Interpreting the $W$-mass anomaly in the vectorlike quark models

Junjie Cao, Lei Meng, Liangliang Shang, Shiyu Wang, Bingfang Yang

School of Physics, Henan Normal University, Xinxiang 453007, China

###### Abstract

The new measurement of $W$-boson mass by the CDF collaboration revealed a
remarkable $7\sigma$ disagreement with the Standard Model (SM) prediction. If
confirmed by other experiments, then the disagreement strongly indicates the
existence of new physics beyond the SM. In this work, seven vectorlike quark
(VLQ) extensions of the SM are investigated to interpret the anomaly, and it
is found that three can explain the anomaly in broad parameter space. The
explanations are consistent with the constraints from oblique parameters, the
LHC search for VLQs, the measurements of the properties for the top quark,
bottom quark, and Higgs boson, and the perturbativity criterion. The typical
size of the involved Yukawa coupling is around 1, which is comparable to the
top quark Yukawa coupling in the SM. The other extensions, however, either
predict a negative correction to the mass in reasonable parameter space or
explain the anomaly by unnatural theoretical input parameters.

##  1 INTRODUCTION

Recently, the CDF collaboration at Fermilab reported their measured $W$-boson
mass, $m_{W}^{CDF}=80.4335\pm 0.0094$ GeV[1], which deviates from the Standard
Model (SM) prediction $m_{W}^{SM}=80.357\pm 0.006$ GeV[2] by more than
$7\sigma$. Even if all known and unknown theoretically higher-order
corrections are included in the uncertainty estimation [3], there is still a
discrepancy of about $5\sigma$. Such a large discrepancy, if confirmed by
other measurements, strongly indicates the existence of new physics beyond the
SM (BSM). Thus, it has attracted considerable research attention [4, 5, 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, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52].

Among the solutions to the discrepancy, extending the SM with vectorlike
quarks (VLQs) is one of the most economic theoretical frameworks. Unlike the
chiral quarks in the SM, the left- and right-handed components of the VLQs
have the same transformation properties under the SM gauge group. As the VLQs
are non-chiral, their mass terms do not arise from the Yukawa coupling to the
SM Higgs field, which avoids the tight constraints on heavy fourth-generation
quarks set by the Higgs boson data of the Large Hadron Collider (LHC) [53,
54]. In addition, the vectorlike top quark partner among the VLQs was usually
designed to cancel the largest quadratic divergence in the Higgs mass, which
is caused by the top quark loop, so the fine-tuning problem of the SM can be
alleviated (see, e.g., Refs. [55, 56, 57]). Correspondingly, VLQs have been
widely adopted in building more complete models, such as the little Higgs [58,
59] and composite Higgs models [60, 61], to break the electroweak symmetry
naturally.

In the VLQ extensions, the VLQs may mix with the SM quarks to form mass
eigenstates and thereby modify their couplings to the $Z$, $W$, and Higgs
bosons. Confronted with the atomic parity violation experiments, the
electroweak precision observables (EWPOs) extracted from the large electron
positron experiments, and the measurements of various low-energy flavor-
conserving and flavor-violating processes, the VLQ mixings with the first two
generations of quarks have been tightly limited, and only mixings with third-
generation quarks may be sizable [62]. Fortunately, such a mixing pattern is
favored from the theoretical perspective: The large Yukawa coupling of the top
quark suggests a possible close connection of the top quark (and the left-
handed component of the bottom quark due to the weak isospin symmetry) with
any new physics related to the symmetry breaking and/or to the fermion mass
hierarchy, which distinguishes the third-generation quarks from other quarks
in the SM. It is emphasized that regardless of whether the VLQs couple
directly with the SM gauge bosons, such mixings can affect significantly
oblique parameters [63, 64] and the $W$-boson mass. The focus of this work is
to study the constraints of the oblique parameters on the VLQ extensions and
their prediction of the $W$-boson mass.

This paper is organized as follows. In Secs. II and III, seven VLQ extensions
and the calculation of the oblique parameters are introduced, respectively. In
Sec. IV, experimental and theoretical constraints on the extensions are
scrutinized. It is emphasized that the global fit of the EWPOs plays an
important role in this aspect. The capability of the VLQ extensions to explain
the $W$-mass discrepancy is investigated in Sec. V, and conclusions are
presented in Sec. VI.

##  2 THE VLQ EXTENSIONS

In this section, the key features of the VLQ extensions [65, 66, 67, 68] are
recapitulated. This brief introduction is restricted to the case where VLQ
multiplets have renormalizable couplings to the SM Higgs field and, for the
sake of simplicity, only one multiplet is involved in each extension. These
multiplets are categorized into seven types by their charges of the SM gauge
group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ in Table 1, where the
$SU(2)_{L}$ singlets $U$ and $D$, the doublets $Q_{1}$, $Q_{5}$, and $Q_{7}$,
and the triplets $T_{1}$ and $T_{2}$ are distinguished by their hypercharge
quantum number. Correspondingly, the component fields $T^{\prime}$,
$B^{\prime}$, $X$, and $Y$ carry the electric charge of $2/3$, $-1/3$, $5/3$,
and $-4/3$, respectively. Neglecting the small Yukawa couplings to the first
two-generation quarks, the Higgs field ($H$) has the following
interactions111In principle, the $Q_{1}$ multiplet may also couple with $H$ by
the form $\bar{Q}_{1}Hb_{R}$. To avoid unnecessary complexity, this
interaction is neglected in this work, and this does not change the obtained
conclusions much.:

$\displaystyle-\mathcal{L}_{H}$ $\displaystyle=$ $\displaystyle
Y_{t^{\prime}}\bar{q^{\prime}}_{L}\tilde{H}t_{R}^{\prime}+Y_{b^{\prime}}\bar{q^{\prime}}_{L}Hb_{R}^{\prime}+\xi_{U}\bar{U}\tilde{H}^{\dagger}q_{L}^{\prime}+\xi_{D}\bar{D}\tilde{H}^{\dagger}q_{L}^{\prime}+\xi_{Q_{1}}\bar{Q}_{1}\tilde{H}t_{R}^{\prime}+\xi_{Q_{5}}\bar{Q}_{5}\tilde{H}b_{R}^{\prime}$
(1)
$\displaystyle+\xi_{Q_{7}}\bar{Q}_{7}Ht_{R}^{\prime}+\frac{1}{2}\xi_{T_{1}}H^{\dagger}\tau\cdot\bar{T}_{1}q_{L}^{\prime}+\frac{1}{2}\xi_{T_{2}}\tilde{H}^{\dagger}\tau\cdot\bar{T}_{2}q_{L}^{\prime}+\text{
H.c. },$

where $q_{L}^{\prime}=(t^{\prime},b^{\prime})_{L}$ and
$t^{\prime}_{R}/b^{\prime}_{R}$ are the left-handed and right-handed third-
generation quark fields in the SM, respectively; $\tau$ denotes the Pauli
matrix; and $\tilde{H}\equiv{\it{i}}\tau_{2}H^{\ast}$. The coefficients
$Y_{i}$ ($i=t^{\prime},b^{\prime}$) and $\xi_{j}$ ($j=U$, $D$, $Q_{1}$,
$Q_{5}$, $Q_{7}$, $T_{1}$, $T_{2}$) parametrize the Yukawa coupling strength,
and only one of $\xi_{j}$ appears in this study of VLQ extensions.

VLQ multiplet | $U$ | $D$ | $Q_{1}$ | $Q_{5}$ | $Q_{7}$ | $T_{1}$ | $T_{2}$
---|---|---|---|---|---|---|---
Component fields | $T^{\prime}$ | $B^{\prime}$ | $(T^{\prime},B^{\prime})$ | $(B^{\prime},Y)$ | $(X,T^{\prime})$ | $(T^{\prime},B^{\prime},Y)$ | $(X,T^{\prime},B^{\prime})$
$SU(3)_{C}$ | 3 | 3 | 3 | 3 | 3 | 3 | 3
$SU(2)_{L}$ | 1 | 1 | 2 | 2 | 2 | 3 | 3
$U(1)_{Y}$ | $2/3$ | $-1/3$ | $1/6$ | $-5/6$ | $7/6$ | $-1/3$ | $2/3$
Table 1: Component fields of VLQ multiplets and their quantum number under the
SM gauge groups.

The interactions in Eq. (1) imply that the VLQs mix with the third-generation
quarks to form mass eigenstates. Assuming the presence of $T^{\prime}$ and
$B^{\prime}$ fields, the mass terms in the weak interaction bases
$\Psi^{t}\equiv(t^{\prime},T^{\prime})$ and
$\Psi^{b}\equiv(b^{\prime},B^{\prime})$ are given by

$\displaystyle-\mathcal{L}_{\rm
mass}=\sum_{i,j=1,2}\left(\bar{\Psi}_{Li}^{t}{\cal{M}}^{t}_{ij}\Psi^{t}_{Rj}+\bar{\Psi}_{Li}^{b}{\cal{M}}^{b}_{ij}\Psi^{b}_{Rj}\right)+\text{H.c.}$
(2) $\displaystyle=$
$\displaystyle\left(\\!\begin{array}[]{cc}\bar{t}^{\prime}_{L}&\bar{T}^{\prime}_{L}\end{array}\\!\right)\left(\\!\begin{array}[]{cc}Y_{t^{\prime}}\frac{v}{\sqrt{2}}&{\cal{M}}^{t}_{12}\\\
{\cal{M}}^{t}_{21}&M_{0}\end{array}\\!\right)\left(\\!\begin{array}[]{c}t^{\prime}_{R}\\\
T^{\prime}_{R}\end{array}\\!\right)+\left(\\!\begin{array}[]{cc}\bar{b}^{\prime}_{L}&\bar{B}^{\prime}_{L}\end{array}\\!\right)\left(\\!\begin{array}[]{cc}Y_{b^{\prime}}\frac{v}{\sqrt{2}}&{\cal{M}}^{b}_{12}\\\
{\cal{M}}^{b}_{21}&M_{0}\end{array}\\!\right)\left(\\!\begin{array}[]{c}b^{\prime}_{R}\\\
B^{\prime}_{R}\end{array}\\!\right)+\text{H.c.}.$ (13)

In this formula, the multiplet-universal bare mass $M_{0}$ is not related to
the Higgs mechanism of the SM. Instead, it may be generated by a Yukawa
coupling to a $SU(2)$-singlet scalar, such as the dilaton or radion field in
Refs. [69, 70], which acquires a vacuum expectation value much larger than
$246~{}{\rm GeV}$. This mechanism can also solve the vacuum stability problem
of the VLQ extensions and thus, it is of theoretical interest [71].

The mass matrices ${\cal{M}}^{t}$ and ${\cal{M}}^{b}$ in Eq. (2) can be
diagonalized by biunitary transformations:

$V^{q\dagger}_{L}\,{\cal{M}}^{q}\,V^{q}_{R}={\cal{M}}^{q}_{\text{diag}}={\rm
diag}(m_{q},m_{Q})\,,$ (14)

where $m_{q}$ and $m_{Q}$ are the masses of physical states with $(q,Q)=(t,T)$
or $(q,Q)=(b,B)$, respectively. Assuming no $CP$ violation in ${\cal{M}}^{q}$,
the $2\times 2$ unitary matrices $V_{L}^{q}$ and $V_{R}^{q}$ are parametrized
by one mixing angle, respectively. That is

$\displaystyle
V_{L,R}^{q}\equiv\left(\\!\begin{array}[]{cc}\cos\theta^{q}_{L,R}&\sin\theta^{q}_{L,R}\\\
-\sin\theta^{q}_{L,R}&\cos\theta^{q}_{L,R}\end{array}\\!\right).$ (17)

The following equations are obtained in the diagonalization:

$\displaystyle{\cal{M}}_{12}^{q}$ $\displaystyle=$ $\displaystyle-
s_{R}^{q}c_{L}^{q}m_{q}+s_{L}^{q}c_{R}^{q}m_{Q},\quad{\cal{M}}_{21}^{q}=-s_{L}^{q}c_{R}^{q}m_{q}+c_{L}^{q}s_{R}^{q}m_{Q},$
(18) $\displaystyle M_{0}$ $\displaystyle=$ $\displaystyle
s_{L}^{q}s_{R}^{q}m_{q}+c_{L}^{q}c_{R}^{q}m_{Q},$
$\displaystyle({\cal{M}}_{21}^{q})^{2}+M_{0}^{2}$ $\displaystyle=$
$\displaystyle(s_{L}^{q})^{2}m_{q}^{2}+(c_{L}^{q})^{2}m_{Q}^{2},\quad({\cal{M}}_{12}^{q})^{2}+M_{0}^{2}=(s_{R}^{q})^{2}m_{q}^{2}+(c_{R}^{q})^{2}m_{Q}^{2},$
$\displaystyle\frac{Y_{q^{\prime}}v}{\sqrt{2}}{\cal{M}}_{21}^{q}+M_{0}{\cal{M}}_{12}^{q}$
$\displaystyle=$ $\displaystyle
s_{L}^{q}c_{L}^{q}\left(m_{Q}^{2}-m_{q}^{2}\right),\quad\frac{Y_{q^{\prime}}v}{\sqrt{2}}{\cal{M}}_{12}^{q}+M_{0}{\cal{M}}_{21}^{q}=s_{R}^{q}c_{R}^{q}\left(m_{Q}^{2}-m_{q}^{2}\right),$

where $s_{L,R}^{q}\equiv\sin\theta_{L,R}^{q}$ and
$c_{L,R}^{q}\equiv\cos\theta_{L,R}^{q}$. These equations imply that $m_{b}$,
$m_{t}$, and one of the four mixing angles can replace the Yukawa couplings of
$Y_{b^{\prime}}$, $Y_{t^{\prime}}$, and $\xi_{j}$ as theoretical inputs, and
$m_{T}$ or $m_{B}$ substitutes $M_{0}$ as an input. They decide the other
mixing angles and masses.

Specifically, the VLQ extensions have the following distinct characteristics
if the bottom quark mass is consistently neglected:

  1. 1.

$U$ extension: There is no $B^{\prime}$ field, and

$\displaystyle{\cal{M}}_{12}^{t}$ $\displaystyle=$
$\displaystyle\frac{\xi_{U}v}{\sqrt{2}},\quad{\cal{M}}_{21}^{t}=0,\quad\tan\theta^{t}_{R}=\frac{m_{t}}{m_{T}}\tan\theta_{L}^{t},\quad
M_{0}^{2}=(s_{L}^{t})^{2}m_{t}^{2}+(c_{L}^{t})^{2}m_{T}^{2}.$

  2. 2.

$D$ extension: There is no $T^{\prime}$ field, and

$\displaystyle{\cal{M}}_{12}^{b}$ $\displaystyle=$
$\displaystyle\frac{\xi_{D}v}{\sqrt{2}},\quad{\cal{M}}_{21}^{b}=0,\quad\tan\theta^{b}_{R}=0,\quad
M_{0}^{2}=(c_{L}^{b})^{2}m_{B}^{2}.$

  3. 3.

$Q_{1}$ extension:

$\displaystyle{\cal{M}}_{12}^{t}$ $\displaystyle=$ $\displaystyle
0,\quad{\cal{M}}_{21}^{t}=\frac{\xi_{Q_{1}}v}{\sqrt{2}},\quad{\cal{M}}_{12}^{b}={\cal{M}}_{21}^{b}=0,\quad\theta_{L}^{b}=\theta_{R}^{b}=0,$
$\displaystyle\tan\theta^{t}_{L}$ $\displaystyle=$
$\displaystyle\frac{m_{t}}{m_{T}}\tan\theta_{R}^{t},\quad
m_{B}^{2}=M_{0}^{2}=(s_{R}^{t})^{2}m_{t}^{2}+(c_{R}^{t})^{2}m_{T}^{2}.$

  4. 4.

$Q_{5}$ extension: There is no $T^{\prime}$ field, and

$\displaystyle{\cal{M}}_{12}^{b}$ $\displaystyle=$ $\displaystyle
0,\quad{\cal{M}}_{21}^{b}=\frac{\xi_{Q_{5}}v}{\sqrt{2}},\quad\tan\theta^{b}_{L}=0,\quad
m_{Y}^{2}=M_{0}^{2}=(c_{R}^{b})^{2}m_{B}^{2}.$

  5. 5.

$Q_{7}$ extension: There is no $B^{\prime}$ field, and

$\displaystyle{\cal{M}}_{12}^{t}$ $\displaystyle=$ $\displaystyle
0,\quad{\cal{M}}_{21}^{t}=\frac{\xi_{Q_{7}}v}{\sqrt{2}},\quad\tan\theta^{t}_{L}=\frac{m_{t}}{m_{T}}\tan\theta_{R}^{t},\quad
m_{X}^{2}=M_{0}^{2}=(s_{R}^{t})^{2}m_{t}^{2}+(c_{R}^{t})^{2}m_{T}^{2}.$

  6. 6.

$T_{1}$ extension:

$\displaystyle{\cal{M}}_{12}^{t}$ $\displaystyle=$
$\displaystyle\xi_{T_{1}}v,\quad{\cal{M}}_{21}^{t}=0,\quad{\cal{M}}_{12}^{b}=\frac{\xi_{T_{1}}v}{\sqrt{2}},\quad{\cal{M}}_{21}^{b}=0,$
$\displaystyle\tan\theta^{t}_{R}$ $\displaystyle=$
$\displaystyle\frac{m_{t}}{m_{T}}\tan\theta_{L}^{t},\quad\theta_{R}^{b}=0,\quad
m_{Y}^{2}=M_{0}^{2}=(s_{L}^{t})^{2}m_{t}^{2}+(c_{L}^{t})^{2}m_{T}^{2},$
$\displaystyle m_{B}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(c_{L}^{t})^{2}m_{T}^{2}+(c_{R}^{t})^{2}m_{T}^{2}+(s_{L}^{t})^{2}m_{t}^{2}+(s_{R}^{t})^{2}m_{t}^{2}\right],\quad
c_{L}^{b}=\frac{m_{Y}}{m_{B}}.$

  7. 7.

$T_{2}$ extension:

$\displaystyle{\cal{M}}_{12}^{t}$ $\displaystyle=$
$\displaystyle\frac{\xi_{T_{2}}v}{\sqrt{2}},\quad{\cal{M}}_{21}^{t}=0,\quad{\cal{M}}_{12}^{b}=\xi_{T_{2}}v,\quad{\cal{M}}_{21}^{b}=0,$
$\displaystyle\tan\theta^{t}_{R}$ $\displaystyle=$
$\displaystyle\frac{m_{t}}{m_{T}}\tan\theta_{L}^{t},\quad\theta_{R}^{b}=0,\quad
m_{X}^{2}=M_{0}^{2}=(s_{L}^{t})^{2}m_{t}^{2}+(c_{L}^{t})^{2}m_{T}^{2},$
$\displaystyle m_{B}^{2}$ $\displaystyle=$ $\displaystyle
2(c_{R}^{t})^{2}m_{T}^{2}-(c_{L}^{t})^{2}m_{T}^{2}+2(s_{R}^{t})^{2}m_{t}^{2}-(s_{L}^{t})^{2}m_{t}^{2},\quad
c_{L}^{b}=\frac{m_{X}}{m_{B}}.$

Note that the Yukawa interaction in Eq. (1) for the $T_{1}$ and $T_{2}$
extensions determines both the $t^{\prime}-T^{\prime}$ mixing and the
$b^{\prime}-B^{\prime}$ mixing simultaneously. As a result, the mixing angles
in the top and bottom sectors are related by
$s_{L}^{t}\simeq\sqrt{2}s_{L}^{b}$ for the $T_{1}$ extension and
$s_{L}^{t}\simeq s_{L}^{b}/\sqrt{2}$ for the $T_{2}$ extension. Given that a
sizable $s_{L}^{b}$ can significantly alter the $Zb\bar{b}$ couplings of the
SM, the mixings are tightly limited (see the discussion later).

In the following, $s_{L}^{t}$ and $m_{T}$ for the $U$, $T_{1}$, and $T_{2}$
extensions, $s_{L}^{b}$ and $m_{B}$ for the $D$ extension, $s_{R}^{t}$ and
$m_{T}$ for the $Q_{1}$ and $Q_{7}$ extensions, and $s_{R}^{b}$ and $m_{B}$
for the $Q_{5}$ extension are used as theoretical inputs to study the impacts
of the VLQ extensions on the oblique parameters and $W$-boson mass.

##  3 OBLIQUE PARAMETERS

In the VLQ extensions, the physical states $q$ and $Q$ contribute to the
transverse component of the vacuum polarization for the gauge bosons in the SM
through loop Feynman diagrams. This effect is formulated as follows [72]

$\displaystyle\Sigma_{V^{\prime}V}\left(p^{2}\right)$ $\displaystyle=$
$\displaystyle\sum_{i,j=q,Q}\frac{2}{16\pi^{2}}\left\\{(g_{L}^{\overline{\psi}_{j}\psi_{i}V^{\prime}}g_{L}^{\overline{\psi}_{i}\psi_{j}V^{*}}+g_{R}^{\overline{\psi}_{j}\psi_{i}V^{\prime}}g_{R}^{\overline{\psi}_{i}\psi_{j}V^{*}})(2p^{2}B_{3}-B_{4})\left(p,m_{\psi_{i}},m_{\psi_{j}})\right.\right.$
(19)
$\displaystyle\left.+\left(g_{L}^{\bar{\psi}_{j}\psi_{i}V^{\prime}}g_{R}^{\bar{\psi}_{i}\psi_{j}V^{*}}+g_{R}^{\bar{\psi}_{j}\psi_{i}V^{\prime}}g_{L}^{\bar{\psi}_{i}\psi_{j}V^{*}}\right)m_{\psi_{i}}m_{\psi_{j}}B_{0}\left(p,m_{\psi_{i}},m_{\psi_{j}}\right)\right\\},$

where $p^{2}$ is the squared momentum of the incoming gauge boson, $\psi_{i}$
and $\psi_{j}$ denote the quark states entering the loop,
$g_{L}^{\bar{\psi}_{j}\psi_{i}V}$ and $g_{R}^{\bar{\psi}_{j}\psi_{i}V}$ are
their chiral couplings to the vector boson $V$ presented in the Appendix of
this work, and contributions from different configurations of the states are
summed.222Note that in this work, $\Sigma_{V^{\prime}V}$ is defined as the
coefficient of the $-g_{\mu\nu}$ term in the vacuum-polarization tensor of the
gauge bosons [73, 74]. It differs from the quantity $A_{VV^{\prime}}$ defined
in Refs. [63, 64] by a minus sign. The loop functions $B_{3}$ and $B_{4}$ are
related to the standard loop functions $B_{1}$ and $B_{21}$ by [72]

$\displaystyle B_{3}\left(p,m_{1},m_{2}\right)$ $\displaystyle=$
$\displaystyle-
B_{1}\left(p,m_{1},m_{2}\right)-B_{21}\left(p,m_{1},m_{2}\right),$
$\displaystyle B_{4}\left(p,m_{1},m_{2}\right)$ $\displaystyle=$
$\displaystyle-
m_{1}^{2}B_{1}\left(p,m_{2},m_{1}\right)-m_{2}^{2}B_{1}\left(p,m_{1},m_{2}\right).$
(20)

In addition, the singlet scalar field responsible for the bare mass $M_{0}$ in
Eq. (2) can contribute to the vacuum polarization. Such a contribution,
however, is induced by the mixing of the scalar and the SM Higgs and usually
neglected. This is because the LHC Higgs data as well as the vacuum stability
constrain the mixing to make it small (see the studies in Refs. [69, 70] and
Ref. [71], respectively), and also because the scalar-mediated loops are
usually less important than fermionic loops in contributing to the vacuum
polarization.

At the point $p^{2}=0$, $\Sigma_{V^{\prime}V}$ in Eq.(19) can be simplified as
follows

$\displaystyle\Sigma_{V^{\prime}V}\left(0\right)$ $\displaystyle=$
$\displaystyle\sum_{i,j}\frac{2}{16\pi^{2}}\left\\{(g_{L}^{\overline{\psi}_{j}\psi_{i}V^{\prime}}g_{L}^{\overline{\psi}_{i}\psi_{j}V^{*}}+g_{R}^{\overline{\psi}_{j}\psi_{i}V^{\prime}}g_{R}^{\overline{\psi}_{i}\psi_{j}V^{*}})F_{1}\left(m_{\psi_{i}},m_{\psi_{j}}\right)\right.$
(21)
$\displaystyle\left.+\left(g_{L}^{\bar{\psi}_{j}\psi_{i}V^{\prime}}g_{R}^{\bar{\psi}_{i}\psi_{j}V^{*}}+g_{R}^{\bar{\psi}_{j}\psi_{i}V^{\prime}}g_{L}^{\bar{\psi}_{i}\psi_{j}V^{*}}\right)m_{\psi_{i}}m_{\psi_{j}}F_{2}\left(m_{\psi_{i}},m_{\psi_{j}}\right)\right\\},$
$\displaystyle\frac{\partial\Sigma_{V^{\prime}V}\left(p^{2}\right)}{\partial
p^{2}}|_{p^{2}=0}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\frac{2}{16\pi^{2}}\left\\{(g_{L}^{\overline{\psi}_{j}\psi_{i}V^{\prime}}g_{L}^{\overline{\psi}_{i}\psi_{j}V^{*}}+g_{R}^{\overline{\psi}_{j}\psi_{i}V^{\prime}}g_{R}^{\overline{\psi}_{i}\psi_{j}V^{*}})F_{3}\left(m_{\psi_{i}},m_{\psi_{j}})\right.\right.$
(22)
$\displaystyle\left.+\left(g_{L}^{\bar{\psi}_{j}\psi_{i}V^{\prime}}g_{R}^{\bar{\psi}_{i}\psi_{j}V^{*}}+g_{R}^{\bar{\psi}_{j}\psi_{i}V^{\prime}}g_{L}^{\bar{\psi}_{i}\psi_{j}V^{*}}\right)m_{\psi_{i}}m_{\psi_{j}}F_{4}\left(m_{\psi_{i}},m_{\psi_{j}}\right)\right\\},$

where

$\displaystyle F_{1}\left(m_{1},m_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{m_{1}^{2}}{2}\ln\frac{m_{2}^{2}}{\mu^{2}}+\frac{m_{2}^{2}}{2}\ln\frac{m_{1}^{2}}{\mu^{2}}-\frac{m_{1}^{2}+m_{2}^{2}}{4}-\frac{m_{1}^{2}-m_{2}^{2}}{2}\ln\frac{m_{2}^{2}}{m_{1}^{2}}-\frac{m_{1}^{2}m_{2}^{2}}{2\left(m_{1}^{2}-m_{2}^{2}\right)}\ln\frac{m_{2}^{2}}{m_{1}^{2}},$
$\displaystyle F_{2}\left(m_{1},m_{2}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\ln\frac{m_{1}^{2}}{\mu^{2}}+\ln\frac{m_{2}^{2}}{\mu^{2}}\right)+1+\frac{m_{1}^{2}+m_{2}^{2}}{2\left(m_{1}^{2}-m_{2}^{2}\right)}\ln\frac{m_{2}^{2}}{m_{1}^{2}},$
$\displaystyle F_{3}\left(m_{1},m_{2}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\left(\ln\frac{m_{1}^{2}}{\mu^{2}}+\ln\frac{m_{2}^{2}}{\mu^{2}}\right)+\frac{1}{9}-\frac{2m_{1}^{2}m_{2}^{2}}{3\left(m_{1}^{2}-m_{2}^{2}\right)^{2}}$
$\displaystyle+\frac{m_{1}^{6}+m_{2}^{6}-3m_{1}^{2}m_{2}^{2}\left(m_{1}^{2}+m_{2}^{2}\right)}{6\left(m_{1}^{2}-m_{2}^{2}\right)^{3}}\ln\frac{m_{2}^{2}}{m_{1}^{2}},$
$\displaystyle F_{4}\left(m_{1},m_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{m_{1}^{4}}{2\left(m_{1}^{2}-m_{2}^{2}\right)^{3}}-\frac{m_{2}^{4}}{2\left(m_{1}^{2}-m_{2}^{2}\right)^{3}}+\frac{m_{1}^{2}m_{2}^{2}}{\left(m_{1}^{2}-m_{2}^{2}\right)^{3}}\ln\frac{m_{2}^{2}}{m_{1}^{2}},$
(23)

with $\mu$ denoting the renormalization scale. Note that $F_{i}(m_{1},m_{2})$
is symmetric under the exchange of $m_{1}$ and $m_{2}$, and if $m_{2}\simeq
m_{1}$, then they can be approximated by

$\displaystyle F_{1}(m_{1},m_{2})$ $\displaystyle\simeq$
$\displaystyle\frac{m_{1}^{2}}{2}\ln\frac{m_{2}^{2}}{\mu^{2}}+\frac{m_{2}^{2}}{2}\ln\frac{m_{1}^{2}}{\mu^{2}}+\frac{5}{12}\frac{(m_{2}^{2}-m_{1}^{2})^{2}}{m_{1}^{2}},$
$\displaystyle F_{2}(m_{1},m_{2})$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{2}\left(\ln\frac{m_{1}^{2}}{\mu^{2}}+\ln\frac{m_{2}^{2}}{\mu^{2}}\right)-\frac{1}{12}\frac{(m_{2}^{2}-m_{1}^{2})^{2}}{m_{1}^{4}},$
$\displaystyle F_{3}(m_{1},m_{2})$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{6}\left(\ln\frac{m_{1}^{2}}{\mu^{2}}+\ln\frac{m_{2}^{2}}{\mu^{2}}\right)-\frac{1}{6},\quad
F_{4}(m_{1},m_{2})\simeq\frac{1}{4m_{1}^{2}}-\frac{m_{2}^{2}}{12m_{1}^{4}},$

when $\mu$ is set at the electroweak scale.

The oblique parameters $S$, $T$, and $U$ are defined in terms of the weak
isospin current $J^{\mu}_{1,2,3}$ and the electromagnetic current
$J^{\mu}_{Q}=J^{\mu}_{3}+J^{\mu}_{Y}$ by their vacuum-polarization amplitudes
[63, 64]

$\displaystyle S$ $\displaystyle\equiv$
$\displaystyle-\frac{16\pi}{m_{Z}^{2}}\left\\{\Sigma_{33}(m_{Z}^{2})-\Sigma_{33}(0)-\Sigma_{3Q}(m_{Z}^{2})\right\\}$
(24) $\displaystyle=$
$\displaystyle\frac{16\pi}{m_{Z}^{2}}\left\\{\Sigma_{3Y}(m_{Z}^{2})-\Sigma_{3Y}(0)\right\\},$
$\displaystyle T$ $\displaystyle\equiv$
$\displaystyle\frac{4\pi}{s_{W}^{2}c_{W}^{2}m_{Z}^{2}}\left\\{\Sigma_{33}(0)-\Sigma_{11}(0)\right\\},$
(25) $\displaystyle U$ $\displaystyle\equiv$
$\displaystyle\frac{16\pi}{m_{Z}^{2}}\left\\{\Sigma_{33}(m_{Z}^{2})-\Sigma_{33}(0)\right\\}-\frac{16\pi}{m_{W}^{2}}\left\\{\Sigma_{11}(m_{Z}^{2})-\Sigma_{11}(0)\right\\},$
(26)

where $s_{W}\equiv\sin\theta_{W}$ and $c_{W}\equiv\cos\theta_{W}$ are the sine
and cosine, respectively, of the weak mixing angle $\theta_{W}$, and $m_{W}$
and $m_{Z}$ denote the mass for $W$ and $Z$ boson, respectively. Evidently,
$T$ and $U$ receive nonzero contributions from the violation of the weak
isospin, and they are finite because of the weak isospin symmetric nature of
the divergence terms. $S$ originates from the mixing between the weak
hypercharge and the third component of the weak isospin, which results from
the spontaneous symmetry breakdown. It involves only soft operators and thus
possesses no divergences [63, 64, 75]. Note that the above definitions are
complete in the sense of including contributions from the SM and any possible
new physics. As the vacuum-polarization amplitudes $\Sigma_{ij}$
($i,j=1,2,3,Q,Y$) receive contributions from different sources additively at
the one-loop level, alternative $S$, $T$, and $U$ can be defined by Eqs.
(24-26), respectively, with $\Sigma^{\rm new}(p^{2})\equiv\Sigma^{\rm
NP}(p^{2})-\Sigma^{\rm SM}(p^{2})$. In this case, $S$, $T$, and $U$ only
contain new physics effects.

Given that the $Z$-boson current is equal to
$\frac{e}{s_{W}c_{W}}(J^{\mu}_{3}-s_{W}^{2}J^{\mu}_{Q})$, with $e$ related to
the fine-structure constant $\alpha$ by $e^{2}\equiv 4\pi\alpha$, the oblique
parameters can be reexpressed by the vacuum polarizations of the SM gauge
bosons as

$\displaystyle\frac{\alpha}{4s_{W}^{2}c_{W}^{2}}\,S$ $\displaystyle=$
$\displaystyle-\frac{\Sigma_{ZZ}^{\rm
new}\left(m_{Z}^{2}\right)-\Sigma_{ZZ}^{\rm
new}\left(0\right)}{m_{Z}^{2}}+\left.\frac{\partial\Sigma_{\gamma\gamma}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}\right|_{p^{2}=0}+\frac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}\left.\frac{\partial\Sigma_{\gamma
Z}^{\rm new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0}$ (27)
$\displaystyle\simeq$ $\displaystyle\left.-\frac{\partial\Sigma_{ZZ}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}\right|_{p^{2}=0}+\left.\frac{\partial\Sigma_{\gamma\gamma}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}\right|_{p^{2}=0}+\frac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}\left.\frac{\partial\Sigma_{\gamma
Z}^{\rm new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0},$
$\displaystyle\alpha T$ $\displaystyle=$ $\displaystyle\frac{\Sigma_{ZZ}^{\rm
new}\left(0\right)}{m_{Z}^{2}}-\frac{\Sigma_{WW}^{\rm
new}\left(0\right)}{m_{W}^{2}},$ (28)
$\displaystyle\frac{\alpha}{4s_{W}^{2}}\,U$ $\displaystyle=$
$\displaystyle-\frac{\Sigma_{WW}^{\rm
new}\left(m_{W}^{2}\right)-\Sigma_{WW}^{\rm
new}\left(0\right)}{m_{W}^{2}}+c_{W}^{2}\,\frac{\Sigma_{ZZ}^{\rm
new}\left(m_{Z}^{2}\right)-\Sigma_{ZZ}^{\rm new}\left(0\right)}{m_{Z}^{2}}$
(29) $\displaystyle+s_{W}^{2}\left.\frac{\partial\Sigma_{\gamma\gamma}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}\right|_{p^{2}=0}+2c_{W}s_{W}\left.\frac{\partial\Sigma_{\gamma Z}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0}$
$\displaystyle\simeq$ $\displaystyle-\frac{\partial\Sigma_{WW}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}|_{p^{2}=0}+c_{W}^{2}\frac{\partial\Sigma_{ZZ}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}|_{p^{2}=0}+s_{W}^{2}\frac{\partial\Sigma_{\gamma\gamma}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}|_{p^{2}=0}$
$\displaystyle+2c_{W}s_{W}\frac{\partial\Sigma_{\gamma Z}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}|_{p^{2}=0}.$

In this study, the latter formulas are used in the calculation of the oblique
parameters. We check that the specific forms of the quark mass matrix listed
at the end of the last section play a vital role in canceling out the UV
divergence of the loop functions. As a result, the oblique parameters are free
of the divergence. With the aid of the oblique parameters, the new physics
correction to $W$-boson mass is given by [73, 74, 76, 77]

$\displaystyle\frac{\delta
m_{W}}{m_{W}}=\frac{\alpha}{2(c_{W}^{2}-s_{W}^{2})}\left(-\frac{1}{2}S+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right).$
(30)

##  4 EXPERIMENTAL AND THEORETICAL CONSTRAINTS

So far, the VLQ extensions have been restricted nontrivially by collider data
and theoretical preferences. In attempting to interpret the mass anomaly, the
following constraints should be considered.

###  4.1 Constraints from the EWPOs

The theories under study affect the EWPOs not only via the vacuum polarization
of the gauge bosons, which is parametrized by the oblique parameters, but also
by modifying the $Zb\bar{b}$ couplings at the tree level by the
$b^{\prime}-B^{\prime}$ mixing. A systematic study of the EWPOs’ constraints
includes formulating the EWPOs as linear functions of $S$, $T$, $U$, $\delta
g_{L}^{b}$, and $\delta g_{R}^{b}$ [78], where $\delta g_{L}^{b}$ and $\delta
g_{R}^{b}$ denote the correction to the $Zb_{L}\bar{b}_{L}$ and
$Zb_{R}\bar{b}_{R}$ couplings, respectively, and sequentially fitting them to
low-energy experimental measurements [79]. It is found that the correlation
between the oblique parameters and $\delta g_{L,R}^{b}$ is much weaker than
those among the oblique parameters and between $\delta g_{L}^{b}$ and $\delta
g_{R}^{b}$ [79]. This conclusion implies that as a good approximation, one may
fit separately the oblique parameters to the EWPOs and $\delta g_{L}^{b}$ and
$\delta g_{R}^{b}$ to the measurements $R_{b}^{\rm exp}$, $A_{\rm FB}^{b,\rm
exp}$, $A_{b}^{\rm exp}$ and $R_{c}^{\rm exp}$ in implementing the
constraints. This strategy was adopted in Ref. [65].

| $U$ | $D$ | $Q_{1}$ | $Q_{5}$ | $Q_{7}$ | $T_{1}$ | $T_{2}$
---|---|---|---|---|---|---|---
${\rm BR}(T\to bW)$ | 50% | $\cdots$ | $<1\%$ | $\cdots$ | $<1\%$ | $<1\%$ | 50%
${\rm BR}(T\to tZ)$ | 25% | $\cdots$ | 50% | $\cdots$ | 50% | 50% | 25%
${\rm BR}(T\to tH)$ | 25% | $\cdots$ | 50% | $\cdots$ | 50% | 50% | 25%
${\rm BR}(B\to tW)$ | $\cdots$ | 50% | 100% | $<1\%$ | $\cdots$ | 50% | $<1\%$
${\rm BR}(B\to bZ)$ | $\cdots$ | 25% | $<1\%$ | 50% | $\cdots$ | 25% | 50%
${\rm BR}(B\to bH)$ | $\cdots$ | 25% | $<1\%$ | 50% | $\cdots$ | 25% | 50%
Table 2: Approximate decay branching ratios of the states $T$ and $B$ in
different VLQ extensions with the Lagrangian given in Eq. (1).

In this study, results from the latest global fit of the oblique parameters in
Ref. [12] and the $Zb\bar{b}$ coupling fit in Ref. [65] are used to constrain
the VLQ theories. The former fit includes the recent CMS measurement of top
quark mass and the latest CDF measurement of $W$-boson mass to obtain the
preference of the oblique parameters. It considers the standard average
scenario and the conservative average scenario. The first scenario adopts the
averaged (or combined) value of $m_{t}$ and $m_{W}$ from different
experimental collaborations, $m_{t}^{\rm Cb}=171.79\pm 0.38~{}{\rm GeV}$ and
$m_{W}^{\rm Cb}=80.4133\pm 0.008~{}{\rm GeV}$, as the fit inputs. The second
scenario, however, considers the tensions among individual measurements of
$m_{t}$ and those of $m_{W}$ (e.g., the tensions between the measurement of
$m_{W}$ in Ref. [1] and Refs. [80, 81]), and thus adopts larger error bars,
that is, $m_{t}^{\rm Cb}=171.79\pm 1.0~{}{\rm GeV}$ and $m_{W}^{\rm
Cb}=80.4133\pm 0.015~{}{\rm GeV}$, in performing the fit. Correspondingly, the
standard average scenario imposes much stronger constraints than the
conservative average scenario [12]. In numerical calculation, the $\chi^{2}$
function for the oblique parameter fit is required to be less than 3.53 and
8.02 for three degrees of freedom to compute the $1\sigma$ and $2\sigma$
confidence level limits, respectively. Concerning the $Zb\bar{b}$ coupling
fit, note that there has been no significant improvement on the measurements
of $R_{b}$, $A_{\rm FB}^{b}$, $A_{b}$ and $R_{c}$ in recent years. Thus, the
$95\%$ confidence level bounds in Ref. [65] can be directly used in this
study. After comparing the constraints from the two separate fits, it is
concluded that the oblique fit limits the parameter space of the $U$, $Q_{1}$,
$Q_{5}$, and $Q_{7}$ extensions more tightly than the $Zb\bar{b}$ coupling
fit, and the situation is reversed for the $D$, $T_{1}$, and $T_{2}$
extensions. The underlying reason is that the $Zb\bar{b}$ couplings are
different from their SM prediction only at the loop level for the $U$,
$Q_{1}$, and $Q_{7}$ extensions, while they are modified at the tree level for
the $D$, $T_{1}$, and $T_{2}$ extensions. The situation of the $Q_{5}$
extension is somewhat subtle, in that it provides a positive correction to
$\delta g_{R}^{b}$ at the tree level, and such a $\delta g_{R}^{b}$ is favored
by experimental data. Consequently, the constraints from the $Zb\bar{b}$
coupling fit on the $Q_{5}$ extension is relaxed greatly [65].

###  4.2 The LHC search for VLQs

In the VLQ extensions, the VLQ-dominated state $Q$ decays into the
$t^{\prime}$\- and $b^{\prime}$-dominated states $t$ and $b$ by the following
channels: $T\to bW^{+},tZ,tH$, $B\to tW^{-},bZ,bH$, $X\to tW^{+}$, and $Y\to
bW^{-}$. Given the Lagrangian in Eq. (1), the decay branching ratios are
roughly fixed when $m_{Q}\gg v$ [65], and they are presented in Table 2. This
feature is utilized to search for $Q$ at the LHC with $\sqrt{s}=13~{}{\rm
TeV}$.

####  4.2.1 Single VLQ productions

The state $T$ may be singly produced at the LHC by the parton processes $qg\to
q^{\prime}(W^{+}b)\bar{b}\to q^{\prime}(T)\bar{b}$, and its cross section is
proportional to $\sin^{2}\theta_{L}^{t}$ for the $U$ extension. The following
analyses are conducted for this production:

  * •

Search for the decays $T\to tH\to(bW^{+})(b\bar{b})\to(bl^{+}\nu)(b\bar{b})$
and $T\to tZ\to(bl^{+}\nu)(q\bar{q})$ [82, 83]. Assuming
$\sin\theta_{L}^{t}\geq 0.5$ $(0.41)$, a lower bound of $m_{T}\geq 1.8{\rm
TeV}$ $(1.6{\rm TeV})$ is obtained, with an integrated luminosity of
$139~{}{\rm fb^{-1}}$.

  * •

Search for the decay $T\to tW^{+}\to(bl^{+}\nu)(q\bar{q}^{\prime})$ [84]. It
is found that $m_{T}\geq 800~{}{\rm GeV}$ for $\sin\theta_{L}^{t}=0.18$, with
an integrated luminosity of $36.1~{}{\rm fb^{-1}}$.

Note that these bounds cannot be applied to this study because, as indicated
below, $\sin\theta_{L}^{t}$ is tightly limited.

####  4.2.2 VLQ pair productions

The state $Q$ may also be pair produced at the LHC via the QCD process $gg\to
Q\bar{Q}$. The cross section of this process is independent of any mixing
angles even though it is suppressed by the phase space when $Q$ is massive.
This feature, together with the branching ratios in Table 2, enables the LHC
search results to be directly applied to this study.

Specifically, the following conclusions are drawn based on analyses of the
pair productions:

  * •

The lower bounds of $m_{T}\geq 1.31~{}{\rm TeV}$ for the $U$ extension,
$m_{B}\geq 1.22~{}{\rm TeV}$ for the $D$ extension, and $m_{T},m_{B}\geq
1.37~{}{\rm TeV}$ for the $Q_{1}$ extension are obtained after combining all
the decay channels of $T$ and $B$ with $36.1~{}{\rm fb^{-1}}$ integrated
luminosity [85].

  * •

The lower bounds of $m_{T}\geq 1.27~{}{\rm TeV}$ for the $U$ extension,
$m_{B}\geq 1.20~{}{\rm TeV}$ for the $D$ extension, $m_{T}\geq 1.46~{}{\rm
TeV}$ and $m_{B}\geq 1.32~{}{\rm TeV}$ for the $Q_{1}$ extension, and
$m_{T}\geq 1.46~{}{\rm TeV}$ for the $Q_{7}$ extension are obtained with
$139~{}{\rm fb^{-1}}$ integrated luminosity [86]. The analyses require one of
$T$ ($B$) to decay by $T\to tZ$ ($B\to bZ$) with $Z\to\ell\bar{\ell}$ and
concentrate on two topologies: exactly the two-lepton final state and at
least, the three-lepton final states.

  * •

The lower bounds of $m_{X}\geq 1.33~{}{\rm TeV}$ and $m_{X}\geq 1.30~{}{\rm
TeV}$ are set for the cases of a purely right-handed and left-handed coupling
to $W$ boson, respectively, with $35.9~{}{\rm fb^{-1}}$ integrated luminosity
[87]. The analyses require, at least, one of the four $W$ bosons from
$X\bar{X}\to tW^{+}\bar{t}W^{-}\to(2b)(4W)$ to decay leptonically and focus on
the single-lepton final state and same-sign dilepton final state.

Given the experimental status, a conservative bound of $m_{T},m_{B}\geq
1.27~{}{\rm TeV}$ is used to present the results of this work. It is
emphasized that a higher bound does not change the conclusions much.

###  4.3 Measurements of the top quark property

As shown in the Appendix, the couplings of the top quark to the SM gauge
bosons are modified at the tree level in the VLQ extensions, which affect top
quark properties such as the width and the single top production rate at the
LHC [65]. In the $U$ extension, the modification is proportional to
$\sin^{2}\theta_{L}^{t}$ in leading approximation and thus at the order of
$1\%$ for $\sin\theta_{L}^{t}\sim 0.1$, which is preferred to explain the mass
anomaly (see the following discussion). The modification is much below the
precision of current experimental measurements and has no impact on this
study. Similar conclusions are obtained for the other extensions.

###  4.4 The Higgs data collected at the LHC

The Lagrangian in Eq.(1) changes the Higgs couplings to the third-generation
quarks. It also introduces $H\bar{Q}Q$ interaction. In the $U$ extension, both
the modification of the $H\bar{t}t$ coupling from its SM prediction and the
new Yukawa coupling coefficient $Y_{T}$ are proportional to
$\sin^{2}\theta_{L}^{t}$ [65]. Consequently, the modifications of the
$H\gamma\gamma$ and $Hgg$ couplings are proportional to
$\sin^{2}\theta_{L}^{t}$, too [88]. For $\sin\theta_{L}^{t}\sim 0.1$, the
deviations of these couplings from their SM values are at the order of $1\%$
even when $m_{T}$ and $m_{t}$ are comparable. However, they only slightly
alter the result of the SM Higgs fit to the LHC data. This conclusion holds
for other extensions.

###  4.5 Perturbativity

In the VLQ extensions, the perturbativity of a theory at the electroweak scale
requires $\xi_{j}$ in Eq.(1) to satisfy $|\xi_{j}|\lesssim\sqrt{4\pi}$. This
condition sets an upper bound on the mixing angles. For example, $\tan
2\theta_{L}^{t}$ in the $U$ extension is given by [65]

$\displaystyle\tan 2\theta_{L}^{t}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}|\xi_{U}|vM_{0}}{M_{0}^{2}-y_{t^{\prime}}^{2}v^{2}/2-\xi_{U}^{2}v^{2}/2}.$
(31)

In the limit $M_{0}\gg v$, $m_{T}\simeq M_{0}$, and the perturbativity implies
that

$\displaystyle\sin\theta_{L}^{t}\lesssim\sqrt{2\pi}v/m_{T},\quad{\rm or\
equivalently,}\quad\sin\theta_{L}^{t}\lesssim 0.62\times\left(\frac{\rm
TeV}{m_{T}}\right).$ (32)

Similar requirements can be set for $\sin\theta_{L}^{b}$ in the $D$ extension,
$\sin\theta_{R}^{t}$ in the $Q_{1}$ and $Q_{7}$ extensions,
$\sin\theta_{R}^{b}$ in the $Q_{5}$ extension, and $\sin\theta_{L}^{t}$ in the
$T_{1}$ and $T_{2}$ extensions. As shown below, most of them are significantly
weaker than the oblique parameters in limiting the parameter space of the VLQ
theories for $m_{T}\lesssim 2~{}{\rm TeV}$.

In addition to the perturbativity, a moderately large $\xi_{j}$ can decrease
significantly the Higgs quartic coupling at high energies to exacerbate the
vacuum stability problem of the SM. Avoiding such a problem imposes special
requirements on the theory [71, 89]. This issue will be discussed at the end
of Sec. V.

##  5 $W$-BOSON MASS IN THE VLQ MODELS

In this section, the predictions of $W$-boson mass in the VLQ extensions are
presented. The oblique parameters are calculated to limit the parameter space
of the theories. In addition, other important supplementary constraints are
considered.

###  5.1 $U$ extension

Figure 1: Parameter space predicting the combined experimental value of
$m_{W}$ in the $U$ extension, which is projected onto the
$m_{T}-\sin\theta_{L}^{t}$ plane. The black contour corresponds to
$m_{W}=80.4133~{}{\rm GeV}$, the central value of $m_{W}^{\rm Cb}$, and the
regions in the green and yellow bands can explain $m_{W}^{\rm Cb}$ at the
$1\sigma$ and $2\sigma$ levels, respectively. Constraints from the oblique
parameters are also plotted, which are bound by the blue-dashed contours at
the $1\sigma$ confidence level and the red-dashed contours at the $2\sigma$
confidence level. In the left panel, the constraints are implemented in the
standard average scenario where $m_{t}^{\rm Cb}=171.39\pm 0.38~{}{\rm GeV}$
and $m_{W}^{\rm Cb}=80.4133\pm 0.008~{}{\rm GeV}$ were obtained, while the
right panel shows the results for the conservative average scenario where
different uncertainties of $m_{t}$ and $m_{W}$, i.e., $m_{t}^{\rm
Cb}=171.39\pm 1.0~{}{\rm GeV}$ and $m_{W}^{\rm Cb}=80.4133\pm 0.015~{}{\rm
GeV}$, were adopted [79]. Bounds from the LHC search for VLQs are represented
by the vertical black-dashed line. All of the observables are calculated by
the exact formulas listed in Sec. 3 even though the simple analytic
expressions in Eqs. (33-35) are good approximations. Note that there is no
$1\sigma$ region for the oblique parameters in the standard average scenario,
which is basically because of the specific theoretical structure of the VLQ
theory.

To understand the intrinsic physics of the $U$ extension, it is helpful to
obtain the leading contributions to the oblique parameters and $m_{W}$. Up to
the $\sin^{2}\theta_{L}^{t}$ order for the $S$ and $U$ parameters and the
$m_{t}^{2}\sin^{2}\theta_{L}^{t}$ order for the $T$ parameter, these
observables are approximated by

S $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left(\frac{2}{3}(s_{L}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}-\frac{5}{3}(s_{L}^{t})^{2}\right),\quad\text{U}\simeq\frac{N_{C}}{6\pi}\times\frac{5}{3}\times(s_{L}^{t})^{2},$
(33) T $\displaystyle\simeq$ $\displaystyle\frac{N_{C}m_{t}^{2}}{8\pi
s_{W}^{2}m_{W}^{2}}\left((s_{L}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}-(s_{L}^{t})^{2}+(s_{L}^{t})^{4}\frac{m_{T}^{2}}{2m_{t}^{2}}\right),$
(34) $\displaystyle\frac{\delta m_{W}}{m_{W}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha N_{C}}{288\pi
s_{W}^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}\times\frac{\left(s_{L}^{t}\right)^{2}}{m_{W}^{2}}\times\left\\{\left(18c_{W}^{2}m_{t}^{2}-8s_{W}^{2}m_{W}^{2}\right)\ln\frac{m_{T}^{2}}{m_{t}^{2}}\right.$
(35)
$\displaystyle\left.+10m_{W}^{2}-18c_{W}^{2}m_{t}^{2}+9\left(s_{L}^{t}\right)^{2}c_{W}^{2}m_{T}^{2}\right\\}$
$\displaystyle\simeq$ $\displaystyle
10^{-4}\times\left(\frac{s_{L}^{t}}{0.1}\right)^{2}\times\left\\{1.30\ln\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}+3.46+0.22\left(\frac{s_{L}^{t}}{0.1}\right)^{2}\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}\right\\},$

where $N_{C}=3$ is the color factor, and the latest combined top quark mass,
$m_{t}^{\rm Cb}=171.79~{}{\rm GeV}$ [79], is used to obtain the semianalytic
approximation of $\delta m_{W}/m_{W}$. These formulas reveal the following
features:

  * •

The VLQ contributions arise from the $t^{\prime}-T^{\prime}$ mixing, which is
reflected by the fact that they are all proportional to $(s_{L}^{t})^{2}$ at
the leading order. Evidently, the larger value the mixing takes, the more
significant the effects become.

  * •

The contributions contain both logarithmic and nonlogarithmic terms. The
logarithmic terms reflect the renormalization group running effect in the
effective Lagrangian framework, and they are associated with the divergence of
the vacuum-polarization diagrams. The nonlogarithmic terms, however, come from
the matching between the full theory and the effective theories below the
scales of $m_{T}$ and $m_{t}$. Computationally speaking, there are two sources
for the nonlogarithmic terms in the $U$ extension. One is from the $t$\- and
$b$-mediated loop contributions subtracted by their corresponding SM
predictions. The other is from the $W$ self-energy diagram induced by the
$W\bar{T}b$ interaction and the $Z$ self-energy diagram induced by the
$Z\bar{T}t$ interaction.

  * •

As indicated by Eq. (31), $s_{L}^{t}\simeq\sqrt{2}\xi_{U}v/(2m_{T})$, and thus
$(s_{L}^{t})^{4}m_{T}^{2}\simeq(s_{L}^{t})^{2}\xi_{U}^{2}v^{2}/2$. It is then
inferred that the $(s_{L}^{t})^{4}m_{T}^{2}/m_{t}^{2}$ term in the $T$
parameter is comparable in magnitude to the $(s_{L}^{t})^{2}$ term if
$\xi_{U}\sim 1$, and therefore, it cannot be neglected in the approximation.
Our numerical calculations verify this conclusion. In addition, it is verified
that the $(s_{L}^{t})^{4}m_{T}^{2}/m_{t}^{2}$ term comes from the
$Z\bar{T}t$-induced self-energy diagram of $Z$ boson.

  * •

Generally, the magnitude of the $T$ parameter is much larger than that of the
$S$ and $U$ parameters. So $\delta m_{W}$ is mainly contributed by the $T$
parameter.

In Fig. 1, the capability of the $U$ extension to explain the mass anomaly is
studied. Relevant parameter space is projected onto the
$m_{T}-\sin\theta_{L}^{t}$ plane. Constraints from the oblique parameters and
the LHC searches for VLQs are also implemented. It is found that the theory
can explain the combined experimental value of $m_{W}$ in broad parameter
space. The explanation is consistent with the constraints from the oblique
parameters at the $2\sigma$ level in the standard average scenario and at the
$1\sigma$ level in the conservative average scenario. In addition, given the
relation $\xi_{U}\simeq\sqrt{2}m_{T}s_{L}^{t}/v\simeq 5.75\times(m_{T}/{\rm
TeV})\times s_{L}^{t}$ and the results in Fig. 1, it is inferred
$\xi_{U}\simeq 1.3$ for $m_{T}=2~{}{\rm TeV}$ to predict the central value of
$m_{W}^{\rm Cb}$. Such a $\xi_{U}$ is consistent with the perturbativity
criterion. It is checked that $\delta m_{W}^{T}/\delta m_{W}\simeq 0.996$ for
this parameter point, where $\delta m_{W}^{T}$ denotes the $T$-term
contribution in Eq. (30).

Figure 2: Similar to Fig.1, but for the $D$ extension with the additional
$Zb\bar{b}$ coupling constraints imposing an upper bound on
$\sin\theta_{L}^{b}$, denoted by the blue-dashed lines. Since explaining the
$W$-boson mass anomaly requires a relatively large $\sin\theta_{L}^{b}$,
constraints from the perturbativity are also plotted, which are labeled by the
blue curves. In addition, given that the extension is hardly consistent with
the constraints from the oblique parameters at the $2\sigma$ level, only the
$3\sigma$ boundaries of the parameters are presented as red-dashed lines in
this figure.

###  5.2 $D$ extension

Similar to the $U$ extension, the oblique parameters and $m_{W}$ in the $D$
extension are approximated by

S $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left\\{\frac{4}{3}(s_{L}^{b})^{2}\ln\frac{m_{B}^{2}}{m_{b}^{2}}-\frac{5}{3}(s_{L}^{b})^{2}\right\\},$
(36) T $\displaystyle\simeq$ $\displaystyle\frac{N_{C}m_{t}^{2}}{8\pi
s_{W}^{2}m_{W}^{2}}\left(-(s_{L}^{b})^{2}\ln\frac{m_{B}^{2}}{m_{t}^{2}}+(s_{L}^{b})^{4}\frac{m_{B}^{2}}{2m_{t}^{2}}\right),$
(37) U $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left\\{-2(s_{L}^{b})^{2}\ln\frac{m_{t}^{2}}{m_{b}^{2}}+\frac{5}{3}(s_{L}^{b})^{2}\right\\},$
(38) $\displaystyle\frac{\delta m_{W}}{m_{W}}$ $\displaystyle\simeq$
$\displaystyle-\frac{\alpha N_{C}}{288\pi
s_{W}^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}\times\frac{\left(s_{L}^{b}\right)^{2}}{m_{W}^{2}}\times\left\\{\left[12\left(c_{W}^{2}-s_{W}^{2}\right)m_{W}^{2}-18c_{W}^{2}m_{t}^{2}\right]\ln\frac{m_{t}^{2}}{m_{b}^{2}}\right.$
(39)
$\displaystyle\left.+\left(16s_{W}^{2}m_{W}^{2}+18c_{W}^{2}m_{t}^{2}\right)\ln\frac{m_{B}^{2}}{m_{b}^{2}}-10m_{W}^{2}-9\left(s_{L}^{b}\right)^{2}c_{W}^{2}m_{B}^{2}\right\\}$
$\displaystyle\simeq$
$\displaystyle-10^{-4}\times\left(\frac{s_{L}^{b}}{0.1}\right)^{2}\times\left\\{1.42\ln\left(\frac{m_{B}}{{\rm
TeV}}\right)^{2}+6.36-0.23\left(\frac{s_{L}^{b}}{0.1}\right)^{2}\left(\frac{m_{B}}{{\rm
TeV}}\right)^{2}\right\\}$ $\displaystyle\simeq$
$\displaystyle-10^{-2}\times\frac{\xi_{D}^{2}}{4\pi}\times\left(\frac{{\rm
TeV}}{m_{B}}\right)^{2}\times\left\\{0.54\ln\left(\frac{m_{B}}{{\rm
TeV}}\right)^{2}+2.42-3.25\times\frac{\xi_{D}^{2}}{4\pi}\right\\}.$

In getting the last approximation of $\delta m_{W}/m_{W}$, the relation
$\sin\theta_{L}^{b}\simeq\sqrt{2}\xi_{D}v/(2m_{B})$ is used, and it shows that
the mass correction decreases as the state $B$ becomes heavy for fixed
$\xi_{D}$.

These approximations reveal the following facts:

  * •

For $s_{L}^{b}\lesssim 0.04$ required by the $Zb\bar{b}$ coupling measurements
[65], the mass correction is negative for $m_{B}\lesssim 20~{}{\rm TeV}$,
which can be learned by the first semianalytic form.

  * •

For $m_{B}\gtrsim 20~{}{\rm TeV}$ and $\xi_{D}^{2}\leq 4\pi$, the correction
remains negative in the second semianalytic form.

They reflect that the extension cannot explain the $W$-boson mass anomaly on
the premise of satisfying all the constraints.

In the following, the implications of the mass anomaly on the $D$ extension
are studied numerically. In Fig. 2, the prediction of $m_{W}$ in the $D$
extension along with various constraints are presented. It shows that,
although the extension can explain the anomaly in very narrow parameter space,
the explanation always conflicts with the constraints from $Zb\bar{b}$
couplings. It also conflicts with the oblique parameters for $m_{B}\lesssim
2~{}{\rm TeV}$ and the perturbativity for $m_{B}\gtrsim 2~{}{\rm TeV}$. These
features are consistent with the analyses of the approximations.

Figure 3: Similar to Fig.1 but for the $Q_{1}$ extension.

###  5.3 $Q_{1}$ extension

The oblique parameters and $m_{W}$ in the $Q_{1}$ extension are approximated
by

S $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left(\frac{4}{3}(s_{R}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}-\frac{7}{3}(s_{R}^{t})^{2}\right),\quad\text{U}\simeq\frac{N_{C}}{6\pi}(s_{R}^{t})^{2},$
(40) T $\displaystyle\simeq$ $\displaystyle\frac{N_{C}m_{t}^{2}}{8\pi
s_{W}^{2}m_{W}^{2}}\left(2(s_{R}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}-3(s_{R}^{t})^{2}+(s_{R}^{t})^{4}\frac{2m_{T}^{2}}{3m_{t}^{2}}\right),$
(41) $\displaystyle\frac{\delta m_{W}}{m_{W}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha N_{C}}{288\pi
s_{W}^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}\times\frac{\left(s_{R}^{t}\right)^{2}}{m_{W}^{2}}\times\left\\{\left(36c_{W}^{2}m_{t}^{2}-16s_{W}^{2}m_{W}^{2}\right)\ln\frac{m_{T}^{2}}{m_{t}^{2}}\right.$
(42)
$\displaystyle\left.+\left(6+16s_{W}^{2}\right)m_{W}^{2}-54c_{W}^{2}m_{t}^{2}+12\left(s_{R}^{t}\right)^{2}c_{W}^{2}m_{T}^{2}\right\\}$
$\displaystyle\simeq$ $\displaystyle
10^{-4}\times\left(\frac{s_{R}^{t}}{0.1}\right)^{2}\times\left\\{2.60\ln\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}+5.35+0.31\left(\frac{s_{R}^{t}}{0.1}\right)^{2}\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}\right\\}.$

In Fig. 3, the $W$-boson mass anomaly is investigated, and roughly the same
conclusions as those in Fig. 1 are drawn. A slight difference comes from the
fact that this model prefers a smaller mixing angle than the $U$ extension,
such as $s_{R}^{t}\simeq 0.095$ (or equivalently $\xi_{Q_{1}}\simeq 1.1$), for
$m_{T}=2~{}{\rm TeV}$ to predict the central value of $m_{W}^{\rm CDF}$. In
addition, it is checked that $\delta m_{W}^{T}/\delta m_{W}\simeq 1.02$ for
this parameter point.

Figure 4: Similar to Fig.1 but for the $Q_{5}$ extension. One new feature is
that the $Zb\bar{b}$ couplings can impose an upper bound on
$\sin\theta_{R}^{b}$ (the blue-dashed line), which is weaker than the oblique
parameters in limiting the theory for $m_{B}\gtrsim 1.6~{}{\rm TeV}$ in the
standard average scenario. Note that compared with the predictions of the $U$
and $Q_{1}$ extensions, the $2\sigma$ region for the oblique parameters in the
standard average scenario is narrowed, and the $1\sigma$ region in the
conservative average scenario is absent. These behaviors reflect that the
$Q_{5}$ extension is less compatible with the latest EWPOs.

###  5.4 $Q_{5}$ extension

The oblique parameters and $m_{W}$ in the $Q_{5}$ extension are approximated
by

S $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left\\{-\frac{2}{3}(s_{R}^{b})^{2}\ln\frac{m_{B}^{2}}{m_{b}^{2}}+\frac{11}{3}(s_{R}^{b})^{2}\right\\},\quad\text{U}\simeq-\frac{N_{C}}{6\pi}(s_{R}^{b})^{2},$
(43) T $\displaystyle\simeq$ $\displaystyle\frac{N_{C}m_{t}^{2}}{8\pi
s_{W}^{2}m_{W}^{2}}(s_{R}^{b})^{4}\frac{2m_{B}^{2}}{3m_{t}^{2}},$ (44)
$\displaystyle\frac{\delta m_{W}}{m_{W}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha N_{C}}{288\pi
s_{W}^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}\times\frac{\left(s_{R}^{b}\right)^{2}}{m_{W}^{2}}\times\left\\{8s_{W}^{2}m_{W}^{2}\ln\frac{m_{B}^{2}}{m_{b}^{2}}\right.$
(45)
$\displaystyle\left.-\left(6+32s_{W}^{2}\right)m_{W}^{2}+12\left(s_{R}^{b}\right)^{2}c_{W}^{2}m_{B}^{2}\right\\}$
$\displaystyle\simeq$ $\displaystyle
10^{-4}\times\left(\frac{s_{R}^{b}}{0.1}\right)^{2}\times\left\\{0.04\ln\left(\frac{m_{B}}{{\rm
TeV}}\right)^{2}+0.13+0.31\left(\frac{s_{R}^{b}}{0.1}\right)^{2}\left(\frac{m_{B}}{{\rm
TeV}}\right)^{2}\right\\}.$

Similar to the $U$ extension, its capability to interpret the anomaly is
studied in Fig. 4. This figure shows that $\sin\theta_{R}^{b}\simeq 0.165$, or
equivalently $\xi_{Q_{5}}\simeq 1.9$, for $m_{B}=2~{}{\rm TeV}$ is needed to
predict the central value of $m_{W}^{\rm Cb}$. In this case, $\delta
m_{W}^{T}/\delta m_{W}\simeq 0.93$. At this stage, it should be clarified that
$\xi_{Q_{5}}$ parametrizes the coupling of the $b_{R}^{\prime}$ field to VLQ
multiplets, and $\xi_{Q_{5}}>y_{t^{\prime}}\gg Y_{b^{\prime}}$ looks
unnatural. In fact, the possibility that $b_{R}^{\prime}$ couples strongly to
new physics lacks solid theoretical motivations in model building [90].

Figure 5: Similar to Fig.2 but for the $Q_{7}$ extension, the $Zb\bar{b}$
couplings are unable to set tight constraints due to the absence of the tree-
level $b^{\prime}-B^{\prime}$ mixing.

###  5.5 $Q_{7}$ extension

The oblique parameters and $m_{W}$ in the $Q_{7}$ extension are approximated
by

S $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left(-\frac{4}{3}(s_{R}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}+5(s_{R}^{t})^{2}\right),\quad\text{U}\simeq-\frac{N_{C}}{6\pi}(s_{R}^{t})^{2},$
(46) T $\displaystyle\simeq$ $\displaystyle\frac{N_{C}m_{t}^{2}}{8\pi
s_{W}^{2}m_{W}^{2}}\left(-2(s_{R}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}+3(s_{R}^{t})^{2}+(s_{R}^{t})^{4}\frac{2m_{T}^{2}}{3m_{t}^{2}}\right),$
(47) $\displaystyle\frac{\delta m_{W}}{m_{W}}$ $\displaystyle\simeq$
$\displaystyle-\frac{\alpha N_{C}}{288\pi
s_{W}^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}\times\frac{\left(s_{R}^{t}\right)^{2}}{m_{W}^{2}}\times\left\\{\left(36c_{W}^{2}m_{t}^{2}-16s_{W}^{2}m_{W}^{2}\right)\ln\frac{m_{T}^{2}}{m_{t}^{2}}\right.$
(48)
$\displaystyle\left.+\left(6+48s_{W}^{2}\right)m_{W}^{2}-54c_{W}^{2}m_{t}^{2}-12\left(s_{R}^{t}\right)^{2}c_{W}^{2}m_{T}^{2}\right\\}$
$\displaystyle\simeq$
$\displaystyle-10^{-4}\times\left(\frac{s_{R}^{t}}{0.1}\right)^{2}\times\left\\{2.61\ln\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}+5.50-0.31\left(\frac{s_{R}^{t}}{0.1}\right)^{2}\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}\right\\}$ $\displaystyle\simeq$
$\displaystyle-10^{-2}\times\frac{\xi_{Q_{7}}^{2}}{4\pi}\times\left(\frac{{\rm
TeV}}{m_{T}}\right)^{2}\times\left\\{0.99\ln\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}+2.09-4.40\times\frac{\xi_{Q_{7}}^{2}}{4\pi}\right\\}.$

Similar to the analyses of the $D$ extension, the correction to $m_{W}$ is
always negative for $s_{R}^{t}\lesssim 0.1$, and the $W$-boson mass anomaly
can be explained only in very narrow parameter space characterized by a large
$\sin\theta_{R}^{t}$. These features are shown in Fig. 5 where the prediction
of $m_{W}$ as well as different constraints are presented. It should be
pointed out that, although the explanation for $m_{T}\lesssim 2~{}{\rm GeV}$
is consistent with all the constraints, it is disfavored by the vacuum
stability discussed at the end of this section.

Figure 6: Similar to Fig.2 but for the $T_{1}$ extension, the perturbativity
is tighter in limiting the $m_{T}-\sin\theta_{L}^{t}$ plane.

###  5.6 $T_{1}$ extension

The oblique parameters and $m_{W}$ in the $T_{1}$ extension are approximated
by

S $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left\\{\frac{2}{3}(s_{L}^{t})^{2}\left(\ln\frac{m_{B}^{2}}{m_{b}^{2}}-\ln\frac{m_{T}^{2}}{m_{t}^{2}}\right)+\frac{13}{6}(s_{L}^{t})^{2}\right\\}$
(49) $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left(\frac{2}{3}(s_{L}^{t})^{2}\ln\frac{m_{t}^{2}}{m_{b}^{2}}+\frac{13}{6}(s_{L}^{t})^{2}\right),$
T $\displaystyle\simeq$ $\displaystyle\frac{N_{C}m_{t}^{2}}{8\pi
s_{W}^{2}m_{W}^{2}}\left(-\frac{3}{2}(s_{L}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}+3(s_{L}^{t})^{2}+(s_{L}^{t})^{4}\frac{19m_{T}^{2}}{24m_{t}^{2}}\right),$
(50) U $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left\\{-(s_{L}^{t})^{2}\left(\ln\frac{m_{B}^{2}}{m_{b}^{2}}-\ln\frac{m_{T}^{2}}{m_{t}^{2}}\right)-\frac{5}{6}(s_{L}^{t})^{2}\right\\}$
(51) $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left(-(s_{L}^{t})^{2}\ln\frac{m_{t}^{2}}{m_{b}^{2}}-\frac{5}{6}(s_{L}^{t})^{2}\right),$
$\displaystyle\frac{\delta m_{W}}{m_{W}}$ $\displaystyle\simeq$
$\displaystyle-\frac{\alpha N_{C}}{288\pi
s_{W}^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}\times\frac{\left(s_{L}^{t}\right)^{2}}{m_{W}^{2}}\times\left\\{\left(6-4s_{W}^{2}\right)m_{W}^{2}\ln\frac{m_{t}^{2}}{m_{b}^{2}}\right.$
(52)
$\displaystyle\left.+27c_{W}^{2}m_{t}^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}+\left(5+16s_{W}^{2}\right)m_{W}^{2}-c_{W}^{2}54m_{t}^{2}-\frac{57}{4}\left(s_{L}^{t}\right)^{2}c_{W}^{2}m_{T}^{2}\right\\}$
$\displaystyle\simeq$
$\displaystyle-10^{-4}\times\left(\frac{s_{L}^{t}}{0.1}\right)^{2}\times\left\\{2.01\ln\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}+4.05-0.36\left(\frac{s_{L}^{t}}{0.1}\right)^{2}\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}\right\\}$ $\displaystyle\simeq$
$\displaystyle-10^{-2}\times\frac{\xi_{T_{1}}^{2}}{4\pi}\times\left(\frac{{\rm
TeV}}{m_{T}}\right)^{2}\times\left\\{0.77\ln\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}+1.53-5.20\times\frac{\xi_{T_{1}}^{2}}{4\pi}\right\\}.$

Given that the $Zb\bar{b}$ coupling constraints require $s_{L}^{t}<0.06$ [65],
the mass correction is always negative, which can be obtained in the same way
as the discussion of the $D$ extension. Numerical results are presented in
Fig. 6, which are similar to those of Fig. 2 except that the perturbativity
constraint becomes tighter.

Figure 7: Similar to Fig.4 but for the $T_{2}$ extension, this figure
indicates that the oblique parameters and the $Zb\bar{b}$ couplings prefer
different parameter regions for $m_{T}\lesssim 5~{}{\rm TeV}$ in the standard
average scenario, so the range of $m_{T}$ is extended up to $8~{}{\rm TeV}$ in
studying the mass anomaly.

###  5.7 $T_{2}$ extension

The oblique parameters and $m_{W}$ in this model are approximated by

S $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left\\{\frac{16}{3}\left(\ln\frac{m_{B}^{2}}{m_{b}^{2}}-\ln\frac{m_{t}^{2}}{m_{b}^{2}}-\ln\frac{m_{T}^{2}}{m_{t}^{2}}\right)-\frac{8}{3}(s_{L}^{t})^{2}\ln\frac{m_{B}^{2}}{m_{b}^{2}}+\frac{2}{3}(s_{L}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}+\frac{13}{3}(s_{L}^{t})^{2}\right\\}$
(53) $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left(-\frac{8}{3}(s_{L}^{t})^{2}\ln\frac{m_{t}^{2}}{m_{b}^{2}}-2(s_{L}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}+\frac{29}{3}(s_{L}^{t})^{2}\right),$
T $\displaystyle\simeq$ $\displaystyle\frac{N_{C}m_{t}^{2}}{8\pi
s_{W}^{2}m_{W}^{2}}\left(3(s_{L}^{t})^{2}\ln\frac{m_{T}^{2}}{m_{t}^{2}}-5(s_{L}^{t})^{2}+(s_{L}^{t})^{4}\frac{19m_{T}^{2}}{6m_{t}^{2}}\right),$
(54) U $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left\\{4\left(\ln\frac{m_{T}^{2}}{m_{t}^{2}}+\ln\frac{m_{t}^{2}}{m_{b}^{2}}-\ln\frac{m_{B}^{2}}{m_{b}^{2}}\right)+2(s_{L}^{t})^{2}\left(\ln\frac{m_{B}^{2}}{m_{b}^{2}}+\ln\frac{m_{t}^{2}}{m_{b}^{2}}-\ln\frac{m_{T}^{2}}{m_{t}^{2}}\right)+\frac{7}{3}(s_{L}^{t})^{2}\right\\}$
(55) $\displaystyle\simeq$
$\displaystyle\frac{N_{C}}{6\pi}\left(4(s_{L}^{t})^{2}\ln\frac{m_{t}^{2}}{m_{b}^{2}}-\frac{5}{3}(s_{L}^{t})^{2}\right),$
$\displaystyle\frac{\delta m_{W}}{m_{W}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha N_{C}}{288\pi
s_{W}^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}\times\frac{\left(s_{L}^{t}\right)^{2}}{m_{W}^{2}}\times\left\\{\left(24-16s_{W}^{2}\right)m_{W}^{2}\ln\frac{m_{t}^{2}}{m_{b}^{2}}\right.$
(56)
$\displaystyle\left.+\left(24s_{W}^{2}m_{W}^{2}+54c_{W}^{2}m_{t}^{2}\right)\ln\frac{m_{T}^{2}}{m_{t}^{2}}-\left(10+96s_{W}^{2}\right)m_{W}^{2}-90c_{W}^{2}m_{t}^{2}+57\left(s_{L}^{t}\right)^{2}c_{W}^{2}m_{T}^{2}\right\\}$
$\displaystyle\simeq$ $\displaystyle
10^{-4}\times\left(\frac{s_{L}^{t}}{0.1}\right)^{2}\times\left\\{4.13\ln\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}+10.38+1.44\left(\frac{s_{L}^{t}}{0.1}\right)^{2}\left(\frac{m_{T}}{{\rm
TeV}}\right)^{2}\right\\}.$

In Fig. 7, the predictions of the $W$-boson mass, as well as various
limitations, are presented. It is shown that although the theory can explain
the mass anomaly in the parameter space, consistent with the constraints from
the oblique parameters, the $Zb\bar{b}$ constraints prefer a significantly
smaller $s_{L}^{t}$ for $m_{T}\lesssim 5~{}{\rm TeV}$ in the standard average
scenario. As a result, a feasible solution to the anomaly sets a clear lower
bound on $m_{T}$, i.e., $m_{T}\gtrsim 7~{}{\rm TeV}$ ($m_{T}\gtrsim 6~{}{\rm
TeV}$) if the theory is required to explain the mass anomaly at $1\sigma$
($2\sigma$) level. Evidently, this value is much larger than the LHC bound.

###  5.8 Other issues of the VLQ extensions

Up to present, there are many studies of the oblique parameters in VLQ
extensions (see, e.g., Refs. [91, 65, 92, 68, 52]) and some of them, however,
are discrepant. So it is essential to clarify the subtleties in the involved
calculations. The oblique parameters were first computed in the framework of
the singlet and doublet VLQ extensions in Ref. [91], and their expressions
were universally formulated in terms of generalized Cabibbo-Kobayashi-Maskawa
mixing matrices. As far as the doublet VLQ extensions are concerned, these
expressions were proved to be correct through numerical calculations by both
us and A. Arhrib, the first author of Ref. [92]. They were also reproduced
analytically in Ref. [52]. In particular, the author of Ref. [52] pointed out
that the derivation of the universal expressions relies on the relations in
Eqs. (7-10) of Ref. [91]. This observation implies that the expressions,
especially those for $S$ and $U$ parameters, cannot be applied to the triplet
extensions $T_{1}$ and $T_{2}$ since the relations are not satisfied in these
extensions. Following this logic, the results of Ref. [68], where the oblique
parameters for the triplet extensions were calculated by the formula in Ref.
[91], may be problematic333In fact, there is another evident mistake in Ref.
[68], i.e., the invalid relations in its Eq. (A.20),
$m_{X}^{2}=(c_{R}^{b})^{2}m_{B}^{2}+(s_{R}^{b})^{2}m_{b}^{2}=(c_{R}^{t})^{2}m_{T}^{2}+(s_{R}^{t})^{2}m_{t}^{2}$,
were used in calculation.. This point was recently verified by A. Arhrib.
Besides these details, it should be noted that the $T_{2}$ extension was also
considered to explain the mass anomaly [52]. Although that work obtained
independently the same formula for the $S$ and $T$ parameters as ours, its
Fig. 3 showed that the extension could explain the anomaly even for
$m_{T}<3~{}{\rm TeV}$, which is opposite to our conclusions. The main reason
for the difference is that the $Zb\bar{b}$ coupling constraints were not
considered in limiting $\sin\theta_{L}^{t}$ in Ref. [52]. In addition, it
neglected the contribution of the $U$ parameter to $m_{W}$, which is sizable
by our calculation.

Concerning our approximations of the oblique parameters, the following
observations are in order:

  * •

They take into account all the dominant contributions of the exact formulas,
which are calculated by the loop functions in Eq. (23). Numerically, the
induced difference for all the quantities is found to be less than $10\%$ for
$s_{L,R}^{t}<0.2$ and $m_{T}>1~{}{\rm TeV}$.

  * •

The formulas for the $T$ parameter agree with those in Ref. [68], except the
$s_{L,R}^{4}m_{T}^{2}$ terms are neglected for the $U$, $D$, $Q_{1}$, $Q_{7}$,
$T_{1}$, and $T_{2}$ extensions in Ref. [68]. Concerning the expression of the
$T$ parameter for the $Q_{5}$ extension, there is a minor typo in Eq. (3.9) of
Ref. [68]; namely, the factor $(s_{R}^{b})^{5}$ should be replaced by
$(s_{R}^{b})^{4}$.

  * •

The formulas for the $S$ and $U$ parameters in the $U$, $D$, $Q_{1}$, $Q_{5}$,
and $Q_{7}$ extensions agree with those in Ref. [68] except for the $S$
parameter in the $Q_{5}$ and $Q_{7}$ extensions. This difference does not
affect $\delta m_{W}$ essentially, as it is dominated by the $T$-term
contribution.

  * •

This study cannot reproduce the formulas of Ref. [68] for the $S$ and $U$
parameters in the $T_{1}$ and $T_{2}$ extensions due to the reasons discussed
above.

Finally, the following related aspects of the VLQ extensions are commented on
briefly.

  * •

It would be pretty reasonable to expect that, if VLQs exist, then so could be
vectorlike leptons (VLLs). In this case, the oblique parameters are also
contributed to by the VLL loops. In the extension of the SM with only one VLL
multiplet, the form of this additional contribution to $T$ parameter is
similar to the results of this work. It has the following property [19]

$\displaystyle T_{\rm
VLL}\propto\frac{m_{L}^{2}}{m_{Z}^{2}}\sin^{4}\theta_{L}\propto\frac{\xi_{L}^{2}v^{2}}{m_{Z}^{2}}\sin^{2}\theta_{L},$
(57)

if terms proportional to SM lepton mass are neglected, where $m_{L}$ denotes
the VLL mass and $\theta_{L}$ is the VLL-lepton mixing angle. Since the VLL-
lepton Yukawa coupling $\xi_{L}$ is naturally much smaller than $\xi_{j}$ in
Eq. (1) and $\sin\theta_{L}$ has been tightly limited by the EWPOs, $T_{\rm
VLL}$ is negligibly small [19]. It is emphasized that this conclusion is valid
only for the minimal VLL extension. To be specific, if the SM is extended by
more than one multiplet, then the VLL contribution may get enhanced when the
mixing and mass splitting between VLLs are large [93]. This feature was
recently utilized in Refs. [42, 94, 95] to explain the $W$-boson mass anomaly
and the muon $g-2$ anomaly simultaneously.

  * •

As mentioned before, the introduction of the VLQs will exacerbate the vacuum
stability problem of the SM when the running of the Higgs quartic coupling by
the renormalization group equation (RGE) is considered. This problem can be
solved or at least alleviated, by adding bosonic freedom into the theory. As
far as the minimal VLQ extensions are concerned, the most economic way is
using the singlet scalar field, which is responsible for the bare mass of
VLQs, to change the RGE behavior of the coupling. It has been shown that this
strategy works well for the $U$, $D$, $Q_{1}$, and $T_{2}$ extensions in broad
parameter space, but it is inefficient for the $Q_{7}$ extension [71]. The
latter situation may be improved by invoking more complex frameworks.

##  6 CONCLUSIONS

In many attractive BSM theories, VLQs are used to solve some fundamental
problems in particle physics. They may couple directly with gauge bosons in
the SM, and their mixings with the third-generation quarks can significantly
affect EWPOs. Motivated by this feature, seven economic VLQ extensions that
predict renormalizable $H\bar{Q}q$ interactions are considered to interpret
the recently observed $W$-boson mass anomaly. For each extension, the
$W$-boson mass are calculated, and the $S$, $T$, and $U$ parameters are used
to limit the parameter space of the extensions. Other important constraints,
such as those from the $Zb\bar{b}$ couplings on the $D$, $Q_{5}$, $T_{1}$, and
$T_{2}$ extensions, are also implemented. The following conclusions are
obtained:

  * •

The $U$ and $Q_{1}$ extensions can interpret the anomaly in their broad
parameter space. The explanations are consistent with the constraints from the
oblique parameters at the $2\sigma$ level in the standard average scenario and
at the $1\sigma$ level in the conservative average scenario. They also satisfy
other experimental constraints, such as the measurements of the properties for
the top quark, bottom quark, and Higgs boson, the LHC search for VLQs, and the
perturbativity of the theories. The typical size of the involved Yukawa
coupling is around 1, which is comparable to the top quark Yukawa coupling in
the SM.

  * •

In the $T_{2}$ extension, the $b^{\prime}-B^{\prime}$ mixing is correlated
with the $t^{\prime}-T^{\prime}$ mixing. Since the former mixing can
significantly alter the $Zb_{L}\bar{b}_{L}$ coupling and is thus tightly
limited by the coupling measurements, the $t^{\prime}-T^{\prime}$ mixing must
be small. As a result, although the $T_{2}$ extension can explain the anomaly
in the parameter space, consistent with the constraints from the oblique
parameters, the $Zb\bar{b}$ couplings exclude the explanation when
$m_{T}\lesssim 5~{}{\rm TeV}$. Fortunately, they become compatible if
$m_{T}\gtrsim 6~{}{\rm TeV}$ in the standard average scenario, a value much
larger than the LHC constraints on $m_{T}$.

  * •

Although the $Q_{5}$ extension can explain the mass anomaly in the parameter
space, consistent with the constraints from the oblique parameters, it needs
an unnaturally large coupling of the right-handed bottom quark to new physics.
Thus, it is theoretically less appealing.

  * •

Owing to the assignment of the charges for the $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}$ gauge groups, the $D$ extension predicts a negative
correction to $W$-boson mass for small VLQ-quark mixing, so it cannot explain
the anomaly after considering the $Zb\bar{b}$ coupling constraints. This
situation is also applied to the $T_{1}$ extension.

  * •

Concerning the $Q_{7}$ extension, there remains very narrow parameter space
that can explain the mass anomaly and at the same time, keep consistent with
the constraints from the oblique parameters and the $Zb\bar{b}$ couplings.
This parameter space, however, is characterized by an excessively large Yukawa
coupling, $\xi_{Q_{7}}\sim 3$, which exacerbates greatly the vacuum stability
problem of the SM.

It is emphasized that if the $W$-boson mass anomaly is confirmed by other
experiments in the future, these obtained conclusions may serve as a useful
guide for model building, which is the main merit of this study.

## ACKNOWLEDGMENTS

We thank Professors Abdesslam Arhrib, Yi Liao, and Jianghao Yu and Dr. Shi-
Ping He for their helpful discussion. This work is supported by the National
Natural Science Foundation of China (NNSFC) (Grants No. 11705048 and No.
12075076), the National Research Project Cultivation Foundation of Henan
Normal University (Grants No. 2020PL16 and No. 2021PL10), and the High
Performance Computing Center of Henan Normal University.

## APPENDIX: W/Z COUPLINGS IN THE VLQ EXTENSIONS

The SM Lagrangian for the W/Z couplings to the third-generation quarks is
modified by their mixing with VLQs [65]. In the following, the third-
generation-quark-dominated mass eigenstates are denoted as $t/b$, and the VLQ-
dominated states are denoted as $Q/Q^{\prime}=X,T,B,Y$. They are collectively
called light quarks and heavy quarks, respectively, for brevity.

The renewed Lagrangian for light quarks takes the following form:

$\displaystyle\mathcal{L}_{W}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\bar{t}\gamma^{\mu}\left(V_{tb}^{L}P_{L}+V_{tb}^{R}P_{R}\right)bW_{\mu}^{+}+\text{H.c.},$
$\displaystyle\mathcal{L}_{Z}$ $\displaystyle=$
$\displaystyle-\frac{g}{2c_{W}}\bar{t}\gamma^{\mu}\left(X_{tt}^{L}P_{L}+X_{tt}^{R}P_{R}\right)tZ_{\mu}-\frac{g}{2c_{W}}\bar{b}\gamma^{\mu}\left(X_{bb}^{L}P_{L}+X_{bb}^{R}P_{R}\right)bZ_{\mu},$

where the charged current form factors ($V$) and the neutral current form
factors ($X$) are listed in Table 3 and 4, respectively.

| $V_{tb}^{L}$ | $V_{tb}^{R}$
---|---|---
$U$ | $c_{L}^{t}$ | 0
$D$ | $c_{L}^{b}$ | 0
$Q_{1}$ | $c_{L}^{t}c_{L}^{b}+s_{L}^{t}s_{L}^{b}$ | $s_{R}^{t}s_{R}^{b}$
$Q_{5}$ | $c_{L}^{b}$ | 0
$Q_{7}$ | $c_{L}^{t}$ | 0
$T_{1}$ | $c_{L}^{t}c_{L}^{b}+\sqrt{2}s_{L}^{t}s_{L}^{b}$ | $\sqrt{2}s_{R}^{t}s_{R}^{b}$
$T_{2}$ | $c_{L}^{t}c_{L}^{b}+\sqrt{2}s_{L}^{t}s_{L}^{b}$ | $\sqrt{2}s_{R}^{t}s_{R}^{b}$
Table 3: Light-light quark couplings to the $W$ boson. | $X_{tt}^{L}$ | $X_{tt}^{R}$ | $X_{bb}^{L}$ | $X_{bb}^{R}$
---|---|---|---|---
$U$ | $(c_{L}^{t})^{2}-2Q_{t}s_{W}^{2}$ | $-2Q_{t}s_{W}^{2}$ | $-1-2Q_{b}s_{W}^{2}$ | $-2Q_{b}s_{W}^{2}$
$D$ | $1-2Q_{t}s_{W}^{2}$ | $-2Q_{t}s_{W}^{2}$ | $-(c_{L}^{b})^{2}-2Q_{b}s_{W}^{2}$ | $-2Q_{b}s_{W}^{2}$
$Q_{1}$ | $1-2Q_{t}s_{W}^{2}$ | $(s_{R}^{t})^{2}-2Q_{t}s_{W}^{2}$ | $-1-2Q_{b}s_{W}^{2}$ | $-(s_{R}^{b})^{2}-2Q_{b}s_{W}^{2}$
$Q_{5}$ | $1-2Q_{t}s_{W}^{2}$ | $-2Q_{t}s_{W}^{2}$ | $-(c_{L}^{b})^{2}+(s_{L}^{b})^{2}-2Q_{b}s_{W}^{2}$ | $(s_{R}^{b})^{2}-2Q_{b}s_{W}^{2}$
$Q_{7}$ | $(c_{L}^{t})^{2}-(s_{L}^{t})^{2}-2Q_{t}s_{W}^{2}$ | $-(s_{R}^{t})^{2}-2Q_{t}s_{W}^{2}$ | $-1-2Q_{b}s_{W}^{2}$ | $-2Q_{b}s_{W}^{2}$
$T_{1}$ | $1+(s_{L}^{t})^{2}-2Q_{t}s_{W}^{2}$ | $2(s_{R}^{t})^{2}-2Q_{t}s_{W}^{2}$ | $-(c_{L}^{b})^{2}-2Q_{b}s_{W}^{2}$ | $-2Q_{b}s_{W}^{2}$
$T_{2}$ | $(c_{L}^{t})^{2}-2Q_{t}s_{W}^{2}$ | $-2Q_{t}s_{W}^{2}$ | $-1-(s_{L}^{b})^{2}-2Q_{b}s_{W}^{2}$ | $-2(s_{R}^{b})^{2}-2Q_{b}s_{W}^{2}$
Table 4: Light-light quark couplings to the $Z$ boson.

Similarly, the Lagrangian for the heavy states, $Q,Q^{\prime}=X,T,B,Y$, is
given by

$\displaystyle\mathcal{L}_{W}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\bar{Q}\gamma^{\mu}\left(V_{QQ^{\prime}}^{L}P_{L}+V_{QQ^{\prime}}^{R}P_{R}\right)Q^{\prime}W_{\mu}^{+}+\text{H.c.},$
$\displaystyle\mathcal{L}_{Z}$ $\displaystyle=$
$\displaystyle-\frac{g}{2c_{W}}\bar{Q}\gamma^{\mu}\left(X_{QQ}^{L}P_{L}+X_{QQ}^{R}P_{R}\right)QZ_{\mu},$

where the charged current form factors ($V$) are presented in Table 5, and the
neutral current form factors ($X$) are given in Tables 6, 7, and 8.

| $V_{XT}^{L}$ | $V_{XT}^{R}$ | $V_{TB}^{L}$ | $V_{TB}^{R}$ | $V_{BY}^{L}$ | $V_{BY}^{R}$
---|---|---|---|---|---|---
$Q_{1}$ | $\cdots$ | $\cdots$ | $c_{L}^{t}c_{L}^{b}+s_{L}^{t}s_{L}^{b}$ | $c_{R}^{t}c_{R}^{b}$ | $\cdots$ | $\cdots$
$Q_{5}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $c_{L}^{b}$ | $c_{R}^{b}$
$Q_{7}$ | $c_{L}^{t}$ | $c_{R}^{t}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$T_{1}$ | $\cdots$ | $\cdots$ | $s_{L}^{t}s_{L}^{b}+\sqrt{2}c_{L}^{t}c_{L}^{b}$ | $\sqrt{2}c_{R}^{t}c_{R}^{b}$ | $\sqrt{2}c_{L}^{b}$ | $\sqrt{2}c_{R}^{b}$
$T_{2}$ | $\sqrt{2}c_{L}^{t}$ | $\sqrt{2}c_{R}^{t}$ | $s_{L}^{t}s_{L}^{b}+\sqrt{2}c_{L}^{t}c_{L}^{b}$ | $\sqrt{2}c_{R}^{t}c_{R}^{b}$ | $\cdots$ | $\cdots$
Table 5: Heavy-heavy quark couplings to the $W$ boson. | $X_{XX}^{L}$ | $X_{XX}^{R}$
---|---|---
$Q_{7}$ | $1-2Q_{X}s_{W}^{2}$ | $1-2Q_{X}s_{W}^{2}$
$T_{2}$ | $2-2Q_{X}s_{W}^{2}$ | $2-2Q_{X}s_{W}^{2}$
Table 6: $ZXX$ couplings in the VLQ extensions.  | $X_{TT}^{L}$ | $X_{TT}^{R}$
---|---|---
$U$ | $(s_{L}^{t})^{2}-2Q_{T}s_{W}^{2}$ | $-2Q_{T}s_{W}^{2}$
$Q_{1}$ | $1-2Q_{T}s_{W}^{2}$ | $(c_{R}^{t})^{2}-2Q_{T}s_{W}^{2}$
$Q_{7}$ | $(s_{L}^{t})^{2}-(c_{L}^{t})^{2}-2Q_{T}s_{W}^{2}$ | $-(c_{R}^{t})^{2}-2Q_{T}s_{W}^{2}$
$T_{1}$ | $1+(c_{L}^{t})^{2}-2Q_{T}s_{W}^{2}$ | $2(c_{R}^{t})^{2}-2Q_{T}s_{W}^{2}$
$T_{2}$ | $(s_{L}^{t})^{2}-2Q_{T}s_{W}^{2}$ | $-2Q_{T}s_{W}^{2}$
Table 7: $ZTT$ couplings in the VLQ extensions. | $X_{BB}^{L}$ | $X_{BB}^{R}$ | $X_{YY}^{L}$ | $X_{YY}^{R}$
---|---|---|---|---
$D$ | $-(s_{L}^{b})^{2}-2Q_{B}s_{W}^{2}$ | $-2Q_{B}s_{W}^{2}$ | $\cdots$ | $\cdots$
$Q_{1}$ | $-1-2Q_{B}s_{W}^{2}$ | $-(c_{R}^{b})^{2}-2Q_{B}s_{W}^{2}$ | $\cdots$ | $\cdots$
$Q_{5}$ | $-(s_{L}^{b})^{2}+(c_{L}^{b})^{2}-2Q_{B}s_{W}^{2}$ | $(c_{R}^{b})^{2}-2Q_{B}s_{W}^{2}$ | $-1-2Q_{Y}s_{W}^{2}$ | $-1-2Q_{Y}s_{W}^{2}$
$T_{1}$ | $-(s_{L}^{b})^{2}-2Q_{B}s_{W}^{2}$ | $-2Q_{B}s_{W}^{2}$ | $-2-2Q_{Y}s_{W}^{2}$ | $-2-2Q_{Y}s_{W}^{2}$
$T_{2}$ | $-1-(c_{L}^{b})^{2}-2Q_{B}s_{W}^{2}$ | $-2(c_{R}^{b})^{2}-2Q_{B}s_{W}^{2}$ | $\cdots$ | $\cdots$
Table 8: $ZBB$ and $ZYY$ couplings in the VLQ extensions. | $V_{Xt}^{L}$ | $V_{Xt}^{R}$ | $V_{Tb}^{L}$ | $V_{Tb}^{R}$
---|---|---|---|---
$U$ | $\cdots$ | $\cdots$ | $s_{L}^{t}$ | 0
$Q_{1}$ | $\cdots$ | $\cdots$ | $s_{L}^{t}c_{L}^{b}-c_{L}^{t}s_{L}^{b}$ | $-c_{R}^{t}s_{R}^{b}$
$Q_{7}$ | $-s_{L}^{t}$ | $-s_{R}^{t}$ | $s_{L}^{t}$ | 0
$T_{1}$ | $\cdots$ | $\cdots$ | $s_{L}^{t}c_{L}^{b}-\sqrt{2}c_{L}^{t}s_{L}^{b}$ | $-\sqrt{2}c_{R}^{t}s_{R}^{b}$
$T_{2}$ | $-\sqrt{2}s_{L}^{t}$ | $-\sqrt{2}s_{R}^{t}$ | $s_{L}^{t}c_{L}^{b}-\sqrt{2}c_{L}^{t}s_{L}^{b}$ | $-\sqrt{2}c_{R}^{t}s_{R}^{b}$
Table 9: Charged current for heavy-light quark transition in VLQ extensions. |  Heavy-light coupling to $W$ boson. |  Heavy-light coupling to $Z$ boson.
---|---|---
| $V_{tB}^{L}$ | $V_{tB}^{R}$ | $V_{bY}^{L}$ | $V_{bY}^{R}$ | $X_{tT}^{L}$ | $X_{tT}^{R}$ | $X_{bB}^{L}$ | $X_{bB}^{R}$
$U$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $s_{L}^{t}c_{L}^{t}$ | 0 | $\cdots$ | $\cdots$
$D$ | $s_{L}^{b}$ | 0 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $-s_{L}^{b}c_{L}^{b}$ | 0
$Q_{1}$ | $c_{L}^{t}s_{L}^{b}-s_{L}^{t}c_{L}^{b}$ | $-s_{R}^{t}c_{R}^{b}$ | $\cdots$ | $\cdots$ | 0 | $-s_{R}^{t}c_{R}^{t}$ | 0 | $s_{R}^{b}c_{R}^{b}$
$Q_{5}$ | $s_{L}^{b}$ | 0 | $-s_{L}^{b}$ | $-s_{R}^{b}$ | $\cdots$ | $\cdots$ | $-2s_{L}^{b}c_{L}^{b}$ | $-s_{R}^{b}c_{R}^{b}$
$Q_{7}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $2s_{L}^{t}c_{L}^{t}$ | $s_{R}^{t}c_{R}^{t}$ | $\cdots$ | $\cdots$
$T_{1}$ | $c_{L}^{t}s_{L}^{b}-\sqrt{2}s_{L}^{t}c_{L}^{b}$ | $-\sqrt{2}s_{R}^{t}c_{R}^{b}$ | $-\sqrt{2}s_{L}^{b}$ | $-\sqrt{2}s_{R}^{b}$ | $-s_{L}^{t}c_{L}^{t}$ | $-2s_{R}^{t}c_{R}^{t}$ | $-s_{L}^{b}c_{L}^{b}$ | 0
$T_{2}$ | $c_{L}^{t}s_{L}^{b}-\sqrt{2}s_{L}^{t}c_{L}^{b}$ | $-\sqrt{2}s_{R}^{t}c_{R}^{b}$ | $\cdots$ | $\cdots$ | $s_{L}^{t}c_{L}^{t}$ | 0 | $s_{L}^{b}c_{L}^{b}$ | $2s_{R}^{b}c_{R}^{b}$
Table 10: Neutral current for heavy-light quark transitions in VLQ extensions.

Finally, the Lagrangian involving heavy-light quark transition is

$\displaystyle\mathcal{L}_{W}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\bar{Q}\gamma^{\mu}\left(V_{Qq}^{L}P_{L}+V_{Qq}^{R}P_{R}\right)qW_{\mu}^{+}+\text{H.c.}$
$\displaystyle-\frac{g}{\sqrt{2}}\bar{q}\gamma^{\mu}\left(V_{qQ}^{L}P_{L}+V_{Qq}^{R}P_{R}\right)QW_{\mu}^{+}+\text{H.c.},$
$\displaystyle\mathcal{L}_{Z}$ $\displaystyle=$
$\displaystyle-\frac{g}{2c_{W}}\bar{q}\gamma^{\mu}\left(X_{qQ}^{L}P_{L}+X_{qQ}^{R}P_{R}\right)QZ_{\mu}+\text{H.c.},$

where the factors $V$ and $X$ are listed in Table 9 and Table 10,
respectively.

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