aainstitutetext: KEK Theory Center, Tsukuba, Ibaraki 305–0801,
Japanbbinstitutetext: Graduate Institute for Advanced Studies, SOKENDAI,
Tsukuba, Ibaraki 305–0801, Japan

# Higgs Probes of axion-like particles

Masashi Aiko a,b and Motoi Endo

###### Abstract

We study axion-like particle contributions to the Higgs boson decays. The
particle is assumed to couple with the standard model electroweak gauge
bosons. Although direct productions of axion-like particles have often been
discussed, we investigate indirect contributions to the Higgs boson decays
into fermions, photons, $W$, and $Z$ bosons at the one-loop level. It is found
that the corrections to the fermions are suppressed, whereas precise
measurements of the di-photon channel of the Higgs boson decay can provide a
significant probe of the model especially when the axion-like particle is
heavy and its coupling to di-photon is suppressed.

††preprint: KEK–TH–2548

##  1 Introduction

The introduction of an axion-like particle (ALP) is one of the simplest
extensions of the Standard Model (SM) and has been attracting wide interest.
It is often motivated in models with spontaneous violations of global
symmetries and is characterized by interactions with gauge bosons.

We consider ALP primarily coupled with the SM ${\rm U(1)}_{Y}$ and ${\rm
SU(2)}_{L}$ gauge bosons. It then interacts with the photon ($\gamma$), $Z$
boson, and charged $W$ boson ($W^{\pm}$) after the electroweak (EW) symmetry
breaking. Such a particle affects experimental observables. For example,
productions of ALPs at colliders have been studied extensively 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 . They are
produced in heavy flavor decays if they are light enough Izaguirre:2016dfi ;
Alonso-Alvarez:2018irt ; Gavela:2019wzg ; Guerrera:2021yss ; Bauer:2021mvw ;
Guerrera:2022ykl . Moreover, they can affect electroweak precision observables
through direct ALP production in the $Z$ boson decay or indirectly via vacuum
polarizations of the EW gauge bosons Bauer:2017ris ; Aiko:2023trb .

In this paper, we study ALP contributions to decay widths of the Higgs boson
($h$). In literature, direct ALP productions from Higgs boson decays, e.g.,
$h\to Za$ and $h\to aa$, have often been discussed Bauer:2017ris ;
Evans:2019xer ; Davoudiasl:2021haa ; Alves:2021puo ; Cepeda:2021rql . Those
channels are open when the ALP is lighter than the Higgs boson. However, if it
is heavier, the direct production channels are forbidden kinematically, and
the ALP affects the Higgs boson decays via radiative corrections. In this
paper, we will investigate ALP-loop contributions to the Higgs boson decays:
$h\to f\bar{f}$ with $f$ denoting SM fermions, $h\to\gamma\gamma$, $h\to
Z\gamma$, $h\to ZZ^{*}$, and $h\to WW^{*}$. They will be compared with the
direct ALP productions from Higgs boson decays. We will also take collider
constraints into account. It will be shown that the contribution to the
fermion channels is suppressed, while $h\to\gamma\gamma$ provides a
significant probe of the ALP compared with other channels such as $h\to
Z\gamma$, $h\to ZZ^{*}$, $h\to WW^{*}$, and $h\to aa$.

This paper is organized as follows. In Sec. 2, we introduce the ALP model. In
Sec. 3, the ALP contributions to Higgs boson decays are explained. Numerical
results are given in Sec. 4, and Sec. 5 is devoted to the conclusion. In
Appendix A, the analytic expressions for the ALP contributions at the one-loop
level are summarized.

##  2 Model

Let us introduce ALP $(a)$. It is assumed to couple with the ${\rm SU(2)}_{L}$
gauge boson $(W_{\mu}^{a})$ and the ${\rm U(1)}_{Y}$ gauge boson $(B_{\mu})$.
We do not consider ALP couplings with other SM particles. The Lagrangian is
given by

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

where $m_{a}$ and $f_{a}$ are the mass and decay constant of the ALP,
respectively. In this paper, the coupling constants, $c_{WW}$ and $c_{BB}$, as
well as $m_{a}$ and $f_{a}$ are regarded as free parameters but $m_{a}<f_{a}$
is satisfied. Field strengths of the gauge bosons and their dual are shown 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)
$\displaystyle\widetilde{X}_{\mu\nu}$
$\displaystyle=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}X_{\rho\sigma},\quad(X=W^{a},B)$
(4)

where $g$ is the ${\rm SU(2)}_{L}$ gauge coupling constant. The totally
antisymmetric tensors are defined with $\epsilon^{012}=1$ and
$\epsilon^{0123}=1$. After the EW symmetry breaking, the above interactions
are expressed as

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

where quartic interaction terms are omitted, and the field strengths denote
$X_{\mu\nu}=\partial_{\mu}X_{\nu}-\partial_{\nu}X_{\mu}$. Here and hereafter,
$A_{\mu}$, $Z_{\mu}$, and $W_{\mu}$ represent the photon, $Z$, and charged $W$
bosons, respectively. Those coupling constants are related to $c_{WW}$ and
$c_{BB}$ as

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

Here and hereafter, $c_{W}=\cos{\theta_{W}}$, $s_{W}=\sin{\theta_{W}}$,
$c_{2W}=\cos{2\theta_{W}}$, and $s_{2W}=\sin{2\theta_{W}}$ with the Weinberg
angle $\theta_{W}$. It is stressed that the original ALP interactions in our
setup are governed by two coupling constants, ${c}_{BB}$ and ${c}_{WW}$,
though there are four couplings, $g_{a\gamma\gamma},g_{aZ\gamma},g_{aZZ}$, and
$g_{aWW}$, after the EW symmetry breaking. It is often useful to represent
$g_{aZ\gamma}$ and $g_{aZZ}$ in terms of $g_{a\gamma\gamma}$ and $g_{aWW}$,

$\displaystyle g_{aZ\gamma}$
$\displaystyle=\frac{s_{W}(g_{aWW}-g_{a\gamma\gamma})}{c_{W}},$ (10)
$\displaystyle g_{aZZ}$
$\displaystyle=\frac{c_{2W}g_{aWW}+s_{W}^{2}g_{a\gamma\gamma}}{c_{W}^{2}}.$
(11)

##  3 Higgs boson decay

In this section, we explain ALP contributions to the Higgs boson decays at the
one-loop level. The decay channels in interest are $h\to f\bar{f}$, $h\to
ZZ^{*}$, $h\to WW^{*}$, $h\to\gamma\gamma$, and $h\to Z\gamma$ as well as
$h\to aa$.#1#1#1 It is noted that there is no ALP contribution to $h\to gg$ at
the one-loop level in our setup. Also, $h\to Za$ is not induced at this level
because gauge-boson loop contributions vanish Bauer:2017ris .  Within the SM,
the first three of them proceed at the tree level, and the following two are
induced radiatively. Also, the last channel is absent. The ALP affects all of
them via radiative corrections.

For the radiative corrections, we adopt the on-shell renormalization to the EW
sector by following Refs. Bohm:1986rj ; Hollik:1988ii #2#2#2Since there is no
ALP mixing with the CP-odd Nambu-Goldstone boson at the one-loop level, we use
the gauge fixing terms, which are the same as in the SM. . The counterterms
relevant to ALP corrections to the Higgs boson decays are introduced in Sec.
3.1, and the decay width for each channel will be given in Sec. 3.2, 3.3, and
3.4.

###  3.1 Renormalization

The bare gauge fields (described with the subscript $B$) are related to the
renormalized ones by means of the counterterms (denoted as $\delta X$) as

$\displaystyle W_{B\mu}^{\pm}$ $\displaystyle=\quantity(1+\frac{1}{2}\delta
Z_{W})W_{\mu}^{\pm},$ (12) $\displaystyle\left(\begin{array}[]{c}Z_{B\mu}\\\
A_{B\mu}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}1+\cfrac{1}{2}\delta
Z_{Z}&\cfrac{1}{2}\left(\delta Z_{Z\gamma}-\cfrac{\delta
s_{W}^{2}}{c_{W}s_{W}}\right)\\\ \cfrac{1}{2}\left(\delta
Z_{Z\gamma}+\cfrac{\delta s_{W}^{2}}{c_{W}s_{W}}\right)&1+\cfrac{1}{2}\delta
Z_{\gamma}\end{array}\right)\left(\begin{array}[]{c}Z_{\mu}\\\
A_{\mu}\end{array}\right).$ (19)

The renormalized $W,Z$ masses and the measured value of the fine-structure
constant are related to bare parameters as

$\displaystyle m_{W,B}^{2}=m_{W}^{2}+\delta m_{W}^{2},\quad
m_{Z,B}^{2}=m_{Z}^{2}+\delta m_{Z}^{2},\quad\alpha_{{\rm em},B}=\alpha_{\rm
em}+\delta\alpha_{\rm em}.$ (20)

The Weinberg angle and the Higgs vacuum expectation value (VEV) satisfy the
following tree-level relations,

$\displaystyle s_{W,B}^{2}=1-\cfrac{m_{W,B}^{2}}{m_{Z,B}^{2}},\quad
v_{B}^{2}=\cfrac{m_{W,B}^{2}s_{W,B}^{2}}{\pi\alpha_{\mathrm{em},B}}.$ (21)

Then, their counterterms are obtained as

$\displaystyle\frac{\delta s_{W}^{2}}{s_{W}^{2}}$
$\displaystyle=\frac{c_{W}^{2}}{s_{W}^{2}}\left(\frac{\delta
m_{Z}^{2}}{m_{Z}^{2}}-\frac{\delta m_{W}^{2}}{m_{W}^{2}}\right),$ (22)
$\displaystyle\frac{\delta v}{v}$
$\displaystyle=\frac{1}{2}\quantity(\frac{\delta
m_{W}^{2}}{m_{W}^{2}}+\frac{\delta
s_{W}^{2}}{s_{W}^{2}}-\frac{\delta\alpha_{\rm em}}{\alpha_{\rm em}}).$ (23)

Let us represent the renormalized inverse propagators of gauge bosons as

$\displaystyle\widehat{\Gamma}^{\mu\nu}_{VV^{\prime}}=-ig^{\mu\nu}(p^{2}-m_{V}^{2})\delta_{VV^{\prime}}+i\widehat{\Pi}_{VV^{\prime}}^{\mu\nu}(p^{2}),$
(24)

with $V,V^{\prime}=W,Z,\gamma$. The self-energies
$\widehat{\Pi}_{VV^{\prime}}^{\mu\nu}(p^{2})$ are decomposed into longitudinal
$(L)$ and transverse $(T)$ polarization components,

$\displaystyle\widehat{\Pi}_{VV^{\prime}}^{\mu\nu}(p^{2})=\left(-g^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\widehat{\Pi}_{VV^{\prime}}^{T}(p^{2})+\frac{p^{\mu}p^{\nu}}{p^{2}}\widehat{\Pi}_{VV^{\prime}}^{L}(p^{2}).$
(25)

Since the longitudinal component is irrelevant to the following discussion, we
focus on the transverse one and omit the subscript, $T$ and $L$, for
simplicity.#3#3#3 Since the ALP is not coupled with the Higgs boson and
fermions at the tree level, there are no ALP contributions to the Higgs and
fermion self-energies.  In terms of the 1PI contributions and counterterms,
they are given by#4#4#4 Since there is no ALP contribution to tadpoles, we
omit the tadpole terms in this paper.

$\displaystyle\widehat{\Pi}_{WW}(p^{2})$ $\displaystyle=\Pi_{WW}^{{\rm
1PI}}(p^{2})-\delta m_{W}^{2}+\delta Z_{W}(p^{2}-m_{W}^{2}),$ (26)
$\displaystyle\widehat{\Pi}_{ZZ}(p^{2})$ $\displaystyle=\Pi_{ZZ}^{{\rm
1PI}}(p^{2})-\delta m_{Z}^{2}+\delta Z_{Z}(p^{2}-m_{Z}^{2}),$ (27)
$\displaystyle\widehat{\Pi}_{\gamma\gamma}(p^{2})$
$\displaystyle=\Pi_{\gamma\gamma}^{{\rm 1PI}}(p^{2})+\delta Z_{\gamma}p^{2},$
(28) $\displaystyle\widehat{\Pi}_{Z\gamma}(p^{2})$
$\displaystyle=\Pi_{Z\gamma}^{{\rm 1PI}}(p^{2})+\delta
Z_{Z\gamma}\left(p^{2}-\frac{1}{2}m_{Z}^{2}\right)+\frac{m_{Z}^{2}}{2}\frac{\delta
s_{W}^{2}}{c_{W}s_{W}}.$ (29)

In the renormalization scheme explored by Refs. Bohm:1986rj ; Hollik:1988ii ,
the residue of the two-point function for the weak gauge bosons is not unity.
Hence, we need to add a derivative of the self-energy,
$\widehat{\Pi}_{VV}(m_{V}^{2})^{\prime}$, to transition amplitudes with
external $W^{\pm}$ and $Z$ bosons,

$\displaystyle\widehat{\Pi}_{VV}(m_{V}^{2})^{\prime}$
$\displaystyle=\Pi_{VV}^{\rm 1PI}(m_{V}^{2})^{\prime}+\delta
Z_{V},\quad(V=W^{\pm},Z).$

Let us focus on the ALP contributions. They arise via $\Pi_{VV^{\prime}}^{\rm
1PI}$, and the results are obtained in Appendix A.2. In the following, the
counterterms denote those from ALP, $\delta X=\delta X_{\rm ALP}$. In
addition, we use the relation, $\Pi_{VV^{\prime}}^{\rm 1PI}(0)_{\rm ALP}=0$
(see Appendix A.2) to simplify the expressions. Then, we obtain

$\displaystyle\delta m_{W}^{2}$ $\displaystyle=\real\quantity[\Pi^{{\rm
1PI}}_{WW}(m_{W}^{2})_{\rm ALP}],$ (30) $\displaystyle\delta m_{Z}^{2}$
$\displaystyle=\real\quantity[\Pi^{{\rm 1PI}}_{ZZ}(m_{Z}^{2})_{\rm ALP}],$
(31) $\displaystyle\frac{\delta\alpha_{\rm em}}{\alpha_{\rm em}}$
$\displaystyle=\Pi_{\gamma\gamma}^{{\rm 1PI}}(0)^{\prime}_{\rm ALP},$ (32)

yielding $\delta s_{W}^{2}$ and $\delta v$ (see Eqs. (22) and (23)). The wave
function renormalization constants are shown as

$\displaystyle\delta Z_{\gamma}$ $\displaystyle=-\frac{\delta\alpha_{\rm
em}}{\alpha_{\rm em}},\quad\delta Z_{Z\gamma}=\frac{\delta
s_{W}^{2}}{c_{W}s_{W}},$ (33) $\displaystyle\delta Z_{V}$
$\displaystyle=2\frac{\delta v}{v}-\frac{\delta
m_{V}^{2}}{m_{V}^{2}},\quad(V=W^{\pm},Z).$ (34)

In our calculation, the Higgs VEV $v$ is represented by the measured value of
the Fermi coupling constant $G_{F}$,

$\displaystyle v^{2}=\frac{1}{\sqrt{2}G_{F}(1-\Delta r)},$ (35)

where $\Delta r$ is obtained from the muon decay rate evaluated at the one-
loop level Sirlin:1980nh . In our setup, it satisfies the following relation,

$\displaystyle\Delta r=-2\frac{\delta v}{v}.$ (36)

###  3.2 $h\to f\bar{f}$

Figure 1: Momentum assignments to the Higgs boson decays.

The decay proceeds at the tree level in the SM, and the ALP affects it via
radiative corrections. In general, the renormalized $hf\bar{f}$ vertex is
decomposed as

$\displaystyle\widehat{\Gamma}_{hf\bar{f}}(p_{1},p_{2},q)$
$\displaystyle=\widehat{\Gamma}_{hf\bar{f}}^{S}+\gamma_{5}\widehat{\Gamma}_{hf\bar{f}}^{P}+\not{p}_{1}\widehat{\Gamma}_{hf\bar{f}}^{V_{1}}+\not{p}_{2}\widehat{\Gamma}_{hf\bar{f}}^{V_{2}}$
$\displaystyle\quad+\not{p}_{1}\gamma_{5}\widehat{\Gamma}_{hf\bar{f}}^{A_{1}}+\not{p}_{2}\gamma_{5}\widehat{\Gamma}_{hf\bar{f}}^{A_{2}}+\not{p}_{1}\not{p}_{2}\widehat{\Gamma}_{hf\bar{f}}^{T}+\not{p}_{1}\not{p}_{2}\gamma_{5}\widehat{\Gamma}_{hf\bar{f}}^{PT},$
(37)

where $p_{1}\ (p_{2})$ denotes the incoming four-momentum of the fermion
(anti-fermion), and $q$ is the outgoing four-momentum of the Higgs boson (see
the left panel of Fig. 1). The form factors are composed of the tree and one-
loop contributions as

$\displaystyle\widehat{\Gamma}_{hf\bar{f}}^{X}={\Gamma}_{hf\bar{f}}^{X,\mathrm{tree}}+{\Gamma}_{hf\bar{f}}^{X,\mathrm{loop}},\quad(X=S,P,V_{1},V_{2},A_{1},A_{2},T,PT).$
(38)

The former is given by

$\displaystyle\Gamma_{hf\bar{f}}^{S,{\rm
tree}}=-\frac{m_{f}}{v},\quad\Gamma_{hf\bar{f}}^{X,{\rm tree}}=0\quad(X\neq
S).$ (39)

The latter is composed of the 1PI and counterterm contributions,

$\displaystyle\Gamma^{X,{\rm loop}}_{hf\bar{f}}=\Gamma^{X,{\rm
1PI}}_{hf\bar{f}}+\delta\Gamma^{X}_{hf\bar{f}}.$ (40)

In our setup, there is no ALP contribution to the 1PI terms at the one-loop
level, and the counterterms do not vanish only for
$\delta\Gamma^{S}_{hf\bar{f}}$ in the presence of the Higgs VEV counterterm.
Thus, the ALP contributions are obtained as

$\displaystyle\Gamma_{hf\bar{f},\,{\rm ALP}}^{S,{\rm
loop}}=\delta\Gamma_{hf\bar{f}}^{S}=-\Gamma_{hf\bar{f}}^{S,{\rm
tree}}\frac{\delta v}{v},\quad\Gamma_{hf\bar{f},\,{\rm ALP}}^{X,{\rm
loop}}=0\quad(X\neq S).$ (41)

As a result, the decay rate for $h\to f\bar{f}$ $(f\neq t)$ is expressed as

$\displaystyle\Gamma(h\to f\bar{f})=\Gamma_{\mathrm{LO}}(h\to
f\bar{f})\quantity[1+\Delta_{\mathrm{SM}}^{f}+\Delta_{\mathrm{ALP}}^{f}].$
(42)

At the leading order, the rate is obtained at the tree level as

$\displaystyle\Gamma_{\mathrm{LO}}(h\to
f\bar{f})=N_{c}^{f}\frac{G_{F}m_{h}m_{f}^{2}}{4\sqrt{2}\pi}\lambda^{3/2}\quantity(\frac{m_{f}^{2}}{m_{h}^{2}},\frac{m_{f}^{2}}{m_{h}^{2}}),$
(43)

with $N_{c}^{f}=3\,(1)$ for $f$ being quarks (leptons). The kinetic factor is
defined as

$\displaystyle\lambda\quantity(x,y)=(1-x-y)^{2}-4xy.$ (44)

The SM correction $\Delta_{\rm{SM}}^{f}$ is found in Refs. Kniehl:1991ze ;
Dabelstein:1991ky , and the ALP term $\Delta_{\rm{ALP}}^{f}$ at the one-loop
level is obtained from Eqs. (36) and (41) as

$\displaystyle\Delta_{\text{ALP}}^{f}$
$\displaystyle=\frac{2}{\absolutevalue{\Gamma_{hf\bar{f}}^{S,{\rm
tree}}}^{2}}\real\quantity[\Gamma_{hf\bar{f}}^{S,{\rm
tree}}\quantity(\Gamma^{S,{\rm loop}}_{hf\bar{f},\,{\rm ALP}})^{*}]-\Delta
r=0.$ (45)

Therefore, $\Delta_{\rm{ALP}}^{f}$ is canceled out, and there is no correction
from ALP to $h\to f\bar{f}$ at this level.

###  3.3 $h\to VV^{*}~{}(V=Z,W^{\pm})$

The decay proceeds at the tree level in the SM, and the ALP affects it via
radiative corrections. In general, the renormalized $hVV$ vertex is decomposed
as

$\displaystyle\widehat{\Gamma}_{hVV}^{\mu\nu}(p_{1},p_{2},q)$
$\displaystyle=\quantity[g^{\mu\nu}\widehat{\Gamma}_{hVV}^{1}+\frac{p_{1}^{\nu}p_{2}^{\mu}}{m_{V}^{2}}\widehat{\Gamma}_{hVV}^{2}+i\epsilon^{\mu\nu\rho\sigma}\frac{p_{1\rho}p_{2\sigma}}{m_{V}^{2}}\widehat{\Gamma}_{hVV}^{3}](p_{1}^{2},p_{2}^{2},q^{2}),$
(46)

where $p_{1}^{\mu}$ and $p_{2}^{\mu}$ are the incoming four-momenta of the
weak bosons, and $q^{\mu}$ is the outgoing four-momentum of the Higgs boson
(see the right panel of Fig. 1). The form factors are composed of the tree and
one-loop contributions as

$\displaystyle\widehat{\Gamma}^{X}_{hVV}={\Gamma}^{i,{\rm
tree}}_{hVV}+{\Gamma}^{i,{\rm loop}}_{hVV},\quad(i=1,2,3).$ (47)

The former is given by

$\displaystyle\Gamma^{1,\mathrm{tree}}_{hVV}=\frac{2m_{V}^{2}}{v},\quad\Gamma^{2,\mathrm{tree}}_{hVV}=\Gamma^{3,\mathrm{tree}}_{hVV}=0.$
(48)

The latter is decomposed into the 1PI and counterterm contributions as#5#5#5
The ALP contribution to the tadpole is absent and omitted in the expression.

$\displaystyle{\Gamma}^{i,\mathrm{loop}}_{hVV}=\Gamma^{i,\mathrm{1PI}}_{hVV}+\delta\Gamma^{i}_{hVV}.$
(49)

The ALP contributes to $\Gamma^{1,\mathrm{1PI}}_{hVV}$ and
$\Gamma^{2,\mathrm{1PI}}_{hVV}$ at the one-loop level. The results are given
in Appendix A.3. The ALP contributions to the counterterms become non-zero
only for $\delta\Gamma^{1}_{hVV}$ in the presence of the Higgs VEV counterterm
as

$\displaystyle\delta\Gamma^{1}_{hVV}=\Gamma^{1,\mathrm{tree}}_{hVV}\frac{\delta
v}{v},\quad\delta\Gamma^{2}_{hVV}=\delta\Gamma^{3}_{hVV}=0.$ (50)

####  3.3.1 $h\to ZZ^{*}$

Since the Higgs boson mass is smaller than $2m_{Z}$, one of the final-state
$Z$ bosons is off-shell and decays into a pair of SM fermions. The decay rate
for $h\to ZZ^{\ast}\to Zf\bar{f}$ is written as

$\displaystyle\Gamma(h\to Zf\bar{f})=\Gamma_{\mathrm{LO}}(h\to
Zf\bar{f})\quantity[1+\Delta_{\rm SM}^{Z}+\Delta_{\rm ALP}^{Z}].$ (51)

In the following, we neglect masses of the fermions. At the leading order, the
rate is obtained at the tree level as

$\displaystyle\Gamma_{\mathrm{LO}}(h\to Zf\bar{f})$
$\displaystyle=N_{c}^{f}\int_{0}^{s_{\rm max}^{Z}}|\overline{{\cal
M}_{0}^{Z}}|^{2}\,\differential s,$ (52)

with $s_{\rm max}^{Z}=(m_{h}-m_{Z})^{2}$. The squared amplitude
$|\overline{{\cal M}_{0}^{Z}}|^{2}$ is shown as

$\displaystyle|\overline{{\cal
M}_{0}^{Z}}|^{2}=\frac{g_{Z}^{2}}{256\pi^{3}m_{h}^{3}}\absolutevalue{\Gamma_{hZZ}^{1,\text{tree}}}^{2}\frac{v_{f}^{2}+a_{f}^{2}}{(x_{s}-x_{Z})^{2}}\frac{\lambda(x_{Z},x_{s})+12x_{Z}x_{s}}{3x_{Z}}\lambda^{1/2}(x_{Z},x_{s}),$
(53)

where $x_{Z}=m_{Z}^{2}/m_{h}^{2}$, $x_{s}=s/m_{h}^{2}$,
$s=(p_{f}+p_{\bar{f}})^{2}$, $v_{f}=I_{3}^{f}/2-Q_{f}s_{W}^{2}$, and
$a_{f}=I_{3}^{f}/2$ with the weak isospin $I_{3}^{f}$ and the electric charge
$Q_{f}$. The radiative corrections $\Delta_{\rm SM/ALP}^{Z}$ can be written in
terms of the renormalized quantities Kniehl:1993ay ; Kanemura:2019kjg . The
ALP correction $\Delta_{\rm ALP}^{Z}$ is obtained as#6#6#6In general, there
are also corrections from renormalized $Zf\bar{f}$ and $hf\bar{f}$ vertices
and 1PI box diagrams (see Refs. Kniehl:1993ay ; Kanemura:2019kjg ). However,
they do not receive ALP contributions in our setup and are omitted in the
expression.

$\displaystyle\Delta_{\rm ALP}^{Z}$
$\displaystyle=\frac{2}{\Gamma_{\mathrm{LO}}}\int_{0}^{s_{\rm
max}^{Z}}\\!\differential s\,|\overline{{\cal
M}_{0}^{Z}}|^{2}\real\\{\left[\frac{\Gamma_{hZZ}^{1,{\rm
loop}}}{\Gamma_{hZZ}^{1,{\rm
tree}}}+\frac{\bar{\lambda}(x_{Z},x_{s})}{x_{Z}}\frac{\Gamma_{hZZ}^{2,{\rm
loop}}}{\Gamma_{hZZ}^{1,{\rm tree}}}\right](m_{Z}^{2},s,m_{h}^{2})$
$\displaystyle\quad+\frac{v_{f}Q_{f}c_{W}s_{W}}{v_{f}^{2}+a_{f}^{2}}\frac{s-m_{Z}^{2}}{s}\left[\frac{\Gamma_{hZ\gamma}^{1,{\rm
1PI}}}{\Gamma_{hZZ}^{1,{\rm
tree}}}+\bar{\lambda}(x_{Z},x_{s})\frac{m_{h}^{2}\Gamma_{hZ\gamma}^{2,{\rm
1PI}}}{\Gamma_{hZZ}^{1,{\rm tree}}}\right](m_{Z}^{2},s,m_{h}^{2})$
$\displaystyle\quad-\frac{\widehat{\Pi}_{ZZ}(s)}{s-m_{Z}^{2}}-\frac{v_{f}Q_{f}s_{W}c_{W}}{v_{f}^{2}+a_{f}^{2}}\frac{\widehat{\Pi}_{Z\gamma}(s)}{s}-\Delta
r-\frac{1}{2}\widehat{\Pi}_{ZZ}(m_{Z}^{2})^{\prime}\Bigg{\\}}.$ (54)

The kinematic factor $\bar{\lambda}(x,y)$ is defined by

$\displaystyle\bar{\lambda}(x,y)=\frac{1-x-y}{2}\frac{\lambda(x,y)}{\lambda(x,y)+12xy}.$
(55)

####  3.3.2 $h\to WW^{*}$

Similarly to $h\to ZZ^{*}$, one of the final-state $W$ bosons is off-shell and
decays into a pair of SM fermions. The decay rate for $h\to WW^{*}\to
Wf\bar{f^{\prime}}$ is given by

$\displaystyle\Gamma(h\to Wf\bar{f^{\prime}})$
$\displaystyle=\Gamma_{\mathrm{LO}}(h\to
Wf\bar{f^{\prime}})\quantity[1+\Delta_{\rm SM}^{W}+\Delta_{\rm ALP}^{W}].$
(56)

In the following, we neglect masses of the fermions. At the leading order, the
rate is obtained at the tree level as

$\displaystyle\Gamma_{\mathrm{LO}}(h\to Wf\bar{f^{\prime}})$
$\displaystyle=N_{c}^{f}\int_{0}^{s_{\rm max}^{W}}\\!|\overline{{\cal
M}_{0}^{W}}|^{2}\,\differential s,$ (57)

with $s_{\rm max}^{W}=(m_{h}-m_{W})^{2}$. The squared tree-level amplitude
$|\overline{{\cal M}_{0}^{W}}|^{2}$ is obtained as

$\displaystyle|\overline{{\cal
M}_{0}^{W}}|^{2}=\frac{g^{2}}{512\pi^{3}m_{h}^{3}}\absolutevalue{\Gamma_{hWW}^{1,\text{tree}}}^{2}\frac{\lambda(x_{W},x_{s})+12x_{W}x_{s}}{3x_{W}(x_{s}-x_{W})^{2}}\lambda^{1/2}(x_{W},x_{s}),$
(58)

with $x_{W}=m_{W}^{2}/m_{h}^{2}$ and $s=(p_{f}+p_{\bar{f^{\prime}}})^{2}$. The
radiative corrections $\Delta_{\rm SM/ALP}^{W}$ can be written in terms of the
renormalized quantities Kanemura:2019kjg . The ALP correction
$\Delta_{\rm{ALP}}^{W}$ is obtained as#7#7#7Similarly to $h\to ZZ^{*}$, there
are generally corrections from renormalized $Wf\bar{f^{\prime}}$ and
$hf\bar{f}$ vertices and 1PI box diagrams. However, ALP contributions vanish
in our setup, and only non-zero terms are shown.

$\displaystyle\Delta_{\rm ALP}^{W}$
$\displaystyle=\frac{2}{\Gamma_{\mathrm{LO}}}\int_{0}^{s_{\rm
max}^{W}}\\!\differential s\,|\overline{{\cal
M}_{0}^{W}}|^{2}\real\\{\left[\frac{\Gamma_{hWW}^{1,{\rm
loop}}}{\Gamma_{hWW}^{1,{\rm
tree}}}+\frac{\bar{\lambda}(x_{W},x_{s})}{x_{W}}\frac{\Gamma_{hWW}^{2,{\rm
loop}}}{\Gamma_{hWW}^{1,{\rm tree}}}\right](m_{W}^{2},s,m_{h}^{2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad-\frac{\widehat{\Pi}_{WW}(s)}{s-m_{W}^{2}}-\Delta
r-\frac{1}{2}\widehat{\Pi}_{WW}(m_{W}^{2})^{\prime}\Bigg{\\}}.$ (59)

Note that since there is no photon-mediated diagram, we have no infrared (IR)
divergences in ALP contributions. Thus, the treatment of IR regularization in
the numerical analysis is the same as that in the SM Kanemura:2019kjg .

Figure 2: Feynman diagram of ALP contributions to $h\to\gamma\gamma,Z\gamma$
and $h\to aa$. In the right panel, the gauge bosons denote $W/Z/\gamma$.

###  3.4 $h\to V\gamma$ $(V=\gamma,Z)$

In contrast to the previous channels, $h\to V\gamma$ proceeds via radiative
corrections in the SM as well as the ALP model. The renormalized
$h\mathcal{V}\mathcal{V^{\prime}}$ vertex is expressed as

$\displaystyle\widehat{\Gamma}_{h\mathcal{V}\mathcal{V^{\prime}}}^{\mu\nu}(p_{1},p_{2},q)$
$\displaystyle=\quantity[g^{\mu\nu}\widehat{\Gamma}_{h\mathcal{V}\mathcal{V^{\prime}}}^{1}+p_{1}^{\nu}p_{2}^{\mu}\widehat{\Gamma}_{h\mathcal{V}\mathcal{V^{\prime}}}^{2}+i\epsilon^{\mu\nu\rho\sigma}p_{1\rho}p_{2\sigma}\widehat{\Gamma}_{\mathcal{V}\mathcal{V^{\prime}}}^{3}](p_{1}^{2},p_{2}^{2},q^{2}),$
(60)

with $\mathcal{V}=\gamma,Z$ and $\mathcal{V^{\prime}}=\gamma$. Here,
$p_{1}^{\mu}$ and $p_{2}^{\mu}$ are the incoming four-momenta of the gauge
bosons, and $q^{\mu}$ is the outgoing four-momentum of the Higgs boson (see
the left panel of Fig. 2). The form factors are composed only of the one-loop
contributions, which are generally decomposed into the 1PI and counterterm
components as

$\displaystyle\widehat{\Gamma}^{X}_{h\mathcal{V}\mathcal{V^{\prime}}}={\Gamma}^{i,{\rm
loop}}_{h\mathcal{V}\mathcal{V^{\prime}}}=\Gamma^{i,\mathrm{1PI}}_{h\mathcal{V}\mathcal{V^{\prime}}}+\delta\Gamma^{i}_{h\mathcal{V}\mathcal{V^{\prime}}},\quad(i=1,2,3).$
(61)

Since the $Z\text{--}\gamma$ mixing cannot induce $h\to\gamma\gamma$ at the
one-loop level, the counterterms for the $h\gamma\gamma$ vertex are zero.
However, those for $hZ\gamma$ generally exist as

$\displaystyle\delta\Gamma^{1}_{hZ\gamma}=\frac{\Gamma_{hZZ}^{1,{\rm
tree}}}{2}\quantity(\delta Z_{Z\gamma}-\frac{\delta
s_{W}^{2}}{s_{W}c_{W}}),\quad\delta\Gamma^{2}_{hZ\gamma}=\delta\Gamma^{3}_{hZ\gamma}=0.$
(62)

In the present setup, it is found from Eq. (33) that the ALP contribution to
$\delta\Gamma^{1}_{hZ\gamma}$ is canceled out. Therefore, the ALP
contributions are obtained by the 1PI terms as

$\displaystyle\widehat{\Gamma}^{X}_{h\mathcal{V}\mathcal{V^{\prime}}}=\Gamma^{i,\mathrm{1PI}}_{h\mathcal{V}\mathcal{V^{\prime}}},\quad(i=1,2,3).$
(63)

The results are obtained in Appendix A.3.

####  3.4.1 $h\to\gamma\gamma$

The decay rate for $h\to\gamma\gamma$ is given by

$\displaystyle\Gamma(h\to\gamma\gamma)=\frac{1}{16\pi
m_{h}}\absolutevalue{\widehat{\Gamma}_{h\gamma\gamma}^{1}(0,0,m_{h}^{2})}^{2}$
$\displaystyle=\frac{\sqrt{2}G_{F}\alpha_{\mathrm{em}}^{2}m_{h}^{3}}{256\pi^{3}}\absolutevalue{I_{W}(\tau_{W})+\sum_{f}N_{c}^{f}Q_{f}^{2}I_{F}(\tau_{f})(1+\Delta_{\rm
QCD}^{f})+I_{\rm ALP}(\tau_{a})}^{2},$ (64)

with $\tau_{i}=m_{h}^{2}/(4m_{i}^{2})$ for $i=W,f,a$. The SM loop functions
are given by Ellis:1975ap ; Shifman:1979eb ; Djouadi:2005gi

$\displaystyle I_{W}(\tau)$
$\displaystyle=\quantity[2\tau^{2}+3\tau+3(2\tau-1)f(\tau)]\tau^{-2},$ (65)
$\displaystyle I_{F}(\tau)$
$\displaystyle=-2\quantity[\tau+(\tau-1)f(\tau)]\tau^{-2}.$ (66)

with $f(\tau)$ defined as

$\displaystyle f(\tau)=\begin{dcases}\arcsin^{2}(\sqrt{\tau})&(\tau\leq 1),\\\
-\frac{1}{4}\quantity[\ln{\frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}}}-i\pi]^{2}\quad&(\tau>1).\end{dcases}$
(67)

The ALP loop function is given by

$\displaystyle I_{\rm ALP}(\tau)$ $\displaystyle=\frac{4\pi v}{\alpha_{\rm
em}m_{h}^{2}}\Gamma_{h\gamma\gamma}^{1,{\rm 1PI}}(0,0,m_{h}^{2})_{\rm ALP}.$
(68)

The term $\Delta_{\rm QCD}^{f}$ denotes QCD corrections to the quark-loop
contributions. We include it up to next-to-next-to-leading order, and whose
expressions in the large top-mass limit ($\tau_{t}\ll 1)$ are given in Refs.
Dawson:1993qf ; Spira:1995rr .

####  3.4.2 $h\to Z\gamma$

The decay rate for $h\to Z\gamma$ is given by

$\displaystyle\Gamma_{\mathrm{LO}}(h\to Z\gamma)=\frac{1}{8\pi
m_{h}}\quantity(1-\frac{m_{Z}^{2}}{m_{h}^{2}})\absolutevalue{\widehat{\Gamma}_{hZ\gamma}^{1}(m_{Z}^{2},0,m_{h}^{2})}^{2}$
$\displaystyle=\frac{\sqrt{2}G_{F}\alpha_{\mathrm{em}}^{2}m_{h}^{3}}{128\pi^{3}}\quantity(1-\frac{m_{Z}^{2}}{m_{h}^{2}})^{3}$
$\displaystyle\quad\times\absolutevalue{J_{W}(\tau_{W},\lambda_{W})+\sum_{f}N_{c}^{f}Q_{f}v_{f}J_{F}(\tau_{f},\lambda_{f})(1+\Delta_{\rm
QCD}^{f})+J_{\rm ALP}(\tau_{a},\lambda_{a})}^{2},$ (69)

with $\tau_{i}=4m_{i}^{2}/m_{h}^{2}$ and $\lambda_{i}=4m_{i}^{2}/m_{Z}^{2}$.
The SM loop functions are given by Djouadi:2005gi ; Cahn:1978nz ;
Bergstrom:1985hp

$\displaystyle J_{W}(\tau,\lambda)$
$\displaystyle=\frac{c_{W}}{s_{W}}\bigg{\\{}\quantity[\quantity(1+\frac{2}{\tau})\frac{s_{W}^{2}}{c_{W}^{2}}-\quantity(5+\frac{2}{\tau})]I_{1}(\tau,\lambda)$
$\displaystyle\quad+4\quantity(3-\frac{s_{W}^{2}}{c_{W}^{2}})I_{2}(\tau,\lambda)\bigg{\\}},$
(70) $\displaystyle J_{F}(\tau,\lambda)$
$\displaystyle=\frac{4}{s_{W}c_{W}}\quantity[I_{1}(\tau,\lambda)-I_{2}(\tau,\lambda)],$
(71)

with $I_{1}(\tau,\lambda)$ and $I_{2}(\tau,\lambda)$ defined as

$\displaystyle I_{1}(\tau,\lambda)$
$\displaystyle=\frac{\tau\lambda}{2(\tau-\lambda)}+\frac{\tau^{2}\lambda^{2}}{2(\tau-\lambda)^{2}}\quantity[f(\tau^{-1})-f(\lambda^{-1})]$
$\displaystyle\quad+\frac{\tau^{2}\lambda}{(\tau-\lambda)^{2}}\quantity[g(\tau^{-1})-g(\lambda^{-1})],$
(72) $\displaystyle I_{2}(\tau,\lambda)$
$\displaystyle=-\frac{\tau\lambda}{2(\tau-\lambda)}\quantity[f(\tau^{-1})-f(\lambda^{-1})].$
(73)

The function $f(\tau)$ is found in Eq. (67), while $g(\tau)$ is defined as

$\displaystyle
g(\tau)=\begin{dcases}\sqrt{\tau^{-1}-1}\arcsin(\sqrt{\tau})&(\tau\geq 1),\\\
\frac{\sqrt{1-\tau^{-1}}}{2}\quantity[\ln{\frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}}}-i\pi]\quad&(\tau<1).\end{dcases}$
(74)

The ALP loop function is given by

$\displaystyle J_{\rm ALP}(\tau,\lambda)$ $\displaystyle=\frac{4\pi
v}{\alpha_{\rm
em}m_{h}^{2}}\quantity(1-\frac{m_{Z}^{2}}{m_{h}^{2}})^{-1}\Gamma_{hZ\gamma}^{1,{\rm
1PI}}(m_{Z}^{2},0,m_{h}^{2})_{\rm ALP}.$ (75)

The term $\Delta_{\rm QCD}^{f}$ denotes QCD corrections to the quark-loop
contributions. In our analysis, it is included up to the next-to-leading
order, and its expression is the same as that for the di-photon decay
Spira:1991tj .

###  3.5 $h\to aa$

The last channel in interest is $h\to aa$, i.e., a direct production of ALPs
from the Higgs boson decay. It proceeds via radiative corrections because the
ALP is not coupled directly with the Higgs boson. Therefore, the ALP
contributions are obtained by the 1PI diagrams as

$\displaystyle\widehat{\Gamma}_{haa}(p_{1}^{2},p_{2}^{2},q^{2})=\Gamma_{haa}^{\rm
1PI}(p_{1}^{2},p_{2}^{2},q^{2}).$ (76)

Here, $p_{1}^{\mu}$ and $p_{2}^{\mu}$ are the incoming four-momenta of ALPs,
and $q^{\mu}$ is the outgoing four-momentum of the Higgs boson (see the right
panel of Fig. 2). The results are given in Appendix A.3. As a result, the
decay rate for $h\to aa$ is given by

$\displaystyle\Gamma(h\to aa)=\frac{1}{32\pi
m_{h}}\sqrt{1-\frac{4m_{a}^{2}}{m_{h}^{2}}}\absolutevalue{\Gamma_{haa}^{\rm
1PI}(m_{a}^{2},m_{a}^{2},m_{h}^{2})}^{2}.$ (77)

##  4 Results

In this section, we show numerical results of the ALP contributions to the
Higgs boson decays. For the channels whose final-states are composed only of
the SM particles, the scaling factor of Higgs boson couplings is defined as

$\displaystyle\kappa_{XY}=\sqrt{\frac{\Gamma(h\to XY)}{\Gamma(h\to XY)_{\rm
SM}}},$ (78)

where the denominator is SM predictions, and the numerator includes both the
SM and ALP contributions. In our analysis, the former is evaluated at the
next-leading order by using the H-COUP program Kanemura:2017gbi ;
Kanemura:2019slf , and the ALP contributions are evaluated at the one-loop
level.

Among the decay channels, the ALP corrections to $h\to f\bar{f}$ vanish at the
one-loop level, as shown in Sec. 3.2. Hence, we focus on those into
$\gamma,Z$, and $W$ as well as $h\to aa$ in the following. As mentioned in
Sec. 2, the ALP interactions are determined by two coupling constants. Thus,
let us regard $g_{a\gamma\gamma}$ and $g_{aWW}$ as free parameters, while
$g_{aZ\gamma}$ and $g_{aZZ}$ are represented by them. Also, the decay constant
and cutoff scale are set as $f_{a}=1\,\mathrm{TeV}$ and $\Lambda=4\pi f_{a}$,
respectively.

Figure 3: Scaling factors,
$\kappa_{W},\kappa_{Z},\kappa_{\gamma},\kappa_{Z\gamma}$, as a function of
$g_{a\gamma\gamma}$ with fixing $g_{aW}=0$. The ALP mass is chosen to be
$m_{a}=5,100,500,800\,\mathrm{GeV}$ (solid blue, yellow, green, and red lines,
respectively). The horizontal dotted/dashed lines are experimental bounds and
future sensitivities.

In Fig. 3, we show $\kappa_{W},\kappa_{Z},\kappa_{\gamma},\kappa_{Z\gamma}$ as
a function of $g_{a\gamma\gamma}$ with setting $g_{aWW}=0$, where $\kappa_{X}$
denotes $\kappa_{XX}$. The ALP mass is chosen as
$m_{a}=5,100,500,800\,\mathrm{GeV}$. It is found that the ALP contributions
are less sensitive to the value of $m_{a}$. Moreover, there is no ALP
contribution to $h\to WW^{*}$ because of $g_{aWW}=0$. On the other hand, the
other channels receive corrections; they become effective for
$g_{a\gamma\gamma}\gtrsim 1\,\mathrm{TeV}^{-1}$ in $h\to\gamma\gamma,Z\gamma$,
and $10\,\mathrm{TeV}^{-1}$ in $h\to ZZ^{*}$. It is noted that the ALP
contribution interferes destructively with the SM one for
$h\to\gamma\gamma,ZZ^{*}$, and its magnitude for $h\to\gamma\gamma$ becomes
twice as large as or even larger than the SM contribution for
$g_{a\gamma\gamma}\sim 10\,\mathrm{TeV}^{-1}$.

In the plots, experimental bounds and future prospects are shown by horizontal
lines. The dashed lines denote the bounds obtained by analyzing the full
dataset of the ATLAS Run II ATLAS:2022vkf . The CMS experiment provides almost
the same results CMS:2022dwd . As for the prospects, the HL-LHC sensitivities
Cepeda:2019klc  are shown by the dot-dashed lines. We also show the dotted
lines that represent far-future sensitivities expected by combining data from
the FCC-ee/eh/hh experiments deBlas:2019rxi .

By comparing the ALP contributions with the experimental bound/sensitivities,
it is found that $h\to\gamma\gamma$ provides the best probe to the ALP among
the decay channels. The model is excluded for $g_{a\gamma\gamma}\gtrsim
2.8-3.9\,\mathrm{TeV}^{-1}$, depending on the ALP mass, except for the narrow
region at $g_{a\gamma\gamma}\sim 10\,\mathrm{TeV}^{-1}$. The HL-LHC experiment
can probe the ALP with $g_{a\gamma\gamma}\gtrsim 1.6-2.2\,\mathrm{TeV}^{-1}$,
and the sensitivity may reach $g_{a\gamma\gamma}=0.7-0.9\,\mathrm{TeV}^{-1}$
at FCC-ee/eh/hh.

Figure 4: Same as Fig 3 but $g_{aWW}$ is varied with $g_{a\gamma\gamma}=0$.

In Fig. 4, we study the case with $g_{aWW}\neq 0$ instead of
$g_{a\gamma\gamma}$. Here,
$\kappa_{W},\kappa_{Z},\kappa_{\gamma},\kappa_{Z\gamma}$ are shown as a
function of $g_{aWW}$ with setting $g_{a\gamma\gamma}=0$. It is found that the
ALP contributions are less sensitive to $m_{a}$ and become effective for
$g_{aWW}\gtrsim 1\,\mathrm{TeV}^{-1}$ in $h\to\gamma\gamma,Z\gamma$ and
$10\,\mathrm{TeV}^{-1}$ in $h\to ZZ^{*},WW^{*}$. The ALP contributions
decrease the decay width of $h\to WW^{*}$ and increase that of $h\to ZZ^{*}$.
The difference between these two behaviors is understood by noticing that the
$hZ\gamma$ vertex and the $Z\text{--}\gamma$ mixing give positive corrections
in $h\to ZZ^{*}$. On the other hand, the ALP contributions decrease the decay
widths of $h\to\gamma\gamma$ and $h\to Z\gamma$ for $g_{aWW}\lesssim 10~{}{\rm
TeV}^{-1}$. However, the magnitude can become comparable to or larger than the
SM corrections when $g_{aWW}\gtrsim 10~{}{\rm TeV}^{-1}$ and increase the
decay widths compared with the SM predictions.

By comparing the ALP contributions with the experimental bound/sensitivities,
it is found that $h\to\gamma\gamma$ provides the best probe to the ALP,
similarly to the case with $g_{a\gamma\gamma}\neq 0$. Since
$g_{aZ\gamma}\propto(g_{aWW}-g_{a\gamma\gamma})$ is satisfied (see Eq.(10)),
the values of $\kappa_{\gamma}$ in Fig. 4 become exactly the same as those in
Fig. 3. Therefore, we obtain the same conclusion as the case with
$g_{a\gamma\gamma}\neq 0$.

Figure 5: Experimental bound (red) and future sensitivities (green and blue)
of ALP contributions to $h\to\gamma\gamma$ (solid lines) on the
$(g_{a\gamma\gamma},g_{aWW})$ plane. The gray regions surrounded by dashed
lines are excluded by the collider constraints.

Let us next compare the Higgs results with collider constraints. In Fig. 5, we
show the collider limits on the $(g_{a\gamma\gamma},g_{aWW})$ plane. Here, we
consider $m_{a}=5,195$, and $600\,\mathrm{GeV}$, where the constraints on
$g_{a\gamma\gamma}$ is relatively weak (see the discussion in Ref.
Aiko:2023trb ). The regions excluded by the collider limits are filled with
gray color surrounded by dashed lines. For $m_{a}\lesssim 100\,\mathrm{GeV}$,
the LEP experiments give strong bounds, while heavier ALPs are constrained by
the LHC results. The details of the collider constraints are given in Ref.
Aiko:2023trb . It is commented that $h\to aa$ may have a sensitivity to the
parameter space for $m_{a}=5\,\mathrm{GeV}$. However, we found that the effect
is so weak that no lines appear in the parameter range of Fig. 5.

In the plots, the experimental bound and future sensitivities to the ALP
contributions to $h\to\gamma\gamma$ are shown by solid red (ATLAS Run II
bound), green (HL-LHC), and blue (FCC-ee/eh/hh) lines. For
$m_{a}=5\,\mathrm{GeV}$, it is found that the $h\to\gamma\gamma$ sensitivities
are almost weaker than the LEP constraints even at FCC-ee/eh/hh. By contrast,
for $m_{a}=195\,\mathrm{GeV}$ the future sensitivity at HL-LHC can compete
with the LHC constraint from the measurement of $pp\to\nu\nu\gamma\gamma$, and
the model could be probed better by FCC-ee/eh/hh. For
$m_{a}=600\,\mathrm{GeV}$, it is noticed that the collider constraints are
relaxed especially for $g_{a\gamma\gamma}\sim 0$, and thus, $h\to\gamma\gamma$
provides a significant probe of the model.

Figure 6: Experimental bound (red) and future sensitivities (green and blue)
of ALP contributions to $h\to\gamma\gamma$ (solid lines) and $h\to aa$ (dot-
dashed lines) as a function of the ALP mass. The gray regions surrounded by
dashed lines are excluded by the collider constraints and/or flavor limits.

Since the model is much less constrained for $g_{a\gamma\gamma}\sim 0$, let us
discuss the ALP mass dependence of the results for $g_{a\gamma\gamma}=0$ and
compare them with the collider constraints. We also take flavor constraints
into account for $m_{a}\lesssim 4.8\,\mathrm{GeV}$ (see Ref. Aiko:2023trb  for
details). In Fig. 6, we show the experimental bound and future sensitivities
on $g_{aWW}$ by the measurements of $\kappa_{\gamma}$ (solid lines) as a
function of $m_{a}$. It is found that the current bound on $\kappa_{\gamma}$
(red line) provides a stronger constraint than the colliders; the region of
$g_{aWW}\gtrsim 3.2-4.1\,\mathrm{TeV}^{-1}$ is excluded for
$m_{a}>300\,\mathrm{GeV}$. It is noted that there is a narrow allowed region
above $g_{aWW}=10\,\mathrm{TeV}^{-1}$ between the solid lines, where the ALP
contribution is almost twice as large as the SM one with an opposite sign. As
for future prospects, $g_{aWW}>1.7-2.3\,\mathrm{TeV}^{-1}$ can be probed for
$m_{a}>180\,\mathrm{GeV}$ by HL-LHC (green line). If FCC-ee/eh/hh would be
constructed, the sensitivity could reach $g_{aWW}=0.7-1\,\mathrm{TeV}^{-1}$
(blue line) and might compete with or even be better than the collider
constraints at LEP and LHC.

In the figure, we also show the result of $h\to aa$, which is open
kinematically for $m_{a}<m_{h}/2$. The current bound and future sensitivities
are shown by the dot-dashed red, green, and blue lines, respectively. It is
found that the decay is less sensitive to the ALP model compared with
$h\to\gamma\gamma$.

Before closing this section, let us comment on the case for $g_{aWW}=0$. In
Fig. 6, $g_{aWW}$ was varied with fixing $g_{a\gamma\gamma}=0$. If
$g_{a\gamma\gamma}$ is varied with $g_{aWW}=0$, we obtain the same lines for
$h\to\gamma\gamma$ as those in Fig. 6 (see Eq.(10)). However, the collider
constraints are much more severe, and almost the whole parameter regions
accessible by future $h\to\gamma\gamma$ measurements are already excluded.
Similarly, if we consider the case for $g_{aZ\gamma}=0$, i.e.,
$g_{a\gamma\gamma}=g_{aWW}$ is fixed, the ALP contributions to
$h\to\gamma\gamma$ and $h\to Z\gamma$ are suppressed. Then, the ALP is probed
by $h\to ZZ^{*}$ and $h\to WW^{*}$. Since the ALP contributions become
effective only for $g_{a\gamma\gamma}\sim 10\,\mathrm{TeV}^{-1}$, such a
parameter region is excluded by the collider constraints.

##  5 Conclusion

In this paper, we have studied the Higgs boson decays in the ALP model. The
ALP is assumed to couple with the SM ${\rm SU(2)}_{L}$ and ${\rm U(1)}_{Y}$
gauge bosons. We have provided the formulae for the ALP contributions to $h\to
f\bar{f}$, $h\to\gamma\gamma$, $h\to Z\gamma$, $h\to ZZ$, and $h\to
W^{+}W^{-}$ as well as $h\to aa$ at the one-loop level. It was found that
those to $h\to f\bar{f}$ vanish at this level, while the other decay widths
receive ALP contributions via radiative corrections.

Among the decay channels, it was concluded that $h\to\gamma\gamma$ can provide
a significant probe of the model. Compared with the collider constraints, the
decay becomes relevant especially when the ALP is heavy and
$g_{a\gamma\gamma}$ is suppressed. For $g_{a\gamma\gamma}=0$, it was shown
that the current LHC result of $h\to\gamma\gamma$ excludes the ALP with
$g_{aWW}\gtrsim 3.2-4.1\,\mathrm{TeV}^{-1}$ and $m_{a}>300\,\mathrm{GeV}$. In
the future, the HL-LHC experiment can probe
$g_{aWW}>1.7-2.3\,\mathrm{TeV}^{-1}$ for $m_{a}>180\,\mathrm{GeV}$. Also, the
sensitivity could reach $g_{aWW}=0.7-1\,\mathrm{TeV}^{-1}$ at the FCC-ee/eh/hh
experiments; it might compete with or even be better than the collider
constraints on the direct ALP productions at LEP and LHC. Moreover,
$h\to\gamma\gamma$ has a better sensitivity than $h\to aa$. Therefore, the
precision measurements of $h\to\gamma\gamma$ could provide a powerful tool in
searching for the ALP model.

## 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. 22KJ3126
[MA]) and Scientific Research B (No. 21H01086 [ME]).

##  Appendix A ALP contributions

In this section, we give the analytic expressions for the 1PI diagram
contributions of the ALP to two-point and three-point functions. The
calculations were performed in the $R_{\xi}$ gauge by using FeynCalc
Mertig:1990an ; Shtabovenko:2016sxi ; Shtabovenko:2020gxv  and FeynArts
Hahn:2000kx  with a model file generated with FeynRules Alloul:2013bka . We
have checked that all the results are shown to be gauge invariant.

###  A.1 Loop functions

The loop integrals with a tensor structure can be decomposed into linear
combinations of the metric tensor and a set of the momenta with scalar
functions Passarino:1978jh . The loop integrals are defined as

$\displaystyle A_{0}$
$\displaystyle=\int\frac{\overline{\differential^{d}k}}{i\pi^{2}}\frac{1}{N_{0}},$
(79) $\displaystyle B_{0}$
$\displaystyle=\int\frac{\overline{\differential^{d}k}}{i\pi^{2}}\frac{1}{N_{0}N_{1}},$
(80) $\displaystyle[C_{0},C^{\mu},C^{\mu\nu}]$
$\displaystyle=\int\frac{\overline{\differential^{d}k}}{i\pi^{2}}\frac{[1,k^{\mu},k^{\mu}k^{\nu}]}{N_{0}N_{1}N_{2}},$
(81)

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 . The propagator
factor is defined as $N_{i}=(k+r_{i})^{2}-m_{i}^{2}+i\epsilon$ where $r_{0}=0$
and $r_{N}=\sum_{i=1}^{N}p_{i}$ with incoming four-momenta $p_{i}$. We take
$\mu=\Lambda$ in this paper. The $C^{\mu}$ and $C^{\mu\nu}$ functions are
decomposed as Denner:1991kt

$\displaystyle C^{\mu}$
$\displaystyle=\sum_{i=1}^{2}r_{i}^{\mu}C_{i}(p_{1}^{2},p_{2}^{2},q^{2};m_{0}^{2},m_{1}^{2},m_{2}^{2}),$
(82) $\displaystyle C^{\mu\nu}$
$\displaystyle=\quantity[g^{\mu\nu}C_{00}+\sum_{i,j=1}^{2}r_{i}^{\mu}r_{j}^{\nu}C_{ij}](p_{1}^{2},p_{2}^{2},q^{2};m_{0}^{2},m_{1}^{2},m_{2}^{2}).$
(83)

In this paper, we use LoopTools Hahn:1998yk  for the evaluation of scalar
functions. Also, we use the following abbreviations for scalar functions.

$\displaystyle A_{0}(X)$ $\displaystyle=A_{0}(m_{X}^{2}),$ (84) $\displaystyle
B_{0}(p^{2};X,Y)$ $\displaystyle=B_{0}(p^{2};m_{X}^{2},m_{Y}^{2}),$ (85)
$\displaystyle C_{i,\,ij}(X,Y,Z)$
$\displaystyle=C_{i,\,ij}(p_{1}^{2},p_{2}^{2},q^{2};m_{X}^{2},m_{Y}^{2},m_{Z}^{2}).$
(86)

###  A.2 Two-point functions

The ALP contributions to two-point functions of the EW gauge boson are given
as Aiko:2023trb

$\displaystyle 16\pi^{2}\Pi_{WW}^{{\rm 1PI},T}(p^{2})_{\rm ALP}$
$\displaystyle=\frac{1}{18}g_{aWW}^{2}F(p^{2};a,W),$ (87) $\displaystyle
16\pi^{2}\Pi_{\gamma\gamma}^{{\rm 1PI},T}(p^{2})_{\rm ALP}$
$\displaystyle=\frac{1}{18}\quantity[g_{a\gamma\gamma}^{2}F(p^{2};a,\gamma)+g_{aZ\gamma}^{2}F(p^{2};a,Z)],$
(88) $\displaystyle 16\pi^{2}\Pi_{Z\gamma}^{{\rm 1PI},T}(p^{2})_{\rm ALP}$
$\displaystyle=\frac{1}{18}\quantity[g_{a\gamma\gamma}g_{aZ\gamma}F(p^{2};a,\gamma)+g_{aZ\gamma}g_{aZZ}F(p^{2};a,Z)],$
(89) $\displaystyle 16\pi^{2}\Pi_{ZZ}^{{\rm 1PI},T}(p^{2})_{\rm ALP}$
$\displaystyle=\frac{1}{18}\quantity[g_{aZ\gamma}^{2}F(p^{2};a,\gamma)+g_{aZZ}^{2}F(p^{2};a,Z)],$
(90)

where the loop function is defined as

$\displaystyle F(p^{2};a,V)$
$\displaystyle=3\quantity[p^{2}-\quantity(m_{a}+m_{V})^{2}]\quantity[p^{2}-\quantity(m_{a}-m_{V})^{2}]\quantity[B_{0}(p^{2};a,V)-B_{0}(0;a,V)]$
$\displaystyle\quad-3p^{2}\quantity[A_{0}(a)+A_{0}(V)+\quantity(2m_{a}^{2}+2m_{V}^{2}-p^{2})B_{0}(0;a,V)]$
$\displaystyle\quad+7p^{2}(3m_{a}^{2}+3m_{V}^{2}-p^{2}).$ (91)

The derivative of the loop function is explicitly shown as

$\displaystyle F^{\prime}(p^{2};a,V)$
$\displaystyle=6(p^{2}-m_{a}^{2}-m_{V}^{2})\quantity[B_{0}(p^{2};a,V)-B_{0}(0;a,V)]$
$\displaystyle\quad+3\quantity[p^{2}-\quantity(m_{a}+m_{V})^{2}]\quantity[p^{2}-\quantity(m_{a}-m_{V})^{2}]B^{\prime}_{0}(p^{2};a,V)$
$\displaystyle\quad-3\quantity[A_{0}(a)+A_{0}(V)+2\quantity(m_{a}^{2}+m_{V}^{2}-p^{2})B_{0}(0;a,V)]$
$\displaystyle\quad+7(3m_{a}^{2}+3m_{V}^{2}-2p^{2}).$ (92)

###  A.3 Three-point functions

The ALP contributions to the form factors of the $hWW$ vertices are given by

$\displaystyle 16\pi^{2}\Gamma^{1,{\rm
1PI}}_{hWW}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$
$\displaystyle=\frac{gm_{W}g_{aWW}^{2}}{8}C_{hVV1}^{VSV}(W,a,W)_{\rm ALP},$
(93) $\displaystyle 16\pi^{2}\Gamma^{2,{\rm
1PI}}_{hWW}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$
$\displaystyle=\frac{gm_{W}^{3}g_{aWW}^{2}}{2}C_{hVV2}^{VSV}(W,a,W)_{\rm
ALP},$ (94) $\displaystyle\Gamma^{3,{\rm
1PI}}_{hWW}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$ $\displaystyle=0.$ (95)

where the loop functions are obtained as

$\displaystyle C_{hVV1}^{VSV}(X,a,X)_{\rm ALP}$
$\displaystyle=(p_{1}^{2}+p_{2}^{2}-3q^{2}-2m_{a}^{2}+2m_{X}^{2})B_{0}(q^{2};X,X)$
$\displaystyle\quad+2(p_{1}^{2}+m_{a}^{2}-m_{X}^{2})B_{0}(p_{1}^{2};a,X)+2(p_{2}^{2}+m_{a}^{2}-m_{X}^{2})B_{0}(p_{2}^{2};a,X)$
$\displaystyle\quad-2\quantity[p_{1}^{2}p_{2}^{2}-p_{1}^{2}(m_{a}^{2}+m_{X}^{2})-p_{2}^{2}(m_{a}^{2}+m_{X}^{2})+2m_{a}^{2}q^{2}+\quantity(m_{a}^{2}-m_{X}^{2})^{2}]C_{0}(X,a,X)$
$\displaystyle\quad+4(q^{2}-p_{1}^{2}-p_{2}^{2})C_{00}(X,a,X)+4(q^{2}-p_{1}^{2}-p_{2}^{2})-2A_{0}(a)+2A_{0}(X),$
(96) $\displaystyle C_{hVV2}^{VSV}(X,a,X)_{\rm ALP}$
$\displaystyle=B_{0}(q^{2};X,X)+2m_{a}^{2}C_{0}(X,a,X)$
$\displaystyle\quad+(p_{1}^{2}+m_{a}^{2}-m_{X}^{2})C_{1}(a,X,X)+(p_{2}^{2}+m_{a}^{2}-m_{X}^{2})C_{1}(a,X,X)$
$\displaystyle\quad+\quantity(p_{1}^{2}+p_{2}^{2}-q^{2})C_{12}(a,X,X)-2.$ (97)

The ALP contributions to the form factors of the $hZZ$ vertices are given by

$\displaystyle 16\pi^{2}\Gamma^{1,{\rm
1PI}}_{hZZ}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$
$\displaystyle=\frac{g_{Z}m_{Z}g_{aZZ}^{2}}{8}C_{hVV1}^{VSV}(Z,a,Z)_{\rm
ALP},$ (98) $\displaystyle 16\pi^{2}\Gamma^{2,{\rm
1PI}}_{hZZ}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$
$\displaystyle=\frac{g_{Z}m_{Z}^{3}g_{aZZ}^{2}}{2}C_{hVV2}^{VSV}(Z,a,Z)_{\rm
ALP},$ (99) $\displaystyle\Gamma^{3,{\rm
1PI}}_{hZZ}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$ $\displaystyle=0.$ (100)

The ALP contributions to the form factors of the $hZ\gamma$ vertices are given
by

$\displaystyle 16\pi^{2}\Gamma^{1,{\rm
1PI}}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$
$\displaystyle=\frac{g_{Z}m_{Z}g_{aZ\gamma}g_{aZZ}}{8}C_{hVV1}^{VSV}(Z,a,Z)_{\rm
ALP},$ (101) $\displaystyle 16\pi^{2}\Gamma^{2,{\rm
1PI}}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$
$\displaystyle=\frac{g_{Z}m_{Z}g_{aZ\gamma}g_{aZZ}}{2}C_{hVV2}^{VSV}(Z,a,Z)_{\rm
ALP},$ (102) $\displaystyle\Gamma^{3,{\rm
1PI}}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$ $\displaystyle=0.$
(103)

The ALP contributions to the form factors of the $h\gamma\gamma$ vertices are
given by

$\displaystyle 16\pi^{2}\Gamma^{1,{\rm
1PI}}_{h\gamma\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$
$\displaystyle=\frac{g_{Z}m_{Z}g_{aZ\gamma}^{2}}{8}C_{hVV1}^{VSV}(Z,a,Z)_{\rm
ALP},$ (104) $\displaystyle 16\pi^{2}\Gamma^{2,{\rm
1PI}}_{h\gamma\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$
$\displaystyle=\frac{g_{Z}m_{Z}g_{aZ\gamma}^{2}}{2}C_{hVV2}^{VSV}(Z,a,Z)_{\rm
ALP},$ (105) $\displaystyle\Gamma^{3,{\rm
1PI}}_{h\gamma\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm ALP}$ $\displaystyle=0.$
(106)

The gauge boson contributions to the $haa$ vertex are given by

$\displaystyle 16\pi^{2}\Gamma^{\rm 1PI}_{haa}(p_{1}^{2},p_{2}^{2},q^{2})_{\rm
ALP}$
$\displaystyle=\frac{gm_{W}g_{aWW}^{2}}{2}C_{haa}^{VVV}(W,W)+\frac{g_{Z}m_{Z}g_{aZZ}^{2}}{4}C_{haa}^{VVV}(Z,Z)$
$\displaystyle\quad+\frac{g_{Z}m_{Z}g_{aZ\gamma}^{2}}{4}C_{haa}^{VVV}(Z,\gamma),$
(107)

where the loop function is obtained as

$\displaystyle C_{haa}^{VVV}(V_{1},V_{2})$
$\displaystyle=2\quantity[p_{1}^{2}p_{2}^{2}+2m_{V_{2}}^{2}q^{2}-(m_{V_{1}}^{2}+m_{V_{2}}^{2})(p_{1}^{2}+p_{2}^{2})+(m_{V_{1}}^{2}-m_{V_{2}}^{2})^{2}]C_{0}(V_{1},V_{2},V_{1})$
$\displaystyle\quad+(3q^{2}-p_{1}^{2}-p_{2}^{2}-2m_{V_{1}}^{2}+2m_{V_{2}}^{2})B_{0}(q^{2};V_{1},V_{1})$
$\displaystyle\quad-2(p_{1}^{2}-m_{V_{1}}^{2}+m_{V_{2}}^{2})B_{0}(p_{1}^{2};V_{1},V_{2})-2(p_{2}^{2}-m_{V_{1}}^{2}+m_{V_{2}}^{2})B_{0}(p_{2}^{2};V_{1},V_{2})$
$\displaystyle\quad-2A_{0}(V_{1})+2A_{0}(V_{2})-9(q^{2}-p_{1}^{2}-p_{2}^{2}).$
(108)

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