# Analysis of the $Z_{b}(10650)$ state based on electromagnetic properties

Ulaş Özdem  0000-0002-1907-2894<EMAIL_ADDRESS>Health Services
Vocational School of Higher Education, Istanbul Aydin University, Sefakoy-
Kucukcekmece, 34295 Istanbul, Türkiye

###### Abstract

In this study, the magnetic and quadrupole moments of the $Z_{b}(10650)$ state
are determined using the compact diquark-antidiquark interpolating current
through the QCD light-cone sum rule. The values that are obtained as a result
of the analysis are as follows: $\mu_{Z_{b}}=2.35^{+0.34}_{-0.33}~{}\mu_{N}$
and $\mathcal{D}_{Z_{b}}=(1.82^{+0.35}_{-0.31})\times
10^{-2}~{}\mbox{fm}^{2}$. Examining the results obtained, it can be seen that
the magnetic moments are large enough to be measured experimentally, while the
quadrupole moment is obtained as a small but non-zero value, corresponding to
a prolate charge distribution. The magnetic moment is the leading-order
response of a bound system to a weak external magnetic field. It therefore
provides an excellent platform to probe the internal structures of hadrons
governed by the quark-gluon dynamics of QCD.

Hidden-bottom tetraquarks, diquark-antidiquark picture, electromagnetic
multipole moments, QCD light-cone sum rules

##  I Motivation

Over the past decade, a series of heavy quarkonium-like states, known as the
XYZ states, have been discovered through ongoing experimental efforts. Charged
states such as $Z_{c}(3900)$, $Z_{c}(4430)$ and $Z_{c}(4020)$ provide strong
evidence for the existence of exotic hadrons since the presence of light
quarks explains their non-zero electric charge. These electrically charged
particles have hidden-heavy flavor (hidden-charm or hidden-bottom), excluding
the pure $Q\bar{Q}$, opening a new era in hadron physics. In 2011, two charged
bottomonium-like states, $Z_{b}(10610)$ and $Z_{b}(10650)$, have been reported
by the Belle Collaboration in the processes
$\Upsilon(5S)\rightarrow\pi\pi\Upsilon(nS)$, and
$\Upsilon(5S)\rightarrow\pi\pi h_{b}(kP)$ [1] with n = 1, 2, 3 and k = 1, 2.
The masses and widths of these states were measured as

$\displaystyle M_{Z_{b}(10610)}$ $\displaystyle=10607.2\pm 2~{}\mbox{MeV},$
$\displaystyle\Gamma_{Z_{b}(10610)}$ $\displaystyle=18.4\pm 2.4~{}\mbox{MeV},$
(1) $\displaystyle M_{Z_{b}(10650)}$ $\displaystyle=10652.2\pm
1.5~{}\mbox{MeV},$ $\displaystyle\Gamma_{Z_{b}(10650)}$ $\displaystyle=11.5\pm
2.2~{}\mbox{MeV}.$ (2)

The neutral partner of the $Z_{b}^{0}(10610)$ was also observed in the
$\Upsilon(5S)\to\Upsilon(nS)\pi\pi$ decay by the Belle Collaboration [2]. The
quantum numbers $I^{G}(J^{P})=1^{+}(1^{+})$ are favored by the analysis of the
angular distribution. Both $Z_{b}(10610)$ and $Z_{b}(10650)$ states are
members of a family of charged hidden-bottom states. These states have
attracted the attention of many theoretical groups since they were the first
charged bottomonium-like states observed and because they are very close to
the thresholds of $B\bar{B}^{*}$ and $B^{*}\bar{B}^{*}$. Various models and
approaches have been used to study the spectroscopic parameters and decays of
both states, including compact tetraquark states, molecular states, threshold
cusps, re-scattering effects, etc. It is a challenge to understand these new
charmonium/bottomonium-like states as exotic since it is relatively easy to
reproduce the properties of these states using the models mentioned above.
Many comprehensive reviews of this topic can be found in the literature [3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. All possible configurations
considered in the various studies yielded mass and decay width determinations
in agreement with experimental observations, which indicates that other
properties of this state need to be further investigated to shed light on its
underlying structure and reach a definitive conclusion. Therefore, to
elucidate the internal organization of these states, it is also important to
study the decay channels, such as weak, strong, and electromagnetic, along
with the spectroscopic parameters of these states.

The magnetic and quadrupole moments are other intrinsic parameters of hadrons
that may contain important information about their quark-gluon organization
and underlying dynamics. Such a study would therefore deepen our knowledge of
tetraquarks and help us understand the underlying dynamics that govern how
they form, as well as the geometric shapes of these tetraquarks. The magnetic
and quadrupole moments of the hidden-heavy tetraquarks were extracted in
several studies [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In our study, the
magnetic and quadrupole moments of the $Z_{b}(10650)$ state (for brevity,
hereinafter often referred to as $Z_{b}$) are determined using the compact
diquark-antidiquark interpolating current through the QCD light-cone sum rule.
In the QCD light-cone sum rule method [28, 29, 30], we describe the
correlation function in two different representations: One is based on the
hadronic degrees of freedom and is called the hadronic representation, and the
other is based on the quark-gluon degrees of freedom and is called the QCD
representation. Double Borel transformations on $p^{2}$ and $(p+q)^{2}$ are
then applied to both representations to suppress the contributions of the
higher states and the continuum. The quark-hadron duality ansatz is also
carried out to further suppress the contributions of the higher states and the
continuum, and to enhance the ground state contribution. The magnetic and
quadrupole moments are obtained by matching the coefficients of the same
Lorentz structures of both representations of the correlation function.

This article is organized as follows. After the introduction, we introduce our
theoretical framework explicitly in section II. The numerical results and
conclusions for the magnetic and quadrupole moments of the $Z_{b}$ state are
presented in section III. Finally, the manuscript ends with a summary in
section IV. Explicit expressions for the magnetic moment of the $Z_{b}$ state
and the photon distribution amplitudes are listed in the appendices A and B,
respectively.

##  II Electromagnetic multipole moments of the $Z_{b}(10650)$ state from QCD
light-cone sum rules

In the QCD light-cone sum rule, one initiates the calculations for the
magnetic and quadrupole moments using the following correlation function:

$\Pi_{\mu\nu}(p,q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\mu}(x)J_{\nu}^{\dagger}(0)\\}|0\rangle_{\gamma},$ (3)

where $q$ is the momentum of the photon, $\gamma$ denotes the external
electromagnetic background field. The four-quark current operator
$J_{\mu(\nu)}(x)$ with spin-parity quantum numbers $J^{P}=1^{+}$ is written as

$\displaystyle J_{\mu}(x)$
$\displaystyle=\frac{\epsilon\tilde{\epsilon}}{\sqrt{2}}\Big{\\{}[{u^{bT}}(x)C\sigma_{\alpha\mu}\gamma_{5}b^{c}(x)][\bar{d}^{d}(x)\gamma^{\alpha}C{\bar{b}^{eT}}(x)]$
$\displaystyle-[{u^{bT}}(x)C\gamma^{\alpha}b^{c}(x)][\bar{d}^{d}(x)\gamma_{5}\sigma_{\alpha\mu}C{\bar{b}^{eT}}(x)]\Big{\\}},$
(4)

where $\epsilon=\epsilon_{abc}$, $\tilde{\epsilon}=\epsilon_{ade}$ with color
indices the $a$, $b$, $c$, $d$, and $e$, C being the charge conjugation
operator, and $\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. It
should be noted that magnetic and quadrupole moment calculations have been
done under the assumption that the $Z_{b}$ state can also be in the
$B^{*}\bar{B}^{*}$ molecular configuration, however, these results are not
reported in the manuscript because a reliable sum rule could not be obtained.

In the hadronic representation, a complete set of hadronic states with the
same quantum numbers as the state of interest and the corresponding
interpolating current is plugged into the correlation function, which gives
the hadronic representation in terms of the electromagnetic multipole moments
as

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$ $\displaystyle={\frac{\langle 0\mid
J_{\mu}(x)\mid Z_{b}(p,\varepsilon^{i})\rangle}{p^{2}-m_{Z_{b}}^{2}}}$
$\displaystyle\times\langle Z_{b}(p,\varepsilon^{i})\mid
Z_{b}(p+q,\varepsilon^{f})\rangle_{\gamma}$ $\displaystyle\times\frac{\langle
Z_{b}(p+q,\varepsilon^{f})\mid{J^{\dagger}}_{\nu}(0)\mid
0\rangle}{(p+q)^{2}-m_{Z_{b}}^{2}}+\cdots\,,$ (5)

with dots representing continuum and higher state contributions.

The matrix elements $\langle 0\mid J_{\mu}(x)\mid
Z_{b}(p,\varepsilon^{i})\rangle$, $\langle
Z_{b}(p+q,\varepsilon^{f})\mid{J^{\dagger}}_{\nu}(0)\mid 0\rangle$ and
$\langle Z_{b}(p,\varepsilon^{i})\mid
Z_{b}(p+q,\varepsilon^{f})\rangle_{\gamma}$ have been described regarding
hadronic parameters such as residues, polarization vectors, and form factors,
by the following expressions

$\displaystyle\langle 0\mid J_{\mu}(x)\mid Z_{b}(p,\varepsilon^{i})\rangle$
$\displaystyle=\lambda_{Z_{b}}\varepsilon_{\mu}^{i}\,,$ (6)
$\displaystyle\langle Z_{b}(p+q,\varepsilon^{f})\mid{J^{\dagger}}_{\nu}(0)\mid
0\rangle=$ $\displaystyle=\lambda_{Z_{b}}\varepsilon_{\nu}^{*f}\,,$ (7)
$\displaystyle\langle Z_{b}(p,\varepsilon^{f})\mid
Z_{b}(p+q,\varepsilon^{i})\rangle_{\gamma}$
$\displaystyle=-\varepsilon^{\gamma}(\varepsilon^{i})^{\alpha}(\varepsilon^{f})^{\beta}$
$\displaystyle\bigg{\\{}G_{1}(Q^{2})(2p+q)_{\gamma}~{}g_{\alpha\beta}$
$\displaystyle+G_{2}(Q^{2})(g_{\gamma\beta}~{}q_{\alpha}-g_{\gamma\alpha}~{}q_{\beta})$
$\displaystyle-\frac{1}{2m_{Z_{b}}^{2}}G_{3}(Q^{2})~{}(2p+q)_{\gamma}$
$\displaystyle\times q_{\alpha}q_{\beta}\bigg{\\}},$ (8)

where $\varepsilon^{\gamma}$ represents the photon’s polarization vector,
while $\varepsilon^{i}$ and $\varepsilon^{f}$ represent the polarization
vectors of the initial and final $Z_{b}$ state, respectively.

Using the Eqs (II)-(8), the hadronic representation of the correlation
function is,

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$
$\displaystyle=\frac{\varepsilon_{\rho}\,\lambda_{Z_{b}}^{2}}{[m_{Z_{b}}^{2}-(p+q)^{2}][m_{Z_{b}}^{2}-p^{2}]}\bigg{\\{}G_{1}(Q^{2})$
$\displaystyle\times(2p+q)_{\rho}\bigg{(}g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m_{Z_{b}}^{2}}-\frac{(p+q)_{\mu}(p+q)_{\nu}}{m_{Z_{b}}^{2}}$
$\displaystyle+\frac{(p+q)_{\mu}p_{\nu}}{2m_{Z_{b}}^{4}}(Q^{2}+2m_{Z_{b}}^{2})\bigg{)}$
$\displaystyle+G_{2}(Q^{2})\bigg{(}q_{\mu}g_{\rho\nu}-q_{\nu}g_{\rho\mu}-\frac{p_{\nu}}{m_{Z_{b}}^{2}}\big{(}q_{\mu}p_{\rho}$
$\displaystyle-\frac{1}{2}Q^{2}g_{\mu\rho}\big{)}+\frac{(p+q)_{\mu}}{m_{Z_{b}}^{2}}\big{(}q_{\nu}(p+q)_{\rho}+\frac{1}{2}Q^{2}g_{\nu\rho}\big{)}$
$\displaystyle-\frac{(p+q)_{\mu}p_{\nu}p_{\rho}}{m_{Z_{b}}^{4}}\,Q^{2}\bigg{)}$
$\displaystyle-\frac{G_{3}(Q^{2})}{m_{Z_{b}}^{2}}(2p+q)_{\rho}\bigg{(}q_{\mu}q_{\nu}-\frac{p_{\mu}q_{\nu}}{2m_{Z_{b}}^{2}}Q^{2}$
$\displaystyle+\frac{(p+q)_{\mu}q_{\nu}}{2m_{Z_{b}}^{2}}Q^{2}-\frac{(p+q)_{\mu}q_{\nu}}{4m_{Z_{b}}^{4}}Q^{4}\bigg{)}\bigg{\\}}\,.$
(9)

The magnetic and quadrupole moments of hadrons are related to their magnetic
($F_{M}(Q^{2})$) and quadrupole ($F_{\mathcal{D}}(Q^{2})$) form factors. The
form factors $F_{M}(Q^{2})$ and $F_{\mathcal{D}}(Q^{2})$, which are more
directly accessible experimentally, are given by the form factors
$G_{i}(Q^{2})$

$\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,,$ $\displaystyle
F_{\mathcal{D}}(Q^{2})=G_{1}(Q^{2})-G_{2}(Q^{2})+(1+\beta)G_{3}(Q^{2})\,,$
(10)

where $\beta=Q^{2}/4m_{Z_{b}}^{2}$ with $Q^{2}=-q^{2}$. At zero momentum
transfer, the magnetic ($\mu_{Z_{b}}$) and quadrupole ($\mathcal{D}_{Z_{b}}$)
moments are described by the form factors $F_{M}(Q^{2}=0)$ and
$F_{\mathcal{D}}(Q^{2}=0)$ as follows

$\displaystyle\mu_{Z_{b}}=\frac{e}{2m_{Z_{b}}}F_{M}(Q^{2}=0)\,,$
$\displaystyle\mathcal{D}_{Z_{b}}=\frac{e}{m_{Z_{b}}^{2}}F_{\cal
D}(Q^{2}=0)\,.$ (11)

The QCD representation of the evaluations requires the use of the
interpolating field explicitly in the correlation function, Eq. (3). This is
followed by all the possible contractions of the quark operators according to
Wick’s theorem, which turns the result into the one that contains the quark
propagators in the form of

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,q)$
$\displaystyle=i\,\frac{\epsilon\tilde{\epsilon}\epsilon^{\prime}\tilde{\epsilon}^{\prime}}{2}\int
d^{4}xe^{ipx}\langle
0|\Big{\\{}\mathrm{Tr}\Big{[}\gamma^{\alpha}{\tilde{S}}_{b}^{e^{\prime}e}(-x)\gamma^{\beta}$
$\displaystyle\times
S_{d}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\sigma_{\mu\alpha}\gamma_{5}{S}_{b}^{cc^{\prime}}(x)\gamma_{5}\sigma_{\nu\beta}\tilde{S}_{u}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle-\mathrm{Tr}\Big{[}\gamma^{\alpha}{\tilde{S}}_{b}^{e^{\prime}e}(-x)\sigma_{\nu\beta}\gamma_{5}S_{d}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\sigma_{\mu\alpha}\gamma_{5}$
$\displaystyle\times{S}_{b}^{cc^{\prime}}(x)\gamma^{\beta}\tilde{S}_{u}^{bb^{\prime}}(x)\Big{]}-\mathrm{Tr}\Big{[}\gamma_{5}\sigma_{\mu\alpha}{\tilde{S}}_{b}^{e^{\prime}e}(-x)$
$\displaystyle\times\gamma^{\beta}S_{d}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma^{\alpha}{S}_{b}^{cc^{\prime}}(x)\gamma_{5}\sigma_{\nu\beta}\tilde{S}_{u}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{5}\sigma_{\mu\alpha}{\tilde{S}}_{b}^{e^{\prime}e}(-x)\sigma_{\nu\beta}\gamma_{5}S_{d}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma^{\alpha}$
$\displaystyle\times{S}_{b}^{cc^{\prime}}(x)\gamma^{\beta}\tilde{S}_{u}^{bb^{\prime}}(x)\Big{]}\Big{\\}}|0\rangle_{\gamma},$
(12)

where $S_{b}(x)$ and $S_{q}(x)$ are the propagators for heavy and light
quarks, respectively. The explicit formulas of these propagators are given in
the following form [31, 32]

$\displaystyle S_{q}(x)$
$\displaystyle=S_{q}^{free}(x)-\frac{\langle\bar{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\langle\bar{q}\sigma.Gq\rangle}{192}x^{2}$
$\displaystyle\times\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\bigg{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\bigg{]},$ (13) $\displaystyle S_{b}(x)$
$\displaystyle=S_{b}^{free}(x)-\frac{g_{s}m_{b}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}(\sigma_{\mu\nu}{x\\!\\!\\!/}$
$\displaystyle+{x\\!\\!\\!/}\sigma_{\mu\nu})\frac{K_{1}\Big{(}m_{b}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\Big{(}m_{b}\sqrt{-x^{2}}\Big{)}\Bigg{]}.$
(14)

where

$\displaystyle S_{q}^{free}(x)$
$\displaystyle=\frac{1}{2\pi^{2}x^{2}}\Big{(}i\frac{{x\\!\\!\\!/}}{x^{2}}-\frac{m_{q}}{2}\Big{)},$
(15) $\displaystyle S_{b}^{free}(x)$
$\displaystyle=\frac{m_{b}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{b}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{b}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}.$
(16)

The correlation functions specified in Eq. (II) contain both perturbative and
non-perturbative contributions from short and long distances, respectively. To
derive formulas for the perturbative contributions, i.e. when the photon is
radiated at a short-distance, it is sufficient to replace one of the light or
heavy propagators in Eq. (II) by the following

$\displaystyle S^{free}(x)\rightarrow\int d^{4}z\,S^{free}(x-z)\,\hbox
to0.0pt{/\hss}{\\!A}(z)\,S^{free}(z)\,,$ (17)

where the three surviving propagators in eq. (II) are considered to be the
free ones. To derive the formulas for the nonperturbative contributions, i.e.
where the photon is radiated at a long distance, replace one of the light
quark propagators in the correlation function given in Eq. (II) as follows

$\displaystyle
S_{\mu\nu}^{ab}(x)\rightarrow-\frac{1}{4}\big{[}\bar{q}^{a}(x)\Gamma_{i}q^{b}(0)\big{]}\big{(}\Gamma_{i}\big{)}_{\mu\nu},$
(18)

where the remaining propagators in Eq. (II) are taken into account as full
propagators, and
$\Gamma_{i}=\textbf{1},\gamma_{5},\gamma_{\mu},i\gamma_{5}\gamma_{\mu},\sigma_{\mu\nu}/2$.
When a photon interacts nonperturbatively with light-quark fields, the matrix
elements of the nonlocal operators
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)\right|0\rangle$
appear between the vacuum and the photon state, expressed in terms of photon
distribution amplitudes (DAs) (for details see Ref. [33]). Eqs. (II)-(18) are
used for evaluation the QCD representation of the correlation function. The
expression of the correlation function in the x-space is then transferred to
the momentum space by means of the Fourier transform.

The physical quantities that we search for in this section, namely the
magnetic and quadrupole moments of the $Z_{b}$ state, are obtained by matching
the coefficients of the same Lorentz structures,
$(\varepsilon.p)(p_{\mu}q_{\nu}-p_{\nu}q_{\mu})$ and
$(\varepsilon.p)q_{\mu}q_{\nu}$, acquired in both the hadronic and QCD
representations. These matches are written as

$\displaystyle\mu_{Z_{b}}$
$\displaystyle=\frac{e^{\frac{m_{Z_{b}}^{2}}{M^{2}}}}{\lambda_{Z_{b}}^{2}}\,\,\Delta_{1}^{QCD}(M^{2},s_{0}),$
(19) $\displaystyle\mathcal{D}_{Z_{b}}$
$\displaystyle=\frac{m_{Z_{b}}^{2}e^{\frac{m_{Z_{b}}^{2}}{M^{2}}}}{\lambda_{Z_{b}}^{2}}\,\,\Delta_{2}^{QCD}(M^{2},s_{0}),$
(20)

where $M^{2}$ and $s_{0}$ represent the Borel mass the continuum threshold
parameter, respectively. For brevity, since the
$\Delta_{1}^{QCD}(M^{2},s_{0})$ and $\Delta_{2}^{QCD}(M^{2},s_{0})$ functions
have a similar form, only the explicit expressions of the
$\Delta_{1}^{QCD}(M^{2},s_{0})$ function are given in the Appendix A.

##  III Numerical analysis and conclusion

In this section, we conduct a numerical analysis of the QCD light-cone sum
rules for the magnetic and quadrupole moments of the $Z_{b}$ tetraquark state,
which were derived in the previous section. In the numerical analysis
following input parameters are used: $m_{u}=m_{d}=0$,
$m_{b}=4.18^{+0.03}_{-0.02}\,$GeV, $m_{Z_{b}}=10652.2\pm 1.5\,$MeV [34],
$\lambda_{Z_{b}}=(2.12\pm 0.31)\times 10^{-1}$ GeV5 [35],
$\langle\bar{u}u\rangle$ = $\langle\bar{d}d\rangle$=$(-0.24\pm
0.01)^{3}\,$GeV3 [36], and $\langle g_{s}^{2}G^{2}\rangle=0.88~{}$GeV4 [37].
The photon DAs and corresponding parameters can be found in Appendix B.

The sum rules are obtained for the magnetic and quadrupole moments given in
Eqs. (19) and (20), which also depend on the Borel and continuum threshold
parameters $M^{2}$ and $s_{0}$. The choice of working regions for $M^{2}$ and
$s_{0}$ has to fulfill standard restrictions imposed on the pole contribution
(PC) and the convergence of the operator product expansion (OPE). In order to
quantify these constraints, it is appropriate to use the expressions

PC $\displaystyle=\frac{\Delta(M^{2},s_{0})}{\Delta(M^{2},\infty)}\geq 30\%,$
(21) OPE Convergence
$\displaystyle=\frac{\Delta^{\mbox{Dim7}}(M^{2},s_{0})}{\Delta(M^{2},s_{0})}\leq
5\%,$ (22)

where $\Delta^{\mbox{Dim7}}(M^{2},s_{0})$ is the contribution of the highest
dimensional term in the OPE. Because of the aforementioned constraints, the
obtained working regions for $M^{2}$ and $s_{0}$ are provided as follows:
$11~{}\mbox{GeV}^{2}\leq M^{2}\leq 15~{}\mbox{GeV}^{2}$ and
$121~{}\mbox{GeV}^{2}\leq s_{0}\leq 125~{}\mbox{GeV}^{2}$. Results obtained
for the PC values and the OPE convergence in the working intervals obtained
for auxiliary parameters: $31.2\%\leq\mbox{PC}\leq 53.4\%$ and $\mbox{OPE}\leq
3.69\%$. From these values, we can see that the working regions determined for
$M^{2}$ and $s_{0}$ meet the above requirements. For the sake of completeness,
it is worth examining how the magnetic and quadrupole moments depend on
$M^{2}$ for different values of $s_{0}$. It is seen that the magnetic and
quadrupole moments exhibit relatively mild dependence on the variation of
$M^{2}$ within its working region.

Figure 1:  Magnetic and quadrupole moments of the $Z_{b}$ state versus $M^{2}$
for various $s_{0}$ values.

The magnetic and quadrupole moment results, taking into account the
uncertainties in the input parameters and the variation in the $M^{2}$ and
$s_{0}$ working regions, are given below:

$\displaystyle\mu_{Z_{b}}=2.35^{+0.34}_{-0.33}~{}\mu_{N}\,,$ (23)
$\displaystyle\mathcal{D}_{Z_{b}}=(1.82^{+0.35}_{-0.31})\times
10^{-2}~{}\mbox{fm}^{2}.$ (24)

The order of the numerical results of the magnetic moments can also give an
insight into the experimental measurement of them. By examining the magnetic
moment result, one can assume that the magnetic moment of $Z_{b}$ is large
enough to be measured in future experiments with the increased luminosity. In
the case of the quadrupole moment, we get a non-zero but small value for the
$Z_{b}$ state, indicating a non-spherical charge distribution. It is well
known that the sign of the quadrupole moment contains information about the
geometric shape of the hadron under study. For the $Z_{b}$ state, the
quadrupole moment has a positive sign, corresponding to a prolate charge
distribution. To the best of our knowledge, this is the first study of the
magnetic and quadrupole moments of the $Z_{b}(10650)$ state, so there are no
theoretical or experimental results to compare. However, to give an idea of
the results obtained, a comparison can be made with the magnetic moment
results obtained by the QCD light-cone sum rules method in Ref. [18] for the
$Z_{b}(10610)$ state in the molecular and compact tetraquark pictures. The
result obtained are $\mu_{Z_{b}(10610)}=1.73\pm 0.63~{}\mu_{N}$ and
$\mu_{Z_{b}(10610)}=1.59\pm 0.58~{}\mu_{N}$ for the compact diquark-
antidiquark and molecular configuration, respectively. Based on the presented
results, it can be seen that the magnetic moment of the $Z_{b}(10650)$ state
is of the same order as that of the $Z_{b}(10610)$ state.

It is useful to consider the individual quark sector contributions to the
magnetic and quadrupole moments to gain a deeper understanding of the
underlying quark-gluon dynamics. This can be achieved by selecting the
appropriate charge factors $e_{u}$, $e_{d}$, and $e_{b}$. When it has been
done magnetic moment we see that the terms proportional to the $e_{u}$
contribute about $\%67$ to the total results, $e_{d}$ about $\%33$, and
$e_{b}$ is zero. In the case of the quadrupole moment, we observe that the
proportional terms of $e_{u}$ contribute about $\%66$ to the total results,
$e_{d}$ contributes about $\%34$, and $e_{b}$ is zero. A closer look reveals
that the missing $e_{b}$ contribution is because the terms containing this
term exactly cancel each other out.

##  IV Summary and outlook

In the present article, we have explored the magnetic and quadrupole moments
of the $Z_{b}(10650)$ state with the spin-parity $J^{P}=1^{+}$ in the
framework of the QCD light-cone sum rule by modeling this state as a compact
diquark-antidiquark configuration. Examining the results obtained, it can be
seen that the magnetic moments are large enough to be measured experimentally,
while the quadrupole moment is obtained as a small but non-zero value,
corresponding to a prolate charge distribution. Individual quark sector
contributions to the magnetic and quadrupole moments have also been analyzed,
and it has been found that only light quarks contribute to these quantities.
Our predictions should also be checked through other phenomenological methods
as well. The results of this study on the magnetic and quadrupole moments of
the $Z_{b}(10650)$ state can be used in future experimental studies of the
exotic states.

##  V Acknowledgements

The author thanks A. Özpineci for useful discussions, comments and
suggestions.

##  Appendix A  The explicit expressions of the
$\Delta_{1}^{QCD}(M^{2},s_{0})$

The explicit forms of the $\Delta_{1}^{QCD}(M^{2},s_{0})$ functions appearing
in the above sum rules are

$\displaystyle\Delta_{1}^{QCD}(M^{2},s_{0})$
$\displaystyle=\frac{27(e_{d}-e_{u})}{1310720\pi^{5}}\Bigg{[}I[0,5,3,1]-3I[0,5,3,2]+3I[0,5,3,3]-I[0,5,3,4]-3I[0,5,4,1]$
$\displaystyle+6I[0,5,4,2]-3I[0,5,4,3]+3I[0,5,5,1]-3I[0,5,5,2]-I[0,5,6,1]\Bigg{]}$
$\displaystyle+\frac{m_{b}^{2}(e_{d}-e_{u})}{65536\pi^{5}}\Bigg{[}I[0,4,2,2]-2I[0,4,2,3]+I[0,4,2,4]-2I[0,4,3,2]+2I[0,4,3,3]+I[0,4,4,2]\Bigg{]}$
$\displaystyle-\frac{m_{b}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{7077888\pi^{3}}\Bigg{[}e_{d}\Big{(}-92I_{4}[\mathcal{S}]+60I_{4}[\mathcal{T}_{1}]+159I_{4}[\mathcal{T}_{2}]-21I_{4}[\mathcal{T}_{3}]+78I_{4}[\mathcal{T}_{4}]+11I_{4}[\mathcal{\tilde{S}}]\Big{)}-e_{u}\Big{(}117I_{3}[\mathcal{S}]$
$\displaystyle+60I_{3}[\mathcal{T}_{1}]+159I_{3}[\mathcal{T}_{2}]-21I_{3}[\mathcal{T}_{3}]+78I_{3}[\mathcal{T}_{4}]+160I_{3}[\mathcal{\tilde{S}}]\Big{)}-192(e_{d}-e_{u})I_{6}[h_{\gamma}]\Bigg{]}I[0,1,3,0]$
$\displaystyle+\frac{f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle}{28311552\pi^{3}}\Bigg{[}-128m_{b}^{2}(e_{d}-e_{u})(52I[0,1,2,0]-3I[0,1,3,0])I_{6}[\psi_{\gamma}^{\nu}]-\Big{(}27e_{u}I_{1}[\mathcal{A}]-27e_{d}I_{2}[\mathcal{A}]$
$\displaystyle+128(3e_{d}-e_{u})I_{6}[\psi_{\gamma}^{\nu}]\Big{)}\Bigg{]}$
$\displaystyle+\frac{m_{b}\langle\bar{q}q\rangle}{393216\pi^{3}}\Bigg{[}4\Bigg{(}-e_{u}\Big{(}5I_{3}[\mathcal{S}]-23I_{3}[\mathcal{T}_{1}]-23I_{3}[\mathcal{T}_{2}]+I_{3}[\mathcal{\tilde{S}}]\Big{)}+e_{d}\Big{(}23I_{4}[\mathcal{T}_{1}]+23I_{4}[\mathcal{T}_{2}]+22I_{4}[\mathcal{\tilde{S}}]\Big{)}\Bigg{)}$
$\displaystyle\times
I[0,3,4,0]-3\Bigg{(}3e_{u}\Big{(}I_{3}[\mathcal{S}]+I_{3}[\mathcal{T}_{1}]+I_{3}[\mathcal{T}_{2}]+I_{3}[\mathcal{\tilde{S}}]\Big{)}-e_{d}\Big{(}2I_{4}[\mathcal{S}]-3I_{4}[\mathcal{T}_{1}]-3I_{4}[\mathcal{T}_{2}]+I_{4}[\mathcal{\tilde{S}}]\Big{)}$
$\displaystyle+96(e_{d}-e_{u})I_{6}[h_{\gamma}]\Bigg{)}I[0,3,5,0]\Bigg{]}$
$\displaystyle+\frac{f_{3\gamma}}{6291456\pi^{3}}\Bigg{[}4\Big{(}512m_{b}^{2}(e_{d}-e_{u})I_{6}[\psi_{\gamma}^{\nu}]I[0,3,4,0]-3\big{(}-e_{u}I_{1}[\mathcal{V}]+9e_{d}I_{2}[\mathcal{V}]\big{)}I[0,4,5,0]\Big{)}$
$\displaystyle+9\Big{(}e_{u}I_{1}[\mathcal{V}]-e_{d}I_{2}[\mathcal{V}]-64(3e_{d}-e_{u})I_{6}[\psi_{\gamma}^{\nu}]\Big{)}I[0,4,6,0]\Bigg{]},$
(26)

where the functions $I[n,m,l,k]$, $I_{1}[\mathcal{F}]$, $I_{2}[\mathcal{F}]$,
$I_{3}[\mathcal{F}]$, $I_{4}[\mathcal{F}]$, $I_{5}[\mathcal{F}]$, and
$I_{6}[\mathcal{F}]$ are defined as:

$\displaystyle I[n,m,l,k]$
$\displaystyle=\int_{4m_{c}^{2}}^{s_{0}}ds\int_{0}^{1}dt\int_{0}^{1}dw~{}e^{-s/M^{2}}~{}s^{n}\,(s-4\,m_{c}^{2})^{m}\,t^{l}\,w^{k},$
$\displaystyle I_{1}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{2}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{3}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{4}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{5}[\mathcal{F}]$
$\displaystyle=\int_{0}^{1}du~{}F(u)\delta^{\prime}(u-u_{0}),$ $\displaystyle
I_{6}[\mathcal{F}]$ $\displaystyle=\int_{0}^{1}du~{}F(u),$ (27)

where $\mathcal{F}$ represents the corresponding photon DAs.

##  Appendix B Photon Distribution Amplitudes and Wave Functions

In this appendix, we present the matrix elements
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)\right|0\rangle$ in
terms of the photon DAs and wave functions of different twists. The expansion
of the matrix element is an expansion in increasing twists of the DAs. The
twist of a DA is defined as the dimension minus the spin of the operators
contributing to a given DA. The DAs $\varphi_{\gamma}(u)$ have twist two,
$\psi^{v}(u)$, $\psi^{a}(u)$, ${\cal A}(\alpha_{i})$ and ${\cal
V}(\alpha_{i})$ have twist 3, and $h_{\gamma}(u)$, $\mathbb{A}(u)$, ${\cal
S}(\alpha_{i})$, ${\cal{\tilde{S}}}(\alpha_{i})$, ${\cal T}_{1}(\alpha_{i})$,
${\cal T}_{2}(\alpha_{i})$, ${\cal T}_{3}(\alpha_{i})$ and ${\cal
T}_{4}(\alpha_{i})$ have twist 4. The matrix elements
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)\right|0\rangle$ are
parameterized in terms of the photon DAs as follows [33]

$\displaystyle\langle\gamma(q)|\bar{q}(x)\gamma_{\mu}q(0)|0\rangle=e_{q}f_{3\gamma}\left(\varepsilon_{\mu}-q_{\mu}\frac{\varepsilon
x}{qx}\right)\int_{0}^{1}due^{i\bar{u}qx}\psi^{v}(u),$
$\displaystyle\langle\gamma(q)|\bar{q}(x)\gamma_{\mu}\gamma_{5}q(0)|0\rangle=-\frac{1}{4}e_{q}f_{3\gamma}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\nu}q^{\alpha}x^{\beta}\int_{0}^{1}due^{i\bar{u}qx}\psi^{a}(u),$
$\displaystyle\langle\gamma(q)|\bar{q}(x)\sigma_{\mu\nu}q(0)|0\rangle=-ie_{q}\langle\bar{q}q\rangle(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu})\int_{0}^{1}due^{i\bar{u}qx}\left(\chi\varphi_{\gamma}(u)+\frac{x^{2}}{16}\mathbb{A}(u)\right)$
$\displaystyle-\frac{i}{2(qx)}e_{q}\langle\bar{q}q\rangle\left[x_{\nu}\left(\varepsilon_{\mu}-q_{\mu}\frac{\varepsilon
x}{qx}\right)-x_{\mu}\left(\varepsilon_{\nu}-q_{\nu}\frac{\varepsilon
x}{qx}\right)\right]\int_{0}^{1}due^{i\bar{u}qx}h_{\gamma}(u),$
$\displaystyle\langle\gamma(q)|\bar{q}(x)g_{s}G_{\mu\nu}(vx)q(0)|0\rangle=-ie_{q}\langle\bar{q}q\rangle\left(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu}\right)\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal S}(\alpha_{i}),$
$\displaystyle\langle\gamma(q)|\bar{q}(x)g_{s}\tilde{G}_{\mu\nu}(vx)i\gamma_{5}q(0)|0\rangle=-ie_{q}\langle\bar{q}q\rangle\left(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu}\right)\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}\tilde{\cal S}(\alpha_{i}),$
$\displaystyle\langle\gamma(q)|\bar{q}(x)g_{s}\tilde{G}_{\mu\nu}(vx)\gamma_{\alpha}\gamma_{5}q(0)|0\rangle=e_{q}f_{3\gamma}q_{\alpha}(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu})\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal A}(\alpha_{i}),$
$\displaystyle\langle\gamma(q)|\bar{q}(x)g_{s}G_{\mu\nu}(vx)i\gamma_{\alpha}q(0)|0\rangle=e_{q}f_{3\gamma}q_{\alpha}(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu})\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal V}(\alpha_{i}),$
$\displaystyle\langle\gamma(q)|\bar{q}(x)\sigma_{\alpha\beta}g_{s}G_{\mu\nu}(vx)q(0)|0\rangle=e_{q}\langle\bar{q}q\rangle\left\\{\left[\left(\varepsilon_{\mu}-q_{\mu}\frac{\varepsilon
x}{qx}\right)\left(g_{\alpha\nu}-\frac{1}{qx}(q_{\alpha}x_{\nu}+q_{\nu}x_{\alpha})\right)\right.\right.q_{\beta}$
$\displaystyle-\left(\varepsilon_{\mu}-q_{\mu}\frac{\varepsilon
x}{qx}\right)\left(g_{\beta\nu}-\frac{1}{qx}(q_{\beta}x_{\nu}+q_{\nu}x_{\beta})\right)q_{\alpha}-\left(\varepsilon_{\nu}-q_{\nu}\frac{\varepsilon
x}{qx}\right)\left(g_{\alpha\mu}-\frac{1}{qx}(q_{\alpha}x_{\mu}+q_{\mu}x_{\alpha})\right)q_{\beta}$
$\displaystyle+\left.\left(\varepsilon_{\nu}-q_{\nu}\frac{\varepsilon
x}{q.x}\right)\left(g_{\beta\mu}-\frac{1}{qx}(q_{\beta}x_{\mu}+q_{\mu}x_{\beta})\right)q_{\alpha}\right]\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal T}_{1}(\alpha_{i})$
$\displaystyle+\left[\left(\varepsilon_{\alpha}-q_{\alpha}\frac{\varepsilon
x}{qx}\right)\left(g_{\mu\beta}-\frac{1}{qx}(q_{\mu}x_{\beta}+q_{\beta}x_{\mu})\right)\right.q_{\nu}$
$\displaystyle-\left(\varepsilon_{\alpha}-q_{\alpha}\frac{\varepsilon
x}{qx}\right)\left(g_{\nu\beta}-\frac{1}{qx}(q_{\nu}x_{\beta}+q_{\beta}x_{\nu})\right)q_{\mu}$
$\displaystyle-\left(\varepsilon_{\beta}-q_{\beta}\frac{\varepsilon
x}{qx}\right)\left(g_{\mu\alpha}-\frac{1}{qx}(q_{\mu}x_{\alpha}+q_{\alpha}x_{\mu})\right)q_{\nu}$
$\displaystyle+\left.\left(\varepsilon_{\beta}-q_{\beta}\frac{\varepsilon
x}{qx}\right)\left(g_{\nu\alpha}-\frac{1}{qx}(q_{\nu}x_{\alpha}+q_{\alpha}x_{\nu})\right)q_{\mu}\right]\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal T}_{2}(\alpha_{i})$
$\displaystyle+\frac{1}{qx}(q_{\mu}x_{\nu}-q_{\nu}x_{\mu})(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal T}_{3}(\alpha_{i})$
$\displaystyle+\left.\frac{1}{qx}(q_{\alpha}x_{\beta}-q_{\beta}x_{\alpha})(\varepsilon_{\mu}q_{\nu}-\varepsilon_{\nu}q_{\mu})\int{\cal
D}\alpha_{i}e^{i(\alpha_{\bar{q}}+v\alpha_{g})qx}{\cal
T}_{4}(\alpha_{i})\right\\},$ (28)

where the integral measure ${\cal D}\alpha_{i}$ is written as

$\displaystyle\int{\cal
D}\alpha_{i}=\int_{0}^{1}d\alpha_{\bar{q}}\int_{0}^{1}d\alpha_{q}\int_{0}^{1}d\alpha_{g}\delta(1-\alpha_{\bar{q}}-\alpha_{q}-\alpha_{g})~{}.$
(29)

The explicit equations for the photon DAs with different twists are provided
as

$\displaystyle\varphi_{\gamma}(u)$ $\displaystyle=$ $\displaystyle
6u\bar{u}\left(1+\varphi_{2}(\mu)C_{2}^{\frac{3}{2}}(u-\bar{u})\right),$
$\displaystyle\psi^{v}(u)$ $\displaystyle=$ $\displaystyle
3\left(3(2u-1)^{2}-1\right)+\frac{3}{64}\left(15w^{V}_{\gamma}-5w^{A}_{\gamma}\right)\left(3-30(2u-1)^{2}+35(2u-1)^{4}\right),$
$\displaystyle\psi^{a}(u)$ $\displaystyle=$
$\displaystyle\left(1-(2u-1)^{2}\right)\left(5(2u-1)^{2}-1\right)\frac{5}{2}\left(1+\frac{9}{16}w^{V}_{\gamma}-\frac{3}{16}w^{A}_{\gamma}\right),$
$\displaystyle h_{\gamma}(u)$ $\displaystyle=$
$\displaystyle-10\left(1+2\kappa^{+}\right)C_{2}^{\frac{1}{2}}(u-\bar{u}),$
$\displaystyle\mathbb{A}(u)$ $\displaystyle=$ $\displaystyle
40u^{2}\bar{u}^{2}\left(3\kappa-\kappa^{+}+1\right)+8(\zeta_{2}^{+}-3\zeta_{2})\left[u\bar{u}(2+13u\bar{u})\right.$
$\displaystyle+\left.2u^{3}(10-15u+6u^{2})\ln(u)+2\bar{u}^{3}(10-15\bar{u}+6\bar{u}^{2})\ln(\bar{u})\right],$
$\displaystyle{\cal A}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
360\alpha_{q}\alpha_{\bar{q}}\alpha_{g}^{2}\left(1+w^{A}_{\gamma}\frac{1}{2}(7\alpha_{g}-3)\right),$
$\displaystyle{\cal V}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
540w^{V}_{\gamma}(\alpha_{q}-\alpha_{\bar{q}})\alpha_{q}\alpha_{\bar{q}}\alpha_{g}^{2},$
$\displaystyle{\cal T}_{1}(\alpha_{i})$ $\displaystyle=$
$\displaystyle-120(3\zeta_{2}+\zeta_{2}^{+})(\alpha_{\bar{q}}-\alpha_{q})\alpha_{\bar{q}}\alpha_{q}\alpha_{g},$
$\displaystyle{\cal T}_{2}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
30\alpha_{g}^{2}(\alpha_{\bar{q}}-\alpha_{q})\left((\kappa-\kappa^{+})+(\zeta_{1}-\zeta_{1}^{+})(1-2\alpha_{g})+\zeta_{2}(3-4\alpha_{g})\right),$
$\displaystyle{\cal T}_{3}(\alpha_{i})$ $\displaystyle=$
$\displaystyle-120(3\zeta_{2}-\zeta_{2}^{+})(\alpha_{\bar{q}}-\alpha_{q})\alpha_{\bar{q}}\alpha_{q}\alpha_{g},$
$\displaystyle{\cal T}_{4}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
30\alpha_{g}^{2}(\alpha_{\bar{q}}-\alpha_{q})\left((\kappa+\kappa^{+})+(\zeta_{1}+\zeta_{1}^{+})(1-2\alpha_{g})+\zeta_{2}(3-4\alpha_{g})\right),$
$\displaystyle{\cal S}(\alpha_{i})$ $\displaystyle=$ $\displaystyle
30\alpha_{g}^{2}\\{(\kappa+\kappa^{+})(1-\alpha_{g})+(\zeta_{1}+\zeta_{1}^{+})(1-\alpha_{g})(1-2\alpha_{g})+\zeta_{2}[3(\alpha_{\bar{q}}-\alpha_{q})^{2}-\alpha_{g}(1-\alpha_{g})]\\},$
$\displaystyle\tilde{\cal S}(\alpha_{i})$ $\displaystyle=$
$\displaystyle-30\alpha_{g}^{2}\\{(\kappa-\kappa^{+})(1-\alpha_{g})+(\zeta_{1}-\zeta_{1}^{+})(1-\alpha_{g})(1-2\alpha_{g})+\zeta_{2}[3(\alpha_{\bar{q}}-\alpha_{q})^{2}-\alpha_{g}(1-\alpha_{g})]\\},$
(30)

where $f_{3\gamma}=-0.0039~{}$GeV2, $\zeta_{2}=0.3$, $\varphi_{2}(1~{}GeV)=0$,
$w^{V}_{\gamma}=3.8\pm 1.8$, $w^{A}_{\gamma}=-2.1\pm 1.0$, $\kappa=0.2$,
$\kappa^{+}=0$, $\zeta_{1}=0.4$, and $\chi=-2.85\pm 0.5~{}\mbox{GeV}^{-2}$
[38].

## References

  * [1] A. Bondar, et al., Observation of two charged bottomonium-like resonances in Y(5S) decays, Phys. Rev. Lett. 108 (2012) 122001.  arXiv:1110.2251, doi:10.1103/PhysRevLett.108.122001.
  * [2] P. Krokovny, et al., First observation of the $Z\frac{0}{b}$(10610) in a Dalitz analysis of $\Upsilon$(10860) $\to\Upsilon$(nS)$\pi^{0}\pi^{0}$, Phys. Rev. D 88 (5) (2013) 052016.  arXiv:1308.2646, doi:10.1103/PhysRevD.88.052016.
  * [3] R. Faccini, A. Pilloni, A. D. Polosa, Exotic Heavy Quarkonium Spectroscopy: A Mini-review, Mod. Phys. Lett. A 27 (2012) 1230025.  arXiv:1209.0107, doi:10.1142/S021773231230025X.
  * [4] A. Esposito, A. L. Guerrieri, F. Piccinini, A. Pilloni, A. D. Polosa, Four-Quark Hadrons: an Updated Review, Int. J. Mod. Phys. A 30 (2015) 1530002\.  arXiv:1411.5997, doi:10.1142/S0217751X15300021.
  * [5] H.-X. Chen, W. Chen, X. Liu, S.-L. Zhu, The hidden-charm pentaquark and tetraquark states, Phys. Rept. 639 (2016) 1–121.  arXiv:1601.02092, doi:10.1016/j.physrep.2016.05.004.
  * [6] A. Ali, J. S. Lange, S. Stone, Exotics: Heavy Pentaquarks and Tetraquarks, Prog. Part. Nucl. Phys. 97 (2017) 123–198.  arXiv:1706.00610, doi:10.1016/j.ppnp.2017.08.003.
  * [7] A. Esposito, A. Pilloni, A. D. Polosa, Multiquark Resonances, Phys. Rept. 668 (2017) 1–97.  arXiv:1611.07920, doi:10.1016/j.physrep.2016.11.002.
  * [8] S. L. Olsen, T. Skwarnicki, D. Zieminska, Nonstandard heavy mesons and baryons: Experimental evidence, Rev. Mod. Phys. 90 (1) (2018) 015003.  arXiv:1708.04012, doi:10.1103/RevModPhys.90.015003.
  * [9] R. F. Lebed, R. E. Mitchell, E. S. Swanson, Heavy-Quark QCD Exotica, Prog. Part. Nucl. Phys. 93 (2017) 143–194.  arXiv:1610.04528, doi:10.1016/j.ppnp.2016.11.003.
  * [10] F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, B.-S. Zou, Hadronic molecules, Rev. Mod. Phys. 90 (1) (2018) 015004, [Erratum: Rev.Mod.Phys. 94, 029901 (2022)].  arXiv:1705.00141, doi:10.1103/RevModPhys.90.015004.
  * [11] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, C.-Z. Yuan, The $XYZ$ states: experimental and theoretical status and perspectives, Phys. Rept. 873 (2020) 1–154.  arXiv:1907.07583, doi:10.1016/j.physrep.2020.05.001.
  * [12] Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, S.-L. Zhu, Pentaquark and Tetraquark states, Prog. Part. Nucl. Phys. 107 (2019) 237–320.  arXiv:1903.11976, doi:10.1016/j.ppnp.2019.04.003.
  * [13] S. Agaev, K. Azizi, H. Sundu, Four-quark exotic mesons, Turk. J. Phys. 44 (2) (2020) 95–173.  arXiv:2004.12079, doi:10.3906/fiz-2003-15.
  * [14] X.-K. Dong, F.-K. Guo, B.-S. Zou, A survey of heavy-antiheavy hadronic molecules, Progr. Phys. 41 (2021) 65–93.  arXiv:2101.01021, doi:10.13725/j.cnki.pip.2021.02.001.
  * [15] W. Chen, T. G. Steele, H.-X. Chen, S.-L. Zhu, Mass spectra of Zc and Zb exotic states as hadron molecules, Phys. Rev. D 92 (5) (2015) 054002.  arXiv:1505.05619, doi:10.1103/PhysRevD.92.054002.
  * [16] L. Meng, B. Wang, G.-J. Wang, S.-L. Zhu, Chiral perturbation theory for heavy hadrons and chiral effective field theory for heavy hadronic molecules, Phys. Rept. 1019 (2023) 1–149.  arXiv:2204.08716, doi:10.1016/j.physrep.2023.04.003.
  * [17] H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, S.-L. Zhu, An updated review of the new hadron states, Rept. Prog. Phys. 86 (2) (2023) 026201.  arXiv:2204.02649, doi:10.1088/1361-6633/aca3b6.
  * [18] U. Ozdem, K. Azizi, Magnetic dipole moment of $Z_{b}(10610)$ in light-cone QCD, Phys. Rev. D 97 (1) (2018) 014010.  arXiv:1709.09714, doi:10.1103/PhysRevD.97.014010.
  * [19] U. Ozdem, K. Azizi, Magnetic and quadrupole moments of the $Z_{c}(3900)$, Phys. Rev. D 96 (7) (2017) 074030.  arXiv:1707.09612, doi:10.1103/PhysRevD.96.074030.
  * [20] Y.-J. Xu, Y.-L. Liu, M.-Q. Huang, The magnetic moment of $Z_{c}(3900)$ as an axial-vector molecular state, Eur. Phys. J. C 80 (10) (2020) 953.  arXiv:2007.15214, doi:10.1140/epjc/s10052-020-08515-5.
  * [21] U. Özdem, Magnetic moments of the vector hidden-charm tetraquark states, Phys. Rev. D 105 (11) (2022) 114030.  arXiv:2206.05196, doi:10.1103/PhysRevD.105.114030.
  * [22] Z.-G. Wang, The magnetic moment of the $Z_{c}(3900)$ as an axialvector tetraquark state with QCD sum rules, Eur. Phys. J. C 78 (4) (2018) 297.  arXiv:1712.05664, doi:10.1140/epjc/s10052-018-5794-0.
  * [23] U. Özdem, K. Azizi, Magnetic dipole moment of the $Z_{cs}(3985)$ state: diquark–antidiquark and molecular pictures, Eur. Phys. J. Plus 136 (9) (2021) 968.  arXiv:2102.09231, doi:10.1140/epjp/s13360-021-01977-w.
  * [24] Y.-J. Xu, Y.-L. Liu, C.-Y. Cui, M.-Q. Huang, $\bar{D}^{(*)}_{s}D^{(*)}$ molecular state with $J^{P}$= $1^{+}$, Phys. Rev. D 104 (9) (2021) 094028.  arXiv:2011.14313, doi:10.1103/PhysRevD.104.094028.
  * [25] U. Özdem, A. K. Yıldırım, Magnetic dipole moments of the Zc(4020)+, Zc(4200)+, Zcs(4000)+ and Zcs(4220)+ states in light-cone QCD, Phys. Rev. D 104 (5) (2021) 054017.  arXiv:2104.13074, doi:10.1103/PhysRevD.104.054017.
  * [26] Y.-H. Wang, J. Wei, C.-S. An, C.-R. Deng, $Z_{cs}$(4000)+ and $Z_{cs}$(4220)+ in a Multiquark Color Flux-Tube Model, Chin. Phys. Lett. 40 (2) (2023) 021201.  doi:10.1088/0256-307X/40/2/021201.
  * [27] U. Özdem, Magnetic and quadrupole moments of the $Z_{c}(4020)^{+}$, $Z_{c}(4050)^{+}$ and $Z_{c}(4600)^{+}$ states in the diquark-antidiquark picture (7 2023).  arXiv:2307.05028.
  * [28] V. L. Chernyak, I. R. Zhitnitsky, B meson exclusive decays into baryons, Nucl. Phys. B 345 (1990) 137–172.  doi:10.1016/0550-3213(90)90612-H.
  * [29] V. M. Braun, I. E. Filyanov, QCD Sum Rules in Exclusive Kinematics and Pion Wave Function, Z. Phys. C 44 (1989) 157.  doi:10.1007/BF01548594.
  * [30] I. I. Balitsky, V. M. Braun, A. V. Kolesnichenko, Radiative Decay Sigma+ —$>$ p gamma in Quantum Chromodynamics, Nucl. Phys. B 312 (1989) 509–550.  doi:10.1016/0550-3213(89)90570-1.
  * [31] K.-C. Yang, W. Y. P. Hwang, E. M. Henley, L. S. Kisslinger, QCD sum rules and neutron proton mass difference, Phys. Rev. D 47 (1993) 3001–3012.  doi:10.1103/PhysRevD.47.3001.
  * [32] V. M. Belyaev, B. Y. Blok, CHARMED BARYONS IN QUANTUM CHROMODYNAMICS, Z. Phys. C 30 (1986) 151.  doi:10.1007/BF01560689.
  * [33] P. Ball, V. M. Braun, N. Kivel, Photon distribution amplitudes in QCD, Nucl. Phys. B 649 (2003) 263–296.  arXiv:hep-ph/0207307, doi:10.1016/S0550-3213(02)01017-9.
  * [34] R. L. Workman, et al., Review of Particle Physics, PTEP 2022 (2022) 083C01.  doi:10.1093/ptep/ptac097.
  * [35] Z.-G. Wang, Analysis of the hidden-bottom tetraquark mass spectrum with the QCD sum rules, Eur. Phys. J. C 79 (6) (2019) 489.  arXiv:1903.10895, doi:10.1140/epjc/s10052-019-7019-6.
  * [36] B. L. Ioffe, QCD at low energies, Prog. Part. Nucl. Phys. 56 (2006) 232–277.  arXiv:hep-ph/0502148, doi:10.1016/j.ppnp.2005.05.001.
  * [37] R. D. Matheus, S. Narison, M. Nielsen, J. M. Richard, Can the X(3872) be a 1++ four-quark state?, Phys. Rev. D 75 (2007) 014005.  arXiv:hep-ph/0608297, doi:10.1103/PhysRevD.75.014005.
  * [38] J. Rohrwild, Determination of the magnetic susceptibility of the quark condensate using radiative heavy meson decays, JHEP 09 (2007) 073.  arXiv:0708.1405, doi:10.1088/1126-6708/2007/09/073.


  *

# Impact of the $\mathrm{Ce}~{}4f$ states in the electronic structure of the
intermediate-valence superconductor CeIr3

Shin-ichi Fujimori1, Ikuto Kawasaki1, Yukiharu Takeda1, Hiroshi Yamagami1,2,
Norimasa Sasabe3, Yoshiki J. Sato4,5, Ai Nakamura4, Yusei Shimizu4, Arvind
Maurya4,6, Yoshiya Homma4, Dexin Li4, Fuminori Honda4,7, and Dai Aoki4 1
Materials Sciences Research Center, Japan Atomic Energy Agency, Sayo, Hyogo
679-5148, Japan  2 Department of Physics, Faculty of Science, Kyoto Sangyo
University, Kyoto 603-8555, Japan  3 Japan Synchrotron Radiation Research
Institute, SPring-8, Sayo, Hyogo 679-5148, Japan  4 Institute for Materials
Research, Tohoku University, Oarai, Ibaraki 311-1313, Japan  5 Department of
Physics, Faculty of Science and Technology, Tokyo University of Science, Noda,
Chiba 278-8510, Japan  6 Department of Physics, School of Physical Sciences,
Mizoram University, Aizawl 796 004, India  7 Central Institute of Radioisotope
Science and Safety Management, Kyushu University, Motooka 744, Fukuoka Nishi,
Fukuoka 819-0395, Japan<EMAIL_ADDRESS>
###### Abstract

The electronic structure of the $f$-based superconductor $\mathrm{CeIr_{3}}$
was studied by photoelectron spectroscopy. The energy distribution of the
$\mathrm{Ce}~{}4f$ states were revealed by the $\mathrm{Ce}~{}3d-4f$ resonant
photoelectron spectroscopy. The $\mathrm{Ce}~{}4f$ states were mostly
distributed in the vicinity of the Fermi energy, suggesting the itinerant
character of the $\mathrm{Ce}~{}4f$ states. The contribution of the
$\mathrm{Ce}~{}4f$ states to the density of states (DOS) at the Fermi energy
was estimated to be nearly half of that of the $\mathrm{Ir}~{}5d$ states,
implying that the $\mathrm{Ce}~{}4f$ states have a considerable contribution
to the DOS at the Fermi energy. The $\mathrm{Ce}~{}3d$ core-level and
$\mathrm{Ce}~{}3d$ X-ray absorption spectra were analyzed based on a single-
impurity Anderson model. The number of the $\mathrm{Ce}~{}4f$ states in the
ground state was estimated to be $0.8-0.9$, which is much larger than the
values obtained in the previous studies (i.e., $0-0.4$).

††: Electronic Structure

##  1 Introduction

$\mathrm{CeIr_{3}}$ is a $f$-based superconductor with a relatively high
transition temperature ($T_{\mathrm{SC}}=3.1~{}\mathrm{K}$), and its
superconducting properties have attracted much attention in recent years [1,
2, 3, 4]. Its thermodynamical properties and $\mu$SR measurements suggest
$\mathrm{CeIr_{3}}$ is a weak-coupling superconductor with an anisotropic
$s$-wave gap [1, 4]. One of the key issues of this compound is the role of the
$\mathrm{Ce}~{}4f$ states in superconductivity. Isostructural analogue
compounds $\mathrm{LaIr_{3}}$ ($T_{\mathrm{SC}}=2.5~{}\mathrm{K}$ [5, 6]) and
$\mathrm{ThIr_{3}}$ ($T_{\mathrm{SC}}=4.41~{}\mathrm{K}$ [7]) also exhibit the
superconductivity with similar or higher transition temperatures; hence, the
role of the $\mathrm{Ce}~{}4f$ states in the superconductivity of
$\mathrm{CeIr_{3}}$ is considered to be negligible [8]. Experimentally, the
$\mathrm{Ce}~{}4f$ states in $\mathrm{CeIr_{3}}$ are suggested as an
intermediate-valence state, but the occupancy of the $4f$ states $n_{4f}$ is a
matter of controversies. Based on the relationship between the transition
temperature and the valence states in $\mathrm{(La,Th)Ir_{3}}$ alloys,
$n_{4f}$ is estimated as $n_{4f}\sim 0.4$ [9]. Górnicka et alrecently
suggested that $n_{4f}\sim 0$ based on the weak-temperature-dependent magnetic
susceptibility of $\mathrm{CeIr_{3}}$. They argued that the contributions from
the $\mathrm{Ce}~{}4f$ states at the Fermi energy are negligible in this
compound [2]. On the contrary, the experimental specific heat coefficient
($\gamma_{\mathrm{exp}}=25.1~{}\mathrm{mJ/mol\cdot K^{2}}$) is much larger
than that obtained by the GGA calculation
($\gamma_{\mathrm{GGA}}=10.2~{}\mathrm{mJ/mol\cdot K^{2}}$) [2], suggesting
that there should be a considerable contribution from the correlated
$\mathrm{Ce}~{}4f$ states at the Fermi energy. Gutowska et alrecently
calculated the electronic structure of $\mathrm{CeIr_{3}}$ based on the
dynamical mean field theory (DMFT) and reproduced the experimental specific
heat coefficient and the electron–phonon coupling [10]. Their calculation
showed that the $\mathrm{Ce}~{}4f$ states had a finite contribution at the
Fermi energy, and its occupation number was $n_{4f}\sim 0.8$. Thus, the nature
of the $\mathrm{Ce}~{}4f$ states of this compound is controversial.

In this study, we have investigated the $\mathrm{Ce}~{}4f$ states of
$\mathrm{CeIr_{3}}$ by the $\mathrm{Ce}~{}3d-4f$ resonant photoelectron
spectroscopy (RPES), X-ray absorption spectroscopy (XAS), and core-level
photoelectron spectroscopy. Accordingly, the RPES study shows that the
$\mathrm{Ce}~{}4f$ states form a sharp peak at the Fermi energy, suggesting
that the $\mathrm{Ce}~{}4f$ states in $\mathrm{CeIr_{3}}$ have a considerable
contribution to the Fermi energy despite the stronger contribution from the
$\mathrm{Ir}~{}5d$ states. The results of the analysis of the
$\mathrm{Ce}~{}3d$ XAS and core-level spectra based on the single-impurity
Anderson model reveal that $n_{4f}=0.8-0.9$, contrast to the predictions of
the previous studies [2, 9]. These results argue that the $\mathrm{Ce}~{}4f$
states are in an intermediate-valence state, and the contribution from the
$\mathrm{Ce}~{}4f$ states to superconductivity is not negligible in
$\mathrm{CeIr_{3}}$.

##  2 Experimental Procedure

Single crystals of $\mathrm{CeIr_{3}}$ were prepared using the Czochralski
method in a tetra arc furnace under Ar atmosphere. The details of the sample
growth and their characterizations were presented in Refs. [1, 3, 11].
Photoemission experiments were performed at the soft X-ray beamline BL23SU of
SPring-8 [12]. The total energy resolution at the on-resonant condition
($h\nu\sim 881~{}\mathrm{eV}$) was approximately $65~{}\mathrm{meV}$, while
that in the $\mathrm{Ce}~{}3d$ core-level measurements was approximately
$190~{}\mathrm{meV}$. Clean sample surfaces were obtained by cleaving high-
quality single crystals in situ under an ultra-high vacuum condition. The
sample temperature was kept at $20~{}\mathrm{K}$ during the measurements.

##  3 Results and Discussion

Figure 1: $\mathrm{Ce}~{}3d$ XAS and core-level spectra of
$\mathrm{CeIr_{3}}$. The schematic figures of the ground and final states in
the XAS and core-level spectra are shown on the right. (a) $\mathrm{Ce}~{}3d$
XAS spectrum of $\mathrm{CeIr_{3}}$ consisting of $3d_{5/2}$ and $3d_{3/2}$
components. Each has $4f^{1}$ and $4f^{2}$ final-state peaks. The dashed
square corresponds to the region shown in Fig. 2(a). (b) $\mathrm{Ce}~{}3d$
core-level spectrum of $\mathrm{CeIr_{3}}$ consisting of $3d_{5/2}$ and
$3d_{3/2}$ components. Each has $4f^{0}$, $4f^{1}$, and $4f^{2}$ final-state
peaks.

Figure 1 shows the $\mathrm{Ce}~{}3d$ XAS and core-level spectra of
$\mathrm{CeIr_{3}}$ together with the schematic figures of their ground and
final states in the XAS and core-level spectra. Figure 1 (a) depicts the
$\mathrm{Ce}~{}3d$ XAS spectrum of $\mathrm{CeIr_{3}}$ consisting of
$3d_{5/2}$ and $3d_{3/2}$ components. Each had $4f^{1}$ and $4f^{2}$ final-
state peaks. The $4f^{1}$ and $4f^{2}$ peaks originated from the transitions
from the $4f^{0}$ and $4f^{1}$ configurations in the ground state,
respectively. The existence of these two peaks indicated that the
$\mathrm{Ce}~{}4f$ states in $\mathrm{CeIr_{3}}$ are in an intermediate-
valence state. Figure 1(b) shows the $\mathrm{Ce}~{}3d$ core-level spectrum of
$\mathrm{CeIr_{3}}$. The photon energy was $h\nu=1200~{}\mathrm{eV}$. The
spectrum was essentially consistent with that in a previous XPS study [2];
however, each peak was well resolved by the better energy resolution in the
present experiment. The $4f^{0}$ peak originates from the $4f^{0}$
configuration in the ground state. The enhanced intensity of the $4f^{0}$ peak
again suggested that the $\mathrm{Ce}~{}4f$ states were in a strong mixed
valence state. The calculation based on the single-impurity Anderson model
suggested that $1-n_{4f}\approx I(f^{0})/[I(f^{0})+I(f^{1})+I(f^{2})]$, where
$I(f^{n})$ is the $4f^{n}$ final-state peak intensity [13]. Thus, $n_{4f}$ of
$\mathrm{CeIr_{3}}$ should be close to, but less than unity in the present
case. The $4f^{2}$ final-state peak intensity was enhanced in the spectrum and
was much stronger than in the case of heavy fermion compounds
$\mathrm{Ce}T\mathrm{In}_{5}(T=\mathrm{Co},\mathrm{Rh},\mathrm{Ir})$ as an
example [14]. The $4f^{2}$ final-state peak intensity is more enhanced in a
more hybridized compound [15]; hence, the enhanced intensity in
$\mathrm{CeIr_{3}}$ indicates the existence of a strong $f-d$ hybridization in
$\mathrm{CeIr_{3}}$. The later part of this section will present a detailed
analysis of these spectra.

Figure 2: Resonant photoemission spectra of $\mathrm{CeIr_{3}}$. (a) On-
resonance and off-resonance photoemission spectra of $\mathrm{CeIr_{3}}$
measured at $h\nu=881.3$ and $870~{}\mathrm{eV}$, which reflect the
contributions mainly from the $\mathrm{Ce}~{}4f$ and $\mathrm{Ir}~{}5d$
states, respectively. (b) On- and off-resonance spectra in the very vicinity
of $E_{\mathrm{F}}$. The areas of the spectra are normalized to the nominal
numbers of electrons of the $\mathrm{Ce}~{}4f$ and $\mathrm{Ir}~{}5d$ states
($n_{4f}:n_{5d}=0.8:7\times 3$).

Figure 2 summarizes the RPES measurement of $\mathrm{CeIr_{3}}$. Figure 2 (a)
shows the on- and off-resonance photoemission spectra of $\mathrm{CeIr_{3}}$
measured at $h\nu=881.3~{}\mathrm{eV}$ and $h\nu=870~{}\mathrm{eV}$, which
reflects contributions mainly from the $\mathrm{Ce}~{}4f$ and
$\mathrm{Ir}~{}5d$ states, respectively [16]. The $\mathrm{Ce}~{}4f$ states
form a very sharp peak at the Fermi energy, indicating its considerable
contribution to this energy. Meanwhile, a weak hump structure was found to
exist around $E_{\mathrm{B}}=2.3~{}\mathrm{eV}$. This was assigned to the
contribution from the $4f^{0}$ configuration in the final state. In contrast
to the $\mathrm{Ce}~{}4f$ states, the $\mathrm{Ir}~{}5d$ states form a broad
band structure distributed in a wide energy range, with an overall spectral
shape consistent with the GGA [2] and DFT + DMFT calculations [10]. We
attempted herein to estimate the practical contributions from each orbital at
the Fermi energy. Figure 2 (b) depicts the on- and off-resonance spectra in
the very vicinity of the Fermi energy. The areas of the spectra were
normalized to the nominal numbers of the electrons of the $\mathrm{Ce}~{}4f$
and $\mathrm{Ir}~{}5d$ states in $\mathrm{CeIr_{3}}$
($n_{4f}\\!:\\!n_{5d}=0.8\\!:\\!7\times 3$). The $\mathrm{Ce}~{}4f$ and the
$\mathrm{Ir}~{}5d$ spectra intensities can be directly compared. In practice,
the occupation number of the $\mathrm{Ce}~{}4f$ states in $\mathrm{CeIr_{3}}$
is estimated to be approximately 0.8 as we shall discuss later. Both the
$\mathrm{Ir}~{}5d$ and $\mathrm{Ce}~{}4f$ states have considerable
contributions to the Fermi energy. The contribution from the
$\mathrm{Ce}~{}4f$ states at the Fermi energy was approximately 44 % of that
from the $\mathrm{Ir}~{}5d$ states; thus, the contribution from the
$\mathrm{Ce}~{}4f$ states at the Fermi energy was estimated to be
approximately 31 % of the total DOS. Note that the $\mathrm{Ce}~{}4f$ states
could form a very narrow peak at the Fermi energy, but might be smeared out by
the present energy resolution ($65~{}\mathrm{meV}$). In this case, the
substantial contribution from the $\mathrm{Ce}~{}4f$ states may be higher than
this estimation.

A further interesting point was that we found no trace of the spin-orbit-
coupled $4f_{7/2}$ multiplet, located at $E_{\mathrm{B}}\sim
280~{}\mathrm{meV}$. The $4f_{7/2}$ multiplet originates from the final state
effect, and its intensity becomes weaker than that of the peak at
$E_{\mathrm{F}}$ in the Ce compound with higher Kondo temperature
($T_{\mathrm{K}}$) [17]. Thus, the existence of the sharp peak at
$E_{\mathrm{F}}$ and the absence of the $4f_{7/2}$ multiplet peak in the
$\mathrm{Ce}~{}4f$ spectrum indicate that $T_{\mathrm{K}}$ is very high in
this compound. The spectral shape of the $\mathrm{Ce}~{}4f$ spectrum is very
similar not with the GGA calculation [2], but with the DMFT calculation [10].
This is consistent with the itinerant but correlated nature of the
$\mathrm{Ce}~{}4f$ states in $\mathrm{CeIr_{3}}$.

We tried qualitatively understand the $\mathrm{Ce}~{}3d$ core-level and XAS
spectra by analyzing them based on the single-impurity Anderson model (SIAM)
[18, 19, 20]. The Hamiltonian of the system is given as

$H=H_{1}+H_{2}$ (1)

where

$\displaystyle H_{1}$
$\displaystyle=\sum_{k,\nu}\varepsilon_{v}(k)a^{\dagger}_{v}(k,\nu)a_{v}(k,\nu)+\varepsilon^{0}_{f}\sum_{\nu}{a_{f}}^{\dagger}(\nu)a_{f}(\nu)$
(2)
$\displaystyle+\varepsilon_{d}\sum_{\mu}{a_{d}}^{\dagger}(\mu)a_{d}(\mu)+U_{ff}\sum_{\nu>\nu^{\prime}}{a_{f}}^{\dagger}(\nu)a_{f}(\nu){a_{f}}^{\dagger}(\nu^{\prime})a_{f}(\nu^{\prime})$
$\displaystyle+\frac{V}{\sqrt{N}}\sum_{k,\nu}[{a_{v}}^{\dagger}(k,\nu)a_{f}(\nu)+{a_{f}}^{\dagger}(\nu)a_{v}(k,\nu)]$
$\displaystyle-
U_{fc}\sum_{\mu,\nu}a_{d}(\mu)^{\dagger}{a_{d}}(\mu)a_{f}^{\dagger}(\nu)a_{f}(\nu)$

and

$\displaystyle H_{2}$
$\displaystyle=\frac{1}{2}\sum_{\nu_{1},\nu_{2},\nu_{3},\nu_{4}}g_{ff}(\nu_{1},\nu_{2},\nu_{3},\nu_{4}){a_{f}}^{\dagger}(\nu_{1}){a_{f}}^{\dagger}(\nu_{2})a_{f}(\nu_{3})a_{f}(\nu_{4})$
(3)
$\displaystyle+\sum_{\nu_{1},\nu_{2},\mu_{1},\mu_{2}}g_{fd}(\nu_{1},\nu_{2},\mu_{1},\mu_{2}){a_{f}}^{\dagger}(\nu_{1}){a_{d}}^{\dagger}(\mu_{1})a_{d}(\mu_{2})a_{f}(\nu_{2})$
$\displaystyle+\zeta_{d}\sum_{\mu_{1},\mu_{2}}\alpha_{\mu_{1},\mu_{2}}{a_{d}}^{\dagger}(\mu_{1})a_{d}(\mu_{2})+\zeta_{f}\sum_{\nu_{1},\nu_{2}}\beta_{\nu_{1},\nu_{2}}{a_{f}}^{\dagger}(\nu_{1})a_{f}(\nu_{2})$

$H_{1}$ describes the SIAM with the valence band level $\varepsilon_{v}(k)$,
$4f$ level $\varepsilon^{0}_{f}$ and $3d$ level $\varepsilon_{d}$.
${a_{v}}^{\dagger}(k,\nu)$, ${a_{f}}^{\dagger}(\nu)$, and
${a_{d}}^{\dagger}(\mu)$ are the electron creation operators in these states.
$k$ denotes the energy level index ($k=1,2,\cdots,N$) in the valence band.
$\nu$ and $\mu$ denote the combined indices to specify both the spin and
orbital degeneracies of the $f$ ($\nu=1,2,\cdots,N_{f}$) and $d$
($\mu=1,2,\cdots,N_{d}$) states with the numbers of the degeneracy of
$N_{f}=14$ and $N_{d}=10$, respectively. $H_{2}$ represents the multiplet
coupling effect described by Slater integrals $F^{2}$, $F^{4}$, and $F^{6}$
between $4f$ electrons ($g_{ff}$), the Slater integrals $F^{2}$, $F^{4}$,
$G^{1}$, $G^{3}$, and $G^{5}$ between the $4f$ electrons and the $3d$ core
hole ($g_{fd}$), and the spin-orbit interactions of the $3d$ states
($\zeta_{d}$) and the $4f$ states ($\zeta_{f}$). The Slater integrals were
reduced down to 80% of its atomic Hartree–Fock values, which are calculated by
using the Cowan program [21]. The valence band energies $\varepsilon_{v}(k)$
were taken as

$\varepsilon_{v}(k)=\varepsilon^{0}_{v}-\frac{W}{2}+\frac{W}{N}(k-\frac{1}{2}),\;k=1,2,\cdots
N,$ (4)

where $\varepsilon^{0}_{v}$ and $W$ are the center and width of the valence
band, respectively. The $N$ and $W$ values were $N=20$ and
$W=5~{}\mathrm{eV}$, respectively. The $3d$ core-level
($F_{3d~{}\mathrm{XPS}}(\omega)$) and $3d$ XAS
($F_{3d~{}\mathrm{XAS}}(\Omega)$) spectral functions are given as follows:

$F_{3d~{}\mathrm{XPS}}(\omega)=\sum_{f,\mu}|\braket{f}{a_{d}(\mu)}{g}|^{2}\frac{\Gamma/\pi}{(E_{g}+\omega-
E_{f})^{2}+\Gamma^{2}}$ (5)

$F_{3d~{}\mathrm{XAS}}(\Omega)=\sum_{f}|\braket{f}{T_{1}}{g}|^{2}\frac{\Gamma/\pi}{(E_{g}+\Omega-
E_{f})^{2}+\Gamma^{2}}$ (6)

where $\omega$, $\Omega$, and $T_{1}$ are the binding energy of the
photoelectrons, photon energy, and optical dipole transition from the $3d$ to
$4f$ states, respectively. Note that $\ket{f}$ in Eqs. (5) and (6) represent
$N-1$ and $N$ electron states, respectively

Figure 3: (a) $\mathrm{Ce}~{}3d$ XAS and (b) core-level spectra of
$\mathrm{CeIr_{3}}$ and their analysis results based on the SIAM. The
histogram represents the matrix elements in the calculation. The curves depict
the spectra with lifetime broadening represented in Eqs. (5) and (6).

Figure 3 shows the analysis result of the (a) $\mathrm{Ce}~{}3d$ core-level
and (b) XAS spectra. We used the following parameters here:
$E_{\mathrm{F}}-\varepsilon^{0}_{f}=-2.3~{}\mathrm{eV}$;
$U_{ff}=7.0~{}\mathrm{eV}$; $U_{fc}=10.5~{}\mathrm{eV}$; and
$V=0.5~{}\mathrm{eV}$. The ratios of the $4f^{0}$, $4f^{1}$, and $4f^{2}$
configurations in the ground state were 0.111, 0.812, and 0.077, respectively.
The occupation number of the $\mathrm{Ce}~{}4f$ states in the ground state was
$n_{4f}=0.966$. Figure 3 (a) compares the experimental $\mathrm{Ce}~{}3d$
core-level spectrum of $\mathrm{CeIr_{3}}$ and the analysis result. The
histogram represents the matrix elements in the calculation. The curves depict
the spectra with the life–time broadening represented in Eqs. (5) and (6). The
experimental spectrum was reproduced well by the SIAM using these parameters.
Figure 3(b) displays the result of the $\mathrm{Ce}~{}3d$ XAS spectrum
calculated by the same parameter set. The overall spectral shape was mostly
reproduced by the calculation [18], but some discrepancies were found between
the experiment and the calculation. In particular, the peak intensity of
$4f^{1}$ peak was weaker, and its peak position was located at the
deeper–photon energy side in the calculation when compared to the experiment.
To reproduce the XAS spectrum, we took a different parameter set, especially
for $E_{\mathrm{F}}-\varepsilon^{0}_{f}=-2.3~{}\mathrm{eV}$ [22]. This problem
is presumably caused by the presence of two different $\mathrm{Ce}$ sites in
this compound and not considered in the present analysis. Even in the case of
the XAS spectrum parameters, the occupation number of the $\mathrm{Ce}~{}4f$
states $n_{4f}$ was estimated to be 0.766; and thus, $n_{4f}$ should be
$0.8-0.9$ in both cases.

Previous studies have suggested that the DOS at $E_{\mathrm{F}}$ are dominated
by the $\mathrm{Ir}~{}5d$ states, and $\mathrm{CeIr_{3}}$ has been considered
a superconductor dominated by the $\mathrm{Ir}~{}5d$ bands. However, the
present study revealed that the $\mathrm{Ce}~{}4f$ states contribute
significantly to the DOS at $E_{\mathrm{F}}$ and thus to the
superconductivity. Therefore, $\mathrm{CeIr_{3}}$ is a $s$-wave superconductor
with a considerable $\mathrm{Ce}~{}4f$ contribution as in the case of
$\mathrm{CeRu}_{2}$ [23].

Accordingly, all these results suggest that the $\mathrm{Ce}~{}4f$ states have
an itinerant character, and a considerable contribution to the Fermi energy.
The occupation number of the $\mathrm{Ce}~{}4f$ states $n_{4f}$ in the ground
state is estimated to be $0.8-0.9$, which was larger than that in the previous
studies.

##  4 Conclusion

We studied the electronic structure of $\mathrm{CeIr_{3}}$ by the
$\mathrm{Ce}~{}3d-4f$ resonant photoelectron, $\mathrm{Ce}~{}3d$ XAS, and
$\mathrm{Ce}~{}3d$ core-level spectroscopies. All experimental data suggested
the well-hybridized nature of the $\mathrm{Ce}~{}4f$ states with the
occupation number of $n_{f}=0.8-0.9$, which was consistent with the DMFT
calculation result [10]. The comparison of the intensities at the Fermi energy
of the on- and off-resonance spectra suggested that more than 30 % of the DOS
at the Fermi energy was contributed by the $\mathrm{Ce}~{}4f$ states. These
results claimed a finite contribution of the $\mathrm{Ce}~{}4f$ states on the
superconductivity of $\mathrm{CeIr_{3}}$. Thus, $\mathrm{CeIr_{3}}$ is an
$s$-wave-type superconductor with a finite contribution from the
$\mathrm{Ce}~{}4f$ states.

The experiment was performed under Proposal Nos. 2019A3811, 2019B3811, and
2020A3811 at SPring-8 BL23SU. The present work was financially supported by
JSPS KAKENHI Grant Numbers JP16H01084, JP18K03553, JP19J20539, JP20KK0061, and
JP 22H03874.

##  Appendix A Influence of Auger processes in RPES spectra

Figure 4: Resonant photoemission spectra of $\mathrm{CeIr_{3}}$. (a) Blowup of
the $\mathrm{Ce}~{}4f_{5/2}$ XAS spectrum of $\mathrm{CeIr_{3}}$. (b) Density
plot of the RPES spectra of $\mathrm{CeIr_{3}}$ as a function of the photon
energy. (c) Comparison of the RPES spectra of $\mathrm{CeIr_{3}}$, with
intensities normalized to their maxima.

The contribution from the Auger signals in the resonant photoemission spectrum
of the $\mathrm{Ce}$-based compounds was recognized [24, 25, 26], and an
appropriate selection of excitation energies is required for the measurement
of the on-resonant photoemission spectrum. In this appendix, we describe the
influence of Auger signals in the resonant photoemission spectra. Figures 4
(a) and (b) show the $\mathrm{Ce}~{}3d_{5/2}$ XAS and the density plot of the
valence band spectra measured at $h\nu=880-884~{}\mathrm{eV}$, respectively.
The gigantic resonant enhancement of the $\mathrm{Ce}~{}4f$ signals at the
$\mathrm{Ce}~{}3d_{5/2}$ absorption edge was recognized at approximately
$h\nu=881-882~{}\mathrm{eV}$. Meanwhile, two diagonal lines were recognized in
Fig. 4(b), as indicated by the dotted lines, which depict the contributions
from the $NVV$ Auger signals. The resonant photoemission spectra measured at
$h\nu=881-882~{}\mathrm{eV}$ normalized by the peak height of each spectrum is
displayed in Fig. 4(c) to examine the impact of the Auger signals on the
$\mathrm{Ce}~{}4f$ spectral profiles. The spectral shapes of the resonance
spectra measured at $h\nu=881-881.4~{}\mathrm{eV}$ were almost identical, but
the higher-binding-energy side of the spectrum was broadened as the photon
energy was further increased. This broadening was the contribution of the
Auger signals; thus, the on-resonant spectrum should be measured at
$h\nu\lesssim 881.4~{}\mathrm{eV}$. We have used the incident photon energy of
$h\nu=881.3~{}\mathrm{eV}$ for the on-resonance spectrum to avoid the
contribution from the Auger signals in the valence-band spectrum.

## References

## References

  * [1] Sato Y J, Nakamura A, Shimizu Y, Maurya A, Homma Y, Li D, Honda F and Aoki D 2018 J. Phys. Soc. Jpn. 87 053704
  * [2] Górnicka K, Carnicom E M, Gołab S, Łapiński M, Wiendlocha B, Xie W, Kaczorowski D, Cava R J and Klimczuk T 2019 Supercond. Sci. Technol. 32 025008
  * [3] Sato Y J, Honda F, Shimizu Y, Nakamura A, Homma Y, Maurya A, Li D, Koizumi T and Aoki D 2020 Phys. Rev. B 102 174503
  * [4] Adroja D T, Bhattacharyya A, Sato Y J, Lees M R, Biswas P K, Panda K, Anand V K, Stenning G B G, Hillier A D and Aoki D 2021 Phys. Rev. B 103 104514
  * [5] Haldolaarachchige N, Schoop L, Khan M A, Huang W, Ji H, Hettiarachchilage K and Young D P 2017 J. Phys. Condens. Matter 29 475602
  * [6] Bhattacharyya A, Adroja D T, Biswas P K, Sato Y J, Lees M R, Aoki D and Hillier A D 2019 J. Phys. Condens. Matter 32 065602
  * [7] Górnicka K, Das D, Gutowska S, Wiendlocha B, Winiarski M J, Klimczuk T and Kaczorowski D 2019 Phys. Rev. B 100 214514
  * [8] Huber J 1990 Physica B 163 219
  * [9] Hakimi M and Huber J 1985 Physica B+C 135 434
  * [10] Gutowska S and Wiendlocha B 2022 J. Magn. Magn. Mater. 547 168917
  * [11] Sato Y J, Shimizu Y, Honda F, Nakamura A, Homma Y, Maurya A, Li D and Aoki D 2020 JPS Conf. Proc. 29 011007
  * [12] Saitoh Y, Fukuda Y, Takeda Y, Yamagami H, Takahashi S, Asano Y, Hara T, Shirasawa K, Takeuchi M, Tanaka T and Kitamura H 2012 J. Synchrotron Rad. 19 388
  * [13] Gunnarsson O and Schönhammer K 1983 Phys. Rev. B 28 4315
  * [14] Treske U, Khoshkhoo M S, Roth F, Knupfer M, Bauer E D, Sarrao J L, Büchner B and Koitzsch A 2014 J. Phys. Condens. Matter 26 205601
  * [15] Fuggle J C, Hillebrecht F U, Zołnierek Z, Lässer R, Freiburg C, Gunnarsson O and Schönhammer K 1983 Phys. Rev. B 27 7330
  * [16] Yeh J and Lindau I 1985 Atomic Data and Nuclear Data Tables 32 1
  * [17] Kasai S, Imada S, Yamasaki A, Sekiyama A, Iga F, Kasaya M and Suga S 2007 J. Electron Spectrosc. Relat. Phenom. 156-158 441
  * [18] Nakazawa M, Tanaka S, Uozumi T and Kotani A 1996 J. Phys. Soc. Jpn. 65 2303
  * [19] Kotani A and Shin S 2001 Rev. Mod. Phys. 73 203
  * [20] De Groot F and Kotani A 2008 Core level spectroscopy of solids (CRC press)
  * [21] Cowan R D 1981 The theory of atomic structure and spectra (Univ of California Press)
  * [22] See supplementary materials for detailed information.
  * [23] Kiss T, Kanetaka F, Yokoya T, Shimojima T, Kanai K, Shin S, Ōnuki Y, Togashi T, Zhang C, Chen C T and Watanabe S 2005 Phys. Rev. Lett. 94(5) 057001
  * [24] Cho E J, Jung R J, Choi B H, Oh S J, Iwasaki T, Sekiyama A, Imada S, Suga S, Muro T, Park J G and Kwon Y S 2003 Phys. Rev. B 67 155107
  * [25] Jung R J, Choi B H, Oh S J, Kim H D, Cho E J, Iwasaki T, Sekiyama A, Imada S, Suga S and Park J G 2003 Phys. Rev. Lett. 91 157601
  * [26] Fujimori S i 2016 J. Phys. Condens. Matter 28 153002