#  Scalar Democracy

Christopher T. Hill<EMAIL_ADDRESS>Fermi National Accelerator Laboratory
P.O. Box 500, Batavia, Illinois 60510, USA  Pedro A. N. Machado
<EMAIL_ADDRESS>Fermi National Accelerator Laboratory
P.O. Box 500, Batavia, Illinois 60510, USA  Anders E. Thomsen
<EMAIL_ADDRESS>CP3-Origins, University of Southern Denmark
Campusvej 55, DK-5230 Odense M, Denmark  Jessica Turner<EMAIL_ADDRESS>Fermi National Accelerator Laboratory
P.O. Box 500, Batavia, Illinois 60510, USA

(August 27, 2024)

###### Abstract

We conjecture that there exists a scalar bound state for every pair of
fundamental fermions at a UV (“composite”) scale, $\Lambda\gg
v_{\text{weak}}$. This implies a large number of universally coupled, sub-
critical Higgs doublets. All but the Standard Model Higgs are “dormant,” with
large positive squared masses and each receives a small vacuum expectation
values via mixing with the Standard Model Higgs. Universal couplings, modulo
renormalization group running effects, flips the flavor problem into the
masses and mixings of the Higgs system. Doublets associated with heavy fermion
masses, $b,c,\tau$ likely lie in the multi-TeV range, but may be observable at
the current LHC, or a high-luminosity and/or an energy-upgraded LHC. In the
lepton sector we are lead to a Higgs seesaw for neutrino masses, and corollary
processes of observable flavor violation. The observation of the first
sequential doublet coupled to $\overline{b}b$ with masses $\lesssim 3.5$ TeV
would lend credence to the hypothesis.

###### pacs:

14.80.Bn,14.80.-j,14.80.-j,14.80.Da

††preprint: []FERMILAB-PUB-19-068-T
CP3-Origins-2019-003 DNRF90

##  I Introduction

In the present paper we propose that every fermion pair binds to form a
complex scalar boson, due to a universal attractive interaction at a very high
scale, $\Lambda$. Amongst many new states, including lepto-quarks, colored
isodoublets and singlets, etc., this hypothesis implies the existence of a
large number of Higgs bosons.

We assume the lightest of the these is the Standard Model Higgs (SMH). The
remaining doublets are sequentially heavier with positive $M^{2}$’s, i.e.,
“dormant.”111We use the term “dormant” as distinct from “inert”, which to us
implies electroweak sterile scalars. They will have universal couplings to
their constituent fermions at $\Lambda$, but renormalized couplings at the
electroweak scale that are $g\simeq 1$ for quarks, and $g_{\ell}\simeq 0.7$
for leptons. Each Standard Model (SM) fermion acquires its mass through its
coupling to the particular Higgs doublet it comprises, which in turn mixes
with the SMH to acquire a perturbative “tadpole” mass. In particular our
present model is “subcritical:” the negative $M^{2}$ of the SMH arises from
mixing with dormant Higgses.

The associated Higgs bosons thus become lighter for the heavier fermions, and
very heavy for the neutrinos. Though we have no theory of the masses and
mixings of the large array of composite scalars, we can make use of
phenomenology. The model then becomes predictive, essentially because the
Yukawa couplings are determined. We call this system “Scalar Democracy” as it
harkens back to the “Nuclear Democracy” of the late 1960’s.

Sequential Higgs bosons have certainly been considered previously, and we
cannot review the vast literature. Nonetheless, few theorists venture beyond a
few Higgs bosons. Many Higgs bosons arise in the context of an extended
gravity, such as a scheme described to us by Bjorken Bjorken , which first
inspired the idea of Scalar Democracy. A model that presages some of our
present discussion is the “Private Higgs” Porto and Zee (2008); BenTov and Zee
(2013). A Coleman-Weinberg mechanism and Higgs portal interactions lead one of
us to a second Higgs doublet coupled to $\overline{b}b$ with $g=1$ at about
$400$ GeV, (see Hill (2014) and a long list of references therein). In this
model, the SMH is a $\overline{t}t$ composite state Miransky  _et al._ (1989);
Bardeen  _et al._ (1990); Hill and Simmons (2003), but the predictions of
minimal composite $\overline{t}t$ are significantly modified by mixing. We
note that recent work attempting to construct an “asymptotically safe” UV
completion of the SM arrives at a similar structure in the Higgs sector to
ours Litim and Sannino (2014); Bond  _et al._ (2017); Abel  _et al._ (2018),
but for different reasons. The phenomenology of many Higgs bosons is, to our
knowledge, essentially unexplored.

In our particular scenario, we count 18 sequential colorless doublets in the
quark sector, and 18 in the lepton sector. In the quark sector this includes
the lightest doublet, $H_{0}$, associated with the top quark,
$(\overline{t}_{\mathrm{L}},\,\overline{b}_{\mathrm{L}})\tilde{H}_{0}t_{\mathrm{R}}$
(where $\tilde{H}_{0}$ is the charge conjugated Higgs), which is identified
with the SM Higgs doublet. This establishes the universal quark sector Yukawa
coupling to be $g=1$. The quark masses and mixings are then determined by the
spectrum of the 18 quark-sector Higgs doublets. We will see the Higgs doublet
associated with the $b$-quark,
$g(\overline{t}_{\mathrm{L}},\,\overline{b}_{\mathrm{L}})H_{b}b_{\mathrm{R}}$,
is expected to have a mass $\lesssim 3.5$ TeV. We also have 18 doublets in the
charged lepton sector. With the exception of $H_{\tau}$ these tend to be much
heavier as lepton masses are small. This framework provides three alternatives
for the neutrino mass generation. Interestingly, neutrinos could be Dirac or
Majorana via a type I or II seesaw mechanism.

Many of the scalars providing quark and lepton masses are well beyond present
collider reach. The lighter ones, associated with the $b$ and $c$ quarks, as
well as the $\tau$ lepton may have thresholds in the sub-10-TeV range and thus
accessible to the LHC and future upgrades and higher energy machines. However,
the heavier states may leave indirect imprints on flavor-changing observables
in the quark and lepton sectors.

Scalar Democracy for us is new dynamics with subcritical compositeness due to
a universal interaction in the far UV. We thus blend a few key ideas from
compositeness and mainly emphasize that the Yukawa couplings are universal at
$\Lambda$, subject only to renormalization group evolution from $\Lambda$ to
the weak scale. This, together with the input masses and mixings of the Higgs
bosons, flips the quark and lepton flavor problems away from the issue of
understanding a fundamental Yukawa-coupling matrix. The puzzle of fermion mass
hierarchy becomes one of understanding and disentangling the multi-Higgs mass
spectrum. This conversion of the flavor problem is an interesting exercise in
its own right.

It should be emphasized that it is hard to understand the small Yukawa
couplings in the SM, such as $g_{\mathrm{electron}}\sim 10^{-6}$. This cannot
be generated perturbatively from zero, owing to the chiral symmetry of the
electron. It is natural that $g_{\mathrm{electron}}\sim 1$ in its coupling to
a new Higgs $H_{e}$, but through mass mixing, the induced coupling to the SMH
becomes small. This is a key motivation for a scheme such as the one presented
here.

As we have emphasized, the present scheme does not provide any explanation of
the multiple Higgs masses and mixings. We will not concern ourselves with the
overall naturalness, and treat symmetry-breaking effects as inherent in the
Higgs mass terms. This is analogous to “soft-symmetry-breaking” in chiral
Lagrangians and resonance models of the 1960’s. We do impose fine-tuning
constraints in the low-energy effective theory of $H_{b}$, where the mixing
generates the largest feedback on the SMH, $H_{0}$ (level repulsion), and the
negative $M^{2}$ of the SMH can arise from this effect.

It is our conclusion, at least at a first pass with various simplifying
assumptions made along the way, that such a theory can exist. New phenomena
may show up at a high luminosity and/or energy-doubled LHC, and certainly at a
$100$ TeV collider. $H_{b}$ in the lightest mass limit $\sim 1$ TeV is already
accessible to the LHC, while it would be seen in the higher mass range $\sim
3$ TeV in upgraded LHC runs. In our opinion, a robust theoretical spectroscopy
of multiple Higgs bosons offers a rationale for luminosity and energy upgrades
of the LHC and future ultra energetic machines. If the upgraded LHC were to
fail to discover a pair of isodoublet Higgs bosons with universal coupling,
such as the lightest $H_{b}$ or $H_{\tau}$ states in our model, then this
scheme would be disfavored.

Analyzing, at least schematically, the phenomenological consequences and
constraints of the Scalar Democracy hypothesis is our main goal. We begin in
Section II with a theoretical “motivation” for this perspective. Then in
Section III, we summarize the dynamics of our model at low energies and arrive
at a fairly simple effective Lagrangian describing the couplings to quarks and
leptons of the multi-Higgs spectrum. This is followed by a more detailed
discussion of phenomenology in the subsequent section, Section IV. Here we
discuss the main production channels and collider prospects for the dormant
Higgses. We will also discuss the implications of the dormant Higgses on
flavor physics and the resulting bounds on their masses.

A reader interested in a summary of the observable features of the model,
including neutrino masses, may skip directly to Section IV.

##  II Theoretical motivation

###  II.1 Are There Many Scalars in Nature?

Gravity is a universal attractive interaction. All pairs of fundamental
fermions must have attractive gravitational scattering amplitudes. Near the
Planck scale this may involve exotic, new strong dynamics, new condensates,
instantons, etc., and perhaps a new way to generate hierarchies.

For example, enhanced gravitational interactions may trigger condensates,
dynamically generating Majorana masses for the right-handed neutrinos
Barenboim (2010). Similar effects may arise through gravitational instantons
Dvali and Funcke (2016). Alternatively, intriguing extensions of gravity, such
as brane-world models, extra dimensions Randall and Sundrum (1999), or
“bigravity” Arkani-Hamed  _et al._ (2003); de Rham and Gabadadze (2010), have
ingredients that may likewise produce universal attractive interactions at
various scales. We can also generate a large subset of these scalars by
postulating new, strong gauge dynamics, which is more concrete but will not be
developed presently.

We assume that a universal attractive interaction generates bound state scalar
fields at a high energy scale $\Lambda$ (which may be of order
$M_{\text{Planck}}$, but could be lower). Our hypothesis is general and
transcends a wide class of possible models. While we invoke a universal
pairing force, such as gravity, this is nonetheless a schematic proposal.
However, when we tie this to the SMH it becomes predictive.

We suppose the new scalar bosons are field-theoretic bound states of pairs of
SM fermions. In analogy to the Nambu-Jona-Lasinio (NJL) model Nambu and Jona-
Lasinio (1961), the constituent fermions are free (unconfined) and any fermion
pair couples with a common Yukawa coupling $g$ at scale $\Lambda$ to its
bound-state scalar (the usual non-relativistic intuition does not apply.). If
the new interaction is medium-strong (subcritical), then bound state scalars
will form with masses which will be below $\Lambda$ and positive. The
formation of these bound states does not break any symmetries since a
composite scalar inherits the quantum numbers of its constituents. Hence the
scalars, in the absence of mixing effects, are presumably degenerate with mass
$M^{2}<\Lambda^{2}$ and cannot break flavor or gauge symmetries which would
trigger proton decay, etc.

We do not have a theory of the scalar mass spectrum, in particular, and do not
provide a fine-tuning mechanism to generate a large hierarchy. Moreover, we
require symmetry-breaking and mixing effects to further empower the scenario.
All of these effects will be considered to be “soft-symmetry-breaking” or
“relevant” operators, and simply inserted by hand.

While the dynamics determining the spectrum is unknown, we know that it must
respect the SM gauge symmetries. Rather than trying to concoct a theory of
masses and mixing angles amongst a large number of Higgs doublets over a large
range of scales, we will assume such a theory exists and let phenomenology
guide us.

We connect the hypothesis with the SM by postulating that the SMH is the
lightest scalar doublet. This would then be the
$(\overline{t}_{\mathrm{L}},\,\overline{b}_{\mathrm{L}})t_{\mathrm{R}}$ bound-
state element of the scalar complex Miransky  _et al._ (1989); Bardeen  _et
al._ (1990). The top-quark Yukawa coupling, $g$, is then the universal
coupling for all quark pairs to their particular Higgs doublets. Leptons will
likewise have a universal coupling to their Higgs bosons, $g_{\ell}$, but we
expect that the renormalization group (RG) running will yield a ratio
$g_{\ell}/g\sim 0.7$, due mainly to QCD (this is the analogue of the
$\mathrm{SU}(5)$ relation for $m_{b}/m_{\tau}$ Georgi and Glashow (1974)). In
fact $g=1$ is close to the RG fixed point for the top quark Pendleton and Ross
(1981); Hill (1981), which is not far from the prediction of compositeness
Bardeen  _et al._ (1990). In fact, this scheme can bring the RG fixed point
into concordance with $m_{t}=173$ GeV (see Eq. (16) and discussion below).

###  II.2 Sketch of the Dynamics

The dynamics of the lowest-lying sequential Higgs bosons is rather simple. Let
us anticipate the dynamical implications in the case of the top and bottom
quarks, and the three generations of neutrino masses.

The SMH, $H_{0}$, in isolation has the usual potential:

$V_{\text{Higgs}}=-M_{0}^{2}H_{0}^{\dagger}H_{0}+\frac{\lambda}{2}(H_{0}^{\dagger}H_{0})^{2},$
(1)

where $-M_{0}^{2}$ is negative, and phenomenologically $M_{0}\simeq 88$ GeV,
$\lambda\simeq 1/4$ (the observed physical Higgs boson mass is
$\sqrt{2}M_{0}\simeq 125$ GeV).

New sequential Higgs doublets, $H_{x}$, are “dormant,” meaning they have the
usual SMH electroweak quantum numbers, but owing to large, positive mass
terms, $M_{x}^{2}H_{x}^{\dagger}H_{x}$, they do not directly undergo
condensation. However, in order to generate the light quark and lepton masses,
they must have small mixings to the SMH,

$\displaystyle V$ $\displaystyle=$ $\displaystyle
M_{H}^{2}H_{0}^{\dagger}H_{0}+\frac{\lambda}{2}(H_{0}^{\dagger}H_{0})^{2}$ (2)
$\displaystyle+\sum_{x}\left(M_{x}^{2}H_{x}^{\dagger}H_{x}-\mu_{x}^{2}H_{0}^{\dagger}H_{x}\;+\;\mathrm{h.c.}\;\right),$

parametrized by $\mu_{x}$. The vacuum expectation values (VEV) of the dormant
Higgses are small, so the quartic terms should generally be negligible, and we
ignore it.

The mass mixing causes each $H_{x}$ to acquire a small VEV (“tadpole”) of
order

$\left\langle H_{x}\right\rangle=\binom{0}{\mu_{x}^{2}v/M_{x}^{2}},$ (3)

where $v\simeq 174$ GeV is the electroweak VEV.

Due to the mixing $H_{0}$ is then “level-repelled” down by an amount or order
$\mu^{4}_{x}/M^{2}_{x}$:

$-M_{0}^{2}=M_{H}^{2}-\sum_{x}\frac{|\mu_{x}^{2}|^{2}}{M_{x}^{2}}.$ (4)

We refer the interested reader to Ref. Dobrescu and Fox (2010) for a similar
implementation of the mechanism in supersymmetric models. Therefore, starting
from a positive mass term, $M_{H}^{2}$, in Eq. (2), this cumulative effect can
explain why the SMH boson has the tachyonic, or negative, $-M_{0}^{2}$, in Eq.
(1).

Hence, in this scenario there is only a single condensate, associated with a
conventional “Mexican hat potential,” i.e., the SMH. The rest of the
sequential Higgs bosons remain approximately pure doublets acquiring small
tadpoles via mixing, which is generally suppressed by $1/M_{x}^{2}$.

Let us presently anticipate the main discussion and illustrate how this setup
operates for the top-bottom sub-system. There we have the Yukawa interactions:

$\displaystyle\mathcal{L}_{\mathrm{yuk}}=-g(\overline{t}_{\mathrm{L}},\,\overline{b}_{\mathrm{L}})\tilde{H}_{0}t_{\mathrm{R}}-g(\overline{t}_{\mathrm{L}},\,\overline{b}_{\mathrm{L}})H_{b}b_{\mathrm{R}}.$
(5)

The top mass determines the common Yukawa coupling to be $g=m_{t}/v\simeq 1$.

The $b$-quark then receives its mass from $H_{b}$. By assuming mixing
$M_{b}^{2}H_{b}^{\dagger}H_{b}-\mu_{b}^{2}H_{0}^{\dagger}H_{b}+\text{h.c.}$,
we find the induced tadpole VEV to be $\left\langle
H_{b}\right\rangle=(0,\,v_{b})$ where $v_{b}=v{\mu_{b}^{2}}/{M_{b}^{2}}$. This
implies that the $b$-quark mass is

$m_{b}=gv\frac{\mu_{b}^{2}}{M_{b}^{2}}=m_{t}\frac{\mu_{b}^{2}}{M_{b}^{2}},$
(6)

while the Higgs boson mass is222Here and throughout the paper we will use
$M_{x}$ to denote Higgs masses and $m_{x}$ to denote fermion masses.

$-M_{0}^{2}=M_{H}^{2}-\frac{\mu_{b}^{4}}{M_{b}^{2}}.$ (7)

In order to avoid fine-tuned cancellations between the two terms on the RHS of
the above equation we anticipate that

$\frac{\mu_{b}^{2}}{M_{b}}\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}100\;\makebox{GeV},$
(8)

providing us with the estimate for an upper “naturalness” bound on $M_{b}$:
333Note that, if we assumed $M_{H}^{2}=0$, then we could generate all of the
tachyonic Higgs mass from the mixing with $H_{b}$ and we would predict:
$M_{b}=(88\;\text{GeV})m_{t}/m_{b}=3.36$ TeV.

$M_{b}\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}\frac{m_{t}}{m_{b}}\cdot
100\;\text{GeV}\simeq 3.5\;\text{TeV}.$ (9)

This is not a firm upper bound, as it is based on a tuning argument. Another
possible way of parametrizing the low scale fine-tuning would be to assume
that all $\mu$ parameters are at the electroweak scale, for instance $\mu=100$
GeV. We will use these two tuning criteria later as benchmarks for comparison.
Furthermore, we can certainly have a lighter $M_{b}$ with less fine-tuning. We
note that a mass $M_{b}\sim 380$ GeV was previously obtained in a scheme in
which the $H_{b}$ with $g=1$ drives a Coleman-Weinberg potential for the SMH,
and this has not to our knowledge been ruled out by the LHC Hill (2014).

We will find that this simple scheme is consistent with flavor-physics
constraints. Interestingly, flavor dynamics is not then a fundamental
consequence of the Yukawa coupling matrices as in the SM, but rather the low-
energy suppression of flavor-changing neutral currents (FCNCs) is a
consequence of the heaviness of these states—we also find some natural limits
in which the structure of the theory is simplified and in which the
phenomenological constraints are easy to understand.

The lightest doublets beyond SMH are $H_{b}$ associated with the $b$-quark,
$H_{\tau}$ associated with $\tau$, and $H_{c}$ associated with charm (and
possibly $H_{ct}$ and $H_{tc}$, depending on how the CKM matrix is generated,
as we will see later). Depending on mixing assumptions, $H_{\tau}$ could in
principle be lighter than $H_{b}$. We thus encourage LHC experimentalists to
consider searching for these objects, and we discuss collider phenomenology in
Section IV.1 and Section IV.2.

At the other end of the spectrum, we have the physics of neutrinos and charged
leptons. In the Scalar Democracy, there are three alternatives for the
neutrino mass mechanism. For example, where neutrinos are Dirac their masses
are generated like any other fermion in the present model. The Yukawa
couplings for the neutrinos are similar to those for quarks and leptons, and
the appropriate neutrino mass terms are generated by mixing of the dormant and
SM Higgses. We generate neutrino mixings and small neutrino masses by having
ultra-large and positive $M^{2}$ for the neutrino-Higgs fields in the matrix,
e.g.,

$m_{\nu_{\alpha}}\sim g_{\ell}v\frac{\mu_{\alpha}^{2}}{M_{\nu_{\alpha}}^{2}}.$
(10)

For neutrino masses of order $10^{-2}$ eV we thus have $M_{\nu_{\alpha}}\sim
10^{11}$ GeV, too heavy for any observable collider or flavor signature.
Notwithstanding, the Scalar Democracy scenario may also lead to Majorana
neutrinos via type-I and type-II seesaw mechanisms. In the type-I seesaw case,
a light sterile neutrino is predicted, with mass in the keV$-$GeV range. A
general discussion of all these possibilities will be provided in Section V.1.

###  II.3 Scalar Democracy and Counting Scalars

If we assume one boundstate per fermion pair at some high scale $\Lambda$,
then we can count the number of composite scalars in the theory.

All of the SM matter fields can be represented by $48$ two-component left-
handed spinors, $\psi_{A}^{i}$. This includes all the left-handed and anti-
right-handed fermions. We can collect these into a large global
$\mathrm{SU}(48)\times\mathrm{U}(1)$ multiplet, corresponding to the global
symmetry assuming that the new dynamics are blind to the SM gauge
interactions. We emphasize that this is a dynamical symmetry, and familiar GUT
theories that contain only the SM fermions will be gauged subgroups of this
$\mathrm{SU}(48)$. Here the indices $\left(i,j\right)$ run over all the $48$
flavor, doublet, and color degrees of freedom of the SM fermions.

The most general scalar-field bilinear interaction we can construct of these
fields is

$\displaystyle\epsilon^{AB}\psi_{A}^{i}\psi_{B}^{j}\Theta_{ij}\;+\;\mathrm{h.c.}\;,$
(11)

where $\Theta_{ij}$ transforms as the symmetric $\mathbf{1176}$ representation
of the $\mathrm{SU}(48)\times\mathrm{U}(1)$ (this is analogue to the sextet
representation of $\mathrm{SU}(3)$). The field $\Theta_{ij}$ contains many
complex scalar fields with assorted quantum numbers, including baryon and
lepton number, color, and weak charges.

To make contact with the SM fields, we consider the usual $24$ left-handed
quarks and leptons, $\Psi_{\mathrm{L},i}$, and the $24$ right-handed
counterparts, $\Psi_{\mathrm{R},\widehat{i}}$. The index $i$ now runs over the
chiral $\mathrm{SU}(24)_{\mathrm{L}}$ and $\widehat{i}$ over the chiral
$\mathrm{SU}(24)_{\mathrm{R}}$ subgroups of $\mathrm{SU}(48)$. With this
notation, we can construct three interactions with bilinear fermion fields,

$\displaystyle\Phi_{i\widehat{j}}\overline{\Psi}_{L}^{i}\Psi_{R}^{\widehat{j}}+\Omega_{ij}\overline{\Psi}_{L}^{i}\Psi_{R}^{jC}+\widehat{\Omega}_{\widehat{ij}}\overline{\Psi}_{R}^{\widehat{i}}\Psi_{L}^{\widehat{j}C}+\text{h.c}.,$
(12)

where $\Phi_{i\widehat{j}}$ is the
$(\mathbf{24}_{\mathrm{L}},\,\mathbf{24}_{\mathrm{R}})$ complex scalar field
with $24^{2}=576$ complex degrees of freedom. $\Omega$ and $\widehat{\Omega}$
are the symmetric $\mathbf{300}$ representation of
$\mathrm{SU}(24)_{\mathrm{L}}$ and $\mathrm{SU}(24)_{\mathrm{R}}$ respectively
444 Notation: If $\Psi_{\mathrm{L}}^{j}=\frac{1-\gamma^{5}}{2}\Psi^{j}$ is a
left-handed spinor transforming as a $\mathbf{24}$ under
$\mathrm{SU}(24)_{\mathrm{L}}$, then
$(\Psi_{\mathrm{L}}^{j})^{C}=i\gamma^{2}\gamma^{0}\frac{(1-\gamma^{5})^{\ast}}{2}(\Psi^{j})^{\ast}=\left(i\gamma^{2}\gamma^{0}\right)\frac{1-\gamma^{5}}{2}\left(-i\gamma^{2}\gamma^{0}\right)\Psi^{jC}=\frac{1+\gamma^{5}}{2}\Psi^{jC}=(\Psi^{jC})_{\mathrm{R}}\equiv\Psi_{\mathrm{R}}^{jC}$
(in the notation of Bjorken and Drell) transforms as a
$\overline{\mathbf{24}}$ under $\mathrm{SU}(24)_{\mathrm{L}}$ and has a
$\frac{1+\gamma^{5}}{2}$ projection. . Thus we have
$\Phi\left(576\right)+\Omega\left(300\right)+\widehat{\Omega}\left(300\right)=\allowbreak\Theta\left(1176\right)$,
matching the degrees of freedom of $\Theta_{ij}$. Here $\Omega_{ij}$ and
$\widehat{\Omega}_{ij}$ are the analogues of Majorana masses and carry fermion
number, while $\Phi$ contains fermion number neutral fields, such as Higgs
fields, in addition to ($B-L$) leptoquark multiplets.

The $\Phi,\Omega$ and $\widehat{\Omega}$ fields can be viewed as the
“composite fields” arising from a NJL model effective description of the new
forces. Consider just the
$\mathrm{SU}(24)_{\mathrm{L}}\times\mathrm{SU}(24)_{\mathrm{R}}\times\mathrm{U}(1)\times\mathrm{U}(1)_{A}$
invariant NJL model:

$\displaystyle-\frac{g^{2}}{M^{2}}(\overline{\Psi}_{\mathrm{L}}^{i}\Psi_{\mathrm{R}}^{j})(\overline{\Psi}_{\mathrm{R},i}\Psi_{\mathrm{L},j}),$
(13)

where the negative sign denotes an attractive interaction in the potential. It
should be noted that we can equally well write current-current (and tensor-
tensor) interactions, mediated by heavy spin-1 bosons (or Pauli-Fierz spin-2
gravitons); these will generally contain scalar channels and will Fierz
rearrange to effectively reduce to Eq. (13) with the attractive signs. There
also exist the possibility of the following NJL models:

$\displaystyle-\frac{g^{2}}{M^{2}}(\overline{\Psi}_{\mathrm{L}}^{i}\Psi_{\mathrm{R}}^{Cj})(\overline{\Psi}_{\mathrm{R},i}^{C}\Psi_{\mathrm{L},j})\;\;\makebox{or}\;\;\left(\mathrm{R}\leftrightarrow\mathrm{L}\right),$
(14)

which lead to the composite bosons $\Omega$ and $\widehat{\Omega}$. Such
universal master interactions may arise as sub-sectors of more general
gravitational scattering amplitudes with many other effects near the Planck
scale, $M\sim M_{\text{Planck}}$, including gravitational instantons or, at
lower energy scales, e.g., from a strong bigravity force. Eq. (13) by itself
can therefore be a starting point of a discussion of a dynamically generated
extended Higgs boson spectrum.

The first step to solving an NJL theory would be to factorize the interaction
of Eq. (13) by introducing auxiliary scalar fields. This leads to the equation
we started with, Eq. (12), where $\Phi,\Omega$ and $\widehat{\Omega}$ are
auxiliary fields. The universal interaction will bind fermion pairs into
scalars that are bound states of ordinary quarks and leptons and will generate
a plethora of Higgs doublets. These bound states will have a universal Yukawa
coupling $g$ at the scale $M^{2}$. Moreover, with $g$ taking on a near-but-
subcritical value, these bound states will generally have large positive
masses but can be tuned to be lighter than $M$.

Symmetry-breaking effects will be required to split the spectroscopy,
including the SMH down to its observed negative mass term. All other doublets
remain heavy, but will mix. The problem of solving an NJL model in the
large-$N$ fermion loop approximation, or equivalently by the RG, is discussed
in detail in Refs. Bardeen  _et al._ (1990); Hill and Simmons (2003).

If $g$ is supercritical then some or all multiplets will acquire negative
renormalized masses, $M^{2}(\mu\rightarrow 0)<0$ and the theory develops a
vacuum instability. For example, the field $\Phi^{ij}$ with a supercritical
coupling will generally condense into a diagonal VEV,
$\left\langle\Phi_{ij}\right\rangle=V\delta_{ij}$, and this would become a
spontaneously broken $\Sigma$-model of
$\mathrm{SU}(24)_{\mathrm{L}}\times\mathrm{SU}(24)_{\mathrm{R}}\times\mathrm{U}(1)\times\mathrm{U}(1)_{A}$.
In this supercritical case, all the fermions would acquire large, diagonal
constituent masses of order $gV$, inconsistent with observation.

The structure we have just outlined, even if subcritical, will contain many
composite Higgs doublets. While we want to have a reasonably deep binding of
these scalars, the system must be near-to-but-subcritical such that no large
VEVs will form. We would then expect a positive, diagonal mass-squared matrix
amongst the many composite scalar states and all fermions would be massless.
However, the effect of mixing, i.e., off-diagonal mass terms, can arise from
“extended interactions” in analogy to “extended technicolor” or latticized
extra dimensions.

Exactly how the scalar mass spectrum is generated is beyond the scope of our
present discussion. We will simply assume such a spectrum of masses and
mixings between the bound state scalars that allows for a light sector from
the SMH to multi-TeV scales exists and extends up to the highest scales. We
assume that $\Omega_{ij}$, $\widehat{\Omega}_{ij}$ and the color-carrying weak
doublets have very large positive $M^{2}$ and therefore we will ignore them.

Let us examine the quantum numbers of the spectrum of states in the
$\Phi^{ij}$ system. There are $24\times 24=\allowbreak 576\ $ composite
complex scalars, and these devolve into the following states upon gauging the
fermions:

  * •

$9\times(\mathbf{1},\,\mathbf{2},\,\tfrac{1}{2})\sim\overline{Q}_{\mathrm{L}}U_{\mathrm{R}}$;
$3^{2}\times 1\times 2=18$ complex degrees of freedom (DoFs),

  * •

$9\times(\mathbf{1},\,\mathbf{2},\,-\tfrac{1}{2})\sim\overline{Q}_{\mathrm{L}}D_{\mathrm{R}}$;
$3^{2}\times 1\times 2=18$ complex DoFs,

  * •

$9\times(\mathbf{1},\,\mathbf{2},\,\tfrac{1}{2})\sim\overline{L}_{\mathrm{L}}N_{\mathrm{R}}$
leptonic; $3^{2}\times 1\times 2=18$ complex DoFs,

  * •

$9\times(\mathbf{1},\,\mathbf{2},\,-\tfrac{1}{2})\sim\overline{L}_{\mathrm{L}}E_{\mathrm{R}}$
leptonic; $3^{2}\times 1\times 2=18$ complex DoFs,

  * •

$9\times(\mathbf{8},\,\mathbf{2},\,\pm\tfrac{1}{2})\sim\overline{Q}_{\mathrm{L}}\lambda^{a}U_{\mathrm{R}}[D_{\mathrm{R}}]$;
$3^{2}\times 8\times 2\times 2=288$ complex DoFs,

  * •

$9\times(\mathbf{3},\,\mathbf{2},\,\tfrac{1}{6}[-\tfrac{5}{6}])\sim\overline{L}_{\mathrm{L}}U_{\mathrm{R}}[D_{\mathrm{R}}]$;
$3^{2}\times 3\times 2\times 2=108$ complex DoFs,

  * •

$9\times(\overline{\mathbf{3}},\,\mathbf{2},\,-\tfrac{1}{6}[-\tfrac{7}{6}])\sim\overline{Q}_{\mathrm{L}}N_{\mathrm{R}}[E_{\mathrm{R}}]$;
$3^{2}\times 3\times 2\times 2=108$ complex DoFs,

where the brackets denote the SM quantum numbers. The first four entries in
the above list are the $36$ Higgs doublets in the quark and lepton sectors
respectively.

If we consider the $18$ scalars in the quark sector, ignoring their masses and
EW charges, we will have a Yukawa interaction at the scale $\Lambda$ that is
$\mathrm{SU}(6)_{\mathrm{L}}\times\mathrm{SU}(6)_{\mathrm{R}}$ invariant and
of the form:

$\displaystyle
g\overline{\Psi}_{\mathrm{L}}\Sigma\Psi_{\mathrm{R}}\;+\;\mathrm{h.c.}\;,$
(15)

where $\Psi=(u,d,c,s,t,b)$ and $\Sigma$ is a $6\times 6$ complex matrix
composed of $18$ doublets. The renormalization group equation for $g$ is then
determined to be

$\displaystyle(16\pi^{2})\frac{\mathrm{d}g}{\mathrm{d}\ln(\mu)}=g(9g^{2}-8g_{3}^{2}-\kappa)$
(16)

at one-loop order, where $\kappa$ includes the smaller electroweak corrections
(which breaks the
$\mathrm{SU}(6)_{\mathrm{L}}\times\mathrm{SU}(6)_{\mathrm{R}}$ invariance).
This describes the running of $g$ down to scales at which the various Higgs
doublets decouple. Likewise, we have an equation for the lepton sector,
$g\rightarrow g_{\ell}$, where the $-8g_{3}^{2}$ term is dropped. Assuming the
Planck mass corresponds to a Landau pole in $g$ and that all Higgs bosons are
active down to the electroweak scale, and then we derive $g\simeq 0.93$ and
$g_{\ell}\simeq 0.71$ (this also leads to some “fine-structure” as the
electroweak terms in $\kappa$ split degeneracy of between the Yukawa couplings
for the up- and down-type quarks; the full details of this are beyond the
scope of this paper).

This result implies a top quark mass of $\simeq 161$ GeV. This is the
prediction of the modified RG fixed point (equivalent to a focus point) of
Hill (1981) including additional Higgs bosons, and represents a significant
improvement over the original minimal top condensation models Bardeen  _et
al._ (1990). This prediction is robust with respect to the precise values,
$\Lambda$ and $g(\Lambda)$. If we include the masses of the heavier Higgs
bosons (as discussed below) and decouple them at their thresholds, the
prediction will increase and we expect it to converge on the observed top
quark mass.

Grand unification is greatly complicated by such dynamics. We relegate such an
investigation to future work.

##  III Low Energy Effective Theory

The observed Higgs boson must reside amongst the color singlet
$\overline{Q}_{\mathrm{L}}q_{\mathrm{R}}$ doublets. We proceed under the
simplifying assumption that the SM Higgs doublet, $H_{0}$, can be identified
with the doublet that couples to the fermionic combination of a top quark pair

$(\overline{t}_{\mathrm{L}},\,\overline{b}_{\mathrm{L}})t_{\mathrm{R}}\sim
H_{0}.$ (17)

This is the unique logical choice, as it has the largest Yukawa coupling in
the Standard Model and our theory dictates that all quarks will have this
universal coupling. This therefore recovers in part the top-condensation
models Miransky  _et al._ (1989); Bardeen  _et al._ (1990); Hill and Simmons
(2003).

We further reduce the scope of the problem by assuming all of the dormant
Higgs doublets apart from the 36 color singlet
$\overline{Q}_{\mathrm{L}}q_{\mathrm{R}}$ and
$\overline{L}_{\mathrm{L}}\ell_{\mathrm{R}}$ doublets are arbitrarily heavy
and therefore decoupled from the low-energy effective theory. Although the
phenomenology of these other scalars could be very interesting, this framework
does not provide any insight on their mass scale. Besides, we will assume for
concreteness that the neutrino mass mechanism is the same as for charged
fermions, and comment on alternatives in Section V.1.

At this point, it is convenient to resort to a common notation for the
individual Higgs bosons. Below the scale $\Lambda$, the effective theory we
are considering is a modification of the SM where the Higgs sector has been
replaced by

$\mathcal{L}\supset\left|D_{\mu}H^{\prime}\right|^{2}-V(H^{\prime})-g\,\overline{Q}_{\mathrm{L}}^{\prime
i}H_{ij}^{\prime d}D_{\mathrm{R}}^{\prime
j}-g\,\overline{Q}_{\mathrm{L}}^{\prime i}\tilde{H}_{ij}^{\prime
u}U_{\mathrm{R}}^{\prime j}\\\ -g_{\ell}\,\overline{L}_{\mathrm{L}}^{\prime
i}H_{ij}^{\prime e}E_{\mathrm{R}}^{\prime
j}-g_{\ell}\,\overline{L}_{\mathrm{L}}^{\prime
i}\tilde{H}_{ij}^{\prime\nu}N_{\mathrm{R}}^{\prime j}\;+\;\mathrm{h.c.}\;$
(18)

The generational indices of the fermions are labeled by indices $i,j$ and the
primes indicate that we are working in the gauge eigenbasis. The Higgs
doublets are denoted by $H^{\prime f}_{ij}$ in their flavor eigenbasis, where
$f=u,d,\nu,e$ represents the fermion type which acquire mass from the doublet
and $i,j$ the generations they couple to. Each doublet has an upper charged
component and a lower neutral component, $H_{x}=(h_{x}^{+},h^{0}_{x})$ and we
employ the charge conjugation convention $\tilde{H}=i\sigma_{2}H^{\ast}$. All
fermionic masses and mixings are due to the VEVs of the Higgs bosons.
Universality implies that all quarks have a Yukawa coupling $g\simeq 1$ and
leptons $g_{\ell}\simeq 0.7$.

###  III.1 The Higgs Potential

Formally, we can define the 36 Higgs doublets as a “vector,” where we separate
out the (mostly) SM-like Higgs: $\left(H_{0}^{\prime},H_{a}^{\prime}\right)$.
Thus, in addition to the SM-like Higgs, $H_{0}^{\prime}$, we have 35 doublets
represented as,
$H^{\prime}_{a}=\left(H_{1}^{\prime},\ldots,H_{35}^{\prime}\right)$. Here, we
define new notation and apologize for that. Nevertheless, this index notation
will only be applied in this subsection. In the remainder of the paper, we
revert to the notation $H^{f}_{ij}$.

We may write out the multi-Higgs boson potential as

$\begin{split}V=\,&M_{H}^{2}H_{0}^{\prime\dagger}H_{0}^{\prime}+\frac{\lambda}{2}\left|H_{0}^{\prime}\right|^{4}+H^{\prime\dagger}_{a}M_{ab}^{2}H^{\prime}_{b}\\\
&\quad-\left(H^{\prime\dagger}_{a}\mu^{2}_{a}H_{0}^{\prime}\;+\;\mathrm{h.c.}\;\right),\end{split}$
(19)

where the positive quartic interactions of the dormant Higgses are taken to be
negligible due to their large masses555A universal quartic coupling is another
option for the quartic terms that confine Higgs mixing to the mass matrix..
Thus the Higgs mixing is determined by the mass matrix. The mass terms that
mix $H_{a}^{\prime}$ with $H_{0}^{\prime}$ can be viewed as vector,
$\mu^{2}_{a}=\left(\mu_{1}^{2},...,\mu_{35}^{2}\right)$, and $M^{2}_{ab}$ is a
$35\times 35$ hermitian mass matrix amongst the heavy scalars, which is taken
to have positive eigenvalues.

It is important to note that there are two distinct types of mixing in this
model sourced by $\mu^{2}_{a}$ and the off-diagonal part of $M_{ab}^{2}$
respectively. The former is responsible for mixing the SMH into all the Higgs
flavor states, while the latter introduces mixing between the dormant Higgses.
We will regard the off-diagonal elements, $\mu^{2}_{a}$, as perturbatively
small when compared to $M_{ab}^{2}$.

The negligible quartic couplings leads to a mass degeneracy of the charged and
neutral Higgs components of the dormant doublets. Therefore, retaining the
symmetry-breaking mass terms, the dormant Higgs fields will acquire tadpole
VEVs via the $\mu^{2}_{a}H^{\prime\dagger}_{a}H_{0}^{\prime}$ mixings, but
their fluctuating fields will be degenerate doublets.

The Higgs mass terms are diagonalized at leading order (LO) in
$M_{H}^{2},\mu_{a}^{2}/M^{2}_{ab}$ by going to the basis defined by

$\begin{split}H^{\prime}_{a}&=H_{a}+M^{-2}_{ab}\mu_{b}^{2}H_{0}+\ldots\,,\\\
H_{0}^{\prime}&=H_{0}-\mu_{a}^{2\ast}M_{ab}^{-2}H_{b}+\ldots\,,\end{split}$
(20)

where $H_{0},H_{a}$ defines the physical Higgs doublets before EWSB. After
rotating away the $\mu_{a}$ mixing, the equations of motion for the VEVs of
the Higgs fields read

$\begin{split}-M_{0}^{2}\left\langle
H_{0}\right\rangle+\lambda\left|\left\langle
H_{0}\right\rangle\right|^{2}\left\langle H_{0}\right\rangle&=0,\\\
M^{2}_{ab}\left\langle H_{b}\right\rangle&=0,\end{split}$ (21)

where

$-M_{0}^{2}=\left(M_{H}^{2}-\mu^{2}_{a}M^{-2}_{ab}\mu^{2}_{b}\right)\simeq-\left(88\,\mathrm{GeV}\right)^{2},$
(22)

is the mass term for the SMH. The Higgs tadpoles are thus given by

$\left\langle
H_{0}\right\rangle=\binom{0}{v}\quad\mathrm{and}\quad\left\langle
H_{a}\right\rangle=\bm{0},$ (23)

where $v^{2}=-M_{0}^{2}/\lambda\simeq(174\,\mathrm{GeV})^{2}$ is the SM VEV.

From our assumptions of the Higgs potential, we ignore further mixing between
the massive Higgs bosons. The dormant Higgses constitute degenerate doublets
and $H_{0}$ is directly identified with the SMH, its components being the
physical Higgs and the 3 EW Goldstone Bosons. Eq. (20) shows how $H_{0}$ is
mixed into all the Higgs flavor states. Accordingly, the dormant Higgses
acquire small VEVs which in turn produces the fermion masses in our model.

###  III.2 Fermion masses and mixing

The fermions acquire their masses from the Yukawa couplings as shown in Eq.
(18). In general, the mass matrices for the fermions are determined by the
VEVs of the corresponding Higgs flavor eigenstates;

$m_{ij}^{f}\equiv\left[\mathcal{L}_{f}^{\dagger}\,\mathrm{diag}(m_{1}^{f},\,m_{2}^{f},\,m_{3}^{f})\mathcal{R}_{f}\right]_{ij}=g_{f}\left\langle
h^{\prime f,0}_{ij}\right\rangle,$ (24)

where $g_{f}$ is the Higgs coupling to quarks ($g$) or leptons ($g_{\ell}$),
$\mathcal{L}_{f},\mathcal{R}_{f}$ are unitary matrices which rotate left- and
right-handed fermions respectively from the gauge to the mass basis. We will
proceed to determine the physical couplings between the fermions and the new
Higgses assuming the Higgs potential outlined in the previous section and
furthermore take the limit where the mass matrix, $M_{ab}^{2}$, of the dormant
Higgses is diagonal thereby eliminating dormant-Higgs mixing.

First, we focus on the quarks whose mass eigenstates are given by

$\begin{split}U_{\mathrm{L}}=\begin{pmatrix}u_{\mathrm{L}}\\\
c_{\mathrm{L}}\\\ t_{\mathrm{L}}\end{pmatrix}=\mathcal{L}_{u}Q^{\prime
u}_{\mathrm{L}},\quad U_{\mathrm{R}}=\begin{pmatrix}u_{\mathrm{R}}\\\
c_{\mathrm{R}}\\\
t_{\mathrm{R}}\end{pmatrix}=\mathcal{R}_{u}U^{\prime}_{\mathrm{R}},\\\
D_{\mathrm{L}}=\begin{pmatrix}d_{\mathrm{L}}\\\ s_{\mathrm{L}}\\\
b_{\mathrm{L}}\end{pmatrix}=\mathcal{L}_{d}Q^{\prime d}_{\mathrm{L}},\quad
D_{\mathrm{R}}=\begin{pmatrix}d_{\mathrm{R}}\\\ s_{\mathrm{R}}\\\
b_{\mathrm{R}}\end{pmatrix}=\mathcal{R}_{d}D^{\prime}_{\mathrm{R}}.\end{split}$
(25)

The couplings between the down-type quarks and the neutral components of the
dormant Higgses, in the mass eigenbasis, are given by 666For convenience we
have identified e.g. $H_{23}^{d}=H_{sb}$. We apologize for the notational
inefficiencies.

$\mathcal{L}\supset-\overline{D}_{\mathrm{L}}\left[g\,\mathcal{L}_{d}\begin{pmatrix}h^{0}_{d}&h^{0}_{ds}&h^{0}_{db}\\\
h^{0}_{sd}&h^{0}_{s}&h^{0}_{sb}\\\
h^{0}_{bd}&h^{0}_{bs}&h^{0}_{b}\end{pmatrix}\mathcal{R}_{d}^{\dagger}\right.\\\
\quad+\left.\begin{pmatrix}m_{d}&&\\\ &m_{s}&\\\
&&m_{b}\end{pmatrix}\left(1+\dfrac{h}{v}\right)\right]D_{\mathrm{R}},$ (26)

and likewise for the up-type quarks

$\mathcal{L}\supset-\overline{U}_{\mathrm{L}}\left[g\,\mathcal{L}_{u}\begin{pmatrix}h^{0}_{u}&h^{0}_{uc}&h^{0}_{ut}\\\
h^{0}_{cu}&h^{0}_{c}&h^{0}_{ct}\\\
h^{0}_{tu}&h^{0}_{tc}&[h]\end{pmatrix}\mathcal{R}_{u}^{\dagger}\right.\\\
\quad+\left.\begin{pmatrix}m_{u}&&\\\ &m_{c}&\\\
&&m_{t}\end{pmatrix}\left(1+\dfrac{h}{v}\right)\right]U_{\mathrm{R}}.$ (27)

where $[h]=-\mu^{2\ast}_{a}M^{-2}_{ab}h_{b}^{0}$ arises from the feedback on
the SMH. Similarly in the lepton sector we set the mass eigenstates to be

$\begin{split}N_{\mathrm{L}}=\begin{pmatrix}\nu_{1\mathrm{L}}\\\
\nu_{2\mathrm{L}}\\\
\nu_{3\mathrm{L}}\end{pmatrix}=\mathcal{L}_{\nu}L^{\prime\nu}_{\mathrm{L}},\quad
N_{\mathrm{R}}=\begin{pmatrix}\nu_{1\mathrm{R}}\\\ \nu_{2\mathrm{R}}\\\
\nu_{3\mathrm{R}}\end{pmatrix}=\mathcal{R}_{\nu}N^{\prime}_{\mathrm{R}},\\\
E_{\mathrm{L}}=\begin{pmatrix}e_{\mathrm{L}}\\\ \mu_{\mathrm{L}}\\\
\tau_{\mathrm{L}}\end{pmatrix}=\mathcal{L}_{e}L^{\prime e}_{\mathrm{L}},\quad
E_{\mathrm{R}}=\begin{pmatrix}e_{\mathrm{R}}\\\ \mu_{\mathrm{R}}\\\
\tau_{\mathrm{R}}\end{pmatrix}=\mathcal{R}_{e}E^{\prime}_{\mathrm{R}},\end{split}$
(28)

which gives interactions with the neutral Higgs bosons of the following form:

$\mathcal{L}\supset-\overline{E}_{\mathrm{L}}\left[g_{\ell}\,\mathcal{L}_{e}\begin{pmatrix}h^{0}_{e}&h^{0}_{e\mu}&h^{0}_{e\tau}\\\
h^{0}_{\mu e}&h^{0}_{\mu}&h^{0}_{\mu\tau}\\\ h^{0}_{\tau
e}&h^{0}_{\tau\mu}&h^{0}_{\tau}\end{pmatrix}\mathcal{R}_{e}^{\dagger}\right.\\\
\quad+\left.\begin{pmatrix}m_{e}&&\\\ &m_{\mu}&\\\
&&m_{\tau}\end{pmatrix}\left(1+\dfrac{h}{v}\right)\right]E_{\mathrm{R}}$ (29)

and

$\mathcal{L}\supset-\overline{N}_{\mathrm{L}}\left[g_{\ell}\,\mathcal{L}_{\nu}\begin{pmatrix}h^{0}_{1}&h^{0}_{12}&h^{0}_{13}\\\
h^{0}_{21}&h^{0}_{2}&h^{0}_{23}\\\
h^{0}_{31}&h^{0}_{32}&h^{0}_{3}\end{pmatrix}\mathcal{R}_{\nu}^{\dagger}\right.\\\
\quad+\left.\begin{pmatrix}m_{1}&&\\\ &m_{2}&\\\
&&m_{3}\end{pmatrix}\left(1+\dfrac{h}{v}\right)\right]N_{\mathrm{R}}.$ (30)

A similar construction for the coupling of the fermions to the charged Higgses
follows straightforwardly. We observe that $h$, the neutral component of
$H_{0}$, is completely indistinguishable from the SMH.

From the previously outlined assumptions on the Higgs sector we are able to
estimate the masses of the dormant Higgses. Using Eq. (24) and Eq. (20), along
with diagonal dormant mass matrix, we find that the fermion mass matrix is
related to the Higgs mass terms by

$m^{f}_{ij}=\left(\dfrac{g_{f}}{g}\right)\dfrac{\mu_{ij}^{f\,2}}{M^{f\,2}_{ij}}m_{t}.$
(31)

To avoid a large fine-tuning in the SMH mass term, we expect
$\mu^{4}/M^{2}\lesssim\left(100\,\mathrm{GeV}\right)^{2}$. We thus arrive at
the estimate

$M^{f}_{ij}\lesssim\left(\dfrac{g_{f}}{g}\right)\dfrac{m_{t}}{m^{f}_{ij}}\cdot
100\,\mathrm{GeV}.$ (32)

We therefore expect the dormant Higgs masses to be inversely proportional to
the corresponding entry in the fermion mass matrix. Once we have made an
ansatz for the fermion rotation matrices, $\mathcal{L}_{f},\mathcal{R}_{f}$,
this provides us with an estimate for the Higgs masses. The bound shown in Eq.
(32) should merely be viewed as a guideline, as it is based on a fine-tuning
argument.

Before we proceed to review two important limiting cases in the next section,
we first define our notation: the Cabibbo-Kobayashi-Maskawa (CKM) matrix is
given by $V_{\text{CKM}}=\mathcal{L}_{u}\mathcal{L}^{\dagger}_{d}$, and
likewise the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix is given by
$V_{\text{PMNS}}=\mathcal{L}_{\nu}\mathcal{L}^{\dagger}_{e}$. In the limit of
negligible dormant Higgs mixing, the required input to determine the Yukawa
couplings is the set of mass rotations, $\mathcal{L}_{f}$ and
$\mathcal{R}_{f}$. These rotation matrices are implicitly generated by the
unknown $\mu_{a}^{2}$ which gives rise to the VEV structure we have encoded
into the above expressions. Examining Eqs. (26) and (30), we find that the
individual matrices $\mathcal{L}_{f},\mathcal{R}_{f}$ are unobservable in the
Standard Model. However, their combinations form the mixing matrices,
$V_{\mathrm{CKM}}$ and $V_{\mathrm{PMNS}}$, which enter in the charged
currents of the quark and lepton sectors respectively. Given sufficient
dormant Higgs data, the structure of $\mathcal{L}$ and $\mathcal{R}$ could be
determined, but at present cannot be determined directly. We must therefore
make an ansatz for their form.

###  III.3 $\mathcal{R}_{f}=\mathds{1}_{3}$ and No-Mixing Limits

We proceed by considering the various symmetries that arise in the special
limit $\mathcal{R}_{f}\rightarrow\mathbb{1}_{3}$ and $m_{q}=0$. These are
useful for relaxing rather stringent flavor mixing constraints.

Consider Eq. (26) with $\mathcal{R}_{d}=\mathbb{1}_{3}$ and the strange quark
mass set to zero, $m_{s}=0$. Furthermore, decompose the mass matrix of the
dormant Higgs fields, $H_{a}$, as a sum of a diagonal $M_{\mathrm{diag}}^{2}$
and an off-diagonal $\delta M^{2}$ Hermitian matrices:

$M^{2}=M_{\mathrm{diag}}^{2}+\delta M^{2}.$ (33)

In the case $\delta M^{2}=0$, we find there exists a discrete symmetry777This
symmetry can only be approximately realized in a realistic part of the
parameter space; it is broken by the quark mass terms and the Yukawa coupling
between the top quark and the dormant Higgses., e.g., reflection of the right-
handed strange quark, $s_{R}\rightarrow-s_{R}$ and the corresponding Higgs
fields $(H_{bs},H_{s},H_{ds})\rightarrow-(H_{bs},H_{s},H_{ds})$. This symmetry
is a generalization of the Glashow-Weinberg symmetry Glashow and Weinberg
(1977), and in our present case of many Higgs bosons this is restrictive.

First, we observe that in general $H_{s}$ will mediate an interaction of the
form,

$\frac{1}{M^{2}_{s}}(\overline{s}^{\prime}_{\mathrm{L}}s^{\prime}_{\mathrm{R}})(\overline{s}^{\prime}_{\mathrm{R}}s^{\prime}_{\mathrm{L}}),$
(34)

in the gauge basis. In the case $\mathcal{R}_{d}$ were not unity, this would
contain mixed combinations such as

$\frac{1}{M^{2}_{s}}|(\mathcal{L}_{d,22}\overline{s}_{\mathrm{L}}+\mathcal{L}_{d,12}\overline{d}_{\mathrm{L}})(\mathcal{R}_{d,22}^{\ast}s_{\mathrm{R}}+\mathcal{R}_{d,12}^{\ast}{d}_{\mathrm{R}})|^{2},\\\
\supset\frac{1}{M^{2}_{s}}\mathcal{L}_{d,22}\mathcal{L}_{d,12}\mathcal{R}_{d,22}^{\ast}\mathcal{R}_{d,12}^{\ast}(\overline{s}_{\mathrm{L}}d_{\mathrm{R}})(\overline{s}_{\mathrm{R}}d_{\mathrm{L}})+...$
(35)

in the flavor basis, which results in $\Delta S=2$ transitions. The
$K_{L}$–$K_{S}$ mass splitting places severe limits on
$M_{s}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}10^{3}$ TeV for
left- and right-handed mixings of the order of the CKM matrix, while the mass
estimate of Eq. (32) suggests $M_{s}\lesssim 100$ TeV. With the discrete
symmetry, $\mathcal{R}_{d}=\mathbb{1}_{3}$, no such interactions are generated
at tree-level, and the tension is substantially alleviated. Moreover, the
discrete symmetry forbids the similarly dangerous Higgs mixing term $\delta
M_{ds,sd}^{2}H^{\dagger}_{ds}H_{sd}$, which mediates interactions such as
$\frac{1}{M^{2}}(\overline{d}_{\mathrm{L}}{s}_{\mathrm{R}})(\overline{d}_{\mathrm{R}}s_{\mathrm{L}})$
directly.

Higgs field |  Fermion mass |  Case (1) [TeV] | Case (2) [TeV]
---|---|---|---
$H^{\prime}_{0}=v+\frac{h}{\sqrt{2}}$ |  $m_{t}=gv=175$ GeV |  $m_{H}=0.125$ | $m_{H}=0.125$
$H^{\prime}_{b}=v\frac{\mu^{2}}{M_{b}^{2}}+H_{b}$ |  $m_{b}=gv\frac{\mu^{2}}{M_{b}^{2}}=4.5$ GeV |  $M_{b}=3.9$ | $M_{b}=0.620$
$H^{\prime}_{\tau}=v\frac{\mu^{2}}{M_{\tau}^{2}}+H_{\tau}$ |  $m_{\tau}=g_{\ell}v\frac{\mu^{2}}{M_{\tau}^{2}}=1.8$ GeV |  $M_{\tau}=6.8$ | $M_{\tau}=0.825$
$H^{\prime}_{c}=v\frac{\mu^{2}}{M_{b}^{2}}+H_{c}$ |  $m_{c}=gv\frac{\mu^{2}}{M_{c}^{2}}=1.3$ GeV |  $M_{c}=13.5$ | $M_{c}=1.2$
$H^{\prime}_{\mu}=v\frac{\mu^{2}}{M_{\mu}^{2}}+H_{\mu}$ |  $m_{\mu}=g_{\ell}v\frac{\mu^{2}}{M_{\mu}^{2}}=106$ MeV |  $M_{\mu}=1.2\times 10^{2}$ | $M_{\mu}=3.4$
$H^{\prime}_{s}=v\frac{\mu^{2}}{M_{s}^{2}}+H_{s}$ |  $m_{s}=gv\frac{\mu^{2}}{M_{s}^{2}}=95$ MeV |  $M_{s}=1.8\times 10^{2}$ | $M_{s}=4.3$
$H^{\prime}_{d}=v\frac{\mu^{2}}{M_{d}^{2}}+H_{d}$ |  $m_{d}=gv\frac{\mu^{2}}{M_{d}^{2}}=4.8$ MeV |  $M_{d}=3.6\times 10^{3}$ | $M_{d}=19$
$H^{\prime}_{u}=v\frac{\mu^{2}}{M_{u}^{2}}+H_{u}$ |  $m_{u}=gv\frac{\mu^{2}}{M_{u}^{2}}=2.3$ MeV |  $M_{u}=7.6\times 10^{3}$ | $M_{u}=27$
$H^{\prime}_{e}=v\frac{\mu^{2}}{M_{e}^{2}}+H_{e}$ |  $m_{e}=g_{\ell}v\frac{\mu^{2}}{M_{e}^{2}}=0.5$ MeV |  $M_{e}=2.45\times 10^{4}$ | $M_{e}=49$
Table 1: The non-mixing estimates for heavy dormant Higgs bosons masses
assuming (1) the level-repulsion feedback on the Higgs mass term is limited to
$(100\,\text{GeV})^{2}$ for each of the quarks and leptons, hence $M_{q}=(100$
GeV$)(m_{t}/m_{q})$ and $M_{\ell}=(100$ GeV$)(g_{\ell}m_{t}/m_{\ell})$. (2)
$\mu=100$ GeV for all mixings, hence $M_{q}=\mu(m_{t}/m_{q})^{1/2}$ and
$M_{\ell}=\mu(m_{t}g_{\ell}/gm_{q})^{1/2}$. Here $g=1$, $g_{\ell}=0.7$ and
$v=175$ GeV. The scalar spectrum relevant to the neutrino mass generation is
discussed in more detail in Sec. V.1.

At the one-loop level the Higgs-Higgs box diagrams, with exchange of the full
doublet $H_{s}$, produces the effective interaction

$\frac{1}{32\pi^{2}}\frac{1}{M_{s}^{2}}\left(\overline{s}^{\prime}_{\mathrm{L}}\gamma_{\mu}s^{\prime}_{\mathrm{L}}\right)\left(\overline{s}^{\prime}_{\mathrm{L}}\gamma_{\mu}s^{\prime}_{\mathrm{L}}\right).$
(36)

This operator is not restricted by $\mathcal{R}_{d}=\mathds{1}_{3}$, but
yields a $\Delta S=2$ operator for nontrivial left-handed rotations,
$\mathcal{L}_{q}$,

$\frac{1}{32\pi^{2}}\frac{1}{M_{s}^{2}}(\mathcal{L}_{d,22})^{2}(\mathcal{L}_{d,12}^{\ast})^{2}\left(\overline{s}_{\mathrm{L}}\gamma_{\mu}d_{\mathrm{L}}\right)\left(\overline{s}_{\mathrm{L}}\gamma_{\mu}d_{\mathrm{L}}\right).$
(37)

Comparing this effective operator with kaon-mixing bounds on the left-handed
current Isidori (2014), we arrive at the limit $M_{s}\gtrsim 60$ TeV, which is
compatible with the mass estimate.

Our model can thus generate CKM mixing, with multiple flavorful Higgs doublets
and yet present no large FCNC (no tree-level $\mathcal{O}(1/M_{x}^{2})$
operators) if we make $\mathcal{R}=\mathds{1}_{3}$. Typical cases include
small breaking effects, giving $\mathcal{R}\neq\mathds{1}_{3}$, and flavor
physics remains an important probe in a system like this.

In getting a sense of the model, it is useful to consider the limit in which
$\mathcal{L}_{f},\mathcal{R}_{f}$ are set to unity with $\delta M_{ab}^{2}=0$
which turns off the mixing among dormant Higgses at leading order. We still
allow non-zero $\mu^{2}_{a}$ terms that mix the dormant Higgses to the SMH.

Let us focus on the up-quark system c.f. Eq. (27). The top quark has acquired
its mass by direct coupling to the SMH, which defines the universal quark
Yukawa coupling $g=1$. In this limit we can identify the fields that develop
tadpole VEVs and give the light quark masses, Eq. (20):

$\displaystyle H^{\prime}_{c}=H_{c}+\frac{\mu^{2}_{c}}{M_{c}^{2}}H_{0},\qquad
m_{c}=\frac{\mu_{c}^{2}}{M_{c}^{2}}v,$ $\displaystyle
H^{\prime}_{u}=H_{u}+\frac{\mu^{2}_{u}}{M_{u}^{2}}H_{0},\qquad
m_{u}=\frac{\mu^{2}_{u}}{M_{u}^{2}}v,$ (38)

and likewise for the $d$, $e$ and $\nu$ sectors. In this limit we can compute
the diagonal dormant Higgs masses. All other fields $H_{x}$ have no VEVs in
this limit. The full set of dormant Higgs masses are tabulated in Table 1
under two different case assumptions:

  1. 1.

The feedback on the Higgs mass is maximal, but limited to
$(100\,\text{GeV})^{2}$ for each fermion (thus choosing a larger value of
$\mu$ for lighter quarks), hence $M_{q}=(m_{t}/m_{q})\cdot 100$ GeV and
$M_{\ell}=(g_{\ell}m_{t}/m_{\ell})\cdot 100$ GeV.

  2. 2.

$\mu=100$ GeV for each SMH mixing term, $M_{q}=(m_{t}/m_{q})^{1/2}\cdot 100$
GeV and $M_{\ell}=(g_{\ell}m_{t}/m_{\ell})^{1/2}\cdot 100$ GeV, which makes
only the lightest $H_{b}$ , $H_{\tau}$, $H_{c}$ have any significant feedback
on the SMH mass.

Hence, in the zero mixing angle limit, it is easy to see how the flavor
problem is mapped into an inverted mass spectrum of Higgs doublets.

##  IV Quark-sector Phenomenology

In Section IV.1, we begin by asking if relevant current LHC searches can place
limits on the, possibly multi-TeV-scale, lightest dormant Higgses. We follow
with a discussion of the prospects of discovery both at the LHC and at future
colliders of higher luminosities and center of mass (COM) energies in Section
IV.2. Throughout Section IV.1 and Section IV.2, we calculate the leading-order
(LO) production cross section of the NP (new physics) processes using
Universal FeynRules Output (UFO) format of FeynRules Alloul  _et al._ (2014)
implemented into MadGraph5_aMC@NLO Alwall  _et al._ (2014) and applying the
NNPDF2.3 PDF set Ball  _et al._ (2013). We implement the five-flavor scheme in
order to account for the intrinsic $b$-quark content of the proton. In
addition, constraints on the dormant Higgs masses from meson mixing and
dedicated searches for flavor change in the charged lepton sector are outlined
in Section IV.3 and Section V.2 respectively.

###  IV.1 Current Limits from LHC Searches

In this section, we investigate if current limits from the LHC can place
meaningful constraints on the masses of the Higgs spectrum of the Scalar
Democracy. The dormant Higgs most accessible at the LHC, is the lightest (non-
SM) doublet which couples to the $b$-quarks, namely
$H_{b}=(h_{b}^{+},h_{b}^{0})$, where $h_{b}^{0}$ is a complex field. Let us
first focus on the neutral component, $h^{0}_{b}$.

For single $h^{0}_{b}$ production, the dominant contribution is via $b$-quark
fusion as shown in Fig. 1. We remind the reader, that the coupling of the
$b$-quarks to $h^{0}_{b}$ is $g=1$. This implies that $h_{b}^{0}$ will decay
approximately to two $b$-quarks with a branching ratio of one. As a
consequence, a simple comparison of a search for new resonances decaying into
jets containing $b$-hadrons is particularly amenable for us to investigate if
the best suited current searches may be able to exclude the relevant (multi-
TeV) mass regime of $h^{0}_{b}$. The ATLAS analysis we choose to recast
corresponds to a dataset collected at $\sqrt{s}=13$ TeV and searches for two
$b$-tagged jets with an invariant mass ($m_{bb}$) in the $0.57-5$ TeV range
Aaboud  _et al._ (2018). In the “low” ($0.6\leq m_{bb}(\text{TeV})<1.25$) and
“high” ($1.25\leq m_{bb}(\text{TeV})\leq 5.0$) invariant mass regimes, $24.3$
$\text{fb}^{-1}$ and $36.1$ $\text{fb}^{-1}$ of integrated luminosity were
analyzed respectively. The primary aim of this analysis is to search for
$Z^{\prime}$ decays to $b$-quarks. Although this search is designed to place
limits on a vector boson, rather than a scalar, the final states are the same
in both cases so the difference in acceptance between our model and that of
Aaboud  _et al._ (2018) should be small.

Figure 1: The dominant contribution to single $h^{0}_{b}$ production is
$b$-quark fusion. $h^{0}_{b}$ decays with a branching ratio of approximately
one to a two-$b$-quark final state.

We note that single $h^{0}_{b}$ production is also possible via gluon fusion
mediated by a bottom loop888The top loop will also contribute to this process
because of the mixing of the SMH with $h^{0}_{b}$. The modified coupling of
the top-quarks to $h^{0}_{b}$ is approximately $\simeq-gm_{b}/m_{t}$, cf. Eq.
(27).. However, due to the chiral suppression present in the loop mediated
processes, these contributions are subdominant to the tree-level production by
more than two orders of magnitude and therefore will be ignored presently.

The one- and two-sigma regions for the expected upper limit on cross section
of di-$b$-jet production is shown in Fig. 2, as indicated by green and yellow
respectively, while the observed upper limit, at $95\%\,\text{CL}$, is shown
by the black circles. There is no significant deviation between observed and
expected limits. The details of the analysis used to calculate these curves
can be found in the aforementioned reference.

We have not performed a full analysis, including detector effects, to
calculate signal cross section and therefore the uncorrected cross section
will likely be a slight overestimate. To mitigate this issue, the LO
production cross section of $h^{0}_{b}$ was multiplied by a relatively
stringent efficiency factor, $\lvert\epsilon_{2b-\text{jet}}\rvert=0.2$, to
account for the two $b$-jet reconstruction efficiency. To justify this choice
of efficiency, we refer to Aaboud  _et al._ (2018), which details the
$b$-tagging efficiencies for both “low” and “high” mass regions. They found
for the low-dijet-mass regions the efficiency for tagging two $b$-jets
decreased from $0.5$ to $0.2$ as $m_{bb}$ was increased from $0.65$ to $1.25$
TeV. However, in the high-dijet-mass region, the event tagging efficiency for
two $b$-jets ranged from $0.4-0.05$ for masses as $m_{bb}$ was increased from
$1.25$ to $5.0$ TeV.

Figure 2: The green (yellow) region shows the $\pm 1\sigma$ ($\pm 2\sigma$)
expected limit for number of events from $Z^{\prime}$ decaying to two
$b$-quarks. We note that the observed cross section (within 95$\%$ C.L) is
within $2\sigma$ of the expected cross section. The blue lines shows the
predicted LO inclusive cross section of $h^{0}_{b}$ production. The observed
and expected data was analysed by an ATLAS group Aaboud  _et al._ (2018) and
this data was made available via HEPData Maguire  _et al._ (2017). Figure 3:
The left plot displays the total LO cross section for single $h^{0}_{b}$
production as a function of its mass, $M_{b}$, for three fixed COM energies
$\sqrt{s}=13,26$ and $100$ TeV as indicated by the solid fuchsia, dashed green
and dotted purple lines. The right plot shows the LO cross section as a
function of COM energy for three fixed mass $M_{b}$ = $1.0,2.0,3.5$ TeV as
indicated by solid purple, green and red lines.

The total inclusive cross section for charged and neutral Higgs production in
heavy quark annihilation has been calculated at next-to-next-to-leading (NNLO)
accuracy in QCD Harlander (2016). For masses of new Higgses of $600$ GeV
(which couple solely to $b$-quarks), the NNLO corrections are small,
$\sigma_{\text{NNLO}}/\sigma_{\text{LO}}\sim 0.95$. Moreover, it is likely
this K-factor will remain small for larger Higgs masses and therefore we did
not calculate higher order corrections to the cross section.

Figure 4: The left plot displays the significance, $S/\sqrt{B}$ as a function
of integrated luminosity at $\sqrt{s}=13$ TeV for three fixed masses:
$M_{b}=1.0,2.0,3.5$ TeV as indicated by solid purple, green and red. The right
plot shows the analogous information but with a COM energy, $\sqrt{s}=26$ TeV.

Although, our reinterpretation lacks the sophistication of a full experimental
analysis, it allows for a rough comparison of our signal cross section to a
currently available dataset. From Fig. 2, it seems likely that the LHC could
exclude the lower mass region, $M_{b}\lesssim 1.0$ TeV.

###  IV.2 Prospects for Future Discovery

The next most pertinent issue is the prospect for future discovery of
$h^{0}_{b}$ and the required integrated luminosity.

In the left plot of Fig. 3, we show the LO production cross section for single
$h^{0}_{b}$ production as a function of $h^{0}_{b}$ mass for three COM
energies: $13$, $26$ and $100$ TeV. As masses below $1$ TeV are disfavored by
data and the mass regime of $\sim 3.5$ TeV is favored theoretically, we shall
discuss the latter. For masses $M_{b}=3.5$ TeV, the corresponding cross
section is $\sim 2.4\times 10^{-4}$, $0.011$ and $2.0$ pb for $\sqrt{s}=13,26$
and $100$ TeV respectively. In the right plot of Fig. 3 we display the LO
cross section as a function of COM energy for three fixed masses:
$M_{b}=1.0,2.0,3.5$ TeV as shown in purple, green and red respectively.
Naturally, the cross section increases for higher COM energies.

The most crucial plots in addressing the question of observability are the two
plots shown in Fig. 4. The quantity of interest is the significance, defined
as $S/\sqrt{B}$ where $S$ is the expected number of signal counts and $B$ is
the expected number of background (BG) counts. The left and right plots
display the significance, $S/\sqrt{B}$, as a function of integrated luminosity
for COM energies $\sqrt{s}=13,26$ TeV respectively. The significance is
defined to be

$\frac{S}{\sqrt{B}}=0.5(\mathcal{L}\,\lvert\epsilon_{2b-\text{jet}}\rvert)^{1/2}\frac{\sigma_{\text{LO}}\left(pp\rightarrow
h^{0}_{b}\right)}{\left[\sigma_{\text{SM}}\left(pp\rightarrow
b\overline{b}\right)\right]^{1/2}},$ (39)

where $\mathcal{L}$ is the integrated luminosity and
$\lvert\epsilon_{2b-\text{jet}}\rvert=0.2$ is the applied reconstruction
efficiency for two $b$-jets which we assume is the same for both signal and SM
background.

In order to estimate the SM background, we applied a number of cuts on the
di-$b$-quark production:

  * •

$p_{T}\left(b\right)\geq 100$ GeV

  * •

$M_{b}-\Gamma\\!\left(h^{0}_{b}\right)\leq m_{bb}\leq
M_{b}+\Gamma\\!\left(h^{0}_{b}\right)$

  * •

$\lvert y\rvert>3.0$

where $p_{T}\left(b\right)$ is the transverse momentum of each of the
$b$-quarks, $m_{bb}$ is the invariant mass of the $b$-quark system,
$\Gamma\left(h^{0}_{b}\right)=3M_{b}/8\pi$ is the width of $h^{0}_{b}$ and $y$
is the rapidity. We note that the cross sections have been calculated at the
level of the hard matrix element. The above cuts were applied to the invariant
mass system of the two $b$-jets of the BG. We did not apply the cuts to the
signal process but have chosen them in such a manner as to not significantly
reduce the signal strength. In order to approximate the effect of the cuts on
the signal, we assumed a Breit-Wigner distribution for the differential cross
section as a function of the invariant mass such that the chosen range
captures approximately half of the signal events. Subsequently, we multiplied
the signal by a factor of 0.5, c.f Eq. (39).

In both plots of Fig. 4, the $S/\sqrt{B}$ value is shown for three fixed
masses: $1.0$, $2.0$, $3.5$ TeV (as indicated by purple, green and red colors
respectively) and at two COM energies: $13$ and $26$ TeV (solid and dashed
lines respectively). The integrated luminosity takes the range between
$\left(10^{2}-3.5\times 10^{6}\right)\,\text{pb}^{-1}$. The value
$S/\sqrt{B}=5$ is indicated by the dashed black lines and provides an estimate
of the required integrated luminosity for the discovery of $h^{0}_{b}$.

For current COM energies, the masses, $M_{b}=1.0,2.0$ TeV would be
discoverable at integrated luminosities of $\sim 2\times 10^{3}$ and $2\times
10^{5}$ $\text{pb}^{-1}$ respectively. However, for the heavier mass
$M_{b}=3.5$ TeV, this would require greater than $3.5\times
10^{6}\,\text{pb}^{-1}$ of integrated luminosity. With higher COM energies,
such as $26$ TeV, the masses $M_{b}=1.0,2.0,3.5$ TeV would all become
accessible at $\sim 5\times 10^{2},10^{4},2\times 10^{5}\,\text{pb}^{-1}$
respectively. Although we have not shown the projections of the significance
for a COM collider with $\sqrt{s}=100$ TeV, we have found that $h^{0}_{b}$ of
all three masses can be discovered with less than $3\,\text{ab}^{-1}$ of
integrated luminosity.

In the event that $h^{0}_{b}$ were discovered, a search and discovery of its
charged counterpart, $h^{\pm}_{b}$, would complete the doublet, $H_{b}$.
Although, $h^{\pm}_{b}$ is mass degenerate with $h^{0}_{b}$, its production
cross section is smaller by at least an order of magnitude, because it couples
to $\overline{t}b$ or $\overline{b}t$ and therefore cannot be produced from
heavy quark fusion. The leading production channel of $h^{\pm}_{b}$ is in
association with a bottom and a top quark, where the final state will be two
$b$-jets and two further jets. Such a search would be more feasible than
di-$h^{\pm}_{b}$ production, which is mainly mediated via photons or off-shell
Z-boson, as this process is kinematically suppressed from the two Higgses in
the final state.

One could also consider the production of the lightest dormant doublet
associated to the lepton sector, $H_{\tau}$. Due to its large coupling to
$\tau$, $g_{\ell}=0.7$, the neutral $h^{0}_{\tau}$ decays dominantly to
$\tau\tau$. As $H_{\tau}$ does not couple strongly to quarks, it can only be
produced via electroweak interactions, making it challenging to probe at the
LHC. In spite of the fact the $\tau$ reconstruction efficiency is slightly
higher than that of $b$-quarks ($\lvert\epsilon_{\tau}\rvert\sim 0.8$ as
measured at $\sqrt{s}=13$ TeV collaboration (2017)), as the $h^{0}_{\tau}$
production is electroweak in nature, the cross section is significantly
(several orders of magnitude) suppressed compared with the analogous
production of $h^{0}_{b}$.

Although this discussion has been largely schematic, and certainly an in-depth
analysis would be needed to search for the lightest dormant Higgs of the
Scalar Democracy, we find it an encouraging first step in demonstrating that
the lightest dormant Higgs is within reach at current energies and most
certainly future upgrades.

###  IV.3 Meson-antimeson mixing

Having discussed the main production channels for the dormant Higgses, we now
turn to the implications of the Scalar Democracy for flavor phenomenology in
the simple limit discussed in Section III.2, where the dormant Higgses do not
mix with each other. The NP parameters are reduced from several hundreds in
the generic model down to a more manageable set consisting of the dormant
Higgs masses and the fermion rotation matrices,
$\mathcal{L}_{f},\mathcal{R}_{f}$, once we impose the requirement that SM
fermion masses must be reproduced.

Potentially large FCNCs may arise when the quark fields are rotated from their
gauge to their mass eigenbasis and will place stringent constraints on the
mass scale of these dormant Higgses. The constraints from flavor-violating
processes in both the quark and lepton sectors have been studied in depth the
context of two Higgs doublet models, see e.g. Ref. Crivellin  _et al._ (2013).

Integrating out the heavy Higgs mass eigenstates gives NP contributions to the
effective four-fermion operators at the low energy scale relevant for flavor
physics. The neutral Higgses induce new four-fermion operators already at
tree-level, contributing to the effective low-energy interactions among
fermions with flavor indices ($q,p,r,s$), namely

$\mathcal{L}_{\mathrm{eff}}\supset-\sum_{f,ij\neq
u,33}\\!\dfrac{g^{2}}{(M^{f}_{ij})^{2}}\mathcal{L}_{f,qi}\mathcal{L}^{\ast}_{f,si}\mathcal{R}_{f,pj}^{\ast}\mathcal{R}_{f,rj}(\overline{f}^{q}_{\mathrm{L}}f_{\mathrm{R}}^{p})(\overline{f}_{\mathrm{R}}^{r}f_{\mathrm{L}}^{s}).$
(40)

The sum runs over all fermions except the $t$-quark denoted by ($u$,33), in
order to avoid double counting, as this would be the SMH contribution.
Besides, the SMH couplings are aligned with the fermion masses and thus do not
lead to tree-level flavor changing transitions. The tree-level exchange is in
most cases expected to be the leading NP contribution to the $\Delta F=2$
operators, as it is only suppressed by the mass of the dormant Higgses
mediating the process.

The right-handed rotation matrices, $\mathcal{R}_{f}$, are not constrained a
priori as they are unphysical in the SM. The limit
$\mathcal{R}_{f}=\mathds{1}_{3}$ is thus viable and turn off the $\Delta F=2$
operators in Eq. (40), as discussed in Section III.3. Near this limit the
leading NP contributions will not come from tree-level contributions, but will
rather be induced at the one-loop level from box diagrams with neutral or
charged Higgses with or without SM charged currents $W$ bosons.

Figure 5: Experimental constraints and mass estimates for the dormant Higgses
in the quark sector in six different benchmarks (see text); the labels denote
the indices of the corresponding Higgs. The gray lines are the mass estimates.
The colored lines correspond to the most stringent experimental lower bound on
each of the Higgs masses; orange if the constraint is from $D_{0}$ mixing,
green from $K_{0}$ and blue from $B_{s}$. If a mass-estimate entry is not
shown, it is above the scale of the plot. Similarly if a mass bound is not
shown it is below the scale of the plot.

We will focus on meson-antimeson mixing which are $\Delta F=2$ observables
that are loop and GIM suppressed in the SM. These observables are thus
sensitive to NP degrees of freedom which propagate in the mixing amplitudes.
As such, signals of NP may be constrained through comparison of observed meson
mixing and the SM predictions.

Meson mixing have been observed in the $D^{0}$–$\overline{D}^{0}$,
$B_{s}$–$\overline{B}_{s}$, $B^{0}$–$\overline{B}^{0}$ and
$K^{0}$–$\overline{K}^{0}$ systems. In each case, the observed mass splitting,
$\Delta M$, of the resulting meson mass eigenstates constrains the magnitude
of the transition amplitudes. Although the experimental values are known to
high precision Amhis  _et al._ (2014) the SM theory predictions are plagued by
large hadronic uncertainties, and there is still room for NP Brod and Gorbahn
(2012); Lenz (2012). The phases of the transition amplitudes give rise to CP-
violation in the respective meson-antimeson systems as observed through the
parameters $\phi_{D}$, $\epsilon_{K}$, $S_{\psi K_{s}}$ and $S_{\psi\phi}$.

An exact analysis of the flavor constraints are somewhat beyond the scope of
this paper. We will content ourselves with a rough estimate for the allowed
mass range for each of the dormant Higgses. To this end, we apply the bounds
of Isidori (2014) on the Wilson coefficients for the scalar exchange effective
operators of Eq. (40), cf. Table 2.

To assess the parameter space, we will probe the NP constraints using a number
of benchmark cases for the rotation matrices. In each case, flavor bounds will
place a lower limit on the masses of the dormant Higgses, and we will compare
these to the expected masses obtained from the fine-tuning arguments c.f. Eq.
(32).

| Bounds on $C$ [$\mathrm{TeV}^{-2}$]
---|---
Operator | Re | Im
$C(\overline{s}_{\mathrm{R}}d_{\mathrm{L}})(\overline{s}_{\mathrm{L}}d_{\mathrm{R}})$ | $6.9\times 10^{-9}$ | $2.6\times 10^{-11}$
$C(\overline{b}_{\mathrm{R}}d_{\mathrm{L}})(\overline{b}_{\mathrm{L}}d_{\mathrm{R}})$ | $3.9\times 10^{-7}$ | $1.7\times 10^{-7}$
$C(\overline{b}_{\mathrm{R}}s_{\mathrm{L}})(\overline{b}_{\mathrm{L}}s_{\mathrm{R}})$ | $8.8\times 10^{-6}$ | $2.9\times 10^{-6}$
$C(\overline{c}_{\mathrm{R}}u_{\mathrm{L}})(\overline{c}_{\mathrm{L}}u_{\mathrm{R}})$ | $5.7\times 10^{-8}$ | $1.1\times 10^{-8}$
Table 2: Bounds used to constrain the size of the NP contributions to FCNC
operators relevant for meson-antimeson mixing Isidori (2014).

The meson-antimeson mixing bounds on NP contribution to the $\Delta F=2$
operators, can be translated into bounds on the dormant Higgs masses once
$\mathcal{L}_{q}$ and $\mathcal{R}_{q}$ are assumed. For each Higgs we
determine the mass where it saturates each of the bounds in the absence of
contribution from other dormant Higgses; the highest such mass is used as a
lower bound for the corresponding Higgs mass. This is a somewhat simplified
approach to the constraints, as it discounts possible interference between
contributions from multiple Higgses, not to mention the running of the
effective operators below the EW scale. Since we are only exploring a small
region of parameter space of the full model in any case, the bounds we obtain
using these approximations remain a good indication for the scale where the
Higgses become visible in flavor physics.

The benchmark points we consider are informed by the constraint that the left-
handed quark mixings must reproduce the CKM matrix. Hence, it is unavoidable
to have some left-handed mixing, albeit there is still freedom to put all of
the left-handed mixing into either the up or the down sector. A good middle
ground would seem to be taking a half and half approach. One will quickly find
that with this assumption for the left-handed rotation, the meson-antimeson
mixing constraints will favor small mixing of the right-handed quarks. We
consider the following benchmarks:
Benchmarks

$\displaystyle\text{(1)}\quad\mathcal{L}_{u}$
$\displaystyle=\mathcal{L}_{d}^{\dagger}=V_{\mathrm{CKM}}^{1/2},\quad$
$\displaystyle\mathcal{R}_{u}^{\dagger}=\mathcal{R}_{d}=\mathds{1}_{3}$
$\displaystyle\text{(2)}\quad\mathcal{L}_{u}$
$\displaystyle=\mathcal{L}_{d}^{\dagger}=V_{\mathrm{CKM}}^{1/2},\quad$
$\displaystyle\mathcal{R}_{u}=\mathcal{R}_{d}=V(1,1)$
$\displaystyle\text{(3)}\quad\mathcal{L}_{u}$
$\displaystyle=\mathcal{L}_{d}^{\dagger}=V_{\mathrm{CKM}}^{1/2},\quad$
$\displaystyle\mathcal{R}_{u}=\mathcal{R}_{d}=V(3,3)$
$\displaystyle\text{(4)}\quad\mathcal{L}_{u}$
$\displaystyle=\mathcal{L}_{d}^{\dagger}=V_{\mathrm{CKM}}^{1/2},\quad$
$\displaystyle\mathcal{R}_{u}^{\dagger}=\mathcal{R}_{d}=V_{\mathrm{CKM}}^{1/3000}$
$\displaystyle\text{(5)}\quad\mathcal{L}_{u}$
$\displaystyle=\mathcal{L}_{d}^{\dagger}=V_{\mathrm{CKM}}^{1/2},\quad$
$\displaystyle\mathcal{R}_{u}^{\dagger}=\mathcal{R}_{d}=V_{\mathrm{CKM}}^{1/30}$
$\displaystyle\text{(6)}\quad\mathcal{L}_{u}$ $\displaystyle=\mathds{1}_{3},$
$\displaystyle\mathcal{L}_{d}^{\dagger}=V_{\mathrm{CKM}},$
$\displaystyle\mathcal{R}_{u}^{\dagger}$
$\displaystyle=V_{\mathrm{CKM}}^{1/2},$
$\displaystyle\mathcal{R}_{d}=V_{\mathrm{CKM}}^{1/300}$

where

$V(\eta_{1},\eta_{2})=\begin{pmatrix}1&\eta_{1}\cdot 10^{-4}&0\\\
-\eta_{1}\cdot 10^{-4}&1&8\eta_{2}\cdot 10^{-3}\\\ 0&-8\eta_{2}\cdot
10^{-3}&1\end{pmatrix},$ (41)

is an approximately unitary matrix. For each benchmark we have determined the
lower bound individually for each dormant Higgs mass, for all the effective
operators in Table 2 to satisfy their bounds. The resulting lower bounds are
plotted in Fig. 5. As a comparison the gray lines in the figure show the
expected masses of the Higgses assuming the absence of fine-tuning, cf. Eq.
(32). We proceed to investigate the benchmark points one at a time.

_Benchmark 1_ shows the mass spectrum of the theory in the absence of right-
handed rotation. The Higgs mass estimates in Fig. 5 exhibits the inverse
hierarchy of the quark masses detailed in Table 1 with the Higgses associated
to the lighter quarks tending to be the heaviest. The non-trivial left-handed
rotation will lower the mass estimates for the off-diagonal mass. In
particular, $H_{ct}$, must be almost as light as $H_{b}$ in order to generate
a large off diagonal mass element in the fermion mass matrix. The
corresponding plot in Fig. 5 does not show any bounds on the Higgs masses, as
we have not included the loop-induced effects here.

_Benchmarks 2 and 3_ illustrate a marginal case, for how much right-handed
rotation can be allowed without tension between mass estimates and flavor
bounds. The matrix $V(\eta_{1},\eta_{2})$ used to parametrize the right-handed
matrix is applied for both $\mathcal{R}_{u}$ and $\mathcal{R}_{d}$ for
simplicity. The $(1,2)$ entry governs the mixing of the first two generations
of quarks. As $\eta_{1}$ is increased as do the flavor bounds on $H_{u}$,
$H_{uc}$, $H_{cu}$ and $H_{c}$, because they are governed by $D^{0}$ mixing
(between the $u$ and $c$ quarks). Similarly the bounds on the $H_{d}$,
$H_{ds}$, $H_{sd}$ and $H_{s}$ from $K^{0}$ are pushed up, though they are
less problematic. The entry (1,3) of $V(\eta_{1},\eta_{2})$ governs the bound
on $H_{b}$ from $B_{s}$ mixing in a similar manner. Going from the marginal
case $\eta_{1}=\eta_{2}=1$ in Benchmark 2, to the case $\eta_{1}=\eta_{2}=3$
in Benchmark 3, we observe how all the corresponding flavor bounds are
increased by a factor $\sqrt{3}$, giving a slight tension between mass
estimates and flavor bounds. We would like to emphasize that this tension only
exists in the complete absence of fine-tuning of the Higgs mass, $-M_{0}^{2}$.
If one were to tolerate fine-tuning to the percent level, the Higgs mass
estimates would increase by a factor 10.

_Benchmarks 4 and 5_ parametrize $\mathcal{R}_{q}$ in terms of powers of CKM
matrix, to illustrate how large the right-handed mixings are allowed to be in
terms of a more familiar matrix. Due to the relatively large mixing,
$(V_{\mathrm{CKM}})_{12}\simeq(V_{\mathrm{CKM}})_{21}\simeq 0.23$, one has to
go to the case
$\mathcal{R}_{u}^{\dagger}=\mathcal{R}_{d}=V_{\mathrm{CKM}}^{1/3000}$ of
Benchmark 4, before the flavor bounds decrease below the mass estimates of
$H_{c}$. On the other hand, if one is willing to accept a percent-level fine
tuning of the Higgs mass we can go to the case
$\mathcal{R}_{u}^{\dagger}=\mathcal{R}_{d}=V_{\mathrm{CKM}}^{1/30}$ of
Benchmark 5. We stress that, the mass estimates shown in Fig. 5 all are with
the complete absence of fine-tuning.

_Benchmark 6_ puts the entire left-handed rotation into $\mathcal{L}_{d}$, and
gives an alternative example of evading the flavor bounds. In this case a
large $\mathcal{R}_{u}$ can be tolerated without generating $D^{0}$ mixing at
tree level. The main constraints in this case, is from $K^{0}$ mixing.

Though our selection of a few benchmark points does not constitute an in-depth
analysis of the flavor physics of the Scalar Democracy, it demonstrates that
there is room for the framework to avoid tension with flavor physics. Yet,
care has to be taken to ensure that one works in an allowed regions of
parameter space.

##  V Lepton-sector phenomenology

###  V.1 Neutrino sector

In our framework, the mass spectrum of SM fermions is mapped onto a scalar
mass spectrum where lighter scalars develop larger VEVs and thus leads to
heavier fermion masses, see Eq. (32). As the masses of charged fermions have
been precisely measured, a sharper prediction of the scalar spectrum can be
made. We do not know the absolute value of neutrino masses, nor the mass
generation mechanism, resulting in looser possibilities for the scalar bound
states associated to neutrino masses. The goal of this section is to examine
these possibilities.

Although the absolute values of neutrino masses are still to be measured,
neutrino oscillations give us valuable information on their masses. From the
KamLAND experiment and observations of the solar neutrino spectrum Gando  _et
al._ (2013); Abe  _et al._ (2016), the solar mass splitting 999The three
neutrino mass eigenstates are labeled such that $\nu_{1}$ ($\nu_{3}$) has the
largest (smallest) admixture of $\nu_{e}$. has been measured to be $\Delta
m^{2}_{21}\equiv m_{2}^{2}-m_{1}^{2}\simeq+7.4\times 10^{-5}$ eV2. The sign of
$\Delta m^{2}_{21}$ is known due to matter potential effects on neutrino
oscillations. Besides, atmospheric and accelerator neutrino oscillation
experiments Abe  _et al._ (2018); Adamson  _et al._ (2017a); Abe  _et al._
(2017a) are compatible with $|\Delta
m^{2}_{31}|\equiv|m^{2}_{3}-m^{2}_{1}|\simeq 2.5\times 10^{-3}$ eV2.
Therefore, the following two neutrino mass orderings are still viable
experimentally, $m_{1}<m_{2}<m_{3}$ or $m_{3}<m_{1}<m_{2}$ (recent experiments
exhibit a slight preference towards the former scenario Adamson  _et al._
(2017b); Abe  _et al._ (2017b)). In addition, the sum of neutrino masses is
bounded from above by $0.12$ eV from cosmological observations Aghanim  _et
al._ (2018). We can thus identify three extreme scenarios for the neutrino
mass spectrum $(m_{1},\,m_{2},\,m_{3})$: normal hierarchy
$(0,\,0.008\leavevmode\nobreak\ {\rm eV},\,0.05\leavevmode\nobreak\ {\rm
eV})$, inverted hierarchy $(0.05\leavevmode\nobreak\ {\rm
eV},\,0.05\leavevmode\nobreak\ {\rm eV},\,0)$ and semi-degenerate neutrinos
$\sim(0.03\leavevmode\nobreak\ {\rm eV},\,0.03\leavevmode\nobreak\ {\rm
eV},\,0.06\leavevmode\nobreak\ {\rm eV})$. As the heaviest neutrino needs to
have a mass above $0.05$ eV, that will typically translate into an upper bound
for some of the scalars (see Table 3).

Scenario |  Higgs field |  Neutrino mass |  Case (1) | Case (2)
---|---|---|---|---
Dirac |  $H_{\nu}=v\frac{\mu^{2}}{M_{\nu}^{2}}+H^{\prime}_{\nu}$ |  $m_{\nu}=g_{\ell}\frac{\mu^{2}}{M_{\nu}}<0.06$ eV |  $M_{\nu}>2\times 10^{14}$ GeV |  $M_{\nu}>1.4\times 10^{8}$ GeV
|  |  $m_{\nu}^{\rm heaviest}>0.05$ eV |  $M_{\nu}^{\rm lightest}<2.4\times 10^{14}$ GeV |  $M_{\nu}^{\rm lightest}<1.5\times 10^{8}$ GeV
Type-I |  $S_{N}=\mu\frac{v^{2}}{M_{S_{N}}^{2}}+S^{\prime}_{N}$ |  $m_{N}=g_{\ell}\mu\frac{v^{2}}{M_{S_{N}}^{2}}$ |  No sharp prediction | see Fig. 6
Type-II |  $\Delta=\mu\frac{v^{2}}{M_{\Delta}^{2}}+\Delta^{\prime}$ |  $m_{\nu}=g_{\ell}\mu\frac{v^{2}}{M_{\Delta}^{2}}<0.06$ eV |  $M_{\Delta}>2\times 10^{14}$ GeV |  $M_{\Delta}>1.8\times 10^{8}$ GeV
|  |  $m_{\nu}^{\rm heaviest}>0.05$ eV |  $M_{\Delta}^{\rm lightest}<2.4\times 10^{14}$ GeV |  $M_{\Delta}^{\rm lightest}<2\times 10^{8}$ GeV
Table 3:  Dormant Higgs bosons masses, for different neutrino mass generation
mechanisms, assuming (1) the level-repulsion feedback on the Higgs is limited
to $(100\leavevmode\nobreak\ {\rm GeV})^{2}$ for each neutrino, and (2)
$\mu=100$ GeV for all mixings (see text for details). Here, $v=175$ GeV and
$g_{\ell}=0.7$. The requirement of a minimum mass for the heaviest neutrino
typically translates into a low bound on the mass of some scalar.

In the Scalar Democracy there are three viable alternatives for the generation
of neutrino masses. The neutrino mass generation can be similar to the charged
fermion mass generation, as outlined in Eq. (30). In this case, neutrinos
would be Dirac fermions. Nevertheless, due to the quantum numbers of the
lepton doublet, $L_{\mathrm{L}}$, and the right-handed neutrino,
$N_{\mathrm{R}}$, two other bound states can contribute to neutrino masses,
namely, $S_{N}\sim\overline{N}_{\mathrm{R}}N_{\mathrm{L}}^{C}$, and
$\Delta\sim\overline{L}_{\mathrm{L}}L_{\mathrm{R}}^{C}$. While $H_{\nu}$ may
lead to Dirac neutrino masses, an accompanying non-zero expectation value for
$S_{N}$ would constitute a Majorana mass term for right-handed neutrinos,
realizing a type-I seesaw mechanism, whereas $\Delta$ may be identified as the
scalar triplet in type-II seesaw models (see Refs. Minkowski (1977); Mohapatra
and Valle (1986); Gell-Mann  _et al._ (1979); Mohapatra and Senjanovic (1980,
1981); Yanagida (1979); Schechter and Valle (1980); Lazarides  _et al._
(1981)). Thus, the mechanism of neutrino masses depends on which dormant
Higgses acquire a VEV.

In the type-I seesaw realization, the relevant terms in the Lagrangian are
(generation indices have been suppressed)

$\mathcal{L}\supset-
g_{\ell}\overline{L}_{\mathrm{L}}\tilde{H}_{\nu}N_{\mathrm{R}}-g_{\ell}S_{N}\overline{N}_{\mathrm{R}}N_{\mathrm{L}}^{C},$
(42)

and hence neutrino masses are given by

$m_{\nu}^{\text{type-I}}=g_{\ell}\frac{\left\langle
H_{\nu}\right\rangle^{2}}{\left\langle S_{N}\right\rangle},\qquad
m_{N}=g_{\ell}\left\langle S_{N}\right\rangle.$ (43)

Because $S_{N}$ is a SM singlet, it will not acquire its VEV from mixing with
the SMH as in Eq. (2), but rather from the term
$\mu_{S}H_{0}^{\dagger}H_{0}S_{N}$, which gives, $\left\langle
S_{N}\right\rangle=\mu_{S}v^{2}/M_{S}^{2}$. Requiring electroweak values for
the dimensionful parameter $\mu_{S}=100$ GeV, and imposing constraints on
singlet-Higgs mixing and active-sterile neutrino mixing leads to the allowed
region in the $(M_{S},M_{\nu})$ plane shown in Fig. 6. The gray regions are
excluded due to mixings while the blue region predicts too heavy neutrinos
101010Note that the constraint on the active-sterile mixing strongly depends
on the mass $m_{N}$ and the active flavor de Gouvêa and Kobach (2016). The
value chosen here is extremely conservative.. The atmospheric mass splitting
requires at least one neutrino to be heavier than $0.05$ eV. A dashed black
line corresponding to $m_{\nu}=0.01$ eV is drawn to guide the reader. Notice
that a sort of seesaw mechanism between the sterile neutrino and the singlet
scalar is in place: the heavier is $S$, the smaller its VEV and thus the
lighter is $N$ (see arrows in Fig. 6). If instead we require the level-
repulsion feedback to the SMH mass to be small, no sharp prediction can be
made about $M_{S}$, as it can always be made arbitrarily large leading to
pseudo-Dirac neutrinos.

Figure 6: Mass of scalar singlet $S$ versus scalar doublet $H_{\nu}$ in the
Scalar Democracy type-I seesaw scenario, under the assumption that
$\mu_{S}=\mu_{\nu}=100$ GeV (see text for details). The gray regions are rule
out by constraints on singlet-Higgs mixing (leftmost region) and active-
sterile neutrino mixing (rightmost region). The blue region predicts too large
neutrino masses and is excluded by cosmological observations. The heaviest
neutrino is above 0.05 eV, and thus at least one pair of scalars should have
masses between the dashed black line and the blue region. The mass of the
sterile neutrino $m_{N}$ is also indicated for two values of $M_{S}$.

In the case of type-II seesaw, in the absence of $N_{R}$, the $\Delta$ would
get its VEV from the usual term
$\mu_{\Delta}H_{0}^{T}i\sigma_{2}\Delta^{\dagger}H_{0}$. Neutrino masses would
be given by

$m_{\nu}^{\text{type-II}}=0.42\leavevmode\nobreak\ {\rm
eV}\left(\frac{g_{\ell}}{0.7}\right)\left(\frac{\mu_{\Delta}}{100\leavevmode\nobreak\
{\rm GeV}}\right)\left(\frac{10^{8}\leavevmode\nobreak\ {\rm
GeV}}{M_{\Delta}}\right)^{2},$ (44)

such that electroweak values of $\mu_{\Delta}$ would correspond to scalar
triplets around the $10^{8}$ GeV scale, as can be seen in Table 3, “Case (2)”
column. Restricting the level repulsion feedback on the Higgs potential leads
to exactly the same mass constraints as in the Dirac neutrino case. Obtaining
scalar triplet masses accessible at the LHC, $M_{\Delta}\sim{\rm TeV}$ would
require $\mu_{\Delta}\sim 10\leavevmode\nobreak\ {\rm eV}$, as it happens in
usual type-II seesaw scenarios. For the three mechanisms discussed, a summary
of the masses of the scalar bound states responsible for the neutrino spectrum
can be found in Table 3 (labeled as “Dirac”, “Type-I” and “Type-II”).

###  V.2 Lepton flavor violation

Any charged lepton flavor violating (CLFV) decay would immediately indicate
the presence of new physics and as a consequence much effort has been put into
detecting such effects. The result of this work places severe constraints on
the branching ratios for the different CLFV decays of the charged leptons.

In the absence of mixing among dormant Higgses, the CLFV decays must be
mediated by the Higgses associated to the lepton mass generation. As a result
the CLFV transitions induced in our framework do not involve quarks. The
relevant constraints are the radiative and the three-body decays are shown in
Table 4.

Process | Experimental Upper Limit
---|---
$\text{Br}\left(\mu^{+}\rightarrow e^{+}e^{+}e^{-}\right)$ |  $1.0\times 10^{-12}$ Bellgardt  _et al._ (1988)
$\text{Br}\left(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}\right)$ |  $2.1\times 10^{-8}$ Hayasaka  _et al._ (2010)
$\text{Br}\left(\tau^{-}\rightarrow e^{-}\mu^{+}\mu^{-}\right)$ |  $2.7\times 10^{-8}$ Hayasaka  _et al._ (2010)
$\text{Br}\left(\tau^{-}\rightarrow\mu^{-}e^{+}e^{-}\right)$ |  $1.8\times 10^{-8}$ Hayasaka  _et al._ (2010)
$\text{Br}\left(\tau^{-}\rightarrow e^{-}e^{-}e^{-}\right)$ |  $2.7\times 10^{-8}$ Hayasaka  _et al._ (2010)
$\text{Br}\left(\tau^{-}\rightarrow e^{+}\mu^{-}\mu^{-}\right)$ |  $1.7\times 10^{-8}$ Hayasaka  _et al._ (2010)
$\text{Br}\left(\tau^{-}\rightarrow\mu^{+}e^{-}e^{-}\right)$ |  $1.5\times 10^{-8}$ Hayasaka  _et al._ (2010)
$\text{Br}\left(\tau^{\pm}\rightarrow\mu^{\pm}\gamma\right)$ |  $4.4\times 10^{-8}$ Hayasaka  _et al._ (2008)
$\text{Br}\left(\tau^{\pm}\rightarrow e^{\pm}\gamma\right)$ |  $3.3\times 10^{-8}$ Hayasaka  _et al._ (2008)
$\text{Br}\left(\mu^{+}\rightarrow e^{+}\gamma\right)$ |  $4.2\times 10^{-13}$ Baldini  _et al._ (2016)
Table 4: The $90\%$ C.L. upper limit on decay branching rates for LFV
processes.

The NP contribution to the decay of a charged lepton to another through the
emission of a photon is governed, in the low-energy theory, by the dipole
operator

$\mathcal{L}_{\mathrm{eff}}\supset
C_{ij}\overline{E}_{i\mathrm{L}}\sigma^{\mu\nu}E_{j\mathrm{R}}F^{\mu\nu}\;+\;\mathrm{h.c.}\;,$
(45)

where $i$ and $j$ are flavor indices and $C_{ij}$ has inverse mass dimension.
The decay width is given by

$\Gamma(e_{i}\to
e_{j}\gamma)=\dfrac{m_{e,i}^{3}}{4\pi}\left(\left|C_{ij}\right|^{2}+\left|C_{ji}\right|^{2}\right).$
(46)

In the Scalar Democracy, NP effects are induced by diagrams where a dormant
Higgs propagates in the loop. The coefficients $C_{ij}$ depend only on the
rotation matrices of the external fermions and the Higgs mass.

For concreteness, we will assume that the Dirac neutrino mass mechanism is at
work (see Table 3). The NP contribution to the coefficients of the dipole
operators, from both neutral and charged Higgses, is given by

$C_{ij}=\dfrac{-eg_{\ell}^{2}}{384\pi^{2}}\sum_{kl}\Bigg{(}\mathcal{L}_{e,ik}\mathcal{L}_{e,jk}^{\ast}\left[\dfrac{2m_{e,j}}{(M^{e}_{kl})^{2}}-\dfrac{m_{e,j}}{(M^{\nu}_{kl})^{2}}\right]\\\
\qquad\qquad+\mathcal{R}_{e,ik}\mathcal{R}^{\ast}_{e,jk}\dfrac{m_{e,i}}{(M^{e}_{lk})^{2}}\Bigg{)},$
(47)

to leading order in $m_{e}^{2}/(M^{e})^{2}$, where $M^{e}_{kl}$ denotes the
mass of the scalar that generates the $(kl)$ entry of the corresponding
leptonic mass matrix. Due to the smallness of neutrino masses, the neutrino
Higgs contribution is completely irrelevant to the radiative decay. As an
aside, we note that a quick estimate of the NP contribution to
$\mu\to\mu\gamma$ will show that even in the most optimistic scenario $C_{22}$
is several orders of magnitude too small to significantly influence the SM
prediction for the muon $g-2$.

The charged lepton decay to three lighter charged leptons is mediated at tree-
level by neutral Higgses giving the effective operators of Eq. (40). The
resulting decay width is

$\Gamma(e_{p}^{-}\to
e_{r}^{-}e_{s}^{+}e_{t}^{-})\simeq\dfrac{7g_{\ell}^{4}m_{e,p}^{5}}{6144\pi^{3}}\left(\left|\sum_{ij}\dfrac{\mathcal{L}_{e,ri}\mathcal{L}_{e,si}^{\ast}\mathcal{R}_{e,pj}^{\ast}\mathcal{R}_{e,tj}}{(M^{e}_{ij})^{2}}\right|^{2}+\left|\sum_{ij}\dfrac{\mathcal{L}_{e,ri}\mathcal{L}_{e,pi}^{\ast}\mathcal{R}_{e,sj}^{\ast}\mathcal{R}_{e,tj}}{(M^{e}_{ij})^{2}}\right|^{2}+(r\longleftrightarrow
t)\right).$ (48)

Many of these decays only violate flavor by one unit, and so, in contrast to
the NP contribution to meson-antimeson mixing, the operator does not vanish
for all dormant Higgses in the limit where $\mathcal{R}_{e}=\mathds{1}_{3}$.

Figure 7: Experimental constraints and mass estimates for the dormant Higgses
in the lepton sector in three different benchmarks; the labels denote the
indices of the corresponding Higgs. The gray lines are the mass estimates. The
colored lines correspond to the most stringent experimental constraint on each
of the Higgs masses; orange if the constraint is from $\mu\to 3e$ decays, blue
from $\tau\to 3\ell$ and green from $\mu\to e\gamma$.

Similarly to the quark sector, the only direct constraint on the lepton
rotation matrices stem from the requirement
$V_{\mathrm{PMNS}}=\mathcal{L}_{\nu}\mathcal{L}_{e}^{\dagger}$. Therefore, we
have the same freedom of choice in their parametrization. In contrast to the
quark sector, the lepton observables do not significantly favor small rotation
angles. For our benchmark points we have therefore chosen large rotation
matrices to illustrate the case where NP will soon become detectable in LFV
observables.
Benchmarks

$\displaystyle\text{(1)}\quad\mathcal{L}_{e}$
$\displaystyle=\mathcal{R}_{\nu}=V_{\mathrm{PMNS}}^{\dagger\;1/2},\quad$
$\displaystyle\mathcal{L}_{\nu}=\mathcal{R}_{e}=V_{\mathrm{PMNS}}^{1/2}$
$\displaystyle\text{(2)}\quad\mathcal{L}_{e}$
$\displaystyle=\mathcal{R}_{\nu}=U_{\mathrm{TBM}},\quad$
$\displaystyle\mathcal{L}_{\nu}=\mathcal{R}_{e}=V_{\mathrm{PMNS}}U_{\mathrm{TBM}}$
$\displaystyle\text{(3)}\quad\mathcal{L}_{e}$
$\displaystyle=\mathcal{R}_{\nu}=V_{\mathrm{PMNS}}^{\dagger\;1/2}U_{\mathrm{TBM}}^{1/2},\quad$
$\displaystyle\mathcal{L}_{\nu}=\mathcal{R}_{e}=V_{\mathrm{PMNS}}^{1/2}U_{\mathrm{TBM}}^{1/2}$

We apply the PDG convention for the PMNS matrix Tanabashi  _et al._ (2018)
using the best-fit values of the three mixing angles and phase,
$\delta_{\text{CP}}$ Esteban  _et al._ (2017). The use of the tribimaximal
(TBM) mixing matrix Harrison  _et al._ (2002); Xing (2002); Harrison and Scott
(2002); He and Zee (2003) is motivated from work on discrete flavor model and
its structure is given by

$U_{\text{TBM}}=\begin{pmatrix}-\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}&0\\\
\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\ \end{pmatrix}.$
(49)

For each of the benchmark points we have determined the strongest individual
limit on each $e$-type dormant Higgs by comparing the NP contribution to the
decay bounds in Table 4. We have plotted the leading constraints on the
individual Higgs masses resulting from our benchmark points in Fig. 7 together
with the corresponding mass estimates resulting from Eq. (32). In the
benchmarks almost all the Higgs mass constraints come from the experimental
bounds on the muon decays—even for the dormant Higgses coupling primarily to
the third generation leptons. This is due to the somewhat anarchic structure
of the PMNS mixing matrix, meaning that all the Higgses mediate LFV muon
decays. The PMNS mixing angle suppression is insufficient to compensate the
weaker bounds on the LFV tau branching ratios. Despite the slightly stronger
experimental bound on $\mathop{}\\!\mathrm{Br}(\mu\to e\gamma)$ than
$\mathop{}\\!\mathrm{Br}(\mu^{-}\to e^{-}e^{+}e^{-})$, the loop suppression in
the leading dormant Higgs contribution to the radiative decay is enough that
almost all the $H^{e}$ Higgses get their strongest bound from the decay to
three charged leptons. For at least some of the Higgses it is possible to
engineer the rotation matrices in such a way as to avoid the experimental
bound on $\mathop{}\\!\mathrm{Br}(\mu^{-}\to e^{-}e^{+}e^{-})$.

Regardless of the anarchic flavor structure chosen for the benchmark points,
Fig. 7 shows that there is only severe tension between the mass estimates of
$H_{e}$, $H_{\mu}$, $H_{e\mu}$ and $H_{\mu e}$ and their corresponding
constraints in Benchmark 2. We therefore expect that most of the parameter
space (consistent with our assumptions) will pass the lepton-flavor
constraints. We also anticipate that future CLFV experiments, with
significantly improved sensitivities, will be important probes of this
scenario.

The Higgses associated with neutrinos in the Dirac mass scenario, $H^{\nu}$,
are only constrained by the radiative decays. In no case, regardless of the
left-handed mixing, are the bounds on the $H^{\nu}$ masses stronger than
$5\,\mathrm{TeV}$, much below what is expected from the fine-tuning argument.
For this reason the corresponding constraints have not been displayed in Fig.
7.

##  VI Conclusions

While our principle of Scalar Democracy lacks a detailed theoretical
underpinning, it is nonetheless experimentally testable in the coming years.
If some predictions are confirmed, such as the observation of $H_{b}$, and
possibly $H_{\tau}$, $H_{ct}$ and $H_{c}$, with expected ${\cal{O}}(1)$ Yukawa
couplings, it could reshape our view of the UV completion of the Standard
Model.

We propose that a plethora of new scalar bosons exists in nature. We argue
schematically that every fermion pair binds to form a boundstate scalar boson,
due to a universal attractive interaction at a very high scale, $\Lambda$.
Amongst many new states, including lepto-quarks, colored isodoublets and
singlets, etc., this hypothesis implies the existence of a large number of
sequentially more massive Higgs bosons.

We argue that the SM Higgs boson is the first of the sequence, and therefore
must have a dynamical origin, essentially as a $\overline{t}t$ boundstate, but
it is now part of the constellation of composite scalars that influence its
mass. One immediate intrigue is that the top quark mass, predicted by the
renormalization group fixed point Pendleton and Ross (1981); Hill (1981),
which is the prediction of top condensation models Bardeen  _et al._ (1990),
now comes within a few percent of the observed value. This is expected to be
improved by including the multi-Higgs boson masses and decouplings and is a
first indicator that this could be a pathway to new UV physics.

The universal binding we invoke may be intimately associated with gravity,
since strong scattering amplitudes may exist near the Planck-scale that are
parameterized by $d=6$ operators and scale as $1/M_{\text{Planck}}^{2}$.
Amongst these amplitudes we would expected a general four-fermion structure
such as
$(g^{2}/M^{2})(\overline{\psi}_{\mathrm{L}}^{i}\psi_{\mathrm{R}}^{j})(\overline{\psi}_{\mathrm{R}}^{i}\psi_{\mathrm{L}}^{j})$,
where $i,j$ run over the conventional quark and lepton flavors and colors of
the SM. In fact, since gravity does not distinguish between particle and anti-
particle, we would expect similar operators with a Majorana-like structure.
The operators must be gauge invariant, and with the usual SM fields this
contains a subsector of a global $\mathrm{SU}(48)\times\mathrm{U}(1)$ chiral
Lagrangian.

We therefore argue that there exists a sequence of Higgs doublets leading
upwards to large mass scales. These are “dormant” isodoublets, each having
only small “tadpole” VEVs that arise from mixing with the observed Higgs and
that scale as $1/M^{2}$ for heavy Higgs fields of mass $M$. Our main point in
invoking gravity is to motivate that many Higgs doublets may exist in nature.
They could also arise from strong gauge interactions. The masses of these
scalars are lacking a theory–we view them at present as soft-symmetry-
breaking, relevant operators that we insert by hand in accord with known
phenomenolgy.

The virtue of the model is that the Yukawa interactions are “known,” and all
are of order unity. This is interesting as it transforms the problem of flavor
dynamics in a fundamental way: we have only a single Yukawa coupling at
$\Lambda$, and the derived couplings at low energies are subject to relatively
small renormalization corrections. The leptons may see the most significant
renormalization to a value of about $0.7$ (smaller ‘fine-structure” effects
will be generated by the decoupling of heavy Higgs fields). The observed
spectrum of quarks and leptons and their mixing angles can be codified in
terms of the the mass-mixing problem in the extended Higgs sector.

Most interesting is the top-bottom system. The top mass and the SM Higgs
calibrate the overall Yukawa coupling for all quarks,
$g=g_{\mathrm{top}}\simeq 1$. Here the first sequential dormant Higgs $H_{b}$,
couples essentially to $\overline{b}b$, and has a positive mass, $M_{b}^{2}$.
It has a mixing with the lightest Higgs $H_{0}$ causes it to acquire a small
VEV that feeds the $b$-quark its mass. We can view the SM Higgs as having
initially positive $M_{0}^{2}$ (or zero), and the effect of “level repulsion”
by $H_{b}$ drives it to become tachyonic. To avoid significant fine tuning we
would require $\mu^{4}/M_{b}^{2}\sim M_{H}^{2}$, which leads to a soft
“naturalness” upper limit on $M_{H_{b}}\lesssim(v_{\text{weak}}/m_{b})(\sim
100$ GeV$)\sim 3.5$ TeV.

We discuss in some detail a particular search strategy to address the neutral
component of $H_{b}=(h_{b}^{+},h_{b}^{0})$, through the process $pp\rightarrow
h_{b}^{0}\rightarrow\overline{b}b$. This is available at the current LHC
energy and luminosity, and possible upgrades. We find that the LHC already has
the capability of excluding sequential $H_{b}$ with ${\cal{O}}(1)$ Yukawa
coupling to $\overline{b}b$ in a mass range of order
$\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}$ 1 TeV (as in the
model of Hill (2014)). Our conservative (rough) estimates indicate that the
full range of $H_{b}$ masses may be accessible to the high luminosity and/or
energy-doubled LHC, and should help justify the case for such future machines.
Our estimates are preliminary and likely will bound above what can be done by
improved detector based studies and more detailed deployment of cuts and
search strategies. We strongly encourage our experimental colleagues to pursue
this. We thus defer details of possible searches $h_{b}^{+}\rightarrow
t\overline{b}$ and $H_{\tau}$ elsewhere.

The challenge to Scalar Democracy is to avoid unwanted $d>4$ flavor transition
operators. The rare processes we focus on are mainly neutral meson mixing, and
we find that such processes are binding on the Higgs spectroscopy. However,
avoiding these constraints in this model is possible since most of the new
sequential Higgs bosons are very heavy. Conceivably, though we have not
explored it, the extended Higgs sector may provide sources of observable rare
processes in heavy meson decays. Discovery of the lighter doublets could thus
provide impetus for partially unraveling the flavor problem.

We also discuss the leptons. Our model most naturally leads to a “neutrino
Higgs seesaw” for neutrino Dirac masses in analogy to the quark-sector Higgs
structure. However, we have additional fermionic boundstates that can, in
principle, mix to develop tadpole VEVs and produce Majorana masses, realizing
a type-I or type-II seesaw mechanism. Hence the double-$\beta$ decay
experiments will be important probes of the far UV Higgsology of this
framework. Our model can also drive rare lepton number violation experiments,
and we focus discussion on a subset of these,
$\ell_{i}\rightarrow\ell_{j}+\gamma$ and $\ell\rightarrow 3\ell^{\prime}$. We
certainly do not exclude other possible processes as probes of the leptonic
Higgs system.

One view of the future evolution of fundamental physics with energy scale,
argues that the couplings are small and asymptotically free, and that the
theory “fades away” into a linear scale-invariance. We have arrived at a
contrary point of view, that a rich spectrum of new scalar states lies
immediately beyond current energy scales and is within reach of the LHC and
its upgrade path. In part this is motivated by chirality: it is very unlikely
that one can generate a small Yukawa coupling constant from zero (certainly
not perturbatively). Hence the tiny $g_{\mathrm{electron}}\sim 10^{-6}$ is
most likely due to a power law suppression of a coupling that is of order
unity, such as $g_{\mathrm{electron}}\sim g_{\ell}\mu_{e}^{2}/M^{2}_{e}$ where
$g_{\ell}\sim 1$. The power-law suppression then demands new mass scales, such
as $\mu$ and $M$, as realized in our model.

We have argued that a natural way to achieve this “democratically” throughout
the entire flavor system of the SM is with a grand enlargement of the scalar
system. This flips the flavor problem: the lightest (heaviest) fermions are
coupled to the heaviest (lightest) scalars. The resulting spectrum of fermions
is due to the inverted spectrum of associated scalars. This underlies our
“Scalar Democracy.” It focuses urgent attention to the top-bottom (and perhaps
the $\tau$–$\nu_{\tau}$ subsystem) and is testable at the LHC and future
upgrades. We urge our experimental colleagues and other theorists to take up
the cause.

## Acknowledgements

We thank John Campbell, Andre de Gouvea, Roni Harnik, Josh Isaacson and Tobias
Neumann for helpful discussions and Bogdan Dobrescu and Matthew Low for
reading the preliminary draft. One of us recalls a conversation with Bj, in
which he noticed a plethora of Higgs bosons arising in conjunction with
extensions of gravity, and which helped to inspire this work Bjorken . This
work was done at Fermilab, operated by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the United States Department of Energy.
AET would like to thank Fermilab for hosting him during the writing of this
paper, and gratefully acknowledges financial support from the Danish Ministry
of Higher Education and Science through an EliteForsk Travel Grant. The
CP3-Origins centre is partially funded by the Danish National Research
Foundation, grant number DNRF90.

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