Anatomy of Vector-Like Bottom-Quark Models in the
Alignment Limit of the 2-Higgs Doublet Model Type-II

A<EMAIL_ADDRESS>R<EMAIL_ADDRESS>M.
<EMAIL_ADDRESS>S.
<EMAIL_ADDRESS><EMAIL_ADDRESS>

1Abdelmalek Essaadi University, Faculty of Sciences and Techniques, Tangier,
Morocco
2Polydisciplinary Faculty, Laboratory of Fundamental and Applied Physics, Cadi
Ayyad University, Sidi Bouzid, B.P. 4162, Safi, Morocco
3Department of Physics & Astronomy, Uppsala University, Box 516, SE-751 20
Uppsala, Sweden

4School of Physics & Astronomy, University of Southampton, Southampton, SO17
1BJ, United Kingdom

###### Abstract

Expanding upon our ongoing investigation of Vector-Like Quark (VLQ)
phenomenology within a 2-Higgs Doublet Model (2HDM) framework, in this paper,
we complement a previous one dedicated to Vector-Like Top-quarks (VLTs) by
studying Vector-Like Bottom-quarks (VLBs), specifically focusing on their
behavior in the alignment limit of a Type-II Yukawa structure. We examine the
potential for detecting VLBs at the Large Hadron Collider (LHC) and analyze
their decay signatures, encompassing both Standard Model (SM) processes and
exotic decays. The objective is to differentiate among singlet, doublet, and
triplet configurations of VLBs by identifying distinct decay patterns, thereby
providing insights into the structure of Beyond the SM (BSM) physics.

##  1 Introduction

Building on the pivotal discovery of the Higgs boson during the LHC Run 1 at
CERN and the subsequent detailed analyses of its properties by the ATLAS and
CMS collaborations [1, 2], herein, we delve deeper into the exploration of
Vector-Like Quarks (VLQs), particularly focusing on the Bottom-type VLQs
(VLBs), in this continuation of our previous research [3]. The identification
of the Higgs boson has largely validated the Standard Model (SM) predictions,
yet certain anomalies observed across various production and decay channels
signal the potential for Beyond the Standard Model (BSM) physics, inviting
further investigation into the role of VLBs.

In the backdrop of the 2-Higgs Doublet Model (2HDM) [4, 5], which introduces
additional Higgs bosons ($H$, $A$, and $H^{\pm}$) in addition to the
discovered SM-like one ($h$), our study is aimed at uncovering the
implications of including VLBs with unique Electro-Weak (EW) quantum numbers
and non-Yukawa type couplings. This research not only seeks to address the
anomalies noted in Higgs boson measurements but also to extend the analysis
conducted on Top-type VLQs (VLTs) [3], by examining the exotic decay channels
of VLBs into these new Higgs states, a subject that remains underexplored at
the LHC.

Our motivation is driven by the potential of VLBs to decay into 2HDM
additional Higgs bosons, which could evade detection by current LHC search
strategies, which are focused on SM decay modes, and to assess the model
alignment with EW Precision Observables (EWPOs). Through this work, we aim at
contributing to a deeper understanding of VLBs within the BSM framework,
emphasizing the significance of their exotic decay channels in revealing new
aspects of particle physics.

This paper, advancing our series on VLQs [6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] within a 2HDM
context , is structured to firstly elaborate on the theoretical foundations of
the 2HDM and VLBs, followed by a detailed exploration of the decay patterns of
VLBs and their potential implications for LHC detection and characterization.
By leveraging insights from our previous work on VLTs, specifically on the
2HDM Type-II [3], we endeavor to provide a comprehensive perspective on the
capabilities of VLBs to illuminate new physics phenomena.

The paper is structured as follows. In the next section, we discuss our model.
In the following section we present and discuss our results. Then we summarize
and draw our conclusions.

##  2 Model description

###  2.1 Formalism

In this paper, we extend the work presented in Ref. [3], which established a
CP-conserving 2HDM Type-II also featuring VLQ representations, wherein the
phenomenology of this BSM setup was accessed via VLT decay dynamics, while
here we will establish the one of the VLB sector.

The 2HDM part of our scenario produces the canonical set of Higgs bosons: two
CP-even states, with $h$ being the lightest and $H$ the heaviest, a CP-odd
state, $A$, and a pair of charged Higgs bosons, $H^{\pm}$. As tree-level
Flavor Changing Neutral Currents (FCNCs) are very constrained by experiment,
we impose a $\mathbb{Z}_{2}$ symmetry, such that $\Phi_{1}\to\Phi_{1}$ and
$\Phi_{2}\to-\Phi_{2}$, on the Higgs fields. The resulting Higgs potential
(softly broken by dimension-2 terms $\propto m^{2}_{12}$) can be written as

$\displaystyle\mathcal{V}$ $\displaystyle=$ $\displaystyle
m^{2}_{11}\Phi_{1}^{\dagger}\Phi_{1}+m^{2}_{22}\Phi_{2}^{\dagger}\Phi_{2}-\left(m^{2}_{12}\Phi_{1}^{\dagger}\Phi_{2}+{\rm
H.c.}\right)+\tfrac{1}{2}\lambda_{1}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\tfrac{1}{2}\lambda_{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}$
(1)
$\displaystyle\qquad+\lambda_{3}\Phi_{1}^{\dagger}\Phi_{1}\Phi_{2}^{\dagger}\Phi_{2}+\lambda_{4}\Phi_{1}^{\dagger}\Phi_{2}\Phi_{2}^{\dagger}\Phi_{1}+\left[\tfrac{1}{2}\lambda_{5}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2}+{\rm
H.c.}\right].$

By choosing real Vacuum Expectation Values (VEVs) for the two Higgs doublet
fields, $v_{1}$ and $v_{2}$, and demanding $m_{12}^{2}$ and $\lambda_{5}$ to
be real, the potential is CP-conserving. The free independent parameters are
here taken to be the four masses, $m_{h}$, $m_{H}$, $m_{A}$ and $m_{H^{\pm}}$,
the soft $\mathbb{Z}_{2}$-breaking parameter $m_{12}$, the VEV ratio
$\tan\beta=v_{2}/v_{1}$ and the mixing term $\sin(\beta-\alpha)$, where the
angle $\alpha$ diagonalizes the CP-even mass matrix. When we impose that no
(significant) tree-level FCNCs are present in the theory using the (softly
broken) $\mathbb{Z}_{2}$ symmetry, we end up with four different Yukawa
versions of the model. These are: Type-I, where only $\Phi_{2}$ couples to all
fermions; Type-II, where $\Phi_{2}$ couples to up-type quarks and $\Phi_{1}$
couples to charged leptons and down-type quarks; Type-Y (or Flipped), where
$\Phi_{2}$ couples to charged leptons and up-type quarks and $\Phi_{1}$
couples to down-type quarks; Type-X (or Lepton Specific), where $\Phi_{2}$
couples to quarks and $\Phi_{1}$ couples to charged leptons111In this paper,
we will be discussing only Type-II..

The gauge invariant structures that have multiplets with definite
${SU}(3)_{C}\times{SU}(2)_{L}\times{U}(1)_{Y}$ quantum numbers appear in the
interactions of new VLQs with the SM states via renormalizable couplings. The
set of VLQ representations is indicated by:

$\displaystyle B_{L,R}^{0}\,$ $\displaystyle\text{(singlets)}\,,$
$\displaystyle(B^{0}\,Y)_{L,R}\,,\quad(T^{0}\,B^{0})_{L,R}\,$
$\displaystyle\text{(doublets)}\,,$
$\displaystyle(X\,T^{0}\,B^{0})_{L,R}\,,\quad(T^{0}\,B^{0}\,Y)_{L,R}$
$\displaystyle\text{(triplets)}\,.$ (2)

We use in this section a zero superscript to distinguish the weak from the
mass eigenstates. The Electro-Magnetic (EM) charges of the new VLQs are
$Q_{T}=2/3$, $Q_{B}=-1/3$, $Q_{X}=5/3$ and $Q_{Y}=-4/3$. Note that $T$ and $B$
carry the same EM charge as the SM top and bottom quarks, respectively.

The physical up-type quark mass eigenstates may, in general, contain non-zero
$Q_{L,R}^{0}$ (with $Q$ being the VLQ field) components, when new fields
$T_{L,R}^{0}$ of charge $2/3$ and non-standard isospin assignments are added
to the SM. This situation leads to a deviation in their couplings to the $Z$
boson. Atomic parity violation experiments and the measurement of $R_{c}$ at
LEP give some constraints on these deviations for the up and charm quarks
which are far stronger than for the top quark. In the Higgs basis, the Yukawa
Lagrangian contains the following terms:

$-\mathcal{L}\,\,\supset\,\,y^{u}\bar{Q}^{0}_{L}\tilde{H}_{2}u^{0}_{R}+y^{d}\bar{Q}^{0}_{L}H_{1}d^{0}_{R}+M^{0}_{u}\bar{u}^{0}_{L}u^{0}_{R}+M^{0}_{d}\bar{d}^{0}_{L}d^{0}_{R}+\rm{H.c.}$
(3)

Here, $u_{R}$ runs over $(u_{R},c_{R},t_{R},T_{R})$ while $d_{R}$ runs over
$(d_{R},s_{R},b_{R},B_{R})$.

We now turn to the mixing of the new partners to the third generation, $y_{u}$
and $y_{d}$, which are $3\times 4$ Yukawa matrices. In fact, in the light of
the above constraints, it is very reasonable to assume that only the top quark
$t$ “mixes” with $T$. In this case, the $2\times 2$ unitary matrices
$U_{L,R}^{u}$ define the relation between the charge $2/3$ weak and mass
eigenstates:

$\left(\\!\begin{array}[]{c}t_{L,R}\\\
T_{L,R}\end{array}\\!\right)=U_{L,R}^{u}\left(\\!\begin{array}[]{c}t^{0}_{L,R}\\\
T^{0}_{L,R}\end{array}\\!\right)=\left(\\!\begin{array}[]{cc}\cos\theta_{L,R}^{u}&-\sin\theta_{L,R}^{u}e^{i\phi_{u}}\\\
\sin\theta_{L,R}^{u}e^{-i\phi_{u}}&\cos\theta_{L,R}^{u}\end{array}\\!\right)\left(\\!\begin{array}[]{c}t^{0}_{L,R}\\\
T^{0}_{L,R}\end{array}\\!\right)\,.$ (4)

In contrast to the up-type quark sector, the addition of new fields
$B_{L,R}^{0}$ of charge $-1/3$ in the down-type quark sector results in four
mass eigenstates, $d,s,b,B$. Measurements of $R_{b}$ at LEP set tight
constraints on the $b$ mixing with the new fields that are stronger than for
mixing with the lighter quarks $d,s$. In this case, then, $2\times 2$ unitary
matrices $U_{L,R}^{d}$ define the dominant $b-B$ mixing as

$\left(\\!\begin{array}[]{c}b_{L,R}\\\
B_{L,R}\end{array}\\!\right)=U_{L,R}^{d}\left(\\!\begin{array}[]{c}b^{0}_{L,R}\\\
B^{0}_{L,R}\end{array}\\!\right)=\left(\\!\begin{array}[]{cc}\cos\theta_{L,R}^{d}&-\sin\theta_{L,R}^{d}e^{i\phi_{d}}\\\
\sin\theta_{L,R}^{d}e^{-i\phi_{d}}&\cos\theta_{L,R}^{d}\end{array}\\!\right)\left(\\!\begin{array}[]{c}b^{0}_{L,R}\\\
B^{0}_{L,R}\end{array}\\!\right)\,.$ (5)

To ease the notation, we have dropped the superscripts $u$($d$) whenever the
mixing occurs only in the up(down)-type quark sector. Additionally, we
sometime use the shorthand notations
$s_{L,R}^{u,d}\equiv\sin\theta_{L,R}^{u,d}$,
$c_{L,R}^{u,d}\equiv\cos\theta_{L,R}^{u,d}$, etc. (More details on this
Lagrangian formalism are shown in the Appendix.)

This Lagrangian contains all the relevant phenomenological information, as
follows:

  * (i)

the modifications of the SM couplings that might show indirect effects of new
quarks can be found in the terms that do not contain VLQ fields;

  * (ii)

the terms relevant for LHC phenomenology (i.e., VLQ production and decay) are
those involving a heavy and a light quark;

  * (iii)

terms with two VLQs are relevant for their contribution to oblique corrections
(i.e., to the $S,T$ and $U$ parameters of the EWPOs).

In the weak eigenstate basis, the diagonalization of the mass matrices makes
the Lagrangian of the third generation and heavy quark mass terms such as

$\displaystyle\mathcal{L}_{\text{mass}}$ $\displaystyle=$
$\displaystyle-\left(\\!\begin{array}[]{cc}\bar{t}_{L}^{0}&\bar{T}_{L}^{0}\end{array}\\!\right)\left(\\!\begin{array}[]{cc}y_{33}^{u}\frac{v}{\sqrt{2}}&y_{34}^{u}\frac{v}{\sqrt{2}}\\\
y_{43}^{u}\frac{v}{\sqrt{2}}&M^{0}\end{array}\\!\right)\left(\\!\begin{array}[]{c}t^{0}_{R}\\\
T^{0}_{R}\end{array}\\!\right)$ (17)
$\displaystyle-\left(\\!\begin{array}[]{cc}\bar{b}_{L}^{0}&\bar{B}_{L}^{0}\end{array}\\!\right)\left(\\!\begin{array}[]{cc}y_{33}^{d}\frac{v}{\sqrt{2}}&y_{34}^{d}\frac{v}{\sqrt{2}}\\\
y_{43}^{d}\frac{v}{\sqrt{2}}&M^{0}\end{array}\\!\right)\left(\\!\begin{array}[]{c}b^{0}_{R}\\\
B^{0}_{R}\end{array}\\!\right)+\text{H.c.},$

with $M^{0}$ a bare mass term222Note that this bare mass term is not related
to the Higgs mechanism: it is gauge invariant and can appear as such in the
Lagrangian, or it can be generated by a Yukawa coupling to a scalar multiplet
that acquires a VEV $v^{\prime}\gg v$., $y_{ij}^{q}$, $q=u,d$, Yukawa
couplings and $v=246$ GeV the Higgs VEV in the SM. Using the standard
techniques of diagonalization, the mixing matrices are obtained by

$U_{L}^{q}\,\mathcal{M}^{q}\,(U_{R}^{q})^{\dagger}=\mathcal{M}^{q}_{\text{diag}}\,,$
(18)

with $\mathcal{M}^{q}$ the two mass matrices in Eq. (17) and
$\mathcal{M}^{q}_{\text{diag}}$ the diagonals ones. To check the consistency
of our calculation, the corresponding $2\times 2$ mass matrix reduces to the
SM quark mass term if either the $T$ or $B$ quarks are absent.

Notice also that, in multiplets with both $T$ and $B$ quarks, the bare mass
term is the same for the up- and down-type quark sectors. For singlets and
triplets one has $y_{43}^{q}=0$ whereas for doublets $y_{34}^{q}=0$. Moreover,
for the $(XTB)$ triplet, one has $y_{34}^{d}=\sqrt{2}y_{34}^{u}$ and for the
$(TBY)$ triplet one has $y_{34}^{u}=\sqrt{2}y_{34}^{d}$333We write the
triplets in the spherical basis, hence, the $\sqrt{2}$ factors stem from the
relation between the Cartesian and spherical coordinates of an irreducible
tensor operator of rank 1 (vector)..

The mixing angles in the left- and right-handed sectors are not independent
parameters. From the mass matrix bi-unitary diagonalization in Eq. (18) one
finds:

$\displaystyle\tan 2\theta_{L}^{q}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}|y_{34}^{q}|vM^{0}}{(M^{0})^{2}-|y_{33}^{q}|^{2}v^{2}/2-|y_{34}^{q}|^{2}v^{2}/2}\quad\text{(singlets,
triplets)}\,,$ $\displaystyle\tan 2\theta_{R}^{q}$ $\displaystyle=$
$\displaystyle\
\frac{\sqrt{2}|y_{43}^{q}|vM^{0}}{(M^{0})^{2}-|y_{33}^{q}|^{2}v^{2}/2-|y_{43}^{q}|^{2}v^{2}/2}\quad\text{(doublets)}\,,$
(19)

with the relations

$\displaystyle\tan\theta_{R}^{q}$ $\displaystyle=$
$\displaystyle\frac{m_{q}}{m_{Q}}\tan\theta_{L}^{q}\quad\text{(singlets,
triplets)}\,,$ $\displaystyle\tan\theta_{L}^{q}$ $\displaystyle=$
$\displaystyle\frac{m_{q}}{m_{Q}}\tan\theta_{R}^{q}\quad\text{(doublets)}\,,$
(20)

with $(q,m_{q},m_{Q})=(u,m_{t},m_{T})$ and $(d,m_{b},m_{B})$, so one of the
mixing angles is always dominant, especially in the down-type quark sector. In
addition, for the triplets, the relations between the off-diagonal Yukawa
couplings lead to relations between the mixing angles in the up- and down-type
quark sectors,

$\displaystyle\sin 2\theta_{L}^{d}$ $\displaystyle=$
$\displaystyle\sqrt{2}\,\frac{m_{T}^{2}-m_{t}^{2}}{m_{B}^{2}-m_{b}^{2}}\sin
2\theta_{L}^{u}\quad\quad(X\,T\,B)\,,$ $\displaystyle\sin 2\theta_{L}^{d}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\frac{m_{T}^{2}-m_{t}^{2}}{m_{B}^{2}-m_{b}^{2}}\sin
2\theta_{L}^{u}\quad\quad(T\,B\,Y)\,.$ (21)

The masses of the heavy VLQs deviate from $M^{0}$ due to the non-zero mixing
with the SM quarks plus, for doublets and triplets, the masses of the
different components of the multiplet are related. Altogether, these relations
show that all multiplets except the $(TB)$ doublet can be parametrized by a
mixing angle, a heavy quark mass and a CP-violating phase that enters some
couplings, with the latter being ignored for the observables considered in
this paper. In the case of the $(TB)$ doublet, there are two independent
mixing angles and two CP-violating phases for the up- and down-type quark
sectors, with - again - the latter set to zero in our analysis. Hereafter, we
refer to such a construct as the 2HDM+VLQ scenario, distinguishing between the
singlet, doublet and triplet cases. In the present paper, though, given the
emphasis on VLBs, as opposed to VLTs, we will treat the $(B)$ singlet, $(BY)$
and $(TB)$ doublets as well as $(XTB)$ and $(TBY)$ triplets whereas we will
not deal with the $(T)$ singlet and $(XT)$ doublet representations, as their
study was tackled in Ref. [3]. Finally, as discussed in the abstract, we
decided to work in the so-called “alignment limit” of the 2HDM, wherein we fix
$m_{h}=125$ GeV (so that the lightest neutral Higgs state of the 2HDM is the
discovered one) and we have further taken
$m^{2}_{12}=m_{A}^{2}\frac{\tan^{2}\beta}{1+\tan^{2}\beta}$ for the soft
$\mathbb{Z}_{2}$-breaking parameter.

###  2.2 Implementation and validation

In this subsection, we briefly describe our implementation of the
aforementioned BSM scenario. We have used 2HDMC-1.8.0 [31] as a base platform
for our 2HDM+VLQ setup444A public release of it is in progress: herein, the
analytical expressions for the Feynman rules of the interaction vertices of
our 2HDM+VLQ model have been implemented as a new class while several new
tree-level VLQ decays have explicitly been coded alongside those of Higgs
bosons into VLQs themselves.. As a first step, the above Lagrangian components
were implemented into FeynRules-2.3 [32] to generate the proper spectrum of
masses and couplings. With the help of this program, we have then generated
FeynArts-3.11 [33, 34] and FormCalc-9.10 [35, 36] model files as well as
Universal FeynRules Output (UFO) interfaces to be used in MadGraph-3.4.2 [37].
As consistency checks, we have verified the cancellation of Ultra-Violet (UV)
divergences as well as the renormalization scale independence of all relevant
loop-level processes.

###  2.3 EWPOs in VLQ

EWPOs can restrict severely the parameter space of BSM physics scenarios. In
our 2HDM+VLQ setup (Such analytic results for the $S$ and $T$ parameters have
been presented in [38].), for both $S$ and $T$, we have two additive
contributions: one from VLQs and the other from 2HDM Higgses. The latter is
very well known and we have taken it from [13] while the former can be
calculated using the approach of Ref. [14], as it has been done in Ref. [15].
However, the approach of Ref. [14] works only for the singlet and doublet
representations of the new fermions and cannot be applied to the triplet
cases. For all various VLQ representations considered here, we have computed
the $S$ and $T$ parameters with the use of the aforementioned FeynArts and
FormCalc packages using dimensional regularization. The analytic results were
obtained as a combination of standard Passarino-Veltman (PV) functions. We
checked both analytically and numerically that our results for $S$ and $T$ are
UV finite and renormalization scale independent. For the $T$ parameter, we
have compared our results to Refs. [14, 15] and found good agreement in the
case of singlet and doublet. In the case of the triplet representations
$(XTB)$ and $(TBY)$, our result disagrees with Ref. [15]. Furthermore, for the
$S$ parameter, our results disagree with those of Ref. [15]. In fact, Ref.
[15] simply extends the result of the singlet and doublet cases from Ref. [14]
to the triplet one. The author of Ref. [39] derives the correct expression for
$S$ and $T$ in the case of the $(XTB)$ triplet using the approach of Ref.
[14]. We recomputed from scratch the results for the $(XTB)$ and $(TBY)$
triplet cases, then rederived $S$ and $T$ in terms of PV functions and,
crucially, did not neglect the external momentum of the gauge bosons. We are
confident of the correctness of our findings, as they were derived in two
different ways. A major source of disagreement would be the fact that Ref.
[15] uses a simple approximation for the $S$ parameter, which consists in
neglecting the external momentum of the gauge bosons. We also cross-checked
with a new calculation [40, 39] and found good agreement for the $T$
parameter. However, we should mention that, for the $S$ parameter, we use the
complete analytical result and, again, did not neglect any external momentum
as it is usually done [40].

###  2.4 Constraints

In this section, we list the constraints that we have used to check the
validity of our results. From the theoretical side, we have the following
requirements:

  * •

Unitarity constraints require the $S$-wave component of the various
(pseudo)scalar-(pseudo)scalar, (pseudo)scalar-gauge boson, and gauge-gauge
bosons scatterings to be unitary at high energy [41].

  * •

Perturbativity constraints impose the following condition on the quartic
couplings of the scalar potential: $|\lambda_{i}|<8\pi$ ($i=1,...5$) [4].

  * •

Vacuum stability constraints require the potential to be bounded from below
and positive in any arbitrary direction in the field space, as a consequence,
the $\lambda_{i}$ parameters should satisfy the conditions as [42, 43]:

$\displaystyle\lambda_{1}>0,\quad\lambda_{2}>0,\quad\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}},$
$\displaystyle\lambda_{3}+\lambda_{4}-|\lambda_{5}|>-\sqrt{\lambda_{1}\lambda_{2}}.\hskip
14.22636pt$ (22)

  * •

Constraints from EWPOs, implemented through the oblique parameters, $S$ and
$T$ [44], require that, for a parameter point of our model to be allowed, the
corresponding
$\chi^{2}(S^{\mathrm{2HDM\text{-}II}}+S^{\mathrm{VLQ}},~{}T^{\mathrm{2HDM\text{-}II}}+T^{\mathrm{VLQ}})$
is within 95% Confidence Level (CL) in matching the global fit results [45]:

$\displaystyle S=0.05\pm 0.08,\quad T=0.09\pm 0.07,\quad \rho_{S,T}=0.92\pm
0.11.\hskip 14.22636pt(\text{For~{}~{}}U=0)$

Note that unitarity, perturbativity, vacuum stability, as well as $S$ and $T$
constraints, are enforced through the public code 2HDMC-1.8.0[31].

From the experimental side, we evaluated the following:

  * •

Constraints from the SM-like Higgs-boson properties are taken into account by
using HiggsSignal-3 [46, 47] via HiggsTools [48]. We require that the relevant
quantities (signal strengths, etc.) satisfy
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\mathrm{min}}$ for these measurements at
95% CL ($\Delta\chi^{2}\leq 6.18$).

  * •

Constraints from direct searches at colliders, i.e., LEP, Tevatron, and LHC,
are taken at the 95% CL and are tested using HiggsBouns-6[49, 50, 51, 52] via
HiggsTools. Including the most recent searches for neutral and charged
scalars.

  * •

Constraints from $b\to s\gamma$: To align with the $b\to s\gamma$ limit, the
mass of the charged Higgs boson is set to exceed 600 GeV. This is based on the
analysis in Ref [27], which indicates that incorporating VLQs into the Type-II
2HDM could relax this limit through larger mixing angles. However, EWPOs
constrain these angles to smaller values, consequently maintaining the charged
Higgs mass close to the typical 580 GeV in the 2HDM Type-II.

###  2.5 Direct search constraints

In this section, we explore the implications of experimental constraints on
the properties of additional Higgs states and VLQs within our 2HDM+VLQ
framework. The oblique parameters $S$ and $T$ play a crucial role by imposing
stringent limits on the VLQ mixing angles, as elaborated in our previous work
[27] and in the previous subsection. This limitation leads to small mixing
angles, allowing VLQ masses to start from as low as 1000 GeV without
conflicting with existing LHC exclusions. This pattern holds true across
singlet, doublet and triplet VLQ configurations.

###  2.6 VLQs contribution to Higgs production and decay

VLQs contribute to the loop diagrams for Higgs production via gluon-gluon
fusion and Higgs decay into two photons. However, in our study, their
contributions are relatively small. This is partly because VLQs tend to
decouple from the process as their masses increase and also due to the mixing
angles terms $({s^{d}_{L,R}})^{2}$ appear in the coupling $hB\overline{B}$
(see Table XIV of the appendix) are restricted by the EWPOs resulting in small
values for these terms (The largest allowed mixing angle is approximately
$s^{d}_{L,R}\sim 0.2$), further suppressed by their squares. However, it’s
noteworthy that the primary, albeit small, contribution arises from
modifications in the $hb\bar{b}$ coupling.

For the processes $gg\to H$ (or $H\to gg$) and $H\to\gamma\gamma$, the
contribution of all quarks of the same charge to the corresponding amplitudes
is determined by the expression:

$F_{q}=\sum_{i}Y_{ii}A_{1/2}\left(\frac{M_{H}^{2}}{4m_{i}^{2}}\right),$ (23)

Here, the sum runs over $t,T$ for $q=u$ and over $b,B$ for $q=d$. The
couplings to the Higgs $Y_{ii}$ are defined in Table XIV of the appendix, and
the function $A_{1/2}$ is defined in [53].

Our analysis reveals that, across all considered scenarios, the branching
ratio $\mathcal{BR}(h\to gg)$ experiences a decrease of up to 10%, primarily
due to modifications in the $hb\bar{b}$ coupling. Similarly, the branching
ratio $\mathcal{BR}(h\to\gamma\gamma)$ decreases by up to 3%, also attributed
to modifications in the $hb\bar{b}$ coupling. Meanwhile, the contributions of
the new terms $hB\overline{B}$ are considered negligible, as discussed
earlier.

##  3 Results and discussions

In discussing our results, we deal with these in turn for the different VLQ
representations.

###  3.1 2HDM with $(B)$ singlet

Parameters |  Scanned ranges
---|---
$m_{h}$ |  $125$
$m_{A}$ |  [$300$, $1000$]
$m_{H}$ |  [$300$, $1000$]
$m_{H^{\pm}}$ |  [$600$, $1000$]
$\tan\beta$ |  [$1$, $20$]
$m_{B}$ |  [$1000$, $2000$]
$\sin\theta_{L}^{u,d}$ |  [$-0.5$, $0.5$]
$\sin\theta_{R}^{u,d}$ |  [$-0.5$, $0.5$]
Table 1: 2HDM and VLQs parameters for all scenarios with their scanned ranges.
Masses are in GeV. The phases $\phi_{u}$ and $\phi_{d}$ are set to zero.

In Fig. 1, we conduct an analysis across the 2HDM parameters, including the
Higgs boson masses, $\tan\beta$, $\sin(\beta-\alpha)$ and $m_{12}$, along with
the mass of the singlet VLB, $m_{B}$, and the fermionic mixing angle,
$\sin\theta_{L}$, as outlined in Tab. 1.

Figure 1: (Left) Scatter plots of randomly generated points superimposed onto
the fit limits in the $(\Delta S,\Delta T)$ plane from EWPO data at 95% CL
with a correlation of 92%. Here, we illustrate the 2HDM and VLQ contributions
separately and also the total one. (Right) The $B$ width
$\Gamma(B)(\equiv\Gamma_{B})$ as a function of $s_{L}^{d}\equiv\sin\theta_{L}$
with $m_{B}$ indicated as a color gauge.

In the left section of Fig. 1, we present our findings on how the 2HDM scalars
and the sole VLQ in this 2HDM+VLQ scenario affect the $S$ and $T$ parameters.
We detail these influences both individually and collectively. The $S$
parameter contributions from the VLQ and 2HDM sectors, denoted as $S_{\rm
VLQ,2HDM}$, respectively, are relatively small. In contrast, for the $T$
parameter, the $T_{\rm VLQ,2HDM}$ effects can be substantial and often exhibit
opposite signs. This variance allows for significant cancellations, especially
since $T_{\rm 2HDM}$ can have either sign while $T_{\rm VLQ}$ is almost
invariably negative. Specifically, a large negative value for $T_{\rm VLQ}$
can arise with high $\sin\theta_{L}$ values and a correspondingly large
positive $T_{\rm 2HDM}$ term can occur in the 2HDM when there is a
considerable mass splitting among the heavy Higgs bosons. It is important to
note that, within this representation, the constraints imposed by the $S$ and
$T$ parameters are more stringent than those from $R_{b}$ limits [8].

The right panel of Fig. 1 displays our analysis projected onto the
$(\sin\theta_{L},\Gamma(B))$ plane555Here, $\Gamma(B)$, at times also denoted
by $\Gamma_{B}$ is the VLB total width., illustrating that the mixing angle
$|\sin\theta_{L}|$ typically does not exceed 0.15 across the entire $m_{B}$
range. For minimal mixing ($|\sin\theta_{L}|<0.1$), the total width is
relatively small, often a few GeV or less. In contrast, for more large mixing
($|\sin\theta_{L}|\approx 0.1-0.15$) combined with large $m_{B}$ values, the
total width may vary between 10 and 100 GeV. These $\Gamma(B)$ values are
likely to be comparable to the experimental resolution of reconstructed VLBs,
suggesting that the width of the VLB could be limited to a maximum of 10% of
its mass.

This detailed analysis underscores the critical impact of incorporating
additional degrees of freedom from both the VLQ sector and 2HDM one. Their
integration enables a diverse array of solutions, accommodating lighter
$m_{B}$ masses ($\sim 1$ TeV) and extensive mixing angles $|\sin\theta_{L}|$.
This, in turn, could significantly enhance $pp\to B\overline{B}$ production
rates at the LHC (see Fig. 2) as well as the likelihood of $B$ decays into
heavy (pseudo)scalar Higgs states obtained from the 2HDM sector with notable
frequencies, as we shall see.

In Fig. 2, we depict the pair production cross-section666Note that the pair
production of VLQs is model independent and relies solely on their mass.
$\sigma(pp\to Q\bar{Q})$ at the next-to-next-to-leading order (NNLO) using the
sets CTEQ6L1 [54] parton densities, as a function of their mass $m_{Q}$. The
curves are depicted in blue and orange, corresponding to centre of mass
energies of 13 TeV and 14 TeV, respectively.

Figure 2: Pair production cross-section $\sigma(pp\to Q\bar{Q})$, computed at
NNLO using Top++[55] for center-of-mass energies $\sqrt{s}$ = 13 TeV (blue)
and $\sqrt{s}$ = 14 TeV (orange), respectively.

Advancing our discussion to the Branching Ratios (${\cal BR}$) associated with
the $B$ state, we find the scenario within the SM, supplemented by an
additional singlet bottom, mirrors closely the dynamics outlined for the
singlet VLT scenario tackled in Ref. [3]. The distribution of ${\cal BR}$s
among $B\to W^{-}t$, $B\to Zb$ and $B\to hb$ remains approximately at 50%, 25%
and 25%, respectively, as for a VLT. Remarkably, these distributions show
limited sensitivity to variations in the $\sin\theta_{L}$ mixing angle and are
bound by a sum rule that ensures their collective contribution to the $B$
state decay processes into SM final states respects

$\displaystyle{\cal BR}(B\to{\rm SM})={\cal BR}(B\to W^{-}t)+{\cal BR}(B\to
Zb)+{\cal BR}(B\to hb)=1.$ (24)

The introduction of additional decay pathways through (pseudo)scalar channels
such as $B\to H^{-}t$, $B\to Ab$ and $B\to Hb$ introduces a significant shift.
This evolution not only diversifies the decay mechanisms but also modifies the
constraints previously established by direct $B$ searches at the LHC,
prompting an updated sum rule:

$\displaystyle{\cal BR}(B\to{\rm SM})+{\cal BR}(B\to{\rm non~{}SM})=1,$
$\displaystyle{\cal BR}(B\to{\rm non~{}SM})={\cal BR}(B\to H^{-}t)+{\cal
BR}(B\to Ab)+{\cal BR}(B\to Hb).$ (25)

In Fig. 3, we depict the correlation between the decay processes of the $B$
state into SM particles and through novel pathways, organized as follows:
$B\to{hb,Hb}$ (left), $B\to{Zb,Ab}$ (middle), and $B\to{W^{-}t,H^{-}t}$
(right). It is observed that when non-SM decay channels become kinematically
open, the ${\cal BR}(B\to H^{-}t)$ begins to rival the ${\cal BR}(B\to
W^{-}t)$ especially for low $\tan\beta\sim 1$. A similar competitive
relationship is noted between ${\cal BR}(B\to Hb)$ and ${\cal BR}(B\to Ab)$
but for medium $\tan\beta$.

Figure 3: The correlation between ${\cal BR}(B\to hb)$ and ${\cal BR}(B\to
Hb)$ (left), ${\cal BR}(B\to Zb)$ and ${\cal BR}(B\to Ab)$ (middle) as well as
${\cal BR}(B\to W^{-}t)$ and ${\cal BR}(B\to H^{-}t)$ (right) with $\tan\beta$
indicated in the color gauge. Figure 4: The ${\cal BR}(B\to$ SM) (left) and
${\cal BR}(B\to$ non SM) (right) mapped onto the $(m_{B},\tan\beta)$ plane,
with $\sin\theta_{L}=0.045$, $\sin(\beta-\alpha)=1$, $m_{h}=125$ GeV,
$m_{H}=585$, $m_{A}=582$ GeV, and $m_{H^{\pm}}=650$ GeV (recall that
$m_{12}^{2}=m_{A}^{2}\tan\beta/(1+\tan^{2}\beta)$). Here, the shaded areas are
excluded by HiggsBounds ($H^{+}\to t\bar{b}$ [56] for $\tan\beta<2$ and
$H\to\tau\tau$ [57] for $\tan\beta>6$). All other constraints ($S$, $T$,
HiggsSignals and theoretical ones) are also checked.

In Fig. 4, we display the cumulative ${\cal BR}$s of the of the $B$ state into
SM particles (left panel) and non-SM particles (right panel), plotted as
functions of the singlet bottom mass $m_{B}$ and the parameter $\tan\beta$. It
is evident that, for high values of $\tan\beta$, the decay rates into non-SM
particles increase significantly, reaching up to 50% (38% within the
permissible region). In contrast, at medium and low $\tan\beta$ values,
regardless of $m_{B}$, the scenario reverses, with decays into SM particles
becoming fully predominant.

Given this context, we propose three Benchmark Points (BPs) tailored for
investigating the Type-II 2HDM extended by a singlet VLQ carrying the bottom
EM charge. The proposed BPs are as follows777Should these conditions not be
fully satisfied, we will proceed with only two BPs instead of the proposed
three.:

  * i)

BP1: where ${\cal BR}(B\to$ SM) is similar to ${\cal BR}(B\to$ non SM);

  * ii)

BP2: where ${\cal BR}(B\to$ SM) is rather small, which makes ${\cal BR}(B\to$
non SM) substantial;

  * iii)

BP3: where ${\cal BR}(B\to$ SM) is rather large, which makes ${\cal BR}(B\to$
non SM) marginal.

Input parameters for these BPs are given in Tab. 2.

Parameters | BP1 | BP2
---|---|---
Masses are in GeV
$m_{h}$ | 125 | 125
$m_{H}$ | 645.79 | 774.80
$m_{A}$ | 406.54 | 754.38
$m_{H\pm}$ | 677.91 | 786.51
$\tan\beta$ | 1.04 | 1.51
$m_{B}$ | 1946.63 | 1049.94
$\sin(\theta^{d})_{L}$ | 0.061 | $-0.023$
$\mathcal{BR}(H^{\pm}\to XY)$ in %
${\cal BR}(H^{+}\to t\bar{b})$ | 54.31 | 99.80
${\cal BR}(H^{+}\to\tau\nu)$ | 0.00 | 0.0243
${\cal BR}(H^{+}\to W^{+}A)$ | 45.58 | 0.00
$\mathcal{BR}(B\to XY)$ in %
${\cal BR}(B\to W^{-}t)$ | 27.41 | 41.13
${\cal BR}(B\to Zb)$ | 13.98 | 22.31
${\cal BR}(B\to hb)$ | 13.86 | 21.69
${\cal BR}(B\to Hb)$ | 11.07 | 4.62
${\cal BR}(B\to Ab)$ | 12.78 | 5.22
${\cal BR}(B\to H^{-}t)$ | 20.86 | 5.00
$\Gamma$ in $\mathrm{GeV}$
$\Gamma(B)$ | 32.33 | 27.33
Table 2: The full description of our BPs for the $(B)$ singlet case.

###  3.2 2HDM with $(TB)$ doublet

We now discuss the case of the $(TB)$ doublet. In the SM extended with such a
VLQ multiplet, both mixing angles in the up-and down-type quark sectors enter
the phenomenology of the model888Hereafter, we replace the $d$ and $u$
superscripts in the mixing angles with $b$ and $t$, respectively.. For given
$\theta_{R}^{b}$, $\theta_{R}^{t}$ and $m_{B}$ mass, the relationship between
the mass eigenstates and the mixing angles reads as [8]:

$\displaystyle
m_{T}^{2}=(m_{B}^{2}\cos^{2}\theta_{R}^{b}+m_{b}^{2}\sin^{2}\theta_{R}^{b}-m_{t}^{2}\sin^{2}\theta_{R}^{t})/\cos^{2}\theta_{R}^{t},$
(26)

from where one can compute the $m_{T}$ value. Furthermore, using Eqs.
(19)–(20), one can also compute the left mixing $\theta_{L}^{b}$ and
$\theta_{L}^{t}$.

Figure 5: (Left) Scatter plots of randomly generated points superimposed onto
the fit limits in the $(\Delta S,\Delta T)$ plane from EWPO data at 95% CL
with a correlation of 92%. Here, we illustrate the 2HDM and VLQ contributions
separately and also the total one. (Right) The same points are mapped onto the
$(m_{B},\delta)$ plane, where $\delta$ is the mass difference between $B$ and
$T$. Here, we only present the VLQ contribution and the total one. Further,
all constraints have been taken into account.

In our analysis of the 2HDM+$(TB)$ doublet, we use the scan ranges described
in Tab. 1. Fig. 5 depicts the $S$ and $T$ parameters, distinguishing the
contributions from the VLQ and the 2HDM separately. As seen in the left panel
of Fig. 5, similar to the earlier case involving a heavy bottom ($B$) singlet,
the VLQ $(TB)$ doublet and 2HDM states can contribute with opposite signs to
the oblique parameters. This aspect could significantly alter the constraints
on the 2HDM+VLQ parameter space derived from EWPOs. Moreover, as illustrated
in the right panel of the same figure, within the SM extended by a VLQ (green
points), the mass splitting between $B$ and $T$ must not exceed 25 GeV in the
positive direction and 5 GeV in the negative direction. However, when
introducing 2HDM states alongside VLQs, this allowed mass splitting widens,
reaching up to 40 GeV positively and 10 GeV negatively.

In the scenario of the SM extended by a VLQ $(TB)$ doublet, the $B$ state can
decay through multiple modes: $B\rightarrow{W^{-}t,Zb,th}$. Notably, if the
mixing angle $\theta^{b}_{R}$ is zero ($\theta^{b}_{R}=0$), the $bBZ$ and
$bBh$ couplings disappear, leading to a dominant decay mode of $B\rightarrow
W^{-}t$. With non-zero mixing, these decay modes become viable, with their
${\cal BR}$s depending on the mixing angles and VLQ masses. The introduction
of additional Higgs particles opens new decay channels:
$B\rightarrow{Ab,Hb,H^{-}t}$999It is crucial to consider that constraints on
the mass splitting between $B$ and $T$, as imposed by the $S$ and $T$
parameters, restrict the potential for VLQ decays into each other, such as
$B\rightarrow W^{-}/H^{-}T$. Therefore, we will concentrate solely on the
kinematically open $1\to 2$ channels (on-shell decays)..

Figure 6: The $\Gamma(B)/m_{B}$ ratio $(\Gamma(B)$ mapped over the
$(m_{B},\tan\beta)$ plane (left) and
$(s^{d}_{R}\equiv\sin\theta^{d}_{R},\tan\beta)$ plane (right), with
$\sin\theta^{u}_{R}=0.042$ in the left panel and $m_{B}=1600$ GeV in the right
panel (the 2HDM parameters are the same as in Fig. 4). Here, the shaded areas
are excluded by HiggsBounds. The regions between the vertical lime green lines
are allowed by the $S$, $T$ parameter constraints, all other constraints
(HiggsSignals and theoretical ones) are also checked.

To further explore the phenomenology of this BSM setup, we perform a
comprehensive scan over both the 2HDM and VLQ $(TB)$ doublet parameters as
specified in Tab. 1, adhering to all theoretical and experimental constraints.
Fig. 6 displays the total width of the $B$ state as a function of its mass
$m_{B}$ (left panel) and the down-quark sector mixing angle $s^{d}_{R}$ (right
panel), with both dimensions correlated to $m_{B}$ and showing insensitivity
to $\tan\beta$. The analysis reveals that $\Gamma(B)/m_{B}$ significantly
increases with $m_{B}$, and, as indicated in the right panel, this trend is
particularly pronounced for large mixing values ($|s^{d}_{R}|\geq 2$) which
are already beyond the permissible bounds set by the $S$ and $T$ parameters
(green line).

Fig. 7 explores the decay patterns into SM and non-SM particles, the latter
involving one additional Higgs state, such as $B\to H^{-}t$, $B\to Ab$ and
$B\to Hb$101010In principle, in the definition of these two ${\cal BR}$s, one
should account for the $B\to W^{-}T$ and $B\to H^{-}T$ decays, however,
because both the $W^{-}$ and $H^{-}$ are off-shell, they are always small, so
we will not discuss these here (recall the degeneracy $m_{B}\sim m_{T}$)..
This analysis reveals that non-SM decay channels of the $B$ state can account
for up to 40% of decays for higher $m_{B}$ values, with negligible dependence
on $\tan\beta$.

Figure 7: The ${\cal BR}(B\to$ SM) (left) and ${\cal BR}(B\to$ non SM) (right)
mapped onto the $(m_{B},\tan\beta)$ plane, with the same description as in
Fig. 6 (left). Figure 8: The correlation between ${\cal BR}(B\to hb)$ and
${\cal BR}(B\to Hb)$ (left), ${\cal BR}(B\to Zb)$ and ${\cal BR}(B\to Ab)$
(middle) as well as ${\cal BR}(B\to W^{-}t)$ and ${\cal BR}(B\to H^{-}t)$
(right) with $\tan\beta$ indicated in the color gauge.

In Fig. 8, we further analyze the correlation between the individual SM and
non-SM decay channels of the $B$ state. This examination reveals that, across
both neutral and charged current scenarios, non-SM decay channels generally
exhibit lower ${\cal BR}$s compared to their SM ones. Specifically, the ${\cal
BR}$ for $B\to Zb$ and $B\to hb$ can reach approximately 45% while for $B\to
Ab$ and $B\to Hb$ can touch around 25% at most. Furthermore, the analysis
indicates a notable variation in the charged current decays: the non-SM decay
$B\to H^{-}t$ can achieve a maximum ${\cal BR}$ of 34% at low $\tan\beta$
values. In contrast, the SM decay $B\to W^{-}t$ can reach up to 100% at medium
$\tan\beta$ levels. This discrepancy highlights the impact of $\tan\beta$ on
the decay behavior of the $B$ state, with the couplings $bBH$ and $bBA$
showing a slight increase proportional to $\tan\beta$ at medium levels of this
parameter.

Parameters | BP1 | BP2
---|---|---
Masses are in GeV
$m_{h}$ | 125 | 125
$m_{H}$ | 598.48 | 700.23
$m_{A}$ | 449.60 | 470.00
$m_{H\pm}$ | 613.8 | 685.86
$\tan\beta$ | 1.79 | 1.17
$m_{T}$ | 1898.63 | 1071.66
$m_{B}$ | 1916.73 | 1068.72
$\sin(\theta^{u})_{L}$ | -0.000059 | 0.012241
$\sin(\theta^{d})_{L}$ | 0.000345 | -0.000047
$\sin(\theta^{u})_{R}$ | -0.000653 | 0.075789
$\sin(\theta^{d})_{R}$ | 0.137099 | -0.010480
$\mathcal{BR}(H^{\pm}\to{XY})$ in %
${\cal BR}(H^{+}\to t\bar{b})$ | 62.73 | 63.8
${\cal BR}(H^{+}\to W^{+}A)$ | 37.13 | 36.08
$\mathcal{BR}(B\to{XY})$ in %
${\cal BR}(B\to W^{-}t)$ | 0.0017 | 79.11
${\cal BR}(B\to Zb)$ | 27.03 | 0.79
${\cal BR}(B\to hb)$ | 26.80 | 0.77
${\cal BR}(B\to Hb)$ | 22.01 | 0.26
${\cal BR}(B\to Ab)$ | 24.14 | 0.52
${\cal BR}(B\to H^{-}t)$ | 0.00 | 18.53
$\Gamma$ in $\mathrm{GeV}$
$\Gamma(B)$ | 78.82 | 2.75
Table 3: The full description of our BPs for the $(TB)$ doublet case.

Building on the approach of the previous section, we conclude this part by
introducing in Tab. 3 two BPs that are conducive to experimental exploration.
These BPs are designed with a focus on $B$ quark masses and feature varying
widths ($\Gamma_{B}$) to span a range of possible detection scenarios.

###  3.3 2HDM with $(BY)$ doublet

In the case of the SM extended with a $(BY)$ doublet, the ensuing VLQ
structure is fully described by the $\theta_{R}^{b}$ mixing angle and the new
bottom mass $m_{B}$. In fact, for a given $\theta_{R}^{b}$ value,
$\theta_{L}^{b}$ is computed using Eq. (20). The mass of the new VLQ with
exotic EM charge ($+5/3$), the $Y$ state, is given as a function of such a
mixing angle $m_{B}$ as well as $m_{b}$ by [8].

$m_{Y}^{2}=m_{B}^{2}\cos\theta_{R}^{2}+m_{b}^{2}\sin\theta_{R}^{2}.$ (27)

This is independent (at tree level) from the additional parameters entering
the 2HDM Higgs sector, however, the latter impinges on the viability of this
BSM construct against EWPO data.

Expanding on the analysis presented in Tab. 1, Fig. 9 (left) shows that while
the combined (VLQ+2HDM) contribution is within the $2\sigma$ CL of the $\Delta
S$ and $\Delta T$ parameters (blue points), the two separate contributions
from the VLQ and 2HDM sectors frequently surpass the boundaries established by
EWPO data ellipses.

In the right panel of Fig. 9, we further illustrate the size of the mass
splitting $\delta=m_{B}-m_{Y}$ allowed by EWPO data. In the case of the VLQ
only structure, the splitting is always very small. Instead, in the case of
the full 2HDM+VLQ scenario, one can see that the splitting could be very
large, of the order of 40 GeV, the more so the larger $m_{B}$.

Figure 9:  (Left) Scatter plots of randomly generated points superimposed onto
the fit limits in the $(\Delta S,\Delta T)$ plane from EWPO data at 95% CL
with a correlation of 92%. Here, we illustrate the 2HDM and VLQ contributions
separately and also the total one. (Right) The same points are here mapped
onto the $(m_{B},\delta)$ plane, where $\delta$ is the mass difference between
$B$ and $Y$. Here, we only present the VLQ contribution and the total one.
Further, all constraints have been taken into account.

In Fig. 10, we compare again ${\cal BR}(B\to{\rm SM})$ to ${\cal BR}(B\to{\rm
non~{}SM})$, mapped over $m_{B}$ and $\tan\beta$. In contrast to the previous
VLQ doublet realization, here, the non-SM decays of the $B$ state can
dominate, potentially reaching up to 100% within the permissible parameter
space, which notably expands with increasing $\tan\beta$. Interestingly, the
decay behavior of the heavy bottom partner shows minimal sensitivity to the
mass $m_{B}$.

We now discuss the size of the individual ${\cal BR}$s of $B$ decays. As
usual, alongside the SM decays of new bottom $B$, into $W^{-}t,Zb$ and $hb$,
one only has the non-SM decays $B\to Hb$ and $B\to Ab$: in fact, the $B\to
H^{-}t$ channel is not available, as the intervening coupling is identically
zero, hence it is excluded from our discussion.

Figure 10: The ${\cal BR}(B\to$ SM) (left) and ${\cal BR}(B\to$ non SM)
(right) mapped onto the $(m_{B},\tan\beta)$ plane. with
$\sin\theta_{R}^{u}=0.057$ (the 2HDM parameters are the same as in Fig. 4).
Here, the shaded areas are excluded by HiggsBounds, and all other constraints
($S$, $T$, HiggsSignals and theoretical ones) are also checked. Figure 11: The
correlation between ${\cal BR}(B\to hb)$ and ${\cal BR}(B\to Hb)$ (left),
${\cal BR}(B\to Zb)$ and ${\cal BR}(B\to Ab)$ (right) with $\tan\beta$ as
indicated in the color gauge.

In Fig. 11, we present the usual correlations between ${\cal BR}(B\to hb)$ and
${\cal BR}(B\to Hb)$ (left) as well as ${\cal BR}(B\to Zb)$ and ${\cal
BR}(B\to Ab)$ (right). At medium $\tan\beta$ values (noting that large
$\tan\beta$ values are ruled out by $H\to\tau\tau$ [57] searches), ${\cal
BR}(B\to Hb)$ and ${\cal BR}(B\to Ab)$ significantly rise, each reaching up to
50% as their respective couplings increase proportionally to $\tan\beta$. This
results in a notable elevation of ${\cal BR}(B\to Zb)$ and ${\cal BR}(B\to
hb)$, positioning medium $\tan\beta$ as an ideal range for investigating
exotic $B$ decays within the established parameter space. Conversely, at lower
$\tan\beta$ values, ${\cal BR}(B\to hb)$ and ${\cal BR}(B\to Zb)$ receive a
boost, each approaching approximately 38%, which in turn mildly suppresses
${\cal BR}(B\to Hb)$ and ${\cal BR}(B\to Ab)$. This nuanced behavior across
different $\tan\beta$ ranges underscores the complex interplay between these
parameters and the decay modes of the $B$ state in this BSM framework. At
small $\tan\beta$, instead, one can see that both ${\cal BR}(B\to hb)$ and
${\cal BR}(B\to Zb)$ are somewhat enhanced with each nearing 38% and therefore
${\cal BR}(B\to Hb)$ and ${\cal BR}(B\to Ab)$ are somewhat suppressed.

Parameters | BP1 | BP2 | BP3
---|---|---|---
Masses are in GeV
$m_{h}$ | 125 | 125 | 125
$m_{H}$ | 710.03 | 608.02 | 782.90
$m_{A}$ | 629.51 | 604.74 | 765.13
$m_{H\pm}$ | 748.28 | 602.00 | 773.11
$\tan\beta$ | 1.17 | 5.94 | 1.12
$m_{B}$ | 1266.05 | 1686.13 | 1042.89
$m_{Y}$ | 1262.22 | 1686.10 | 1040.30
$\sin(\theta^{d})_{L}$ | 0.000294 | 0.000018 | -0.000324
$\sin(\theta^{d})_{R}$ | 0.077721 | 0.006235 | -0.070470
$\mathcal{BR}(H^{\pm}\to{XY})$ in %
${\cal BR}(H^{+}\to t\bar{b})$ | 94.54 | 95.74 | 99.82
${\cal BR}(H^{+}\to\tau\nu)$ | 0.00 | 4.06 | 4.50.00
${\cal BR}(H^{+}\to W^{+}A)$ | 5.28 | 0.00 | 0.00
$\mathcal{BR}(B\to{XY})$ in %
${\cal BR}(B\to W^{-}t)$ | 0.00 | 0.00 | 0.0015
${\cal BR}(B\to Zb)$ | 25.24 | 0.05 | 38.34
${\cal BR}(B\to hb)$ | 24.75 | 0.05 | 37.26
${\cal BR}(B\to Hb)$ | 22.67 | 49.86 | 11.50
${\cal BR}(B\to Ab)$ | 27.33 | 50.02 | 12.87
${\cal BR}(B\to H^{-}t)$ | - | - |
$\Gamma$ in $\mathrm{GeV}$
$\Gamma(B)$ | 7.92 | 57.94 | 2.397
Table 4: The full description of our BPs for the $(BY)$ doublet case.

Concluding our discussion, for further exploration of this BSM scenario, we
propose three BPs as detailed in Tab. 4. These BPs encapsulate the conditions
elucidated earlier.

###  3.4 2HDM with $(XTB)$ triplet

We discuss here the $(XTB)$ triplet case. Before presenting our numerical
results, though, let us first introduce our parametrization. This model is
fixed by giving the new bottom mass and one mixing angle, let us say
$\theta_{L}^{t}$, the other parameters are then computable. In fact,
$\theta_{R}^{t}$ is derived from Eq. (20) while $m_{X}$ is given by [8]:

$\displaystyle
m_{X}^{2}=m_{T}^{2}\cos\theta^{u}_{L}+m_{t}^{2}\sin\theta^{u}_{L}=m_{B}^{2}\cos^{2}\theta_{L}^{b}+m_{b}^{2}\sin^{2}\theta_{L}^{b}.$
(28)

Using the above relation between $m_{T}$ and $m_{X}$ together with the one
between up- and down-type quark mixing in Eq. (21), one can derive the mass of
the new bottom quark as follows:

$\displaystyle
m_{B}^{2}=\frac{1}{2}\sin^{2}(2\theta^{u}_{L})(m_{T}^{2}-m_{t}^{2})^{2}/(m_{X}^{2}-m_{b}^{2})+m_{X}^{2}.$
(29)

The down-type quark mixing is then given by.

$\displaystyle\sin(2\theta_{L}^{d})=\sqrt{2}\frac{m_{T}^{2}-m_{t}^{2}}{m_{B}^{2}-m_{b}^{2}}\sin(2\theta^{u}_{L}).$
(30)

Figure 12:  Scatter plots for randomly generated points superimposed onto the
fit limits in the $(\Delta S,\Delta T)$ plane from EWPO data at 95% CL with a
correlation of 92%. Here, we illustrate the 2HDM and VLQ contributions
separately and also the total one. Further, all constraints have been taken
into account.

As usual, we conduct a systematic scan over the parameters of both the 2HDM
and VLQ sectors, as detailed in Tab. 1. In Fig. 12, it is evident that, while
the combined contribution of 2HDM and VLQ falls within the 95% CL of
$\Delta\chi^{2}$ for $\Delta S$ and $\Delta T$ , the individual contributions
of 2HDM and VLQ origin mostly lie outside the $2\sigma$ level.

Furthermore, the splitting between $m_{B}$, $m_{T}$ and $m_{X}$ is severely
constrained by the EWPOs, as depicted in Fig. 13, where mass differences
mapped against $m_{B}$ are given. It is noticeable that such a splitting is
relatively small in the case of the SM with a VLQ construct, approximately of
the order of a few GeV. However, in the case of the full 2HDM with VLQ
scenario, such a splitting becomes significantly larger, reaching up to 11 or
22 GeV, indicating that inter VLQ decays are likely to play a negligible role
in this BSM realization.

Figure 13: Scatter plots of randomly generated points mapped onto the
$(m_{B},\delta)$ plane, where $\delta$ is the mass difference between $B$ and
$T$, $B$ and $X$, and $T$ and $X$. Here, we illustrate the VLQ and 2HDM+VLQ
contributions separately. Further, all constraints have been taken into
account. Figure 14: The ${\cal BR}(B\to$ SM) (left) and ${\cal BR}(B\to$ non
SM) (right) mapped onto the $(m_{B},\tan\beta)$ plane, with
$\sin\theta_{L}^{u}=0.0093$ (the 2HDM parameters are the same as in Fig. 4).
Here, the shaded areas are excluded by HiggsBounds, and all other constraints
($S$, $T$, HiggsSignals and theoretical ones) are also checked.

Regarding the 2HDM with a $(XTB)$ triplet, the VLB can decay into the SM
channels (${Zb,hb}$ and ${W^{-}t}$) as well as extra channels ${Hb,Ab,H^{-}t}$
involving additional Higgs states of the 2HDM. The relative importance of
these channels is depicted in Fig. 14, revealing the dominance of non-SM
channels, particularly evident at medium $\tan\beta$ values where it can reach
100%. However, SM channels can still contribute significantly, reaching up to
60% in cumulative branching ratios within the allowed range.

Figure 15: The correlation between ${\cal BR}(B\to hb)$ and ${\cal BR}(B\to Hb)$ (left), ${\cal BR}(B\to Zb)$ and ${\cal BR}(B\to Ab)$ (middle) as well as ${\cal BR}(B\to W^{-}t)$ and ${\cal BR}(B\to H^{-}t)$ (right) with $\tan\beta$ indicated in the color gauge. Parameters | BP3 | BP2 | BP1
---|---|---|---
Masses are in GeV
$m_{h}$ | 125 | 125 | 125
$m_{H}$ | 782.99 | 630.00 | 599.77
$m_{A}$ | 719.97 | 629.92 | 586.59
$m_{H\pm}$ | 788.20 | 686.76 | 602.36
$\tan\beta$ | 1.23 | 5.64 | 2.08
$m_{T}$ | 1024.61 | 1833.36 | 1022.66
$m_{B}$ | 1024.98 | 1833.60 | 1022.69
$m_{X}$ | 1024.2 | 1833.12 | 1022.64
$\sin(\theta^{u})_{L}$ | 0.028 | $-0.016$ | 0.0071
$\sin(\theta^{d})_{L}$ | 0.038 | $-0.022$ | 0.0097
$\mathcal{BR}(H^{\pm}\to{XY})$ in %
${\cal BR}(H^{+}\to t\bar{b})$ | 99.80 | 96.19 | 99.73
${\cal BR}(H^{+}\to\tau\nu)$ | 0.011 | 3.45 | 0.08
${\cal BR}(H^{+}\to W^{+}A)$ | 0.014 | 0.07 | 0.00
$\mathcal{BR}(B\to{XY})$ in %
${\cal BR}(B\to W^{-}t)$ | 1.72 | 0.06 | 1.10
${\cal BR}(B\to Zb)$ | 46.64 | 5.43 | 29.89
${\cal BR}(B\to hb)$ | 29.55 | 3.51 | 18.94
${\cal BR}(B\to Hb)$ | 5.28 | 2.75 | 8.40
${\cal BR}(B\to Ab)$ | 11.93 | 88.06 | 38.00
${\cal BR}(B\to H^{-}t)$ | 4.86 | 0.16 | 3.64
$\Gamma$ in $\mathrm{GeV}$
$\Gamma(B)$ | 0.85 | 15.000 | 0.086
Table 5: The full description of our BPs for the $(XTB)$ triplet case.

Now, we proceed to discuss these individual $B$ decay channels. (Note that,
due to the small splitting between $X$ and $B$, the decay $B\to W^{+}Y$ is
closed for a real $W^{+}$ and is highly suppressed for an off-shell one, so we
do not discuss it here.)

As previously mentioned, the introduction of additional channels associated
with the extra 2HDM states has implications for the SM framework. While the SM
predicts the decay of the new bottom quark into final states, such as
${Zb,hb}$ and ${W^{-}t}$, the full 2HDM+VLQ scenario introduces new channels
$B\to{Hb,Ab,H^{-}t}$ that could be of significant importance and in the
present multiplet structure. Specifically, the decay $B\to Ab$ could exhibit a
${\cal BR}$ as high as 90% for large $\tan\beta$ values, owing to its coupling
proportionality to $\tan\beta$. In contrast, the relevance of other exotic
decays is limited, typically reaching up to 25%. These patterns are
illustrated in Fig. 15.

Once again, we offer an overview of our proposed BPs in Tab. 5, all of which
meet the conditions discussed previously.

###  3.5 2HDM with $(TBY)$ triplet

Figure 16:  Scatter plots for randomly generated points superimposed onto the
fit limits in the $(\Delta S,\Delta T)$ plane from EWPO data at 95% CL with a
correlation of 92%. Here, we illustrate the 2HDM and VLQ contributions
separately and also the total one. Further, all constraints have been taken
into account.

We finally discuss the $(TBY)$ triplet case. In the 2HDM with such a VLQ
representation, the situation is very similar to the case of the ($T$) singlet
and $(TB)$ doublet. Before discussing our numerical results, though, let us
again first introduce our parametrization. This model is fixed by giving the
new top mass and one mixing angle, let us say $\theta_{L}^{t}$, the other
parameters are then computable. In fact, $\theta_{R}^{t}$ is derived from Eq.
(20) while $m_{Y}$ is given by [8]:

$\displaystyle m_{Y}^{2}$ $\displaystyle=$ $\displaystyle
m_{T}^{2}\cos^{2}\theta_{L}^{t}+m_{t}^{2}\sin^{2}\theta_{L}^{t}=m_{B}^{2}\cos^{2}\theta_{L}^{b}+m_{b}^{2}\sin^{2}\theta_{L}^{b}.$
(31)

Using the above relation between $m_{T}$ and $m_{Y}$ together with the one
between up- and down-type quark mixing in Eq. (21), one can derive the mass of
the new bottom quark as:

$\displaystyle m_{B}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\sin^{2}2\theta_{L}^{t}\frac{(m_{T}^{2}-m_{t}^{2})^{2}}{m_{Y}^{2}-m_{b}^{2}}+m_{Y}^{2}.$
(32)

With this in hand, one can then derive the down-type quark mixing
$\theta_{L,R}^{d}$ using Eqs. (20)–(21).

Based on the scan listed in Tab. 1, we first illustrate in Fig. 16 the allowed
95% CL regions from the $S$ and $T$ parameter constraints, which shows that
both the 2HDM only and VLQ only contributions could well be out of the allowed
ranges but, when adding these together, one indeed finds viable solutions
because of cancellations. As mentioned previously, when adding VLQs alongside
extra Higgs states, the phenomenology of the $S$ and $T$ can change
drastically, with respect to the SM case.

Figure 17: Scatter plots of randomly generated points mapped onto the
$(m_{T/B},\delta)$ plane, where $\delta$ is the mass difference between $T$
and $B$, $T$ and $Y$, and $B$ and $Y$. Here, we illustrate the VLQ and
2HDM+VLQ contributions separately. Further, all constraints have been taken
into account.

Following Fig. 17, in the SM with this considered VLQ representation, the
splitting $\delta$ between the masses of $T$ and $B$ (left), $T$ and $Y$
(middle) as well as $B$ and $Y$ (right) is rather small, of the order a
fraction of GeV at small $m_{T}$. However, for high $m_{T}$ values, there
exists a narrow strip of parameter space where these splittings could be of
the order 1 GeV. This is very little. In contrast, in the 2HDM+VLQ scenario,
the dynamics change significantly. The splitting between $B$ and $T$ could
reach $-15$ GeV for small $m_{B}$ (around 1000 GeV), while the splittings
between $B$ and $Y$, as well as $T$ and $Y$, could extend up to 30 GeV in the
same mass region. For large $m_{B}$, the mass splitting of both the individual
contributions of VLQ and the combined 2HDM+VLQ scenario could reach up to
$-28$ GeV between $B$ and $T$, and up to $-58$ GeV between $B$ and $Y$ as well
as $T$ and $Y$. Once again, the significant cancellations between additional
Higgs and VLQ states, naturally occurring in loop contributions due to their
different spin statistics, enable the expansion of significant parameter
space. This parameter space will be further explored in line with the analyses
presented in the preceding two subsections to study 2HDM decays of VLQ states.
Nonetheless, the small values of $\delta$ observed in these plots underscore
the fact that VLQ decays into each other are once again mostly negligible.

Figure 18: The ${\cal BR}(T\to$ SM) (left) and ${\cal BR}(T\to$ non SM)
(right) mapped onto the $(m_{T},\tan\beta)$ plane, with
$\sin\theta_{L}^{u}=0.02$ (the 2HDM parameters are the same as in Fig. 4).
Here, the shaded areas are excluded by HiggsBounds, and all other constraints
($S$, $T$, HiggsSignals and theoretical ones) are also checked. Figure 19: The
correlation between ${\cal BR}(B\to hb)$ and ${\cal BR}(B\to Hb)$ (left)
(right), ${\cal BR}(B\to Zb)$ and ${\cal BR}(B\to Ab)$ (middle) as well as
${\cal BR}(B\to W^{-}t)$ and ${\cal BR}(B\to H^{-}t)$(left) with $\tan\beta$
indicated in the color gauge.

In the SM with a $(TBY)$ triplet, the decay patterns of $B$ states are
essentially the same as in the $(TB)$ doublet case. Therefore, in Fig. 18, we
investigate the decays of the new bottom quark into both SM channels, such as
${Zb,hb}$ and ${W^{-}t}$ as well exotic ones, such as ${Hb,Ab}$ and
${H^{-}t}$. The cumulative ${\cal BR}$s of non-SM decays are notably
pronounced, particularly evident at small $\tan\beta$ values and large
$m_{B}$, where they can reach up to 35% within the allowed region. However, SM
channels still dominate, accounting for approximately 80% of cumulative ${\cal
BR}$s for small $m_{B}$ values around $1000$ GeV within the allowed range.

We move now on to delve into the individual $B$ decay channels depicted in
Fig. 19. important to note that, due to the small splitting between $B$ and
$Y$, the decay $B\to W^{-}Y$ is closed for a real $W^{-}$ and highly
suppressed for an off-shell one, thus we will not discuss it here. Clearly,
one can observe from the figure a strong anti-correlation between standard and
exotic channels. Regarding charged currents, the $\tan\beta$ dependence
reveals that large values of it favor SM decays whereas small values favor
2HDM ones. Concerning neutral currents, both large and medium $\tan\beta$
values result in both the SM decays and 2HDM ones achieving their maximum,
around 21% for the SM and 15% for the 2HDM.

Figure 20: The ratio $\Gamma(B)/m_{B}$ mapped over the $(m_{T},\tan\beta)$
plane. with $\sin\theta_{L}^{u}=0.02$ (the 2HDM parameters are the same as in
Fig. 4). Here, the shaded areas are excluded by HiggsBounds, and all other
constraints ($S$, $T$, HiggsSignals and theoretical ones) are also checked.

Proceeding to our BPs, we initially analyze the ratio $\Gamma(B)/m_{B}$
illustrated in Fig. 20. This visualization underscores the consistently narrow
nature of the $B$ state within our BSM framework. Notably, the quantity
$\Gamma(B)/m_{B}$ reaches its peak of 0.008, primarily observed for high
values of $m_{B}$. Subsequently, in Tab. 6, we present our BP explicitly,
which include both light (BP1) and heavy (BP2) $B$ states. This diverse
selection allows for a comprehensive exploration of exotic $B$ decays in
phenomenological analyses.

Parameters | BP1 | BP2
---|---|---
Masses are in GeV
$m_{h}$ | 125 | 125
$m_{H}$ | 772.39 | 656.89
$m_{A}$ | 770.13 | 384.936
$m_{H\pm}$ | 777.26 | 683.82
$\tan\beta$ | 5.22 | 1.04
$m_{T}$ | 1075.63 | 1813.76
$m_{B}$ | 1075.48 | 1809.04
$m_{Y}$ | 1075.33 | 1804.39
$\sin(\theta^{u})_{L}$ | $-0.023$ | 0.102
$\sin(\theta^{d})_{L}$ | $-0.016$ | 0.0716
$\mathcal{BR}(H^{\pm}\to{XY})$ in %
${\cal BR}(H^{+}\to t\bar{b})$ | 97.09 | 48.26
${\cal BR}(H^{+}\to\tau\nu)$ | 2.71 | 0.00
${\cal BR}(H^{+}\to W^{+}A)$ | 0.00 | 51.64
$\mathcal{BR}(B\to{XY})$ in %
${\cal BR}(B\to W^{-}t)$ | 47.69 | 29.31
${\cal BR}(B\to Zb)$ | 21.26 | 14.09
${\cal BR}(B\to hb)$ | 20.69 | 13.96
${\cal BR}(B\to Hb)$ | 4.98 | 10.62
${\cal BR}(B\to Ab)$ | 5.04 | 12.85
${\cal BR}(B\to H^{-}t)$ | 0.30 | 19.14
$\Gamma$ in $\mathrm{GeV}$
$\Gamma(B)$ | 0.25 | 35.21
Table 6: The full description of our BPs for the $(TBY)$ triplet case.

##  4 Conclusions

The third generation quarks of the SM, top and bottom, are often associated to
SM extensions involving VLQs with the same EM charge, with which they can mix
and thus interact, thereby mutually affecting their phenomenology. While this
approach has been commonly exploited in the case of a SM-like Higgs sector,
the relevance of these new states of Nature could actually be more important
in the case of BSM scenarios with an extended Higgs sector, as VLQs and
companion Higgs states can also interact with each other rather strongly.
Furthermore, VLQs with exotic charges can belong to the same representation,
thereby also entering the model phenomenology.

Specifically, in this paper, we have considered a 2HDM of Type-II supplemented
by a VLB companion, alongside other new fermionic states ($T$, $X$ and $Y$),
of the SM falling in a singlet, doublet or triplet representation, in
appropriate combinations. After constraining the parameter space of this model
for all VLQ multiplet cases against the latest theoretical and experimental
constraints, we have eventually proceeded to tension the ‘standard’ decays
$B\to W^{-}t,Zb$ and $hb$ (with $h$ being the discovered SM-like Higgs state)
against the ‘exotic’ ones $B\to H^{-}t,Ab$ and $Hb$ (where $H^{-},A$ and $H$
are the heavy Higgs states of the 2HDM). Also notice that $T$ states were not
considered amongst the decay products of the $B$ ones, because of the strong
mass degeneracy between these (also involving, in part, the exotic VLQ states
$X$ and $Y$).

In doing so, we have come to the following broad conclusions.

  * •

The simultaneous presence of a VLB and additional heavy Higgs boson loops as
well as of their mixing effects with $b$ and $h$ states of the SM in the
calculation of the EWPOs (i.e., the $S,T$ and $U$ parameters) has enabled us
to access a much wider portion of parameter space than would otherwise been
possible if only the separate effects of, on the one hand, $B$ and, on the
other hand, $H^{-},A$ and $H$ states had been considered. In particular, this
opens up parameter space wherein $B$, as well as $H^{-},A$ and $H$ particles,
can be light enough to be pursued in direct searches at the LHC, following
$B\overline{B}$ pair production via QCD.

  * •

The ability to access the aforementioned $B$ exotic decays (and measure their
relative rates) could allow one, in the presence of mass measurements of the
Higgs states, to disentangle the underlying VLQ multiplet structure, whether
singlet, doublet or triplet, thereby greatly charactering the underlying BSM
scenario, including confirming its Type-II nature and measure the key
parameters of our 2HDM+VLQ model.

We have come to these conclusions by performing an inclusive analysis at the
cross-section and ${\cal BR}$ level, without any MC simulations. We advocate
the latter as the next step to carry out a vigorous research programme testing
the possibility of both an extended Higgs sector and an enlarged fermionic
spectrum being simultaneously accessible at the LHC through non-SM like
signatures of a VLB produced in pairs in QCD collisions. To this end, we have
proposed several BPs amenable to experimental investigation during Run 3 of
the LHC and/or its high luminosity phase.

## Acknowledgments

The work of AB, RB and MB is supported by the Moroccan Ministry of Higher
Education and Scientific Research MESRSFC and CNRST Project PPR/2015/6. SM is
supported in part through the NExT Institute, STFC Consolidated Grant
ST/L000296/1 and Knut and Alice Wallenberg foundation under the grant KAW
2017.0100 (SHIFT).

## Appendix

###  4.1 Lagrangian in the mass basis

After EWSB, we are left with five Higgs bosons that are two-CP even $h$ and
$H$, one CP-odd $A$, and then a pair of charged Higgs $H^{\pm}$. We now
collect the Lagrangian on the mass basis in the general 2HDM Type-II.

###  4.2 Light-light interactions

$\displaystyle\mathcal{L}_{W}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\overline{t}\gamma^{\mu}(V^{L}_{tb}P_{L}+V^{R}_{tb}P_{R})bW^{+}_{\mu}+H.c.,$
$\displaystyle\mathcal{L}_{Z}$ $\displaystyle=$
$\displaystyle-\frac{g}{2c_{W}}\overline{t}\gamma^{\mu}(X^{L}_{tt}P_{L}+X^{R}_{tt}P_{R}-2Q_{t}s_{W}^{2})tZ_{\mu}$
$\displaystyle-\frac{g}{2c_{W}}\overline{b}\gamma^{\mu}(-X^{L}_{bb}P_{L}-X^{R}_{bb}P_{R}-2Q_{b}s_{W}^{2})bZ_{\mu}+H.c.,$
$\displaystyle\mathcal{L}_{h^{0}}$ $\displaystyle=$
$\displaystyle-\frac{gm_{t}}{2M_{W}}Y_{tt}^{h}\overline{t}th^{0}-\frac{gm_{b}}{2M_{W}}Y_{bb}^{h}\overline{b}bh^{0}+H.c.,$
$\displaystyle\mathcal{L}_{H^{0}}$ $\displaystyle=$
$\displaystyle-\frac{gm_{t}}{2M_{W}}Y_{tt}^{H}\overline{t}tH^{0}-\frac{gm_{b}}{2M_{W}}Y_{bb}^{H}\overline{b}bH^{0}+H.c.,$
$\displaystyle\mathcal{L}_{A}$ $\displaystyle=$
$\displaystyle-i\frac{gm_{t}}{2M_{W}}Y_{tt}^{A}\overline{t}\gamma_{5}tA+i\frac{gm_{b}}{2M_{W}}Y_{bb}^{A}\overline{b}\gamma_{5}bA+H.c.,$
$\displaystyle\mathcal{L}_{H^{+}}$ $\displaystyle=$
$\displaystyle-\frac{gm_{t}}{\sqrt{2}M_{W}}\overline{t}(\cot\beta
Z^{L}_{tb}P_{L}+\tan\beta Z^{R}_{tb}P_{R})bH^{+}+H.c.$ (33)
$\displaystyle\begin{array}[]{c|ccc}&V^{L}_{tb}&&V^{R}_{tb}\\\
\hline\cr(B)&c_{L}&&0\\\
(TB)&c_{L}^{u}c_{L}^{d}+s_{L}^{u}s_{L}^{d}e^{i(\phi_{u}-\phi_{d})}&&s_{R}^{u}s_{R}^{d}e^{i(\phi_{u}-\phi_{d})}\\\
(BY)&c_{L}&&0\\\
(XTB)&c_{L}^{u}c_{L}^{d}+\sqrt{2}s_{L}^{u}s_{L}^{d}&&\sqrt{2}s_{R}^{u}s_{R}^{d}\\\
(TBY)&c_{L}^{u}c_{L}^{d}+\sqrt{2}s_{L}^{u}s_{L}^{d}&&\sqrt{2}s_{R}^{u}s_{R}^{d}\end{array}$
(40)

Table VII: Light-light couplings to the $W$ boson.

$\displaystyle\begin{array}[]{c|cccccc}&\quad X_{tt}^{L}&&\quad
X_{tt}^{R}&\quad X_{bb}^{L}&&\quad X_{bb}^{R}\\\
\hline\cr(B)&1&&0&c_{L}^{2}&&0\\\
(TB)&1&&(s_{R}^{u})^{2}&1&&(s_{R}^{d})^{2}\\\
(BY)&1&&0&c_{L}^{2}-s_{L}^{2}&&-s_{R}^{2}\\\
(XTB)&(c_{L}^{u})^{2}&&0&1+(s_{L}^{d})^{2}&&2(s_{R}^{d})^{2}\\\
(TBY)&1+(s_{L}^{u})^{2}&&2(s_{R}^{u})^{2}&(c_{L}^{d})^{2}&&0\end{array}$ (47)

Table VIII: Light-light couplings to the $Z$ boson.

$\displaystyle\begin{array}[]{c|ccc }&Y_{tt}^{h}&Y_{tt}^{H}&Y_{tt}^{A}\\\
\hline\cr(B)&s_{\beta\alpha}+c_{\beta\alpha}\cot\beta&c_{\beta\alpha}-s_{\beta\alpha}\cot\beta&-\cot\beta\\\
(TB)&(s_{\beta\alpha}+c_{\beta\alpha}\cot\beta)(c_{R}^{u})^{2}&(c_{\beta\alpha}-s_{\beta\alpha}\cot\beta)(c_{R}^{u})^{2}&-\cot\beta(c_{R}^{u})^{2}\\\
(BY)&s_{\beta\alpha}+c_{\beta\alpha}\cot\beta&c_{\beta\alpha}-s_{\beta\alpha}\cot\beta&-\cot\beta\\\
(XTB)&(s_{\beta\alpha}+c_{\beta\alpha}\cot\beta)(c_{L}^{u})^{2}&(c_{\beta\alpha}-s_{\beta\alpha}\cot\beta)(c_{L}^{u})^{2}&-\cot\beta(c_{L}^{u})^{2}\\\
(TBY)&(s_{\beta\alpha}+c_{\beta\alpha}\cot\beta)(c_{L}^{u})^{2}&(c_{\beta\alpha}-s_{\beta\alpha}\cot\beta)(c_{L}^{u})^{2}&-\cot\beta(c_{L}^{u})^{2}\end{array}$
(54)

Table IX: Light-light top quark couplings to the triplets Higgs {$h,H,A$}.

$\displaystyle\begin{array}[]{c|ccc }&Y_{bb}^{h}&Y_{bb}^{H}&Y_{bb}^{A}\\\
\hline\cr(B)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)c_{L}^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)c_{L}^{2}&\tan\beta
c_{L}^{2}\\\
(TB)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)(c_{R}^{d})^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)(c_{R}^{d})^{2}&\tan\beta(c_{R}^{d})^{2}\\\
(BY)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)c_{R}^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)c_{R}^{2}&\tan\beta
c_{R}^{2}\\\
(XTB)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)(c_{L}^{d})^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)(c_{L}^{d})^{2}&\tan\beta(c_{L}^{d})^{2}\\\
(TBY)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)(c_{L}^{d})^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)(c_{L}^{d})^{2}&\tan\beta(c_{L}^{d})^{2}\\\
\end{array}$ (61)

Table X: Light-light bottom quark couplings to the triplets Higgs {$h,H,A$}.

$\displaystyle\begin{array}[]{c|ccc}&Z^{L}_{tb}&&Z^{R}_{tb}\\\
\hline\cr(B)&c_{L}&&\frac{m_{t}}{m_{B}}c_{L}\\\
(TB)&c_{L}^{d}c_{L}^{u}+\frac{s_{L}^{d}}{s_{L}^{u}}(s_{L}^{u}{}^{2}-s_{R}^{u}{}^{2})e^{i(\phi_{u}-\phi_{d})}&&\frac{m_{b}}{m_{t}}\left[c_{L}^{u}c_{L}^{d}+\frac{s_{L}^{u}}{s_{L}^{d}}(s_{L}^{d}{}^{2}-s_{R}^{d}{}^{2})e^{i(\phi_{u}-\phi_{d})}\right]\\\
(BY)&c_{R}&&c_{L}\\\ (XTB)&c_{L}^{u}&&c_{L}^{d}\\\
(TBY)&c_{L}^{u}&&c_{L}^{d}\end{array}$ (68)

Table XI: Light-light couplings to the charged Higgs.

###  4.3 Heavy-heavy interactions

$\displaystyle\mathcal{L}_{W}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\overline{Q}\gamma^{\mu}(V^{L}_{QQ^{\prime}}P_{L}+V^{R}_{QQ^{\prime}}P_{R})Q^{\prime}W^{+}_{\mu}+H.c.,$
$\displaystyle\mathcal{L}_{Z}$ $\displaystyle=$
$\displaystyle-\frac{g}{2c_{W}}\overline{Q}\gamma^{\mu}(\pm X^{L}_{QQ}P_{L}\pm
X^{R}_{QQ}P_{R}-2Q_{Q}s_{W}^{2})QZ_{\mu}$ $\displaystyle\ \mathcal{L}_{h^{0}}$
$\displaystyle=$
$\displaystyle-\frac{gm_{Q}}{2M_{W}}Y_{QQ}^{h}\overline{Q}Qh^{0}+H.c.,$
$\displaystyle\mathcal{L}_{H^{0}}$ $\displaystyle=$
$\displaystyle-\frac{gm_{Q}}{2M_{W}}Y_{QQ}^{H}\overline{Q}QH^{0}+H.c.,$
$\displaystyle\mathcal{L}_{A}$ $\displaystyle=$ $\displaystyle\pm
i\frac{gm_{Q}}{2M_{W}}Y_{QQ}^{A}\overline{Q}\gamma_{5}QA+H.c.,$
$\displaystyle\mathcal{L}_{H^{+}}$ $\displaystyle=$
$\displaystyle-\frac{gm_{Q}}{\sqrt{2}M_{W}}\overline{Q}(\cot\beta
Z^{L}_{QQ}P_{L}+\tan\beta Z^{R}_{QQ}P_{R})QH^{+}+H.c.$ (69)

$\displaystyle\begin{array}[]{c|cccc}&V_{{T}B}^{L}&\quad
V_{{T}B}^{R}&V_{BY}^{L}&V_{BY}^{R}\\\ \hline\cr(TB)&\quad
s_{L}^{u}s_{L}^{d}e^{-i(\phi_{u}-\phi_{d})}+c_{L}^{u}c_{L}^{d}&\quad
c_{R}^{u}c_{R}^{d}&-&-\\\ (BY)&-&\quad-&c_{L}&c_{R}\\\
(XTB)&s_{L}^{u}s_{L}^{d}+\sqrt{2}c_{L}^{u}c_{L}^{d}&\quad\sqrt{2}c_{R}^{u}c_{R}^{d}&-&-\\\
(TBY)&s_{L}^{u}s_{L}^{d}+\sqrt{2}c_{L}^{u}c_{L}^{d}&\quad\sqrt{2}c_{R}^{u}c_{R}^{d}&\sqrt{2}c_{L}^{d}&\sqrt{2}c_{R}^{d}\end{array}$
(75)

Table XII: Heavy-heavy couplings to the $W$ boson.

$\displaystyle\begin{array}[]{c|ccccc}&X_{BB}^{L}&X_{BB}^{R}\\\
\hline\cr(B)&(s_{L})^{2}&&&&0\\\ (TB)&1&&&&(c_{R}^{d})^{2}\\\
(BY)&s_{L}^{2}-c_{L}^{2}&&&&-c_{R}^{2}\\\
(XTB)&1+(c_{L}^{d})^{2}&&&&2(c_{R}^{d})^{2}\\\
(TBY)&(s_{L}^{d})^{2}&&&&0\end{array}$ (82)

Table XIII: Heavy-heavy couplings to the $Z$ boson.

$\displaystyle\begin{array}[]{c|cccc
}&Y_{BB}^{h}&Y_{BB}^{H}&Y_{BB}^{A}&Y_{YY}^{\\{h,H,A\\}}\\\
\hline\cr(B)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)s_{L}^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)s_{L}^{2}&\tan\beta
s_{L}^{2}&-\\\
(TB)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)(s_{R}^{d})^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)(s_{R}^{d})^{2}&\tan\beta(s_{R}^{d})^{2}&-\\\
(BY)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)s_{R}^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)s_{R}^{2}&\tan\beta
s_{R}^{2}&0\\\
(XTB)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)(s_{L}^{d})^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)(s_{L}^{d})^{2}&\tan\beta(s_{L}^{d})^{2}&-\\\
(TBY)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)(s_{L}^{d})^{2}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)(s_{L}^{d})^{2}&\tan\beta(s_{L}^{d})^{2}&0\end{array}$
(89)

Table XIV: Heavy-heavy Bottom VLQ couplings to the triplets Higgs {$h,H,A$}.

$\displaystyle\begin{array}[]{c|ccccc}&Z^{L}_{TB}&&Z^{R}_{TB}&Z^{L}_{BY}&Z^{R}_{BY}\\\
\hline\cr(TB)&s_{L}^{d}s_{L}^{u}e^{i(\phi_{d}-\phi_{u})}+\frac{c_{L}^{d}}{c_{L}^{u}}(s_{R}^{u}{}^{2}-s_{L}^{u}{}^{2})&&\frac{m_{B}}{m_{T}}\left[s_{L}^{u}s_{L}^{d}e^{i(\phi_{d}-\phi_{u})}+\frac{c_{L}^{u}}{c_{L}^{d}}(s_{R}^{d}{}^{2}-s_{L}^{d}{}^{2})\right]&-&-\\\
(XTB)&-&&-&-&-\\\ (TBY)&-&&-&s_{L}e^{-i\phi}&0\end{array}$ (94)

Table XV: Heavy-heavy couplings to the charged Higgs.

###  4.4 Light-heavy interactions

$\displaystyle\mathcal{L}_{W}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\overline{Q}\gamma^{\mu}(V^{L}_{Qq}P_{L}+V^{R}_{Qq}P_{R})qW^{+}_{\mu}$
$\displaystyle-\frac{g}{\sqrt{2}}\overline{q}\gamma^{\mu}(V^{L}_{qQ}P_{L}+V^{R}_{qQ}P_{R})QW^{+}_{\mu}+H.c.$
$\displaystyle\mathcal{L}_{Z}$ $\displaystyle=$
$\displaystyle-\frac{g}{2c_{W}}\overline{q}\gamma^{\mu}(\pm X^{L}_{qQ}P_{L}\pm
X^{R}_{qQ}P_{R})QZ_{\mu}+H.c$ $\displaystyle\mathcal{L}_{h^{0}}$
$\displaystyle=$
$\displaystyle-\frac{gm_{T}}{2M_{W}}\overline{t}(Y^{L}_{htT}P_{L}+Y^{R}_{htT}P_{R})Th^{0}$
$\displaystyle-\frac{gm_{B}}{2M_{W}}\overline{b}(Y^{L}_{hbB}P_{L}+Y^{R}_{hbB}P_{R})Bh^{0}+H.c.$
$\displaystyle\mathcal{L}_{H^{0}}$ $\displaystyle=$
$\displaystyle-\frac{gm_{T}}{2M_{W}}\overline{t}(Y^{L}_{HtT}P_{L}+Y^{R}_{HtT}P_{R})TH^{0}$
$\displaystyle-\frac{gm_{B}}{2M_{W}}\overline{b}(Y^{L}_{HbB}P_{L}+Y^{R}_{HbB}P_{R})Bh^{0}+H.c.$
$\displaystyle\mathcal{L}_{A}$ $\displaystyle=$ $\displaystyle
i\frac{gm_{T}}{2M_{W}}\overline{t}(Y^{L}_{AtT}P_{L}-Y^{R}_{AtT}P_{R})TA$
$\displaystyle-i\frac{gm_{B}}{2M_{W}}\overline{b}(Y^{L}_{AbB}P_{L}-Y^{R}_{AbB}P_{R})BA+H.c.$
$\displaystyle\mathcal{L}_{H^{+}}$ $\displaystyle=$
$\displaystyle-\frac{gm_{T}}{\sqrt{2}M_{W}}\overline{T}(\cot\beta
Z^{L}_{Tb}P_{L}+\tan\beta Z^{R}_{Tb}P_{R})bH^{+}$ (95)
$\displaystyle-\frac{gm_{B}}{\sqrt{2}M_{W}}\overline{t}(\cot\beta
Z^{L}_{tB}P_{L}+\tan\beta Z^{R}_{tB}P_{R})BH^{+}+H.c,$

$\displaystyle\begin{array}[]{c|cccc}&V_{tB}^{L}&V_{tB}^{R}&V_{bY}^{L}&V_{bY}^{R}\\\
\hline\cr(B)&s_{L}e^{i\phi}&0&-&-\\\
(TB)&c_{L}^{u}s_{L}^{d}e^{i\phi_{d}}-s_{L}^{u}c_{L}^{d}e^{i\phi_{u}}&-s_{R}^{u}c_{R}^{d}e^{i\phi_{u}}&-&-\\\
(BY)&s_{L}e^{i\phi}&0&-s_{L}e^{i\phi}&-s_{R}e^{i\phi}\\\
(XTB)&(c_{L}^{u}s_{L}^{d}-\sqrt{2}s_{L}^{u}c_{L}^{d})e^{i\phi}&-\sqrt{2}s_{R}^{u}c_{R}^{d}e^{i\phi}&-&-\\\
(TBY)&(c_{L}^{u}s_{L}^{d}-\sqrt{2}s_{L}^{u}c_{L}^{d})e^{i\phi}&-\sqrt{2}s_{R}^{u}c_{R}^{d}e^{i\phi}&-\sqrt{2}s_{L}^{d}e^{i\phi}&-\sqrt{2}s_{R}^{d}e^{i\phi}\end{array}$
(102)

Table XVI: Light-heavy couplings to the $W$ boson.

$\displaystyle\begin{array}[]{c|cc}&X_{bB}^{L}&X_{bB}^{R}\\\
\hline\cr(B)&c_{L}s_{L}e^{i\phi}&0\\\
(TB)&0&-s_{R}^{d}c_{R}^{d}e^{i\phi_{d}}\\\
(BY)&2c_{L}s_{L}e^{i\phi}&c_{R}s_{R}e^{i\phi}\\\
(XTB)&-s_{L}^{d}c_{L}^{d}e^{i\phi}&-2s_{R}^{d}c_{R}^{d}e^{i\phi}\\\
(TBY)&s_{L}^{d}c_{L}^{d}e^{i\phi}&0\end{array}$ (109)

Table XVII: Light-heavy couplings to the $Z$ boson.

$\displaystyle\begin{array}[]{c|ccc }&Y_{hbB}^{L}&Y_{HbB}^{L}&Y_{AbB}^{L}\\\
\hline\cr(B)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}c_{L}s_{L}e^{i\phi}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}c_{L}s_{L}e^{i\phi}&\tan\beta\frac{m_{b}}{m_{B}}c_{L}s_{L}e^{i\phi}\\\
(TB)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)s_{R}^{d}c_{R}^{d}e^{i\phi_{d}}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)s_{R}^{d}c_{R}^{d}e^{i\phi_{d}}&\tan\beta
s_{R}^{d}c_{R}^{d}e^{i\phi_{d}}\\\
(BY)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)c_{R}s_{R}e^{i\phi}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)c_{R}s_{R}e^{i\phi}&\tan\beta
c_{R}s_{R}e^{i\phi}\\\
(XTB)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}s_{L}^{d}c_{L}^{d}e^{i\phi}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}s_{L}^{d}c_{L}^{d}e^{i\phi}&\tan\beta\frac{m_{b}}{m_{B}}s_{L}^{d}c_{L}^{d}e^{i\phi}\\\
(TBY)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}s_{L}^{d}c_{L}^{d}e^{i\phi}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}s_{L}^{d}c_{L}^{d}e^{i\phi}&\tan\beta\frac{m_{b}}{m_{B}}s_{L}^{d}c_{L}^{d}e^{i\phi}\end{array}$
(116)

Table XVIII: Light-heavy left couplings of Bottom quarks to the triplets Higgs
{$h,H,A$}.

$\displaystyle\begin{array}[]{c|ccc }&Y_{hbB}^{R}&Y_{HbB}^{R}&Y_{AbB}^{R}\\\
\hline\cr(B)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)c_{L}s_{L}e^{i\phi}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)c_{L}s_{L}e^{i\phi}&\tan\beta
c_{L}s_{L}e^{i\phi}\\\
(TB)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}s_{R}^{d}c_{R}^{d}e^{i\phi_{d}}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}s_{R}^{d}c_{R}^{d}e^{i\phi_{d}}&\tan\beta\frac{m_{b}}{m_{B}}s_{R}^{d}c_{R}^{d}e^{i\phi_{d}}\\\
(BY)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}c_{R}s_{R}e^{i\phi}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)\frac{m_{b}}{m_{B}}c_{R}s_{R}e^{i\phi}&\tan\beta\frac{m_{b}}{m_{B}}c_{R}s_{R}e^{i\phi}\\\
(XTB)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)s_{L}^{d}c_{L}^{d}e^{i\phi}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)s_{L}^{d}c_{L}^{d}e^{i\phi}&\tan\beta
s_{L}^{d}c_{L}^{d}e^{i\phi}\\\
(TBY)&(s_{\beta\alpha}-c_{\beta\alpha}\tan\beta)s_{L}^{d}c_{L}^{d}e^{i\phi}&(c_{\beta\alpha}+s_{\beta\alpha}\tan\beta)s_{L}^{d}c_{L}^{d}e^{i\phi}&\tan\beta
s_{L}^{d}c_{L}^{d}e^{i\phi}\end{array}$ (123)

Table XIX: Light-heavy right couplings of Bottom quarks to the triplets Higgs
{$h,H,A$}.

$\displaystyle\begin{array}[]{c|ccccc}&Z^{L}_{bY}&Z^{R}_{bY}&Z^{L}_{tB}&&Z^{R}_{tB}\\\
\hline\cr(B)&-&-&s_{L}&&0\\\ (BY)&s_{R}e^{-i\phi}&0&0&&0\\\
(TB)&-&-&\frac{m_{t}}{m_{B}}\left[c_{L}^{u}s_{L}^{d}e^{i\phi_{d}}+(s_{R}^{u}{}^{2}-s_{L}^{u}{}^{2})\frac{c_{L}^{d}}{s_{L}^{u}}e^{i\phi_{u}}\right]&&c_{L}^{u}s_{L}^{d}e^{i\phi_{d}}+(s_{L}^{d}{}^{2}-s_{R}^{d}{}^{2})\frac{s_{L}^{u}}{c_{L}^{d}}e^{i\phi_{u}}\\\
(TBY)&c_{L}^{u}&0&s_{L}^{d}&&0\end{array}$ (129)

Table XX: Light-heavy couplings to the charged Higgs.

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where $\mathcal{C}^{i}(\cdot)$ is consumer $i$’s complexity cost function.
Here a consumption vector can be view as an act. Preference relations are
defined on consumption vectors as what we discussed in section 3. With
complexity cost function, a complexity averse consumer $i$ is unable to choose
every consumption vector that a standard consumer can choose. Note that this
embeds the standard model as a special case, where $\mathcal{C}^{i}(c^{i})=0$
for all $c^{i}\in\mathbb{R}^{LS}_{+}=C$ and $i\in\\{1,\cdots,I\\}$. This also
embeds CCE (Gul et al., (2017)) as a special case, where
$\mathcal{C}^{i}(c^{i})=0$ for all crude comsumption vectors and
$\mathcal{C}^{i}(c^{i})=\infty$ for all others.

######  Definition 9.

Given an economy $\mathcal{E}$ specified by
$\\{u^{i},e^{i},\mathcal{C}^{i},\mu^{i}\\}^{I}_{i=1}$, an allocation
$(\hat{c}^{1},\cdots,\hat{c}^{I})$, and a price vector
$p=(p_{1\omega_{1}},\cdots,p_{L\omega_{n}})$ constitute a complexity averse
competitive equilibrium(CACE) if

  1. (1)

For every $i$, $\hat{c}^{i}\in C^{i}$ is maximal for $\succsim^{i}$ in the
budget set

$\\{c^{i}\in C^{i}:p\cdot c^{i}\leq p\cdot e^{i}\\};$

  2. (2)

$\sum_{i}\hat{c}^{i}=\sum_{i}e^{i}$.

To illustrate the idea, consider the following example of consumption under
uncertainty. Consider an economy $\mathcal{E}$ with $I=3,\ L=1$, and
$\Omega=\\{\omega_{1},\omega_{2},\omega_{3}\\}$. Suppose we have
$e^{1}=(1,0,0)$, $e^{2}=(0,2,0)$, $e^{3}=(0,0,3)$, and utility index of the
form $u^{i}=\ln c^{i}_{\omega}$. Thus, an agent’s utility of her consumption
vector $c^{i}$ is $U(c^{i})=\mu^{i}_{\omega_{1}}\ln
c^{i}_{\omega_{1}}+\mu^{i}_{\omega_{2}}\ln
c^{i}_{\omega_{2}}+\mu^{i}_{\omega_{3}}\ln c^{i}_{\omega_{3}}$, where
$(\mu^{i}_{\omega_{1}},\mu^{i}_{\omega_{2}},\mu^{i}_{\omega{3}})=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$
are the agent $i$’s subjective probabilities of the three states.

Since there are three states, consumers could have five different information
structures

$(\\{1,2,3\\}),(\\{1\\},\\{2\\},\\{3\\}),(\\{1,2\\},\\{3\\}),(\\{1\\},\\{2,3\\}),(\\{1,3\\},\\{2\\}).$

Thus, every agent has three types of consumption vectors
$C\in\\{C_{1},C_{2},C_{3}\\}$. $C_{1}$ means that the agent chooses to make a
most complex consumption vector where $c^{i}_{\omega}\neq
c^{i}_{\omega^{\prime}}$ for any $\omega,\omega^{\prime}\in\Omega$
corresponding to information structure
$(\\{\omega_{1}\\},\\{\omega_{2}\\},\\{\omega_{3}\\})$; $C_{2}$ is the
coarsest consumption vector where $c^{i}_{\omega}=c^{i}_{\omega^{\prime}}$ for
any $\omega,\omega^{\prime}\in\Omega$ corresponding to information structure
$(\\{\omega_{1},\omega_{2},\omega_{3}\\})$; $C_{3}$ indicates that the agent
is able to make some complex consumption vectors where
$c^{i}_{\omega}=c^{i}_{\omega^{\prime}}\neq c^{i}_{\omega^{\prime\prime}}$ for
any $\omega,\omega^{\prime},\omega^{\prime\prime}\in\Omega$. Preferences are
well defined, so the agent is capable of identifying the cost for all
consumption vectors. By assumption, we have
$\mathcal{C}(\hat{c}^{ij})=\mathcal{C}(\hat{c}^{ik})\
\forall\hat{c}^{ij},\hat{c}^{ik}\in C_{1}$,
$\mathcal{C}(\bar{c}^{ij})=\mathcal{C}(\bar{c}^{ik})\
\forall\bar{c}^{ij},\bar{c}^{ik}\in C_{3}$, and
$\mathcal{C}(\tilde{c}^{ij})=\mathcal{C}(\tilde{c}^{ik})=0\
\forall\hat{c}^{ij},\tilde{c}^{ik}\in C_{2}$. Moreover,
$\mathcal{C}(\hat{c}^{i})>\mathcal{C}(\bar{c}^{i})>\mathcal{C}(\tilde{c}^{i})$
for any $\hat{c}^{i}\in C_{1},\bar{c}^{i}\in C_{3},\tilde{c}^{i}\in C_{2}$.

If all consumers choose consumption vectors in $C_{1}$, this is the standard
competitive equilibrium (SCE), we have
$c^{*}=(c^{i}_{1},c^{i}_{2},c^{i}_{3})=(\frac{1}{3},\frac{2}{3},1)$ for all
$i\in I$, $\delta=(1,0,0)$, and the prices are given by
$p^{*}=(p_{1},p_{2},p_{3})=(1,\frac{1}{2},\frac{1}{3})$. Note that when
consumer 1 chooses a plan in $C_{1}$, her utility is
$\frac{1}{3}\ln\big{(}\frac{(p_{1})^{2}}{27p_{2}p_{3}}\big{)}-\mathcal{C}(C_{1})$.
If consumer 1 chooses a consumption vector where $c^{1}_{1}=c^{1}_{2}\neq
c^{1}_{3}$, it is easy to verify that her utility is
$\frac{1}{3}\ln\big{(}\frac{4(p_{1})^{3}}{27(p_{1}+p_{2})^{2}p_{3}}\big{)}-\mathcal{C}(C_{3})$.
If we let
$\mathcal{C}(C_{1})-\mathcal{C}(C_{3})>\frac{1}{3}\ln(\frac{(p_{1}+p_{2})^{2}}{4p_{1}p_{2}})$,
then $(c^{*},p^{*})$ is not an equilibrium any more. Suppose consumer 1 is
complexity averse and her best choice is to pick consumption vectors in
$C_{3}$ with specific cost function $\mathcal{C}^{1}$, and consumer 2 and 3
are standard agents. Then, we can find a CACE
$(\hat{c},\hat{p})=\Big{(}\big{(}(\frac{1}{2},\frac{1}{2},1.2),(\frac{2}{9},\frac{2}{3},\frac{4}{5}),(\frac{5}{18},\frac{5}{6},1)\big{)};\big{(}1,\frac{1}{3},\frac{5}{18}\big{)}\Big{)}$.
There is a significant difference between SCE price $p^{*}$ and CACE price
$\hat{p}$. We can see that
$\frac{p^{*}_{1}}{p^{*}_{3}}<\frac{\hat{p}_{1}}{\hat{p}_{3}}$, which indicates
that CACE prices might be more extreme than SCE prices.