmetry with
the generator

$U(1)_{F}:\frac{1}{2}\begin{pmatrix}\mathbb{I}_{2\times 2}&0&0&0\\\ 0&0&0&0\\\
0&0&-\mathbb{I}_{2\times 2}&0\\\ 0&0&0&0\\\ \end{pmatrix}.$ (7)

Under $U(1)_{F}$, the complex scalar field $s$ has charge 1, both Higgs
doublets $H$ have charge 1/2, and other pNGB fields have charge 0. The complex
singlet $s$ can then be identified as the composite flavon field. We then get
the charge assignment for all pNGBs as

$s:1,\quad H_{1},~{}H_{2}:1/2,\quad\phi,~{}\eta:0~{},$ (8)

which is a little different from the normal FN mechanism since Higgs also
carries flavor charges 111In fact, this global symmetry is more similar to the
$U(1)$ Peccei-Quinn (PQ) symmetry [19]. Models that identify $U(1)_{PQ}$ as
flavor symmetry had been studied in axiflavon models [12, 13]. However, in
this paper, we will not deal with the strong CP problem and axions, so we
would like to call it $U(1)_{F}$ flavor symmetry.. So far, we get the desired
scalar sector with the flavon and Higgs doublets. We can then move on to the
fermion sector.

##  III Yukawa coupling

In CHMs, the SM Yukawa couplings can arise from the partial compositeness
mechanism [21]. That is, elementary fermions mix with composite operators of
the same SM quantum numbers from the strong dynamics,

$\mathcal{L}=\lambda_{L}\bar{q}_{L}{O}_{R}+\lambda_{R}\bar{q}_{R}{O}_{L},$ (9)

where $q_{L}$, $q_{R}$ are elementary fermions and ${O}_{L}$, ${O}_{R}$ are
composite operators of some representations of $SU(6)$.

To be able to mix with the elementary fermions, the representations of the
composite operators must contain states with the same SM quantum numbers as
the SM fermions. To account for the correct hypercharge, e.g., $q_{L}=2_{1/6}$
for left-handed quarks, $q_{R}=1_{2/3}$ for right-handed up-type quarks, and
$q_{R}=1_{-1/3}$ for right-handed down-type quarks, the composite operators
need to carry additional charges under the $U(1)_{X}$ outside $SU(6)$, and the
SM hypercharge is a linear combination of the $SU(6)$ generator
$\text{Diag}(0,0,1/2,0,0,-1/2)$ and $X$.

Let us start with the top quark. To get the top Yukawa coupling, the suitable
and economical choice of composite operators is $\mathbf{6}$ with $X=1/6$. The
composite operator as a $\mathbf{6}_{1/6}$ of $SU(6)$ (where the subscript
$1/6$ denotes its $U(1)_{X}$ charge) can be decomposed under the SM gauge
group as

$O_{L,R}^{i}\sim{\xi^{i}}_{\alpha}Q_{L,R}^{\alpha}\sim\mathbf{6}_{1/6}=\mathbf{2}_{1/6}\oplus\mathbf{1}_{2/3}\oplus\mathbf{\bar{2}}_{1/6}\oplus\mathbf{1}_{-1/3},$
(10)

where $Q_{L,R}$ are the corresponding composite resonances. The composite
states $Q_{L,R}$ belong to the $\mathbf{6}$ representations of $Sp(6)$ and
play the roles of SM fermion composite partners. For $SU(2)$, $\mathbf{2}$ and
$\mathbf{\bar{2}}$ are equivalent and related by the $\epsilon$ tensor. We
make the distinction to keep track of the order of the fermions in a doublet.
We see that the composite states have the appropriate quantum numbers to mix
with the SM quarks.

The left-handed top quark can mix with the first two components of the sextet.
The mixing term can be express as

$\lambda_{L}\bar{q}_{La}{\Lambda^{a}}_{i}O_{R}^{i}=\lambda_{L}\bar{q}_{La}{\Lambda^{a}}_{i}\left({\xi^{i}}_{\alpha}Q_{R}^{\alpha}\right),$
(11)

where $a$ represents an $SU(2)_{W}$ index, and

${(\Lambda)^{a}}_{i}=\Lambda=\begin{pmatrix}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
\end{pmatrix}$ (12)

is the spurion which keeps track of the symmetry breaking.

To get the complete top Yukawa coupling, we couple the elementary right-handed
top quark to the $\mathbf{\bar{6}}_{1/6}$, which decomposes under
$SU(2)_{W}\times U(1)_{Y}$ as

$O^{\prime}_{L,Rj}\sim{{\xi^{*}}_{j}}^{\beta}\Sigma_{0\beta\alpha}Q_{L,R}^{\alpha}\sim\mathbf{\bar{6}}_{1/6}=\mathbf{\bar{2}}_{1/6}\oplus\mathbf{1}_{-1/3}\oplus\mathbf{2}_{1/6}\oplus\mathbf{1}_{2/3}~{}.$
(13)

The right-handed top quark mixes with the last component of the
$\mathbf{\bar{6}}_{1/6}$, which can be written as

$\lambda_{t_{R}}\bar{t}_{R}{\Gamma_{t_{R}}}^{j}O^{\prime}_{Lj}=\lambda_{t_{R}}\bar{t}_{R}{\Gamma_{t_{R}}}^{j}\left({{\xi^{*}}_{j}}^{\beta}\Sigma_{0\beta\alpha}Q_{L}^{\alpha}\right),$
(14)

where $\Gamma_{t_{R}}=\left(0~{}0~{}0~{}0~{}0~{}1\right)$ is the corresponding
spurion.

Combining $\lambda_{L}$ and $\lambda_{t_{R}}$ couplings, we can generate the
SM Yukawa coupling for the top quark

$\displaystyle\sim\lambda_{L}\lambda_{t_{R}}\bar{q}_{La}{\Lambda^{a}}_{i}{\xi^{i}}_{\alpha}\Sigma_{0}^{\alpha\beta}{{\xi^{T}}_{\beta}}^{j}{\Gamma}^{\dagger}_{t_{R}j}t_{R}\supset\lambda_{L}\lambda_{t_{R}}\left(\bar{q}_{L}H_{2}t_{R}\right)~{}.$
(15)

The top quark gets its mass from the vacuum of $H_{2}$ as

$\displaystyle
m_{t}=\frac{\lambda_{L}\lambda_{t_{R}}}{g_{T}}\frac{v_{2}}{\sqrt{2}}~{},$ (16)

where $g_{T}$ is a coupling of the composite top partners.

Similarly, for the bottom quark, we can couple $b_{R}$ to the third component
of $\mathbf{\bar{6}}_{1/6}$ with the coupling $\lambda_{b_{R}}$ and spurion
$\Gamma_{b_{R}}=\left(0~{}0~{}1~{}0~{}0~{}0\right)$. This generates a bottom
Yukawa coupling

$\displaystyle\sim\lambda_{L}\lambda_{b_{R}}\bar{q}_{La}{\Lambda^{a}}_{i}{\xi^{i}}_{\alpha}\Sigma_{0}^{\alpha\beta}{{\xi^{T}}_{\beta}}^{j}{\Gamma}^{\dagger}_{b_{R}j}b_{R}\supset\lambda_{L}\lambda_{b_{R}}\left(\bar{q}_{L}H_{1}b_{R}\right)~{},$
(17)

where the bottom quark gets its mass from the vacuum of $H_{1}$ instead.

In this paper, we will not address the lepton sector, so there are only two
types of 2HDMs satisfying the natural flavor conservation [22, 23]. They are
categorized by Type-I and Type-II based on the Yukawa couplings of the quarks.
So far, the Yukawa couplings of the third generation quarks come from
different Higgs doublets, which implies a Type-II 2HDM. The smallness of the
bottom quark mass can be achieved by a small VEV of $H_{1}$, i.e. a large
tan$\beta$ Type-II 2HDM. However, the parameter space with a large tan$\beta$
is strongly constrained by direct searches, and it is also not what we want.
To get mass hierarchy between the top and bottom through the FN mechanism, we
want an insertion of the flavon field $s$ in these Yukawa coupling terms.

##  IV Froggatt-Nielsen mechanism

###  IV.1 FN mechanism: The first taste

Before we move on to the correct FN mechanism setup, let us first look at the
flavor charges of quarks. In the previous section, all the quarks are embedded
in $\mathbf{{6}}_{1/6}$ and $\mathbf{\bar{6}}_{1/6}$ of $SU(6)$ without
additional flavor charges, which are decomposed under $SU(2)_{W}\times
U(1)_{F}$ as

$\displaystyle\mathbf{{6}}_{0}$
$\displaystyle=\mathbf{{2}}_{1/2}\oplus\mathbf{1}_{0}\oplus\mathbf{\bar{2}}_{-1/2}\oplus\mathbf{1}_{0}~{},$
(18) $\displaystyle\mathbf{\bar{6}}_{0}$
$\displaystyle=\mathbf{\bar{2}}_{-1/2}\oplus\mathbf{1}_{0}\oplus\mathbf{2}_{1/2}\oplus\mathbf{1}_{0}~{}.$
(19)

It means that the flavor charges of fermions are set as

$q_{L}=(t_{L},~{}b_{L})^{T}:1/2,\quad t_{R},~{}b_{R}:0~{},$ (20)

where both right-handed quarks have no flavor charge.

Within this assignment, we can already write down a suppressed bottom quark
mass through the FN mechanism by the term like

$\displaystyle\frac{1}{f}\left(\bar{q}_{L}s\tilde{H}_{2}b_{R}\right)\sim\left(\frac{v_{s}v_{2}}{2f}\right)\bar{b}_{L}b_{R}~{},$
(21)

where $v_{s}$ is the VEV of the flavon field. The term satisfies the flavor
symmetry. The reason it is possible is that the top quark gets mass from ${H}$
with flavor charge 1/2, but the bottom quark can get mass from $\tilde{H}$
with flavor charge $-1/2$. However, it turns out that this term can not
successfully realize the FN mechanism in this model.

To see that, we can go back to the term we derived for the bottom quark mass
in Eq. (17). In the non-linear Sigma model, if we expand the $\Sigma$ field to
the next order, it becomes

$\displaystyle\bar{q}_{L}(H_{1}+\frac{i}{2f}s\tilde{H_{2}})b_{R}\supset\frac{i}{2f}\left(\bar{q}_{L}s\tilde{H}_{2}b_{R}\right)~{},$
(22)

which already contains the term in Eq. (21). That means, due to the shift
symmetry of pNGBs, the term $s\tilde{H}_{2}$ can only show up following
$H_{1}$. That also means we can always transfer the nontrivial vacuum of
$\langle s\tilde{H}_{2}\rangle$ to the leading order $\langle H_{1}\rangle$ by
shift symmetry. Therefore, the bottom quark mass still comes from $\langle
H_{1}\rangle$, and it is equivalent to the Type-II 2HDM we have already
gotten.

If we define $\Delta_{f}\equiv[f_{L}]-[f_{R}]$ as the difference between
flavor charges of left-handed and right-handed fermions. Fixing the top quark
charge as in Eq. (20) with $\Delta_{t}=1/2$, we find that $\Delta_{b}=1/2$
gives us the bottom quark mass through $H_{1}$, which leads to a Type-II 2HDM.
$\Delta_{b}=-1/2$, instead, generates the bottom quark mass through
$\tilde{H}_{2}$ and makes it a Type-I 2HDM. Either case is just normal 2HDM.
To realize the FN mechanism, we need to have a larger $|\Delta_{b}|$, which
would allow us to generate the bottom Yukawa coupling term with the insertion
of two pNGBs, $s$ and $H$, at the same time. That also requires us to embed
the bottom quark into a larger representation, which will generate a term with
the insertion of two $\Sigma$ fields.

###  IV.2 Antisymmetric tensor representation $\mathbf{15}$ and
$\mathbf{\bar{15}}$

The minimal choice is to have a bit larger $|\Delta_{b}|=3/2$. There are two
cases, case (1) with $\Delta_{b}=3/2$ and case (2) with $\Delta_{b}=-3/2$. By
analyzing the quantum numbers, we expect to generate bottom Yukawa coupling
terms as

$\displaystyle(1)~{}\bar{q}_{L}s{H}_{1}b_{R}\quad\text{and}\quad(2)~{}\bar{q}_{L}s^{*}\tilde{H}_{2}b_{R}.$
(23)

To realize such $|\Delta_{b}|$, the minimal choice is to use antisymmetric
tensor representation $\mathbf{15}$ and $\mathbf{\bar{15}}$. To mix the SM
quarks with composite operators, we first analyze their SM quantum numbers. To
have operators sharing the same quantum numbers with the SM quarks, additional
gauge $U(1)_{X}$ and global $U(1)_{R}$ are required. With additional $x$ and
$r$ charges, the representation $\mathbf{15}_{x,r}$ can be decomposed under
$SU(2)_{W}\times U(1)_{Y}\times U(1)_{F}$ as

$\displaystyle\mathbf{15}_{x,r}=$ $\displaystyle\mathbf{(3\oplus
1)}_{x,r}\oplus\mathbf{2}_{x+\frac{1}{2},r+\frac{1}{2}}\oplus\mathbf{2}_{x-\frac{1}{2},r+\frac{1}{2}}\oplus\mathbf{\bar{2}}_{x+\frac{1}{2},r-\frac{1}{2}}$
$\displaystyle\oplus\mathbf{\bar{2}}_{x-\frac{1}{2},r-\frac{1}{2}}\oplus\mathbf{1}_{x,r+1}\oplus\mathbf{1}_{x,r}\oplus\mathbf{1}_{x,r-1},$
(24)

where the first subscript denotes its hypercharge and the second subscript
denotes its flavor charge. Or we can write them in matrix form as

$\displaystyle\mathbf{15}_{x,r}=\begin{pmatrix}1_{x,r+1}&2_{x+\frac{1}{2},r+\frac{1}{2}}&(3\oplus
1)_{x,r}&2_{x-\frac{1}{2},r+\frac{1}{2}}\\\
\cdot&0&\bar{2}_{x+\frac{1}{2},r-\frac{1}{2}}&1_{x,r}\\\
\cdot&\cdot&1_{x,r-1}&\bar{2}_{x-\frac{1}{2},r-\frac{1}{2}}\\\
\cdot&\cdot&\cdot&0\\\ \end{pmatrix},$ (25)

and also for it complex conjugate $\mathbf{\bar{15}}_{x,r}$ as

$\displaystyle\mathbf{\bar{15}}_{x,r}=\begin{pmatrix}1_{x,r-1}&\bar{2}_{x-\frac{1}{2},r-\frac{1}{2}}&(3\oplus
1)_{x,r}&\bar{2}_{x+\frac{1}{2},r-\frac{1}{2}}\\\
\cdot&0&{2}_{x-\frac{1}{2},r+\frac{1}{2}}&1_{x,r}\\\
\cdot&\cdot&1_{x,r+1}&{2}_{x+\frac{1}{2},r+\frac{1}{2}}\\\
\cdot&\cdot&\cdot&0\\\ \end{pmatrix}.$ (26)

Since they are antisymmetric, we only put the numbers on the up-right triangle
for simplicity.

###  IV.3 Two ways to embed the bottom quark

Next, we want to mix the left-handed bottom quark with $\mathbf{15}$ and the
right-handed bottom quark with $\mathbf{\bar{15}}$. The goal is to find a pair
with $|\Delta_{b}|=3/2$. From the previous decomposition, we found two pairs
that satisfy our requirement:

$\left(2_{x+\frac{1}{2},r+\frac{1}{2}},1_{x,r-1}\right)\quad\text{and}\quad\left(\bar{2}_{x+\frac{1}{2},r-\frac{1}{2}},1_{x,r+1}\right),$
(27)

which correspond to case (1) with $\Delta_{b}=3/2$ and case (2) with
$\Delta_{b}=-3/2$ respectively.

Let us start with case (1) by taking the first pair with $x=-1/3$ and $r=0$.
Just as we have done before, we first write down the composite operators and
the corresponding composite resonances as

$O_{L,R}^{ij}\sim{\xi^{i}}_{\alpha}{\xi^{j}}_{\beta}Q_{L,R}^{\alpha\beta}\sim\mathbf{15}_{-\frac{1}{3},0}=\mathbf{14}_{-\frac{1}{3},0}\oplus\mathbf{1}_{-\frac{1}{3},0},$
(28)

where $Q_{L,R}$ are the corresponding composite resonances. $Q_{L,R}$ are
$\mathbf{14}$ and $\mathbf{1}$ of $Sp(6)$ and play the roles of the SM fermion
composite partners.

The mixing term for the left-handed quark can be expressed as

$\lambda_{b_{L}}\bar{q}_{La}{\Lambda^{a}}_{ij}O_{R}^{ij}=\lambda_{b_{L}}\bar{q}_{La}{\Lambda^{a}}_{ij}\left({\xi^{i}}_{\alpha}{\xi^{j}}_{\beta}Q_{L,R}^{\alpha\beta}\right),$
(29)

where

${(\Lambda)^{a}}_{ij}=(\begin{pmatrix}0&0&1&0&0&0\\\ 0&0&0&0&0&0\\\
-1&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
\end{pmatrix},\begin{pmatrix}0&0&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&-1&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ \end{pmatrix})$ (30)

is the spurion that can help us keep track of symmetry breaking.

We still need to mix the right-handed bottom quark with the composite
operators and the corresponding composite resonances as
$O^{\prime}_{L,Rij}\sim{{\xi^{*}}_{i}}^{\alpha}{{\xi^{*}}_{j}}^{\beta}\Sigma_{0\alpha\rho}\Sigma_{0\beta\sigma}Q_{L,R}^{\rho\sigma}\sim\mathbf{\bar{15}}_{-\frac{1}{3},0}.$
The right-handed bottom quark need to mix with the
$\mathbf{1}_{-\frac{1}{3},-1}$ of the $\mathbf{\bar{15}}_{-\frac{1}{3},0}$,
which can be written as

$\lambda_{b_{R}}\bar{b}_{R}{\Gamma}^{ij}O^{\prime}_{Lij}=\lambda_{b_{R}}\bar{b}_{R}{\Gamma}^{ij}\left({\xi^{*}_{i}}^{\alpha}{\xi^{*}_{j}}^{\beta}\Sigma_{0\alpha\rho}\Sigma_{0\beta\sigma}Q_{L,R}^{\rho\sigma}\right),$
(31)

where

${(\Gamma)}^{ij}=\begin{pmatrix}0&1&0&0&0&0\\\ -1&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ \end{pmatrix}$ (32)

is the corresponding spurion.

Combining $\lambda_{b_{L}}$ and $\lambda_{b_{R}}$ couplings, we can generate
the bottom quark Yukawa coupling as

$\displaystyle\sim\lambda_{b_{L}}\lambda_{b_{R}}\bar{q}_{La}{\Lambda^{a}}_{ij}{\xi^{i}}_{\alpha}{\xi^{j}}_{\beta}\Sigma_{0}^{\alpha\rho}\Sigma_{0}^{\beta\sigma}{{\xi^{T}}_{\rho}}^{k}{{\xi^{T}}_{\sigma}}^{l}{{\Gamma}^{\dagger}}_{kl}b_{R}$
$\displaystyle=\lambda_{b_{L}}\lambda_{b_{R}}\bar{q}_{La}{\Lambda^{a}}_{ij}\Sigma^{ik}\Sigma^{jl}{{\Gamma}^{\dagger}}_{kl}b_{R}\supset\lambda_{b_{L}}\lambda_{b_{R}}\left(\bar{q}_{L}s{H}_{1}b_{R}\right)~{},$
(33)

which is exactly what we expect in Eq. (23). The bottom quark gets mass from
$H_{1}$ but with additional suppression from the FN mechanism as

$\displaystyle m_{b}=\frac{\langle
is\rangle}{f}\frac{\lambda_{b_{L}}\lambda_{b_{R}}}{g_{B}}\frac{v_{1}}{\sqrt{2}}=\frac{\lambda_{b_{L}}\lambda_{b_{R}}}{g_{B}}\frac{v_{s}v_{1}}{{2f}},$
(34)

where $g_{B}$ is a coupling of the composite bottom partners. This is like a
Type-II 2HDM but with smaller tan$\beta$ due to the suppression by small
$v_{s}/f$.

Therefore, for case (1), we can get the top-bottom mass hierarchy. Assuming
all the $\lambda$ and $g$ are $\mathcal{O}(1)$ couplings, the mass ratio
bocomes 222We consider the running of quark masses up to 1 TeV [25], which
gives $m_{b}/m_{t}\sim 2.43\text{ GeV}/150\text{ GeV}\sim 1/60$. The ratio
might be larger because the VEV of the flavon field is below the TeV scale.

$\displaystyle\frac{m_{b}}{m_{t}}\sim\frac{v_{s}}{\sqrt{2}f}\frac{v_{1}}{v_{2}}=\frac{\epsilon}{\text{tan}\beta}\sim\frac{1}{60}~{},$
(35)

where $\epsilon\equiv{v_{s}}/{\sqrt{2}f}$. The hierarchy comes from both
$\epsilon$ and tan$\beta$. Taking the symmetry breaking scale $f\sim 1$ TeV,
we get

$\displaystyle v_{s}\sim 25~{}\text{tan}\beta\text{ GeV,}$ (36)

If $\epsilon$ (namely $v_{s}$) is small, we can get a Type-II 2HDM with a
smaller tan$\beta$.

Similarly, consider case (2) by taking the second pair with $x=-1/3$ and
$r=1$, i.e.
$\left(\bar{2}_{\frac{1}{6},\frac{1}{2}},1_{-\frac{1}{3},2}\right)$. The
spurion for the left-handed quark becomes

${(\Lambda)^{a}}_{ij}=(\begin{pmatrix}0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&0\\\ 0&0&-1&0&0&0\\\ 0&0&0&0&0&0\\\
\end{pmatrix},\begin{pmatrix}0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&-1&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ \end{pmatrix}),$ (37)

and for the right-handed bottom quark is

${(\Gamma)}^{ij}=\begin{pmatrix}0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&-1&0&0\\\ 0&0&0&0&0&0\\\ \end{pmatrix}~{}.$ (38)

Combining $\lambda_{b_{L}}$ and $\lambda_{b_{R}}$ couplings in case (2), we
get the bottom Yukawa coupling as

$\displaystyle\sim\lambda_{b_{L}}\lambda_{b_{R}}\bar{q}_{La}{\Lambda^{a}}_{ij}{\xi^{i}}_{\alpha}{\xi^{j}}_{\beta}\Sigma_{0}^{\alpha\rho}\Sigma_{0}^{\beta\sigma}{{\xi^{T}}_{\rho}}^{k}{{\xi^{T}}_{\sigma}}^{l}{{\Gamma}^{\dagger}}_{kl}b_{R}$
$\displaystyle=\lambda_{b_{L}}\lambda_{b_{R}}\bar{q}_{La}{\Lambda^{a}}_{ij}\Sigma^{ik}\Sigma^{jl}{{\Gamma}^{\dagger}}_{kl}b_{R}\supset\lambda_{b_{L}}\lambda_{b_{R}}\left(\bar{q}_{L}s^{*}\tilde{H}_{2}b_{R}\right)~{}.$
(39)

Again it is what we expect in Eq. (23). This case will lead to a Type-I 2HDM
with the small bottom Yukawa coupling merely due to the FN mechanism as

$\displaystyle m_{b}=\frac{\langle
is\rangle}{f}\frac{\lambda_{b_{L}}\lambda_{b_{R}}}{g_{B}}\frac{v_{2}}{\sqrt{2}}=\frac{\lambda_{b_{L}}\lambda_{b_{R}}}{g_{B}}\frac{v_{s}v_{2}}{{2f}}.$
(40)

Assuming all the $\lambda$ are $\mathcal{O}(1)$ couplings. The mass ratio

$\displaystyle\frac{m_{b}}{m_{t}}\sim\frac{v_{s}}{\sqrt{2}f}={\epsilon}\sim\frac{1}{60}\quad\implies\quad
v_{s}\sim 25\text{ GeV, }$ (41)

if the symmetry breaking scale $f\sim 1$ TeV.

###  IV.4 Composite resonances and spaghetti diagrams

In the last section, we see how the FN mechanism can be realized and create
the hierarchy between the top and bottom mass. The composite resonances, which
carry the same quantum number but different flavor charges, play the role of
the Froggatt-Nielsen fields in the FN mechanism. We can write down all the
composite resonances in matrix form as

$\displaystyle\mathbf{15}_{-\frac{1}{3},0}$
$\displaystyle=\mathbf{14}_{-\frac{1}{3},0}\oplus\mathbf{1}_{-\frac{1}{3},0}$
$\displaystyle=\begin{pmatrix}0&\tilde{B}_{1}&T_{\frac{1}{2}}&\tilde{B}^{\prime}_{0}&\tilde{T}_{0}&\tilde{B}_{\frac{1}{2}}\\\
\cdot&0&B_{\frac{1}{2}}&Y_{0}&\tilde{B}^{\prime\prime}_{0}&Y_{\frac{1}{2}}\\\
\cdot&\cdot&0&B_{-\frac{1}{2}}&T_{-\frac{1}{2}}&0\\\
\cdot&\cdot&\cdot&0&\tilde{B}_{-1}&Y_{-\frac{1}{2}}\\\
\cdot&\cdot&\cdot&\cdot&0&\tilde{B}_{-\frac{1}{2}}\\\
\cdot&\cdot&\cdot&\cdot&\cdot&0\\\ \end{pmatrix}\oplus\tilde{B}_{0},$ (42)

where $T$ and $B$ are composite resonances with the same quantum numbers as
the SM top and bottom quarks but with different flavor charges as labeled in
the subscript, $\tilde{T}$ and $\tilde{B}$ are resonances with the same
hypercharges as the SM top and bottom quarks but under different $SU(2)_{W}$
representations, and $Y$ are exotic resonances with hypercharge $-4/3$.

The FN mechanism can also be expressed through the “spaghetti diagrams”, which
looks like a 2 to 2 scattering in this case with only one flavon inserted.
Spaghetti diagrams that generate the suppressed bottom quark mass are shown in
Fig. 1. These diagrams give us the bottom mass we expect after integrating out
the heavy Froggatt-Nielsen fields, which are composite fermionic resonances in
this model, and replacing the scalar fields with their VEVs.

Figure 1: Spaghetti diagrams that generate the bottom Yukawa coupling through
the Froggatt-Nielsen mechanism in case (1). Diagrams for case (2) are similar.

###  IV.5 Comparison between two cases

So far, we see two different flavor charge assignments for the right-handed
bottom quark, which lead to two different bottom Yukawa coupling terms. Both
of them successfully generate a suppressed bottom Yukawa coupling through the
FN mechanism. The difference between these two cases is listed in Table 1.

| Case (0) | Case (1) | Case (2)
---|---|---|---
$\Delta_{b}\equiv[q_{L}]-[b_{R}]$ | $1/2$ | $3/2$ | $-3/2$
Flavor charge of $b_{R}$ | 0 | -1 | 2
Coupling term | $\bar{q}_{L}{H}_{1}b_{R}$ | $\bar{q}_{L}s{H}_{1}b_{R}$ | $\bar{q}_{L}s^{*}\tilde{H}_{2}b_{R}$
Type of 2HDM | Type-II | Type-II | Type-I
Suppression of $m_{b}/m_{t}$ | $1/{\text{tan}\beta}$ | ${\epsilon}/{\text{tan}\beta}$ | ${\epsilon}$
Table 1: The comparison between two cases with suppressed bottom Yukawa
couplings through the FN mechanism. Case (0) for the unsuccessful first taste
is also shown. In the last row, $\epsilon\equiv{v_{s}}/{\sqrt{2}f}$ is the
suppression by the FN mechanism.

Here we assume the flavor charge of $q_{L}$ is 1/2 and $t_{R}$ is 0, such that
the top quark mass comes from $H_{2}$. We can see the two cases represent
different signs of $\Delta_{b}$. It will affect the way we extend our model to
include lighter quarks, which will be discussed next. The difference between
the types of 2HDM results in different Higgs phenomenology. The second Higgs
doublet is expected to be the main target among the exotic states in the
model. The results of the direct searches will be shown in the next section.
The factor of suppression is also related to the experimental constraint. The
smaller $\epsilon$ required for the correct mass ratio implies a smaller VEV
$v_{s}$ of the flavon field, which will end up with a larger deviation in
flavor observables and thus is strongly constrained.

###  IV.6 Include all the generations

As yet, we only get the hierarchy between the top and bottom quarks, which
belong to the third generation. To include the lighter quarks, more
suppression is needed, which means more insertion of the flavon field $s$ and
a larger difference in flavor charges. This will require the lighter quarks to
be embedded in even larger representations.

Take case (1) for example. We have already gotten the flavor charges of the
third generation quarks. To extend to the first and the second generations,
one possible flavor charges assignment 333The assignment is borrowed from Eq.
(2.15) of [14]. The resulting mass ratios and CKM matrix can partially
reproduce observed values with $\mathcal{O}(1)$ correction. is listed in Table
2. It implies that we need even larger representations to have flavor charges
different by $7/2$. That would require representations with more than 4
indices for the quark sector.

| $U(1)_{F}$ |  | $U(1)_{F}$ |  | $U(1)_{F}$
---|---|---|---|---|---
$q_{3,L}=(t_{L},b_{L})^{T}$ | 1/2 | $t_{R}$ | 0 | $b_{R}$ | -1
$q_{2,L}=(c_{L},s_{L})^{T}$ | 3/2 | $c_{R}$ | 0 | $s_{R}$ | -1
$q_{1,L}=(u_{L},d_{L})^{T}$ | 3/2 | $u_{R}$ | -2 | $d_{R}$ | -2
Table 2: A possible flavor charge assignment of all elementary quarks for case
(1) setup.

For case (2), it is more difficult to get a consistent flavor charges
assignment for the desired CKM matrix. For the flavor charge of the third
generation quarks, we find that they follow the order
$[b_{R}]>[q_{L}]>[t_{R}]$, which is also applied to the extension. From the
relation, the left-handed quarks should always sit in the middle. This
requirement restricts the flavor charge difference we can have. For example,
$q_{1,L}$ and $q_{2,L}$ can only be either $1/2$ or $3/2$, which will lead to
unsuppressed entries in the CKM matrix. Therefore, considering the flavor
charge assignment for the light quarks, case (1) is preferred over case (2).
However, we will still discuss the constraints on parameter space of case (2)
assuming that it can generate a similar Yukawa matrix as case (1).

The exact embedding will be explored in future work. To discuss the
experimental constraints of flavons in the following section, we will assume
this mechanism can be extended to all the generations and is responsible for
all the light quark masses in both cases. Also, for flavon phenomenology, the
results are mainly determined by two parameters, the flavon mass $M_{s}$ and
the flavon VEV $v_{s}$.

##  V Collider Signature

The phenomenology of this model is similar to other CHMs based on
$SU(6)/Sp(6)$ coset with partial compositeness [18], which includes 14 pNGBs
and composite partners of the SM particles. The main targets would be on the
particles that couple to SM particles at leading order. In our setup, the most
important search modes include the second Higgs doublet, flavons, and
fermionic composite resonances.

###  V.1 The second Higgs Doublet

The phenomenology of 2HDM has been well-studied, and we can directly borrow
the results from [27]. For case (2) as a Type-I 2HDM, there is no further
constraint since the second Higgs doublet is decoupled from the fermion
sector. But for case (1), a Type-II 2HDM, the constraints are important
because the suppression of the bottom mass comes partially from the FN
mechanism and partially from tan$\beta$. Therefore, the value of tan$\beta$
will decide the $\epsilon$ we need from the FN mechanism. The strongest
constraint for a Type-II 2HDM comes from the $\tau\tau$ search, which
restricts tan$\beta<6-10$ for a wide mass scale. If we make it a Flipped 2HDM
instead, where the charged leptons get masses from $H_{2}$ instead of $H_{1}$,
the coupling between heavy Higgs and $\tau\tau$ will become much smaller. Then
the main constraint comes from $\bar{b}b$ search, and tan$\beta\sim 20$ is
still allowed. However, we would like to stick to a normal Type-II 2HDM for
case (1) and set the benchmark with

$\displaystyle\text{tan}\beta\sim 6\quad\implies\quad v_{s}\sim 150\text{
GeV}$ (43)

for the following discussion.

###  V.2 Flavons

The physical flavon fields include a scalar component $s$ and a pseudoscalar
component $a$. The masses of two types of flavons depend on the complete
flavon potential, which is discussed in the appendix B. If flavor symmetry is
exact and spontaneously broken by flavor symmetry conserving potential, then
the pseudoscalar flavon should be massless, which is not acceptable.
Therefore, the explicit breaking of flavor symmetry in the flavon potential is
needed. For simplicity, we will assume the mass of scalar, $M_{s}$, and the
mass of pseudoscalar, $M_{a}$, are the same. This spectrum can be achieved if
flavor symmetry is broken by a tadpole term in the flavon potential as shown
in appendix B. Therefore, from now on, we will use flavon $s$ for both the
scalar and pseudoscalar components and $M_{s}$ for the flavon mass, which is
expected to be at the sub-TeV scale.

The production and decay of flavons have already been comprehensively
discussed in [28, 29]. Although the flavon coupling terms in these papers
might look different from ours, the exact values are determined by the
observed quark masses and the CKM matrix. Therefore, the flavon couplings with
the form $m/v_{s}$ should have similar values in all kinds of flavon models up
to an $\mathcal{O}(1)$ factor. The numerical values in these two sections are
derived based on their analysis with additional adjustments from our setup.

The main production for sub-TeV flavons come from the single production
process $b\bar{b}\to s$. The cross section for flavons with $M_{s}=500$ GeV is

$\displaystyle\sigma(b\bar{b}\to s)\sim 9.8\times 10^{-3}\left(\frac{150\text{
GeV}}{v_{s}}\right)^{2}\text{ pb}$ (44)

in 14 TeV LHC. Taking $v_{s}=150$ GeV, around $2.2\times 10^{4}$ flavons will
be produced in the HL-LHC era with an integrated luminosity of 3 ab-1. In case
(2) with smaller $v_{s}$, the number is multiplied by a factor of 36.

The decay branching ratios for flavons are independent of $v_{s}$ but only
depend on the flavor structure. If flavons only couple to the third
generation, the dominate decay channel will be $b\bar{b}$ channel and
$\tau\tau$ channel with roughly $\sim 85\%$ and $\sim 15\%$ branching ratio.
If the FN mechanism is extended to all SM particles and responsible for the
full Yukawa matrix, then there will be exotic final states like $tc$ and $tu$.
It turns out that the $tc$ channel will be the dominant one due to the large
mixing required to reproduce the desired CKM matrix. The ratios depend on
tan$\beta$, too. Under the benchmark values, we get the branching ratios for
each channels as $tc~{}(96.8\%)$, $b\bar{b}~{}(2.7\%)$, and
$\tau\tau~{}(0.5\%)$. However, the hadronic channels suffer from large
backgrounds. The leptonic channel can reach $\sigma\times BR\sim 10^{-3}$ pb
for sub-TeV flavon in HL-LHC, but it is still above the benchmark value. The
discovery can be made in a future 100 TeV collider, where the cross section is
expected to be $\sim 100$ times larger, and the integrated luminosity is also
higher. In that case, the distinct $tc$ channel search will provide strong
evidence for the origin of the Yukawa matrix.

###  V.3 Fermionic Resonances

The top partners in the $SU(6)/Sp(6)$ CHM are vector-like fermionic resonances
that form a sextet of the $Sp(6)$ global symmetry. Their quantum numbers under
the SM gauge symmetry are $(3,2,1/6)[\times 2]$, $(3,1,2/3)$, and
$(3,1,-1/3)$, which are identical to those of the SM quarks. There are no
exotic states with higher or lower hypercharges. These states are degenerate
in the limit of unbroken $Sp(6)$ global symmetry. Only small splittings arise
from the explicit symmetry breaking effects. Their mass $M_{T}\sim g_{T}f$
plays the important role of cutting off the quadratic contribution from the
top quark loop to the Higgs potential. The generic expectation of the
composite fermionic resonances is $M_{F}=5-10$ TeV with $g_{F}=5-10$. However,
naturalness prefers a smaller $M_{T}$ to minimize the required fine-tuning,
which usually requires $g_{T}\gtrsim 1$. The current bound on the top partner
mass has reached $\sim 1.2$ TeV [30, 31]. The HL-LHC can further constrain the
mass up to $\sim 1.5$ TeV [32]. A future 100 TeV collider will cover the
entire interesting mass range of the top partners if no severe tuning
conspires.

For the bottom partners, they form a
$\mathbf{14}_{-1/3}\oplus\mathbf{1}_{-1/3}$ under $Sp(6)$ global symmetry. The
quantum numbers for the total of 15 fields under the SM gauge group are
$(3,2,1/6)[\times 2]$, $(3,2,-5/6)[\times 2]$, $(3,1,-1/3)[\times 4]$, and
$(3,3,-1/3)$, which include exotic resonances with EM charge $-4/3$. The
states are not degenerate, and the singlet is expected to be lighter. The
masses of the bottom partners $M_{B}\sim g_{B}f$, unlike the top partners,
don’t have a large effect on the fine-tuning due to the small bottom Yukawa
coupling. Therefore, they could be around the compositeness scale with
$M_{B}=5-10$ TeV, which is beyond the LHC searches. The heavier $M_{B}\sim
g_{B}f$ also leads to additional suppression $g_{T}/g_{B}$ on the mass ratio
between the top and bottom quarks, which can relieve the required $\epsilon$
we need.

If we extend the FN mechanism to the light generations, a larger
representation is required to get a larger flavor charge difference, which
also implies a larger EM charge difference within the multiplet. Therefore,
there could be more exotic resonances with EM charges like $-7/3$ or $5/3$,
which are important in identifying the correct representation. These heavy
fermionic resonances can be found in a future 100 TeV collider. If the exotic
spectrum corresponding to the large representation shows up, it might unveil
the nature of SM fermion partners and the origin of Yukawa couplings.

##  VI Flavor constraints

Compared to the collider signatures, the flavor constraints usually probe a
higher scale and place stronger bounds on the models. Assume that the FN
mechanism can be extended to all elementary quarks and leptons with suitable
Yukawa coupling matrices. Then we can discuss the flavor constraints through a
similar analysis as in [29].

The new flavor processes can be mediated through flavons or the second Higgs
doublet. The flavon contributions strongly depend on the couplings and
spectrum of flavons. As we mention above, there are a scalar component and a
pseudoscalar component. We will assume the scalar and pseudoscalar components
share the same mass $M_{s}$. This assumption will give us the weakest flavor
constraints because, for some flavor processes, the contributions from a
scalar and a pseudoscalar will cancel exactly if they are degenerate. It can
also be understood that the assumption raises an $U(1)$ symmetry for the
flavon field around the vacuum, which forbids these flavor processes. However,
we will see even the weakest constraints from flavor are much stronger than
the direct searches.

###  VI.1 Meson Decay

The new particles might enhance some rare processes that are suppressed within
SM. The measurements of rare decays of neutral mesons can give strong
constraints on the new physics scale. In this model, flavons can mediate some
rare decays of neutral mesons. For example, the branching ratio of
$B_{s}\to\mu^{+}\mu^{-}$ provides a constraint on dimension-6 operators
induced by flavons, which include

$\displaystyle
C_{S}^{ij}(\bar{q}_{i}P_{L}q_{j})(\bar{\ell}\ell)\quad\text{and}\quad\tilde{C}_{S}^{ij}(\bar{q}_{i}P_{R}q_{j})(\bar{\ell}\ell)$
(45)

from a scalar flavon with coefficient

$\displaystyle
C_{S}^{ij}=g_{\ell\ell}g_{ji}\left(\frac{1}{M_{s}^{2}}\right)\quad\text{and}\quad\tilde{C}_{S}^{ij}=g_{\ell\ell}g_{ij}\left(\frac{1}{M_{s}^{2}}\right)$
(46)

and

$\displaystyle
C_{P}^{ij}(\bar{q}_{i}P_{L}q_{j})(\bar{\ell}\gamma_{5}\ell)\quad\text{and}\quad\tilde{C}_{P}^{ij}(\bar{q}_{i}P_{R}q_{j})(\bar{\ell}\gamma_{5}\ell)$
(47)

from a pseudoscalar flavon with coefficient

$\displaystyle
C_{P}^{ij}=g_{\ell\ell}g_{ji}\left(\frac{1}{M_{a}^{2}}\right)\quad\text{and}\quad\tilde{C}_{P}^{ij}=g_{\ell\ell}g_{ij}\left(\frac{1}{M_{a}^{2}}\right).$
(48)

The difference between $C$ and $\tilde{C}$ will modify the predicted SM
values. The leading order deviation comes from the pseudoscalar flavon
exchange, which interferes with the SM contribution. The coupling $g_{ij}$ is
determined by the observed fermion masses over the flavon VEV $v_{s}$.
Therefore, once we take the mass $M_{s}=M_{a}$, the measurement can put a
constraint on the $C_{P}^{ij}-\tilde{C}_{P}^{ij}$ and thus the product of
$v_{s}M_{s}$. Later we will find that most of the flavor constraints can be
transferred into the constraint on the value of $v_{s}M_{s}$.

The latest result of $B_{s}\to\mu^{+}\mu^{-}$ measurement by LHCb [33]
requires $v_{s}M_{s}\geq 5\times 10^{4}\text{ (GeV)}^{2}$, which give a
$M_{s}$ lower bound under the benchmark value as

$\displaystyle\text{case (1) }M_{s}\geq 400\text{ (GeV)},\quad\text{case (2)
}M_{s}\geq 2000\text{ (GeV)}.$

There is a stronger constraint for case (2) flavon model with smaller $v_{s}$.
The reason is, though we want to have a small $v_{s}$ to generate the
hierarchy, a small $v_{s}$ also implies a larger coupling between flavons and
the SM quarks, which is disfavored by flavor physics. We also find that case
(1) as a Type-II 2HDM has a looser bound due to the assistance from
tan$\beta$. The improvement in the measurement of BR($B_{s}\to\mu^{+}\mu^{-}$)
will further constraints the allowed values in the future. The interesting
parameter space might be ruled out by LHCb and Belle-II.

Meson decays also put strong constraints on the second Higgs doublet. A light
charged Higgs boson can induce a significant contribution to the branching
ratio BR$(B\to X_{s}\gamma)$ [34, 35, 36, 37, 38, 39]. In the Type-II or
flipped 2HDM, this gives a strong lower bound on the charged Higgs boson mass
$M_{H^{\pm}}>600$ GeV [40, 41], which would require a tuning or an additional
symmetry in the 2HDM potential in case (1) model.

###  VI.2 Neutral Meson Mixing

The strongest bounds for flavons come from the neutral meson mixing,
especially from the light mesons. The relevant $\Delta F=2$ interaction terms
include

$\displaystyle
C_{2}^{ij}(\bar{q}^{i}_{R}q^{j}_{L})^{2},\quad\tilde{C}_{2}^{ij}(\bar{q}^{i}_{L}q^{j}_{R})^{2},\quad\text{and}\quad
C_{4}^{ij}(\bar{q}^{i}_{R}q^{j}_{L})(\bar{q}^{i}_{L}q^{j}_{R}).$

In this paper, since we assume that the scalar and pseudoscalar flavons share
the same mass $M_{s}$, there is an $U(1)$ symmetry that forbids $C_{2}^{ij}$
and $\tilde{C}_{2}^{ij}$ terms. That is, the contributions from scalars and
pseudoscalars will cancel exactly. The only relevant dimension-6 operator is

$\displaystyle
C_{4}^{ij}(\bar{q}^{i}_{R}q^{j}_{L})(\bar{q}^{i}_{L}q^{j}_{R})\quad\text{
with}\quad C_{4}^{ij}=-g_{ij}g^{*}_{ji}\left(\frac{1}{M_{s}^{2}}\right).$ (49)

The coefficients as a function of $v_{s}M_{s}$ are strongly constrained by
experiments.

In Table 3, we conclude the flavor constraints on the product $v_{s}M_{s}$
from all neutral meson systems, including those with the first generation
quarks. The numbers are extracted from [29]. The corresponding lower bounds on
flavon mass $M_{s}$ are also shown based on the benchmark value of each case.

|  $v_{s}M_{s}$ (GeV2) | Case (1) (GeV) | Case (2) (GeV)
---|---|---|---
$C_{B_{s}}$ | 32000 | 210 | 1280
$\varphi_{B_{s}}$ | 128000 | 850 | 5120
$C_{B_{d}}$ | 183000 | 1220 | 7320
$\varphi_{B_{d}}$ | 250000 | 1670 | 10000
$\Delta m_{K}$ | 255000 | 1700 | 10200
$\epsilon_{K}$ | 2550000 | 17000 | 102000
Table 3: Flavor constraints from all kinds of neutral meson mixing
observables, including the lower bounds on the value of $v_{s}M_{s}$ and
flavon mass $M_{s}$ of each case.

From the constraints of neutral meson mixing, we again find that case (1) is
preferred because case (2) has a smaller $v_{s}$ and thus larger couplings to
the SM fermions. The lower bounds for case (2) have reached multi-TeV, which
might be too heavy to be treated as pNGBs. The flavor symmetry is hardly
broken, and the sigma model might not be an appropriate way to describe it.
Even for case (1) with milder bounds, constraints from the CP phases are also
high. If we assume that the flavon preserves CP-symmetry and ignore the
constraints from the CP phase, the current bounds for case (1) become
$M_{s}\geq 1.2-1.7$ TeV, and the future experiments will raise the bounds by a
factor of 2. If the FN mechanism is not responsible for the first generation
quarks, then the only constraint is from $C_{B_{s}}$, and a sub-TeV flavon is
still allowed. The bounds can also be relieved if the bottom partner is
heavier than the top partner, where $g_{B}>g_{T}$ can give another
suppression, and the required $v_{s}$ can be larger. Nevertheless, the most
interesting mass region for flavons as pNGBs of the TeV scale confinement will
be covered in the near future by LHCb and Belle-II.

##  VII Conclusions

The Froggatt-Nielsen mechanism is an appealing solution to the Flavor Puzzle.
However, the scale of flavor dynamics and the flavon field can be arbitrarily
high. The predictive flavon models require the dynamics to stabilize the
flavon potential. One way, analogous to the composite Higgs models, is to
introduce the flavon field as a pseudo-Nambu-Goldstone boson. In this paper,
we construct a non-linear sigma model with pNGBs, including both the Higgs
doublets and the flavon field.

The flavon field as a pNGB provides a possibility to have the origin of flavor
hierarchy at the TeV scale. The shift symmetry is slightly broken, which leads
to the flavon mass and VEV. The non-linear nature of the flavon also
constraints the interactions we can write down. In this paper, we show two
possible ways to generate suppressed bottom Yukawa coupling terms through the
Froggatt-Nielsen mechanism, where the composite resonances play the role of
the FN fields. The derivation and explanation of the process are presented in
detail.

Two cases lead to different phenomenology and receive different constraints.
Case (1) as a Type-II 2HDM with small tan$\beta$ has a larger $v_{s}$ and
smaller couplings to the SM fermions. Some parameter space with the sub-TeV
flavon is still allowed if the constraints from the neutral meson of the first
generation quarks are not taken into account. Case (2) as a Type-I 2HDM has a
weaker bound on the Higgs sector. However, the requirements of small $v_{s}$
and the strong couplings with the SM particles are disfavored. Future
measurements of neutral meson systems by LHCb and Belle-II will keep probing
the scenario with the light flavon. Either push the mass bound to a much
higher scale or find the existence of the pNGB flavon.

###### Acknowledgements.

We thank Hsin-Chia Cheng for useful discussions. This work is supported by the
Department of Energy Grant number DE-SC-0009999.

##  Appendix A The SM gauge sector

The SM electroweak gauge group $SU(2)_{W}\times U(1)_{Y}$ is embedded in
$SU(6)\times U(1)_{X}$ with generators given by

$\displaystyle\frac{1}{2}$ $\displaystyle\begin{pmatrix}\sigma^{a}&0&0&0\\\
0&0&0&0\\\ 0&0&-\sigma^{a*}&0\\\ 0&0&0&0\\\ \end{pmatrix}\text{ and
}\frac{1}{2}\begin{pmatrix}0_{2\times 2}&0&0&0\\\ 0&1&0&0\\\ 0&0&0_{2\times
2}&0\\\ 0&0&0&-1\\\ \end{pmatrix}+X\mathbf{I}~{}.$

The extra $U(1)_{X}$ factor accounts for the different hypercharges of the SM
fermions but is not relevant for the bosonic fields. These generators belong
to $Sp(6)\times U(1)_{X}$ and are not broken by $\Sigma_{0}$.

Using the $\Sigma$ field, the Lagrangian for kinetic terms of Higgs boson is
given by

$\mathcal{L}_{h}=\frac{f^{2}}{4}\text{tr}\left[(D_{\mu}\Sigma)(D^{\mu}\Sigma)^{\dagger}\right]+\cdots,$
(50)

where $D_{\mu}$ is the electroweak covariant derivative. Expanding this term,
we get

$\displaystyle\mathcal{L}_{h}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}h_{1})(\partial^{\mu}h_{1})+\frac{1}{2}(\partial_{\mu}h_{2})(\partial^{\mu}h_{2})$
$\displaystyle+\frac{f^{2}}{2}g_{W}^{2}\left(\text{sin}^{2}\frac{\sqrt{h_{1}^{2}+h_{2}^{2}}}{\sqrt{2}f}\right)\left[W^{+}_{\mu}W^{-\mu}+\frac{Z_{\mu}Z^{\mu}}{2\text{cos}\theta_{W}}\right]~{}.$
(51)

The non-linear behavior of the Higgs boson in CHMs is apparent from the
dependence of trigonometric functions.

The $W$ boson acquires a mass when $h_{1}$ and $h_{2}$ obtain nonzero VEVs
$V_{1}$ and $V_{2}$ of

$m_{W}^{2}=\frac{f^{2}}{2}g_{W}^{2}\left(\text{sin}^{2}\frac{\sqrt{V_{1}^{2}+V_{2}^{2}}}{\sqrt{2}f}\right)=\frac{1}{4}g_{W}^{2}(v_{1}^{2}+v_{2}^{2}),$
(52)

where

$v_{i}\equiv\sqrt{2}f\frac{V_{i}}{\sqrt{V_{1}^{2}+V_{2}^{2}}}\,\text{sin}\frac{\sqrt{V_{1}^{2}+V_{2}^{2}}}{\sqrt{2}f}\approx
V_{i}=\langle h_{i}\rangle~{}.$ (53)

The parameter that parametrizes the nonlinearity of the CHM is given by

$\xi\equiv\frac{v^{2}}{f^{2}}=2\,\sin^{2}\frac{\sqrt{V_{1}^{2}+V_{2}^{2}}}{\sqrt{2}f}~{},$
(54)

where the VEV $v^{2}=v_{1}^{2}+v_{2}^{2}=(246\text{ GeV})^{2}$. The $\xi$
plays an important role in the phenomenology of CHMs, but it is not of
interest in this study.

##  Appendix B The pNGB potential

The pNGB potential comes from the explicit breaking of $SU(6)$ global
symmetry. Within SM, there are symmetry-breaking sources like the gauge
couplings and Yukawa couplings. Additional sources are also needed to
introduce the flavon potential. Here we will briefly list their contributions
to the pNGB potential one by one.

Starting with the SM gauge interactions, we can derive the pNGB potential by
the generators listed in appendix A. Both $SU(2)_{W}$ and $U(1)_{Y}$ only
break the global symmetry partially and generate the potential for the pNGBs
which are charged. The two Higgs doublets are charged under both gauge
interactions and get

$\displaystyle\Delta
V_{H}=\frac{3}{16\pi^{2}}\left(\frac{3}{4}c_{w}g^{2}+\frac{1}{4}c^{\prime}g^{\prime
2}\right)M_{\rho}^{2}|H|^{2},$ (55)

where $M_{\rho}\sim g_{\rho}f$ is the mass of the vector resonances $\rho$,
which act as the gauge boson partners to cut off the gauge loop contribution
to the pNGB masses, and $c_{w}$ and $c^{\prime}$ are $\mathcal{O}(1)$
constants. The scalar triplet $\phi$ also gets a potential

$\displaystyle\Delta
V_{\phi}=\frac{3}{16\pi^{2}}\left(2c_{w}g^{2}\right)M_{\rho}^{2}(\phi^{a}\phi_{a}),$
(56)

The $SU(2)_{W}\times U(1)_{Y}$ singlets $s$ and $\eta$ do not receive
potentials from the gauge interactions at this order, but they will obtain
potentials elsewhere.

Next, the Yukawa coupling also breaks the $SU(6)$ global symmetry. Take the
top quark loop-induced potential for example, where the required spurions are
already written in section III. We can estimate

$\displaystyle\Delta V_{H}$
$\displaystyle\sim-\frac{N_{c}}{8\pi^{2}}\lambda_{L}^{2}\lambda_{R}^{2}f^{4}\left|{(\Lambda)^{a}}_{i}{(\Gamma^{*})}_{j}\Sigma^{ij}\right|^{2}$
$\displaystyle\supset-\frac{N_{c}}{8\pi^{2}}\lambda_{L}^{2}\lambda_{R}^{2}f^{2}\,|H|^{2}=-\frac{N_{c}}{8\pi^{2}}y_{t}^{2}M_{T}^{2}\,|H|^{2}.$
(57)

The dominant quartic term is also from the top loop as

$\displaystyle\Delta V_{H}\sim\frac{N_{c}}{4\pi^{2}}y_{t}^{4}\,|H|^{4}.$ (58)

Similar potentials also arise for other SM Yukawa interactions.

The real singlet $\eta$ does not get a potential at the leading order, but it
couples quadratically to the Higgs doublets (e.g., from Eq. (B)), so it can
still obtain a potential after the Higgs doublets develop nonzero VEVs.
Through Eq. (B), $\eta$ gets a quadratic potential

$\displaystyle\Delta
V_{\eta}\sim\frac{3}{8\pi^{2}}y_{t}^{2}M_{T}^{2}\cdot\left(\frac{v}{f}\right)^{2}\eta^{2}.$
(59)

So far, we haven’t gotten any potential for the flavon field $s$. Although the
flavon field in our model couples to the bottom quark, which will lead to a
loop-induced pNGB potential. However, we would like to have the potential from
a separate source, so they are independent of the FN mechanism. A nontrivial
potential for the flavon field $s$ is common in models with collective
symmetry breaking [16, 18], where the potential

$\Delta V\sim
M_{s}^{2}\left|s\pm\frac{i}{2f}\tilde{H_{2}}^{\dagger}H_{1}\right|^{2}\supset
M_{s}^{2}|s|^{2}$ (60)

is introduced. For example, in the $SU(6)/Sp(6)$ little Higgs model [16], the
term can be generated by gauging two copies of $SU(2)$. However, it introduces
new heavy gauge bosons $W^{\prime}$ and $Z^{\prime}$, which are strongly
constrained.

Another way, following [18], is using the interactions between the elementary
fermions and the resonances of the strong dynamics. In the section III, we see
that the elementary quark doublets can couple to composite operators of
$SU(6)$ representations $\mathbf{6}$ and $\mathbf{\bar{6}}$ with $x=1/6$ and
$r=0$, which are decomposed under $SU(2)_{W}\times U(1)_{Y}\times U(1)_{F}$ as

$\displaystyle\mathbf{6}_{1/6,0}=\mathbf{2}_{1/6,1/2}\oplus\mathbf{1}_{2/3,0}\oplus\mathbf{\bar{2}}_{1/6,-1/2}\oplus\mathbf{1}_{-1/3,0},$
(61a)
$\displaystyle\mathbf{\bar{6}}_{1/6,0}=\mathbf{\bar{2}}_{1/6,-1/2}\oplus\mathbf{1}_{-1/3,0}\oplus\mathbf{2}_{1/6,1/2}\oplus\mathbf{1}_{2/3,0}.$
(61b)

Both operators create the same resonances, which belong to $\mathbf{6}$ of the
$Sp(6)$ group.

Now consider two elementary quark doublets, $q_{L}$ and $q^{\prime}_{L}$,
couple to the first two components of the composite operators of $\mathbf{6}$
and $\mathbf{\bar{6}}$ respectively, while both representations contain the
same resonances:

$\lambda_{L}\bar{q}_{La}{\Lambda^{a}}_{i}O_{R}^{i}=\lambda_{L}\bar{q}_{La}{\Lambda^{a}}_{i}\left({\xi^{i}}_{\alpha}Q_{R}^{\alpha}\right),$
(62)

where

${(\Lambda)^{a}}_{i}=\Lambda=\begin{pmatrix}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
\end{pmatrix}~{},$ (63)

and

$\lambda^{\prime}_{L}\bar{q}^{\prime}_{La}\epsilon^{ab}{\Omega_{b}}^{i}O^{\prime}_{Ri}=\lambda^{\prime}_{L}\bar{q}^{\prime}_{La}\epsilon^{ab}{\Omega_{b}}^{i}\left({\xi^{*}_{i}}^{\beta}\Sigma_{0\beta\alpha}Q_{R}^{\alpha}\right),$
(64)

where

${(\Omega)_{a}}^{i}=\Omega=\begin{pmatrix}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
\end{pmatrix}~{}.$ (65)

The combination of the two interactions breaks the $SU(6)$ global symmetry
explicitly. It leads to a potential for the pNGBs at
$\mathcal{O}(\lambda_{L}^{2}\lambda_{L}^{\prime 2})$ of the form

$\Delta
V_{s}\propto[{(\Lambda)^{a}}_{i}{(\Omega^{*})^{b}}_{j}\Sigma^{ij}][{{(\Omega)_{b}}^{m}}{(\Lambda^{*})_{a}}^{n}\Sigma^{*}_{mn}]~{},$
(66)

which can easily be checked by drawing a one-loop diagram, with $q_{L}$,
$q^{\prime}_{L}$, $Q_{R}$ running in the loop. After expanding it, we obtain a
flavon potential

$\Delta V_{s}\sim\frac{N_{c}}{8\pi^{2}}\lambda_{L}^{2}\lambda_{L}^{\prime
2}f^{4}\left|{(\Lambda)^{a}}_{i}{(\Omega^{*})^{b}}_{j}\Sigma^{ij}\right|^{2}\supset
M_{s}^{2}|s|^{2},$ (67)

where

$M_{s}^{2}\sim\frac{N_{c}}{8\pi^{2}}\lambda_{L}^{2}\lambda_{L}^{\prime
2}f^{2}~{}.$ (68)

Notice that we have chosen different (generations of) elementary quark
doublets, $q_{L}$ and $q_{L}^{\prime}$, in the two couplings such that the
leading order potential is the quadratic term $|s|^{2}$.

To have a nontrivial flavon VEV, we want to introduce interactions that
explicitly break the $U(1)_{F}$ symmetry. It can be achieved by mixing $q_{L}$
to both resonances, which have the quantum number $\mathbf{2}_{1/6,1/2}$ and
$\mathbf{2}_{1/6,-1/2}$, with coupling $\lambda_{L}$ and
$\lambda_{L}^{\prime\prime}$. In this way, the loop can be closed at
$\mathcal{O}(\lambda_{L}\lambda_{L}^{\prime\prime})$ and generate a $s$
tadpole term

$\Delta
V_{s}\sim\frac{N_{c}}{8\pi^{2}}\lambda_{L}\lambda_{L}^{\prime\prime}f^{4}\left(\epsilon_{ab}{(\Lambda)^{a}}_{i}{(\Omega^{*})^{b}}_{j}\Sigma^{ij}\right)\sim\kappa\,s~{},$
(69)

where

$\kappa\sim\frac{N_{c}}{8\pi^{2}}\lambda_{L}\lambda_{L}^{\prime\prime}g_{\psi}^{2}f^{3}.$
(70)

Combining the two potentials we got, the flavon VEV is given by

$v_{s}\sim\frac{\kappa}{M_{s}^{2}}\sim\frac{\lambda_{L}\lambda_{L}^{\prime\prime}g_{\psi}^{2}}{\lambda_{L}^{2}\lambda_{L}^{\prime
2}}f\propto\lambda_{L}^{\prime\prime}f,$ (71)

which is controlled by the explicit breaking coupling
$\lambda_{L}^{\prime\prime}$. If $\lambda_{L}^{\prime\prime}$ is small, we can
have $v_{s}\ll f$ with the desired value. Although the tadpole term shifts the
vacuum, it preserves the shape of the potential. That is, the masses of the
two flavon degrees of freedom, a scalar component $s$ and a pseudoscalar
component $a$, are the same with

$M_{s}=M_{a}\sim\sqrt{\frac{N_{c}}{8\pi^{2}}}\lambda_{L}\lambda_{L}^{\prime}f.$
(72)

The value is controlled by $\lambda_{L}$ and $\lambda_{L}^{\prime}$, which can
be large and lead to a heavy flavon.

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