The oblique parameters in the 2HDM with Vector-Like Quarks: Confronting
$M_{W}$ CDF-II Anomaly

H<EMAIL_ADDRESS>A<EMAIL_ADDRESS>R.
<EMAIL_ADDRESS>M<EMAIL_ADDRESS>J.
El<EMAIL_ADDRESS>
1 Université Abdelmalek Essaadi, FSTT, B. 416, Tangier, Morocco.
2Polydisciplinary Faculty, Laboratory of Fundamental and Applied Physics, Cadi
Ayyad University, Sidi Bouzid, B.P. 4162, Safi, Morocco.
3 LPTHE, Physics Department, Faculty of Sciences, Ibn Zohr University, P.O.B.
8106 Agadir, Morocco.

###### Abstract

The CDF collaboration has released a new measurement of the $W$ boson mass
using their complete data set with 8.8 fb-1 in $p\bar{p}$ collisions. This
result deviates from the Standard Model prediction by around 7$\sigma$. We
explain how the two Higgs doublet model (2HDM) with vector-like quarks is
affected by the recently discovered W boson mass. In our study, we include
both theoretical constraints such as perturbative unitarity and vacuum
stability as well as a number of experimental constraints. We also look into
how the effective mixing angle, measured by the SLD collaboration in addition
to the CDF W-boson mass, is used to determine the $S$ and $T$ parameters. In
the alignment limit, we investigate the case where the lighter CP-even neutral
Higgs boson of the 2HDM is the one found at the LHC and demonstrate how the
parameter space of the 2HDM type II in the presence of vector-like quarks is
constrained. It is found that in most cases, there is a cancellation between
the 2HDM and vector-like quarks contributions, which enlarges the parameter
space of both models.

##  1 Introduction

Since the advent of the precise measurement programs at SLD, LEP, and
Tevatron, ElectroWeak Precision Observables (EWPOs), such as the W boson mass,
the effective mixing angle $\sin^{2}\theta_{eff}$, and the Z boson width,
among others, have been used to precisely assess the viability of the Standard
Model (SM) of particle physics. It turns out that some Beyond SM (BSM)
extensions can be severely constrained by these EWPOs. Using their whole data
set of 8.8 fb-1 in $p\bar{p}$ collisions at CDF detector, the CDF
collaboration recently released a new measurement of the $W$ boson mass
(Fermilab Tevatron) [1]:

$M^{\mathrm{CDF}}_{W}=80.4335\pm 0.0094~{}\mathrm{GeV}$ (1)

This measurement is more precise than all prior measures put together, and it
is highly accurate. In addition, the new CDF value agrees with some previous W
mass measurements, but there are also some discrepancies [2]. Based on complex
SM radiative correction calculations that relate the W mass to other SM
particles such as the Higgs boson and the top masses, the SM prediction for
the W mass at the per mille level is given by [2]:

$M_{W}^{\mathrm{SM}}=80.357\pm 0.006\mathrm{GeV}$ (2)

It is clear that the above CDF measurement depicts a remarkable 7$\sigma$
disagreement with the SM prediction. Several analyses have been presented in
this direction, including those involving the 2HDM [3], the triplet Higgs
model [4] and other SM extensions [5].

In the past, during the LEP era, it was well known that the global fit of SM
predictions to LEP data was used to predict the existence of a heavy top quark
and a relatively light Higgs boson well before their discovery at the Tevatron
and LHC, respectively. Although the CDF measurement needs to be confirmed, it
is quite likely that the difference between the experimental value and the
expected SM value is due to a non-decoupled new interactions or new particles.
If this is the case and if the scale of the new interaction is not too high,
there is a chance that these new particles will show up in future experiments.

The SM of particle physics has been highly successful in explaining all
observed phenomena at LEP and the LHC. However, there are several indications
that the SM is only an effective theory of a more fundamental and complete
theory that addresses its weaknesses, such as the dark matter puzzle, the need
for additional sources of CP violation to explain the baryon asymmetry in the
universe, and the issues with neutrino mass and mixing. Most BSM models
predict additional vector-like fermions (quarks and leptons) and/or additional
Higgs bosons in their spectrum [6, 7]. It is now well established that SM-like
chiral fermions111SM-like chiral fermions would have the same properties and
interactions as the fermions in the SM, but would be different in terms of
their masses and other properties. In the SM, the left-handed and right-handed
components of the fermions are treated differently, with only the left-handed
components interacting via the weak force. (a fourth generation of quarks)
have been excluded by high precision measurements of Higgs production rates at
the LHC [10, 9]. In the present study, we focus on new states of matter that
are spin 1/2 particles and transform as triplets under color. However, unlike
SM quarks, their left- and right-handed couplings have the same Electro-Weak
(EW) quantum numbers, called Vector-Like Quarks (VLQs) [11, 7, 6, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. This class of models is well
motivated since they show up in many composite Higgs models [27, 28, 30, 29,
31, 32] as well as in various little Higgs models [33, 34, 35]. We will also
consider an extension of the SM that includes both VLQs and an additional
Higgs doublet (as stated in reference [7]).
Several searches for VLQs have been carried out by the ATLAS and CMS
collaborations, both from single production via an electroweak interaction and
from pair production through the strong interaction. No significant deviation
from the expected background of the SM has been observed, and exclusion limits
have been set on the masses of the VLQs. Both the ATLAS and CMS experiments
have searched $T$ and $B$ production at $\sqrt{s}=8$ TeV and $\sqrt{13}$ TeV,
spanning practically all production and decay channels. With an integrated
luminosity of 138 $fb^{-1}$, CMS presented a search [36] for $T\bar{T}$ and
$B\bar{B}$ pair production of VLQs, excluding masses up to 1.54 TeV (1.56 TeV)
for vector-like T and B quarks, respectively, at a 95% confidence level,
depending on their branching ratio scenarios. For vector-like T quarks, masses
below 1.48 TeV are excluded in any branching fraction scenario, i.e. decay to
the final states tZ, bW, and th, where h is the SM Higgs boson. Analogous
exclusion limits by ATLAS [37] at $95\%$ confidence level have been set on the
VLQs masses up to 1030 (1210) GeV for singlet (doublet) vector-like T quark
and 1010 (1140) GeV for singlet (doublet) vector-like B quark. It is important
to note that all these search limits are given with the assumption that VLQs
decay only to third generation SM particles, and they can be lowered if the
VLQs have a non-negligible coupling with the first and second generation SM
quarks. Besides, if we consider 2HDM+VLQs, in this case, we will have the
possibility for other new decay modes of VLQs to BSM Higgs bosons, such as the
decay of vector-like T quark to $tH,tA$, and $bH^{\pm}$, and hence the
previous exclusion limits can be further weakened down [7] where $A$, $H$ and
$H^{\pm}$ are extra Higgs bosons predicted by the 2HDM.

The oblique parameters $S$, $T$ and $U$ represent important tools at the TeV
scale that indicate the level of violation of the custodial isospin symmetry.
They contribute significantly to electroweak radiative corrections. It is very
well known that in the SM, $S=T=U=0$ at a reference point where the top quark
and Higgs boson masses are fixed. Beyond the SM, these parameters are
sensitive to new physics and hence measure their effects. The exact
calculations of S, T and U oblique parameters are tedious and prone to error,
with the help of public codes such as FeynArts, FormCalc, and LoopTools, an
automated code has been introduced to improve the reliability of the
calculations. In the current work, we present to the best of our knowledge,
the first exact calculations of electroweak oblique parameters $S$ and $T$ in
the 2HDM type II in the presence of singlets, doublets, and triplets vector-
like quarks scenarios in the context of CP conserving case. Our results are
given using the well-known scalar Passarino-Veltman functions [8]. We examine
in detail the conditions that may affect the oblique parameters in our model,
then study the impact of the new CDF result on the extra new quarks. The
2HDM+VLQs is a good candidate for analyzing the oblique parameters formalism
because the model has new quarks and their couplings to gauge bosons are
model-dependent.
This paper is organized as follows: In Section 2, we give a brief review of
VLQs and 2HDM in terms of particle content and their mass. We also present how
the W boson mass can be parameterized in terms of the oblique parameters $S$,
$T$ and $U$. In Section 3, we show our numerical analysis within the 2HDM type
II augmented by VLQs. We conclude in Section 4. Finally, in the appendix we
present the contributions of the vector-like quarks to $S$, $T$ and $U$
parameters and we give the relevant couplings to our study and explain the
CDF-II W-boson mass anomaly within the SM extended with VLQs.

##  2 Models description

###  2.1 VLQs

In this section, we introduce our parametrization and notation for VLQs 222In
this work we review both the SM and the 2HDM with VLQs. More detailed
description can be found in Ref. [6].. As low energy physics measurements
severely limit the mixing of VLQs with the first and second generations, we
only consider the scenario where the VLQs couple to the third-generation SM
quarks. There are seven distinct possibilities of Vector-like $SU(2)_{L}$
multiplets with renormalizable couplings that are allowed to mix through
Yukawa couplings with the SM quarks, i.e we have two electroweak singlets,
three electroweak doublets and two electroweak triplets as indicated below:

$\displaystyle T_{L,R}^{0}\,,\quad B_{L,R}^{0}$
$\displaystyle\text{(singlets)}\,,$
$\displaystyle(T^{0}\,B^{0})_{L,R}\,,\quad(X^{0}\,T^{0})_{L,R}\,,\quad(B^{0}\,Y^{0})_{L,R}$
$\displaystyle\text{(doublets)}\,,$
$\displaystyle(X^{0}\,T^{0}\,B^{0})_{L,R}\,,\quad(T^{0}\,B^{0}\,Y^{0})_{L,R}$
$\displaystyle\text{(triplets)}\,.$ (3)

Where the heavy quarks have the following electric charges:
$Q_{T}=\frac{2}{3},Q_{B}=-\frac{1}{3},Q_{X}=\frac{5}{3}$, and
$Q_{Y}=-\frac{4}{3}$. In the Higgs basis, the Yukawa Lagrangian may be
expressed as:

$-\mathcal{L}\,\,\supset\,\,y^{u}\bar{Q}^{0}_{L}\tilde{H}_{2}U^{0}_{R}+y^{d}\bar{Q}^{0}_{L}H_{1}D^{0}_{R}+M^{0}_{u}\bar{U}^{0}_{L}U^{0}_{R}+M^{0}_{d}\bar{D}^{0}_{L}D^{0}_{R}+\rm{h.c}.$
(4)

where, $U_{R}\equiv(u_{R},c_{R},t_{R},T_{R})$ and
$D_{R}\equiv(d_{R},s_{R},b_{R},B_{R})$ while $y^{u,d}$ are 3$\times$4 Yukawa
matrices. In the scenario where the new heavy quarks couple to third
generation, the physical mass eigenstates fields $(t,T)$ and $(b,B)$ can be
obtained by rotating the gauge eigenstates through four unitary mixing
matrices $U_{L,R}^{u}$ and $U_{L,R}^{d}$, which are parametrized as follows:

$\left(\\!\begin{array}[]{c}t_{L,R}\\\
T_{L,R}\end{array}\\!\right)=U_{L,R}^{u}\left(\\!\begin{array}[]{c}t^{0}_{L,R}\\\
T^{0}_{L,R}\end{array}\\!\right)=\left(\\!\begin{array}[]{cc}\cos\theta_{L,R}^{u}&-\sin\theta_{L,R}^{u}\\\
\sin\theta_{L,R}^{u}e^{-i\phi_{u}}&\cos\theta_{L,R}^{u}\end{array}\\!\right)\left(\\!\begin{array}[]{c}t^{0}_{L,R}\\\
T^{0}_{L,R}\end{array}\\!\right)\,.$ (5)
$\left(\\!\begin{array}[]{c}b_{L,R}\\\
B_{L,R}\end{array}\\!\right)=U_{L,R}^{d}\left(\\!\begin{array}[]{c}b^{0}_{L,R}\\\
B^{0}_{L,R}\end{array}\\!\right)=\left(\\!\begin{array}[]{cc}\cos\theta_{L,R}^{d}&-\sin\theta_{L,R}^{d}\\\
\sin\theta_{L,R}^{d}e^{-i\phi_{d}}&\cos\theta_{L,R}^{d}\end{array}\\!\right)\left(\\!\begin{array}[]{c}b^{0}_{L,R}\\\
B^{0}_{L,R}\end{array}\\!\right)\,.$ (6)

In the weak eigenstate basis, the mass terms for the top and bottom quarks and
the VLQs can be extracted from

$\displaystyle\mathcal{L}_{\text{mass}}$ $\displaystyle=$
$\displaystyle-\left(\\!\begin{array}[]{cc}\bar{t}_{L}^{0}&\bar{T}_{L}^{0}\end{array}\\!\right)\left(\\!\begin{array}[]{cc}y_{33}^{u}\frac{v}{\sqrt{2}}&y_{34}^{u}\frac{v}{\sqrt{2}}\\\
y_{43}^{u}\frac{v}{\sqrt{2}}&M^{0}\end{array}\\!\right)\left(\\!\begin{array}[]{c}t^{0}_{R}\\\
T^{0}_{R}\end{array}\\!\right)$ (18)
$\displaystyle-\left(\\!\begin{array}[]{cc}\bar{b}_{L}^{0}&\bar{B}_{L}^{0}\end{array}\\!\right)\left(\\!\begin{array}[]{cc}y_{33}^{d}\frac{v}{\sqrt{2}}&y_{34}^{d}\frac{v}{\sqrt{2}}\\\
y_{43}^{d}\frac{v}{\sqrt{2}}&M^{0}\end{array}\\!\right)\left(\\!\begin{array}[]{c}b^{0}_{R}\\\
B^{0}_{R}\end{array}\\!\right)+\text{h.c.},$

Where $y_{ij}$ are the Yukawa couplings and $M^{0}$ is a bare mass term. For
singlets and triplets, $y_{43}=0$, while for doublets, $y_{34}=0$. The unitary
matrices will be specified in the case where the mass matrices are diagonal in
the mass eigenstate basis.

$U_{L}\,\mathcal{M}\,(U_{R})^{\dagger}=\mathcal{M}^{q}_{\text{diag}}\,,$ (19)

with $\mathcal{M}$ are the mass matrices in Eq. (18) and
$\mathcal{M}^{q}_{\text{diag}}$ are the diagonalized matrices given by.

$\mathcal{M}^{t}_{\text{diag}}=\left(\\!\begin{array}[]{cc}m_{t}&0\\\
0&m_{T}\end{array}\\!\right)\quad\text{and}\quad\mathcal{M}^{b}_{\text{diag}}=\left(\\!\begin{array}[]{cc}m_{b}&0\\\
0&m_{B}\end{array}\\!\right)$ (20)

Note that the two mixing angles $\theta_{L}$ and $\theta_{R}$ are not
independent parameters. By using Eq. (19), and depending on the representation
of the VLQs, one can find that:

$\displaystyle\tan\theta_{R}^{q}$ $\displaystyle=$
$\displaystyle\frac{m_{q}}{m_{Q}}\tan\theta_{L}^{q}\quad\text{(singlets,
triplets)}\,,$ $\displaystyle\tan\theta_{L}^{q}$ $\displaystyle=$
$\displaystyle\frac{m_{q}}{m_{Q}}\tan\theta_{R}^{q}\quad\text{(doublets)}\,,$
(21)

with $(q,m_{q},m_{Q})=(u,m_{t},m_{T})$ and $(d,m_{b},m_{B})$.
Additionally, there are relationships between the mass eigenstates and angles
that depend on the representation of the VLQs [6]

$\displaystyle{Doublets:}$ $\displaystyle(T\,B):$ $\displaystyle
m_{T}^{2}(c_{R}^{u})^{2}+m_{t}^{2}(s_{R}^{u})^{2}=m_{B}^{2}(c_{R}^{d})^{2}+m_{b}^{2}(s_{R}^{d})^{2}$
(22) $\displaystyle(X\,T):$ $\displaystyle
m_{X}^{2}=m_{T}^{2}(c_{R}^{u})^{2}+m_{t}^{2}(s_{R}^{u})^{2}$
$\displaystyle(B\,Y):$ $\displaystyle
m_{Y}^{2}=m_{B}^{2}(c_{R}^{d})^{2}+m_{b}^{2}(s_{R}^{d})^{2}$
$\displaystyle{Triplets:}$ $\displaystyle(X\,T\,B):$ $\displaystyle
m_{X}^{2}=m_{T}^{2}(c_{L}^{u})^{2}+m_{t}^{2}(s_{L}^{u})^{2}$ (23)
$\displaystyle\phantom{m_{X}^{2}}=m_{B}^{2}(c_{L}^{d})^{2}+m_{b}^{2}(s_{L}^{d})^{2}$

From the above equations, it is easy to derive the following relationships:

$\displaystyle{m_{B}^{2}}=\frac{1}{2}\sin^{2}\left(2\theta^{u}_{L}\right)\frac{m_{T}^{2}-m_{t}^{2}}{m_{X}^{2}-m_{b}^{2}}+m_{X}^{2}$
$\displaystyle\sin(2\theta_{L}^{d})=\sqrt{2}\frac{m_{T}^{2}-m_{t}^{2}}{(m_{B}^{2}-m_{b}^{2})}\sin(2\theta_{L}^{u})$
(24)

Similarly for the $(T\,B\,Y)$ triplet, we have:

$\displaystyle m_{Y}^{2}=m_{B}^{2}(c_{L}^{d})^{2}+m_{b}^{2}(s_{L}^{d})^{2}$
$\displaystyle\phantom{m_{Y}^{2}}=m_{T}^{2}(c_{L}^{u})^{2}+m_{t}^{2}(s_{L}^{u})^{2}$
(25)

We may therefore deduce the following relations:

$\displaystyle{m_{B}^{2}}=\frac{1}{8}\sin^{2}\left(2\theta^{u}_{L}\right)\frac{m_{T}^{2}-m_{t}^{2}}{m_{Y}^{2}-m_{b}^{2}}+m_{Y}^{2}$
$\displaystyle\sin(2\theta_{L}^{d})=\frac{m_{T}^{2}-m_{t}^{2}}{\sqrt{2}(m_{B}^{2}-m_{b}^{2})}\sin(2\theta_{L}^{u})$
(26)

where the mixing between the top-quark and down-quark vector-like fermions is
described by $c_{L,R}^{u,d}=\cos\,\theta_{L,R}^{u,d}$ and
$s_{L,R}^{u,d}=\sin\,\theta_{L,R}^{u,d}$.
It can be easily inferred that all representations of the VLQs can be
characterized by only two parameters, except for the $(T,B)$ top partner
doublet model which requires three parameters. In our numerical analysis, the
following input parameters will be used:

Representations Singlets Doublets Triplets $T$ $B$ $T\,B$ $X\,T$ $B\,Y$
$X\,T\,B$ $T\,B\,Y$ Parameters $m_{T},s_{L}^{u}$ $m_{B},s_{L}^{d}$
$m_{T},s_{R}^{u},s_{R}^{d}$ $m_{T},s_{R}^{u}$ $m_{B},s_{R}^{d}$
$m_{T},s_{L}^{u}$ $m_{T},s_{L}^{u}$

###  2.2 2HDM

One of the simplest extensions of the SM’s Higgs sector is the two Higgs
doublet model (2HDM) which consist of adding a new Higgs doublet to the
existing one in the SM. The most general scalar potential which is invariant
under $SU_{L}(2)\times U_{Y}(1)$ and satisfy a discrete $Z_{2}$ symmetry
$\phi_{1}\rightarrow+\phi_{1}$ and $\phi_{2}\rightarrow-\phi_{2}$, can be
written as [38]

$\displaystyle V(\phi_{1},\phi_{2})$ $\displaystyle=$ $\displaystyle
m_{11}^{2}(\phi_{1}^{\dagger}\phi_{1})+m_{22}^{2}(\phi_{2}^{\dagger}\phi_{2})-[m_{12}^{2}(\phi_{1}^{\dagger}\phi_{2})+\text{h.c.}]$
(27) $\displaystyle+$
$\displaystyle\frac{1}{2}\lambda_{1}(\phi_{1}^{\dagger}\phi_{1})^{2}+\frac{1}{2}\lambda_{2}(\phi_{2}^{\dagger}\phi_{2})^{2}+\lambda_{3}(\phi_{1}^{\dagger}\phi_{1})(\phi_{2}^{\dagger}\phi_{2})~{}$
$\displaystyle+$
$\displaystyle\lambda_{4}(\phi_{1}^{\dagger}\phi_{2})(\phi_{2}^{\dagger}\phi_{1})+\frac{1}{2}\left[\lambda_{5}(\phi_{1}^{\dagger}\phi_{2})^{2}+\rm{h.c.}\right],$

where $\lambda_{1,2,3,4,5}$ as well as $m_{11,22,12}^{2}$ are chosen to be
real ensuring that the potential is CP conserving. If both Higgses $\phi_{1}$
and $\phi_{2}$ interact with all SM fermions, like in the SM, we end up with
Flavour Changing Neutral Currents (FCNCs) at the tree-level in the Yukawa
sector. For this purpose, a discrete $Z_{2}$ symmetry has been introduced to
prevent the occurrence of large tree-level FCNCs [39, 38]. Moreover, based on
the $Z_{2}$ symmetry assignment, there are four possible types of Yukawa
sector. In this study, we concentrate on 2HDM Type II, where one doublet
couples to down-type quarks and leptons, while the other couples to up-type
quarks.

After EWSB takes place, we are left with five physical Higgs bosons: two CP-
even $h$ and $H$ with $m_{h}<m_{H}$, one CP-odd $A$, and a pair of charged
Higgs $H^{\pm}$. In our study, we assume that the light CP-even Higgs $h$ will
be identified as the Higgs boson observed at the LHC, and we assume
$m_{h}\approx 125$ GeV. The other parameters of the model are:

  * •

$\tan\beta=v_{2}/v_{1}$ which is the ratio of the vevs.

  * •

$\sin(\beta-\alpha)$ , where $\alpha$ is the mixing angle between the CP-even
components of the Higgs fields

  * •

$m_{12}^{2}$: $Z_{2}$ soft breaking term

  * •

$m_{H,A,H^{\pm}}$: the physical masses of the extra Higgs bosons.

###  2.3 $W$ mass and ($S$, $T$, $U$) parameters

The $W$-boson mass $M_{W}$ is one of the significant EWPOs due to its precise
experimental measurements. It can be calculated as a function of the oblique
parameters $S$, $T$ and $U$ [41, 40]. The oblique parameters in 2HDM differ
from those in the SM because of the additional contributions from the Higgs
boson loops and the modified couplings of the SM-type Higgs bosons.

The W-boson mass can be calculated from the oblique parameters, which can be
expressed as follows [42]:

$\displaystyle\left.M_{W}^{2}=M_{W}^{2}\right\rvert^{SM}{}\left[1+\frac{\alpha(M_{Z})}{\left(c_{W}^{2}-s_{W}^{2}\right)}\left(-\frac{1}{2}S+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right)\right].$
(28)

Where $c_{W}=\cos\theta_{W}$ and $s_{w}=\sin\theta_{W}$, $\theta_{W}$ is
Weimberg angle. When compared to the contributions from $S$ and $T$, the
contribution from the oblique parameter U is expected to be negligible in many
new physics models. Hence, we present the results for $S$ and $T$ assuming
$U=0$.
Additionally, it is possible to assess the impact of changes in the $W$-boson
mass on the effective weak mixing angle $\sin^{2}\theta_{\mathrm{eff}}$, which
is calculated as described in [43]:

$\displaystyle\sin^{2}\theta_{eff}=\left.\sin^{2}\theta_{\mathrm{eff}}\right\rvert^{SM}{}-\alpha\frac{c_{W}^{2}s_{W}^{2}}{c_{W}^{2}-s_{W}^{2}}(T-\frac{s_{W}^{2}}{c_{W}^{2}}S+\frac{1}{2}U)$
(29)

with the SM values $\left.M_{W}\right\rvert^{SM}{}=80.357$ GeV and
$\left.\sin^{2}\theta_{\mathrm{eff}}\right\rvert^{SM}{}=0.231532$.
In the framework of 2HDM with VLQs, the oblique parameters $S$ and $T$ have
two separate contributions: one from the VLQs and the other from the Higgs
bosons in the 2HDM:

$\displaystyle S_{SM+VLQs}=S_{SM}+S_{VLQs},\quad\ T_{SM+VLQs}=S_{SM}+S_{VLQs}$
$\displaystyle S_{2HDM+VLQs}=S_{2HDM}+S_{VLQs},\quad\
T_{2HDM+VLQs}=S_{2HDM}+S_{VLQs}$ (30)

The detailed analytical formulations for $S_{2HDM,VLQs}$ and $T_{2HDM,VLQs}$
in terms of the Passarino-Veltman functions can be found in Appendix
Acknowledgments. With the help of LoopTools [44, 45], the numerical
evaluations of the scalar functions are carried out.

We compared our analytical expression for the contributions of 2HDM to $S$ and
$T$ to those in [15] and [46] and found good agreement. In the case of singlet
and doublet representations, the VLQs contributions can be computed with the
use of Ref. [16]. This has been used for a number of VLQ models, as mentioned
in Ref. [17]. We used dimensional regularization to compute the $S$ and $T$
parameters for each of the different VLQs under examination using FeynArts and
FormCalc packages [47, 48]. We made sure that our calculations for $S$ and $T$
were UV finite and independent of the renormalization scale, both analytically
and numerically. In the case of the $T$ parameter, we compared our findings to
those in Refs.[16, 17] and found good agreement for the singlet and doublet
cases, but our results diverged from those in Ref. [17] for the triplets XTB
and TBY. However, our findings for the $S$ parameter also differed from those
in Ref. [17]. In reality, Ref.[17] merely applied the findings from Ref.[16]
for singlet and doublet situations to the triplet case. The author of Ref.[18]
used the method described in Ref. [16] to derive the correct equation for $S$
and $T$ in the case of an XTB triplet model. We calculated both $S$ and $T$
parameters as functions of the standard Passarino-Veltman functions for the
XTB and TBY triplet models and took into account the gauge bosons external
momentum. This is a major point of contention with Ref.[17], which adopts a
crude approximation that ignores the gauge bosons’ external momentum for the
$S$ parameter. We further cross-checked our results with new computations in
Refs. [19, 18] and found good agreement for the $T$ parameter. However, we
should note that for the $S$ parameter, we used the full analytic calculation
and did not omit any external momentum, as is typically the case in Ref [19].

##  3 Numerical Results

Before presenting our results, we will describe how the theoretical
requirements and experimental measurements were used to constrain the
parameter space of the 2HDM + VLQs models.
From a theoretical perspective, the perturbativity, vacuum stability, and
unitarity constraints are enforced as the following:

  * •

Unitarity: A variety of scattering processes are required to be unitary [49,
50, 51].

  * •

Perturbativity: The quartic couplings of the scalar potential should obey the
following conditions:$|\lambda_{i}|<8\pi$ for each $i=1,..5$ [38].

  * •

Vacuum stability: The scalar potential must be bounded from below and positive
in any direction of the fields $\Phi_{i}$. As a consequence, the parameter
space must satisfy the following conditions [52, 53]:

$\displaystyle\lambda_{1}>0,\quad\lambda_{2}>0,\quad\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}},$
$\displaystyle\lambda_{3}+\lambda_{4}-|\lambda_{5}|>-\sqrt{\lambda_{1}\lambda_{2}}.\hskip
14.22636pt$ (31)

The experimental constraints used in our analysis include:

  * •

SM-like Higgs boson discovery: Compatibility of the SM-like scalar with the
observed Higgs boson. We require that the relevant quantities, calculated with
HiggsSignals-2.6.1 [54, 55, 56, 57, 58, 59, 60, 61], satisfy measurements at
95% C.L.

  * •

BSM Higgs boson exclusions: Exclusion limits at 95% C.L. from direct Higgs
boson searches at LEP, Tevatron and LHC are enforced via HiggsBounds-5.10.1
[54, 55, 56, 57, 58, 59, 60, 61]

We then performed a systematic scan over the following physical parameters
using the public code 2HDMC-1.8.0333We have modified the code as to implement
the analytical expressions of $S_{VLQs}$ and $T_{VLQs}$ given in the Appendix
Acknowledgments.

  * •

The SM parameters : $m_{t}$ = 173.40 GeV, $m_{b}$ =4.2 GeV, $m_{Z}$=91.1876
GeV, $m_{h}$=125 GeV and $\alpha$=1/137.036.

  * •

The 2HDM parameters : $m_{h}$ = 125 GeV, $m_{H,A}\in[300,900]$ GeV,
$m_{H^{\pm}}\in[600,900]$ GeV, $\tan\beta\in[1,15]$ and
$\sin(\beta-\alpha)$=1. Where we have assumed that the charged Higgs boson is
heavier than 600 GeV in order to comply with $B_{S}\to X_{s}\gamma$
measurement.

  * •

The VLQs masses : $m_{T}\in[700,2000]$ GeV and $m_{B}\in[700,2000]$ GeV.

  * •

The VLQs mixing angles are chosen in the range [-0.5,0.5].

It should be noted that, in order to select points within the 2$\sigma$ ranges
of the new CDF measurement, we only consider points that satisfy
$\chi^{2}_{M_{W}^{\mathrm{CDF-II}}}\leq 4$ where:

$\displaystyle\chi^{2}_{M_{W}^{\mathrm{CDF-
II}}}=\frac{\left(M_{W}^{\mathrm{2HDM+VLQ}}-M_{W}^{\mathrm{CDF-
II}}\right)^{2}}{\left(\Delta M_{W}^{\mathrm{CDF-II}}\right)^{2}}$ (32)

###  3.1 2HDM + VLQs

In this subsection, we will illustrate results for the seven VLQ models
studied in this analysis. We start with the singlets, and then move to the
doublets and finish with the triplets.

####  3.1.1 2HDM+T

Figure 1: Left: Scan results of the 2HDM+T singlet in the $(S,~{}T)$ plane.
The blue points indicate the 2HDM, while the VLQ are indicated by orange
points. The combination of the 2HDM and VLQ is illustrated by green points.
Right: the same data points are displayed in the
$(m_{H^{\pm}}-m_{A},m_{H^{\pm}}-m_{H})$ plane, with the color code
representing the value of $M_{W}$.

In Fig. 1 (left panel), we present the results of our analysis in the $(S,T)$
plane, with each contribution displayed separately. The results from the two
Higgs doublet model (2HDM) are represented in blue, the VLQ in orange, and the
combination of the 2HDM and VLQ are shown in green. The plot illustrates that
the $T$ parameter is consistently positive for both the 2HDM and VLQ. On the
other hand, the $S$ parameter is positive in the VLQ, and negative in the
2HDM. Furthermore, it is noteworthy that the cancellation between the 2HDM and
VLQ leads to the $S$ parameter consistently assuming negative values in this
representation. In the right panel of Fig. 1, we show the allowed parameter
space in the $(m_{H^{\pm}}-m_{A},m_{H^{\pm}}-m_{H})$ plane within the 2HDM+T
singlet top partner model. The color coding indicates the value of the W boson
mass. One can observe that the CDF $M_{W}$ measurement value can only be
reached in the case where $m_{H^{\pm}}>m_{H},m_{A}$. Like in the 2HDM, the
degenerate case $m_{H^{\pm}}=m_{A}=m_{H}$ is strongly disfavored here by the
CDF-II data. Morever, it can also be seen that the mass splitting between the
charged Higgs boson and the CP-even Higgs $H$ can go up to 100 GeV, while the
mass splitting between $H^{\pm}$ and the CP-odd Higgs boson $A$ can be at most
of the order of 40 GeV to 320 GeV.

Figure 2: Scan result of the 2HDM+T singlet in the $(T,M_{W})$ plane left and
$(M_{W},\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}})$ right. The light red
band indicates the $M_{W}$ value with the associated 1$\sigma$ uncertainty
measured recently by the CDF collaboration, while the light green band shows
the result of $\sin^{2}\theta_{\mathrm{eff}}$ and its associated 1$\sigma$
uncertainty measured by the SLD collaboration.

In the left panel of Fig. 2, we display the allowed parameters space of our
scan with the predicted values for $M_{W}$ as a function of the $T$ parameter
within the 2HDM+T singlet top partner model. The color coding of the points
represents the values of the $S$ parameter. The red-dashed line shows the SM
prediction for the W boson mass, and the light red region indicates the new
CDF measurement at a level of $\pm 1\sigma$. It is evident from this plot that
there is a positive correlation between $M_{W}^{2HDM+T}$ and the $T$ parameter
as demanded by Eq. (28). The values of the $S$ parameter in the 2HDM+T singlet
top partner model preferred by the CDF-II measurement of $M_{W}$ are in the
range between -0.012 and -0.002. In the right panel of Fig. 2, we display the
parameter points of the 2HDM+T singlet in the
$(M_{W},\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}})$ plane. The color coding
shows the size of $T$ parameter. The light green region indicates the
measurements of $\sin^{2}\theta_{\mathrm{eff}}$ within $\pm 1\sigma$
confidence level reported by the SLD collaboration, while the light red band
shows the W boson mass value at $\pm 1\sigma$ recently measured by the CDF. At
first sight, one can see that there is good agreement between the preferred
parameter space of the CDF $M_{W}$ measurement and the value measured by the
SLD experiment. Furthermore, It can be observed that there is an obvious
strong negative correlation between
$\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}}$ and $M_{W}$ since in the $T$
parameter dominant case, a positive contribution to the W boson mass
prediction corresponds to a negative contribution to the effective weak mixing
angle. We find that the parameter $T$ within the 2HDM+T singlet top partner
model takes values in the range between 0.05 and 0.25.

####  3.1.2 2HDM+B

Figure 3: Scan result of the 2HDM+B singlet in the $(T,M_{W})$ plane left and
$(M_{W},\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}})$ right. The light red
band indicates the $M_{W}$ value with the associated 1$\sigma$ uncertainty
measured recently by the CDF collaboration, while the light green band shows
the result of $\sin^{2}\theta_{\mathrm{eff}}$ and its associated 1$\sigma$
uncertainty measured by the SLD collaboration.

We depict the allowed parameter points in the left panel of Fig. 3. The
predicted values for the W boson mass, $M_{W}$, are displayed as a function of
the $T$ parameter within the 2HDM+B singlet bottom partner model. The color
coding of the points represents the values of the $S$ parameter and indicates
that values of S between $-0.01$ and 0.01 are preferred by the CDF’s
measurement of $M_{W}$. In the right panel, we show the predictions for
$M_{W}$ and $\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}}$, with the color
code indicating the values of the $T$ parameter. We can see that the points
preferred by the CDF’s measurement of $M_{W}$ are consistent with the values
of $\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}}$ measured by the SLD
experiment. Moreover, the $T$ parameter in the 2HDM+B singlet bottom partner
model is found to be in the range between 0.14 and 0.26.

Figure 4: Scan results of the 2HDM + B singlet in the
$(m_{H^{\pm}}-m_{A},m_{H^{\pm}}-m_{H})$ plane , with the colour code
indicating $M_{W}$ GeV.

In Fig. 4, we plot the allowed parameter space in the
$(m_{H^{\pm}}-m_{A},m_{H^{\pm}}-m_{H})$ plane in the 2HDM+B singlet bottom
partner model. The color code represents the value of the W boson mass. We can
see that there is only one mass hierarchy and, similarly to the behavior shown
in the 2HDM+T singlet top partner model, the CDF $M_{W}$ measurement value can
be explained only when $m_{H^{\pm}}>m_{H},m_{A}$. Furthermore, it can be
observed that the mass splitting between the charged Higgs boson and the CP-
even Higgs $H$ can be as high as 140 GeV, while the mass difference between
$H^{\pm}$ and the CP-odd Higgs boson $A$ can reach 320 GeV.

####  3.1.3 2HDM + TB

Figure 5: Left: Scan results of the 2HDM+TB doublet in the $(S,T)$ plane. The
blue points indicate the 2HDM, while the VLQ are indicated by orange points.
The combination of the 2HDM and VLQ is illustrated by green points. Right: the
same data points are displayed in the $(m_{H^{\pm}}-m_{A},m_{H^{\pm}}-m_{H})$
plane, with the color code representing the value of $M_{W}$.

In Fig. 5 (left panel), we present the results of our analysis of the 2HDM+TB
doublet scenario in the $(S,T)$ plane. The contributions of the 2HDM and the
VLQs are depicted separately in blue and orange, respectively, while the
combination of both models is shown in green. The plot indicates that the $T$
parameter has both positive and negative values in the 2HDM, while it is
solely positive in the doublet VLQs. Similarly, the $S$ parameter displays
similar behavior, with positive values in the doublet VLQs and both positive
and negative values in the 2HDM. Moreover, the cancellation between the 2HDM
and VLQs in this scenario results in a consistent positive value for the $T$
parameter and a combination of positive and negative values for the $S$
parameter.

In the right panel of Fig. 5, we present the allowed parameter space in the
$(m_{H^{\pm}}-m_{H})$ versus $(m_{H^{\pm}}-m_{A})$ plane for the 2HDM+TB
doublet. The color code represents the value of the W boson mass. It has been
previously demonstrated in 2HDM studies [3] that the charged Higgs boson mass
$m_{H^{\pm}}$ must be non-degenerate with the masses of the two neutral Higgs
boson $m_{H}$ and $m_{A}$ in order to explain the current anomaly in the W
boson mass. However, the degenerate scenario, where $m_{H^{\pm}}=m_{A}=m_{H}$,
is still allowed in the 2HDM+TB top partner doublet model. Additionally, the
CDF-II $M_{W}$ measurement value can also be explained in the case where
$(m_{H^{\pm}}-m_{A})(m_{H^{\pm}}-m_{H})>0$ and also it is interesting to
observe that the case where $m_{A}<m_{H^{\pm}}<m_{H}$ is not excluded. Note
that, in the 2HDM this mass splitting region is not allowed because the $T$
parameter is negative but here the presence of the top partner doublet $TB$
pushed it to the preferred positive values required by the $M_{W}$ mass
anomaly. Furthermore, we see that the mass splitting between the charged Higgs
boson and the CP-odd Higgs boson can reach up to 360 GeV, while the mass
difference between the charged Higgs boson and the CP-even Higgs boson $H$ can
be at most 100 GeV. The large mass gap between $m_{H^{\pm}}$ and $m_{A}$ can
guarantee the dominant decay mode of $H^{\pm}\rightarrow AW^{\pm}$ which is
just gauge coupling and have no mixing suppression. Future experimental
searches for this decay mode would be able to confirm or rule out the
explanation of the recent $M_{W}$ mass anomaly within this model.

Figure 6: Scan result of the 2HDM+TB doublet in the $(T,M_{W})$ plane left and
$(M_{W},\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}})$ right. The light red
band indicates the $M_{W}$ value with the associated 1$\sigma$ uncertainty
measured recently by the CDF collaboration, while the light green band shows
the result of $\sin^{2}\theta_{\mathrm{eff}}$ and its associated 1$\sigma$
uncertainty measured by the SLD collaboration.

We display in the left panel of Fig. 6 the allowed parameters points in the
plane of $M_{W}$ versus $T$ within the 2HDM+TB top partner doublet model. The
points are color-coded to show the values of the $S$ parameter. The red dashed
line indicates the SM prediction for the W boson mass and the light red region
shows the new CDF-II measurement at $\pm 1\sigma$ CL. It can seen that
$M_{W}^{2HDM+TB}$ is a linear function of the $T$ parameter. Furthermore, the
regions consistent with CDF-II measurement of $M_{W}$ require the values of
the $S$ parameter to be in the range between 0 and 0.06. In the right panel of
Fig. 6, we visualize our results within the 2HDM+TB top partner doublet model
in the $\sin^{2}\theta_{\mathrm{eff}}$ versus $M_{W}$ plane. The color-coding
indicates the allowed values of the $T$ parameter. It is interesting to see
that full agreement is achieved between the SLD measurement of the weak mixing
angle and the favored parameter space required by the CDF-II data. Also, we
see that the preferred range for the $T$ parameter is almost entirely positive
and varies between 0.14 and 0.26.

####  3.1.4 2HDM+BY or 2HDM+XT

Figure 7: Scan result of the 2HDM+BY (2HDM+XT in the $T$ versus $M_{W}$ plane
in the upper left panel (lower left panel) and in the $M_{W}$ versus
$\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}}$ plane in the upper right panel
(lower right panel). The light red band indicates the $M_{W}$ value with the
associated 1$\sigma$ uncertainty measured recently by the CDF collaboration,
while the light green band shows the result of $\sin^{2}\theta_{\mathrm{eff}}$
and its associated 1$\sigma$ uncertainty measured by the SLD collaboration.

We display in the upper left and lower left panels of Fig. 7 the allowed
parameter points in the $M_{W}$ versus $T$ plane in the frameworks of the
2HDM+BY and 2HDM+XT models, respectively. The color coding of the points shows
the allowed values of $S$ parameter. The red-dashed line indicates the SM
prediction for the W boson mass and the light red region displays the new CDF-
II measurement at the level of $\pm 1\sigma$. One can observe that the regions
consistent with CDF-II measurement of $M_{W}$ require the values of the $S$
parameter to be in the range between -0.030 and -0.01 in the 2HDM+BY model,
while the favored values of the $S$ parameter in the 2HMD+XT model are still
negative and range between -0.035 and -0.01. In the upper right and lower
right panels of Fig. 7 we visualize our results in the
$\sin^{2}\theta_{\mathrm{eff}}$ versus $M_{W}$ plane within the 2HDM+BY and
2HDM+XT models, respectively. The color code indicates the allowed values
taken by the $T$ parameter. It is interesting to see that a full agreement is
achieved between the weak mixing angle measured by the SLD collaboration and
the favored parameters space required by the CDF-II data. Also, we see that
the preferred range for the $T$ parameter for both models needed to explain
the anomaly is almost entirely positive and varies between 0.14 and 0.22.

Figure 8: The points from the parameter scan of the 2HDM+BY (2HDM+XT)models in
the $m_{H^{\pm}}-m_{A}$ versus $m_{H^{\pm}}-m_{H}$ plane in the left panel
(right panel), with the color code indicating the W-boson mass $M_{W}$. All
points have $\chi^{2}_{M_{W}^{\mathrm{CDF-II}}}\leq 4$.

In the left panel of Fig. 8, we show the allowed parameter points in the
$(m_{H^{\pm}}-m_{H})$ versus $(m_{H^{\pm}}-m_{A})$ plane within the 2HDM+BY
model. The color code indicates the value of the W boson mass. It is
interesting to see that within this model the charged Higgs boson mass
$m_{H^{\pm}}$ can be degenerate with the two neutral BSM Higgs boson masses
$m_{H}$ and $m_{A}$ to accommodate the CDF-II W-boson mass. Additionally, the
recent anomaly can also be explained in the case where
$(m_{H^{\pm}}-m_{A})(m_{H^{\pm}}-m_{H})>0$ and again it is interesting to
observe that the case where $m_{A}<m_{H^{\pm}}<m_{H}$ is not excluded due to
the presence of (B,Y) doublet which gives the favored large positive values of
the $T$ parameter required to accommodate the W-mass reported by the CDF-II
collaboration. Furthermore, we see that the mass splitting between the charged
Higgs boson and CP-odd Higgs boson can go up to 360 GeV whilst the mass
splitting between $H^{\pm}$ and CP-even Higgs boson $H$ can at most be of the
order of 100 GeV. In the right panel of Fig. 8 we present the allowed
parameter points within the 2HDM+XT doublet model. One can observe that both
splittings $m_{H^{\pm}}-m_{H}$ and $m_{H^{\pm}}-m_{A}$ are required to be
simultaneously positive and always non-zero in order to be consistent with the
CDF-II data. The maximum mass splitting between the charged Higgs boson and
the CP-even Higgs $H$ can reach 200 GeV, while the maximum splitting between
$H^{\pm}$ and the CP-odd Higgs boson $A$ can go up to 340 GeV. It should be
noted that, in this model, the degenerate case where $m_{H^{\pm}}=m_{A}=m_{H}$
is ruled out by the direct Higgs boson searches at colliders, as computed by
HiggsBounds-5.

####  3.1.5 2HDM+TBY or 2HDM+XTB

Figure 9: Scan result of the 2HDM+TBY (upper panels) and (2HDM+XTB) (lower
panels) in the $T$ versus $M_{W}$ plane in the upper left panel (lower left
panel) and in the $M_{W}$ versus
$\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}}$ plane in the upper right panel
(lower right panel). The light red band indicates the $M_{W}$ value with the
associated 1$\sigma$ uncertainty measured recently by the CDF collaboration,
while the light green band shows the result of $\sin^{2}\theta_{\mathrm{eff}}$
and its associated 1$\sigma$ uncertainty measured by the SLD collaboration.

We present in the upper-left and lower-left panels of Fig.9 the allowed points
from the parameter scan in the $M_{W}$ versus $T$ plane in the framework of
the 2HDM+TBY and 2HDM+XTB triplet models, respectively. The color coding of
the points indicates the allowed values of the $S$ parameter. The red-dashed
line shows the SM prediction for the W-boson mass, and the light red region
displays the new CDF-II measurement at the $\pm 1\sigma$ level. We can see
that the regions consistent with the new CDF measurement require the values of
the $S$ parameter to be in the range between -0.014 and 0.11 in the 2HDM+TBY
model, while a negative $S$ parameter contribution is required to accommodate
the anomaly in the 2HDM+XTB model, with favored values of the $S$ parameter
ranging between -0.032 and -0.0012. In the upper-right and lower-right panels
of Fig.9, we visualize, respectively, our results within the 2HDM+TBY and
2HDM+XTB models in the $\sin^{2}\theta_{\mathrm{eff}}$ versus $M_{W}$ plane.
The color code indicates the allowed values taken by the $T$ parameter.
Besides the fact that $M_{W}$ is anti-correlated with the weak mixing angle
$\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}}$, it is interesting to note that
in both models there is full agreement between the weak mixing angle measured
by the SLD collaboration and the points from the parameter space preferred by
the CDF-II data. Also, we see that the favored range for the $T$ parameter
within the 2HDM+TBY model that is required to explain the anomaly is almost
entirely positive and varies between 0.13 and 0.3, while it ranges between
0.13 and 0.19 in the framework of the 2HDM+XTB model.

Figure 10: The points from the parameter scan of the 2HDM+TBY (2HDM+XTB)
models in the $m_{H^{\pm}}-m_{A}$ versus $m_{H^{\pm}}-m_{H}$ plane in the left
panel (right panel), with the color code indicating the W-boson mass $M_{W}$.

In the left panel of Fig. 10, we present the allowed parameter points in the
$(m_{H^{\pm}}-m_{H})$ versus $(m_{H^{\pm}}-m_{A})$ plane within the 2HDM+TBY
triplet model. The color code indicates the value of the W boson mass. At a
glance, it is evident that the CDF value of $M_{W}$ can be reproduced in the
scenario where the three BSM scalars are mass-degenerate, i.e.,
$m_{H^{\pm}}=m_{A}=m_{H}$. Additionally, the CDF-II $M_{W}$ measurement can
also be explained in the case where
$(m_{H^{\pm}}-m_{A})(m_{H^{\pm}}-m_{H})>0$. It is worth noting that the
scenario with $m_{H}<m_{H^{\pm}}<m_{A}$ is still allowed and can accommodate
the anomaly. Besides, we see that the mass splitting between the charged Higgs
boson and the CP-odd Higgs can reach up to 160 GeV, while the mass splitting
between $H^{\pm}$ and the CP-even Higgs $H$ can go up to 350 GeV.
In the right panel of Fig. 10, the allowed parameter space within the 2HDM+XTB
model is presented. It can be seen that the degenerate scenario, in which the
masses of the charged Higgs boson ($m_{H^{\pm}}$), the CP-even Higgs
($m_{H}$), and the CP-odd Higgs ($m_{A}$) are equal, cannot explain the CDF
value of the W boson mass. Furthermore, it is shown that the mass difference
between the charged Higgs boson and the CP-even Higgs $H$ is restricted to a
range of 150 GeV to 110 GeV, while the mass difference between the charged
Higgs boson and the CP-odd Higgs $A$ is limited to a range between -10 GeV and
350 GeV.

##  4 Conclusion

The very recent measurement result of the W-boson mass $M_{W}$ reported by the
CDF collaboration deviates from the SM prediction by about 7$\sigma$. If the
CDF-II result is correct, it would indicate the need for new physics beyond
the Standard Model to explain the observed mass of the W boson that matches
the CDF-II data. In this paper, we consider the implications of the recent W
boson mass anomaly in the framework of 2HDM type II augmented by vector-like
quarks. We studied all seven distinct possibilities of vector-like $SU(2)_{L}$
multiplets including singlets, doublets, and triplets. We computed the
electroweak precision observables S and T using the standard scalar Passarino-
Veltman functions and provided the generic analytic expressions for $S$ and
$T$ and listed the couplings used for specific models such us the singlet,
doublet and triplet VLQs models. In order to explain the CDF measurement we
need a positive $T$ and this is possible if we consider VLQs alone, which is
all time positive. In the case of the 2HDM augmented with VLQs, both $S$ and
$T$ contributions are split into two separate contributions, one from VLQs and
one from 2HDM which do not mix. We found that in most cases, there is a
cancellation between VLQs and 2HDM contributions, which enlarges the parameter
space of both models. After this observation we performed a $\chi^{2}$
analysis and found that it is possible to solve the tension between the new
CDF $M_{W}$ measurement and the SM prediction. We also noticed that the
scenario with 2HDM alone with a degenerate spectrum that is ruled out by the
CDF measurement could survive if we include both 2HDM and VLQs. We also
discussed the consequences of the 2HDM augmented with VLQs on the effective
mixing angle $\sin^{2}\theta_{\text{eff}}$.

## Acknowledgments

We thank Junjie Cao for useful discussions. This work is supported by the
Moroccan Ministry of Higher Education and Scientific Research MESRSFC and
CNRST: Projet PPR/2015/6. M.B is grateful for the technical support of
CNRST/HPC-MARWAN.

##  Appendix A VLQs contributions to the oblique parameters S, T and U

In this appendix we give the analytical expressions for S, T and U in the 2HDM
as well as in the generic expression in the VLQ models as a function of the
well-known scalar Passarino-Veltman functions.

###  A.1 Passarino-Veltman functions

The Passarino-Veltman functions are defined as follow:

$\displaystyle A_{0}(m_{1}^{2})$ $\displaystyle=$
$\displaystyle\frac{(2\pi\mu)^{4-D}}{i\pi^{2}}\int{d^{D}q\frac{1}{d_{1}}},$
(33) $\displaystyle B_{0};B^{\mu};B^{\mu\nu}(p_{1}^{2},m_{1}^{2},m_{2}^{2})$
$\displaystyle=$
$\displaystyle\frac{(2\pi\mu)^{4-D}}{i\pi^{2}}\int{d^{D}q\frac{1;q^{\mu};q^{\mu}q^{\nu}}{d_{1}d_{2}}}$
(34)

With the denominators $d_{i}$ are defined by $d_{1}=q^{2}-m_{1}^{2}$ and
$d_{2}=(q+p_{1})^{2}-m_{2}^{2}$ and $\mu$ is the renormalization scale.
$p_{1}$ is the external momentum and $q$ is the internal momentum which is
integrated out.

The functions $B^{\mu}$ and $B^{\mu\nu}$ can be decomposed as,

$\displaystyle B^{\mu}$ $\displaystyle=$ $\displaystyle
p_{1}^{\mu}B_{1},\quad\
B^{\mu\nu}=g^{\mu\nu}B_{00}+p_{1}^{\mu}p_{1}^{\nu}B_{11}.$ (35)

Thus, the functions $B_{00}$ and $B_{1}$ can be calculated by contracting
$B^{\mu}$ and $B^{\mu\nu}$ with $p_{1}$ and $g_{\mu\nu}$ respectively,

$\displaystyle B_{1}(p^{2},m_{1}^{2},m_{2}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[A_{0}(m_{1}^{2})-A_{0}(m_{2}^{2})-\left(p^{2}+m_{1}^{2}-m_{2}^{2}\right)B_{0}(p^{2},m_{1}^{2},m_{2}^{2})\right]$
(36) $\displaystyle B_{00}(p^{2},m_{1}^{2},m_{2}^{2})$ $\displaystyle=$
$\displaystyle-\frac{p^{2}-3(m_{1}^{2}+m_{2}^{2})}{18}+\frac{m_{1}^{2}B_{0}(p^{2},m_{1}^{2},m_{2}^{2})}{3}$
(37)
$\displaystyle+\frac{A_{0}(m_{2}^{2})+(p^{2}+m_{1}^{2}-m_{2}^{2})B_{1}(p^{2},m_{1}^{2},m_{2}^{2})}{6}$

###  A.2  $S$, $T$ and $U$ parameters

The oblique parameters can be parameterized in terms of the gauge bosons two-
point functions as follows:

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{4s_{W}^{2}c_{W}^{2}}{\alpha}\Re
e\Bigg{[}\frac{\Pi_{\gamma\gamma}(m_{Z}^{2})}{m_{Z}^{2}}-\frac{\Pi_{ZZ}(m_{Z}^{2})-\Pi_{ZZ}(0)}{m_{Z}^{2}}-\Big{(}\frac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}\Big{)}\Big{(}\frac{\Pi_{Z\gamma}(m_{Z}^{2})+\Pi_{Z\gamma}(0)}{m_{Z}^{2}}\Big{)}\Bigg{]}$
$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{\alpha}\Re
e\left[\frac{\Pi_{ZZ}(0)}{m_{Z}^{2}}-\frac{\Pi_{WW}(0)}{m_{W}^{2}}-\frac{2s_{W}}{c_{W}}\frac{\Pi_{Z\gamma}(0)}{m_{Z}^{2}}\right]$
(39) $\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{4s_{W}^{2}}{\alpha}\Re
e\Bigg{[}\frac{\Pi_{WW}(0)-\Pi_{WW}(m_{W}^{2})}{m_{W}^{2}}-c_{W}^{2}\frac{\Pi_{ZZ}(0)-\Pi_{ZZ}(m_{Z}^{2})}{m_{Z}^{2}}-2s_{W}c_{W}\frac{\Pi_{\gamma
Z}(0)-\Pi_{\gamma Z}(m_{Z}^{2})}{m_{Z}^{2}}+$ (40) $\displaystyle
s_{W}^{2}\frac{\Pi_{\gamma\gamma}(m_{Z}^{2})}{m_{Z}^{2}}\Bigg{]}$

Where $\alpha$ is the fine-structure constant and $c_{w}$ ($s_{w}$) denotes
the cosine (sine) of the Weinberg angle.

###  A.3 $S$, $T$ and $U$ parameters in 2HDM

The contributions of the 2HDM to the oblique parameters S, T and U can be
given as follows:

$\displaystyle\Delta S$ $\displaystyle=$ $\displaystyle\frac{c_{W}^{2}}{\pi
m_{W}^{2}}\Bigg{\\{}c_{\beta-\alpha}^{2}\bigg{[}m_{Z}^{2}\Big{(}-B_{0}(0,m_{h}^{2},m_{Z}^{2})+B_{0}(m_{Z}^{2},m_{h}^{2},m_{Z}^{2})+B_{0}(0,m_{H}^{2},m_{Z}^{2})$
(41) $\displaystyle-
B_{0}(m_{Z}^{2},m_{H}^{2},m_{Z}^{2})\Big{)}+B_{00}(m_{Z}^{2},m_{h}^{2},m_{A}^{2})-B_{00}(0,m_{h}^{2},m_{A}^{2})+B_{00}(0,m_{h}^{2},m_{Z}^{2})$
$\displaystyle-
B_{00}(m_{Z}^{2},m_{h}^{2},m_{Z}^{2})-B_{00}(0,m_{H}^{2},m_{Z}^{2})+B_{00}(m_{Z}^{2},m_{H}^{2},m_{Z}^{2})\bigg{]}$
$\displaystyle+s_{\beta-\alpha}^{2}\bigg{[}B_{00}(m_{Z}^{2},m_{H}^{2},m_{A}^{2})-B_{00}(0,m_{H}^{2},m_{A}^{2})\bigg{]}-B_{00}(m_{Z}^{2},m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})$
$\displaystyle+(c_{W}^{2}-s_{W}^{2})^{2}B_{00}(0,m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})+2A_{0}(m_{H^{\pm}}^{2})s_{W}^{2}c_{W}^{2}\Bigg{\\}}$

$\displaystyle\Delta T=$ $\displaystyle\frac{1}{4m_{W}^{2}\pi
s_{W}^{2}}\Big{\\{}m_{W}^{2}c^{2}_{\beta-\alpha}B_{0}(0,m_{h}^{2},m_{W}^{2})+s^{2}_{\beta-\alpha}B_{00}(0,m_{h}^{2},m_{W}^{2})$
$\displaystyle-c^{2}_{\beta-\alpha}\left[m_{Z}^{2}B_{0}(0,m_{h}^{2},m_{Z}^{2})-B_{00}(0,m_{h}^{2},m_{Z}^{2})\right]$
$\displaystyle-c^{2}_{\beta-\alpha}\left[m_{W}^{2}B_{0}(0,m_{H}^{2},m_{W}^{2})-B_{00}(0,m_{H}^{2},m_{W}^{2})\right]+c^{2}_{\beta-\alpha}\left[m_{Z}^{2}B_{0}(0,m_{H}^{2},m_{Z}^{2})-B_{00}(0,m_{H}^{2},m_{Z}^{2})\right]$
$\displaystyle-2m_{H^{\pm}}^{2}s_{W}^{4}B_{0}(0,m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})+(4s_{W}^{4}-1)B_{00}(0,m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})$
$\displaystyle-c^{2}_{\beta-\alpha}\left[B_{00}(0,m_{h}^{2},m_{A}^{2})-B_{00}(0,m_{h}^{2},m_{H^{\pm}}^{2})\right]$
$\displaystyle-s^{2}_{\beta-\alpha}\left[B_{00}(0,m_{H}^{2},m_{A}^{2})-B_{00}(0,m_{H}^{2},m_{H^{\pm}}^{2})\right]+B_{00}(0,m_{A}^{2},m_{H^{\pm}}^{2})-B_{00}(0,m_{W}^{2},m_{h}^{2})$
$\displaystyle-
B_{00}(0,m_{W}^{2},m_{Z}^{2})+B_{00}(0,m_{Z}^{2},m_{W}^{2})-2m_{H^{\pm}}^{2}s_{W}^{4}\Big{\\}}$
(42)

$\displaystyle\Delta U$ $\displaystyle=$ $\displaystyle\frac{1}{\pi
m_{W}^{2}}\Bigg{\\{}c_{\beta-\alpha}\Big{[}m_{Z}^{2}c_{W}^{2}B_{0}(0,m_{h}^{2},m_{Z}^{2})-m_{Z}^{2}c_{W}^{2}B_{0}(0,m_{h}^{2},m_{W}^{2})$
(43) $\displaystyle-
m_{Z}^{2}c_{W}^{2}B_{0}(m_{Z}^{2},m_{h}^{2},m_{Z}^{2})+m_{Z}^{2}c_{W}^{2}B_{0}(m_{W}^{2},m_{h}^{2},m_{W}^{2})-m_{Z}^{2}c_{W}^{2}B_{0}(0,m_{H}^{2},m_{Z}^{2})$
$\displaystyle+m_{Z}^{2}c_{W}^{2}B_{0}(0,m_{H}^{2},m_{W}^{2})+m_{Z}^{2}c_{W}^{2}B_{0}(m_{Z}^{2},m_{H}^{2},m_{Z}^{2})-m_{Z}^{2}c_{W}^{2}B_{0}(m_{W}^{2},m_{H}^{2},m_{W}^{2})$
$\displaystyle-
c_{W}^{2}B_{00}(m_{Z}^{2},m_{h}^{2},m_{A}^{2})+c_{W}^{2}B_{00}(0,m_{h}^{2},m_{A}^{2})-c_{W}^{2}B_{00}(0,m_{h}^{2},m_{Z}^{2})$
$\displaystyle+c_{W}^{2}B_{00}(m_{Z}^{2},m_{h}^{2},m_{Z}^{2})+c_{W}^{2}B_{00}(0,m_{H}^{2},m_{Z}^{2})-c_{W}^{2}B_{00}(m_{Z}^{2},m_{H}^{2},m_{Z}^{2})$
$\displaystyle+B_{00}(m_{W}^{2},m_{h}^{2},m_{H^{\pm}}^{2})-B_{00}(0,m_{h}^{2},m_{H^{\pm}}^{2})-B_{00}(0,m_{H}^{2},m_{W}^{2})$
$\displaystyle+B_{00}(m_{W}^{2},m_{H}^{2},m_{W}^{2})\Big{]}-s_{\beta-\alpha}\Big{[}c_{W}^{2}B_{00}(m_{Z}^{2},m_{H}^{2},m_{A}^{2})+c_{W}^{2}B_{00}(0,m_{H}^{2},m_{A}^{2})$
$\displaystyle-
B_{00}(0,m_{h}^{2},m_{W}^{2})+B_{00}(m_{W}^{2},m_{h}^{2},m_{W}^{2})+B_{00}(m_{W}^{2},m_{H}^{2},m_{H^{\pm}}^{2})-B_{00}(0,m_{H}^{2},m_{H^{\pm}}^{2})\Big{]}$
$\displaystyle+B_{00}(m_{W}^{2},m_{A}^{2},m_{H^{\pm}}^{2})-B_{00}(0,m_{A}^{2},m_{H^{\pm}}^{2})+B_{00}(0,m_{W}^{2},m_{h}^{2})-B_{00}(m_{W}^{2},m_{W}^{2},m_{h}^{2})$
$\displaystyle-
c_{W}^{2}B_{00}(m_{Z}^{2},m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})-\frac{1}{2}c_{W}^{2}(-4c_{2W}+c_{4W}+1)B_{00}(0,m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})-B_{00}(0,m_{Z}^{2},m_{W}^{2})$
$\displaystyle+B_{00}(0,m_{W}^{2},m_{Z}^{2})+B_{00}(m_{W}^{2},m_{Z}^{2},m_{W}^{2})-B_{00}(m_{W}^{2},m_{W}^{2},m_{Z}^{2})+2A_{0}(m_{H^{\pm}}^{2})s_{W}^{4}c_{W}^{2}\Bigg{\\}}$

It is understood that in the above expressions for $\Delta S$, $\Delta T$, and
$\Delta U$, the SM bosonic contribution has been subtracted from the 2HDM
contribution.

###  A.4 $S$, $T$ and $U$ parameters in VLQs

We use FeynArts/FormCalc public codes to calculate the analytical expressions
of the self-energy of the gauge bosons, presented in equations
(LABEL:eq:S_expression) and (39), in a scenario where VLQs interact only with
third generation SM quarks. By taking into account only the contribution of
those heavy quarks (both SM top and bottom as well as the new VLQs) to the
self-energy, as the other contributions are the same as in the SM, the
expressions of the electroweak gauge boson self-energies can be written as:

$\displaystyle-\Pi_{\gamma Z}(q^{2})$ $\displaystyle=$
$\displaystyle\sum_{f}{g_{\gamma
z}(c_{Zff}^{L},c_{Zff}^{R},Q_{f},m_{f}^{2},q^{2})}$ (44)
$\displaystyle-\Pi_{\gamma\gamma}(q^{2})$ $\displaystyle=$
$\displaystyle\sum_{f}{g_{vv}(Q_{f},Q_{f},m_{f}^{2},m_{f}^{2},q^{2})}$ (45)
$\displaystyle-\Pi_{ZZ}(q^{2})$ $\displaystyle=$ $\displaystyle
2\sum_{f_{i}\neq
f_{j}}{\delta_{0}{(|Q_{f_{i}}-Q_{f_{j}}|)}g_{vv}\left(c_{Zf_{i}f_{j}}^{L},c_{Zf_{i}f_{j}}^{R},m_{f_{i}}^{2},m_{f_{j}}^{2},q^{2}\right)}$
(46)
$\displaystyle+\sum_{f}{g_{vv}\left(c_{Zff}^{L},c_{Zff}^{R},m_{f}^{2},m_{f}^{2},q^{2}\right)}$
$\displaystyle-\Pi_{WW}(q^{2})$ $\displaystyle=$ $\displaystyle\sum_{f_{i}\neq
f_{j}}{\delta_{1}{(|Q_{f_{i}}-Q_{f_{j}}|)}g_{vv}\left(c_{Wf_{i}f_{j}}^{L},c_{Wf_{i}f_{j}}^{R},m_{f_{i}}^{2},m_{f_{j}}^{2},q^{2}\right)}$
(47)

Where $c_{Vff}$ are the couplings between the gauge boson Z or W and two
fermions ($f=t,b,T,X,B,Y$) given in App. B , $\delta_{d}$ is the Dirac
function with a non-null value at d, and the functions $g_{ab}$ are defined
as,

$g_{vv}$ and $g_{Z\gamma}$ are given by:

$\displaystyle g_{vv}\left(x,y,m_{1}^{2},m_{2}^{2},k^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{N_{c}}{8\pi^{2}}\Big{[}(x^{2}+y^{2})\big{(}A_{0}(m_{2}^{2})-2B_{00}(k^{2},m_{1}^{2},m_{2}^{2})+k^{2}B_{1}(k^{2},m_{1}^{2},m_{2}^{2})\big{)}$
(48)
$\displaystyle+\big{(}(x^{2}+y^{2})m_{1}^{2}-2xym_{1}m_{2}\big{)}B_{0}(k^{2},m_{1}^{2},m_{2}^{2})\Big{]}$
$\displaystyle g_{\gamma z}\left(x,y,Q,m^{2},k^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{-N_{c}}{8\pi^{2}}\Big{[}(x+y)Q\big{(}A_{0}(m^{2})-2B_{00}(k^{2},m^{2},m^{2})$
(49) $\displaystyle+k^{2}B_{1}(k^{2},m^{2},m^{2})\big{)}\Big{]}$

Where the $N_{c}$ is the color factor with $N_{c}=3$ for quarks. $A_{0}$,
$B_{0}$, $B_{00}$ and $B_{1}$ are the usual Passarino-Veltman functions.

The SM contribution to S and T is calculated by considering the exchange of
only top and bottom quarks in the self-energies of the gauge bosons, and the
SM couplings used are given by Eq.(67).

After computing the expressions of the $S$, $T$ and $U$ parameters for both
the SM and the SM augmented with VLQs, it is necessary to perform a
subtraction to extract the contribution of new physics to the oblique
parameters.

$\displaystyle\Delta S=S_{SM+VLQs}-S_{SM},\ \Delta T=T_{SM+VLQs}-T_{SM},\
\Delta U=U_{SM+VLQs}-U_{SM}$ (50)

It is important to note that we checked that both S, T and U are UV finite and
renormalization scale $\mu$ independent in all the aforementioned models.

The oblique parameters of the extensions with VLQs have been studied in many
works [19, 6, 17, 16]. In our analysis, we have compared our results
numerically with some of those papers and found the following conclusions:

  * •

Our expressions for the oblique parameter $T$ are in agreement with all the
previously mentioned papers in all extensions, except in the SM+XTB case where
there is a disagreement with those of the Ref. [17].

  * •

The approximation proposed in Eqs.(25),(26),(27), and (28) of [16] can lead to
large deviations in the evaluation of the oblique parameter $S$ in the SM
+$B$, $XT$, $BY$ extension, but when using the general terms proposed in Eqs.
(17), (18), (22), and (23) of [16], these deviations disappear in the case of
large $s^{d}_{R}$ in the SM +$TB$ extension.

  * •

The expression for the $S$ parameter in the triplet extensions cannot be
obtained in the same manner as in [17] where they extended the formulations
from Ref.[16] to calculate this oblique parameter. This can be problematic, as
demonstrated in Ref.[18].

##  Appendix B Gauge interactions of the vector-like quarks

The neutral and charged currents are modified by the presence of VLQs, the
couplings between the exotic heavy quarks to the SM third generation quarks
and the electroweak massive gauge bosons can be parametrized as follows:

$\displaystyle Zq^{\prime}q$ $\displaystyle=$
$\displaystyle\frac{e}{2s_{w}c_{w}}\gamma^{\mu}(\lambda^{L}_{Zq^{\prime}q}\mathbb{L}+\lambda^{R}_{Zq^{\prime}q}\mathbb{R}),$
$\displaystyle Wq^{\prime}q$ $\displaystyle=$
$\displaystyle\frac{e}{\sqrt{2}s_{w}}\gamma^{\mu}(\lambda^{L}_{Wq^{\prime}q}\mathbb{L}+\lambda^{R}_{Wq^{\prime}q}\mathbb{R})$
(51)

Where $\lambda_{Vq^{\prime}q}^{L,R}$ are the components for both left- and
right-handed couplings of the Z and W bosons. Note that these couplings depend
on the chosen representations of VLQs as follows:

###  B.1 Vector-like singlet T

In the $SU(2)_{L}$ vector-like singlet T scenario, the left and right handed
charged current couplings $\lambda_{Wq^{\prime}q}^{L,R}$ are given by:

$\displaystyle\lambda_{Wtb}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u},\quad\lambda_{Wtb}^{R}=0,$ $\displaystyle\lambda_{WTb}^{L}$
$\displaystyle=$ $\displaystyle s_{L}^{u},\quad\lambda_{WTb}^{R}=0.$ (52)

the left and right handed neutral current couplings
$\lambda^{L,R}_{Zq^{\prime}q}$ are given by:

$\displaystyle\lambda_{Ztt}^{L}$ $\displaystyle=$
$\displaystyle(c_{L}^{u})^{2}-\frac{4}{3}s_{W}^{2},\quad\lambda_{Ztt}^{R}=-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{Zbb}^{L}$ $\displaystyle=$
$\displaystyle-1+\frac{2}{3}s_{W}^{2},\quad\lambda_{Zbb}^{R}=\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZTT}^{L}$ $\displaystyle=$
$\displaystyle(s_{L}^{u})^{2}-\frac{4}{3}s_{W}^{2},\quad\lambda_{ZTT}^{R}=-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZtT}^{L}$ $\displaystyle=$ $\displaystyle
s_{L}^{u}c_{L}^{u},\quad\lambda_{ZtT}^{R}=0.$ (53)

###  B.2 Vector-like singlet B

In the $SU(2)_{L}$ vector-like B singlet scenario, the left and right-handed
quarks-W boson couplings $\lambda_{Wq^{\prime}q}^{L,R}$ are given by:

$\displaystyle\lambda_{Wtb}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{d},\quad\lambda_{Wtb}^{R}=0,$ $\displaystyle\lambda_{WtB}^{L}$
$\displaystyle=$ $\displaystyle s_{L}^{d}\ ,\quad\lambda_{WtB}^{R}=0.$ (54)

While the neutral current couplings to the Z boson are given by:

$\displaystyle\lambda_{Ztt}^{L}$ $\displaystyle=$ $\displaystyle
1-\frac{4}{3}s_{W}^{2},\quad\lambda_{Ztt}^{R}=-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{Zbb}^{L}$ $\displaystyle=$
$\displaystyle-(c_{L}^{d})^{2}+\frac{2}{3}s_{W}^{2},\quad\lambda_{Zbb}^{R}=\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZBB}^{L}$ $\displaystyle=$
$\displaystyle-(s_{L}^{d})^{2}+\frac{2}{3}s_{W}^{2},\quad\lambda_{ZBB}^{R}=\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZbB}^{L}$ $\displaystyle=$ $\displaystyle-
s_{L}^{d}c_{L}^{d},\quad\lambda_{ZbB}^{R}=0.$ (55)

###  B.3 Vector-like doublet TB

In the case of the $TB$ doublet model, the couplings between T , B and the SM
third generation quarks with the W-boson are given by:

$\displaystyle\lambda_{Wtb}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u}c_{L}^{d}+s_{L}^{u}s_{L}^{d},\quad\lambda_{Wtb}^{R}=s_{R}^{u}s_{R}^{d},$
$\displaystyle\lambda_{WTB}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u}c_{L}^{d}+s_{L}^{u}s_{L}^{d},\quad\lambda_{WTB}^{R}=c_{R}^{u}c_{R}^{d},$
$\displaystyle\lambda_{WTb}^{L}$ $\displaystyle=$ $\displaystyle
s_{L}^{u}c_{L}^{d}-c_{L}^{u}s_{L}^{d},\quad\lambda_{WTb}^{R}=-c_{R}^{u}s_{R}^{d},$
$\displaystyle\lambda_{WtB}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u}s_{L}^{d}-s_{L}^{u}c_{L}^{d},\quad\lambda_{WtB}^{R}=-s_{R}^{u}c_{R}^{d}.$
(56)

The left and right-handed neutral current couplings to the Z boson are given
by:

$\displaystyle\lambda_{Ztt}^{L}$ $\displaystyle=$ $\displaystyle
1-\frac{4}{3}s_{W}^{2},\quad\lambda_{Ztt}^{R}=(s_{R}^{u})^{2}-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{Zbb}^{L}$ $\displaystyle=$
$\displaystyle-1+\frac{2}{3}s_{W}^{2},\quad\lambda_{Zbb}^{R}=-(s_{R}^{d})^{2}+\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZTT}^{L}$ $\displaystyle=$ $\displaystyle
1-\frac{4}{3}s_{W}^{2},\quad\lambda_{ZTT}^{R}=(c_{R}^{u})^{2}-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZBB}^{L}$ $\displaystyle=$
$\displaystyle-1+\frac{2}{3}s_{W}^{2},\quad\lambda_{ZBB}^{R}=-(c_{R}^{d})^{2}+\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZtT}^{L}$ $\displaystyle=$ $\displaystyle
0,\quad\quad\lambda_{ZtT}^{R}=-s_{R}^{u}c_{R}^{u},$
$\displaystyle\lambda_{ZbB}^{L}$ $\displaystyle=$ $\displaystyle
0,\quad\quad\lambda_{ZbB}^{R}=s_{R}^{d}c_{R}^{d}.$ (57)

###  B.4 Vector-like doublet XT

For the XT doublet model, the couplings between the X and T heavy quarks and
the third generation SM quarks with the W boson are defined as:

$\displaystyle\lambda_{Wtb}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u},\quad\lambda_{Wtb}^{R}=0,$ $\displaystyle\lambda_{WTb}^{L}$
$\displaystyle=$ $\displaystyle s_{L}^{u},\quad\lambda_{WTb}^{R}=0,$
$\displaystyle\lambda_{WtX}^{L}$ $\displaystyle=$ $\displaystyle-
s_{L}^{u},\quad\lambda_{WtX}^{R}=-s_{R}^{u},$ $\displaystyle\lambda_{WXT}^{L}$
$\displaystyle=$ $\displaystyle c_{L}^{u},\quad\lambda_{WXT}^{R}=c_{R}^{u}.$
(58)

the left- and right-handed neutral current couplings to the Z boson
$\lambda^{L,R}_{Zq^{\prime}q}$ are defined as:

$\displaystyle\lambda_{Ztt}^{L}$ $\displaystyle=$ $\displaystyle
1-2(s_{L}^{u})^{2}-\frac{4}{3}s_{W}^{2},\quad\lambda_{Ztt}^{R}=-(s_{R}^{u})^{2}-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{Zbb}^{L}$ $\displaystyle=$
$\displaystyle-1+\frac{2}{3}s_{W}^{2},\quad\lambda_{Zbb}^{R}=\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZTT}^{L}$ $\displaystyle=$ $\displaystyle
1-2(c_{L}^{u})^{2}-\frac{4}{3}s_{W}^{2},\quad\lambda_{ZTT}^{R}=-(c_{R}^{u})^{2}-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZXX}^{L}$ $\displaystyle=$ $\displaystyle
1-\frac{10}{3}s_{W}^{2},\quad\lambda_{ZXX}^{R}=1-\frac{10}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZtT}^{L}$ $\displaystyle=$ $\displaystyle
2s_{L}^{u}c_{L}^{u},\quad\lambda_{ZtT}^{R}=c_{R}^{u}s_{R}^{u}.$ (59)

###  B.5 Vector-like doublet BY

In the case of the BY doublet model, the interaction of vector-like B and Y
quarks and the third generation SM quarks with the W boson are defined as:

$\displaystyle\lambda_{Wtb}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{d},\quad\lambda_{Wtb}^{R}=0,$ $\displaystyle\lambda_{WtB}^{L}$
$\displaystyle=$ $\displaystyle s_{L}^{d},\quad\lambda_{WtB}^{R}=0,$
$\displaystyle\lambda_{WbY}^{L}$ $\displaystyle=$ $\displaystyle-
s_{L}^{d},\quad\lambda_{WbY}^{R}=-s_{R}^{d},$ $\displaystyle\lambda_{WBY}^{L}$
$\displaystyle=$ $\displaystyle c_{L}^{d},\quad\lambda_{WBY}^{R}=c_{R}^{d}.$
(60)

The neutral current couplings to the Z boson for both the left- and right-
handed states are given by:

$\displaystyle\lambda_{Ztt}^{L}$ $\displaystyle=$ $\displaystyle
1-\frac{4}{3}s_{W}^{2},\quad\lambda_{Ztt}^{R}=-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{Zbb}^{L}$ $\displaystyle=$
$\displaystyle-1+2(s_{L}^{d})^{2}+\frac{2}{3}s_{W}^{2},\quad\lambda_{Zbb}^{R}=(s_{R}^{d})^{2}+\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZBB}^{L}$ $\displaystyle=$
$\displaystyle-1+2(c_{L}^{d})^{2}+\frac{2}{3}s_{W}^{2},\quad\lambda_{ZBB}^{R}=(c_{R}^{d})^{2}+\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZYY}^{L}$ $\displaystyle=$
$\displaystyle-1+\frac{8}{3}s_{W}^{2},\quad\lambda_{ZYY}^{R}=-1+\frac{8}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZbB}^{L}$ $\displaystyle=$
$\displaystyle-2s_{L}^{d}c_{L}^{d},\quad\quad\lambda_{ZbB}^{R}=-s_{R}^{d}c_{R}^{d}.$
(61)

###  B.6 Vector-like triplet XTB

In the framework of the XTB triplet model, the left- and right-handed charged
current couplings $\lambda_{Wq^{\prime}q}^{L,R}$ are expressed as follows:

$\displaystyle\lambda_{Wtb}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u}c_{L}^{d}+\sqrt{2}s_{L}^{u}s_{L}^{d},\quad\lambda_{Wtb}^{R}=\sqrt{2}s_{R}^{u}s_{R}^{d},$
$\displaystyle\lambda_{WTB}^{L}$ $\displaystyle=$ $\displaystyle
s_{L}^{u}s_{L}^{d}+\sqrt{2}c_{L}^{u}c_{L}^{d},\quad\lambda_{WTB}^{R}=\sqrt{2}c_{R}^{u}c_{R}^{d},$
$\displaystyle\lambda_{WtB}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u}s_{L}^{d}-\sqrt{2}s_{L}^{u}c_{L}^{d},\quad\lambda_{WtB}^{R}=-\sqrt{2}s_{R}^{u}c_{R}^{d},$
$\displaystyle\lambda_{WTb}^{L}$ $\displaystyle=$ $\displaystyle
s_{L}^{u}c_{L}^{d}-\sqrt{2}c_{L}^{u}s_{L}^{d},\quad\lambda_{WTb}^{R}=-\sqrt{2}c_{R}^{u}s_{R}^{d},$
$\displaystyle\lambda_{WtX}^{L}$ $\displaystyle=$
$\displaystyle-\sqrt{2}s_{L}^{u},\quad\lambda_{WtX}^{R}=-\sqrt{2}s_{R}^{u},$
$\displaystyle\lambda_{WXT}^{L}$ $\displaystyle=$
$\displaystyle\sqrt{2}c_{L}^{u},\quad\lambda_{WXT}^{R}=\sqrt{2}c_{R}^{u}.$
(62)

While, the left- and right-handed neutral current couplings
$\lambda^{L,R}_{Zf^{\prime}f}$ are defined as

$\displaystyle\lambda_{Ztt}^{L}$ $\displaystyle=$
$\displaystyle(c_{L}^{u})^{2}-\frac{4}{3}s_{W}^{2},\quad\lambda_{Ztt}^{R}=-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{Zbb}^{L}$ $\displaystyle=$
$\displaystyle-1-(s_{L}^{d})^{2}+\frac{2}{3}s_{W}^{2},\quad\lambda_{Zbb}^{R}=-2(s_{R}^{d})^{2}+\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZTT}^{L}$ $\displaystyle=$
$\displaystyle(s_{L}^{u})^{2}-\frac{4}{3}s_{W}^{2},\quad\lambda_{ZTT}^{R}=-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZBB}^{L}$ $\displaystyle=$
$\displaystyle-1-(c_{L}^{d})^{2}+\frac{2}{3}s_{W}^{2},\quad\lambda_{ZBB}^{R}=-2(c_{R}^{d})^{2}+\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZXX}^{L}$ $\displaystyle=$ $\displaystyle
2-+\frac{10}{3}s_{W}^{2},\quad\lambda_{ZXX}^{R}=2-+\frac{10}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZtT}^{L}$ $\displaystyle=$ $\displaystyle
s_{L}^{u}c_{L}^{u},\quad\quad\lambda_{ZtT}^{R}=0,$
$\displaystyle\lambda_{ZbB}^{L}$ $\displaystyle=$ $\displaystyle
s_{L}^{d}c_{L}^{d},\quad\quad\lambda_{ZbB}^{R}=2s_{R}^{d}c_{R}^{d}.$ (63)

###  B.7 Vector-like triplet TBY

In the TBY triplet model, the interactions between the vector-like T, B, and Y
quarks, as well as the third generation SM quarks, with the W boson are
defined as:

$\displaystyle\lambda_{Wtb}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u}c_{L}^{d}+\sqrt{2}s_{L}^{u}s_{L}^{d},\quad\lambda_{Wtb}^{R}=\sqrt{2}s_{R}^{u}s_{R}^{d},$
$\displaystyle\lambda_{WTB}^{L}$ $\displaystyle=$ $\displaystyle
s_{L}^{u}s_{L}^{d}+\sqrt{2}c_{L}^{u}c_{L}^{d},\quad\lambda_{WTB}^{R}=\sqrt{2}c_{R}^{u}c_{R}^{d},$
$\displaystyle\lambda_{WtB}^{L}$ $\displaystyle=$ $\displaystyle
c_{L}^{u}s_{L}^{d}-\sqrt{2}s_{L}^{u}c_{L}^{d},\quad\lambda_{WtB}^{R}=-\sqrt{2}s_{R}^{u}c_{R}^{d},$
$\displaystyle\lambda_{WTb}^{L}$ $\displaystyle=$ $\displaystyle
s_{L}^{u}c_{L}^{d}-\sqrt{2}c_{L}^{u}s_{L}^{d},\quad\lambda_{WTb}^{R}=-\sqrt{2}c_{R}^{u}s_{R}^{d},$
$\displaystyle\lambda_{WbY}^{L}$ $\displaystyle=$
$\displaystyle-\sqrt{2}s_{L}^{d},\quad\lambda_{WbY}^{R}=-\sqrt{2}s_{R}^{d},$
$\displaystyle\lambda_{WBY}^{L}$ $\displaystyle=$
$\displaystyle\sqrt{2}c_{L}^{d},\quad\lambda_{WBY}^{R}=\sqrt{2}c_{R}^{d}.$
(64)

The left- and right-handed neutral current couplings
$\lambda^{L,R}_{Zf^{\prime}f}$ in this model are given by:

$\displaystyle\lambda_{Ztt}^{L}$ $\displaystyle=$ $\displaystyle
1+(s_{L}^{u})^{2}-\frac{4}{3}s_{W}^{2},\quad\lambda_{Ztt}^{R}=2(s_{R}^{u})^{2}-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{Zbb}^{L}$ $\displaystyle=$
$\displaystyle-(c_{L}^{d})^{2}+\frac{2}{3}s_{W}^{2},\quad\lambda_{Zbb}^{R}=\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZTT}^{L}$ $\displaystyle=$ $\displaystyle
1+(c_{L}^{u})^{2}-\frac{4}{3}s_{W}^{2},\quad\lambda_{ZTT}^{R}=2(c_{R}^{u})^{2}-\frac{4}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZBB}^{L}$ $\displaystyle=$
$\displaystyle-(s_{L}^{d})^{2}+\frac{2}{3}s_{W}^{2},\quad\lambda_{ZBB}^{R}=\frac{2}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZYY}^{L}$ $\displaystyle=$
$\displaystyle-2+\frac{8}{3}s_{W}^{2},\quad\lambda_{ZYY}^{R}=-2+\frac{8}{3}s_{W}^{2},$
$\displaystyle\lambda_{ZtT}^{L}$ $\displaystyle=$ $\displaystyle-
s_{L}^{u}c_{L}^{u},\quad\quad\lambda_{ZtT}^{R}=-2s_{R}^{u}c_{R}^{u},$
$\displaystyle\lambda_{ZbB}^{L}$ $\displaystyle=$ $\displaystyle-
s_{L}^{d}c_{L}^{d},\quad\quad\lambda_{ZbB}^{R}=0.$ (65)

For simplification purposes and avoiding repetition of the global quantities
of the $(Vf^{\prime}f)$ couplings during the manipulation of the expressions
for the oblique parameters. We introduce a new definition coupling given by:

$c^{L,R}_{Zf^{\prime}f}=\frac{e}{2c_{W}s_{W}}\lambda^{L,R}_{Zf^{\prime}f},\text{
and
}c^{L,R}_{Wf^{\prime}f}=\frac{e}{\sqrt{2}s_{W}}\lambda^{L,R}_{Wf^{\prime}f}.$
(66)

Additionally, the couplings in the SM framework are given as

$\displaystyle c_{Wtb}^{L}$ $\displaystyle=$
$\displaystyle\frac{e}{\sqrt{2}s_{w}},\quad c_{Wtb}^{R}=0,$ $\displaystyle
c_{Ztt}^{L}$ $\displaystyle=$
$\displaystyle\frac{e}{2s_{w}c_{w}}(1-\frac{4}{3}s_{w}^{2}),\quad
c_{Ztt}^{R}=-\frac{2es_{w}}{3c_{w}},$ $\displaystyle c_{Zbb}^{L}$
$\displaystyle=$
$\displaystyle\frac{e}{2s_{w}c_{w}}(-1+\frac{2}{3}s_{w}^{2}),\quad
c_{Zbb}^{R}=\frac{es_{w}}{3c_{w}}.$ (67)

##  Appendix C The CDF-II W-boson mass anomaly within the SM+VLQs

In the following, we will emphasize the impact of the new heavy quarks on the
$W$-boson mass in the SM +VLQs models.

Figure 11: Scan result of the SM + (VLQs) in the $(T,M_{W})$. The light red
band indicates the $M_{W}$ value with the associated 1$\sigma$ uncertainty
measured recently by the CDF collaboration, while the light green band shows
the result of $\sin^{2}\theta_{\mathrm{eff}}$ and its associated 1$\sigma$
uncertainty measured by the SLD collaboration.

In Fig. 11, we depict the $W$ mass calculated using Eq.28 as a function of the
T parameter in the context of the SM + VLQs extensions. The oblique parameter
$S$ is displayed on the color map. The orange band represents the new $M_{W}$
value measured by the CDF collaboration, along with its $1\sigma$ uncertainty.
In all VLQs representations, the contribution of the T parameter required to
explain the CDF’s newly measured W-boson mass is positive. While the $S$
parameter is only positively restricted in the SM + T,B,TB and TBY extension,
it is negative in the SM + BY and XTB extension, and takes on both positive
and negative values in the SM + XT extension.

Figure 12: Scan result of the SM + (VLQs) in the
$(M_{W},\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}})$ plane. The light red
band indicates the $M_{W}$ value with the associated 1$\sigma$ uncertainty
measured recently by the CDF collaboration, while the light green band shows
the result of $\sin^{2}\theta_{\mathrm{eff}}$ and its associated 1$\sigma$
uncertainty measured by the SLD collaboration.

In Fig. 12 we present the same points in the
$(M_{W},\sin^{2}\theta^{\mathrm{LEP}}_{\mathrm{eff}})$ plane with the T
parameter in the color bar. The light blue band represents the measurement of
the SLD collaboration for $\sin^{2}\theta_{eff}$ within the $\pm 1\sigma$. It
is interesting to see that the points satisfying the CDF-II data are in good
agreement with the data points preferred by the SLD measurement of the
$\sin^{2}\theta_{eff}$ within 1 $\sigma$ uncertainty in the case of SM+(T, BY
and XTB) while the SM+(TB, XT and TBY) are only partially within the bands.
Furthermore, it can be seen that within the SM+B singlet, there is a
significant tension between the points preferred by the CDF-II data and the
SLD measurement.

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