aainstitutetext: KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305–0801,
Japanbbinstitutetext: The Graduate University of Advanced Studies (Sokendai),
Tsukuba, Ibaraki 305–0801, Japan

# Electroweak precision test of axion-like particles

Masashi Aiko a,b and Motoi Endo

###### Abstract

We study the contributions of an axion-like particle to the electroweak
precision observables. The particle is assumed to couple with the standard
model electroweak gauge bosons. We provide the formulae of the contributions
valid for any mass of the axion-like particle. It is found that the effects
arise not only via the oblique $S$ and $U$ parameters but also via radiative
corrections to the gauge couplings. Besides, the decay of $Z\to a\gamma$
affects the total width of the $Z$ boson. All of those contributions are
considered simultaneously in the global fit analysis of the electroweak
precision observables. Also, we discuss the recent CDF result of the $W$-boson
mass measurement. Since the model is tightly constrained by flavor and
collider constraints, it is found that the discrepancy from the standard model
prediction is solved only when the axion-like particle is heavier than
$500\,\mathrm{GeV}$ and its coupling to di-photon is suppressed.

††preprint: KEK–TH–2497

##  1 Introduction

New light pseudoscalar particles are one of the most popular and simple
extensions of the Standard Model (SM). They are often motivated in models with
spontaneous violations of global symmetries and are called axion-like
particles (ALPs). Their masses are generated by explicit small violations of
the global symmetries, whereas their interactions are characterized by the
symmetries. They affect low-energy observables via the interactions with the
SM particles.

In this paper, we revisit contributions to the electroweak precision
observables (EWPOs). The ALPs are assumed to be coupled primarily with the SM
${\rm U(1)}_{Y}$ and ${\rm SU(2)}_{L}$ gauge bosons, yielding ALP couplings
with the photon ($\gamma$) and $Z$ boson as well as the charged $W$ boson
after the electroweak (EW) symmetry breaking. Then, the EWPOs are affected via
vacuum polarizations of $\gamma$, $Z$, and $W$. Such contributions have been
studied in Ref. Bauer:2017ris . The authors argued that they are expressed by
the oblique parameters, $S,~{}T$, and $U$ Peskin:1991sw ; $S$ and $U$ are
generated, while $T$ is absent at least at the one-loop level. Interestingly,
$U$ becomes comparable to $S$, unlikely to a wide class of new physics models.
Also, the CDF II collaboration recently reported a new result of the $W$ mass
measurement CDF:2022hxs . The result is not consistent with the SM prediction
as well as the previous experimental values. In Ref. Yuan:2022cpw , the CDF
result has been discussed in the ALP models based on Ref. Bauer:2017ris , and
it was concluded that a light ALP can solve the disagreement marginally.

It is noticed that the above studies ignored the other ALP contributions,
i.e., those except for $S,~{}T$, and $U$. In many analyses of the oblique
parameters, e.g., those in Refs. Baak:2014ora ; ParticleDataGroup:2022pth ,
$S,~{}T$, and $U$ have been restricted/determined by fitting them globally to
the EWPOs under the assumption that there are no additional contributions from
new physics. However, we will show in this paper that this assumption is not
valid in the ALP models. These three parameters are not enough to parameterize
the ALP effects, but there are additional contributions via vacuum
polarizations such as the oblique parameters beyond $S$, $T$, and $U$ (cf.,
Refs. Maksymyk:1993zm ; Barbieri:2004qk ). Although these extra contributions
are suppressed in a wide class of models, this is not the case for the ALP;
they can be comparable to $S$ and $U$. Furthermore, the $Z$ boson can decay
into a light ALP and a photon. This decay proceeds at the tree level and
contributes to the total width of the $Z$ boson. Since those contributions
affect the EWPOs simultaneously with $S$, $T$, and $U$, they must be analyzed
collectively in the global fit. It will be shown that the global-fit results
are changed drastically.

The ALPs are subject to experimental constraints. Although cosmological limits
are very severe, they can be avoided if the ALPs are heavier than $\sim
1\,\mathrm{GeV}$ Jaeckel:2010ni ; Cadamuro:2011fd ; Proceedings:2012ulb . Even
in such a case, the ALPs affect meson decays via the interactions with the $W$
boson Izaguirre:2016dfi ; Alonso-Alvarez:2018irt ; Gavela:2019wzg ;
Guerrera:2021yss ; Bauer:2021mvw ; Guerrera:2022ykl . In particular, the
$B$-meson decay into a $K^{(*)}$ meson with photons via $a\to\gamma\gamma$ or
the decay with leptons via $a\to\ell^{+}\ell^{-}$ is very sensitive to the ALP
contributions. Besides, the ALPs have been studied particularly in the LEP and
LHC experiments Mimasu:2014nea ; Jaeckel:2015jla ; Jaeckel:2012yz ;
Bauer:2017ris ; Bauer:2018uxu ; Florez:2021zoo ; Wang:2021uyb ;
dEnterria:2021ljz ; Knapen:2016moh ; CMS:2018erd ; ATLAS:2020hii ;
Craig:2018kne ; Bonilla:2022pxu . In this paper, their results are applied to
the current model setup, and we will compare all those constraints with the
EWPO fit results. The recent CDF result of the $W$ mass measurement
CDF:2022hxs  will also be discussed under those constraints.

This paper is organized as follows. In Sec. 2, we introduce the ALP model and
provide its decay rates. In Sec. 3, we explain the ALP contributions to the
EWPOs and the analysis strategy. The experimental constraints are summarized
in Sec. 4. We show the numerical results in Sec. 5, and Sec. 6 is devoted to
the conclusion. In Appendix A, the Passarino-Veltman function Passarino:1978jh
is given explicitly. In Appendix B, we give the analytic formula of the three-
body decay width for $a\to Z^{*}\gamma$.

##  2 Model

We consider an ALP $(a)$ coupled with the ${\rm SU(2)}_{L}$ gauge boson
$(W_{\mu}^{a})$ and the ${\rm U(1)}_{Y}$ gauge boson $(B_{\mu})$. The
Lagrangian is shown as Georgi:1986df

$\displaystyle\mathcal{L}_{\mathrm{ALP}}=\frac{1}{2}\partial_{\mu}a\partial^{\mu}a-\frac{1}{2}m_{a}^{2}a^{2}-c_{WW}\frac{a}{f_{a}}W_{\mu\nu}^{a}\widetilde{W}^{a\mu\nu}-c_{BB}\frac{a}{f_{a}}B_{\mu\nu}\widetilde{B}^{\mu\nu},$
(1)

where $m_{a}$ is the ALP mass, and $f_{a}$ is the ALP decay constant. The
coefficients, $c_{WW}$ and $c_{BB}$, as well as $m_{a}$ and $f_{a}$ are
regarded as free parameters, though $m_{a}<f_{a}$ is satisfied. We set
$f_{a}=1\,\mathrm{TeV}$ throughout this paper. The field strengths of the
${\rm SU(2)}_{L}$ and ${\rm U(1)}_{Y}$ gauge bosons are defined as

$\displaystyle W_{\mu\nu}^{a}$
$\displaystyle=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}+g\epsilon^{abc}W_{\mu}^{b}W_{\nu}^{c},$
(2) $\displaystyle B_{\mu\nu}$
$\displaystyle=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu},$ (3)

where $g$ is the ${\rm SU(2)}_{L}$ gauge coupling constant. The dual is
expressed by

$\displaystyle\widetilde{X}_{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}X_{\rho\sigma},\quad(X=W^{a},B).$
(4)

The totally antisymmetric tensors are defined with $\epsilon^{012}=1$ and
$\epsilon^{0123}=1$. After the EW symmetry breaking, the above interactions
are rewritten as

$\displaystyle\mathcal{L}_{\mathrm{int}}$
$\displaystyle=-\frac{1}{4}g_{a\gamma\gamma}aF_{\mu\nu}\widetilde{F}^{\mu\nu}-\frac{1}{2}g_{aZ\gamma}aZ_{\mu\nu}\widetilde{F}^{\mu\nu}$
$\displaystyle\quad-\frac{1}{4}g_{aZZ}aZ_{\mu\nu}\widetilde{Z}^{\mu\nu}-\frac{1}{2}g_{aWW}aW_{\mu\nu}^{+}\widetilde{W}^{-\mu\nu}+\ldots,$
(5)

where quartic interaction terms are omitted. Here, the field strengths are
given by

$\displaystyle
X^{\prime}_{\mu\nu}=\partial_{\mu}X^{\prime}_{\nu}-\partial_{\nu}X^{\prime}_{\mu}.$
(6)

where $X^{\prime}_{\mu}=A_{\mu}$ for the photon $(\gamma)$, $Z_{\mu}$ for the
$Z$ boson, and $W_{\mu}^{\pm}$ for the charged $W$ boson. Note that
$F_{\mu\nu}$ corresponds to $A_{\mu}$. Then, the coupling constants are
expressed by $c_{WW}$ and $c_{BB}$ with $f_{a}$ as

$\displaystyle g_{a\gamma\gamma}$
$\displaystyle=\frac{4}{f_{a}}\quantity(s_{W}^{2}c_{WW}+c_{W}^{2}c_{BB}),$ (7)
$\displaystyle g_{aZ\gamma}$
$\displaystyle=\frac{2}{f_{a}}\quantity(c_{WW}-c_{BB})s_{2W},$ (8)
$\displaystyle g_{aZZ}$
$\displaystyle=\frac{4}{f_{a}}\quantity(c_{W}^{2}c_{WW}+s_{W}^{2}c_{BB}),$ (9)
$\displaystyle g_{aWW}$ $\displaystyle=\frac{4}{f_{a}}c_{WW},$ (10)

where $c_{W}=\cos{\theta_{W}}$, $s_{W}=\sin{\theta_{W}}$, and
$s_{2W}=\sin{2\theta_{W}}$ with the Weinberg angle $\theta_{W}$.

###  2.1 Decay of ALP

The ALP decays into a pair of SM gauge bosons. The partial decay widths for
$a\to V_{i}V_{j}\ (V_{i,j}=\gamma,Z,W^{\pm})$ are obtained as Bauer:2017ris ;
Craig:2018kne ; Bonilla:2021ufe

$\displaystyle\Gamma(a\to
V_{i}V_{j})=\frac{m_{a}^{3}}{32\pi(1+\delta_{ij})}\lambda^{3/2}\quantity(\frac{m_{V_{i}}^{2}}{m_{a}^{2}},\frac{m_{V_{j}}^{2}}{m_{a}^{2}})\absolutevalue{g_{aV_{i}V_{j}}^{\rm
eff}}^{2},$ (11)

with $\lambda(x,y)=(1-x-y)^{2}-4xy$. Here $m_{V}$ is the gauge-boson mass
($m_{\gamma}=0$). Note that $\delta_{ij}=0$ for $a\to Z\gamma$ and $a\to
W^{+}W^{-}$.

The ALP coupling to the SM gauge bosons $g^{\rm eff}_{aV_{i}V_{j}}$ is
obtained both at the tree and loop levels. In particular, the coupling to the
di-photon is given at the tree level by $g_{a\gamma\gamma}$ in Eq. (7) and is
generated by loop corrections with $g_{aWW}$ in Eq. (10) as Bauer:2017ris

$\displaystyle g_{a\gamma\gamma}^{\rm
eff}=g_{a\gamma\gamma}+\frac{2\alpha}{\pi}g_{aWW}B_{2}(\tau_{W}),$ (12)

where $\alpha\equiv e^{2}/(4\pi)$ with the QED coupling $e=gs_{W}$. The loop
function $B_{2}$ is defined as

$\displaystyle B_{2}(\tau)=1-(\tau-1)f^{2}(\tau),$ (13)

where $f(\tau)$ is given by

$\displaystyle
f(\tau)=\left\\{\matrixquantity{\arcsin{\frac{1}{\sqrt{\tau}}},\\\
\frac{\pi}{2}+\frac{i}{2}\log{\frac{1+\sqrt{1-\tau}}{1-\sqrt{1-\tau}}},}\right.\qquad\matrixquantity{\text{for}~{}\tau\geq
1\\\ \text{for}~{}\tau<1},$ (14)

with $\tau_{W}=4m_{W}^{2}/m_{a}^{2}$. In the light ALP limit, $\tau_{W}\gg 1$,
the function is approximated as $B_{2}\to m_{a}^{2}/(6m_{W}^{2})$, i.e.,
suppressed by $m_{W}$. In the heavy limit, $\tau_{W}\ll 1$, we obtain
$B_{2}\to 1+\pi^{2}/4-\log^{2}{(m_{a}/m_{W})}$, which is enlarged by the
logarithm. On the other hand, it is sufficient for us to evaluate the decay
widths for $a\to Z\gamma,\,ZZ,\,W^{+}W^{-}$ at the tree level, i.e.,
$g_{aV_{i}V_{j}}^{\rm eff}=g_{aV_{i}V_{j}}$.

When the ALP is lighter than the $Z$ boson, and if $g_{a\gamma\gamma}^{\rm
eff}$ is suppressed, the ALP decays into either a pair of SM fermions, $a\to
f\bar{f}$, or the fermions with a photon, $a\to Z^{*}\gamma\to
f\bar{f}\gamma$, by exchanging an off-shell $Z$ boson. Since the ALP does not
couple with SM fermions directly, the former proceeds via radiative
corrections, as will be explained later. On the other hand, even though the
latter is a three-body decay, it proceeds at the tree level, and thus, its
decay width can be comparable to the former. For $m_{a}\ll m_{Z}$, the decay
width is obtained as

$\displaystyle\Gamma(a\to
f\bar{f}\gamma)=N_{c}^{f}\frac{g_{Z}^{2}g_{aZ\gamma}^{2}}{30720\pi^{3}}\quantity[(g_{V,f})^{2}+(g_{A,f})^{2}]\frac{m_{a}^{7}}{m_{Z}^{4}},$
(15)

where $N_{c}^{f}=1~{}(3)$ is the color factor for leptons (quarks), and
$g_{Z}=g/c_{W}$ is the $Z$-boson coupling constant. The vector and axial form-
factors of $Z$ boson are defined as $g_{V,f}=I_{3}^{f}-2\,Q_{f}s_{W}^{2}$ and
$g_{A,f}=I_{3}^{f}$. The formula valid for any $m_{a}$ is provided in Appendix
B.

The ALP decays into a pair of leptons and heavy quarks, $a\to f\bar{f}$, via
gauge-boson loops. The partial decay widths are obtained as Bauer:2017ris ;
Bauer:2020jbp

$\displaystyle\Gamma(a\to
f\bar{f})=N_{c}^{f}\frac{m_{a}m_{f}^{2}}{8\pi}\absolutevalue{g_{aff}^{\rm
eff}}^{2}\sqrt{1-\frac{4m_{f}^{2}}{m_{a}^{2}}}.$ (16)

The effective coupling induced by radiative corrections is given by

$\displaystyle g_{aff}^{\rm eff}$
$\displaystyle=3Q_{f}^{2}\frac{\alpha}{4\pi}g_{a\gamma\gamma}\ln{\frac{\Lambda^{2}}{m_{f}^{2}}}+\frac{3}{4s_{W}^{2}}\frac{\alpha}{4\pi}g_{aWW}\ln{\frac{\Lambda^{2}}{m_{W}^{2}}}$
$\displaystyle\quad+\frac{3}{s_{W}c_{W}}\frac{\alpha}{4\pi}g_{aZ\gamma}Q_{f}\quantity(I_{3}^{f}-2Q_{f}s_{W}^{2})\ln{\frac{\Lambda^{2}}{m_{Z}^{2}}}$
$\displaystyle\quad+\frac{3}{s_{W}^{2}c_{W}^{2}}\frac{\alpha}{4\pi}g_{aZZ}\quantity(Q_{f}^{2}s_{W}^{4}-I_{3}^{f}Q_{f}s_{W}^{2}+\frac{1}{8})\ln{\frac{\Lambda^{2}}{m_{Z}^{2}}},$
(17)

where $Q_{f}$ and $I_{3}^{f}$ are the electric charge and the weak isospin of
the fermion $f$, and $\tau_{f}=4m_{f}^{2}/m_{a}^{2}$. Here, we keep the terms
enhanced by $\ln{(\Lambda^{2}/m_{f}^{2})}$ or $\ln{(\Lambda^{2}/m_{V}^{2})}$
$(V=Z,W)$. The cutoff scale $\Lambda$ is determined by a UV theory of the ALP
model and treated as a model parameter in this paper. The subleading terms are
given in Refs. Bauer:2017ris ; Bauer:2020jbp ; Bonilla:2021ufe , though they
are irrelevant (cf., Ref. Arias-Aragon:2022iwl ). Note that the above formula
is valid for the decay which conserves the lepton/quark flavors.

The $W$ boson contribution can generate flavor-violating interactions of
quarks. In particular, those for the down-type quarks can be sizable due to
top-quark loops. With keeping the up-type quark masses non-zero, the following
term arises apart from Eq. (17) Izaguirre:2016dfi ; Alonso-Alvarez:2018irt ;
Gavela:2019wzg ; Guerrera:2021yss ; Bauer:2021mvw ; Guerrera:2022ykl :

$\displaystyle g_{ad_{i}d_{j}}^{\rm
eff}=-\frac{3}{4s_{W}^{2}}\frac{\alpha}{4\pi}g_{aWW}\sum_{q=u,c,t}V_{qi}V_{qj}^{*}G(x_{q}),$
(18)

where $V_{ij}$ is the Cabbibo-Kobayashi-Maskawa (CKM) matrix, and the loop
function is defined as

$\displaystyle G(x)=\frac{x\quantity(1-x+x\ln{x})}{(1-x)^{2}}.$ (19)

with $x_{q}=m_{q}^{2}/m_{W}^{2}$.

On the other hand, the inclusive rate for the ALP decaying into light hadrons
is shown as (cf., Ref. Bauer:2017ris )

$\displaystyle\Gamma(a\to{\rm hadrons})$
$\displaystyle=32\pi\alpha_{s}^{2}m_{a}^{3}\quantity(1+\frac{83}{4}\frac{\alpha_{s}}{\pi})\absolutevalue{g_{agg}^{\rm
eff}}^{2}+\sum_{f=u,d,s}\Gamma(a\to f\bar{f}),$ (20)

where $m_{a}\gg\Lambda_{\rm QCD}$ is assumed. The effective coupling with
gluons is given by

$\displaystyle g_{agg}^{\rm eff}=\frac{1}{32\pi^{2}}\sum_{f=u,d,s}g_{aff}^{\rm
eff}.$ (21)

Note that the above expression is not valid for $m_{a}\lesssim 3~{}{\rm GeV}$,
where the perturbative QCD is not applicable, and various hadronic decay
channels appear (cf., Refs. Bauer:2017ris ; Aloni:2018vki ).

Figure 1: Branching ratios of the ALP with $f_{a}=1\,\mathrm{TeV}$ and
$\Lambda=4\pi f_{a}$. Here, $c_{BB}=0$, (left) $c_{WW}=0$ (middle), and
$c_{\gamma\gamma}\propto g_{a\gamma\gamma}=0$ (right) are set. The decay into
light hadrons is denoted as “had” in the legend.

In Fig. 1, we show the branching ratios of the ALP with $\Lambda=4\pi f_{a}$.
Among the ALP couplings, $c_{BB}$ and $c_{WW}$, we set $c_{BB}=0$ on the left
panel, $c_{WW}=0$ on the middle, and $c_{\gamma\gamma}\propto
g_{a\gamma\gamma}=0$ on the right. Here, the decay rate of $a\to Z\gamma$
includes those from the three-body decays, $a\to Z^{*}\gamma\to
f\bar{f}\gamma$. In the case with $c_{\gamma\gamma}\neq 0$, the decay of
$a\to\gamma\gamma$ is dominant for $m_{a}\leq m_{Z}$. When $c_{BB}=0$, the
decay of $a\to WW$ is dominant for $m_{a}>2m_{W}$, and the decays of $a\to
Z\gamma$ and $a\to ZZ$ become relevant above $m_{Z}$ and $2m_{Z}$,
respectively. When $c_{WW}=0$, the decay of $a\to\gamma\gamma$ is still
dominant for $m_{a}>m_{Z}$, while the decays of $a\to Z\gamma$ and $a\to ZZ$
become sub-dominant above the thresholds. In the case with
$c_{\gamma\gamma}=0$, the fermionic decay modes are relevant for
$m_{a}<m_{Z}$, while the bosonic channels become relevant for $m_{a}\gtrsim
m_{Z}$.

Before closing this section, let us comment on the ALP couplings with the $Z$
and Higgs bosons, $g_{aZh}^{\rm eff}$. It is known that gauge-boson loop
contributions to $g_{aZh}^{\rm eff}$ vanish at the one-loop level
Bauer:2017ris . Thus, the ALP decay, $a\to Zh$, is irrelevant for Fig. 1 and
the following analysis.

##  3 Electroweak precision test (EWPT)

New physics models as well as the SM have been tested by using the EWPOs. The
theoretical predictions are compared with the experimental data by performing
global fit analyses. In particular, the former values are determined within
the SM by a set of input observables: the fine-structure constant $\alpha$,
the Fermi coupling constant $G_{F}$, and the $Z$ mass $m_{Z}$.

The ALP contributions arise in vacuum polarizations due to the ALP
interactions with the EW gauge bosons. Such contributions have been studied in
terms of the oblique parameters, $S$, $T$, and $U$ Bauer:2017ris . It has been
known that the ALP generates $S$ and $U$, while $T$ is absent at least at the
one-loop level. Besides, $U$ becomes as large as $S$, in contrast to the case
in a wide class of new physics models. However, $S$ and $U$ are not enough to
represent the ALP contributions. We introduce three additional variables,
$\Delta_{Z}$, $\Delta_{W}$ and $\Delta\alpha$.#1#1#1 No specific parameters
are assigned for these variables in Ref. Hagiwara:1994pw .  The parameters
$\Delta_{Z}$ and $\Delta_{W}$ represent radiative corrections to the gauge
coupling constants of the $Z$ and $W$ bosons, respectively. Also,
$\Delta\alpha$ is those to the QED coupling. Although they are negligible
compared to $S$ and $T$ in a wide class of new physics models, this is not the
case for the ALP models. Moreover, the decay of $Z\to a\gamma$ alters the
$Z$-pole observables, such as the hadronic and total decay widths of the $Z$
boson (see Eq. (11)). Since the decay proceeds at the tree level, its impact
on EWPOs is likely stronger than those via vacuum polarizations. We provide
their formulae as well as those for $S$, $T$, and $U$ in this section.

Since the EW data, except for the recent CDF result of the $W$ mass, are
consistent with the SM predictions, the ALP contributions should be smaller
than the SM ones. Hence, we retain only the leading contributions from the
ALP, which stem from interference terms between the ALP and SM transition
amplitudes. The ALP contributions are estimated up to the one-loop level.
Besides, since we are interested in the observables at the EW scale, we focus
on the leading terms in the $m_{f}/m_{W}$ expansions, where $m_{f}$ and
$m_{W}$ are the SM fermion and $W$ masses, respectively.

###  3.1 Formulation

In this section, we basically follow Ref. Hagiwara:1994pw  for the formulation
of the EWPOs.#2#2#2 It is also checked that the following results are
consistent with the formulae in the literature, e.g., Refs. Hollik:1993cg ;
Hollik:1995dv . See also, Ref. Cirigliano:2013lpa .  We consider the ALP
contributions to vacuum polarizations. They generally appear in the gauge-
boson propagators as

$\displaystyle\Gamma^{ab}_{\mu\nu}(k)=-ig_{\mu\nu}(k^{2}-m_{a,0}^{2})\delta^{ab}-i\left(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\Pi_{T}^{ab}(k^{2})-i\frac{k_{\mu}k_{\nu}}{k^{2}}\Pi_{L}^{ab}(k^{2}),$
(22)

with $a,b=\gamma,W,Z$. The mass parameter $m_{a,0}$ is related to the pole
mass $m_{a}$ as $m_{a,0}^{2}=m_{a}^{2}+{\rm Re}\,\Pi_{T}^{aa}(m_{a}^{2})$.
Here, $\Pi_{T}^{ab}(k^{2})$ and $\Pi_{L}^{ab}(k^{2})$ are the unrenormalized
transverse and longitudinal self-energy corrections, respectively.#3#3#3
Although the radiative corrections are assumed to include the pinch terms
Cornwall:1981zr ; Cornwall:1989gv ; Degrassi:1992ue ; Degrassi:1992ff ;
Binosi:2009qm  in Ref. Hagiwara:1994pw , they are irrelevant in the following
ALP analysis because gauge-dependent terms do not arise.  We focus on
$\Pi_{T}^{ab}(k^{2})$ because the terms proportional to $k_{\mu}k_{\nu}$ are
subdominant with respect to the $m_{f}/m_{W}$ expansion and hereafter
neglected. For convenience, we define

$\displaystyle\Pi_{T,V}^{ab}(k^{2})=\frac{\Pi_{T}^{ab}(k^{2})-\Pi_{T}^{ab}(m_{V}^{2})}{k^{2}-m_{V}^{2}}.$
(23)

It is also useful to parameterize the vacuum polarizations as

$\displaystyle\Pi_{T}^{\gamma\gamma}(k^{2})$
$\displaystyle=e^{2}\,\Pi_{T}^{QQ}(k^{2}),$ (24)
$\displaystyle\Pi_{T}^{Z\gamma}(k^{2})$
$\displaystyle=eg_{Z}\left[\Pi_{T}^{3Q}(k^{2})-s_{W}^{2}\Pi_{T}^{QQ}(k^{2})\right],$
(25) $\displaystyle\Pi_{T}^{ZZ}(k^{2})$
$\displaystyle=g_{Z}^{2}\left[\Pi_{T}^{33}(k^{2})-2s_{W}^{2}\Pi_{T}^{3Q}(k^{2})+s_{W}^{4}\Pi_{T}^{QQ}(k^{2})\right],$
(26) $\displaystyle\Pi_{T}^{WW}(k^{2})$
$\displaystyle=g^{2}\,\Pi_{T}^{11}(k^{2}).$ (27)

Transition amplitudes for charged currents mediated by the $W$ boson, e.g.,
$\mu\to e\bar{\nu}_{e}\nu_{\mu}$, are shifted from the tree-level ones as

$\displaystyle\mathcal{M}^{\rm CC}\to\mathcal{M}^{\rm CC}\left[1-{\rm
Re}\,\Pi_{T,W}^{WW}(k^{2})\right],$ (28)

where $k$ is the momentum carried by the $W$ boson. Here and hereafter, we
ignore imaginary parts of the vacuum polarizations because they correspond to
higher-order corrections. Hence, the extra vacuum-polarization contributions
can be implemented by replacing the ${\rm SU(2)}_{L}$ coupling as

$\displaystyle g^{2}\to\bar{g}^{2}(k^{2})=g^{2}\left[1-{\rm
Re}\,\Pi_{T,W}^{WW}(k^{2})\right].$ (29)

Let us next consider neutral currents, $f\bar{f}\to
f^{\prime}\bar{f}^{\prime}$. Their transition amplitudes are expressed as

$\displaystyle\mathcal{M}^{\rm NC}=\mathcal{M}^{\rm
NC}_{ij}[\bar{\psi}_{f}\gamma_{\mu}P_{a}\psi_{f}][\bar{\psi}_{f^{\prime}}\gamma^{\mu}P_{a^{\prime}}\psi_{f^{\prime}}],$
(30)

where $i=f_{a}$ and $j=f^{\prime}_{a^{\prime}}$ with the chiral projector
$P_{a,a^{\prime}}$. Similar to the charged currents, the leading contributions
from new physics via the vacuum polarizations can be implemented via the
effective coupling and weak mixing angles as

$\displaystyle\mathcal{M}^{\rm NC}_{ij}$
$\displaystyle=\frac{\bar{e}^{2}(s)}{s}Q_{i}Q_{j}+\frac{\bar{g}_{Z}^{2}(s)}{s-m_{Z}^{2}+is\Gamma_{Z}/m_{Z}}[I_{3}^{i}-Q_{i}\bar{s}^{2}(s)][I_{3}^{j}-Q_{j}\bar{s}^{2}(s)],$
(31)

where the first term on the right-hand side has a pole at $s=0$, identified as
the photon-exchange contribution, and the second one peaks at $s=m_{Z}^{2}$,
corresponding to the $Z$ amplitude. Here, $s=k^{2}$ is the squared momentum
carried by the photon or $Z$ boson. The effective parameters are defined as

$\displaystyle\bar{e}^{2}(k^{2})$ $\displaystyle=e^{2}\left[1-{\rm
Re}\,\Pi_{T,\gamma}^{\gamma\gamma}(k^{2})\right],$ (32)
$\displaystyle\bar{g}_{Z}^{2}(k^{2})$ $\displaystyle=g_{Z}^{2}\left[1-{\rm
Re}\,\Pi_{T,Z}^{ZZ}(k^{2})\right],$ (33) $\displaystyle\bar{s}^{2}(k^{2})$
$\displaystyle=s_{W}^{2}\left[1+\frac{c_{W}}{s_{W}}{\rm
Re}\,\Pi_{T,\gamma}^{Z\gamma}(k^{2})\right].$ (34)

Since the ALP couples only with the gauge bosons, there are no contributions
from vertex and box corrections.

The fine-structure constant, $\alpha$, is associated with the photon-exchange
amplitude in the Thomson limit, $k^{2}\to 0$. The effective coupling at a
scale $k^{2}$ satisfies

$\displaystyle\frac{1}{\bar{\alpha}(k^{2})}-\frac{1}{\alpha}$
$\displaystyle=4\pi\,{\rm
Re}\\!\left[\Pi_{T,\gamma}^{QQ}(k^{2})-\Pi_{T,\gamma}^{QQ}(0)\right],$ (35)

where $\bar{\alpha}(k^{2})=\bar{e}^{2}(k^{2})/4\pi$. Hence, the leading ALP
contribution at $k^{2}=m_{Z}^{2}$ gives

$\displaystyle\bar{\alpha}(m_{Z}^{2})$ $\displaystyle=\alpha\left\\{1-{\rm
Re}\left[\Pi_{T,\gamma}^{\gamma\gamma}(m_{Z}^{2})-\Pi_{T,\gamma}^{\gamma\gamma}(0)\right]\right\\}\equiv\alpha\left(1+\Delta\alpha\right).$
(36)

On the other hand, the measured value of the Fermi coupling constant, $G_{F}$,
is related to the effective coupling of the charged current amplitude at
$k^{2}\simeq 0$ as

$\displaystyle G_{F}=\frac{\bar{g}^{2}(0)}{4\sqrt{2}\,m_{W}^{2}}.$ (37)

Here, since we focus on radiative corrections from the ALP, vertex and box
corrections are ignored. It is noted that the $W$ mass is not the input
observable, but its theoretical value is determined from Eq. (37).

The effective couplings and weak mixing angle are expressed in terms of
$\alpha$, $G_{F}$, and $m_{Z}$ with the oblique parameters, $S$, $T$, and $U$,
as

$\displaystyle\frac{1}{\bar{g}_{Z}^{2}(0)}=\frac{1-\alpha
T}{4\sqrt{2}\,G_{F}m_{Z}^{2}},$ (38)
$\displaystyle\bar{s}^{2}(m_{Z}^{2})=\frac{1}{2}-\sqrt{\frac{1}{4}-\bar{\alpha}(m_{Z}^{2})\left[\frac{4\pi}{\bar{g}_{Z}^{2}(0)}+\frac{S}{4}\right]},$
(39)
$\displaystyle\frac{4\pi}{\bar{g}_{W}^{2}(0)}=\frac{\bar{s}^{2}(m_{Z}^{2})}{\bar{\alpha}(m_{Z}^{2})}-\frac{S+U}{4}.$
(40)

In a given model, the oblique parameters are evaluated as

$\displaystyle S=16\pi\,{\rm
Re}\\!\left[\Pi_{T,\gamma}^{3Q}(m_{Z}^{2})-\Pi_{T,Z}^{33}(0)\right],$ (41)
$\displaystyle T=\frac{4\sqrt{2}\,G_{F}}{\alpha}\,{\rm
Re}\\!\left[\Pi_{T}^{33}(0)-\Pi_{T}^{11}(0)\right],$ (42) $\displaystyle
U=16\pi\,{\rm Re}\\!\left[\Pi_{T,Z}^{33}(0)-\Pi_{T,W}^{11}(0)\right].$ (43)

Thus, the leading contributions from new physics are shown as

$\displaystyle\bar{g}_{Z}^{2}(0)$ $\displaystyle=g_{Z}^{2}\left(1+\alpha
T\right),$ (44) $\displaystyle\bar{s}^{2}(m_{Z}^{2})$
$\displaystyle=s_{W}^{2}\left[1+\frac{c_{W}^{2}}{c_{W}^{2}-s_{W}^{2}}\left(\Delta\alpha-\alpha
T\right)+\frac{\alpha S}{4s_{W}^{2}(c_{W}^{2}-s_{W}^{2})}\right],$ (45)
$\displaystyle\bar{g}^{2}(0)$
$\displaystyle=g^{2}\left[1-\frac{s_{W}^{2}\Delta\alpha}{c_{W}^{2}-s_{W}^{2}}-\frac{\alpha
S}{2(c_{W}^{2}-s_{W}^{2})}+\frac{c_{W}^{2}\alpha
T}{c_{W}^{2}-s_{W}^{2}}+\frac{\alpha U}{4s_{W}^{2}}\right].$ (46)

It is noticed that the $Z$\- and $W$-pole observables, where the gauge bosons
carry $k^{2}=m_{Z}^{2}$ and $m_{W}^{2}$ respectively, require the effective
couplings, $\bar{g}_{Z}^{2}(m_{Z}^{2})$ and $\bar{g}^{2}(m_{W}^{2})$. From
Eqs. (29) and (33), they are related to $\bar{g}_{Z}^{2}(0)$ and
$\bar{g}^{2}(0)$ as

$\displaystyle\bar{g}_{Z}^{2}(m_{Z}^{2})$
$\displaystyle=\bar{g}_{Z}^{2}(0)\left\\{1-{\rm
Re}\left[\Pi_{T,Z}^{ZZ}(m_{Z}^{2})-\Pi_{T,Z}^{ZZ}(0)\right]\right\\}\equiv
g_{Z}^{2}(0)\left(1+\Delta_{Z}\right),$ (47)
$\displaystyle\bar{g}^{2}(m_{W}^{2})$
$\displaystyle=\bar{g}^{2}(0)\left\\{1-{\rm
Re}\left[\Pi_{T,W}^{WW}(m_{W}^{2})-\Pi_{T,W}^{WW}(0)\right]\right\\}\equiv
g^{2}(0)\left(1+\Delta_{W}\right).$ (48)

Consequently, the EWPOs are evaluated theoretically in terms of $\Delta_{Z}$,
$\Delta_{W}$, and $\Delta\alpha$ as well as $S$, $T$, and $U$ (see also Sec.
3.3).

###  3.2 ALP contributions

In Eq. (5), the ALP couples with $\gamma$, $Z$, and $W$. In the $R_{\xi}$
gauge, its contributions to the vacuum polarizations at the one-loop level are
derived as

$\displaystyle\Pi_{T}^{WW}(k^{2})$
$\displaystyle=\frac{1}{288\pi^{2}}g_{aWW}^{2}F(k^{2};a,W),$ (49)
$\displaystyle\Pi_{T}^{\gamma\gamma}(k^{2})$
$\displaystyle=\frac{1}{288\pi^{2}}\left[g_{a\gamma\gamma}^{2}F(k^{2};a,\gamma)+g_{aZ\gamma}^{2}F(k^{2};a,Z)\right],$
(50) $\displaystyle\Pi_{T}^{Z\gamma}(k^{2})$
$\displaystyle=\frac{1}{288\pi^{2}}\left[g_{a\gamma\gamma}g_{aZ\gamma}F(k^{2};a,\gamma)+g_{aZ\gamma}g_{aZZ}F(k^{2};a,Z)\right],$
(51) $\displaystyle\Pi_{T}^{ZZ}(k^{2})$
$\displaystyle=\frac{1}{288\pi^{2}}\left[g_{aZ\gamma}^{2}F(k^{2};a,\gamma)+g_{aZZ}^{2}F(k^{2};a,Z)\right].$
(52)

These results are valid for any ALP mass. The loop function is defined as

$\displaystyle F(k^{2};a,V)$
$\displaystyle=3\left[k^{2}-(m_{a}+m_{V})^{2}\right]\left[k^{2}-(m_{a}-m_{V})^{2}\right]$
$\displaystyle\quad\quad\times\left[B_{0}(k^{2};m_{a},m_{V})-B_{0}(0;m_{a},m_{V})\right]$
$\displaystyle\quad-3k^{2}\left[A_{0}(m_{a})+A_{0}(m_{V})+(2m_{a}^{2}+2m_{V}^{2}-k^{2})B_{0}(0;m_{a},m_{V})\right]$
$\displaystyle\quad+7k^{2}(3m_{a}^{2}+3m_{V}^{2}-k^{2}),$ (53)

where $A_{0}(x)$ and $B_{0}(k^{2};x,y)$ are the Passarino-Veltman functions
Passarino:1978jh  and shown explicitly in Appendix A. It is noticed that the
results are independent of the gauge parameters. Then, the ALP contributions
to $S$, $T$, and $U$ are given by

$\displaystyle\alpha S$
$\displaystyle=\frac{2c_{W}^{2}s_{W}^{2}}{9\pi^{2}m_{Z}^{2}}\frac{c_{WW}c_{BB}}{f_{a}^{2}}\quantity[F(m_{Z}^{2};a,\gamma)-F(m_{Z}^{2};a,Z)],$
(54) $\displaystyle\alpha T$ $\displaystyle=0,$ (55) $\displaystyle\alpha U$
$\displaystyle=\frac{2s_{W}^{4}}{9\pi^{2}m_{Z}^{2}}\frac{c_{WW}^{2}}{f_{a}^{2}}\quantity[F(m_{Z}^{2};a,\gamma)+\frac{c_{W}^{2}}{s_{W}^{2}}F(m_{Z}^{2};a,Z)-\frac{1}{s_{W}^{2}c_{W}^{2}}F(m_{W}^{2};a,W)].$
(56)

It is noticed that the ALP does not contribute to $T$. On the other hand, the
contributions to $\Delta\alpha$, $\Delta_{Z}$, and $\Delta_{W}$ are given by

$\displaystyle\Delta\alpha$
$\displaystyle=-\frac{g_{a\gamma\gamma}^{2}}{288\pi^{2}}\quantity[\frac{F(m_{Z}^{2};a,\gamma)}{m_{Z}^{2}}-F^{\prime}(0;a,\gamma)]-\frac{g_{aZ\gamma}^{2}}{288\pi^{2}}\quantity[\frac{F(m_{Z}^{2};a,Z)}{m_{Z}^{2}}-F^{\prime}(0;a,Z)],$
(57) $\displaystyle\Delta_{Z}$
$\displaystyle=\frac{g_{aZ\gamma}^{2}}{288\pi^{2}}\quantity[\frac{F(m_{Z}^{2};a,\gamma)}{m_{Z}^{2}}-F^{\prime}(m_{Z}^{2};a,\gamma)]+\frac{g_{aZZ}^{2}}{288\pi^{2}}\quantity[\frac{F(m_{Z}^{2};a,Z)}{m_{Z}^{2}}-F^{\prime}(m_{Z}^{2};a,Z)],$
(58) $\displaystyle\Delta_{W}$
$\displaystyle=\frac{1}{288\pi^{2}}g_{aWW}^{2}\quantity[\frac{F(m_{W}^{2};a,W)}{m_{W}^{2}}-F^{\prime}(m_{W}^{2};a,W)],$
(59)

where $F^{\prime}(k^{2};a,V)$ is a derivative of $F(k^{2};a,V)$ and expressed
explicitly as

$\displaystyle F^{\prime}(k^{2};a,V)$
$\displaystyle=6\left(k^{2}-m_{a}^{2}-m_{V}^{2}\right)\left[B_{0}(k^{2};m_{a},m_{V})-B_{0}(0;m_{a},m_{V})\right]$
$\displaystyle\quad+3\left[k^{2}-(m_{a}+m_{V})^{2}\right]\left[k^{2}-(m_{a}-m_{V})^{2}\right]B_{0}^{\prime}(k^{2};m_{a},m_{V})$
$\displaystyle\quad-3\left[A_{0}(m_{a})+A_{0}(m_{V})+(2m_{a}^{2}+2m_{V}^{2}-k^{2})B_{0}(0;m_{a},m_{V})\right]$
$\displaystyle\quad+7(3m_{a}^{2}+3m_{V}^{2}-2k^{2}).$ (60)

The relation between $(c_{BB},c_{WW})$ and
$(g_{a\gamma\gamma},g_{aZ\gamma},g_{aZZ})$ is found in Eqs. (7)–(9). It is
noticed that $\Delta\alpha$, $\Delta_{Z}$, and $\Delta_{W}$ can be as large as
$\alpha S$ and $\alpha U$, and thus, must not be neglected.

Let us show the above results in the light ALP-mass limit, $m_{a}\ll
m_{V}\,(V=Z,W)$. They are obtained as

$\displaystyle\alpha S$
$\displaystyle=-\frac{2c_{W}^{2}s_{W}^{2}m_{Z}^{2}}{\pi^{2}}\frac{c_{WW}c_{BB}}{f_{a}^{2}}\quantity(\ln{\frac{m_{Z}^{2}}{\Lambda^{2}}}+1),$
(61) $\displaystyle\alpha U$
$\displaystyle=-\frac{2s_{W}^{4}m_{Z}^{2}}{3\pi^{2}}\frac{c_{WW}^{2}}{f_{a}^{2}}\quantity(\ln{\frac{m_{Z}^{2}}{\Lambda^{2}}}+\frac{1}{3}+\frac{2c_{W}^{2}}{s_{W}^{2}}\ln{\frac{m_{W}^{2}}{m_{Z}^{2}}}),$
(62) $\displaystyle\Delta\alpha$
$\displaystyle=\frac{m_{Z}^{2}}{96\pi^{2}}\quantity[g_{a\gamma\gamma}^{2}\quantity(\ln{\frac{m_{Z}^{2}}{\Lambda^{2}}}+\frac{11}{3})+g_{aZ\gamma}^{2}\quantity(\ln{\frac{m_{Z}^{2}}{\Lambda^{2}}}+\frac{11}{6})],$
(63) $\displaystyle\Delta_{Z}$
$\displaystyle=\frac{m_{Z}^{2}}{96\pi^{2}}\quantity(g_{aZ\gamma}^{2}+g_{aZZ}^{2})\quantity(\ln{\frac{m_{Z}^{2}}{\Lambda^{2}}}+\frac{4}{3}),$
(64) $\displaystyle\Delta_{W}$
$\displaystyle=\frac{m_{W}^{2}}{96\pi^{2}}g_{aWW}^{2}\quantity(\ln{\frac{m_{W}^{2}}{\Lambda^{2}}}+\frac{4}{3}).$
(65)

The results for $S$, $T$, $U$, and $\Delta\alpha$ are consistent with those in
Ref. Bauer:2017ris .

Let us comment on the ALP contributions, $U$, $\Delta\alpha$, $\Delta_{Z}$,
and $\Delta_{W}$. In a wide class of new physics models in a high-energy
scale, they are suppressed compared to $S$ and $T$. However, this is not the
case for the ALP models. This is because the ALP contributions to the vacuum
polarizations vanish at $k^{2}=0$ (see Eq. (53)). Then, the leading
contribution to $S$ arises from the first-order derivative of the vacuum
polarization, which is comparable to $U$, $\Delta\alpha$, $\Delta_{Z}$, and
$\Delta_{W}$.

###  3.3 Observables

The ALP contributions to the $Z$-pole observables are evaluated by the last
term of the transition amplitude in Eq. (31). It is noticed that this term has
the same form as the tree-level one except that the coupling and weak mixing
are replaced by the effective ones. Hence, the ALP contributions are
represented in terms of the following effective charges of the $Z$ coupling,

$\displaystyle g_{V,f}$
$\displaystyle=\sqrt{\rho_{Z}}\left[I_{3}^{f}-2\,Q_{f}\bar{s}^{2}(m_{Z}^{2})\right]\equiv\hat{g}_{V,f}+\Delta
g_{V,f},$ (66) $\displaystyle g_{A,f}$
$\displaystyle=\sqrt{\rho_{Z}}\,I_{3}^{f}\equiv\hat{g}_{A,f}+\Delta g_{A,f},$
(67)

with $\hat{g}_{V,f}=I_{3}^{f}-2\,Q_{f}s_{W}^{2}$ and
$\hat{g}_{A,f}=I_{3}^{f}$. Note that the effective parameters should be
estimated at $k^{2}=m_{Z}^{2}$. Here, $\rho_{Z}$ is defined as

$\displaystyle\rho_{Z}=\bar{g}_{Z}^{2}(m_{Z}^{2})/g_{Z}^{2}=1+\alpha
T+\Delta_{Z}.$ (68)

Here and hereafter, variables with a hat $(\,\hat{}\,)$, e.g., $\hat{g}_{V,f}$
and $\hat{g}_{A,f}$, represent the SM contributions at the tree level, and
those with $\Delta$ such as $\Delta g_{V,f}$ and $\Delta g_{A,f}$ denote the
ALP corrections. On the other hand, those with a subscript “SM” show the SM
predictions including radiative corrections.

When we evaluate the EWPOs, the ALP corrections are dominated by interference
terms between the ALP and SM transition amplitudes. In particular, it is
sufficient for the latter to be evaluated at the tree level when we focus on
the leading ALP contributions. Hence, by expressing the partial decay width
for $Z\to f\bar{f}$ as

$\displaystyle\Gamma_{f}=(\Gamma_{f})_{\rm SM}+\Delta\Gamma_{f},$ (69)

the SM contribution at the tree level, $\hat{\Gamma}_{f}$, is given by

$\displaystyle\hat{\Gamma}_{f}=N_{C}^{f}\frac{G_{F}m_{Z}^{3}}{6\sqrt{2}\pi}\left[\left(\hat{g}_{V,f}\right)^{2}+\left(\hat{g}_{A,f}\right)^{2}\right],$
(70)

and $\Delta\Gamma_{f}$ is obtained as

$\displaystyle\Delta\Gamma_{f}=N_{C}^{f}\frac{G_{F}m_{Z}^{3}}{3\sqrt{2}\pi}\left[\left(\hat{g}_{V,f}\right)^{2}+\left(\hat{g}_{A,f}\right)^{2}\right]\quantity(\frac{\Delta
g_{V,f}}{\hat{g}_{V,f}}+\frac{\Delta g_{A,f}}{\hat{g}_{A,f}}).$ (71)

The radiative corrections to the SM value will be mentioned in Sec. 3.4.

When $m_{a}$ is smaller than $m_{Z}$, the decay channel $Z\to a\gamma$ is
open. Its partial decay width is given by

$\displaystyle\Gamma_{a\gamma}\equiv\Gamma(Z\to
a\gamma)=\frac{m_{Z}^{3}}{96\pi}g_{aZ\gamma}^{2}\quantity(1-\frac{m_{a}^{2}}{m_{Z}^{2}})^{3}.$
(72)

Then, the hadronic and total decay widths of the $Z$ boson are modified as

$\displaystyle\Gamma_{\rm had}$
$\displaystyle=\Gamma_{u}+\Gamma_{d}+\Gamma_{c}+\Gamma_{s}+\Gamma_{b}\equiv(\Gamma_{\rm
had})_{\rm SM}+\Delta\Gamma_{\rm had},$ (73) $\displaystyle\Gamma_{Z}$
$\displaystyle=\Gamma_{e}+\Gamma_{\mu}+\Gamma_{\tau}+3\Gamma_{\nu}+\Gamma_{\rm
had}+\Gamma_{a\gamma}\equiv(\Gamma_{Z})_{\rm SM}+\Delta\Gamma_{Z},$ (74)

where $\Delta\Gamma_{{\rm had},Z}$ are the sums of $\Delta\Gamma_{f}$. It is
noticed that $\Gamma_{a\gamma}$ appears in $\Gamma_{Z}$ and is generated at
the tree level with $g_{aZ\gamma}$. Hence, its impact on the EWPO global fit
is likely stronger than those via vacuum polarizations.

The other $Z$-pole observables relevant for the following analysis are
represented explicitly as follows. First of all, the hadronic cross-section is
shown as

$\displaystyle\sigma_{\rm had}^{0}$ $\displaystyle=(\sigma_{\rm had}^{0})_{\rm
SM}+\frac{12\pi}{m_{Z}^{2}}\frac{\hat{\Gamma}_{e}\hat{\Gamma}_{\rm
had}}{\hat{\Gamma}_{Z}^{2}}\quantity(\frac{\Delta\Gamma_{e}}{\hat{\Gamma}_{e}}+\frac{\Delta\Gamma_{\rm
had}}{\hat{\Gamma}_{\rm had}}-2\frac{\Delta\Gamma_{Z}}{\hat{\Gamma}_{Z}}).$
(75)

Next, the ratios of the partial decay widths become

$\displaystyle R^{0}_{\ell}$ $\displaystyle=(R^{0}_{\ell})_{\rm
SM}+\frac{\hat{\Gamma}_{\rm
had}}{\hat{\Gamma}_{\ell}}\quantity(\frac{\Delta\Gamma_{\rm
had}}{\hat{\Gamma}_{\rm
had}}-\frac{\Delta\Gamma_{\ell}}{\hat{\Gamma}_{\ell}}),$ (76) $\displaystyle
R^{0}_{q}$ $\displaystyle=(R^{0}_{q})_{\rm
SM}+\frac{\hat{\Gamma}_{q}}{\hat{\Gamma}_{\rm
had}}\quantity(\frac{\Delta\Gamma_{q}}{\hat{\Gamma}_{q}}-\frac{\Delta\Gamma_{\rm
had}}{\hat{\Gamma}_{\rm had}}),$ (77)

where $\ell$ denotes the SM leptons. Also, $\sin^{2}\theta^{f}_{\rm eff}$ is
given by

$\displaystyle\sin^{2}\theta^{f}_{\rm eff}$
$\displaystyle=(\sin^{2}\theta^{f}_{\rm eff})_{\rm
SM}-\frac{1}{4|Q_{f}|}\frac{\hat{g}_{V,f}}{\hat{g}_{A,f}}\left(\frac{\Delta
g_{V,f}}{\hat{g}_{V,f}}-\frac{\Delta g_{A,f}}{\hat{g}_{A,f}}\right).$ (78)

The left-right asymmetry is shown as

$\displaystyle\mathcal{A}_{f}$ $\displaystyle=(\mathcal{A}_{f})_{\rm
SM}+\mathcal{\hat{A}}_{f}\quantity(1-\frac{\hat{g}_{V,f}}{\hat{g}_{A,f}}\mathcal{\hat{A}}_{f})\quantity(\frac{\Delta
g_{V,f}}{\hat{g}_{V,f}}-\frac{\Delta g_{A,f}}{\hat{g}_{A,f}}).$ (79)

Here, the SM contribution at the tree level is

$\displaystyle\mathcal{\hat{A}}_{f}=\frac{2\hat{g}_{V,f}/\hat{g}_{A,f}}{1+(\hat{g}_{V,f}/\hat{g}_{A,f})^{2}}.$
(80)

Finally, the forward-backward asymmetry is given by

$\displaystyle A^{0}_{\rm FB}=(A^{0}_{\rm FB})_{\rm
SM}+\frac{3}{4}\mathcal{\hat{A}}_{e}\mathcal{\hat{A}}_{f}\quantity(\frac{\Delta\mathcal{A}_{e}}{\mathcal{\hat{A}}_{e}}+\frac{\Delta\mathcal{A}_{f}}{\mathcal{\hat{A}}_{f}}).$
(81)

On the other hand, the mass and decay widths of the $W$ boson are also
evaluated theoretically. From Eqs. (37) and (46), the mass is obtained as

$\displaystyle m_{W}^{2}$ $\displaystyle=(m_{W}^{2})_{\rm SM}+\frac{\alpha
c_{W}^{2}m_{Z}^{2}}{c_{W}^{2}-s_{W}^{2}}\left[-s_{W}^{2}\frac{\Delta\alpha}{\alpha}-\frac{S}{2}+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right]$
$\displaystyle\equiv(m_{W}^{2})_{\rm SM}+\Delta m_{W}^{2}.$ (82)

Then, the $W$ partial decay width becomes

$\displaystyle\Gamma(W\to f_{i}f_{j})$ $\displaystyle=[(\Gamma_{W})_{ij}]_{\rm
SM}+(\hat{\Gamma}_{W})_{ij}\bigg{[}\frac{3\Delta
m_{W}^{2}}{2c_{W}^{2}m_{Z}^{2}}+\Delta_{W}\bigg{]}.$ (83)

Here, $\Delta_{W}$ arises because the $W$ coupling is estimated at
$k^{2}=m_{W}^{2}$. The SM prediction at the tree level is expressed as

$\displaystyle(\hat{\Gamma}_{W})_{ij}$
$\displaystyle=N_{c}^{f}|V_{ij}|^{2}\frac{g^{2}}{48\pi}m_{W}.$ (84)

where $V_{ij}$ is the CKM matrix when $i,j$ are quarks, while it is the
identity matrix for leptons.

###  3.4 Analysis strategy

| Measurement | Ref. |  | Measurement | Ref.
---|---|---|---|---|---
$\alpha_{s}(m_{Z}^{2})$ | $0.1177\pm 0.0010$ | deBlas:2022hdk  |  $m_{Z}$ [GeV] | $91.1875\pm 0.0021$ | Janot:2019oyi
$\Delta\alpha_{\mathrm{had}}^{(5)}(m_{Z}^{2})$ | $0.02766\pm 0.00010$ | deBlas:2022hdk  |  $\Gamma_{Z}$ [GeV] | $2.4955\pm 0.0023$ |
$m_{t}$ [GeV] | $172.69\pm 0.30$ | ParticleDataGroup:2022pth  |  $\sigma_{h}^{0}$ [nb] | $41.4802\pm 0.0325$ |
$m_{h}$ [GeV] | $125.21\pm 0.17$ | ParticleDataGroup:2022pth  | $R^{0}_{\ell}$ | $20.7666\pm 0.0247$ |
$m_{W}$ [GeV] | $80.377\pm 0.012$ | ParticleDataGroup:2022pth  | $A_{\mathrm{FB}}^{0,\ell}$ | $0.0171\pm 0.0010$ |
| $80.4133\pm 0.0080$ | deBlas:2022hdk  | $R^{0}_{b}$ | $0.21629\pm 0.00066$ | ALEPH:2005ab ; Bernreuther:2016ccf
$\Gamma_{W}$ [GeV] | $2.085\pm 0.042$ | ParticleDataGroup:2022pth  | $R^{0}_{c}$ | $0.1721\pm 0.0030$ |
$\mathcal{B}(W\to\ell\nu)$ | $0.10860\pm 0.00090$ | Schael:2013ita  | $A_{\mathrm{FB}}^{0,b}$ | $0.0996\pm 0.0016$ |
$\mathcal{A}_{\ell}$ (LEP) | $0.1465\pm 0.0033$ | ALEPH:2005ab  | $A_{\mathrm{FB}}^{0,c}$ | $0.0707\pm 0.0035$ |
$\mathcal{A}_{\ell}$ (SLD) | $0.1513\pm 0.0021$ | ALEPH:2005ab  | $\mathcal{A}_{b}$ | $0.923\pm 0.020$ |
|  |  | $\mathcal{A}_{c}$ | $0.670\pm 0.027$ |
Table 1: Experimental data of the SM input parameters and EWPOs.

As a test of the ALP models, we perform a global fit analysis on the EWPOs. A
likelihood function is defined by a multivariate Gaussian, $-2\ln
L=(\mathbf{y}-{\bf\it\mu})^{T}V^{-1}(\mathbf{y}-{\bf\it\mu})$, where
$\mathbf{y}$ is a vector of the measured quantities, ${\bf\it\mu}$ is the
corresponding theoretical predictions, and $V$ is the covariance matrix.

In Table 1, we summarize the measured values of the SM input and the EWPOs.
The covariance matrices are provided in the references for those listed in the
right column. The other SM input parameters, such as $G_{F}$, $\alpha$, and
the masses of the light SM fermions, are fixed to be the observed central
values ParticleDataGroup:2022pth .

In the table, we show two results of the $W$ mass $m_{W}$; the one provided by
Particle Data Group (PDG) ParticleDataGroup:2022pth  and a value combined with
the recent CDF result, for which we adopt the averaged value in Ref.
deBlas:2022hdk . They will be denoted by $m_{W}^{\rm PDG}$ and $m_{W}^{\rm
CDF}$, respectively. In the following analysis, we study both cases in the ALP
models. On the other hand, the SM prediction of $m_{W}$ has been evaluated at
the two-loop level Awramik:2003rn . We also include theoretical uncertainties
from unknown higher-order corrections, $\delta m_{W}=0\pm 4\,\mathrm{MeV}$
(cf., Ref.deBlas:2022hdk ). As a result, based on the SM input in the table,
the SM prediction is derived as

$\displaystyle(m_{W})_{\rm SM}=80.3552\pm 0.0055\,\mathrm{GeV}.$ (85)

The result is consistent with the PDG value, $m_{W}^{\rm PDG}$, but deviates
from the result including the CDF result, $m_{W}^{\rm CDF}$, at the $6\sigma$
level.#4#4#4 The largest uncertainty of the SM prediction originates in
$m_{Z}$. Although the uncertainty from the top quark mass seems to be smaller,
it may involve a potentially larger (theoretical) uncertainty, which reduces
the discrepancy (cf., Ref. deBlas:2022hdk ).

The SM predictions for the $Z$-pole observables as well as the $W$ mass have
been evaluated up to the two-loop level including the electroweak corrections
Awramik:2006uz ; Dubovyk:2019szj . On the other hand, for the $W$ decay
widths, we use the SM predictions provided in Ref. dEnterria:2020cpv . In
particular, those assuming the CKM unitarity are adopted. Theoretical
uncertainties from unknown higher-order corrections are irrelevant for those
observables and neglected in the following analysis unlikely to $m_{W}$.

We implement the ALP contributions to the EWPOs, as explained in Sec. 3.2.
Obviously, $\Delta\alpha$, $\Delta_{Z}$, $\Delta_{W}$ and $\Gamma_{a\gamma}$
must be taken into account at the same as $S$, $T$, and $U$ when we perform
the EWPT. This feature is contrary to the analyses explored in the previous
works Bauer:2017ris ; Bauer:2018uxu ; Yuan:2022cpw , where $\Delta\alpha$,
$\Delta_{Z}$, $\Delta_{W}$ and $\Gamma_{a\gamma}$ were ignored in the analyses
of $S$ and $U$. In particular, they focused on the ALP much lighter than the
$Z$ boson; in such a case $\Gamma_{a\gamma}$ alters $\Gamma_{Z}$ drastically.
We will show the numerical results in Sec. 5.

##  4 Experimental constraints

In this section, we summarize experimental constraints on the ALP models.
Although there are very severe bounds from cosmological measurements
Jaeckel:2010ni ; Cadamuro:2011fd ; Proceedings:2012ulb , the model can avoid
them if the ALP mass is $m_{a}\gtrsim 1\,\mathrm{GeV}$. Then, constraints from
flavor and collider experiments become relevant.

###  4.1 Flavor constraints

Let us first consider the flavor constraints. If the ALPs have an interaction
with the $W$ boson, they generate quark-flavor transitions via $W$ loops (see
Eq. (18)) Izaguirre:2016dfi ; Alonso-Alvarez:2018irt ; Gavela:2019wzg ;
Guerrera:2021yss ; Bauer:2021mvw ; Guerrera:2022ykl . In particular, for
$m_{a}=\mathcal{O}(1)\,\mathrm{GeV}$, the $B$-meson decays provide the best
sensitivities (see, e.g., Ref. Bauer:2021mvw ). The decay rate for $B^{+}\to
K^{+}a$ is obtained as Izaguirre:2016dfi ; Gavela:2019wzg ; Bauer:2021mvw ;
Guerrera:2022ykl

$\displaystyle\Gamma(B^{+}\to
K^{+}a)=\frac{m_{B}^{3}}{64\pi}\absolutevalue{g_{abs}^{\rm
eff}}^{2}f_{0}(m_{a}^{2})\lambda_{Ka}^{1/2}\quantity(1-\frac{m_{K}^{2}}{m_{B}^{2}}),$
(86)

where $\lambda_{Ka}=\lambda(m_{a}^{2}/m_{B}^{2},m_{K}^{2}/m_{B}^{2})$ with the
$B$ $(K)$ meson mass, $m_{B}$ $(m_{K})$. Here, $f_{0}(q^{2})$ is the scalar
form factor, which is evaluated by following Ref.
FlavourLatticeAveragingGroupFLAG:2021npn .

In the ALP parameter region in interest, the ALP produced from the meson is
likely to decay within the detectors. If its lifetime is short enough, it
decays promptly. As the lifetime increases, the ALP propagates inside the
detectors before it decays. Then, the vertex constructed from the final-state
particles of the ALP decay becomes displaced from the primary interaction
vertex. Among various such channels (see e.g., Ref. Bauer:2021mvw ), the
following two studies provide the best sensitivities:

  * •

When $g_{a\gamma\gamma}$ is large enough, the ALP decays predominantly into a
pair of photons. The severest constraint is obtained from $B^{+}\to K^{+}a$,
$a\to\gamma\gamma$ in the mass range of $0.175<m_{a}<4.78\,\mathrm{GeV}$ with
various lifetimes. The analysis was performed by the BaBar collabotation
BaBar:2021ich .

  * •

When $g_{a\gamma\gamma}$ is suppressed, the ALP decays into SM fermions (see
Sec. 2.1). Among the decay channels, the process $B^{+}\to K^{+}a$,
$a\to\mu\mu$, though ${\rm Br}(a\to\mu\mu)$ is $\sim 10^{-3}$, provides a
severe bound for $m_{a}=\mathcal{O}(1)\,\mathrm{GeV}$. The analysis was
performed by the LHCb collaboration in the mass range of
$0.25<m_{a}<4.7\,\mathrm{GeV}$ and the lifetime $0.1<\tau_{a}<1000\,{\rm ps}$
LHCb:2016awg .

These constraints are very severe and inevitable as long as the $B$-meson
decays into the ALP, i.e., for $m_{a}<m_{B}-m_{K}$.#5#5#5 There are several
mass regions in which the SM backgrounds are huge and the flavor constraints
become absent or relaxed drastically. See Refs. BaBar:2021ich ; LHCb:2016awg
for the details.  Thus, the ALPs are favored to be heavier than
$4.8\,\mathrm{GeV}$ to avoid the constraints and contribute to the EWPOs
effectively.

###  4.2 Collider constraints

Even when the ALP is heavier than $4.8\,\mathrm{GeV}$ and the flavor
constraints are avoided, the model is subject to the collider constraints. In
the mass range of $m_{a}\gtrsim 1\,\mathrm{GeV}$, the ALP productions followed
by $a\to\gamma\gamma$ have been studied with the experimental data of LEP
Mimasu:2014nea ; Jaeckel:2015jla ; Knapen:2016moh , Belle II Belle-II:2020jti
, CDF Mimasu:2014nea ; CDF:2013lma , LHC (the proton collisions)
Jaeckel:2012yz ; Bauer:2017ris ; Bauer:2018uxu ; Florez:2021zoo ;
dEnterria:2021ljz ; Wang:2021uyb , and LHC (the Pb collisions) Knapen:2016moh
; CMS:2018erd ; ATLAS:2020hii . There are also studies focusing on the ALP
interaction with the $W$ bosons Craig:2018kne ; Bonilla:2022pxu .

Let us first consider the case that the produced ALP decays into a pair of
photons. The collider constraints are summarized, e.g., in Ref.
dEnterria:2021ljz . For $1\lesssim m_{a}\lesssim 5\,\mathrm{GeV}$, the process
$e^{+}e^{-}\to\gamma a\to 3\gamma$ provides the best sensitivity
Mimasu:2014nea ; Jaeckel:2015jla ; Knapen:2016moh ; Belle-II:2020jti . The
scattering cross section of $e^{+}e^{-}\to\gamma a$ is expressed as
Bauer:2017ris

$\displaystyle\frac{d\sigma(e^{+}e^{-}\to\gamma
a)}{d\Omega}=\frac{\alpha}{128\pi}s^{2}\quantity(1-\frac{m_{a}^{2}}{s})^{3}(1+\cos^{2}\theta)\quantity(\absolutevalue{V}^{2}+\absolutevalue{A}^{2}),$
(87)

where $\theta$ is the photon angle relative to the beam direction, and $V,\ A$
are

$\displaystyle V$
$\displaystyle=\frac{g_{a\gamma\gamma}}{s}+\frac{1-4s_{W}^{2}}{4c_{W}s_{W}}\frac{g_{aZ\gamma}}{s-m_{Z}^{2}+is\Gamma_{Z}/m_{Z}},$
(88) $\displaystyle A$
$\displaystyle=\frac{1}{4c_{W}s_{W}}\frac{g_{aZ\gamma}}{s-m_{Z}^{2}+is\Gamma_{Z}/m_{Z}}.$
(89)

The first term on the right-hand side of $V$ corresponds to a photon-exchange
contribution, and the second one and $A$ to the $Z$ boson. The LEP and LEP II
experiments have measured $e^{+}e^{-}\to\gamma\gamma(\gamma)$ at the center-
or-energy around the $Z$ pole and $200\,\mathrm{GeV}$, respectively. The
results have been used to constraint the ALP:

  * •

The former case, i.e., for $\sqrt{s}\sim m_{Z}$, was analyzed by Ref.
Jaeckel:2015jla . Since the on-shell $Z$ boson contributions dominate the
cross section, the results provided by Ref. Jaeckel:2015jla  can be
interpreted as a bound on $\Gamma(Z\to\gamma a)\times{\rm
Br}(a\to\gamma\gamma)$.#6#6#6 Although the ATLAS collaboration reported a
stronger bound on ${\rm BR}(Z\to 3\gamma)$ ATLAS:2015rsn , the result is
applied to a higher $m_{a}$ region because of photon kinematical cuts
Bauer:2017ris .

On the other hand, when $g_{aZ\gamma}$ is suppressed, the scattering proceeds
by exchanging an off-shell photon. Since the cross section is small, its
constraint is sufficiently weak for the parameter region in interest.

  * •

The LEP II data in the latter case, i.e., for $\sqrt{s}\sim
200\,\mathrm{GeV}$, were analyzed by Ref. Knapen:2016moh . In the above mass
range, since the photons from the ALP decay are likely to be collimated, the
inclusive $e^{+}e^{-}\to 2\gamma$ signal regions were studied. Although the
scattering was supposed to proceed only via an off-shell photon in the
literature, the $Z$ boson contributes generally as well. It is noticed from
Eq. (87) that the latter contribution does not affect the kinematic
distributions of the final-state particles. Thus, we rescale the constraints
provided in Ref. Knapen:2016moh  by $\sigma(e^{+}e^{-}\to\gamma a)\times{\rm
Br}(a\to\gamma\gamma)$.

The LHC results Jaeckel:2012yz ; Bauer:2017ris ; Bauer:2018uxu ;
Florez:2021zoo ; dEnterria:2021ljz ; Wang:2021uyb ; Knapen:2016moh ;
CMS:2018erd ; ATLAS:2020hii  give the best sensitivity for $m_{a}\gtrsim
5\,\mathrm{GeV}$, where the ALPs are assumed to be produced via vector gauge-
boson fusions (VBFs):

  * •

Resonant productions of the ALPs from photon fusions, $\gamma\gamma\to
a\to\gamma\gamma$, have been studied in the Pb-Pb collision at LHC
Knapen:2016moh ; CMS:2018erd ; ATLAS:2020hii . The initial photons are emitted
from the Pb nuclei. Heavy-ion collisions have been considered because the
electric charge is larger than the proton. This process provides the best
sensitivity for $5\lesssim m_{a}\lesssim 100\,\mathrm{GeV}$.

  * •

For $100\lesssim m_{a}\lesssim 160\,\mathrm{GeV}$, resonant productions of the
ALPs via VBFs, $VV\to a\to\gamma\gamma$ $(V=\gamma,Z,W)$, have been studied by
using the ATLAS analyses on Higgs boson productions via VBFs in the p-p
collisions Jaeckel:2012yz . Here, the initial gauge bosons are emitted from
quarks in the incoming protons.

  * •

For $150\lesssim m_{a}\lesssim 400\,\mathrm{GeV}$, VBF productions of the ALPs
in the p-p collisions contribute to the signals Jaeckel:2012yz . It is found
in the literature that a large kinematical cut is imposed on the invariant
mass of the final-state photons, and their constraints can be interpreted as
those on the non-resonant productions of the ALPs via photon fusions,
$\gamma\gamma\to a^{*}\to\gamma\gamma$, where the initial photons are emitted
from the proton.

  * •

More recently, the reference Bauer:2018uxu  has derived a bound on
$\gamma\gamma\to a\to\gamma\gamma$ by recasting the ATLAS result on the search
for spin-0 resonances ATLAS:2017ayi . The result shows the best sensitivity
for $200\lesssim m_{a}<2700\,\mathrm{GeV}$.

The constraints from the on-shell productions of the ALPs via $\gamma\gamma\to
a\to\gamma\gamma$ are interpreted as those on
$\absolutevalue{g_{a\gamma\gamma}^{\rm eff}}^{2}{\rm Br}(a\to\gamma\gamma)$.
On the other hand, those from the non-resonant productions, $\gamma\gamma\to
a^{*}\to\gamma\gamma$, are recast by $\absolutevalue{g_{a\gamma\gamma}^{\rm
eff}}^{4}$.

Next, let us consider the collider constraints for the ALPs decaying into
particles other than a pair of photons. They become significant especially
when $g_{a\gamma\gamma}$ is absent at the tree level. The LEP and LHC bounds,
in this case, have been studied in Ref. Craig:2018kne , where the ALPs are
assumed to be produced at the on-shell. The most relevant ones on each mass
region are summarized as follows:

  * •

For $1\lesssim m_{a}\lesssim 40\,\mathrm{GeV}$, the leading constraint is from
$Z\to\gamma a\to\gamma jj$, where $j$ denotes a jet (gluon and quarks). The
decay was measured on the $Z$ pole at the LEP experiment. The bounds are
understood as those on $\Gamma(Z\to\gamma a)\times\sum_{j}{\rm Br}(a\to jj)$
with $j={\rm light\ hadrons},c,b$.

  * •

For $40\,\mathrm{GeV}\lesssim m_{a}\lesssim m_{Z}$, the search for $Z\to\gamma
a$, $a\to\nu\bar{\nu}\gamma$ at LEP gives a significant constraint. Here, the
neutrinos are produced by exchanging an off-shell $Z$ boson, i.e., $a\to
Z^{*}\gamma\to\nu\bar{\nu}\gamma$. Such a three-body decay can compete with
$a\to ff$ because the former (latter) proceeds at the tree level (via
radiative corrections). The bounds are applied to $\Gamma(Z\to\gamma
a)\times{\rm Br}(a\to\gamma\nu\bar{\nu})$.

  * •

The LHC tri-boson searches give the best sensitivities on the ALP models in
larger mass regions, especially above the $Z$ threshold. In most of the ALP
mass regions, the leading constraint is obtained by
$pp\to\gamma^{*},Z^{*}\to\gamma a$, $a\to
Z\gamma\to\nu\bar{\nu}\gamma$.#7#7#7Although the same final state is yielded
by $pp\to aZ\to a\nu\bar{\nu}$ with $a\to\gamma\gamma$, the process is
unlikely to contribute because the analysis requires a di-photon invariant
mass larger than the ALP mass. Hence, the bounds are imposed on $\sigma(pp\to
a\gamma)\times{\rm Br}(a\to Z\gamma)$. The results are obtained up to
$m_{a}=500\,\mathrm{GeV}$ in the literature.

The same process but with $Z\to\mu\bar{\mu}$ provides the best limit around
$m_{a}=110\,\mathrm{GeV}$. Also, for general choices of $(c_{BB},c_{WW})$ the
constraints from $pp\to W^{*}\to Wa$, $a\to W^{+}W^{-}$ may become significant
especially around $m_{a}=300\,\mathrm{GeV}$, though they should be compared to
the constraints with $g_{a\gamma\gamma}\neq 0$.

These results are based on the assumption that the ALPs are produced directly,
i.e., on-shell. In contrast, the reference Bonilla:2022pxu  has studied non-
resonant productions of the ALPs at the LHC. Here, they analyzed off-shell ALP
contributions to vector-boson scattering processes such as $V_{1}V_{2}\to
a^{*}\to V_{3}V_{4}$ or t-channel ALP exchange, where $V_{i}$ denotes the
gauge boson. The constraints are derived for various $c_{BB}$, $c_{WW}$, and
$m_{a}$. Note that $m_{a}\lesssim 100~{}{\rm GeV}$ is assumed in the
literature, otherwise on-shell ALP contributions cannot be neglected
generally.

##  5 Results

In this section, we show the EWPT results of the ALPs and compare them with
the experimental constraints. Among the ALP contributions, the $Z$-boson decay
into $a+\gamma$ affects the probability likelihood only when the ALP is
lighter than the $Z$ boson. Hence, we will focus on the case of $m_{a}\ll
m_{Z}$ in Sec. 5.1. We will also show how much the overlooked contributions
affect the EWPOs by comparing the results with those based on the analyses in
the previous studies.

In Sec. 5.2, we will study the case when the ALP is not much lighter than the
$Z$ boson. In Sec. 3, we have provided the formulae of the ALP contributions
that can be applied to $m_{a}\gtrsim m_{Z}$. In this case, the ALPs are free
from the flavor constraints but are subject to the collider ones, particularly
from the LHC. We will compare the EWPT results with those bounds.

In Sec. 5.3, we will discuss the goodness of fit by focusing on the $W$ mass.
In particular, the recent CDF result of the $W$ mass measurement is
inconsistent with the SM prediction. It will be shown that the tension may be
solved in the ALP model if the ALP is heavier than $500\,\mathrm{GeV}$, and
thus, the goodness of fit is improved against the SM case.

###  5.1 Light ALP case

Figure 2: EWPO probability distribution on the $(g_{a\gamma\gamma},g_{aWW})$
plane in units of ${\rm TeV}^{-1}$. For the red (blue) colored regions,
$m_{W}^{\rm CDF}$ $(m_{W}^{\rm PDG})$ is adopted as the experimental value of
the $W$ mass. The darker- (lighter-) colored regions enclosed by the dashed
lines represent the 68% (95%) level. The ALP mass is $4\,\mathrm{GeV}$ (left)
and $5\,\mathrm{GeV}$ (right). Also, $f_{a}=1\,\mathrm{TeV}$ and $\Lambda=4\pi
f_{a}$ are taken. The other colored solid lines denote the experimental
constraints. The regions including $(g_{a\gamma\gamma},g_{aWW})=(0,0)$ are
allowed. In the cyan region, all the constraints are satisfied.

Let us first consider the case when the ALP is much lighter than the $Z$
boson. In this case, the ALP contributions to the EWPOs are insensitive to the
ALP mass, as seen from Eqs. (57)–(59). In Fig. 2, the EWPT results are shown
for $f_{a}=1\,\mathrm{TeV}$ and $\Lambda=4\pi f_{a}$. The probability
likelihoods are calculated by globally fitting the ALP model to the EWPOs on
the $(g_{a\gamma\gamma},g_{aWW})$ plane. The red (blue) colored region is
obtained by adopting $m_{W}^{\rm CDF}$ $(m_{W}^{\rm PDG})$ for the
experimental data. The probability distributions are normalized on the
coupling parameter plane. The darker- (lighter-) colored regions enclosed by
the dashed lines correspond to the 68% (95%) level.

On the left panel, the ALP mass is set as $m_{a}=4\,\mathrm{GeV}$. The flavor
constraints impose very severe limits on $|g_{aWW}|$. The vertical axis is
shown on a logarithmic scale to represent its tightness. In particular, the
region above the gray line is excluded by $B^{+}\to K^{+}\gamma\gamma$.
Although this decay is suppressed around $g_{a\gamma\gamma}=0$, such a
parameter region is instead constrained by $B^{+}\to K^{+}\mu\mu$, where the
effective coupling $g_{a\mu\mu}^{\rm eff}$ is as large as or larger than
$g_{a\gamma\gamma}$. Then, the region above the magenta line is excluded. On
the other hand, $g_{a\gamma\gamma}$ is limited by the collider constraints. In
particular, $e^{+}e^{-}\to\gamma\gamma(\gamma)$ at LEP-I provides the best
sensitivity except for $g_{a\gamma\gamma}\simeq 0$, and $e^{+}e^{-}\to\gamma
jj$ on the $Z$ pole (LEP-I) does around $g_{a\gamma\gamma}=0$. These collider
bounds are satisfied in the region between the green lines.

Figure 3: EWPO probability distribution on $(g_{a\gamma\gamma},g_{aWW})$.
Here, $m_{a}=5~{}{\rm GeV}$, $f_{a}=1\,\mathrm{TeV}$, and $\Lambda=4\pi f_{a}$
are taken. The red and blue regions are the same as Fig. 2. On the left
(right) panel, the green (yellow) regions are the EWPT result only via $S$ and
$U$, i.e., ignoring the ALP contributions via $\Delta\alpha$, $\Delta_{Z}$,
$\Delta_{W}$ and $\Gamma_{a\gamma}$. The darker (lighter) regions with the
dashed lines correspond to the 68% (95%) level.

On the right panel, the ALP mass is set as $m_{a}=5\,\mathrm{GeV}$. The EWPT
results are almost the same as those on the left panel: Note that the vertical
axis is shown on a linear (logarithmic) scale on the right (left). For this
ALP mass, the flavor constraints are absent because the heavy-meson decay into
the ALP is kinematically blocked, and the collider bounds become relevant. The
green lines correspond to the constraints from the LEP-I data, as in the case
with $m_{a}=4\,\mathrm{GeV}$. Since this constraint is governed by
$g_{aZ\gamma}$, the region around $g_{aZ\gamma}=0$ is allowed. The scattering
$e^{+}e^{-}\to\gamma\gamma(\gamma)$ at LEP-II also gives an independent
constraint, and the region inside the magenta lines is allowed. In particular,
this constraint works even for $g_{aZ\gamma}\simeq 0$, because a photon-
exchange diagram contributes to the scattering. Although the LHC constraint by
the non-resonant searches for the ALPs is shown by the black line, where the
region inside the line is allowed, the result is weaker than those from LEP I
and II.

In both plots, the regions allowed by all the experimental constraints are
filled by the cyan.#8#8#8 In addition to the constraints discussed in Fig. 2,
the Higgs boson decaying into a pair of ALPs, $h\to aa$, may limit the model.
We have checked that it is weaker than those from the flavor and collider
experiments.  It is found that the EWPT result becomes consistent with the
constraints at the 68% level only when the PDG value is adopted for the $W$
mass. Besides, although the region around $g_{aZ\gamma}=0$ is relatively
favored by the constraints, the probability likelihood of the EWPOs is not
improved well. Therefore, we conclude that the recent CDF result of $m_{W}$
cannot be explained by the ALP as long as the mass is $m_{a}\ll m_{Z}$.

Let us compare our results with those based on the analyses in the previous
studies Bauer:2017ris ; Bauer:2018uxu ; Yuan:2022cpw . They considered the ALP
much lighter than the $Z$ boson, which is the same as the setup in this
subsection. However, as mentioned in Sec. 3, the contributions via
$\Delta\alpha$, $\Delta_{Z}$, $\Delta_{W}$ and $\Gamma_{a\gamma}$ have been
ignored, i.e., the ALP contributions have been supposed to arise only via $S$
and $U$. In Fig. 3, we show the EWPT results with and without including
$\Delta\alpha$, $\Delta_{Z}$, $\Delta_{W}$ and $\Gamma_{a\gamma}$. Here,
$m_{a}=5~{}{\rm GeV}$, $f_{a}=1\,\mathrm{TeV}$ and $\Lambda=4\pi f_{a}$ are
set. The former result is the same as those on the right panel in Fig 2, while
the latter corresponds to the setup in the previous studies.#9#9#9 The “only
$S,U$” regions in the figure are slightly different from those in the previous
papers because the global-fit procedures are not the same. For example, $T=0$
is not fixed in Ref. Yuan:2022cpw  when the parameter regions of $S$ and $U$
are determined. In contrast, this condition is satisfied in our analysis
because the ALP does not contribute to $T$.  On the left (right) panel,
$m_{W}^{\rm CDF}$ $(m_{W}^{\rm PDG})$ is adopted as the experimental value of
the $W$ mass. In both cases, it is obvious that the EWPT results are modified
drastically by the new contributions. In particular, the impact of $Z\to
a\gamma$ is strong, because it proceeds at the tree level, and thus, worsens
the global fit significantly in the region with $g_{aZ\gamma}\neq 0$. In
addition, the effects via $\Delta\alpha$, $\Delta_{Z}$, and $\Delta_{W}$ are
comparable to those via $S$ and $U$. Therefore, it is found that the analyses
based only on $S$ and $U$ are not valid. The ALP contributions via
$\Delta\alpha$, $\Delta_{Z}$, $\Delta_{W}$ and $\Gamma_{a\gamma}$ must be
taken into account in the EWPO analysis.

###  5.2 Heavier ALP case

Figure 4: Same as Fig. 2 but $m_{a}=195\,\mathrm{GeV}$ (left) and
$600\,\mathrm{GeV}$ (right). The collider bounds are shown by the orange and
green solid lines, and both of them are satisfied in the cyan region.

As shown in the previous subsection, the EWPT results for $m_{a}\ll m_{Z}$ are
tightly limited by the flavor and collider constraints. In this subsection, we
consider the case of $m_{a}\gtrsim m_{Z}$. In Fig. 4, we show the EWPT results
on the $(g_{a\gamma\gamma},g_{aWW})$ plane. Here, we take
$m_{a}=195\,\mathrm{GeV}$ and $600\,\mathrm{GeV}$ on the left and right
panels, respectively. Also, $\Lambda=4\pi f_{a}$ with $f_{a}=1\,\mathrm{TeV}$
is set for both masses.

The collider constraints depend on the ALP mass. For $m_{a}=195~{}{\rm GeV}$,
the photon scattering via the off-shell ALP exchange, $\gamma\gamma\to
a^{*}\to\gamma\gamma$, gives the best constraint on $g_{a\gamma\gamma}$. The
region between the green lines is allowed by this bound. On the other hand,
the constraint from $pp\to Z^{*}\to a\gamma$ with $a\to
Z\gamma\to\nu\nu\gamma$ is shown by the orange lines, where the region between
the lines is allowed. Note that the production cross section becomes smaller
as $g_{aZ\gamma}$ is suppressed. The region that satisfies both constraints is
shown by the cyan. It is found that the EWPT result is consistent with them at
the 68% level if the PDG value is adopted for the $W$ mass, while there is no
overlap for the CDF value.

As mentioned in Sec. 4, many of the collider bounds can be avoided if the ALP
is heavier than $500\,\mathrm{GeV}$. From the right panel, where the mass is
$m_{a}=600~{}{\rm GeV}$, only the search for the on-shell ALP production via
the photon fusion, $\gamma\gamma\to a\to\gamma\gamma$, gives a constraint.
This is displayed by the green lines, and the region between them is allowed
(filled by the cyan). It is seen that the ALP coupling is favored to satisfy
$g_{a\gamma\gamma}\simeq 0$, and the constraint becomes weaker as $g_{aWW}$
grows because the branching ratio of $a\to\gamma\gamma$ decreases. We found
that the EWPT result can be consistent with the constraints at the 68% level
even for $m_{W}^{\rm CDF}$ as well as $m_{W}^{\rm PDG}$. The $W$ mass will be
discussed more in Sec. 5.3.

As the ALP coupling to di-photon is favored to be suppressed, let us focus on
the case with $g_{a\gamma\gamma}=0$. In Fig. 5, we show the EWPT result for
various $m_{a}$ with $g_{a\gamma\gamma}=0$ and compare them with the
experimental constraints. Here, we take $\Lambda=4\pi f_{a}$ with
$f_{a}=1\,\mathrm{TeV}$. The probability distribution is normalized on the
$g_{aWW}$ parameter space for each $m_{a}$, and the 68% probability range is
shown. The result for 95% probability range is given in Appendix C. The red
(blue) colored band is the result obtained for $m_{W}^{\rm CDF}$ $(m_{W}^{\rm
PDG})$. The regions excluded by the flavor, LEP, and LHC are shown by the
brown, orange, and green regions, respectively. It is found that the EWPT
result for $m_{W}^{\rm PDG}$ can be consistent with the constraints at the 68%
level except for $50\lesssim m_{a}\lesssim 160\,\mathrm{GeV}$, though a part
of the parameter regions is excluded for $m_{a}\lesssim 50\,\mathrm{GeV}$ (see
also the discussion in Sec. 5.3). In contrast, the result for $m_{W}^{\rm
CDF}$ is already excluded unless the ALP is heavier than $500\,\mathrm{GeV}$.

Figure 5: EWPO probability distribution for various $m_{a}$. Here,
$g_{a\gamma\gamma}=0$, $f_{a}=1\,\mathrm{TeV}$, and $\Lambda=4\pi f_{a}$ are
taken. The distribution function is normalized on the $g_{aWW}$ parameter
region for each $m_{a}$. The red (blue) colored region corresponds to the 68%
level for $m_{W}^{\rm CDF}$ $(m_{W}^{\rm PDG})$. The regions excluded by the
flavor, LEP, and LHC constraints are filled by brown, orange, and green,
respectively.

###  5.3 Goodness of fit and $W$-boson mass

In Secs. 5.1 and 5.2, we showed the EWPT results, where the likelihood
functions were studied by performing the global fit to the EWPOs. It should be
stressed that the probability distribution was normalized on the ALP-coupling
plane, i.e., the total probability is equal to unity on the plane, for each
ALP mass. Therefore, the parameter region corresponding to “68% probability”
does not always mean that all theoretical values are consistent with the
experimental data but indicates that the fit inside the region is relatively
better than the outside. For instance, the tension between the theoretical and
CDF values of the $W$ mass is not always solved even in that region. In this
subsection, let us discuss how well the ALP contributions improve the global
fit, particularly paying attention to the case when $m_{W}^{\rm CDF}$ is
adopted to the likelihood.

Since the largest tension arises in the $W$ mass, the goodness of fit is
governed by $m_{W}$ among the EWPOs listed in Table 1. In Fig. 6, we show the
theoretical values of $m_{W}$ for various $m_{a}$. Here,
$g_{a\gamma\gamma}=0$, $f_{a}=1\,\mathrm{TeV}$ and $\Lambda=4\pi f_{a}$ are
set. The results are obtained by performing the global fit to the EWPOs. In
particular, the indirect prediction of $m_{W}$ is displayed by the black bar,
where $m_{W}$ is not included in the fit observables. On the other hand, the
red bar represents the theoretical value for which $m_{W}$ is included in the
likelihood. We derive the probability distributions as a function of $m_{W}$.
Then, the central value, which is denoted by the dot on the bar, is determined
such that the likelihood becomes maximum, while the 68% uncertainties are
shown by the bars. The results are compared with the SM prediction (green),
PDG (cyan), and CDF-averaged (orange) values, where the $1\sigma$ range is
shown.

Figure 6: ALP predictions of $m_{W}$. Here, $g_{a\gamma\gamma}=0$,
$f_{a}=1\,\mathrm{TeV}$, and $\Lambda=4\pi f_{a}$ are taken. The red (black)
bars represent the results obtained with/without including $m_{W}$ in the
global fit. The orange and cyan bands correspond to $m_{W}^{\rm CDF}$ and
$m_{W}^{\rm PDG}$ with the $1\sigma$ uncertainty, while the green band is the
SM prediction with the $1\sigma$ error. The ALP coupling is restricted by the
experimental constraints during the fit analysis for the results with the
label “w/ Exp.”

For $m_{a}=5\,\mathrm{GeV}$, the black and red bars are obtained without
taking the experimental constraints into account. Even though the constraints
are ignored, the theoretical values cannot be consistent with $m_{W}^{\rm
CDF}$ because of the effects of $Z\to a\gamma$. Since the ALP contribution
deteriorates the fit rapidly, the model cannot change the theoretical values
of the EWPOs against the SM prediction, i.e., does not solve the $m_{W}$
tension. Although the result is shown for $m_{a}=5\,\mathrm{GeV}$, the same
conclusion holds as long as $Z\to a\gamma$ is effective, i.e., for
$m_{a}\lesssim 90\,\mathrm{GeV}$. Similarly, the goodness of fit for the case
with $m_{W}^{\rm PDG}$ is not improved so much against the SM case.

If the ALP is heavier than the $Z$ boson, the decay of $Z\to a\gamma$ is
kinematically blocked. We show the results for $m_{a}=195\,\mathrm{GeV}$ with
and without restring the parameter space by the experimental constraints,
which are shown as the label, “w/o Exp” and “w/ Exp,” respectively. According
to the former result, although the indirect prediction seems to be around the
SM value and does not overlap with the $m_{W}^{\rm CDF}$ range, the
uncertainty is asymmetric between the smaller and larger $m_{W}$ values.
Namely, the probability distribution has a long tail toward larger $m_{W}$.
Consequently, the theoretical prediction with including $m_{W}$ in the fit
(the red bar) becomes consistent with $m_{W}^{\rm CDF}$ at the 68% level.

Once the experimental constraints are taken into account, the parameter space
is restricted tightly for $m_{a}=195\,\mathrm{GeV}$. Then, the ALP
contribution cannot make $m_{W}$ shift from the SM prediction sufficiently, as
shown by the result for “w/ Exp.” It is understood from Fig. 5 that the same
conclusion holds as long as the collider bounds are tight.

The collider constraints as well as those from the flavor are relaxed if the
ALP is heavier than $500\,\mathrm{GeV}$. In the table, we show the theoretical
values for $m_{a}=600\,\mathrm{GeV}$. As shown in Fig. 4, the model is free
from the experimental constraints for $g_{a\gamma\gamma}=0$. Similar to the
case for $m_{a}=195\,\mathrm{GeV}$ without including the experimental
constraints, it is found that the ALP contributions can solve the tension
between the SM and CDF-averaged values of the $W$ mass.

In summary, although the quality of the global fit to the EWPOs can be
improved effectively against the SM case by the ALP with a mass larger than
$m_{Z}$, the model is constrained tightly by the collider measurements for
$m_{a}<500\,\mathrm{GeV}$. Thus, the tension between the SM and CDF-averaged
values of the $W$ mass can be solved only when the ALP is heavier than
$500\,\mathrm{GeV}$.

In Fig. 6, we have focused on the ALP model with $m_{W}^{\rm CDF}$ for the
experimental value of the $W$ mass. Here, let us comment on the case when
$m_{W}^{\rm PDG}$ is adopted. Similar to the above case, the probability
likelihood is less affected by the ALP as long as $Z\to a\gamma$ and/or the
experimental bounds are severe. The goodness of fit is improved effectively
compared to the SM case if the ALP is heavier than about $160\,\mathrm{GeV}$
under the collider constraints (see also Fig. 5).

Figure 7: Same as Fig. 5 but $\Lambda=f_{a}$.

Before closing this section, let us comment on the cutoff dependence of the
results. The cutoff appears in the loop functions and should be determined by
a UV theory of the ALP model. In Fig. 7, we perform the same analysis as Fig.
5 but with $\Lambda=f_{a}$. It is noticed that the EWPT regions drop rapidly
above $m_{a}=400\,\mathrm{GeV}$. This is because the ALP contributions
especially to the $W$ mass flip their signs. As a result, the goodness of fit
is not improved well by the ALP contributions. Although the $m_{W}$ tension
seems to be solved for $90\lesssim m_{a}\lesssim 400\,\mathrm{GeV}$, all the
parameter regions are excluded by the experimental constraints. Therefore, if
the recent CDF result is confirmed in future experiments, a larger cutoff
scale is favored.

##  6 Conclusions and discussion

We have studied the EWPOs in the ALP model. The ALP is assumed to couple with
the SM ${\rm SU(2)}_{L}$ and the ${\rm U(1)}_{Y}$ gauge bosons. We have
provided the formulae of the ALP contributions to the EWPOs valid for any ALP
mass and shown that those to $\Delta\alpha$, $\Delta_{Z}$, and $\Delta_{W}$ as
well as the oblique parameter $U$ can be comparable to $S$ ($T=0$ in the ALP
model). Furthermore, the decay of $Z\to a\gamma$ contributes to the total
width of the $Z$-boson decay significantly.

We have analyzed the probability likelihood generated from the ALP
contributions by performing the global fit to the EWPOs and compared the
results with the flavor and collider constraints. Because of $Z\to a\gamma$,
the ALP contributions to the EWPOs are suppressed as long as the ALP is
lighter than the $Z$ boson. Besides, since the experimental constraints are
tight, the EWPT regions are widely excluded already if the ALP is not so
heavy. Also, even when the ALP is heavy, the parameter regions are limited as
long as the ALP is coupled sizably with a pair of photons. Therefore, a heavy
ALP with interactions satisfying $g_{a\gamma\gamma}\sim 0$ is preferred to
improve the global fit compared to the SM case.

In the numerical analysis, we have studied both of the cases when the recent
CDF result of the $W$ mass measurement is included and excluded in the
experimental data. If the PDG value adopted, i.e., the CDF result is neglected
in the $W$ mass average, the ALP is favored to be heavier than about
$160\,\mathrm{GeV}$ with satisfying $g_{a\gamma\gamma}\sim 0$. On the other
hand, if the recent CDF result, which is inconsistent with the SM prediction,
is included, it has been concluded that the ALP model can solve the tension
only for $m_{a}>500\,\mathrm{GeV}$ with $g_{a\gamma\gamma}\sim 0$.

Let us comment on contributions from higher-dimensional operators that are not
included in the ALP Lagrangian (1). They are likely to be non-negligible when
$m_{a}$ is comparable to $f_{a}$. The higher-dimensional operators should
depend on a UV theory of the ALP model. The theory also determines the cutoff
scale, which has been set as a model parameter in our analysis. Since we have
set $f_{a}=1\,\mathrm{TeV}$, the results might be altered especially around
$m_{a}\sim 1\,\mathrm{TeV}$.

We have argued that the EWPOs are sensitive to the ALP models. In particular,
if the recent CDF result of the $W$ mass measurement would be confirmed in
future experiments, the model could provide an attractive solution. Although
the model parameters are restricted by the experimental constraints, the model
setup for solving the tension has not been explored sufficiently by colliders
such as the LHC. Therefore, further collider studies would be helpful to
establish the scenario.

## Acknowledgements

This work is supported by the Japan Society for the Promotion of Science
(JSPS) Grant-in-Aid for Scientific Research on Innovative Areas (No. 21H00086
[ME] and No. 22J01147 [MA]) and Scientific Research B (No. 21H01086 [ME]).

##  Appendix A Passarino-Veltman functions

The Passarino-Veltman functions Passarino:1978jh  are denoted explicitly as

$\displaystyle A_{0}(m_{0})$
$\displaystyle=\int\frac{\overline{\differential^{d}k}}{i\pi^{2}}\frac{1}{k^{2}-m_{0}^{2}+i\epsilon}=m_{0}^{2}\quantity(1-\ln{\frac{m_{0}^{2}}{\mu^{2}}}),$
(90) $\displaystyle B_{0}(p^{2};m_{0},m_{1})$
$\displaystyle=\int\frac{\overline{\differential^{d}k}}{i\pi^{2}}\frac{1}{[k^{2}-m_{0}^{2}+i\epsilon][(k+p)^{2}-m_{1}^{2}+i\epsilon]}$
$\displaystyle=-\int_{0}^{1}\differential
x\ln{\quantity[\frac{-x(1-x)p^{2}+xm_{0}^{2}+(1-x)m_{1}^{2}}{\mu^{2}}]},$ (91)

where $d=4-2\epsilon$ and
$\overline{\differential^{d}k}=\Gamma(1-\epsilon)(\pi\mu^{2})^{\epsilon}\differential^{d}k$
in the $\overline{\rm MS}$ regularization Hagiwara:1994pw . We take
$\mu=\Lambda$ in this paper. The derivative of the $B_{0}$ function with
respect to $p^{2}$ is obtained as

$\displaystyle B^{\prime}_{0}(p^{2};m_{0},m_{1})$
$\displaystyle=\int_{0}^{1}\differential
x\frac{x(1-x)}{-p^{2}x(1-x)+xm_{0}^{2}+(1-x)m_{1}^{2}}.$ (92)

##  Appendix B Three-body decay of ALP

The decay width for $a\to Z^{*}\gamma\to ff\gamma$ is derived at the tree
level as

$\displaystyle\Gamma(a\to f\bar{f}\gamma)$
$\displaystyle=N_{c}^{f}\frac{g_{Z}^{2}g_{aZ\gamma}^{2}}{1536\pi^{3}m_{a}^{3}}\quantity[(g_{V,f})^{2}+(g_{A,f})^{2}]\int_{0}^{m_{a}^{2}}\differential
x\frac{x(m_{a}^{2}-x)^{3}}{(x-m_{Z}^{2})^{2}+m_{Z}^{2}\Gamma_{Z}^{2}},$ (93)

where the mass of the fermions in the final state is neglected. The integral
is evaluated analytically as

$\displaystyle\int_{0}^{m_{a}^{2}}\differential
x\frac{x(m_{a}^{2}-x)^{3}}{(x-m_{Z}^{2})^{2}+m_{Z}^{2}\Gamma_{Z}^{2}}=\frac{1}{6}\Bigg{\\{}-11m_{a}^{6}+30m_{a}^{4}m_{Z}^{2}-18m_{a}^{2}m_{Z}^{4}+6m_{a}^{2}m_{Z}^{2}\Gamma_{Z}^{2}$
$\displaystyle\quad+3[3m_{Z}^{2}(3m_{Z}^{2}-\Gamma_{Z}^{2})m_{a}^{2}-4m_{Z}^{4}(m_{Z}^{2}-\Gamma_{Z}^{2})+m_{a}^{6}-6m_{a}^{4}m_{Z}^{2}]$
$\displaystyle\qquad\times\ln{\quantity[\frac{(m_{a}^{2}-m_{Z}^{2})^{2}+m_{Z}^{2}\Gamma_{Z}^{2}}{m_{Z}^{2}(m_{Z}^{2}+\Gamma_{Z}^{2})}]}$
$\displaystyle\quad-6\frac{m_{Z}}{\Gamma_{Z}}[\Gamma_{Z}^{4}m_{Z}^{2}-3\Gamma_{Z}^{2}(m_{a}^{4}-3m_{a}^{2}m_{Z}^{2}+2m_{Z}^{4})-(m_{a}^{2}-m_{Z}^{2})^{3}]$
$\displaystyle\qquad\times\quantity[\arctan{\quantity(\frac{m_{Z}}{\Gamma_{Z}})}+\arctan{\quantity(\frac{m_{a}^{2}-m_{Z}^{2}}{m_{Z}\Gamma_{Z}})}]\Bigg{\\}}.$
(94)

##  Appendix C EWPO probability distribution for 95% level.

Figure 8: Upper bound from the EWPT on $g_{aWW}$ at the 95% level (blue line).
Here, the experimental value of the $W$ mass is provided by $m_{W}^{\rm PDG}$.
See Fig. 5 for the model setup and the constraints.

In Fig. 8, we show the upper bound from the EWPT on $g_{aWW}$ at the 95% level
(blue line). Here, the experimental value of the $W$ mass is provided by
$m_{W}^{\rm PDG}$. Also, the model setup and the constraints are the same as
Fig. 5. Note that the EWPT result is consistent with the SM limit, i.e.,
$g_{aWW}=0$, at the 95% level. It is found that the EWPT provides the most
sensitive probe of the ALP for $m_{a}>500\,\mathrm{GeV}$, and the model is
disfavored if $g_{aWW}\gtrsim 4\text{--}6\,\mathrm{TeV}^{-1}$.

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