††thanks: Corresponding author

# Determination of the possible quantum numbers for the newly observed
$\Xi_{b}(6227)^{0}$ state

K. Azizi<EMAIL_ADDRESS>Department of Physics, University of Tehran,
North Karegar Avenue, Tehran 14395-547, Iran  Department of Physics, Doğuş
University, Acıbadem-Kadıköy, 34722 Istanbul, Turkey  School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box
19395-5531, Tehran, Iran  Y. Sarac<EMAIL_ADDRESS>Electrical and
Electronics Engineering Department, Atilim University, 06836 Ankara, Turkey
H. Sundu<EMAIL_ADDRESS>Department of Physics, Kocaeli
University, 41380 Izmit, Turkey

###### Abstract

The LHCb Collaboration recently reported the observation of a new excited
bottom baryon $\Xi_{b}(6227)^{0}$ and announced an improvement in the
measurements related to the previously observed $\Xi_{b}(6227)^{-}$ state. We
conduct an analysis for $\Xi_{b}(6227)^{0}$ state considering it as isospin
partner of the $\Xi_{b}(6227)^{-}$ resonance and possibly $1P$ or $2S$ excited
state with spin $J=\frac{3}{2}$. The corresponding masses for both
possibilities have consistent results with the experimental data, indicating
that only with the mass sum rules, one can not make exact decision on the
nature and quantum numbers of this state. To go further, the decays of these
possible excited states to $\Xi_{b}^{-}\pi^{+}$ final state are also
considered and the relevant strong coupling constants are extracted from the
light cone sum rules. The obtained decay width values support the possibility
of $\Xi_{b}(6227)^{0}$ to be the $1P$ excited state of $\Xi_{b}(5945)^{0}$
baryon.

##  I Introduction

The progress achieved in the experimental investigations has resulted in many
observations recently. Among these observations are the excited states of the
heavy baryons with a single $b$ or $c$ quark. These observations have produced
a subsequent revival of interest in the theoretical investigations of these
excited states. The contribution of the investigations of heavy baryons for
the understanding of the strong interaction has an important place. With the
presence of one heavy quark, they provide an attractive ground to study the
dynamics of light quarks. These studies provide a way not only to improve our
understanding of the confinement mechanism but also a test for the quark model
and heavy quark symmetry. Their investigations gain importance also from the
point of view that by the result of these investigations, one may check the
predictions obtained by different theoretical works on their interactions and
internal organizations and provide insights into the experimental researches.

The quark model predicted the existence of various excited states of heavy
baryons such as their orbital or radial excitations. While almost all the
ground states of the heavy baryon were observed, the number of the observed
excited states of bottom baryons is limited Aaij:2012da ; Chatrchyan:2012ni ;
Aaij:2014yka ; Aaij:2015yoy ; Aaij:2018yqz . In the last few years some
observations Aaij:2020cex ; Aaij:2018tnn ; Aaij:2019amv  were reported
indicating the observed states to be in favor of excited states of the bottom
baryons. The excited states of heavy baryons were investigated via various
theoretical models and some of these investigations dates back to even before
their observations. Among this theoretical models were the various quark
models Capstick:1986bm ; Ebert:2007nw ; Garcilazo:2007eh ; Roberts:2007ni ;
Valcarce:2008dr ; Ebert:2011kk ; Yoshida:2015tia ; Karliner:2015ema ;
Thakkar:2016dna ; Shah:2016mig ; Migura:2006ep  searching their mass spectrum
and decay mechanisms Hussain:1999sp ; Ivanov:1998wj ; Ivanov:1999bk ;
Albertus:2005zy ; Zhong:2007gp ; Hernandez:2011tx ; Liu:2012sj ; Chen:2016iyi
; Wang:2017kfr ; Nagahiro:2016nsx ; Yao:2018jmc ; Wang:2018fjm ; Chen:2018vuc
; Wang:2019uaj . Their masses and decays were investigated via QCD sum rule
method Zhu:2000py ; Wang:2010it ; Mao:2015gya ; Chen:2016phw ; Wang:2017vtv ;
Mao:2017wbz ; Aliev:2018lcs ; Cui:2019dzj ; Azizi:2020tgh , light cone QCD sum
rule method Zhu:1998ih ; Wang:2009cd ; Wang:2009ic ; Aliev:2009jt ;
Aliev:2010yx ; Aliev:2012tb ; Aliev:2014bma ; Aliev:2016xvq ; Chen:2017sci ;
Agaev:2017ywp ; Aliev:2018vye , ${}^{3}P_{0}$ model Chen:2007xf ; Ye:2017yvl ;
Ye:2017dra ; Chen:2017aqm ; Yang:2018lzg ; Guo:2019ytq ; Liang:2019aag ;
Lu:2019rtg , lattice QCD Padmanath:2013bla ; Bahtiyar:2015sga ; Bali:2015lka ;
Bahtiyar:2016dom  and bound state picture Chow:1995nw . For more details about
excited heavy baryon states see also Klempt:2009pi ; Crede:2013kia ;
Chen:2016spr  and the references therein.

In the last decade, we witnessed the observation of various excites bottom
baryon states. The $\Lambda_{b}^{*}(5912){}^{0}$,
$\Lambda_{b}^{*}(5920){}^{0}$ Aaij:2012da , $\Xi_{b}^{\prime}(5935)^{-}$,
$\Xi_{b}(5955)^{-}$ Aaij:2014yka , $\Xi_{b}(6227)^{-}$ Aaij:2018yqz ,
$\Sigma_{b}(6097)^{\pm}$ Aaij:2018tnn , $\Lambda_{b}(6146)^{0}$,
$\Lambda_{b}(6152)^{0}$ Aaij:2019amv , $\Omega_{b}(6316)^{-}$,
$\Omega_{b}(6330)^{-}$, $\Omega_{b}(6340)^{-}$ and $\Omega_{b}(6350)^{-}$
Aaij:2020cex  are among these states. The LHCb’s measurements of the mass and
width for $\Xi_{b}(6227)^{-}$ Aaij:2018yqz  were improved recently in acompany
with a new observation Aaij:2020fxj . The new resonance represented as
$\Xi_{b}(6227)^{0}$ was reported to have the following resonance parameters
Aaij:2020fxj :

$\displaystyle m_{\Xi_{b}(6227)^{0}}=6227.1^{+1.4}_{-1.5}\pm
0.5~{}\mathrm{MeV},~{}~{}~{}~{}~{}~{}~{}~{}\Gamma(\Xi_{b}(6227)^{0})=18.6^{+5.0}_{-4.1}\pm
1.4~{}\mathrm{MeV}.$

On the other hand the improved measurements of the $\Xi_{b}(6227)^{-}$ were
reported as Aaij:2020fxj :

$\displaystyle m_{\Xi_{b}(6227)^{-}}=6227.9\pm 0.8\pm
0.5~{}\mathrm{MeV},~{}~{}~{}~{}~{}~{}~{}~{}\Gamma(\Xi_{b}(6227)^{-})=19.9\pm
2.1\pm 1.5~{}\mathrm{MeV}.$

After the first observation of $\Xi_{b}(6227)^{-}$ Aaij:2018yqz , to enlighten
the quantum numbers of the state, it was investigated via various theoretical
models. In Ref. Wang:2018fjm  its strong decay was considered in constituent
quark model, and possible spin parity assignment was suggested as
$J^{P}=\frac{3}{2}^{-}$ or $J^{P}=\frac{5}{2}^{-}$. Its two-body strong decay
investigation indicted its being a candidate of the $P$-wave state with
$J^{P}=\frac{3}{2}^{-}$ or $J^{P}=\frac{5}{2}^{-}$ Chen:2018orb . QCD sum
rules calculation given in Ref. Aliev:2018lcs  supported its being $1P$
angular-orbital excited state of the $\Xi_{b}(5955)^{-}$ baryon with quantum
numbers $J^{P}=\frac{3}{2}^{-}$. The Ref. Cui:2019dzj  analyzed the state via
the QCD sum rule method within the framework of heavy quark effective theory
and interpreted its quantum numbers as $J^{P}=\frac{3}{2}^{-}$. Refs.
Yang:2019cvw ; Yang:2020zrh  applied the light cone QCD sum rules to study its
decays to ground-state bottom baryons and a meson considering the heavy quark
effective theory. The consistency of the quantum numbers of the
$\Xi_{b}(6227)^{-}$ with a $P$-wave state with $J^{P}=\frac{1}{2}^{-}$,
$\frac{3}{2}^{-}$ and $\frac{5}{2}^{-}$ was also indicated by mass calculation
in the quark-diquark picture Faustov:2020gun ; Faustov:2018vgl . In another
work made in the quark-diquark picture, the preferable spin of this particle
was stated to be $J^{P}=\frac{3}{2}^{-}$ Jia:2019bkr . This state was also
interpreted considering the molecular point of view Yu:2018yxl ; Huang:2018bed
; Nieves:2019jhp ; Zhu:2020lza . These analyses indicate that, to distinguish
different interpretations, we need more investigations providing a deeper
understanding of the substructure and quantum numbers of the considered
resonance.

After the first report of the $\Xi_{b}(6227)^{-}$ Aaij:2018yqz , we performed
a QCD sum rule analyses for this state considering different possible
scenarios for its quantum numbers such as $1P/2S$ excitations of the
$\Xi_{b}^{-}$ and $\Xi_{b}^{\prime}(5935)^{-}$ baryons with
$J^{P}=\frac{1}{2}$ or $1P/2S$ excitations of the $\Xi_{b}(5955)^{-}$ with
$J^{P}=\frac{3}{2}$ Aliev:2018lcs . For all considered cases, we calculated
the corresponding masses using QCD sum rules. All obtained results showed
consistency with the experimental one, which directed us to a further
investigation via their possible decays to $\Lambda_{b}^{0}K^{-}$ and
$\Xi_{b}^{-}\pi$. The obtained decay widths supported the $\Xi_{b}(6227)^{-}$
to be a $1P$ angular-orbital excited state of $\Xi_{b}(5955)^{-}$ with
$J^{P}=\frac{3}{2}^{-}$. In this work, we consider the new observation of
$\Xi_{b}(6227)^{0}$ and its interpretation as an isospin partner of the
$\Xi_{b}(6227)^{-}$ by the LHCb Collaboration Aaij:2020fxj . By considering
also our previous assignment of the $\Xi_{b}(6227)^{-}$ Aliev:2018lcs , we
take into account two possible quantum numbers for $\Xi_{b}(6227)^{0}$ state
as $1P$ or $2S$ excited state of the ground state $\Xi_{b}(5945)^{0}$ with
$J=\frac{3}{2}$. We investigate their masses and decays into the ground states
$\Xi_{b}(5945)^{-}$ and $\pi^{+}$ to shed light on the quantum numbers of
$\Xi_{b}(6227)^{0}$. To this end, we apply two-point QCD sum rule
Shifman:1978bx ; Shifman:1978by ; Ioffe81  and its extension light cone QCD
sum rule (LCSR) Braun:1988qv ; Balitsky:1989ry ; Chernyak:1990ag  for masses
and the coupling constants, respectively. In calculations, a proper
interpolating current for the state is considered based on the quark content
of the state and the considered quantum numbers. We aim to assign probable
quantum numbers to the state based on the comparison of the obtained masses
and decay widths with experimental values.

This article is structured as follows. Section II is set aside for the details
of the QCD sum rule calculations for the mentioned spectroscopic parameters.
In section III the sum rules for the transition of the considered candidate
states to the $\Xi_{b}^{-}$ and $\pi^{+}$ final states are presented. The last
section is devoted to a summary, which also gives a comparison of the obtained
results with experimental data.

##  II QCD sum rule calculations for spectral properties

The spectral properties such as masses and current coupling constants for the
considered candidates $1P$ and $2S$ excited states of $\Xi_{b}(5945)^{0}$ are
calculated in this section. The calculation is performed via the following
two-point correlation function:

$\Pi_{\mu\nu}(k)=i\int d^{4}xe^{ik\cdot x}\langle
0|\mathcal{T}\\{\eta_{\mu}(x){\bar{\eta}_{\nu}}(0)\\}|0\rangle,$ (3)

where $k$ is the four-momentum of the particle and $\eta_{\mu}$ represents the
interpolating current of the considered states, and it carries the quantum
numbers of the considered states. This interpolating current is written in
terms of the quark fields considering the quark content of the state. For the
state of interest, the interpolating current has the following form:

$\displaystyle\eta_{\mu}=\sqrt{\frac{2}{3}}\epsilon^{abc}\Big{\\{}(u^{aT}C\gamma_{\mu}s^{b})b^{c}+(s^{aT}C\gamma_{\mu}b^{b})u^{c}+(b^{aT}C\gamma_{\mu}u^{b})s^{c}\Big{\\}}~{},$
(4)

with $a$, $b$, and $c$ corresponding to the color indices. The operator $C$ is
the charge conjugation.

The above correlator can either be calculated in terms of the hadronic degrees
of freedom such as masses, current coupling constants or in terms of the QCD
degrees of freedom such as quark-gluon condensates, the strong coupling
constant, quark masses, etc. In the analyses, both ways of calculations are
performed, and obtained results are matched considering the coefficients of
the same Lorentz structures for both sides. These matches lead to the QCD sum
rules for the interested physical quantities, which are masses and the
corresponding current coupling constants for this section.

To calculate the correlator in terms of hadronic degrees of freedom, we treat
the interpolating currents as operators annihilating or creating the hadrons.
To proceed in the calculations, we insert complete sets of hadronic states
between the interpolating currents that have the same quantum numbers as the
current itself. This process is followed by integration over $x$, which
results in

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Had}}(k)$ $\displaystyle=$
$\displaystyle\frac{\langle
0|\eta_{\mu}|\Xi_{b}^{0}(k,s;\frac{3}{2})\rangle\langle\Xi_{b}^{0}(k,s;\frac{3}{2})|\bar{\eta}_{\nu}|0\rangle}{m^{2}-k^{2}}+\frac{\langle
0|\eta_{\mu}|\widetilde{\Xi}_{b}^{0}(k,s;\frac{3}{2})\rangle\langle\widetilde{\Xi}_{b}^{0}(k,s;\frac{3}{2})|\bar{\eta}_{\nu}|0\rangle}{\widetilde{m}^{2}-k^{2}}+\ldots,$
(5)

for $1P$ excitation and

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Had}}(k)$ $\displaystyle=$
$\displaystyle\frac{\langle
0|\eta_{\mu}|\Xi_{b}^{0}(k,s;\frac{3}{2})\rangle\langle\Xi_{b}^{0}(k,s;\frac{3}{2})|\bar{\eta}_{\nu}|0\rangle}{m^{2}-k^{2}}+\frac{\langle
0|\eta_{\mu}|\Xi_{b}^{0}{}^{\prime}(k,s;\frac{3}{2})\rangle\langle\Xi_{b}^{0}{}^{\prime}(k,s;\frac{3}{2})|\bar{\eta}_{\nu}|0\rangle}{m^{\prime
2}-k^{2}}+\ldots,$ (6)

for $2S$ excitation. Here we consider the possibilities for the state under
study as $1P$ and $2S$ separately and analyze the results accordingly. In the
last results $m$, $\tilde{m}$ and $m^{\prime}$ represent the masses of the
spin-3/2 ground ($\Xi_{b}(5945)^{0}$ in PDG), $1P$ and $2S$ states,
respectively. Here, $|\Xi_{b}^{0}(k,s;\frac{3}{2})\rangle$,
$|\widetilde{\Xi}_{b}^{0}(k,s;\frac{3}{2})\rangle$ and
$|\Xi_{b}^{0}{}^{\prime}(k,s;\frac{3}{2})\rangle$ also respectively denote the
ground, $1P$ and $2S$ states with $J=\frac{3}{2}$ and dots are used for the
contributions of the higher states and continuum. The matrix elements,
$\langle 0|\eta_{\mu}|\Xi_{b}^{0}(k,s;\frac{3}{2})\rangle$, $\langle
0|\eta_{\mu}|\widetilde{\Xi}_{b}^{0}(k,s;\frac{3}{2})\rangle$ and $\langle
0|\eta_{\mu}|\Xi_{b}^{0}{}^{\prime}(k,s;\frac{3}{2})\rangle$ are defined in
terms of the current coupling constants, $\lambda$ , $\tilde{\lambda}$ and
$\lambda^{\prime}$ as

$\displaystyle\langle 0|\eta_{\mu}|\Xi_{b}^{0}(k,s;\frac{3}{2})\rangle$
$\displaystyle=$ $\displaystyle\lambda u_{\mu}(k,s),$ $\displaystyle\langle
0|\eta_{\mu}|\widetilde{\Xi}_{b}^{0}(k,s;\frac{3}{2})\rangle$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}\gamma_{5}u_{\mu}(k,s),$
$\displaystyle\langle
0|\eta_{\mu}|\Xi_{b}^{0}{}^{\prime}(k,s;\frac{3}{2})\rangle$ $\displaystyle=$
$\displaystyle\lambda^{\prime}u_{\mu}(k,s),$ (7)

where $u_{\mu}(k,s)$ is the Rarita-Schwinger spinor.

Here we should note that the current $\eta_{\mu}$ couples not only to the
obove mentioned spin–3/2 states, but also to the spin–1/2 states of both
parities. To get the contributions of the spin-3/2 states, we should remove
the unwanted pollution coming from the spin-1/2 particles. The general form of
the matrix element of $\eta_{\mu}$ between the spin-1/2 and vacuum states can
be parameterize as

$\displaystyle{\langle}0|\eta_{\mu}|\frac{1}{2}(k,s){\rangle}=\left(C_{1}k_{\mu}+C_{2}\gamma_{\mu}\right)u(k,s),$
(8)

where $C_{1}$ and $C_{2}$ are some constants. Now, we multiply both sides of
the above equation with $\gamma^{\mu}$, and use the condition
$\gamma^{\mu}\eta_{\mu}=0$, to get $C_{1}$ in terms of $C_{2}$. Hence,

$\displaystyle{\langle}0|\eta_{\mu}|\frac{1}{2}^{+}(k,s){\rangle}=C_{2}\left(-\frac{4}{m_{\frac{1}{2}^{+}}}k_{\mu}+\gamma_{\mu}\right)u(k,s),$
(9)

for the positive parity and

$\displaystyle{\langle}0|\eta_{\mu}|\frac{1}{2}^{-}(k,s){\rangle}=C_{2}\gamma_{5}\left(-\frac{4}{m_{\frac{1}{2}^{-}}}k_{\mu}+\gamma_{\mu}\right)u(k,s),$
(10)

for the negative parity states are obtained. From these equations we see that
the unwanted contributions corresponding to the spin-1/2 states with both
parities are proportional to either $k_{\mu}$ or $\gamma_{\mu}$. To remove
these contributions, first we will order the Dirac matrices as
$\gamma_{\mu}\hbox to0.0pt{/\hss}{k}\gamma_{\nu}$ and then set to zero the
terms with $\gamma_{\mu}$ in the beginning and $\gamma_{\nu}$ at the end and
those terms which are proportional to $k_{\mu}$ or $k_{\nu}$. But before that,
using these matrix elements and the summation over spins as

$\displaystyle\sum_{s}u_{\mu}(k,s)\bar{u}_{\nu}(k,s)$ $\displaystyle=$
$\displaystyle-(\\!\not\\!{k}+m)\Big{[}g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2k_{\mu}k_{\nu}}{3m^{2}}+\frac{k_{\mu}\gamma_{\nu}-k_{\nu}\gamma_{\mu}}{3m}\Big{]},$
(11)

in Eqs. (5) and (6) the following results are obtained:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Had}}(k)$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{k^{2}-m^{2}}(\\!\not\\!{k}+m)\Big{[}g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2k_{\mu}k_{\nu}}{3m^{2}}+\frac{k_{\mu}\gamma_{\nu}-k_{\nu}\gamma_{\mu}}{3m}\Big{]}$
(12) $\displaystyle-$
$\displaystyle\frac{\widetilde{\lambda}^{2}}{k^{2}-\widetilde{m}^{2}}(\\!\not\\!{k}-\widetilde{m})\Big{[}g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2k_{\mu}k_{\nu}}{3\widetilde{m}^{2}}+\frac{k_{\mu}\gamma_{\nu}-k_{\nu}\gamma_{\mu}}{3\widetilde{m}}\Big{]}+\ldots,$
$\displaystyle\Pi_{\mu\nu}^{\mathrm{Had}}(k)$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{k^{2}-m^{2}}(\\!\not\\!{k}+m)\Big{[}g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2k_{\mu}k_{\nu}}{3m^{2}}+\frac{k_{\mu}\gamma_{\nu}-k_{\nu}\gamma_{\mu}}{3m}\Big{]}$
(13) $\displaystyle-$
$\displaystyle\frac{\lambda^{\prime}{}^{2}}{k^{2}-m^{\prime}{}^{2}}(\\!\not\\!{k}+m^{\prime})\Big{[}g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2k_{\mu}k_{\nu}}{3m^{\prime}{}^{2}}+\frac{k_{\mu}\gamma_{\nu}-k_{\nu}\gamma_{\mu}}{3m^{\prime}}\Big{]}+\ldots,$

where the first and second equations represent the results for the $1S+1P$ and
$1S+2S$ possiblities, respectively.

As for the calculation of the same correlator, Eq. (3), in terms of QCD
degrees of freedom, the interpolating current is placed inside it explicitly,
and operator product expansion (OPE) is applied. First, the possible
contractions between quark fields are performed, and the result is converted
to the ones given in terms of the light and heavy quark propagators. The
corresponding propagators in coordinate space are used, and via Fourier
transformation, the calculations proceed in momentum space, and finally, the
results in terms of QCD degrees of freedom are achieved.

The results obtained from the hadronic and QCD sides are matched considering
the coefficients of the same Lorentz structures in both sides after the
application of Borel transformation to both sides. Borel transformation
ensures the suppression of the contributions coming from the continuum and
higher states. The mentioned structures in the calculations are the
$\\!\not\\!{k}g_{\mu\nu}$ and $g_{\mu\nu}$. These structures are selected
after the ordering of the Dirac matrices as stated above and removing the
unwanted spin-1/2 pollution. The final results for QCD sum rules obtained from
the matches of the coefficients of these structures are as follows:

$\displaystyle\lambda^{2}e^{-\frac{m^{2}}{M^{2}}}+\widetilde{\lambda}^{2}(\lambda^{\prime}{}^{2})e^{-\frac{\widetilde{m}^{2}(m^{\prime}{}^{2})}{M^{2}}}$
$\displaystyle=$ $\displaystyle\Pi^{\mathrm{QCD}}_{1},$ $\displaystyle
m\lambda^{2}e^{-\frac{m^{2}}{M^{2}}}\mp\widetilde{m}(m^{\prime})\widetilde{\lambda}^{2}(\lambda^{\prime}{}^{2})e^{-\frac{\widetilde{m}^{2}(m^{\prime}{}^{2})}{M^{2}}}$
$\displaystyle=$ $\displaystyle\Pi^{\mathrm{QCD}}_{2},$ (14)

where $-$ and $+$ signs in the last equation are used for the $1P$ excited
$\widetilde{\Xi}_{b}^{0}$ and $2S$ excited $\Xi^{0}_{b}{}^{\prime}$ states,
respectively. The functions $\Pi^{\mathrm{QCD}}_{1,2}$ represent the Borel
transformed coefficients of the structures $\\!\not\\!{k}g_{\mu\nu}$ and
$g_{\mu\nu}$ in the QCD side. These functions are lengthy, hence, as an
example, we only present the explicit expression of the
$\Pi^{\mathrm{QCD}}_{1}$ function in the Appendix.

To get the numerical results for quantities using the obtained sum rules, some
input parameters are required, such as the mass of the ground state
$\Xi_{b}(5945)^{0}$, quark masses, quark-gluon condensates, etc. The values of
these quantities are gathered in Table 1. Note that, in this table, we also
present some other input parameters that are required in the next section.

Parameters | Values
---|---
$m_{\Xi_{b}(5945)^{0}}$ |  $5952.3\pm 0.6~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{\Xi_{b}^{-}}$ |  $5797.0\pm 0.6~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{b}$ |  $4.18^{+0.04}_{-0.03}~{}\mathrm{GeV}$ Zyla:2020zbs
$m_{s}$ |  $128^{+12}_{-4}~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{u}$ |  $2.16^{+0.49}_{-0.26}~{}\mathrm{MeV}$ Zyla:2020zbs
$\lambda_{\Xi_{b}}$ |  $0.054\pm 0.012~{}\mathrm{GeV}^{3}$ Azizi:2016dmr
$\langle\bar{q}q\rangle$ |  $(-0.24\pm 0.01)^{3}$ $\mathrm{GeV}^{3}$ Belyaev:1982sa
$\langle\bar{s}s\rangle$ |  $0.8\langle\bar{q}q\rangle$ Belyaev:1982sa
$m_{0}^{2}$ |  $(0.8\pm 0.1)$ $\mathrm{GeV}^{2}$ Belyaev:1982sa
$\langle g_{s}^{2}G^{2}\rangle$ |  $4\pi^{2}(0.012\pm 0.004)$ $~{}\mathrm{GeV}^{4}$Belyaev:1982cd
$\Lambda$ |  $(0.5\pm 0.1)$ $\mathrm{GeV}$ Chetyrkin:2007vm
Table 1: Some input parameters used in the calculations of masses and coupling
constants.

In addition to the parameters given in Table 1, there are two more auxiliary
parameters required in the analyses. These are the threshold parameter $s_{0}$
and the Borel parameter $M^{2}$. The $s_{0}$ has a relation to the energy of
the excited state with the same quantum numbers of state under question.
However, due to lacking knowledge for these excited states, the corresponding
working intervals are determined considering the standard prescriptions of the
method: namely the dominance of the first two resonances over other excited
states and the convergence of OPE. Besides, the weak dependencies of the
results on the auxiliary parameters are applied. Following these criteria, the
subsequent working intervals are obtained:

$\displaystyle 46~{}\mbox{GeV}^{2}\leq s_{0}\leq 49~{}\mbox{GeV}^{2},$ (15)

and

$\displaystyle 6~{}\mbox{GeV}^{2}\leq M^{2}\leq 9~{}\mbox{GeV}^{2}.$ (16)

Adopting these intervals for $s_{0}$ and $M^{2}$, the masses and the current
coupling constants are obtained for the candidate $1P$ and $2S$ states as

$\displaystyle\tilde{m}=6225.47\pm
106.29~{}\mathrm{MeV},~{}~{}~{}~{}~{}~{}\tilde{\lambda}=0.081\pm
0.010~{}\mathrm{GeV}^{3},$ (17)

and

$\displaystyle m^{\prime}=6225.47\pm
106.29~{}\mathrm{MeV},~{}~{}~{}~{}~{}~{}\lambda^{\prime}=0.546\pm
0.039~{}\mathrm{GeV}^{3},$ (18)

respectively. Note that in these calculations, the ground state mass is used
as given in Table 1 as an input. The errors present in the results arise from
the uncertainties of the input parameters and the auxiliary ones as well. To
depict the dependencies of the results on the auxiliary parameters, we provide
the mass plots of candidate $1P$ state as functions of $M^{2}$ and $s_{0}$ in
Fig. 1. This figure shows the weak variations of the results with respect to
the auxiliary parameters in their working intervals.

Figure 1: Left: The mass $\widetilde{m}$ obtained for the $1P$ excitation of
$\Xi_{b}(5945)^{0}$ baryon vs Borel parameter $M^{2}$. Right: The mass
$\widetilde{m}$ obtained for the $1P$ excitation of $\Xi_{b}(5945)^{0}$ baryon
vs threshold parameter $s_{0}$.

These results indicate that the masses of both the $1P$ and $2S$ excitations
are equal and consistent with the experimentally observed mass value of the
$\Xi_{b}(6227)^{0}$ state. Therefore, to further deepen our prediction, we
extend our analyses to the decays of these candidate states to
$\Xi_{b}^{-}\pi^{+}$ final state. The details for this part of the
investigation are presented in Section III.

##  III Transitions of the excited $\Xi_{b}$ states to $\Xi_{b}^{-}\pi^{+}$

The mass sum rules performed for the experimentally observed
$\Xi_{b}(6227)^{0}$ state in the previous section led to the same mass values
for candidates $1P$ and $2S$ excitations of the $\Xi_{b}(5945)^{0}$ state, but
to different current coupling constants. This prevented us to make an
assignment on the nature of the discovered state. Therefore, to deepen the
analyses and get the exact quantum number predictions, we provide further
investigations that include the decays of the considered candidate states to
the final states $\Xi_{b}^{-}$ and $\pi^{+}$. Here $\Xi_{b}^{-}$ denotes the
ground state spin-1/2 particle, which we will denote it by
$|\Xi_{b}^{-}(p,s;\frac{1}{2})\rangle$ in the following. In this section, the
details of these analyses are presented. To compute the decay widths, we need
to calculate the strong coupling constants for the considered decays, and we
shall calculate them using LCSR starting with the following correlation
function

$\Pi_{\mu}(p,q)=i\int d^{4}xe^{ip\cdot
x}\langle\pi^{+}(q)|\mathcal{T}\\{\eta_{\Xi_{b}^{-}}(x)\bar{\eta}_{\mu}(0)\\}|0\rangle,$
(19)

where $|\pi^{+}(q)\rangle$ in the correlator represents the on-shell
$\pi$-meson state with momentum $q$, $p$ is the momentum of the final baryon
and we assign the momentum $p^{\prime}=p+q$ to the initial particle. In the
calculations, the interpolating current $\eta_{\mu}$ from Eq. (4) for
spin-$\frac{3}{2}$ initial state is used. In this section, both the possible
cases $1P$ and $2S$ are considered for the initial state, as well. The
spin-$\frac{1}{2}$ single heavy $\Xi_{b}^{-}$ baryon in the final state
belongs to the $\bar{3}_{F}$ representation of SU(3) flavor in the family of
the single heavy bottom baryons (for more information see for instance Ref.
Aliev:2010yx ). The current interpolating the spin-$\frac{1}{2}$ $\Xi_{b}^{-}$
state has the following form:

$\displaystyle\eta_{\Xi_{b}^{-}}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\epsilon^{abc}\Big{\\{}2(d^{aT}Cs^{b})\gamma_{5}b^{c}+(d^{aT}Cb^{b})\gamma_{5}s^{c}+(b^{aT}Cs^{b})\gamma_{5}d^{c}+2\beta(d^{aT}C\gamma_{5}s^{b})b^{c}$
(20) $\displaystyle+$
$\displaystyle\beta(d^{aT}C\gamma_{5}b^{b})s^{c}+\beta(b^{aT}C\gamma_{5}s^{b})d^{c}\Big{\\}}~{},$

where $\beta$ is a mixing parameter. This parameter is used to combine the two
possible current to make a general interpolating field. For more information
about how the current in Eq. (20) is obtained using all the quantum numbers of
the $\Xi_{Q}$ ($Q=b$ or $c$), for instance, see Ref. Aliev:2018ube . The
working region for $\beta$, as an auxiliary parameter, is obtained demanding
the condition that the physical quantities depend on it relatively weakly. In
order to scan all regions, we set $\beta=\tan\theta$ and plot the physical
quantities in terms of $\cos\theta$ in the interval $[-1,1]$. Our analyses
show that in the following working windows, the results weakly depend on the
$\cos\theta$:

$\displaystyle-1\leq\cos\theta\leq-0.3~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}0.3\leq\cos\theta\leq
1~{},$ (21)

which will be used in extraction of the numerical values for the quantities
under study in this section.

To proceed in the calculations, we first insert complete sets of hadronic
states having the same quantum numbers with the interpolating currents into
the correlator to attain the hadronic sides as

$\displaystyle\Pi^{\mathrm{Had}}_{\mu}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle
0|\eta_{\Xi_{b}^{-}}|\Xi_{b}^{-}(p,s;\frac{1}{2})\rangle}{p^{2}-m_{\Xi_{b}^{-}}^{2}}\langle\pi^{+}(q)\Xi_{b}^{-}(p,s;\frac{1}{2})|\Xi_{b}^{0}(p^{\prime},s^{\prime};\frac{3}{2})\rangle\frac{\langle\Xi_{b}^{0}(p^{\prime},s^{\prime};\frac{3}{2})|\bar{\eta}_{\mu}|0\rangle}{p^{\prime
2}-m^{2}}$ (22) $\displaystyle+$ $\displaystyle\frac{\langle
0|\eta_{\Xi_{b}^{-}}|\Xi_{b}^{-}(p,s;\frac{1}{2})\rangle}{p^{2}-m_{\Xi_{b}^{-}}^{2}}\langle\pi^{+}(q)\Xi_{b}^{-}(p,s;\frac{1}{2})|\widetilde{\Xi}_{b}^{0}(p^{\prime},s^{\prime};\frac{3}{2})\rangle\frac{\langle\widetilde{\Xi}_{b}^{0}(p^{\prime},s^{\prime};\frac{3}{2})|\bar{\eta}_{\mu}|0\rangle}{p^{\prime
2}-\widetilde{m}^{2}}+\ldots,$

when we consider the $1P$ state, and

$\displaystyle\Pi^{\mathrm{Had}}_{\mu}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle
0|\eta_{\Xi_{b}^{-}}|\Xi_{b}^{-}(p,s;\frac{1}{2})\rangle}{p^{2}-m_{\Xi_{b}^{-}}^{2}}\langle\pi^{+}(q)\Xi_{b}^{-}(p,s;\frac{1}{2})|\Xi_{b}^{0}(p^{\prime},s^{\prime};\frac{3}{2})\rangle\frac{\langle\Xi_{b}^{0}(p^{\prime},s^{\prime};\frac{3}{2})|\bar{\eta}_{\mu}|0\rangle}{p^{\prime
2}-m^{2}}$ (23) $\displaystyle+$ $\displaystyle\frac{\langle
0|\eta_{\Xi_{b}^{-}}|\Xi_{b}^{-}(p,s;\frac{1}{2})\rangle}{p^{2}-m_{\Xi_{b}^{-}}^{2}}\langle\pi^{+}(q)\Xi_{b}^{-}(p,s;\frac{1}{2})|\Xi_{b}^{0}{}^{\prime}(p^{\prime},s^{\prime};\frac{3}{2})\rangle\frac{\langle\Xi_{b}^{0}{}^{\prime}(p^{\prime},s^{\prime};\frac{3}{2})|\bar{\eta}_{\mu}|0\rangle}{p^{\prime
2}-m^{\prime}{}^{2}}+\ldots,$

when we consider the $2S$ state. As is seen, we represent the ground state,
$1P$ and $2S$ excited states of spin-$\frac{3}{2}$ baryon, with the
$\Xi_{b}^{0}$, $\widetilde{\Xi}_{b}^{0}$ and $\Xi_{b}^{0}{}^{\prime}$,
respectively as in the previous section. Here, $\ldots$ corresponds to the
contributions coming from the higher states and continuum again. Besides the
matrix elements given in Eq. (7), we need to define additional matrix
elements:

$\displaystyle\langle
0|\eta_{\Xi_{b}^{-}}|\Xi_{b}^{-}(p,s;\frac{1}{2})\rangle$ $\displaystyle=$
$\displaystyle\lambda_{\Xi_{b}^{-}}u(p,s),$
$\displaystyle\langle\pi^{+}(q)\Xi_{b}^{-}(p,s;\frac{1}{2})|\Xi_{b}^{0}(p^{\prime},s^{\prime};\frac{3}{2})\rangle$
$\displaystyle=$ $\displaystyle
g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\bar{u}(p,s)u_{\mu}(p^{\prime},s^{\prime})q^{\mu},$
$\displaystyle\langle\pi^{+}(q)\Xi_{b}^{-}(p,s;\frac{1}{2})|\widetilde{\Xi}_{b}^{0}(p^{\prime},s^{\prime};\frac{3}{2})\rangle$
$\displaystyle=$ $\displaystyle
g_{\widetilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}\bar{u}(p,s)\gamma_{5}u_{\mu}(p^{\prime},s^{\prime})q^{\mu},$
$\displaystyle\langle\pi^{+}(q)\Xi_{b}^{-}(p,s;\frac{1}{2})|\Xi_{b}^{0}{}^{\prime}(p^{\prime},s^{\prime};\frac{3}{2})\rangle$
$\displaystyle=$ $\displaystyle
g_{\Xi_{b}^{0}{}^{\prime}\Xi_{b}^{-}\pi^{+}}\bar{u}(p,s)u_{\mu}(p^{\prime},s^{\prime})q^{\mu},$
(24)

where $g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}$,
$g_{\widetilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}$ and
$g_{\Xi_{b}^{0}{}^{\prime}\Xi_{b}^{-}\pi^{+}}$ are the strong coupling
constants: the fundamental parameters that help us estimate the widths of the
considering strong decays. In the following, we aim to get sum rules for these
strong coupling constants and obtain their numerical values. Using these new
matrix elements inside Eqs. (22) and (23) as well as the summations over spins
of $\frac{3}{2}$ particles given in Eq. (11) and summations over spins of
$\frac{1}{2}$ states as

$\displaystyle\sum_{s}u(p,s)\bar{u}(p,s)$ $\displaystyle=$
$\displaystyle(\\!\not\\!{p}+m),$ (25)

we obtain the hadronic sides as

$\displaystyle\Pi_{\mu}^{\mathrm{Had}}(p,q)$ $\displaystyle=$
$\displaystyle-\frac{g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda}{(p^{2}-m_{\Xi_{b}^{-}}^{2})(p^{\prime
2}-m{}^{2})}q^{\alpha}(\not{p}+m_{\Xi_{b}^{-}})\left(\not{p}^{\prime}+m\right)T_{\alpha\mu}+\frac{g_{\widetilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\widetilde{\lambda}}{(p^{2}-m_{\Xi_{b}^{-}}^{2})(p^{\prime}{}^{2}-\widetilde{m}^{2})}$
(26) $\displaystyle\times$ $\displaystyle
q^{\alpha}(\not{p}+m_{\Xi_{b}^{-}})\gamma_{5}\left(\not{p}^{\prime}+\widetilde{m}\right)T_{\alpha\mu}\gamma_{5}+\ldots,$
$\displaystyle\Pi_{\mu}^{\mathrm{Had}}(p,q)$ $\displaystyle=$
$\displaystyle-\frac{g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda}{(p^{2}-m_{\Xi_{b}^{-}}^{2})(p^{\prime
2}-m^{2})}q^{\alpha}(\not{p}+m_{\Xi_{b}^{-}})\left(\not{p}^{\prime}+m\right)T_{\alpha\mu}-\frac{g_{\Xi_{b}^{0}{}^{\prime}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda^{\prime}}{(p^{2}-m_{\Xi_{b}^{-}}^{2})(p^{\prime
2}-m^{\prime}{}^{2})}$ (27) $\displaystyle\times$ $\displaystyle
q^{\alpha}(\not{p}+m_{\Xi_{b}^{-}})\left(\not{p}^{\prime}+m^{\prime}\right)T_{\alpha\mu}+\ldots,$

where the expression $T_{\alpha\mu}$ is given as

$\displaystyle
T_{\alpha\mu}(p)=g_{\alpha\mu}-\frac{1}{3}\gamma_{\alpha}\gamma_{\mu}-\frac{2}{3m^{2}}p_{\alpha}p_{\mu}+\frac{1}{3m}\left[p_{\alpha}\gamma_{\mu}-p_{\mu}\gamma_{\alpha}\right].$
(28)

With an application of double Borel transformation with respect to
$p^{\prime}{}^{2}$ and $p^{2}$ with corresponding Borel parameters $M_{1}^{2}$
and $M_{2}^{2}$, the results of the hadronic sides turn into

$\displaystyle{\cal B}\Pi_{\mu}^{\mathrm{Had}}(p,q)$ $\displaystyle=$
$\displaystyle-g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda
e^{-\frac{m^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}}q^{\alpha}(\not{p}+m_{\Xi_{b}^{-}})\left(\not{p}^{\prime}+m\right)T_{\alpha\mu}$
(29) $\displaystyle+$ $\displaystyle
g_{\widetilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\widetilde{\lambda}e^{-\frac{\widetilde{m}{}^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}}q^{\alpha}(\not{p}+m_{\Xi_{b}^{-}})\gamma_{5}\left(\not{p}^{\prime}+\widetilde{m}\right)T_{\alpha\mu}\gamma_{5}.$
$\displaystyle{\cal B}\Pi_{\mu}^{\mathrm{Had}}(p,q)$ $\displaystyle=$
$\displaystyle-g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda
e^{-\frac{m^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}}q^{\alpha}(\not{p}+m_{\Xi_{b}^{-}})\left(\not{p}^{\prime}+m\right)T_{\alpha\mu}$
(30) $\displaystyle-$ $\displaystyle
g_{\Xi_{b}^{0}{}^{\prime}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda^{\prime}e^{-\frac{m^{\prime
2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}}q^{\alpha}(\not{p}+m_{\Xi_{b}^{-}})\left(\not{p}^{\prime}+m^{\prime}\right)T_{\alpha\mu},$

where, to represent the Borel transformed results of the correlation function,
we used ${\cal B}\Pi_{\mu}^{\mathrm{Had}}(p,q)$. Among various Lorentz
structures we consider $\not{p}q_{\mu}$ and $\not{q}q_{\mu}$ to evaluate the
corresponding strong coupling constants. We should also note here that, before
selection of the structures, we order the dirac matrices as
$\not{q}\not{p}\gamma_{\mu}$ and remove the spin-1/2 pollution (in this case
only in the initial states), which couple to the spin-3/2 current, from a
similar way explained in the previous section.

To calculate the QCD side of the calculations, we use the related
interpolating currents explicitly inside the correlator and make the possible
contractions between quark fields via Wick’s theorem. As a result the
expressions are obtained in terms of the heavy and light quark propagators and
the matrix elements of the remaining quark fields after contractions, such as
$\langle\pi^{+}(q)|\bar{q}(x)\Gamma q(y)|0\rangle$ or
$\langle\pi^{+}(q)|\bar{q}(x)\Gamma G_{\mu\nu}q(y)|0\rangle$. The $\Gamma$ and
$G_{\mu\nu}$ present in these matrix elements are the full set of Dirac
matrices and gluon field strength tensor, respectively. These matrix elements
are described in terms of $\pi$-meson distribution amplitudes (DAs), and their
usage in the calculations leads to the nonperturbative contributions. For the
explicit expressions of these matrix elements see the Refs. Ball:2006wn ;
Belyaev:1994zk ; Ball:2004ye ; Ball:2004hn . The results obtained from this
side are quite long mathematical expressions, and therefore we will not write
them here. The coefficients of the same structures from both sides are again
matched, and Borel transformation and continuum subtraction are applied to the
QCD side. After these procedures, the QCD sum rules giving the corresponding
strong coupling constants of the considered decays are attained as

$\displaystyle{\cal B}\Pi_{1}^{\mathrm{QCD,tr}}$ $\displaystyle=$
$\displaystyle-
g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda\frac{[(m+m_{\Xi_{b}^{-}})(2m^{2}+m_{\Xi_{b}^{-}}^{2})-m_{\pi^{+}}^{2}(2m+m_{\Xi_{b}^{-}})]}{3m^{2}}e^{-\frac{m^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}}$
$\displaystyle+$ $\displaystyle
g_{\widetilde{\Xi}b^{0}\Xi_{b}^{-}\pi^{+}}\lambda{\Xi_{b}^{-}}\widetilde{\lambda}\frac{[(\widetilde{m}-m_{\Xi_{b}^{-}})(2\widetilde{m}^{2}+m_{\Xi_{b}^{-}}^{2})-m_{\pi^{+}}^{2}(2\widetilde{m}-m_{\Xi_{b}^{-}})]}{3\widetilde{m}^{2}}e^{-\frac{\widetilde{m}^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}},$
$\displaystyle{\cal B}\Pi_{2}^{\mathrm{QCD,tr}}$ $\displaystyle=$
$\displaystyle-
g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda\frac{[m^{2}+m_{\Xi_{b}^{-}}^{2}-mm_{\Xi_{b}^{-}}-m_{\pi^{+}}^{2}]m_{\Xi_{b}^{-}}}{3m^{2}}e^{-\frac{m^{2}}{M_{1}^{2}}}e^{-\frac{m^{2}_{\Xi_{b}^{-}}}{M_{2}^{2}}}$
(31) $\displaystyle-$ $\displaystyle
g_{\widetilde{\Xi}b^{0}\Xi_{b}^{-}\pi^{+}}\lambda{\Xi_{b}^{-}}\widetilde{\lambda}\frac{[\widetilde{m}^{2}+m_{\Xi_{b}^{-}}^{2}+\widetilde{m}m_{\Xi_{b}^{-}}-m_{\pi^{+}}^{2}]m_{\Xi_{b}^{-}}}{3\widetilde{m}^{2}}e^{-\frac{\widetilde{m}^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}},$

for the ground $+1P$ and

$\displaystyle{\cal B}\Pi_{1}^{\mathrm{QCD,tr}}$ $\displaystyle=$
$\displaystyle-
g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda\frac{[(m+m_{\Xi_{b}^{-}})(2m^{2}+m_{\Xi_{b}^{0}}^{2})-m_{\pi^{+}}^{2}(2m+m_{\Xi_{b}^{-}})]}{3m^{2}}e^{-\frac{m^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}}$
$\displaystyle-$ $\displaystyle
g_{\Xi_{b}^{0}{}^{\prime}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda^{\prime}\frac{[(m^{\prime}+m_{\Xi_{b}^{-}})(2m^{\prime
2}+m_{\Xi_{b}^{-}}^{2})-m_{\pi^{+}}^{2}(2m^{\prime}+m_{\Xi_{b}^{-}})]}{3m^{\prime
2}}e^{-\frac{m^{\prime}{}^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}},$
$\displaystyle{\cal B}\Pi_{2}^{\mathrm{QCD,tr}}$ $\displaystyle=$
$\displaystyle-
g_{\Xi_{b}^{0}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda\frac{[m^{2}+m_{\Xi_{b}^{-}}^{2}-mm_{\Xi_{b}^{-}}-m_{\pi^{+}}^{2}]m_{\Xi_{b}^{-}}}{3m^{2}}e^{-\frac{m^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}}$
(32) $\displaystyle-$ $\displaystyle
g_{\Xi_{b}^{0}{}^{\prime}\Xi_{b}^{-}\pi^{+}}\lambda_{\Xi_{b}^{-}}\lambda^{\prime}\frac{[m^{\prime}{}^{2}+m_{\Xi_{b}^{-}}^{2}-m^{\prime}m_{\Xi_{b}^{-}}-m_{\pi^{+}}^{2}]m_{\Xi_{b}^{-}}}{3m^{\prime}{}^{2}}e^{-\frac{m^{\prime}{}^{2}}{M_{1}^{2}}}e^{-\frac{m_{\Xi_{b}^{-}}^{2}}{M_{2}^{2}}},$

for the ground $+2S$ case. In Eqs. (31) and (32), the ${\cal
B}\Pi_{1}^{\mathrm{QCD,tr}}$ and ${\cal B}\Pi_{2}^{\mathrm{QCD,tr}}$ are used
to represent the results obtained from QCD side of the transition considered
in this section after the Borel transformation. They are the coefficients of
the Lorentz structures $\not{p}q_{\mu}$ and $\not{q}q_{\mu}$, respectively.

To analyze the obtained results we adopt the input parameters given in Table 1
and due to the close masses of the initial and final states we use
$M_{1}^{2}=M_{2}^{2}$ and
$M^{2}=\frac{M_{1}^{2}M_{2}^{2}}{M_{1}^{2}+M_{2}^{2}}$ from which we get
$M_{1}^{2}=M_{2}^{2}=2M^{2}$. For the auxiliary parameters, we exploit the
ones that we used in the mass calculations. Our analyses show that the
considered first two resonances contributions (FTRC) in each case constitute
the main part of the total contribution, satisfying the requirement of the
method. In the case of $\not{q}q_{\mu}$, for instance, the FTRC is obtained to
be $83\%$. The remaining $17\%$ belongs to all the higher states and
continuum. In Fig. 2 the graphs of $g_{\tilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}$
as functions of $M^{2}$ and $s_{0}$ are presented to illustrate its variation
with respect to these auxiliary parameters. From these figures, we see that
the strong coupling constants weakly depend on the $M^{2}$ and $s_{0}$ in
their working intervals.

Figure 2: Left: The coupling constant
$g_{\widetilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}$ obtained for the $1P$
excitation of $\Xi_{b}(5945)^{0}$ vs Borel parameter $M^{2}$. Right: The
coupling constant $g_{\widetilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}$ obtained for
the $1P$ excitation of $\Xi_{b}(5945)^{0}$ vs threshold parameter $s_{0}$.

The coupling constants obtained for each case, that is, for the decays of the
$1P$ and $2S$ initial states, are used to obtain the corresponding decay
widths. To this end, the following decay width formulas are applied

$\displaystyle\Gamma(\widetilde{\Xi}_{b}^{0}$ $\displaystyle\rightarrow$
$\displaystyle\Xi_{b}^{-}\pi^{+})=\frac{g_{\widetilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}^{2}}{24\pi\widetilde{m}^{2}}\left[(\widetilde{m}-m_{\Xi_{b}^{-}})^{2}-m_{\pi^{+}}^{2}\right]f^{3}(\widetilde{m},m_{\Xi_{b}^{-}},m_{\pi^{+}}),$
(33)

for the decay of $1P$ state and

$\displaystyle\Gamma(\Xi_{b}^{0}{}^{\prime}$ $\displaystyle\rightarrow$
$\displaystyle\Xi_{b}^{-}\pi^{+})=\frac{g_{\Xi_{b}^{0}{}^{\prime}\Xi_{b}^{-}\pi^{+}}^{2}}{24\pi
m^{\prime}{}^{2}}\left[(m^{\prime}+m_{\Xi_{b}^{-}})^{2}-m_{\pi^{+}}^{2}\right]f^{3}(m^{\prime},m_{\Xi_{b}^{-}},m_{\pi^{+}}),$
(34)

for the decay of $2S$ state, where $f(x,y,z)$ is

$f(x,y,z)=\frac{1}{2x}\sqrt{x^{4}+y^{4}+z^{4}-2x^{2}y^{2}-2x^{2}z^{2}-2y^{2}z^{2}}.$

The numerical values of the strong coupling constant and partial decay width
for the decay of $1P$ state are obtained as

$\displaystyle g_{\widetilde{\Xi}_{b}^{0}\Xi_{b}^{-}\pi^{+}}=73.82\pm
4.99~{}\mathrm{GeV}^{-1},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Gamma(\widetilde{\Xi}_{b}^{0},\rightarrow\Xi_{b}^{-}\pi^{+})=18.31\pm
2.56~{}\mathrm{MeV},$ (35)

and those of $2S$ state are

$\displaystyle g_{\Xi_{b}^{0}{}^{\prime}\Xi_{b}^{-}\pi^{+}}=52.75\pm
3.71~{}\mathrm{GeV}^{-1},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Gamma(\Xi_{b}^{0}{}^{\prime}\rightarrow\Xi_{b}^{-}\pi^{+})=8.24\pm
1.19~{}\mathrm{MeV}.$ (36)

From these results, among the two possible scenarios, the partial width for
$1P$ excitation case is seen to be consistent with the experimental
observation, $\Gamma(\Xi_{b}(6227)^{0})=18.6^{+5.0}_{-4.1}\pm
1.4~{}\mathrm{MeV}$ Aaij:2020fxj , supporting the possibility that the
observed state’s quantum numbers would be $J^{P}=\frac{3}{2}^{-}$. This
assigned is made based on the comparison of the obtained results for the
partial widths of the decays
$\widetilde{\Xi}_{b}^{0},\rightarrow\Xi_{b}^{-}\pi^{+}$ and
$\Xi_{b}^{0}{}^{\prime}\rightarrow\Xi_{b}^{-}\pi^{+}$ considering them as
dominant decays at different scenarios. If other decay modes are observed, the
results of widths should be taken into account for exact determination of the
quantum numbers for $\Xi_{b}(6227)^{0}$ state.

##  IV Summary and conclusion

This work is performed to shed light on the quantum numbers of the newly
observed $\Xi_{b}(6227)^{0}$ state. Recently, the LHCb announced the
observation of a new excited resonance, $\Xi_{b}(6227)^{0}$, and an
improvement for the mass and width measurements for the $\Xi_{b}(6227)^{-}$
state Aaij:2020fxj . Following the first observation of the
$\Xi_{b}(6227)^{-}$ Aaij:2018yqz , the masses and decay widths for the various
possible scenarios with different possible quantum numbers were considered in
Ref. Aliev:2018lcs , and the obtained results were in favor of its quantum
numbers being $J^{P}=\frac{3}{2}^{-}$. In this study, considering our previous
investigation of $\Xi_{b}(6227)^{-}$ state and the experimental information
that the new state $\Xi_{b}(6227)^{0}$ is the isospin partner of
$\Xi_{b}(6227)^{-}$, we assigned the possible spin quantum number for the
newly observed $\Xi_{b}(6227)^{0}$ state as $J=\frac{3}{2}$. We considered the
two possibilities that this state can be either $1P$ or $2S$ excitation of the
ground state $\Xi_{b}(5945)^{0}$ baryon. With these considerations, we made
QCD sum rules analyses for the corresponding masses and partial widths of the
decays of these excited states. The results of masses and partial decay widths
were obtained as $\tilde{m}=6225.47\pm 106.29~{}\mathrm{MeV}$ and
$\Gamma(\widetilde{\Xi}_{b}^{0},\rightarrow\Xi_{b}^{-}\pi^{+})=18.13\pm
2.35~{}\mathrm{MeV}$ for $1P$ state and as $m^{\prime}=6225.47\pm
106.29~{}\mathrm{MeV}$ and
$\Gamma(\Xi_{b}^{0}{}^{\prime}\rightarrow\Xi_{b}^{-}\pi^{+})=5.43\pm
0.70~{}\mathrm{MeV}$ for $2S$ state. Though both masses are consistent with
the experimental observation, the partial width obtained for
$\widetilde{\Xi}_{b}^{0}$ case, which is in nice agreement with the
experimental value $\Gamma(\Xi_{b}(6227)^{0})=18.6^{+5.0}_{-4.1}\pm
1.4~{}\mathrm{MeV}$ Aaij:2020fxj , supports the possibility to assign the
quantum numbers of $\Xi_{b}(6227)^{0}$ state to be $J^{P}=\frac{3}{2}^{-}$.
Note that this assignment is subject to change: We made it by comparison of
the partial width of the decay mode, and possibly the dominant one, that the
$\Xi_{b}(6227)^{0}$ state has been observed. If another sizable decay mode is
observed, the corresponding width should be added to the one that is
considered in the present study.

## ACKNOWLEDGEMENTS

K. Azizi is thankful to Iran Science Elites Federation (Saramadan) for the
partial financial support provided under the grant number ISEF/M/99171.

## Appendix: The QCD side of the mass sum rules

In this appendix, as an example, we present the explicit expression for the
function $\Pi_{1}^{\mathrm{QCD}}$ (see Eq. (14)), obtained from the mass sum
rules in section II. It is obtained as

$\displaystyle\Pi_{1}^{\mathrm{QCD}}=\int_{(m_{b}+m_{s})^{2}}^{s_{0}}e^{-\frac{s}{M^{2}}}\rho_{1}(s)ds+e^{-\frac{m_{b}^{2}}{M^{2}}}\Gamma_{1}+e^{-\frac{s_{0}}{M^{2}}}\tilde{\Gamma}_{1},$
(37)

where,

$\displaystyle\rho_{1}(s)$ $\displaystyle=$
$\displaystyle-\frac{m_{b}^{3}(m_{s}+m_{u})}{192\pi^{4}}\Big{[}6\psi_{10}-3\psi_{20}+3\psi_{31}-\psi_{32}+6\ln\big{(}\frac{m_{b}^{2}}{s}\big{)}\Big{]}-\frac{m_{b}^{4}}{3840\pi^{4}}\Big{[}-60\psi_{10}+30\psi_{20}-20\psi_{30}+100\psi_{41}$
(38) $\displaystyle+$ $\displaystyle
31\psi_{42}+4\psi_{43}+60\psi_{00}\ln\big{(}\frac{s}{m_{b}^{2}}\big{)}\Big{]}+\langle\bar{u}u\rangle\Big{[}-\frac{m_{0}^{2}}{48m_{b}\pi^{2}}\psi_{02}+\frac{m_{b}}{24\pi^{2}}(2\psi_{10}-\psi_{11}-\psi_{12}-2\psi_{21})+\frac{m_{0}^{2}m_{u}}{96m_{b}^{2}\pi^{2}}(\psi_{02}-4\psi_{03})$
$\displaystyle+$
$\displaystyle\frac{1}{48\pi^{2}}\big{[}4(-\psi_{00}+\psi_{03})m_{u}+(\psi_{11}+\psi_{12})(2m_{s}+3m_{u})\big{]}\Big{]}-\langle\bar{s}s\rangle\Big{[}-\frac{m_{0}^{2}}{48m_{b}\pi^{2}}\psi_{02}+\frac{m_{b}}{24\pi^{2}}(2\psi_{10}-\psi_{11}-\psi_{12}-2\psi_{21})$
$\displaystyle+$
$\displaystyle\frac{m_{0}^{2}m_{s}}{96m_{b}^{2}\pi^{2}}(\psi_{02}-4\psi_{03})+\frac{1}{48\pi^{2}}\big{[}-4\psi_{00}m_{s}+4\psi_{03}m_{s}+(\psi_{11}+\psi_{12})(3m_{s}+2m_{u})\big{]}\Big{]}+\frac{\langle
g^{2}G^{2}\rangle}{1152m_{b}\pi^{4}}\Big{[}\psi_{03}m_{b}+4\psi_{10}m_{b}$
$\displaystyle-$ $\displaystyle
4\psi_{21}m_{b}-3\psi_{10}m_{s}+2\psi_{11}m_{s}+2\psi_{12}m_{s}+3\psi_{21}m_{s}-3\psi_{10}m_{u}+2\psi_{11}m_{u}+2\psi_{12}m_{u}+3\psi_{21}m_{u}$
$\displaystyle+$
$\displaystyle(3\psi_{01}-\psi_{02})(m_{s}+m_{u})-\psi_{00}(m_{b}+3\gamma_{E}(m_{s}+m_{u}))+3\psi_{02}(m_{s}+m_{u})\big{[}\ln\big{(}\frac{\Lambda^{2}}{s}\big{)}+\ln\big{(}\frac{(s-m_{b}^{2})}{\Lambda^{2}}\big{)}$
$\displaystyle+$
$\displaystyle\ln\big{(}\frac{s(s-m_{b}^{2})}{\Lambda^{2}m_{b}^{2}}\big{)}\big{]}\Big{]}+\frac{\langle
g^{2}G^{2}\rangle}{1728m_{b}^{4}M^{10}\pi^{2}}\Big{[}2M^{10}\big{[}-3(2\psi_{01}-2\psi_{12}-5\psi_{13}-11\psi_{14}-2\psi_{23})(m_{u}\langle\bar{s}s\rangle+m_{s}\langle\bar{u}u\rangle)$
$\displaystyle+$
$\displaystyle\psi_{03}(4m_{s}\langle\bar{s}s\rangle-6m_{u}\langle\bar{s}s\rangle-6m_{s}\langle\bar{u}u\rangle+4m_{u}\langle\bar{u}u\rangle)+3\psi_{02}(m_{b}(\langle\bar{s}s\rangle+\langle\bar{u}u\rangle)-9(m_{u}\langle\bar{s}s\rangle+m_{s}\langle\bar{u}u\rangle))\big{]}$
$\displaystyle+$ $\displaystyle
3(m_{u}\langle\bar{s}s\rangle+m_{s}\langle\bar{u}u\rangle)\big{[}\psi_{00}\big{(}m_{0}^{2}m_{b}^{8}-2M^{4}(2m_{b}^{6}+3m_{b}^{4}M^{2}+2m_{b}^{2}M^{4}+6M^{6})\big{)}\ln\big{(}\frac{(s-m_{b}^{2})}{\Lambda^{2}}\big{)}$
$\displaystyle+$ $\displaystyle
12\psi_{04}M^{10}\ln\big{(}\frac{s(s-m_{b}^{2})}{\Lambda^{2}m_{b}^{2}}\big{)}\big{]}\Big{]}-\frac{\langle
g^{2}G^{2}\rangle^{2}}{41472M^{16}\pi^{2}}\Big{[}(m_{u}\langle\bar{s}s\rangle+m_{s}\langle\bar{u}u\rangle)\psi_{00}m_{b}^{2}\big{(}2M^{4}(-2m_{b}^{2}+5M^{2})$
$\displaystyle+$ $\displaystyle
m_{0}^{2}(m_{b}^{4}-7m_{b}^{2}M^{2}+9M^{4})\big{)}\ln\big{(}\frac{(s-m_{b}^{2})}{\Lambda^{2}}\big{)}\Big{]},$
$\displaystyle\Gamma_{1}$ $\displaystyle=$
$\displaystyle\frac{m_{0}^{2}}{288\pi^{2}}(7m_{s}\langle\bar{s}s\rangle-6m_{u}\langle\bar{s}s\rangle-6m_{s}\langle\bar{u}u\rangle+7m_{u}\langle\bar{u}u\rangle)+\frac{\langle\bar{u}u\rangle\langle\bar{s}s\rangle}{216M^{6}}\Big{[}12(m_{b}(m_{s}+m_{u})-2M^{2})M^{4}$
(39) $\displaystyle+$ $\displaystyle
m_{0}^{2}(-5m_{b}^{3}(m_{s}+m_{u})+12m_{b}^{2}M^{2}+12M^{4})\Big{]}+\frac{\langle
g^{2}G^{2}\rangle}{384m_{b}\pi^{4}}\gamma_{E}(m_{s}+m_{u})M^{2}-\frac{\langle
g^{2}G^{2}\rangle^{2}}{82944m_{b}^{3}M^{4}\pi^{4}}\Big{[}2m_{b}^{4}(m_{s}+m_{u})$
$\displaystyle+$ $\displaystyle
m_{b}^{2}\big{(}m_{b}+(-1+3\gamma_{E})(m_{s}+m_{u})\big{)}M^{2}+2\big{(}m_{b}-3(-1+\gamma_{E})(m_{s}+m_{u})\big{)}M^{4}+3(m_{s}+m_{u})\big{(}m_{b}^{4}-m_{b}^{2}M^{2}$
$\displaystyle-$ $\displaystyle
M^{4}\big{)}\ln\big{(}\frac{\Lambda^{2}}{m_{b}^{2}}\big{)}\Big{]}+\frac{\langle
g^{2}G^{2}\rangle}{62208m_{b}^{2}M^{12}\pi^{2}}\Big{[}-72M^{8}\langle\bar{s}s\rangle\big{[}2m_{b}^{4}m_{u}+m_{b}^{2}M^{2}(-2m_{s}+3(2+\gamma_{E})m_{u})+(m_{b}+2m_{s}+6m_{u}$
$\displaystyle+$ $\displaystyle
12\gamma_{E}m_{u})M^{4}\big{]}-24M^{4}\langle\bar{u}u\rangle\big{[}3M^{4}\big{(}2m_{b}^{4}m_{s}+m_{b}^{2}M^{2}(3(2+\gamma_{E})m_{s}-2m_{u})+M^{4}(m_{b}+6m_{s}+12\gamma_{E}m_{s}+2m_{u})\big{)}$
$\displaystyle+$ $\displaystyle
2m_{b}^{3}\pi^{2}\langle\bar{s}s\rangle\big{(}m_{b}^{2}(m_{s}+m_{u})-2m_{b}M^{2}-3(m_{s}+m_{u})M^{2}\big{)}\big{]}+m_{0}^{2}\big{[}9M^{4}\langle\bar{s}s\rangle\big{[}4m_{b}^{6}m_{u}+m_{b}^{4}(-3m_{s}+20m_{u})M^{2}$
$\displaystyle+$ $\displaystyle
2m_{b}^{2}(m_{b}+m_{s}+9m_{u})M^{4}-4(m_{b}-3m_{u})M^{6}\big{]}+\langle\bar{u}u\rangle\big{[}9M^{4}\big{(}4m_{b}^{6}m_{s}+m_{b}^{4}(20m_{s}-3m_{u})M^{2}$
$\displaystyle+$ $\displaystyle
2m_{b}^{2}(m_{b}+9m_{s}+m_{u})M^{4}-4(m_{b}-3m_{s})M^{6}\big{)}+4m_{b}^{3}\pi^{2}\langle\bar{s}s\rangle\big{(}5m_{b}^{4}(m_{s}+m_{u})-6m_{b}^{2}(2m_{b}+5(m_{s}+m_{u}))M^{2}$
$\displaystyle+$ $\displaystyle
6(4m_{b}+5(m_{s}+m_{u}))M^{4}\big{)}\big{]}\big{]}-54M^{4}(m_{u}\langle\bar{s}s\rangle+m_{s}\langle\bar{u}u\rangle)(-m_{0}^{2}m_{b}^{6}+4M^{4}(m_{b}^{4}+2m_{b}^{2}M^{2}+2M^{4}))\ln\big{(}\frac{\Lambda^{2}}{m_{b}^{2}}\big{)}\Big{]}$
$\displaystyle-$ $\displaystyle\frac{\langle
g^{2}G^{2}\rangle^{2}}{248832m_{b}^{2}M^{14}\pi^{2}}(m_{u}\langle\bar{s}s\rangle+m_{s}\langle\bar{u}u\rangle)\Big{[}4M^{4}\big{(}-2m_{b}^{6}+(3\gamma_{E}-8)m_{b}^{4}M^{2}+9m_{b}^{2}M^{4}+3M^{6}\big{)}$
$\displaystyle+$ $\displaystyle
m_{0}^{2}\big{(}2m_{b}^{8}-6(\gamma_{E}-1)m_{b}^{6}M^{2}+3(6\gamma_{E}-19)m_{b}^{4}M^{4}+24m_{b}^{2}M^{6}+6M^{8}\big{)}+3m_{b}^{4}\big{[}m_{0}^{2}(m_{b}^{2}-6M^{2})(m_{b}^{2}-2M^{2})$
$\displaystyle+$ $\displaystyle
4M^{4}(-m_{b}^{2}+3M^{2})\big{]}\ln\big{(}\frac{\Lambda^{2}}{m_{b}^{2}}\big{)}\Big{]},$

and

$\displaystyle\tilde{\Gamma}_{1}$ $\displaystyle=$
$\displaystyle-\frac{\langle
g^{2}G^{2}\rangle}{384m_{b}\pi^{4}}\Big{[}\gamma_{E}(m_{s}+m_{u})M^{2}\Big{]}-\frac{\langle
g^{2}G^{2}\rangle^{2}(m_{s}+m_{u})}{27648m_{b}M^{4}\pi^{4}(m_{b}^{2}-s_{0})^{2}}\Big{[}M^{2}(-2m_{b}^{4}-3M^{2}s_{0}+m_{b}^{2}(M^{2}+2s_{0}))$
(40) $\displaystyle+$
$\displaystyle(2m_{b}^{2}-3M^{2})(m_{b}^{2}-s_{0})^{2}\ln\big{(}\frac{(s_{0}-m_{b}^{2})}{\Lambda^{2}}\Big{)}\Big{]}+\frac{\langle
g^{2}G^{2}\rangle}{576M^{8}\pi^{2}}(m_{u}\langle\bar{s}s\rangle+m_{s}\langle\bar{u}u\rangle)\Big{[}\frac{1}{(m_{b}^{2}-s_{0})^{4}}\big{(}-6m_{0}^{2}m_{b}^{4}M^{8}$
$\displaystyle+$
$\displaystyle(m_{0}^{2}m_{b}^{4}M^{4}-4m_{b}^{2}M^{8})(m_{b}^{2}-s_{0})^{2}+2m_{o}^{2}m_{b}^{4}M^{6}(-m_{b}^{2}+s_{0})+(m_{0}^{2}m_{b}^{4}M^{2}-2M^{6}(2m_{b}^{2}+3M^{2}))(-m_{b}^{2}+s_{0})^{3}\big{)}$
$\displaystyle+$
$\displaystyle(m_{0}^{2}m_{b}^{4}-2M^{4}(2m_{b}^{2}+3M^{2}))\ln\big{(}\frac{(s_{0}-m_{b}^{2})}{\Lambda^{2}}\Big{)}\Big{]}+\frac{\langle
g^{2}G^{2}\rangle^{2}}{41472M^{14}\pi^{2}(m_{b}^{2}-s_{0})^{7}}m_{b}^{2}(m_{u}\langle\bar{s}s\rangle+m_{s}\langle\bar{u}u\rangle)$
$\displaystyle\times$
$\displaystyle\Big{[}-2M^{6}(m_{b}^{2}-s_{0})^{2}\big{[}2m_{b}^{10}-m_{b}^{8}(3M^{2}+8s_{0})+m_{b}^{6}(-M^{4}+14M^{2}s_{0}+12s_{0}^{2})+m_{b}^{4}(2M^{6}+7M^{4}s_{0}-24M^{2}s_{0}^{2}-8s_{0}^{3})$
$\displaystyle+$ $\displaystyle
5M^{2}s_{0}(6M^{6}-2M^{4}s_{0}+M^{2}s_{0}^{2}-s_{0}^{3})+m_{b}^{2}(18M^{8}+8M^{6}s_{0}-11M^{4}s_{0}^{2}+18M^{2}s_{0}^{3}+2s_{0}^{4})\big{]}+m_{0}^{2}M^{2}\big{[}m_{b}^{16}$
$\displaystyle-$ $\displaystyle
6m_{b}^{14}(M^{2}+s_{0})+m_{b}^{12}(4M^{4}+37M^{2}s_{0}+15s_{0}^{2})+m_{b}^{10}(M^{6}-27M^{4}s_{0}-95M^{2}s_{0}^{2}-20s_{0}^{3})+m_{b}^{8}s_{0}(-7M^{6}$
$\displaystyle+$ $\displaystyle
77M^{4}s_{0}+130M^{2}s_{0}^{2}+15s_{0}^{3})+9M^{4}s_{0}^{2}(24M^{8}-6M^{6}s_{0}+2M^{4}s_{0}^{2}-M^{2}s_{0}^{3}+s_{0}^{4})+m_{b}^{2}M^{2}s_{0}(408M^{10}-6M^{8}s_{0}$
$\displaystyle-$ $\displaystyle
30M^{6}s_{0}^{2}+31M^{4}s_{0}^{3}-47M^{2}s_{0}^{4}-7s_{0}^{5})+2m_{b}^{6}(3M^{10}+3M^{8}s_{0}+12M^{6}s_{0}^{2}-59M^{4}s_{0}^{3}-50M^{2}s_{0}^{4}-3s_{0}^{5})+m_{b}^{4}(96M^{12}$
$\displaystyle+$ $\displaystyle
54M^{10}s_{0}+6M^{8}s_{0}^{2}-40M^{6}s_{0}^{3}+102M^{4}s_{0}^{4}+41M^{2}s_{0}^{5}+s_{0}^{6})\big{]}-\big{[}2M^{4}(-2m_{b}^{2}+5M^{2})+m_{0}^{2}(m_{b}^{4}-7m_{b}^{2}M^{2}$
$\displaystyle+$ $\displaystyle
9M^{4})\big{]}(m_{b}^{2}-s_{0})^{7}\ln\big{(}\frac{(s_{0}-m_{b}^{2})}{\Lambda^{2}}\Big{)}\Big{]},$

with

$\displaystyle\psi_{nm}=\frac{(s-m_{b}^{2})^{n}}{s^{m}(m_{b}^{2})^{n-m}}.$
(41)

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