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# Corrections to electroweak precision observables from mixings of an exotic
vector boson in light of the CDF $W$-mass anomaly

Chengfeng Cai  Dayun Qiu  Yi-Lei Tang  Zhao-Huan Yu  Hong-Hao Zhang  School of
Physics, Sun Yat-Sen University, Guangzhou 510275, China

###### Abstract

We enumerate various effective couplings that contribute to the mixings
between an exotic vector boson $Z^{\prime}$ and the neutral electroweak vector
bosons. The miscellaneous mixing patterns can be evaluated perturbatively. The
effective oblique parameters $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$ are
calculated to compare with the electroweak precision test results. With the
contributions to the non-negligible $U^{\prime}$ parameter from the
$\epsilon_{B,W}$ parameters and the aid of some other parameters to cancel the
negative $T^{\prime}$, the recent CDF $W$-mass anomaly can therefore be
explained.

##  I Introduction

Besides searching for new particles by straightforwardly producing them at the
colliders, detecting the tiny deviations of the measured standard model (SM)
parameters from their theoretical predicted values is also an important
approach to new physics (NP) beyond the SM. In the literature, the Peskin-
Takeuchi oblique parameters $S$, $T$, and $U$ Peskin:1990zt ; Peskin:1991sw
are usually applied to test the SM. These parameters are extracted from the
self-energy diagrams of the electroweak (EW) vector bosons, and contribute to
EW precision observables, such as the $Z$-pole parameters and the masses of
the $Z$ and $W$ bosons Burgess:1993vc . Higher-order parameters such as $V$,
$W$, $X$, and $Y$ are introduced in Ref. Barbieri:2004qk . In particular, when
the traditional $S$, $T$, and $U$ parameters are absent, they can dominate the
NP contributions to the EW observables.

Recently, the CDF collaboration published a high-precision measurement of the
$W$ boson mass $m_{W}$ based on the $8.8~{}$\mathrm{f}\mathrm{b}^{-1}$$ of
data collected by the CDF II detector at the Tevatron collider CDF:2022hxs ,

$m_{W}=80.4335\pm 0.0094~{}$\mathrm{G}\mathrm{e}\mathrm{V}$.$ (1)

This result indicates a $\sim 7\sigma$ deviation from the SM prediction
$m_{W}^{\mathrm{SM}}=80.3545\pm 0.0057~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$
given by the global fit of the EW precision measurements deBlas:2021wap . Such
an anomaly had drawn quite a lot of attention and been considered to originate
from NP contributions Fan:2022dck ; Zhu:2022tpr ; Lu:2022bgw ; Athron:2022qpo
; Yuan:2022cpw ; Strumia:2022qkt ; Yang:2022gvz ; deBlas:2022hdk ; Du:2022pbp
; Tang:2022pxh ; Cacciapaglia:2022xih ; Blennow:2022yfm ; Arias-Aragon:2022ats
; Zhu:2022scj ; Sakurai:2022hwh ; Fan:2022yly ; Liu:2022jdq ; Lee:2022nqz ;
Cheng:2022jyi ; Song:2022xts ; Bagnaschi:2022whn ; Paul:2022dds ; Bahl:2022xzi
; Asadi:2022xiy ; DiLuzio:2022xns ; Athron:2022isz ; Gu:2022htv ;
Heckman:2022the ; Babu:2022pdn ; Heo:2022dey ; Du:2022brr ; Cheung:2022zsb ;
Crivellin:2022fdf ; Endo:2022kiw ; Biekotter:2022abc ; Balkin:2022glu ;
Krasnikov:2022xsi ; Ahn:2022xeq ; Han:2022juu ; Zheng:2022irz ;
Kawamura:2022uft ; Peli:2022ybi ; Ghoshal:2022vzo ; Perez:2022uil ;
Kanemura:2022ahw ; Mondal:2022xdy ; Zhang:2022nnh ; Borah:2022obi ;
Chowdhury:2022moc ; Arcadi:2022dmt ; Cirigliano:2022qdm ; Carpenter:2022oyg ;
Popov:2022ldh ; Ghorbani:2022vtv ; Du:2022fqv ; Bhaskar:2022vgk ;
Batra:2022org ; Cao:2022mif ; Zeng:2022lkk ; Baek:2022agi ; Borah:2022zim ;
Almeida:2022lcs ; Cheng:2022aau ; Heeck:2022fvl ; Addazi:2022fbj ; Lee:2022gyf
.

A wide class of NP effects might contribute to oblique parameters, and thus
shift the $W$ boson mass correspondingly. The EW global fits with the CDF
$m_{W}$ data considered have been performed in Refs. Lu:2022bgw ;
deBlas:2022hdk ; Strumia:2022qkt ; Bagnaschi:2022whn ; Fan:2022yly ;
Gu:2022htv ; Balkin:2022glu ; Asadi:2022xiy . An appropriate loop-level NP
contribution implies the corresponding NP scale to be around a few hundred
GeV, easily to conflict with current collider bounds, while the tree-level NP
scale for interpreting it can be as high as multi-TeV Strumia:2022qkt .

Tree-level corrections to the oblique parameters may come from an exotic
neutral vector boson $Z^{\prime}$, which naturally appears in many SM
extensions, such as grand unified theories Robinett:1981yz , little Higgs
models Arkani-Hamed:2001nha , extra dimensions Casalbuoni:1999ns , and a lot
of $\mathrm{U}(1)^{\prime}$ gauge models motivated by various problems
Leike:1998wr ; Langacker:2008yv . These NP models generally introduce kinetic
and mass mixings between the $Z$ and $Z^{\prime}$ bosons, which contribute to
the oblique parameters at tree level Holdom:1990xp ; Babu:1997st . Therefore,
an exotic $Z^{\prime}$ boson could be responsible for the CDF $m_{W}$ anomaly,
as discussed in some recent studies Strumia:2022qkt ; Asadi:2022xiy ;
Zhang:2022nnh ; Du:2022fqv ; Zeng:2022lkk ; Cheng:2022aau ; Alguero:2022est .
Note that from the effective field theory (EFT) point of view, kinetic mixings
between gauge bosons contribute to the $p^{4}$ terms in the vaccuum
polarization amplitudes of the EW gauge fields, leading to higher-order
parameters $V$, $W$, $X$, and $Y$. However, combined with the traditional $S$,
$T$, and $U$, all these parameters are redundant in fitting the EW precision
data at the $Z$-pole, the $W$ mass, and the Fermi constant. Therefore, we can
define three effective oblique parameters $S^{\prime}$, $T^{\prime}$, and
$U^{\prime}$ which include all order effects and then compare them with the
most recent global fit results of $S$, $T$, and $U$ from Ref. deBlas:2022hdk .

In addition to the well-known kinetic mixing between the $\mathrm{U(1)_{Y}}$
vector boson $B$ and the $Z^{\prime}$ boson, loop-level diagrams might also
induce the kinetic mixing terms between the $W^{3}$ and $Z^{\prime}$ bosons as
well as various mass mixing terms. Besides $S^{\prime}$ and $T^{\prime}$, a
non-negligible $U^{\prime}$ parameter due to the mixings could also contribute
to the $W$ mass, as evaluated in Refs. Zeng:2022lkk ; Cheng:2022aau  with some
specific conditions (Note that the effective $U^{\prime}$ can originate from
higher-order corrections such as $V$, $W$, $X$. See Eq.(A) and related
discussion.). In this paper, a more general case described by an effective
Lagrangian is considered. We will present the perturbative corrections to the
$S^{\prime}$, $T^{\prime}$, and $U^{\prime}$ parameters originating from the
kinetic and mass mixings between the $Z^{\prime}$ and the SM gauge bosons, in
addition to three traditional SMEFT operators. Our strategy of calculating the
effective oblique parameters is directly diagonalizing the kinetic and mass
matrices. In principle, this procedure automatically includes the corrections
from all $(p^{2})^{i}$ orders due to the mixings.

This paper is organized as follows. In Sect. II, we briefly introduce the
effective Lagrangian for an exotic $Z^{\prime}$ boson. In Sect. III, the
effective oblique parameters $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$ are
calculated perturbatively, and the analytical results are presented. In Sect.
IV, we estimate the constraints for the $Z^{\prime}$ boson from collider
experiments. The numerical results as well as some corresponding discussions
have been presented in Sect. V, and then we finalize this paper in Sect. VI.
An EFT analysis is also provided in App. A for cross-checking our calculation.

##  II Effective Lagrangian

Besides the SM fields, we introduce an exotic vector field
$\hat{Z}^{\prime\mu}$, with kinetic and mass terms given by

$\displaystyle\mathcal{L}\supset-\frac{1}{4}\hat{Z}^{\prime}_{\mu\nu}\hat{Z}^{\prime\mu\nu}+\frac{1}{2}\hat{m}_{Z^{\prime}}^{2}\hat{Z}^{\prime}_{\mu}\hat{Z}^{\prime\mu},$
(2)

where
$\hat{Z}^{\prime}_{\mu\nu}\equiv\partial_{\mu}\hat{Z}^{\prime}_{\nu}-\partial_{\nu}\hat{Z}^{\prime}_{\mu}$.
The $\hat{Z}^{\prime}$ boson can be either a fundamental gauge boson of an
exotic $\mathrm{U}(1)$ gauge group, or a component from a gauge boson
multiplet in the framework of a non-abelian gauge group. The mass term might
originate from the vacuum expectation value (VEV) of a Higgs sector, or
directly acquired from the Stueckelburg mechanism in the $\mathrm{U}(1)$ case.

The $\hat{Z}^{\prime}$ boson might mix with the electroweak gauge bosons
through the following kinetic mixing terms in the effective Lagrangian,

$\displaystyle\mathcal{L}_{\rm eff}\supset$ $\displaystyle-$
$\displaystyle\frac{\epsilon_{B}}{2}\hat{Z}^{\prime}_{\mu\nu}B^{\mu\nu}-\frac{1}{2\Lambda_{W}^{2}}\hat{Z}^{\prime}_{\mu\nu}W^{a\mu\nu}H^{\dagger}\sigma^{a}H-\frac{1}{2\Lambda_{BW}^{2}}B_{\mu\nu}W^{a\mu\nu}H^{\dagger}\sigma^{a}H$
(3) $\displaystyle-$
$\displaystyle\frac{1}{4\Lambda_{WW}^{4}}W^{a\mu\nu}H^{\dagger}\sigma^{a}HW^{b}_{\mu\nu}H^{\dagger}\sigma^{b}H,$

where $H$ indicates the SM Higgs doublet, and the $\epsilon_{B}$ term can be
created straightforwardly in the $\mathrm{U}(1)$ case, or can arise together
with other $\Lambda_{W}$, $\Lambda_{BW}$, and $\Lambda_{WW}$ terms through
higher-order corrections. The mass terms might also receive corrections,
formulated as

$\displaystyle\mathcal{L}_{\rm eff}$ $\displaystyle\supset$
$\displaystyle\frac{1}{\Lambda_{HD}^{2}}(H^{\dagger}D_{\mu}H)^{\dagger}(H^{\dagger}D^{\mu}H)+\hat{Z}^{\prime\mu}\left[i\lambda_{HZ^{\prime}}(D_{\mu}H)^{\dagger}H+\text{H.c.}\right].$
(4)

All three terms can arise from the loop effects. The $\lambda_{HZ^{\prime}}$
term might originate from something like $\Phi^{\dagger}D^{\prime\mu}\Phi$
where $\Phi$ indicates an exotic Higgs field to break the gauge group
corresponding to $\hat{Z}^{\prime}$, or a dummy $v_{Z^{\prime}}e^{i\phi(x)}$
field in the Stueckelberg mechanism. The
$\mathrm{U(1)_{Y}}\times\mathrm{SU(2)_{L}}$ covariant derivative for the Higgs
doublet is
$D_{\mu}=\partial_{\mu}-i\hat{g}^{\prime}_{Y}B_{\mu}/2-i\hat{g}\sigma^{a}W^{a}_{\mu}/2$,
where the hatted parameters $\hat{g}^{\prime}$ and $\hat{g}$ are the
“original” coupling constants.

Before proceeding, we would like to make some comments on the possible UV
completion of the model. The $S$, $T$, and $U$ parameters corresponds to three
SMEFT operators, $B_{\mu\nu}W^{a\mu\nu}H^{\dagger}\sigma^{a}H$,
$|H^{\dagger}D_{\mu}H|^{2}$, and $|H^{\dagger}W_{\mu\nu}H|^{2}$, which can be
easily generated by introducing some fermionic Cai:2016sjz  or scalar
Cai:2017wdu  EW multiplets. For example, Ref. Cai:2017wdu  had shown that in a
dark matter model with a singlet and a doublet scalars, a significant $T$
parameter can be obtained. In particular, when the ratio of the parameters
$|\lambda_{2}/\lambda_{3}|<0.5$, the $T$ parameter is positive, and thus the
$W$ boson mass can be raised.

The kinetic mixing between $Z^{\prime}_{\mu}$ and the SM $B_{\mu}$ field can
be put by hand since it relates to a gauge-invariant renormalizable operator.
It can also be generated by loops of some NP fields carring both
$\mathrm{U(1)_{Y}}$ and new $\mathrm{U(1)}$ charges. For the kinetic mixings
between $Z^{\prime}_{\mu}$ and the SM $W^{3}_{\mu}$ field, a simple
realization is to generate them by loops of $2n$ scalar EW quadruplet fields,
$X_{1,2,...,2n}$, which carry the same new $\mathrm{U(1)}$ charge. Once we
choose opposite $\mathrm{U(1)_{Y}}$ charges for $X_{1,..,n}$ and
$X_{n+1,..,2n}$, and focus on the custodial symmetric cases (corresponding to
a condition $|\lambda_{-}/\lambda_{3}|=2$ as in Ref. Cai:2017wdu ), then $S$,
$T$, and $U$ are not affected for the same scalar masses and the same
potential parameters. On the other hand, denoting $g_{D}$ to be the new
$\mathrm{U(1)}$ gauge coupling, a relatively large $\epsilon_{W}\sim 0.01\cdot
ng_{D}(v^{2}/m_{X}^{2})$ can be achieved when $ng_{D}$ is large.

After the SM Higgs field $H$ acquires the VEV $\hat{v}$ as usual,

$\displaystyle H=\begin{pmatrix}i\phi^{+}\\\
\frac{\hat{v}+h+i\phi^{0}}{\sqrt{2}}\end{pmatrix},$ (5)

where $\hat{v}\approx 246$ GeV, the kinetic mixing terms can be re-
parametrized to be

$\displaystyle\mathcal{L}_{\rm eff}\supset$ $\displaystyle-$
$\displaystyle\frac{\epsilon_{B}}{2}\hat{Z}^{\prime}_{\mu\nu}B^{\mu\nu}-\frac{\epsilon_{W}}{2}\hat{Z}^{\prime}_{\mu\nu}W^{3\mu\nu}-\frac{\epsilon_{BW}}{2}B_{\mu\nu}W^{3\mu\nu}-\frac{\epsilon_{WW}}{4}W^{3\mu\nu}W^{3}_{\mu\nu},$
(6)

where
$\epsilon_{W}\equiv-\hat{v}^{2}/(2\Lambda_{W}^{2}),~{}\epsilon_{BW}\equiv-\hat{v}^{2}/(2\Lambda_{BW}^{2})$,
and $\epsilon_{WW}\equiv\hat{v}^{4}/(4\Lambda_{WW}^{4})$. Besides the kinetic
mixing terms above, the vector bosons might also receive the mass corrections
induced by

$\displaystyle\mathcal{L}_{\rm eff}\supset\delta
m^{2}(\hat{g}^{\prime}\hat{Z}^{\prime}_{\mu}B^{\mu}-\hat{g}\hat{Z}^{\prime}_{\mu}W^{3\mu})+\frac{1}{4}(\hat{v}^{2}+\delta
v^{2})(\hat{g}^{\prime
2}B_{\mu}B^{\mu}-2\hat{g}\hat{g}^{\prime}B_{\mu}W^{3\mu}+\hat{g}^{\prime
2}W^{3}_{\mu}W^{3\mu}).$ (7)

where $\delta m^{2}\equiv-\lambda_{HZ^{\prime}}\hat{v}^{2}/2$ and $\delta
v^{2}\equiv\hat{v}^{4}/(4\Lambda_{HD}^{2})$. Combined with (2), all the mass
terms are given by

$\displaystyle\mathcal{L}_{mass}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\hat{Z}^{\prime}_{\mu},&B_{\mu},&W^{3}_{\mu}\end{pmatrix}\mathcal{M}_{V}^{2}\begin{pmatrix}\hat{Z}^{\prime\mu}\\\
B^{\mu}\\\ W^{3\mu}\end{pmatrix},$ (8) $\displaystyle\mathcal{M}_{V}^{2}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\hat{m}_{Z^{\prime}}^{2}&\hat{g}^{\prime}\delta
m^{2}&-\hat{g}\delta m^{2}\\\ \hat{g}^{\prime}\delta
m^{2}&\frac{\hat{g}^{\prime 2}}{4}(\hat{v}^{2}+\delta
v^{2})&-\frac{\hat{g}^{\prime}\hat{g}}{4}(\hat{v}^{2}+\delta v^{2})\\\
-\hat{g}\delta m^{2}&-\frac{\hat{g}^{\prime}\hat{g}}{4}(\hat{v}^{2}+\delta
v^{2})&\frac{\hat{g}^{2}}{4}(\hat{v}^{2}+\delta v^{2})\end{pmatrix}.$ (9)

Before diagonalizing the mass-squared matrix (9), we have to diagonalize the
kinetic terms

$\displaystyle\mathcal{L}_{\mathrm{kin}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\begin{pmatrix}\hat{Z}^{\prime}_{\mu\nu},&B_{\mu\nu},&W^{3}_{\mu\nu}\end{pmatrix}\mathcal{K}_{V}\begin{pmatrix}\hat{Z}^{\prime\mu\nu}\\\
B^{\mu\nu}\\\ W^{3\mu\nu}\end{pmatrix},$ (10) $\displaystyle\mathcal{K}_{V}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}1&\epsilon_{B}&\epsilon_{W}\\\
\epsilon_{B}&1+\epsilon_{WW}&\epsilon_{BW}\\\
\epsilon_{W}&\epsilon_{BW}&1\end{pmatrix}.$ (11)

To achieve this, we initially use a congruent transformation matrix composed
of three elementary transformation matrices,

$\displaystyle V_{C}=V_{1}V_{2}V_{3},\quad
V_{C}^{\mathrm{T}}\mathcal{K}_{V}V_{C}=I_{3\times 3},$ (12)

where

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1&-\epsilon_{B}&-\epsilon_{W}\\\ 0&1&0\\\
0&0&1\end{pmatrix},~{}V_{2}=\begin{pmatrix}1&0&0\\\
0&1&\frac{-\epsilon_{BW}+\epsilon_{B}\epsilon_{W}}{1-\epsilon_{B}^{2}}\\\
0&0&1\end{pmatrix},$ $\displaystyle V_{3}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1&0&0\\\
0&\frac{1}{\sqrt{1-\epsilon_{B}^{2}}}&0\\\
0&0&\sqrt{\frac{1-\epsilon_{B}^{2}}{1+\epsilon_{WW}-\epsilon_{B}^{2}-\epsilon_{W}^{2}-\epsilon_{BW}^{2}-\epsilon_{B}^{2}\epsilon_{WW}+2\epsilon_{B}\epsilon_{W}\epsilon_{BW}}}\end{pmatrix}.$
(13)

Correspondingly, the mass-squared matrix becomes

$\displaystyle(V_{1}V_{2}V_{3})^{\mathrm{T}}\mathcal{M}_{V}^{2}V_{1}V_{2}V_{3}.$
(14)

Since

$\displaystyle\det[(V_{1}V_{2}V_{3})^{\mathrm{T}}\mathcal{M}_{V}^{2}V_{1}V_{2}V_{3}]=\det(\mathcal{M}_{V}^{2})\det(V_{1}V_{2}V_{3})^{2}=0,$
(15)

we have one massless eigenstate identified to be exactly the physical photon.

Then we are going to diagonalize the mass-squared matrix (14). Since the
analytic solution is too hard for one to manipulate, though it does exist, we
utilize a perturbative method to deal with it. We use the familiar EW rotation
matrix

$\displaystyle V_{\text{SM}}=\begin{pmatrix}1&0&0\\\
0&-\frac{\hat{g}^{\prime}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}&\frac{\hat{g}}{\sqrt{\hat{g}^{\prime 2}+\hat{g}^{2}}}\\\
0&\frac{\hat{g}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}&\frac{\hat{g}^{\prime}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}\end{pmatrix}$ (16)

to operate (14),

$\displaystyle(V_{1}V_{2}V_{3}V_{\text{SM}})^{\mathrm{T}}\mathcal{M}_{V}^{2}V_{1}V_{2}V_{3}V_{\text{SM}}=\mathcal{M}_{0\text{d}}^{2}+\delta\mathcal{M}^{2},$
(17)

where

$\displaystyle\mathcal{M}_{0\text{d}}^{2}=\operatorname{diag}\left(\hat{m}_{Z^{\prime}}^{2},~{}(\hat{g}^{2}+\hat{g}^{\prime
2})\frac{\hat{v}^{2}+\delta v^{2}}{4},~{}0\right),$ (18)

whose non-zero diagonal elements are much larger than the elements in
$\delta\mathcal{M}^{2}$. Up to the second order of the perturbation theory, we
have

$V_{\text{f},ii}\simeq 1-\sum_{i\neq
j}\frac{\delta\mathcal{M}_{ij}^{2}\delta\mathcal{M}_{ji}^{2}}{2(\mathcal{M}_{0\text{d},jj}^{2}-\mathcal{M}_{0\text{d},ii}^{2})^{2}},$
(19)

and

$\displaystyle V_{\text{f},ij}$ $\displaystyle\simeq$
$\displaystyle\frac{\delta\mathcal{M}_{ji}^{2}}{\mathcal{M}_{0\text{d},ii}^{2}-\mathcal{M}_{0\text{d},jj}^{2}}+\sum_{k\neq
i,j}\frac{\delta\mathcal{M}_{ik}^{2}\delta\mathcal{M}_{ki}^{2}}{(\mathcal{M}_{0\text{d},ii}^{2}-\mathcal{M}_{0\text{d},jj}^{2})(\mathcal{M}_{0\text{d},ii}^{2}-\mathcal{M}_{0\text{d},kk}^{2})}$
(20) $\displaystyle-$
$\displaystyle\frac{\delta\mathcal{M}_{ii}^{2}\delta\mathcal{M}_{ji}^{2}}{(\mathcal{M}_{0\text{d},ii}^{2}-\mathcal{M}_{0\text{d},jj}^{2})^{2}}$

for $i\neq j$. Finally, we acquire a transformation matrix

$\displaystyle V=V_{1}V_{2}V_{3}V_{\text{SM}}V_{\text{f}}$ (21)

to diagonalize $\mathcal{M}_{V}^{2}$,

$\displaystyle
V^{\mathrm{T}}\mathcal{M}_{V}^{2}V=\text{diag}(m_{Z^{\prime}}^{2},~{}m_{Z}^{2},~{}0),$
(22)

where

$\displaystyle m_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\hat{m}_{Z^{\prime}}^{2}+O(\epsilon_{W,B,BW,WW},\delta
v^{2},\delta m^{2}),$ $\displaystyle m_{Z}^{2}$ $\displaystyle=$
$\displaystyle(\hat{g}^{2}+\hat{g}^{\prime 2})\frac{\hat{v}^{2}+\delta
v^{2}}{4}+O(\epsilon_{W,B,BW,WW},\delta v^{2},\delta m^{2})$ (23)

are physical masses squared for the mass eigenstates $Z^{\prime}$ and $Z$.

##  III Evaluations of oblique $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$
parameters

Usually, one regards the Fermi constant $G_{F}$, QCD coupling constant
$\alpha_{s}(m_{Z})$ defined at the $Z$ boson mass scale, the fine structure
constant $\alpha(m_{Z})$, the $Z$ boson pole mass $m_{Z}$, the top quark pole
mass $m_{t}$, and the Higgs boson pole mass $m_{h}$ as the basic input
parameters to the electroweak theories. The $W$ boson mass
$m_{W}^{\mathrm{SM}}$ and the $Z$ boson decay parameters
$R_{e,\mu,\tau}^{\mathrm{SM}}$, $A_{e,\mu,\tau}^{\mathrm{SM}}$, and
$\Gamma_{Z}^{\mathrm{SM}}$ are then predicted for comparing with the
experimental results. The deviation between the theoretical predictions and
experimental results due to EW oblique corrections are usually summarized to
be the three oblique parameters $S$, $T$, and $U$ (see the definitions in Eq.
(38)) Peskin:1990zt ; Peskin:1991sw . However, the original $S$, $T$, $U$ only
include the zeroth and first $p^{2}$-order corrections of the vacuum
polarizations, which cannot fully describe the deviation induced by the
mixings of $Z^{\prime}$. From the EFT point of view, we can integrate out the
$Z^{\prime}$ boson at scales much lower than its mass and generate some dim-6,
dim-8, and even dim-10 SMEFT operators which significantly distort the vacuum
polarizations of EW gauge fields. We leave more detailed discussion of EFT in
App. A.

Before evaluating the effective $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$,
we need to clarify the definition of three “physical” quantities in this work.
The “physical” value of the Weinberg angle $\theta_{w}$ is defined by
Burgess:1993vc ; Babu:1997st

$\displaystyle s_{w}^{2}c_{w}^{2}=\frac{\pi\alpha}{\sqrt{2}m_{Z}^{2}G_{F}},$
(24)

where $s_{w}\equiv\sin\theta_{w}$ and $c_{w}\equiv\cos\theta_{w}$. The
physical value of $m_{Z}$ has been defined in Eq. (23). The fine structure
constant $\alpha$ is extracted from the effective coupling constant between
the massless vector boson (photon) and the charged particles,

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\frac{e^{2}}{4\pi},$
$\displaystyle e$ $\displaystyle=$
$\displaystyle\frac{\hat{g}^{\prime}}{2}V_{23}+\frac{\hat{g}}{2}V_{33}.$ (25)

The Fermi constant is defined as

$\displaystyle G_{F}=\frac{1}{\sqrt{2}v^{2}},$ (26)

which is the only parameter that receives no new physics contribution in this
paper.

Usually the effective $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$ parameters
can be extracted by directly calculating the self-energy diagrams of the EW
gauge bosons. In this paper, we use another equivalent method. The neutral
current (NC) and the charged current (CC) parameters extracted from the
experimental results can be adopted for comparing with the theoretical
predictions to work out the oblique parameter values. Following the steps in
Ref. Burgess:1993vc , one can acquire $S^{\prime}$, $T^{\prime}$, and
$U^{\prime}$ from the NC and CC coefficients. Expressed by the mixing
parameters, the results of the $S^{\prime}$ and $T^{\prime}$ parameters are
given by

$\displaystyle\alpha S^{\prime}$ $\displaystyle=$ $\displaystyle
4\left[-\frac{V_{22}}{V_{23}}\frac{s_{w}c_{w}}{1+{\alpha
T^{\prime}}/{2}}-s_{w}^{2}(c_{w}^{2}-s_{w}^{2})+s_{w}^{2}c_{w}^{2}\alpha
T^{\prime}\right],$ (27) $\displaystyle\alpha T^{\prime}$ $\displaystyle=$
$\displaystyle
2s_{w}c_{w}\left(\frac{V_{32}}{V_{33}}-\frac{V_{22}}{V_{23}}\right)-2.$ (28)

The $U^{\prime}$ parameter should be extracted from the charged current
coupling constants. However, equivalently it is more convenient to look into
the $W$ boson mass Peskin:1991sw

$\displaystyle
m_{W}=m_{W}^{\mathrm{SM}}\left[1-\frac{\alpha}{4(c_{w}^{2}-s_{w}^{2})}\left(S^{\prime}-2c_{w}^{2}T^{\prime}-\frac{c_{w}^{2}-s_{w}^{2}}{2s_{w}^{2}}U^{\prime}\right)\right],$
(29)

where $m_{W}={\hat{g}v}/{2}$ is the physical $W$ boson mass. Neglecting the
loop corrections, the SM prediction is
$m_{W}^{\mathrm{SM}}=\frac{\sqrt{4\pi\alpha}}{2s_{w}\sqrt{\sqrt{2}G_{F}}}$.
With the difference between $m_{W}$ and $m_{W}^{\mathrm{SM}}$, and the
$S^{\prime}$, $T^{\prime}$ parameters acquired in Eqs. (27) and (28), one can
easily derive the $U^{\prime}$ parameter.

Here we list the expressions of $S^{\prime}$, $T^{\prime}$, and $\delta
m_{W}^{2}=m_{W}^{2}-(m_{W}^{\mathrm{SM}})^{2}$ expanded up to the second order
of the parameter set $\epsilon_{B,W,BW,WW}$, $\delta m^{2}$, and $\delta
v^{2}$. The results are given by

$\displaystyle\alpha S^{\prime}$ $\displaystyle=$
$\displaystyle\frac{4gg^{\prime}}{g^{2}+g^{\prime
2}}\epsilon_{BW}-\frac{g^{2}g^{\prime
2}v^{2}(4m_{Z^{\prime}}^{2}-g^{2}v^{2})}{4(g^{2}+g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})^{2}}\epsilon_{B}^{2}+\frac{gg^{\prime}v^{2}[4(g^{2}+g^{\prime
2})m_{Z^{\prime}}^{2}-(g^{4}+g^{\prime 4})v^{2}]}{4(g^{2}+g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})^{2}}\epsilon_{B}\epsilon_{W}$ (30)
$\displaystyle-$ $\displaystyle\frac{g^{2}g^{\prime
2}v^{2}(4m_{Z^{\prime}}^{2}-g^{\prime 2}v^{2})}{4(g^{2}+g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})^{2}}\epsilon_{W}^{2}+\frac{g^{2}g^{\prime}[4m_{Z^{\prime}}^{2}-(g^{2}-g^{\prime
2})v^{2}]}{(g^{2}+g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})^{2}}\epsilon_{B}\delta m^{2}$
$\displaystyle-$ $\displaystyle\frac{gg^{\prime
2}[4m_{Z^{\prime}}^{2}+(g^{2}-g^{\prime 2})v^{2}]}{(g^{2}+g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})^{2}}\epsilon_{W}\delta
m^{2}+\frac{4g^{2}g^{\prime 2}(6g^{2}g^{\prime 2}-g^{4}-g^{\prime
4})}{(g^{4}-g^{\prime 4})^{2}}\epsilon_{BW}^{2}$ $\displaystyle-$
$\displaystyle\frac{4gg^{\prime 3}}{(g^{2}+g^{\prime
2})^{2}}\epsilon_{WW}\epsilon_{BW}+\frac{8g^{3}g^{\prime 3}}{(g^{2}-g^{\prime
2})^{2}(g^{2}+g^{\prime 2})v^{2}}\epsilon_{BW}\delta
v^{2}+\frac{3g^{6}g^{\prime 2}-2g^{4}g^{\prime 4}+3g^{2}g^{\prime
6}}{(g^{4}-g^{\prime 4})^{2}v^{4}}(\delta v^{2})^{2}$ $\displaystyle-$
$\displaystyle\frac{4g^{2}g^{\prime 2}}{(g^{2}+g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})^{2}}(\delta m^{2})^{2},$
$\displaystyle\alpha T^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{\delta
v^{2}}{v^{2}}-\frac{m_{Z^{\prime}}^{2}v^{2}}{4(m_{Z}^{\prime
2}-m_{Z}^{2})^{2}}(g^{\prime}\epsilon_{B}-g\epsilon_{W})^{2}+\frac{2m_{Z}^{2}}{(m_{Z^{\prime}}^{2}-m_{Z}^{2})^{2}}(g^{\prime}\epsilon_{B}-g\epsilon_{W})\delta
m^{2}$ (31) $\displaystyle+$ $\displaystyle\frac{3}{4v^{4}}(\delta
v^{2})^{2}+\frac{4(m_{Z^{\prime}}^{2}-2m_{Z}^{2})}{(m_{Z^{\prime}}^{2}-m_{Z}^{2})^{2}v^{2}}(\delta
m^{2})^{2},$ $\displaystyle\delta m_{W}^{2}$ $\displaystyle=$
$\displaystyle-\frac{g^{3}g^{\prime}v^{2}}{2(g^{2}-g^{\prime
2})}\epsilon_{BW}-\frac{g^{4}}{4(g^{2}-g^{\prime 2})}\delta
v^{2}+\frac{g^{2}v^{2}}{4}\epsilon_{WW}$ (32) $\displaystyle+$
$\displaystyle\frac{g^{4}v^{4}}{16(g^{2}-g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})}(g^{\prime}\epsilon_{B}-g\epsilon_{W})^{2}+\frac{2g^{4}v^{2}}{4(g^{2}-g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})}(g\epsilon_{W}-g^{\prime}\epsilon_{B})\delta
m^{2}$ $\displaystyle-$ $\displaystyle\frac{g^{4}(g^{2}-3g^{\prime
2})(g^{2}+g^{\prime 2})v^{2}}{4(g^{2}-g^{\prime
2})^{3}}\epsilon_{BW}^{2}+\frac{g^{3}g^{\prime}(g^{2}-2g^{\prime
2})v^{2}}{2(g^{2}-g^{\prime 2})^{2}}\epsilon_{BW}\epsilon_{WW}$
$\displaystyle-$
$\displaystyle\frac{g^{2}v^{2}}{4}\epsilon_{WW}^{2}+\frac{g^{4}(g^{2}-2g^{\prime
2})}{4(g^{2}-g^{\prime 2})^{2}}\epsilon_{WW}\delta v^{2}+\frac{g^{4}g^{\prime
4}}{4(g^{2}-g^{\prime 2})^{3}v^{2}}(\delta v^{2})^{2}$ $\displaystyle-$
$\displaystyle\frac{g^{5}g^{\prime}(g^{2}-3g^{\prime 2})}{2(g^{2}-g^{\prime
2})^{3}}\epsilon_{BW}\delta v^{2}+\frac{g^{4}}{(g^{2}-g^{\prime
2})(m_{Z^{\prime}}^{2}-m_{Z}^{2})}(\delta m^{2})^{2}.$

Originally, the above $g$, $g^{\prime}$, $v$, etc. parameters should be
“hatted” and become $\hat{g}$, $\hat{g}^{\prime}$, $\hat{v}$, etc. However,
since the shifts of all these parameters from the physical ones are extremely
small, we can conveniently utilize the physical parameters to evaluate the
$S^{\prime}$, $T^{\prime}$, and $\delta m_{W}^{2}$ instead.

Since the above formulas for evaluating the oblique parameters are rather
complicated, it is not essential to perform a thorough fitting on all these
parameters, so in the rest of this paper we will focus on several specific
cases.

##  IV Phenomenological discussions on $Z^{\prime}$ collider bounds and the
oblique parameters

###  IV.1 $Z^{\prime}$ collider bounds

In order to generate a significant positive $U^{\prime}$ parameter, the
$Z^{\prime}$ mass should lie within a range $m_{Z}<m_{Z^{\prime}}\lesssim 400$
GeV. Since $Z^{\prime}$ couples to both leptons and quarks due to its kinetic
mixing with SM gauge fields, it can be produced both in lepton and hadron
colliders. The neutral current interactions with $Z^{\prime}$ are

$\displaystyle\mathcal{L}_{Z^{\prime}_{\mu}J_{f}^{\mu}}=\sum_{f}\bar{f}\gamma^{\mu}\left[g_{f}^{(V)}+g_{f}^{(A)}\gamma_{5}\right]fZ^{\prime}_{\mu},$
(33)

where $f=u_{i},d_{i},\nu_{i},e_{i}$ are SM fermions, and the couplings are
given by

$\displaystyle g_{f}^{(V)}$ $\displaystyle\approx$ $\displaystyle
e\left[Q_{f}\frac{s_{w}\epsilon_{W}+(c_{w}-r/c_{w})\epsilon_{B}+t_{w}\xi}{r-1}-\frac{r(c_{w}\epsilon_{W}-s_{w}\epsilon_{B})+\xi}{2s_{w}c_{w}(r-1)}T_{f_{L}}^{3}\right],$
(34) $\displaystyle g_{f}^{(A)}$ $\displaystyle\approx$ $\displaystyle
Q_{f}e\frac{r(c_{w}\epsilon_{W}-s_{w}\epsilon_{B})+\xi}{2s_{w}c_{w}(r-1)}T_{f_{L}}^{3},$
(35)

with $t_{w}\equiv\tan\theta_{w}$, $r\equiv m_{Z^{\prime}}^{2}/m_{Z}^{2}$, and
$\xi\equiv\sqrt{g^{2}+g^{\prime 2}}\delta m^{2}/m_{Z}^{2}$.

For $m_{Z^{\prime}}\lesssim 209$ GeV, $Z^{\prime}$ may be directly produced at
the LEP collider with a significant signal. The null result of the on-shell
$Z^{\prime}$ searches at the LEP either pushes the $Z^{\prime}$ mass heavier
than $209$ GeV, or suppresses the couplings $g_{l}^{(V,A)}$ to leptons smaller
than $\mathcal{O}(10^{-2})$ ParticleDataGroup:2020ssz . For
$m_{Z^{\prime}}>209$ GeV, the LEP bound on the off-shell $Z^{\prime}$
production can be interpreted to be
$m_{Z^{\prime}}/\sqrt{g_{e}^{(L)}g_{f}^{(L)}}\gtrsim 4$ TeV Carena:2004xs ,
where $g_{f}^{(L)}=(g_{f}^{(V)}-g_{f}^{(A)})/2$. In order to avoid the
stringent on-shell bound from the LEP, we only concern in the off-shell range
$m_{Z^{\prime}}>220$ GeV, so $g_{f}^{(L)}$ is calculated to be $\lesssim 0.03$
within our interested parameter region $|\epsilon_{B,W}|<0.1$ when $\xi$ is
negligible. Therefore we do not have to worry about the LEP bounds in this
paper.

At the hadron colliders, $Z^{\prime}$ might be probed through the $pp\to
Z^{\prime}\to\text{dijets}$ searches. Current bounds on the universal vector-
current coupling of $Z^{\prime}$ to quarks is $g^{\prime}_{q}\lesssim 0.1$ for
$220~{}\textrm{GeV}\lesssim m_{Z^{\prime}}\lesssim 400$ GeV CMS:2019emo ;
UA2:1993tee ; CMS:2018kcg . A more stringent estimated bound
$g^{\prime}_{q}\lesssim 0.05$ can be acquired around $m_{Z^{\prime}}\sim$TeV
CMS:2016ltu ; ATLAS:2019fgd  (see Figure 88.2 in Ref.
ParticleDataGroup:2020ssz  for a summary of hadron collider bounds). Since our
$Z^{\prime}$-fermion couplings are chiral, in order to compare with the
$g^{\prime}_{q}$ bounds, we define an effective coupling

$\displaystyle
g_{q}^{\text{eff}}=\sqrt{\frac{(g_{q}^{(V)})^{2}+(g_{q}^{(A)})^{2}}{2}}$ (36)

as an estimate of our theoretical value for $g^{\prime}_{q}$. We find that
$g_{q}^{\text{eff}}\lesssim 0.04$ for $|\epsilon_{B,W}|<0.1$ and
$m_{Z^{\prime}}>209$ GeV with negligible $\xi$ is still consistent with the
current LHC bounds. Since the parameter regions of our interest are
sufficiently safe from the analysis of the collider constraints, we shall
neglect them in our following discussions.

###  IV.2 Bounds on the oblique parameters $W$ and $Y$

As will be discussed in the Appendix A, the contributions from the
$\epsilon_{W,B}$ terms to $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$ include
the contributions from $V$, $W$, $X$, and $Y$. (See their definitions in
Appendix A.) To constrain these parameters, one has to study the low-energy
experimental data as in Refs. Barbieri:2004qk ; Cacciapaglia:2006pk , or
directly extract the shape of the vector boson propagators through the
collider data. In this paper we only discuss the latter constraint, which is
more stringent. Refs. Torre:2020aiz ; Panico:2021vav  provided the proposal to
utilize the charged and neutral Drell-Yan differential cross-section
measurements to constrain the $W$ and $Y$ parameters. In Ref. CMS:2022yjm ,
the CMS collaboration published a measurement result
$W=-1.2^{+0.5}_{-0.6}\times 10^{-4}$ through charged Drell-Yan processes. Ref.
Strumia:2022qkt  applied the $l^{+}l^{-}$ data presented in Ref. ATLAS:2017fih
to constrain the $Y$ parameter as $|Y|\lesssim 2\times 10^{-4}$. Such
stringent bounds on $W$ and $Y$ naively exclude nearly the whole interested
parameter space. However, the fitting results are actually based upon some
assumptions which are not the case in this paper.

In the above studies, the $W$ parameter is extracted from the charged Drell-
Yan data, and the corresponding effective operator is
$(D_{\rho}W_{\mu\nu}^{a})^{2}$, which affect $W^{3}$ and $W^{\pm}$
universally. This is eligible when $V=0$. In our paper, however, the kinetic
mixing term $\frac{\epsilon_{W}}{2}\hat{Z}^{\prime}_{\mu\nu}W^{3\mu\nu}$ in
the Lagrangian (6) as well as its corresponding effective operator
$\frac{1}{2\Lambda_{W}^{2}}\hat{Z}^{\prime}_{\mu\nu}W^{a\mu\nu}H^{\dagger}\sigma^{a}H$
in the Lagrangian (2) only affects $W^{3}$ without disturbing $W^{\pm}$,
resulting in $W=V$. (We follow Ref.Barbieri:2004qk  to define $W$ by
$\Pi^{\prime\prime}_{W^{3}W^{3}}(0)$, and define $V$ by
$\Pi^{\prime\prime}_{W^{3}W^{3}}(0)-\Pi^{\prime\prime}_{W^{+}W^{-}}(0)$.)
Since in this case $W$ does not correct the charged Drell-Yan processes
mediated by the off-shell $W^{\pm}$, the $W$ bound presented in Ref.
CMS:2022yjm  can be safely neglected.

Fig. 8 in Ref. Torre:2020aiz  showed the projected exclusion regions of the
$W$ and $Y$ parameters from both the neutral and charged Drell-Yan
measurements at 13 TeV LHC. As we have mentioned, the charged result does not
constrain our case, so only the neutral results are effective. The combined
constraints on $W$ and $Y$ in Ref. Strumia:2022qkt  are also based upon the
$V=0$ assumption, and therefore become invalid again.

In the $Y=0$ case at the 13 TeV LHC with an integrated luminosity
$100~{}$\mathrm{f}\mathrm{b}^{-1}$$, the neutral results in Ref. Torre:2020aiz
predicted a $95\%$ CL bound $|W|\lesssim 0.4\times 10^{-3}$, which is
equivalent to $|U^{\prime}|\lesssim 0.1$. One can verify from the figures in
the next section that such a projected bound is at the brink of our desired
parameter space to accommodate an appropriate $\delta m_{W}^{2}$ when
$\epsilon_{B}=0$. However, this is only a theoretical estimation, and up till
now, we find no extractions of the $W$, $Y$ constraints merely from the
neutral Drell-Yan experimental data in the literature. Moreover, moderate
$\epsilon_{B}$ and $\epsilon_{W}$ should also give rise to a non-negligible
$X$, which modifies the neutral boson propagators as well, but is simply
discarded in all the references above. Therefore all the existing collider
bounds on the $W$ and $Y$ parameters become inapplicable in our case, except
for a small region near $\epsilon_{B}=0$. Thus, we neglect all of them in our
following discussions.

##  V Numerical results

In order to study the space of the parameters, we adopt the “standard average”
result of the $S$, $T$, $U$ parameters from the EW global fit with the recent
CDF $m_{W}$ measurement in Ref. deBlas:2022hdk , as tabulated in Tab. 1. In
Ref. deBlas:2022hdk  only the measurements of the precision observables at the
EW scale and the Fermi constant $G_{F}$ are included, permitting a
straightforward comparison with our effective $S^{\prime}$, $T^{\prime}$ and
$U^{\prime}$. For each of the parameter point, we compute the oblique
parameters and then evaluate the corresponding $\chi^{2}$ based on this
result.

| Result | Correlation
---|---|---
$S$ | $0.005\pm 0.097$ | $1.00$ |  |
$T$ | $0.04\pm 0.12$ | $0.91$ | $1.00$ |
$U$ | $0.134\pm 0.087$ | $-0.65$ | $-0.88$ | $1.00$
Table 1: Global fit results of the oblique parameters $S$, $T$, and $U$
adopted from Ref. deBlas:2022hdk .

The kinetic mixing parameters $\epsilon_{B}$ and $\epsilon_{W}$ contribute
positive values to the $U^{\prime}$ parameter, lifting the mass of the $W$
boson. However, from Eq. (31) we learn that the $T^{\prime}$ parameter
simultaneously acquires a negative contribution. Therefore, a tension arises
when we try to fit with the results in Tab. 1 in the case that all the other
mixing parameters disappear. In Figs. 1 and 2, we plot the fit results on the
$\epsilon_{B}$ versus $\epsilon_{W}$ plain for $m_{Z^{\prime}}=220$ GeV and
300 GeV, respectively. The values of the oblique parameters are also shown as
contours. The non-negligible $U^{\prime}$ parameter plays an important role in
accumulating the predicted $m_{W}$. However, when $m_{Z^{\prime}}$ increases,
the negative $T^{\prime}$ values become harmful in approaching the CDF
measured $m_{W}^{\text{CDF}}$, so the best-fit $\chi^{2}$ arises swiftly,
failing to give a proper fit.

|
---|---
|
Figure 1: Plots for $m_{Z^{\prime}}=220$ GeV and $\delta v^{2}=\epsilon_{WW}=\epsilon_{BW}=\delta m^{2}=0$ on the $\epsilon_{B}$ versus $\epsilon_{W}$ plain. The upper left panel shows the $1\sigma$ (blue), $2\sigma$ (orange) regions calculated according to the global fit result in Ref. deBlas:2022hdk , as well as the contours of $\delta m_{W}=m_{W}-m_{W}^{\mathrm{SM}}$ also displayed. The marks “0.02”, “0.04”, etc. correspond to $\delta m_{W}=0.02,0.04$ GeV, etc. The minimal chi-squared $\chi^{2}_{\min}$ is indicated in the plot title. The remaining three panels display the contours of $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$. In all the panels, the ${\epsilon_{B}}/{\epsilon_{W}}={g}/{g^{\prime}}$ and ${\epsilon_{B}}/{\epsilon_{W}}=-{g}/{g^{\prime}}$ lines indicate the photon-$Z^{\prime}$ and $Z$-$Z^{\prime}$ mixings, respectively. |
---|---
|
Figure 2: Plots for $m_{Z^{\prime}}=300$ GeV and $\delta
v^{2}=\epsilon_{WW}=\epsilon_{BW}=\delta m^{2}=0$ on the $\epsilon_{B}$ versus
$\epsilon_{W}$ plain. The symbols are the same as in Fig. 1.

In the $\epsilon_{B}$ and $\epsilon_{W}$ parameter space there are two
specific combinations of the parameters:
$g^{\prime}\epsilon_{B}=g\epsilon_{W}$ and
$g\epsilon_{B}=-g^{\prime}\epsilon_{W}$. The previous one is equivalent to the
case that $Z^{\prime}$ only mixes with the photon, and
$g^{\prime}\epsilon_{B}-g\epsilon_{W}=0$ results in vanishing $T^{\prime}$ and
$U^{\prime}$ according to Eqs. (31) and (32). On the contrary, when
$g\epsilon_{B}=-g^{\prime}\epsilon_{W}$, $Z^{\prime}$ only mixes with the SM
$Z$ boson, which had been discussed and evaluated in Ref. Zeng:2022lkk ,
although here we perform a more general analytic calculation.

If we turn on other parameters to contribute to $S^{\prime}$ and $T^{\prime}$,
the tension with the global fit can be significantly relieved. Besides the
contribution to $S^{\prime}$, the most urgent task is to hoist the value of
$T^{\prime}$ from the negative abyss. As we have mentioned, a positive
contribution to $T$ can be realized by adding some extra EW multiplets, e.g.,
those in the singlet-doublet scalar dark matter model presented in Ref.
Cai:2017wdu . From the EFT point of view, the new EW multiplets could generate
the $|H^{\dagger}D_{\mu}H|^{2}$ operator and hence $\delta v^{2}$ at loop
level. Both $\delta v^{2}$ and $\delta m^{2}$ contribute positively to
$T^{\prime}$, while their contributions to $S^{\prime}$ and $U^{\prime}$ are
suppressed. Since their impact on $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$
are similar, as an example, we choose to switch on $\delta v^{2}$ and present
the results in Fig. 3.

Figure 3: $1\sigma$ (blue) and $2\sigma$ (orange) regions of the fit and
$\delta m_{W}$ contours for
$m_{Z^{\prime}}=220~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ (upper left),
$300~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ (upper right), and
$400~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ (lower) with $\delta
v^{2}=-50~{}$\mathrm{G}\mathrm{e}\mathrm{V}^{2}$$ and
$\epsilon_{WW}=\epsilon_{BW}=\delta m^{2}=0$. The symbols are the same as in
Fig. 1.

Figure 4: Contours of $S^{\prime}$ (upper left), $T^{\prime}$ (upper right),
and $U^{\prime}$ (lower) for
$m_{Z^{\prime}}=300~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$, $\delta
v^{2}=-50~{}$\mathrm{G}\mathrm{e}\mathrm{V}^{2}$$ and
$\epsilon_{WW}=\epsilon_{BW}=\delta m^{2}=0$. The symbols are the same as in
Fig. 1.

From Fig. 3 one can easily reckon that the best-fit $\chi^{2}$ significantly
lowers, and thus relatively heavier $Z^{\prime}$ with $m_{Z^{\prime}}\gtrsim
400$ GeV can also explain the CDF result very well. Comparing the
$S^{\prime}$, $T^{\prime}$, and $U^{\prime}$ contours for $m_{Z^{\prime}}=300$
GeV in Fig. 4 with $\delta v^{2}=-50~{}$\mathrm{G}\mathrm{e}\mathrm{V}^{2}$$
and Fig. 2 with $\delta v^{2}=0$, we find that a positive $T^{\prime}$ can
significantly improve the fit.

##  VI Summary and future prospect

We have enumerated the possible interactions involving a $Z^{\prime}$ field
that induce its mixings with the neutral EW gauge bosons. Both the
$\epsilon_{B}$ and $\epsilon_{W}$ parameters contribute positively to the non-
negligible $U^{\prime}$ parameter, with the price of lowering the $T^{\prime}$
parameter significantly. Appropriate selections of the parameters such as
$\delta v^{2}$ or $\delta m^{2}$ can accumulate $T^{\prime}$ to relieve the
tension between the global fit results and our theoretical predictions. The
sufficient increase of the $W$-boson mass can be accomplished to explain the
$W$-mass anomaly measured by the CDF II detector within the current collider
bounds on the $Z^{\prime}$ boson.

In this paper, we rely on effective field theory with various non-
renormalizable operators listed in (3) and (4). These operators can arise from
charged particles running inside the loops. Building an ultraviolet-complete
model inducing all these terms with appropriate coupling strengths will become
an important task.

##  Appendix A Definitions and discussions of the effective $S^{\prime}$,
$T^{\prime}$ and $U^{\prime}$

Denote $\Pi_{IJ}(p^{2})$ to be the NP contribution to the $g_{\mu\nu}$
coefficient of the vacuum polarization amplitude for EW gauge fields $I$ and
$J$, and expand it around $p^{2}=0$,

$\Pi_{IJ}(p^{2})\simeq\Pi_{IJ}(0)+p^{2}\Pi^{\prime}_{IJ}(0)+\frac{(p^{2})^{2}}{2!}\Pi^{\prime\prime}_{IJ}(0)+\frac{(p^{2})^{3}}{3!}\Pi^{(3)}_{IJ}(0)+\cdots$
(37)

Then the Peskin-Takeuchi oblique parameters Peskin:1991sw  are defined by

$\displaystyle\alpha S$ $\displaystyle=$ $\displaystyle
4s_{w}^{2}c_{w}^{2}\left[\Pi_{ZZ}^{\prime}(0)-\frac{c_{w}^{2}-s_{w}^{2}}{s_{w}c_{w}}\Pi_{ZA}^{\prime}(0)-\Pi_{AA}^{\prime}(0)\right]=4s_{w}c_{w}\Pi^{\prime}_{W^{3}B}(0),$
$\displaystyle\alpha T$ $\displaystyle=$
$\displaystyle\frac{\Pi_{WW}(0)}{m_{W}^{2}}-\frac{\Pi_{ZZ}(0)}{m_{Z}^{2}}=\frac{1}{m_{W}^{2}}[\Pi_{W^{+}W^{-}}(0)-\Pi_{W^{3}W^{3}}(0)],$
$\displaystyle\alpha U$ $\displaystyle=$ $\displaystyle
4s_{w}^{2}\left[\Pi_{WW}^{\prime}(0)-c_{w}^{2}\Pi_{ZZ}^{\prime}(0)-2s_{w}c_{w}\Pi_{ZA}^{\prime}(0)-s_{w}^{2}\Pi_{AA}^{\prime}(0)\right]$
(38) $\displaystyle=$ $\displaystyle
4e^{2}[\Pi^{\prime}_{W^{+}W^{-}}(0)-\Pi^{\prime}_{W^{3}W^{3}}(0)].$

It is well-known that $S$, $T$, and $U$ correspond to the
$H^{\dagger}W^{a}_{\mu\nu}\sigma^{a}HB^{\mu\nu}$,
$H^{\dagger}D_{\mu}H(D^{\mu}H)^{\dagger}H$, and
$H^{\dagger}W^{a}_{\mu\nu}\sigma^{a}HH^{\dagger}W^{b\mu\nu}\sigma^{b}H$
operators, respectively. In the literature, their effects on various EW
precision observables are evaluated. Comparing the evaluated results with the
experimental data, one can constrain the allowed region of $S$, $T$, and $U$.

Figure 5: Diagrams integrating out the $Z^{\prime}$ boson to accomodate the
effective operators.

However, other operators which do not contribute to Eq. (38) might also shift
exactly the same observables to fake the effects of $S$, $T$, and $U$. The
$\epsilon_{B}$ and $\epsilon_{B}$ terms displayed in the effective Lagrangian
(6) induce the following operators,

$\mathcal{O}_{BB}^{2i}=B_{\mu\nu}\partial^{2i}B^{\mu\nu},\quad\mathcal{O}_{33}^{2i}=W^{3}_{\mu\nu}\partial^{2i}W^{3\mu\nu},\quad\mathcal{O}_{B3}^{2i}=B_{\mu\nu}\partial^{2i}W^{3\mu\nu},$
(39)

by integrating out the heavy $Z^{\prime}$ in the tree-level
$B/W^{3}$-$Z^{\prime}$-$B/W^{3}$ oscillation diagrams shown in Fig. 5. For the
sake of the completeness of this paper, we also list the following operator,

$\displaystyle\mathcal{O}_{WW}^{2i}$ $\displaystyle=$ $\displaystyle
W^{+\mu\nu}\partial^{2i}W^{-}_{\mu\nu}.$ (40)

These set of operators correspond to the higher-order derivatives of the
vacuum polarization functions

$\displaystyle\Pi^{(i)}_{IJ}(p^{2})|_{p^{2}=0}$ $\displaystyle\sim$
$\displaystyle\mathcal{O}_{IJ}^{2i},$ (41)

where $I,J=B,W^{3},W^{\pm}$. When $i=2$, more oblique parameters $V$, $X$,
$Y$, and $W$ can be defined to evaluate the oblique corrections
Barbieri:2004qk . Their definitions are

$\displaystyle V$ $\displaystyle=$
$\displaystyle-\frac{1}{2}m_{W}^{2}[\Pi^{\prime\prime}_{W^{3}W^{3}}(0)-\Pi^{\prime\prime}_{W^{+}W^{-}}(0)],$
$\displaystyle X$ $\displaystyle=$
$\displaystyle-\frac{1}{2}m_{W}^{2}\Pi^{\prime\prime}_{W^{3}B}(0),$
$\displaystyle Y$ $\displaystyle=$
$\displaystyle-\frac{1}{2}m_{W}^{2}\Pi^{\prime\prime}_{BB}(0),$ $\displaystyle
W$ $\displaystyle=$
$\displaystyle-\frac{1}{2}m_{W}^{2}\Pi^{\prime\prime}_{W^{3}W^{3}}(0).$ (42)

In the former studies, the vacuum polarization functions $\Pi_{IJ}(p^{2})$ are
expanded at most to the second order. However, later we will see, the higher
orders account for parts of our results in this paper. Therefore we will give
more general discussions below. We should note that alternative definitions of
the $S$, $T$, $U$, $V$, $W$, $X$ exist in the literature (See Ref.
Burgess:1993mg  as an example) which are somehow but incompletely equivalent
with the definitions that we adopt from Ref. Peskin:1991sw ; Barbieri:2004qk .
More straightforward comparisons between the oblique parameters and the
effective operators are the advantages of our selections of the oblique
parameter definitions.

For brevity of this appendix, let us only preserve the $\epsilon_{B}$ and
$\epsilon_{W}$ terms, and integrating out the $Z^{\prime}$ boson. To the
lowest order, we have

$\Pi_{BB}(p^{2})=\frac{\epsilon_{B}^{2}p^{4}}{p^{2}-m_{Z^{\prime}}^{2}},\quad\Pi_{W^{3}W^{3}}(p^{2})=\frac{\epsilon_{W}^{2}p^{4}}{p^{2}-m_{Z^{\prime}}^{2}},\quad\Pi_{W^{3}B}(p^{2})=\frac{\epsilon_{B}\epsilon_{W}p^{4}}{p^{2}-m_{Z^{\prime}}^{2}}.$
(43)

Immediately we obtain

$\Pi_{BB}^{(i)}(0)=-\frac{i!\epsilon_{B}^{2}}{m_{Z^{\prime}}^{2i-2}},\quad\Pi_{W^{3}W^{3}}^{(i)}(0)=-\frac{i!\epsilon_{W}^{2}}{m_{Z^{\prime}}^{2i-2}},\quad\Pi_{W^{3}B}^{(i)}(0)=-\frac{i!\epsilon_{B}\epsilon_{W}}{m_{Z^{\prime}}^{2i-2}},$
(44)

for $i\geq 2$, and $\Pi_{IJ}(0)=\Pi_{IJ}^{\prime}(0)=0$.

Let us define

$\alpha_{i}=-\frac{\Pi_{BB}^{(i)}}{i!},\quad\beta_{i}=-\frac{\Pi_{W^{3}W^{3}}^{(i)}}{i!},\quad\gamma_{i}=-\frac{\Pi_{W^{3}B}^{(i)}}{i!},$
(45)

and the reciprocal of the resummed propagator (inverse propagator) matrix for
$(B,W^{3})$ regardless of the tensor part is given by

$\displaystyle\Pi^{2\times 2}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}p^{2}-\frac{\hat{g}^{\prime
2}}{4}\hat{v}^{2}-\Pi_{BB}(p^{2})&\frac{\hat{g}\hat{g}^{\prime}}{4}\hat{v}^{2}-\Pi_{W^{3}B}(p^{2})\\\
\frac{\hat{g}\hat{g}^{\prime}}{4}\hat{v}^{2}-\Pi_{W^{3}B}(p^{2})&p^{2}-\frac{\hat{g}^{2}}{4}\hat{v}^{2}-\Pi_{W^{3}W^{3}}(p^{2})\end{pmatrix}$
(46) $\displaystyle=$
$\displaystyle\begin{pmatrix}\sum\limits_{i=2}^{\infty}\alpha_{i}p^{2i}+p^{2}-\frac{\hat{g}^{\prime
2}}{4}\hat{v}^{2}&\sum\limits_{i=2}^{\infty}\gamma_{i}p^{2i}+\frac{\hat{g}\hat{g}^{\prime}}{4}\hat{v}^{2}\\\
\sum\limits_{i=2}^{\infty}\gamma_{i}p^{2i}+\frac{\hat{g}\hat{g}^{\prime}}{4}\hat{v}^{2}&\sum\limits_{i=2}^{\infty}\beta_{i}p^{2i}+p^{2}-\frac{\hat{g}^{2}}{4}\hat{v}^{2}\end{pmatrix}.~{}$

The physical $Z$ and $\gamma$ masses are related to the solution of the
equation $\det(\Pi_{4\times 4})=0$. Similar to the $\alpha_{i}=0$,
$\beta_{i}=0$, $\gamma_{i}=0$ $(i\geq 2)$ case, we can turn to the mass
eigenstates through the EW rotation matrix

$V_{\text{SM}}^{2\times
2}=\begin{pmatrix}-\frac{\hat{g}^{\prime}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}&\frac{\hat{g}}{\sqrt{\hat{g}^{\prime 2}+\hat{g}^{2}}}\\\
\frac{\hat{g}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}&\frac{\hat{g}^{\prime}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}\end{pmatrix},$ (47)

so that

$\displaystyle(V_{\text{SM}}^{2\times 2})^{\mathrm{T}}\Pi^{2\times
2}V_{\text{SM}}^{2\times
2}=\begin{pmatrix}\sum\limits_{i=2}^{\infty}a_{i}p^{2i}+p^{2}-\hat{m}_{Z}^{2}&\sum\limits_{i=2}^{\infty}c_{i}p^{2i}\\\
\sum\limits_{i=2}^{\infty}c_{i}p^{2i}&\sum\limits_{i=2}^{\infty}b_{i}p^{2i}+p^{2}\end{pmatrix},$
(48)

where $\hat{m}_{Z}^{2}=(\hat{g}^{2}+\hat{g}^{\prime 2})\hat{v}^{2}/4$, and

$\displaystyle\begin{pmatrix}a_{i}&c_{i}\\\
c_{i}&b_{i}\end{pmatrix}=(V_{\text{SM}}^{2\times
2})^{\mathrm{T}}\begin{pmatrix}\alpha_{i}&\gamma_{i}\\\
\gamma_{i}&\beta_{i}\end{pmatrix}V_{\text{SM}}^{2\times 2}.$ (49)

Notice that there is a massless solution $p^{2}=0$ with the eigenvector
$(0,1)^{\mathrm{T}}$ corresponding to the massless photon.

Another eigenvector can be parameterized as $(1,t)^{\mathrm{T}}$, so we have
to solve the equation

$\displaystyle\begin{pmatrix}\sum\limits_{i=2}^{\infty}a_{i}p^{2i}+p^{2}-\hat{m}_{Z}^{2}&\sum\limits_{i=2}^{\infty}c_{i}p^{2i}\\\
\sum\limits_{i=2}^{\infty}c_{i}p^{2i}&\sum\limits_{i=2}^{\infty}b_{i}p^{2i}+p^{2}\end{pmatrix}\begin{pmatrix}1\\\
t\end{pmatrix}=0.$ (50)

When all $a_{i},b_{i},c_{i}=0$, we derive $t=0$. If $\alpha_{i}m_{Z}^{2i-2}$,
$\beta_{i}m_{Z}^{2i-2}$, and $\gamma_{i}m_{Z}^{2i-2}$ (or $a_{i}m_{Z}^{2i-2}$,
$b_{i}m_{Z}^{2i-2}$, and $c_{i}m_{Z}^{2i-2}$) are considered to be of the same
order which are much smaller than $1$ regardless of the power index, one can
expect that $t\ll 1$ is also of the same order. Then we can solve Eq. (50)
perturbatively by discarding all the higher-order terms,

$\displaystyle\left\\{\begin{array}[]{l}\sum\limits_{i=2}^{\infty}a_{i}p^{2i}+t\sum\limits_{i=2}^{\infty}c_{i}p^{2i}+p^{2}-\hat{m}_{Z}^{2}=0,\\\
\sum\limits_{i=2}^{\infty}c_{i}p^{2i}+tp^{2}=0,\end{array}\right.\Rightarrow\left\\{\begin{array}[]{l}p^{2}\simeq\hat{m}_{Z}^{2}-\sum\limits_{i=2}^{\infty}a_{i}\hat{m}_{Z}^{2i},\\\
t\simeq-\sum\limits_{i=2}^{\infty}c_{i}m_{Z}^{2i-2}.\end{array}\right.$ (55)

After rotating $\Pi^{2\times 2}$ with

$\displaystyle V_{t}=\begin{pmatrix}1&t\\\ 0&1\end{pmatrix},$ (56)

we acquire the “diagonalized” inverse propagator

$\displaystyle\Pi^{2\times
2}_{\text{M}}=V_{t}^{\mathrm{T}}(V_{\text{SM}}^{2\times
2})^{\mathrm{T}}\Pi^{2\times 2}V_{\text{SM}}^{2\times 2}V_{t}.$ (57)

The element $\Pi^{2\times
2}_{\text{M},00}(p^{2})\simeq\sum\limits_{i=2}^{\infty}a_{i}p^{2i}+\sum\limits_{i=2}^{\infty}c_{i}p^{2i}+p^{2}-\hat{m}_{Z}^{2}$
is the inverse propagator of the physical $Z$ boson near its pole, and the
reciprocal of the residue of the pole is

$\displaystyle\left.\frac{\partial\Pi^{2\times
2}_{\text{M},00}(p^{2})}{\partial
p^{2}}\right|_{p^{2}=\hat{m}_{Z}^{2}-\sum\limits_{i=2}^{\infty}a_{i}\hat{m}_{Z}^{2i}}\simeq
1+\sum\limits_{i=2}^{\infty}ia_{i}p^{2i-2}.$ (58)

A complete calculation of the NC and CC terms requires normalizing out this
factor. Therefore, comparing (55), (56) and (58) with the corresponding terms
of Eq. (5) and (6) in Ref. Burgess:1993vc , we can see that the physical mass
shift $-\sum\limits_{i=2}^{\infty}a_{i}\hat{m}_{Z}^{2i}$, the rotation
parameter $t$, and the field normalization factor
$\sum\limits_{i=2}^{\infty}ia_{i}p^{2i-2}$ are equivalent to
$(z-C)\hat{m}_{Z}^{2}$, $G$, and $C$ defined in Ref. Burgess:1993vc ,
respectively. Straightforwardly casting the symbols there, we have

$(z-C)=-\sum\limits_{i=2}^{\infty}a_{i}\hat{m}_{Z}^{2i-2},\quad
G=-\sum\limits_{i=2}^{\infty}c_{i}\hat{m}_{Z}^{2i-2},\quad
C=\sum\limits_{i=2}^{\infty}ia_{i}\hat{m}_{Z}^{2i-2}.$ (59)

If the operator (40) arises, the inverse propagator of the $W$-boson
regardless of the tensor part is given by

$\displaystyle
p^{2}-\hat{m}_{W}^{2}-\Pi_{W^{+}W^{-}}(p^{2})=\sum\limits_{i=2}^{\infty}d_{i}p^{2i}+p^{2}-\hat{m}_{W}^{2},$
(60)

where $\hat{m}_{W}=\hat{g}v/2$. Again, solving
$p^{2}-\hat{m}_{W}^{2}-\Pi_{W^{+}W^{-}}(p^{2})=0$ gives the solution of the
physical $W$-boson mass,

$\displaystyle
m_{W}^{2}=p^{2}=\hat{m}_{W}^{2}-\sum\limits_{i=2}^{\infty}d_{i}\hat{m}_{W}^{2i}.$
(61)

The reciprocal of the residue of the propagator near the pole accordingly
becomes

$\displaystyle 1-\left.\frac{\partial\Pi_{W^{+}W^{-}}(p^{2})}{\partial
p^{2}}\right|_{p^{2}=\hat{m}_{W}^{2}-\sum\limits_{i=2}^{\infty}d_{i}\hat{m}_{Z}^{2i}}\simeq
1+\sum\limits_{i=2}^{\infty}id_{i}p^{2i-2}.$ (62)

Comparing with the corresponding terms of Eq. (4) and (6) in Ref.
Burgess:1993vc  and casting its symbols again similarly, we obtain

$(w-B)=-\sum\limits_{i=2}^{\infty}d_{i}\hat{m}_{W}^{2i-2},\quad
B=\sum\limits_{i=2}^{\infty}id_{i}\hat{m}_{W}^{2i-2}.$ (63)

Therefore, if we follow Eqs. (27), (28) and (29) to compute the oblique
paramters, we are actually substituting our effective $B$, $C$, $G$, $w$, and
$z$ into Eq. (2) in Ref. Burgess:1993vc  to acquire the effective
$S^{\prime}$, $T^{\prime}$, and $U^{\prime}$ values. Before displaying the
results, we should note that the $w$ parameter appeared in Eq. (15) of Ref.
Burgess:1993vc  should be abolished in our case, since $G_{F}$ is defined in
the low-$p^{2}$ limit so that all the $d_{i}p^{2}$ terms in Eq. (60) become
ineffective. Therefore, all the $w$ symbols corresponding to Eqs. (17) and
(18) in Ref. Burgess:1993vc  should be discarded. Our $S^{\prime}$ and
$T^{\prime}$ in Eqs. (27) and (28) are derived by matching Eq. (23) in Ref.
Burgess:1993vc , and $w$ there comes from Eq. (17), so it completely
disappears. We also utilized Eq. (20) in Ref. Burgess:1993vc  to accommodate
our definition of $U^{\prime}$, and it is easily realized that $w$ within
$A-C-w+z$ disappears again, and the remaining $w$ there was absorbed by
$U^{\prime}$. Therefore, Eq. (2) in Ref. Burgess:1993vc  should be adjusted to

$\displaystyle\alpha S^{\prime}$ $\displaystyle=$ $\displaystyle
4s_{w}^{2}c_{w}^{2}\left(-C-\frac{c_{w}^{2}-s_{w}^{2}}{c_{w}s_{w}}G\right),$
$\displaystyle\alpha T^{\prime}$ $\displaystyle=$ $\displaystyle-z,$
$\displaystyle\alpha U^{\prime}$ $\displaystyle=$ $\displaystyle
4s_{w}^{4}\left[-\frac{1}{s_{w}^{2}}(B-w)+\frac{c_{w}^{2}}{s_{w}^{2}}C-\frac{2c_{w}}{s_{w}}G\right].$
(64)

If $S$, $T$, and $U$ defined in Eq. (38) also exists, they will also
contribute to $S^{\prime}$, $T^{\prime}$ and $U^{\prime}$, and Eq. (64) then
becomes

$\displaystyle\alpha S^{\prime}$ $\displaystyle=$ $\displaystyle\alpha
S+4s_{w}^{2}c_{w}^{2}\left(-C-\frac{c_{w}^{2}-s_{w}^{2}}{c_{w}s_{w}}G\right),$
$\displaystyle\alpha T^{\prime}$ $\displaystyle=$ $\displaystyle\alpha T-z,$
$\displaystyle\alpha U^{\prime}$ $\displaystyle=$ $\displaystyle\alpha
U+4s_{w}^{4}\left[-\frac{1}{s_{w}^{2}}(B-w)+\frac{c_{w}^{2}}{s_{w}^{2}}C-\frac{2c_{w}}{s_{w}}G\right].$
(65)

Expressing with $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$, and $d_{i}$, we
arrive at

$\displaystyle\alpha S^{\prime}-\alpha S$ $\displaystyle=$
$\displaystyle\sum\limits_{i=2}^{\infty}4s_{w}^{2}c_{w}^{2}m_{Z}^{2i-2}\bigg{\\{}[-(i-1)s_{w}^{2}-c_{w}^{2}]\alpha_{i}+[-(i-1)c_{w}^{2}-s_{w}^{2}]\beta_{i}$
$\displaystyle\hskip
74.00005pt+\frac{s_{w}^{4}+c_{w}^{4}+(2i-2)s_{w}^{2}c_{w}^{2}}{s_{w}c_{w}}\gamma_{i}\bigg{\\}},$
$\displaystyle\alpha T^{\prime}-\alpha T$ $\displaystyle=$
$\displaystyle-\sum\limits_{i=2}^{\infty}(i-1)(\alpha_{i}s_{w}^{2}-2\gamma_{i}s_{w}c_{w}+\beta_{i}c_{w}^{2})m_{Z}^{2i-2},$
$\displaystyle\alpha U^{\prime}-\alpha U$ $\displaystyle=$
$\displaystyle-\sum\limits_{i=2}^{\infty}\frac{(i-1)d_{i}m_{W}^{2i-2}}{s_{w}^{2}}+\sum\limits_{i=2}^{\infty}\frac{ic_{w}^{2}}{s_{w}^{2}}m_{Z}^{2i-2}(\alpha_{i}s_{w}^{2}-2\gamma_{i}s_{w}c_{w}+\beta_{i}c_{w}^{2})$
(66)
$\displaystyle+\sum\limits_{i=2}^{\infty}\frac{2c_{w}}{s_{w}}m_{Z}^{2i-2}[\gamma_{i}(c_{w}^{2}-s_{w}^{2})+\beta_{i}s_{w}c_{w}-\alpha_{i}s_{w}c_{w}].$

If we only preserve the $i=2$ terms, with the oblique parameters defined in
Eq. (42), we derive

$\displaystyle\alpha S^{\prime}$ $\displaystyle\simeq$ $\displaystyle\alpha
S+4s_{w}^{2}\left(-Y-W+\frac{s_{w}}{c_{w}}X\right),$ $\displaystyle\alpha
T^{\prime}$ $\displaystyle\simeq$ $\displaystyle\alpha
T-\frac{s_{w}^{2}}{c_{w}^{2}}Y-W+\frac{2s_{w}}{c_{w}}X,$ $\displaystyle\alpha
U^{\prime}$ $\displaystyle\simeq$ $\displaystyle\alpha
U-4s_{w}^{2}\left(-W-V+\frac{2s_{w}}{c_{w}}X\right),$ (67)

which is exactly compatible with the expressions of $\varepsilon_{1}$,
$\varepsilon_{2}$, and $\varepsilon_{3}$ in Ref. Barbieri:2004qk . These
$\varepsilon_{i}$ parameters had originally been suggested in Ref.
Altarelli:1990zd , where they are considered to be equivalent to the $S$ and
$T$ parameters since the contributions from higher derivatives of the gauge
fields are neglected. Ref. Barbieri:2004qk  includes the influence from $V$,
$W$, $X$, and $Y$ without giving the detailed derivations, which are
contributions at the $p^{4}$ order. Besides these, below we will show that the
$p^{6}$ order also arises.

When only the kinetic mixings between $Z^{\prime}$ and the EW gauge bosons are
considered, the oblique parameters are given by

$\displaystyle S=T=U=0,\quad
V=W=\frac{\epsilon_{W}^{2}m_{W}^{2}}{m_{Z^{\prime}}^{2}},\quad
Y=\frac{\epsilon_{B}^{2}m_{W}^{2}}{m_{Z^{\prime}}^{2}},\quad
X=\frac{\epsilon_{B}\epsilon_{W}m_{W}^{2}}{m_{Z^{\prime}}^{2}}.$ (68)

Note that $X$ and $V$ arising from dim-8 and dim-10 SMEFT operators are
usually expected to be negligible, but they have the same order of magnitude
as $W$ and $Y$ in our model, since they are generated by the mediation of a
$Z^{\prime}$ boson which is not much heavier than the EW scale. Substituting
Eq. (68) into Eq. (A), we derive

$\displaystyle\alpha S^{\prime}$ $\displaystyle\simeq$
$\displaystyle-\frac{g^{2}g^{\prime
2}v^{2}(\epsilon_{B}^{2}+\epsilon_{W}^{2})}{(g^{2}+g^{\prime
2})m_{Z^{\prime}}^{2}}+\frac{gg^{\prime}v^{2}\epsilon_{B}\epsilon_{W}}{m_{Z^{\prime}}^{2}},$
$\displaystyle\alpha T^{\prime}$ $\displaystyle\simeq$
$\displaystyle-\frac{(g^{\prime}\epsilon_{B}-g\epsilon_{W})^{2}}{4m_{Z^{\prime}}^{2}},$
$\displaystyle\alpha U^{\prime}$ $\displaystyle\simeq$
$\displaystyle\frac{2g^{\prime
2}v^{2}(g^{2}\epsilon_{W}^{2}-gg^{\prime}\epsilon_{B}\epsilon_{W})}{(g^{2}+g^{\prime
2})m_{Z^{\prime}}^{2}}.$ (69)

These reproduce $S^{\prime}$ and $T^{\prime}$ given by expanding Eq. (30) and
(31) to the $m_{Z^{\prime}}^{-2}$ order. Note that there is no
$\epsilon_{B}^{2}$ term in $U^{\prime}$ if we only include the second order
contribution, $\Pi^{(2)}_{BB}(p^{2})$ ($Y$ parameter). The leading
$\epsilon_{B}^{2}$ contribution to $U^{\prime}$ comes from $\Pi^{(3)}_{BB}(0)$
(coefficient of $p^{6}$ term) and yields $\alpha\delta
U^{\prime}=4s_{w}^{4}c_{w}^{2}\epsilon_{B}^{2}m_{Z}^{4}/m_{Z^{\prime}}^{4}$,
which reproduces the result given in Ref. Holdom:1990xp .

###### Acknowledgements.

We thank to Yu-Pan Zeng for helpful discussions and communications. This work
is supported in part by the National Natural Science Foundation of China under
Grants Nos. 12005312, 11875327, 11905300, and 11805288, the Fundamental
Research Funds for the Central Universities, the Natural Science Foundation of
Guangdong Province, and the Sun Yat-Sen University Science Foundation.

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