# Newly discovered $\Xi_{c}^{0}$ resonances and their parameters

S. S. Agaev  Institute for Physical Problems, Baku State University, Az–1148
Baku, Azerbaijan  K. Azizi  Department of Physics, University of Tehran, North
Karegar Avenue, Tehran 14395-547, Iran  Department of Physics, Doǧuş
University, Acibadem-Kadiköy, 34722 Istanbul, Turkey  School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box
19395-5531, Tehran, Iran  H. Sundu  Department of Physics, Kocaeli University,
41380 Izmit, Turkey

###### Abstract

The aim of the present article is investigation of the newly observed
resonances $\Xi_{c}(2923)^{0}$, $\Xi_{c}(2939)^{0}$, and $\Xi_{c}(2965)^{0}$
which are real candidates to charm-strange baryons. To this end, we calculate
the mass and pole residue of the ground-state and excited $1P$ and $2S$
spin-$1/2$ flavor-sextet baryons $\Xi_{c}^{\prime 0}$, $\Xi_{c}^{\prime
0}(1/2^{-})$ and $\Xi_{c}^{\prime 0}(1/2^{+})$ with quark content $csd$,
respectively. The masses and pole residues of the ground-state and excited
spin-$3/2$ baryons $\Xi_{c}^{\star 0}$ are found as well. Spectroscopic
parameters of these particles are computed in the context of the QCD two-point
sum rule method. Widths of the excited baryons are evaluated through their
decays to final states $\Lambda_{c}^{+}K^{-}$ and $\Xi_{c}^{\prime 0}\pi$.
These processes are explored by means of the full QCD light-cone sum rule
method necessary to determine strong couplings at relevant vertices. Obtained
predictions for the masses and widths of the four excited baryons, as well as
previous results for $1P$ and $2S$ flavor-antitriplet spin-1/2 particles
$\Xi_{c}^{0}$ are confronted with available experimental data on $\Xi_{c}^{0}$
resonances to fix their quantum numbers. Our comparison demonstrates that the
resonances $\Xi_{c}(2923)^{0}$ and $\Xi_{c}(2939)^{0}$ can be considered as
$1P$ excitations of the spin-$1/2$ flavor-sextet and spin-$3/2$ baryons,
respectively. The resonance $\Xi_{c}(2965)^{0}$ may be interpreted as the
excited $2S$ state of either spin-$1/2$ flavor-sextet or antitriplet baryon.

##  I Introduction

The discovery of three new resonances $\Xi_{c}(2923)^{0}$,
$\Xi_{c}(2939)^{0}$, and $\Xi_{c}(2965)^{0}$ by the LHCb collaboration is a
last result of the experiments devoted to investigation of charmed and bottom
baryons with different spin-parities and quark contents Aaij:2020yyt . Five
narrow states $\Omega_{c}^{0}$ fixed in the $\Xi_{c}^{+}K^{-}$ invariant mass
distribution Aaij:2017nav , and four peaks $\Omega_{b}^{-}$ detected recently
in the $\Xi_{b}^{0}K^{-}$ spectrum Aaij:2020cex  were results of previous
measurements performed by LHCb.

Needless to say, that discovery of the resonances $\Omega_{c}^{0}$ stimulated
numerous studies of excited charmed baryons aimed to understand their internal
organizations and quantum numbers. Actually, heavy flavored baryons were
already objects of theoretical analyses, in which spectroscopic parameters of
the ground state and excited particles, their decay channels and strong
couplings, magnetic moments and radiative decays were studied by means of
different models and methods of high energy physics. New experimental
information on $\Omega_{c}^{0}$, besides traditional models, gave rise to
their interpretations as exotic pentaquark states. In our articles
Agaev:2017jyt ; Agaev:2017lip ; Agaev:2017ywp , we investigated the baryons
$\Omega_{c}^{0}$ and $\Omega_{b}^{-}$, where one can find further details and
references to relevant publications.

The baryons from the $\Xi_{c}^{0}=csd$ family are another interesting objects
for both experimental and theoretical analyses. Parameters of the ground-state
$J^{\mathrm{P}}=1/2^{+}$ and $3/2^{+}$ baryons with the content $csd$ were
measured already and included into Particle Data Group (PDG) tables
Tanabashi:2018oca . Thus, the mass and mean lifetime of the flavor-antitriplet
baryon $\Xi_{c}^{0}$ are

$m_{\mathrm{exp}}=2470.90_{-0.29}^{+0.22}~{}\mathrm{MeV},\,\,\tau_{\mathrm{exp}}=(1.53\pm
0.06)\times 10^{-13}~{}\mathrm{s},$ (1)

whereas for the mass of the flavor-sextet $J^{\mathrm{P}}=1/2^{+}$ ground-
state particle $\Xi_{c}^{\prime 0}$ we have

$m_{\mathrm{exp}}=(2579.2\pm 0.5)~{}\mathrm{MeV}.$ (2)

The mass of the $J^{\mathrm{P}}=3/2^{+}$ baryon $\Xi_{c}(2645)^{0}$ is also
known

$m_{\mathrm{exp}}^{\ast}=2645.56_{-0.30}^{+0.24}~{}\mathrm{MeV}.$ (3)

There are a few charged and neutral particles of this family listed in Ref.
Tanabashi:2018oca , some of which will be considered in the last section of
the present work.

As we have noted above, theoretical investigations of heavy flavored baryons,
including $\Xi_{c}$ ones, have long history Capstick:1986bm ; Ebert:2007nw ;
Ebert:2011kk ; Garcilazo:2007eh ; Valcarce:2008dr ; Roberts:2007ni ;
Yoshida:2015tia ; Shah:2016nxi ; Bagan:1992tp ; Huang:2000tn ; Wang:2002ts ;
Wang:2009cr ; Chen:2015kpa ; Chen:2016phw ; Chen:2017sci ; Aliev:2010yx ;
Aliev:2010ev ; Aliev:2011ufa ; Azizi:2015ksa ; Chiladze:1997ev ;
Edwards:2012fx ; Padmanath:2013bla . These particles were explored in the
context of various quark models Capstick:1986bm ; Ebert:2007nw ; Ebert:2011kk
; Garcilazo:2007eh ; Valcarce:2008dr ; Roberts:2007ni ; Yoshida:2015tia ;
Shah:2016nxi , by using the QCD sum rule method Bagan:1992tp ; Huang:2000tn ;
Wang:2002ts ; Wang:2009cr ; Chen:2015kpa ; Chen:2016phw ; Chen:2017sci ;
Aliev:2010yx ; Aliev:2010ev ; Aliev:2011ufa ; Azizi:2015ksa , by means of the
Heavy Quark Effective Theory (HQET) Chiladze:1997ev  and lattice simulations
Edwards:2012fx ; Padmanath:2013bla ; Bahtiyar:2020uuj .

The discovery of three new resonances by LHCb added valuable knowledge about
excited baryons $\Xi_{c}^{0}$, which together with $\Xi_{c}(2930)^{0}$ and
$\Xi_{c}(2970)^{0}$ generated theoretical activities to explain their
parameters. Problem is that LHCb did not inform on spins and parities of these
resonances, which are important topic of continuing theoretical studies. Here,
it is necessary to give some information about the resonance
$\Xi_{c}(2930)^{0}$, which is relatively ”old” member of this family. It was
observed by the BaBar collaboration as the intermediate resonant structure in
the process $B^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}K^{-}$
Aubert:2007eb . Existence of $\Xi_{c}(2930)^{0}$ was confirmed recently by
Belle in Ref. Li:2017uvv , in which the collaboration reported about its
observation as a resonance in the $\Lambda_{c}^{+}K^{-}$ invariant mass
spectrum in the same decay process. The mass and width of this state reported
by Belle are

$\displaystyle m$ $\displaystyle=$ $\displaystyle(2928.9\pm
3.0_{-12.0}^{+0.9})~{}\mathrm{MeV},$ $\displaystyle\,\Gamma$ $\displaystyle=$
$\displaystyle(19.5\pm 8.4_{-7.9}^{+5.9})~{}\mathrm{MeV}.$ (4)

Parameters of $\Xi_{c}(2930)^{0}$, its mass and width were calculated in the
framework of different approaches Chen:2017sci ; Li:2017uvv ; Liu:2012sj ;
Wang:2017kfr ; Chen:2017iyi ; Ye:2017dra ; Ye:2017yvl ; Aliev:2018ube .

The new resonances have masses and widths which do not differ considerably
from ones of $\Xi_{c}(2930)^{0}$. For simplicity of presentation, we label
parameters of $\Xi_{c}(2923)^{0}$, $\Xi_{c}(2939)^{0}$, and
$\Xi_{c}(2965)^{0}$ by subscripts $1$, $2$, and $3$, respectively. The masses
and widths of these states are equal to Aaij:2020yyt

$\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle(2923.04\pm 0.25\pm
0.20\pm 0.14)~{}\mathrm{MeV},$ $\displaystyle\,\Gamma_{1}$ $\displaystyle=$
$\displaystyle(\,7.1\pm 0.8\pm 1.8)~{}\mathrm{MeV},$ (5) $\displaystyle m_{2}$
$\displaystyle=$ $\displaystyle(2938.55\pm 0.21\pm 0.17\pm
0.14)~{}\mathrm{MeV},$ $\displaystyle\,\Gamma_{2}$ $\displaystyle=$
$\displaystyle(\,10.2\pm 0.8\pm 1.1)~{}\mathrm{MeV},$ (6)

and

$\displaystyle m_{3}$ $\displaystyle=$ $\displaystyle(2964.88\pm 0.26\pm
0.14\pm 0.14)~{}\mathrm{MeV},$ $\displaystyle\,\Gamma_{3}$ $\displaystyle=$
$\displaystyle(\,14.1\pm 0.9\pm 1.3)~{}\mathrm{MeV}.$ (7)

These resonances immediately became object of theoretical investigations
Yang:2020zjl ; Wang:2020gkn ; Lu:2020ivo ; Zhu:2020jke , in which they were
studied in a rather detailed form. These states were considered mostly as
conventional flavor-sextet $1P$-wave baryons of different spins Yang:2020zjl ;
Wang:2020gkn  though sextet $2S$ interpretation of the heaviest resonance from
this list is also on agenda Lu:2020ivo . The particles $\Xi_{c}^{0}$ were
described also as molecular $\overline{D}\Lambda-\overline{D}\Sigma$ states
Zhu:2020jke . The mass and width of the excited flavor-antitriplet baryons
$\Xi_{c}^{0}$ were calculated recently in Ref. Aliev:2018ube . Performed
analysis allowed the authors to conclude that the baryon
$\Xi_{c}^{0}(1/2^{-})$ with parameters $\widetilde{m}=(2922\pm
83)~{}\mathrm{MeV}$ and $\widetilde{\Gamma}=(19.4\pm 3.3)~{}\mathrm{MeV}$, and
quantum numbers $(1P,1/2^{-})$ may be interpreted as the state
$\Xi_{c}(2930)^{0}$. The remaining radially excited antitriplet baryon
$\Xi_{c}^{0}(1/2^{+})$ with $m^{\prime}=(2922\pm 83)~{}\mathrm{MeV}$ and
$\Gamma^{\prime}=(13.6\pm 2.3)~{}\mathrm{MeV}$ can be examined as a candidate
to one of three new resonances.

As is seen, various suggestions were made on structures and quantum numbers of
the $csd$ states, and predictions obtained by means of different methods in
the context of these assumptions, sometimes, contradict to each another.
Therefore, additional studies of these baryons are required to clarify
situation with $csd$ resonances. Before detailed analysis, there is a
necessity to establish short-hand notations for different baryons to be
studied in this article. First of all, we omit superscript $0$ for all
baryons. For the flavor-antitriplet spin-$1/2$ baryons, we use standard
notations $\Xi_{c}$, $\Xi_{c}(1/2^{-})$, and $\Xi_{c}(1/2^{+})$ for the
ground-state, first orbitally and radially excited states, respectively. The
flavor-sextet spin-$1/2$ baryons will be presented as $\Xi_{c}^{\prime}$,
$\Xi_{c}^{\prime}(1/2^{-})$, and $\Xi_{c}^{\prime}(1/2^{+})$ in accordance
with their quantum numbers $(1S,1/2^{+})$, $(1P,1/2^{-})$, and $(2S,1/2^{+})$.
For the spin-$3/2$ baryons with $(1S,3/2^{+})$, $(1P,3/2^{-})$, and
$(2S,3/2^{+})$, we introduce brief notations $\Xi_{c}^{{}^{\ast}}$,
$\Xi_{c}^{\ast}(3/2^{-})$, and $\Xi_{c}^{{}^{\ast}}(3/2^{+})$, which cannot
lead to confusions.

In the present article, we explore the ground-state $\Xi_{c}^{\prime}$ and
excited spin-$1/2$ flavor-sextet baryons $\Xi_{c}^{\prime}(1/2^{-})$ and
$\Xi_{c}^{\prime}(1/2^{+})$ , and spin-$3/2$ particles $\Xi_{c}^{{}^{\ast}}$,
$\Xi_{c}^{\ast}(3/2^{-})$, and $\Xi_{c}^{{}^{\ast}}(3/2^{+})$ by computing
their masses and pole residues. These spectroscopic parameters are evaluated
using the QCD sum rule method Shifman:1978bx ; Shifman:1978by , in which
contributions of vacuum condensates up to dimension $10$ are taken into
account. We determine also widths of excited baryons
$\Xi_{c}^{\prime}(1/2^{-})$, $\Xi_{c}^{\prime}(1/2^{+})$,
$\Xi_{c}^{\ast}(3/2^{-})$, and $\Xi_{c}^{{}^{\ast}}(3/2^{+})$ by calculating
partial widths of their strong decays to final states $\Lambda_{c}^{+}K^{-}$
and $\Xi_{c}^{\prime}\pi$. Obtained predictions for partial widths of these
decay modes will allow us to estimate full widths of
$\Xi_{c}^{\prime}(1/2^{-})$, $\Xi_{c}^{\prime}(1/2^{+})$,
$\Xi_{c}^{\ast}(3/2^{-})$, and $\Xi_{c}^{{}^{\ast}}(3/2^{+})$. The strong
decay processes are explored by means of the QCD light-cone sum rule (LCSR)
approach Balitsky:1989ry .

This article is structured in the following way: In Sec. II, we calculate the
spectroscopic parameters of the ground state and excited baryons
$\Xi_{c}^{\prime}$, $\Xi_{c}^{\prime}(1/2^{-})$, and
$\Xi_{c}^{\prime}(1/2^{+})$. Here, we also evaluate the parameters of the
states $\Xi_{c}^{\ast}$, $\Xi_{c}^{\ast}(3/2^{-})$, and
$\Xi_{c}^{{}^{\ast}}(3/2^{+})$. Results extracted from the sum rules in this
section will be compared with the experimental data, but also serve as input
information for the next sections. In Sec. III, we derive the LCSRs for the
strong couplings $g_{1}$ and $g_{2}$ describing the vertices
$\Xi_{c}^{\prime}(1/2^{-})\Lambda_{c}^{+}K^{-}$ and
$\Xi_{c}^{\prime}(1/2^{+})\Lambda_{c}^{+}K^{-}$, that are key ingredients to
evaluate width of the processes
$\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}$ and
$\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}$. In Sec. IV, we
analyze the vertices $\Xi_{c}^{\prime}(1/2^{-})\Xi_{c}^{\prime}\pi$ and
$\Xi_{c}^{\prime}(1/2^{+})\Xi_{c}^{\prime}\pi$, and calculate corresponding
strong couplings $g_{3}$ and $g_{4}$. The partial widths of decays
$\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Xi_{c}^{\prime}\pi$ and
$\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Xi_{c}^{\prime}\pi$ are also found in
this section. Section V is devoted to investigation of the decays
$\Xi_{c}^{\ast}(3/2^{-})\rightarrow\Lambda_{c}^{+}K^{-},\ \Xi_{c}^{\prime}\pi$
and
$\Xi_{c}^{{}^{\ast}}(3/2^{+})\rightarrow\Lambda_{c}^{+}K^{-},\Xi_{c}^{\prime}\pi$.
The last Section VI is reserved for comparison of obtained theoretical
predictions with the LHCb data and, in accordance with this analysis,
assignment of appropriate quantum numbers to new three LHCb resonances. This
section contains also our concluding notes. Appendix contains explicit
expressions some of invariant amplitudes employed to extract parameters of
color-sextet spin-$1/2$ baryons.

##  II Masses and pole residues of the baryons $\Xi_{c}^{\prime}$ and
$\Xi_{c}^{\ast}$

The sum rules required to evaluate the mass and residue of the spin-$1/2$
baryons $\Xi_{c}^{\prime}$, $\Xi_{c}^{\prime}(1/2^{-})$, and
$\Xi_{c}^{\prime}(1/2^{+})$, and spin-$3/2$ baryons $\Xi_{c}^{{}^{\ast}}$,
$\Xi_{c}^{\ast}(3/2^{-})$, and $\Xi_{c}^{{}^{\ast}}(3/2^{+})$ can be obtained
from analysis of the following two-point correlation functions

$\Pi_{(\mu\nu)}(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{\eta_{(\mu)}(x)\overline{\eta}_{(\nu)}(0)\\}|0\rangle,$ (8)

where $\eta(x)$ and $\eta_{\mu}(x)$ are interpolating fields for
$\Xi_{c}^{\prime}$ and $\Xi_{c}^{\ast}$ states with spins $1/2$ and $3/2$,
respectively. In the case of the flavor-sextet spin-$1/2$ baryons the current
$\eta$ is given by the formula

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\epsilon^{abc}\left\\{\left(d_{a}^{T}Cc_{b}\right)\gamma_{5}s_{c}+\beta\left(d_{a}^{T}C\gamma_{5}c_{b}\right)s_{c}\right.$
(9)
$\displaystyle\left.-\left[\left(c_{a}^{T}Cs_{b}\right)\gamma_{5}d_{c}+\beta\left(c_{a}^{T}C\gamma_{5}s_{b}\right)d_{c}\right]\right\\}.$

For spin-$3/2$ baryons, we use

$\displaystyle\eta_{\mu}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\epsilon^{abc}\left\\{\left(d_{a}^{T}C\gamma_{\mu}s_{b}\right)c_{c}+\left(s_{a}^{T}C\gamma_{\mu}c_{b}\right)d_{c}\right.$
(10)
$\displaystyle\left.+\left(c_{a}^{T}C\gamma_{\mu}d_{b}\right)s_{c}\right\\}.$

In formulas for the currents $C$ is the charge conjugation matrix. The current
$\eta(x)$ for the spin-$1/2$ baryons depends on an arbitrary mixing parameter
$\beta$ with $\beta=-1$ corresponding to the Ioffe current.

We begin from the spin-$1/2$ baryons and first compute the mass of the ground-
state particle $\Xi_{c}^{\prime}$. For this purposes, we express the
correlation function $\Pi^{\mathrm{Phys}}(p)$ using the physical parameters of
the ground-state particle $(1S,\,1/2^{+})$. Then, in the ”ground-
state+continuum” approximation $\Pi^{\mathrm{Phys}}(p)$ is given by the simple
formula

$\Pi^{\mathrm{Phys}}(p)=\frac{\langle
0|\eta|\Xi_{c}^{\prime}(p,s)\rangle\langle\Xi_{c}^{\prime}(p,s)|\overline{\eta}|0\rangle}{m^{2}-p^{2}}+\cdots,$
(11)

where $m$ and $s$ are the mass and spin of $\Xi_{c}^{\prime}$, respectively.
Contributions of higher resonances and continuum states are denoted in Eq.
(11) by dots. In expression for $\Pi^{\mathrm{Phys}}(p)$ summation over the
spin $s$ is implied.

We continue our analysis by using the matrix element

$\langle 0|\eta|\Xi_{c}^{\prime}(p,s)\rangle=\lambda u(p,s),$ (12)

where $\lambda$ is the pole residue of $\Xi_{c}^{\prime}$. Carrying out
summation over $s$ in Eq. (11) by employing the matrix element (12) and the
formula

$\sum\limits_{s}u(p,s)\overline{u}(p,s)=\not{p}+m,$ (13)

we get

$\Pi^{\mathrm{Phys}}(p)=\frac{\lambda^{2}(\not{p}+m)}{m^{2}-p^{2}}+\cdots.$
(14)

The function $\Pi^{\mathrm{Phys}}(p)$ contains Lorentz structures proportional
to $\not{p}$ and $I$. To find the sum rule, we can employ invariant amplitudes
that correspond to these structures.

The second component of our investigation is the QCD side of the sum rule
$\Pi^{\mathrm{OPE}}(p)$, which should be computed in the operator product
expansion ($\mathrm{OPE}$) with certain accuracy. To this end, one has to
insert the interpolating current $\eta$ into Eq. (8) and contract the quark
fields. We compute $\Pi^{\mathrm{OPE}}(p)$ using light $q$ and heavy $Q$ quark
$x$-space propagators, explicit expressions of which are presented below

$\displaystyle
S_{q}^{ab}(x)=i\frac{\not{x}\delta_{ab}}{2\pi^{2}x^{4}}-\frac{m_{q}\delta_{ab}}{4\pi^{2}x^{2}}-\frac{\langle\overline{q}q\rangle\delta_{ab}}{12}\left(1-i\frac{m_{q}}{4}\not{x}\right)$
$\displaystyle-\frac{x^{2}\delta_{ab}}{192}\langle\overline{q}g_{s}\sigma
Gq\rangle\left(1-i\frac{m_{q}}{6}\not{x}\right)-\frac{ig_{s}G_{ab}^{\mu\nu}}{32\pi^{2}x^{2}}\left[\not{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\not{x}\right]$
$\displaystyle-\frac{\not{x}x^{2}g_{s}^{2}}{7776}\langle\overline{q}q\rangle^{2}\delta_{ab}-\frac{x^{4}\langle\overline{q}q\rangle\langle
g_{s}^{2}G^{2}\rangle}{27648}\delta_{ab}$
$\displaystyle+\frac{m_{q}g_{s}}{32\pi^{2}}G_{ab}^{\mu\nu}\sigma_{\mu\nu}\left[\ln\left(\frac{-x^{2}\Lambda^{2}}{4}\right)+2\gamma_{E}\right]+\cdots,$
(15)

and

$\displaystyle
S_{Q}^{ab}(x)=\frac{m_{Q}^{2}\delta_{ab}}{4\pi^{2}}\left[\frac{K_{1}\left(m_{Q}\sqrt{-x^{2}}\right)}{\sqrt{-x^{2}}}+i\frac{\not{x}K_{2}\left(m_{Q}\sqrt{-x^{2}}\right)}{\left(\sqrt{-x^{2}}\right)^{2}}\right]$
$\displaystyle-\frac{g_{s}m_{Q}}{16\pi^{2}}\int_{0}^{1}duG_{ab}^{\mu\nu}(ux)\left.\Bigg{\\{}(\sigma_{\mu\nu}\not{x}+\not{x}\sigma_{\mu\nu})\right.$
$\displaystyle\left.\times\frac{K_{1}\left(m_{Q}\sqrt{-x^{2}}\right)}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\left(m_{Q}\sqrt{-x^{2}}\right)\right\\}.$
(16)

Here, $q=u,~{}d$ or $s$, $\gamma_{E}\simeq 0.577$ is the Euler constant, and
$\Lambda$ is the QCD scale parameter. We also introduce the notations
$G_{ab}^{\mu\nu}\equiv G_{A}^{\mu\nu}t_{ab}^{A}$,
$G^{2}=G^{\alpha\beta}G_{\alpha\beta}$, $A=1,2,\cdots,8$, and
$t^{A}=\lambda^{A}/2$, with $\lambda^{A}$ being the Gell-Mann matrices. The
first two terms in Eq. (16) in square brackets are the free part of the heavy
quark propagator in the coordinate representation, and $K_{n}(z)$ are the
modified Bessel functions of the second kind.

After performing required calculations, for $\Pi^{\mathrm{OPE}}(p)$ we get

$\Pi^{\mathrm{OPE}}(p)=\not{p}\Pi_{1}^{\mathrm{OPE}}(p^{2})+I\Pi_{2}^{\mathrm{OPE}}(p^{2}).$
(17)

The function $\Pi^{\mathrm{OPE}}(p)$ expressed in terms of quark-gluon degrees
of freedom has the same Lorentz structure as $\Pi^{\mathrm{Phys}}(p)$. By
equating two representations of the correlation function, performing the Borel
transformation and subtracting contributions due to higher resonances and
continuum states, we extract sum rule equalities.

It is not difficult to see that the Borel transformation of
$\Pi^{\mathrm{Phys}}(p)$ is equal to

$\mathcal{B}\Pi^{\mathrm{Phys}}(p)=\lambda^{2}e^{-\frac{m^{2}}{M^{2}}}(\not{p}+m).$
(18)

To derive the sum rule for $m$, it is enough to use the equality

$\lambda^{2}e^{-\frac{m^{2}}{M^{2}}}=\Pi_{1}^{\mathrm{OPE}}(M^{2},s_{0}),\ $
(19)

where $\Pi_{1}^{\mathrm{OPE}}(M^{2},s_{0})$ is the Borel transformed and
subtracted invariant amplitude $\Pi_{1}^{\mathrm{OPE}}(p^{2})$, and $M^{2}$
and $s_{0}$ are the Borel and continuum threshold parameters, respectively.
The sum rule for the mass of the ground state particle has a simple form and
can be found from the formula

$m^{2}=\frac{\Pi_{1}^{\prime\mathrm{OPE}}(M^{2},s_{0})}{\Pi_{1}^{\mathrm{OPE}}(M^{2},s_{0})}.$
(20)

In Eq. (20), we use the short-hand notation
$\Pi_{1}^{\prime\mathrm{OPE}}(M^{2},s_{0})=d/d(-1/M^{2})\Pi_{1}^{\mathrm{OPE}}(M^{2},s_{0})$.

In order to derive sum rules for parameters of the excited states, let us note
that the current $\eta$ couples not only to ground-state particle
$\Xi_{c}^{\prime}$, but also to its first orbital and radial excitations
$\Xi_{c}^{\prime}(1/2^{-})$, and $\Xi_{c}^{\prime}(1/2^{+})$, respectively. To
write down the phenomenological side of the sum rule, we use the ”ground-
state+excited-state+continuum” scheme. Therefore, we take into account effects
of the baryons $\Xi_{c}^{\prime}$ and $\Xi_{c}^{\prime}(1/2^{-})$, and find

$\displaystyle\Pi^{\mathrm{Phys}}(p)=\frac{\langle
0|\eta|\Xi_{c}^{\prime}(p,s)\rangle\langle\Xi_{c}^{\prime}(p,s)|\overline{\eta}|0\rangle}{m^{2}-p^{2}}$
$\displaystyle+\frac{\langle
0|\eta|\Xi_{c}^{\prime}(1/2^{-},p,\widetilde{s})\rangle\langle\Xi_{c}^{\prime}(1/2^{-},p,\widetilde{s})|\overline{\eta}||0\rangle}{\widetilde{m}^{2}-p^{2}}+\cdots,$
(21)

where $\widetilde{m}$ and $\widetilde{s}$ are the mass and spin of the excited
state $\Xi_{c}^{\prime}(1/2^{-})$.

To simplify $\Pi^{\mathrm{Phys}}(p)$, we employ Eq. (12) and additionally
introduce the matrix element

$\langle
0|\eta|\Xi_{c}^{\prime}(1/2^{-},p,\widetilde{s})\rangle=\widetilde{\lambda}\gamma_{5}\widetilde{u}(p,\widetilde{s}),$
(22)

where $\widetilde{\lambda}$ is the pole residue of the baryon
$\Xi_{c}^{\prime}(1/2^{-})$. Performing summations over $s$ and
$\widetilde{s}$ in Eq. (21) by employing relevant matrix elements and the
formula (13), we get

$\Pi^{\mathrm{Phys}}(p)=\frac{\lambda^{2}(\not{p}+m)}{m^{2}-p^{2}}+\frac{\widetilde{\lambda}^{2}(\not{p}-\widetilde{m})}{\widetilde{m}^{2}-p^{2}}+\cdots.$
(23)

The Borel transformation of $\Pi^{\mathrm{Phys}}(p)$ is equal to

$\displaystyle\mathcal{B}\Pi^{\mathrm{Phys}}(p)=\lambda^{2}e^{-\frac{m^{2}}{M^{2}}}(\not{p}+m)$
$\displaystyle+\widetilde{\lambda}^{2}e^{-\frac{\widetilde{m}^{2}}{M^{2}}}(\not{p}-\widetilde{m}).$
(24)

Then, the sum rule equalities are

$\lambda^{2}e^{-\frac{m^{2}}{M^{2}}}+\widetilde{\lambda}^{2}e^{-\frac{\widetilde{m}^{2}}{M^{2}}}=\Pi_{1}^{\mathrm{OPE}}(M^{2},\widetilde{s}_{0}),$
(25)

and

$\lambda^{2}me^{-\frac{m^{2}}{M^{2}}}-\widetilde{\lambda}^{2}\widetilde{m}e^{-\frac{\widetilde{m}^{2}}{M^{2}}}=\Pi_{2}^{\mathrm{OPE}}(M^{2},\widetilde{s}_{0}).$
(26)

The first of these expressions is extracted from the structure $\sim\not{p}$,
whereas the second one corresponds to terms proportional to $I$.

The derived equalities (25) and (26) contain four parameters $(m,\ \lambda)$
and $(\widetilde{m},\ \widetilde{\lambda})$ of the ground state and first
orbitally excited baryon. As the mass $m$ of the ground-state baryon
$\Xi_{c}^{\prime}$, we use its value evaluated from the sum rule (20).
Therefore, one has to find sum rules for the pole residue of the ground-state
particle, as well as parameters $(\widetilde{m},\ \widetilde{\lambda})$ of the
excited state. Usual way to handle this problem is to act by the operator
$d/d(-1/M^{2})$ to Eqs. (25) and (26), and get missing equations. Then, after
simple manipulations, we obtain

$\displaystyle\widetilde{m}^{2}$ $\displaystyle=$
$\displaystyle\frac{\Pi_{2}^{\prime\mathrm{OPE}}-m\Pi_{1}^{\prime\mathrm{OPE}}}{\Pi_{2}^{\mathrm{OPE}}-m\Pi_{1}^{\mathrm{OPE}}},$
$\displaystyle\lambda^{2}$ $\displaystyle=$
$\displaystyle\frac{\widetilde{m}\Pi_{1}^{\mathrm{OPE}}+\Pi_{2}^{\mathrm{OPE}}}{m+\widetilde{m}}e^{m^{2}/M^{2}}$
$\displaystyle\widetilde{\lambda}^{2}$ $\displaystyle=$
$\displaystyle\frac{m\Pi_{1}^{\mathrm{OPE}}-\Pi_{2}^{\mathrm{OPE}}}{m+\widetilde{m}}e^{\widetilde{m}^{2}/M^{2}}.$
(27)

Expressions written down in Eq. (27) are the QCD two-point sum rules for
parameters of the ground-state and excited baryon, which can be employed to
evaluate their numerical values. In these formulas, for simplicity, we do not
show dependence of the functions
$\Pi_{1,2}^{(\prime)\mathrm{OPE}}(M^{2},\widetilde{s}_{0})$ on $M^{2}$ and
$\widetilde{s}_{0}$. The parameters of the radially excited baryon
$\Xi_{c}^{\prime}(1/2^{+})$ can be extracted using $\Pi^{\mathrm{Phys}}(p)$,
in which the excited $1P$ state is replaced by $2S$ particle. In Eq. (27) this
is equivalent to transformation $\widetilde{m}\rightarrow-m^{\prime}$, and
redefinition of the residue $\widetilde{\lambda}\rightarrow\lambda^{\prime}$,
where $(m^{\prime}$, $\lambda^{\prime})$ are parameters of
$\Xi_{c}^{\prime}(1/2^{+})$.

The sum rules (20) and (27) depend on the vacuum expectations values of the
different quark, gluon, and mixed operators, as well as on the masses of $s$
and $c$-quarks. Values of these universal input parameters are presented below

$\displaystyle\langle\overline{q}q\rangle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3},\
\langle\overline{s}s\rangle=0.8\langle\bar{q}q\rangle,$
$\displaystyle\langle\overline{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\overline{q}q\rangle,\
\langle\overline{s}g_{s}\sigma
Gs\rangle=m_{0}^{2}\langle\overline{s}s\rangle,$ $\displaystyle
m_{0}^{2}=(0.8\pm 0.1)~{}\mathrm{GeV}^{2},$
$\displaystyle\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle=(0.012\pm
0.004)~{}\mathrm{GeV}^{4},$ $\displaystyle
m_{s}=93_{-5}^{+11}~{}\mathrm{MeV},\ m_{c}=1.27\pm 0.2~{}\mathrm{GeV}.$ (28)

The sum rules contain also auxiliary parameters $M^{2}$ and $s_{0}$, which are
not arbitrary, but should meet some restrictions. Thus, inside of working
regions of these parameters convergence of the operator product expansion
should be fulfilled. The dominance of the pole contribution, and prevalence of
the perturbative term in the sum rules are also among constraints of
computations. The extracted predictions should be stable against variations of
$M^{2}$ and $\beta$: the latter is necessary for spin-$1/2$ particles. In
order to explore the dependence on $\beta$, it is convenient to introduce a
parameter $\cos\theta$ through $\beta=\tan\theta$.

Calculations of $\Pi^{\mathrm{OPE}}(p)$ are performed by including into
analysis nonperturbative terms till dimension $10$. In computations we set
$m_{d}=0$, but take into account terms $\sim m_{s}$. Explicit expressions of
the amplitudes $\Pi_{1}^{\mathrm{OPE}}(M^{2},s_{0})$ and
$\Pi_{2}^{\mathrm{OPE}}(M^{2},s_{0})$ in simple case $m_{s}=0$ can be found in
Appendix.

First, we calculate the mass of the ground-state particle $\Xi_{c}^{\prime}$.
The parameters $M^{2}$ and $s_{0}$ necessary for such analysis and prediction
obtained for $m$ are presented in Table 1. Results for the masses and residues
of the baryons $\Xi_{c}^{\prime}(1/2^{-})$ and $\Xi_{c}^{\prime}(1/2^{+})$ are
also collected in Table 1. In this Table, we write down the working regions
for $M^{2}$ and $\widetilde{s}_{0}$ used to evaluate $\widetilde{m},$
$\widetilde{\lambda}$, and $\lambda$. The auxiliary parameter $\cos\theta$ has
been varied inside of the boundaries

$-1.0\leq\cos\theta\leq-0.5,\ 0.5\leq\cos\theta\leq 1.0,$ (29)

where we have attained best stability for our predictions. In numerical
analysis, we use average values of quantities calculated at $\cos\theta=\pm
0.75$ from these regions.

Baryons | $\Xi_{c}^{\prime}$ | $\Xi_{c}^{\prime}(1/2^{-})$ | $\Xi_{c}^{\prime}(1/2^{+})$
---|---|---|---
$(n,J^{\mathrm{P}})$ | $(1S,\frac{1}{2}^{+})$ | $(1P,\frac{1}{2}^{-})$ | $(2S,\frac{1}{2}^{+})$
$M^{2}~{}(\mathrm{GeV}^{2}$) | $3-5$ | $3-5$ | $3-5$
$s_{0}~{}(\mathrm{GeV}^{2}$) | $2.8^{2}-3.0^{2}$ | $3.2^{2}-3.4^{2}$ | $3.2^{2}-3.4^{2}$
$m~{}(\mathrm{MeV})$ | $2576\pm 150$ | $2925\pm 115$ | $2925\pm 115$
$\lambda\times 10^{2}~{}(\mathrm{GeV}^{3})$ | $4.0\pm 0.5$ | $3.9\pm 1.3$ | $15.4\pm 5.0$
Table 1: The sum rule results for the masses and residues of the spin-$1/2$
flavor-sextet baryons $\Xi_{c}^{\prime}$.

The amplitudes $\Pi_{1(2)}^{\mathrm{OPE}}(M^{2},s_{0})$ are formed mainly due
to perturbative contributions. In Fig. 1, as an example, we present different
components of the amplitude $\Pi_{1}^{\mathrm{OPE}}(M^{2},s_{0})$ as functions
of $M^{2}$ at fixed $s_{0}=3.3^{2}~{}\mathrm{MeV}^{2}$. It is seen, that
perturbative term is a dominant contribution to $\Pi_{1}^{\mathrm{OPE}}$.
Nonperturbative effects are small, and only $\mathrm{Dim3}$ term with quark
condensate factor $\langle\overline{s}s\rangle+\langle\overline{d}d\rangle$
can be considered as essential one. Higher dimensional contributions $N>6$ are
very small: Corresponding curves are undistinguishable in the plot, for this
reason, we do not included them into the figure. In the case under
consideration, the operator product expansion demonstrates rapid convergence.
Thus, at $M^{2}=4~{}\mathrm{MeV}^{2}$ already $\mathrm{Dim5}$ term is less
than $0.01$ of the perturbative contribution.

Figure 1:  The different contributions to
$\Pi_{1}^{\mathrm{OPE}}(M^{2},s_{0})$ at fixed
$s_{0}=3.3^{2}~{}\mathrm{MeV}^{2}$.

Reliable predictions for physical quantities imply dominance of the pole
contribution ($\mathrm{PC}$) in the sum rule analysis. In the ”ground-
state+continuum” scheme fixed by $s_{0}\sim 2.9^{2}\ ~{}\mathrm{MeV}^{2}$, a
$\mathrm{PC}$ higher than $50\%$ of the whole result leads to credible
predictions for parameters of the ground-state baryon. In the ”ground-state
+first excited state+continuum” scheme determined by $\widetilde{s}_{0}\sim
3.3^{2}\ ~{}\mathrm{MeV}^{2}$, there are two particles that generate the pole
contribution. In our case, these two baryons constitute $79\%$ of the total
contribution in average. In other words, excited state form approximately
$25\%$ of the whole contribution. This is less than $50\%$ limit necessary for
the ground-state or isolated excited particle. But mass and pole residue of
the excited baryon are obtained from expressions, which contain contributions
of the ground-state baryon as well. For these expressions, as we have noted
above, $\mathrm{PC}\approx 79\%$ which assures correctness of extracted
quantities. It is worth noting that continuum threshold parameters $s_{0}$ and
$\widetilde{s}_{0}$ in this two schemes obey the restriction
$s_{0}<\widetilde{s}_{0}$ which demonstrates the self-consistency of performed
analyses. The central values of the masses of the excited baryons
$\widetilde{m}$ $=m^{\prime}$ are above $\sqrt{s_{0}}$ and below
$\sqrt{\widetilde{s}_{0}}$, as they should be.

In Fig. 2, we plot the mass of the particle $\Xi_{c}^{\prime}(1/2^{-})$ as a
function of $M^{2}$. Here, one can see dependence of the obtained result on
the Borel parameter $M^{2}$, which have been pictured at fixed values of the
continuum threshold parameter $s_{0}$. The residues of the excited baryons
$\Xi_{c}^{\prime}(1/2^{-})$ and $\Xi_{c}^{\prime}(1/2^{+})$ are drawn in Fig.
3, where sensitivity of $\ \widetilde{\lambda}$ and $\lambda^{\prime}$ to a
choice of $M^{2}$ is shown. All these parameters are very stable against
variation of the Borel parameter. The main part of theoretical uncertainties
come from variation of the continuum threshold parameter $s_{0}$, which is
also seen in these figures.

Figure 2:  The mass of the baryon $\Xi_{c}^{\prime}(1/2^{-})$ as a function of
the parameter $M^{2}$ at fixed $s_{0}$.

Figure 3:  The dependence of the residues $\widetilde{\lambda}$ (left panel)
and $\lambda^{\prime}$ (right panel) on the Borel parameter $M^{2}$ and at
fixed $s_{0}$.

The similar analysis with some new technical details can be carried out for
the spin-$3/2$ baryons $\Xi_{c}^{\ast}$ as well. Indeed, in this case, in
order to find the physical side of the sum rules, we use the matrix elements

$\displaystyle\langle
0|\eta_{\mu}|\Xi_{c}^{\ast}(p,s)\rangle=\lambda^{\ast}u_{\mu}(p,s),$
$\displaystyle\langle
0|\eta_{\mu}|\Xi_{c}^{\ast}(3/2^{-},p,\widetilde{s})\rangle=\widetilde{\lambda}^{\ast}\gamma_{5}\widetilde{u}_{\mu}(p,\widetilde{s}),$
(30)

where $u_{\mu}(p,s)$ and $\widetilde{u}_{\mu}(p,\widetilde{s})$ are the
Rarita-Schwinger spinors, and perform the summation over the spins $s$ and
$\widetilde{s}$ using the expression

$\sum\limits_{s}u_{\mu}(p,s)\overline{u}_{\nu}(p,s)=-(\not{p}+m^{\ast})F_{\mu\nu}(m^{\ast},p),$
(31)

where

$\displaystyle F_{\mu\nu}(m^{\ast},p)$ $\displaystyle=$
$\displaystyle\left[g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2}{3m^{\ast
2}}p_{\mu}p_{\nu}\right.$ (32)
$\displaystyle\left.+\frac{1}{3m^{\ast}}(p_{\mu}\gamma_{\nu}-p_{\nu}\gamma_{\mu})\right].$

Here, $m^{\ast}$ is the mass of the spin-$3/2$ baryons $\Xi_{c}^{\ast}$.

In calculations one should take into account that the interpolating current
$\eta_{\mu}$ couples both to spin-$3/2$ and spin-$1/2$ baryons. Therefore, the
sum rules contain contributions of spin-$1/2$ particles as well. These terms
should be removed by using a special ordering of the Dirac matrices. Indeed,
it is easy to demonstrate that structures $\sim\not{p}g_{\mu\nu}$ and $\sim
g_{\mu\nu}$ are formed only due to contributions of spin-$3/2$ baryons.
Therefore, to find the sum rules for the mass and residue of the ground state
particle $\Xi_{c}^{\ast}$, and parameters of the excited baryons
$\Xi_{c}^{\ast}(3/2^{-})$, and $\Xi_{c}^{{}^{\ast}}(3/2^{+})$, we use only
these structures and corresponding invariant amplitudes.

To determine the QCD side of the sum rules, the correlation function
$\Pi_{\mu\nu}(p)$ has to be computed also in terms of the quark propagators.
We calculate $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ by utilizing Eq. (8) and the
current given by Eq. (10). Operations to find $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$
using the quark propagators in the $x$-space and calculation of the Borel
transformed and subtracted invariant amplitudes are well known and were
presented in the literature (for instance see Agaev:2016srl ; Aliev:2011ufa ;
Azizi:2015ksa ; Aliev:2018ube ; Azizi:2018duk ). Thus, we do not go into
further details of these computations, and emphasize only that analysis has
been performed with dimension-$10$ accuracy. We also note that the final
results are very lengthy, so we do not present their explicit expressions
here.

Results obtained for parameters of the spin-$3/2$ baryons $\Xi_{c}^{\ast}$,$\
\Xi_{c}^{\ast}(3/2^{-})$, and $\Xi_{c}^{{}^{\ast}}(3/2^{+})$ are presented in
Table 2. Here, we write down the working regions for parameters $M^{2}$ and
$s_{0}$ used to evaluate $m^{\ast}$ and $\lambda^{\ast}$.

Baryons | $\Xi_{c}^{\star}$ | $\Xi_{c}^{\star}(3/2^{-})$ | $\Xi_{c}^{\star}(3/2^{+})$
---|---|---|---
$(n,J^{\mathrm{P}})$ | $(1S,\frac{3}{2}^{+})$ | $(1P,\frac{3}{2}^{-})$ | $(2S,\frac{3}{2}^{+})$
$M^{2}~{}(\mathrm{GeV}^{2}$) | $3-5$ | $3-5$ | $3-5$
$s_{0}~{}(\mathrm{GeV}^{2}$) | $3.0^{2}-3.2^{2}$ | $3.2^{2}-3.4^{2}$ | $3.2^{2}-3.4^{2}$
$m^{*}~{}(\mathrm{MeV})$ | $2655\pm 102$ | $2960\pm 67$ | $2960\pm 67$
$\lambda^{*}\times 10^{2}~{}(\mathrm{GeV}^{3})$ | $4.7\pm 0.4$ | $2.4\pm 0.3$ | $10.4\pm 1.4$
Table 2: The predictions for spectroscopic parameters of the spin-$3/2$
baryons $\Xi_{c}^{\ast}$.

The masses and residues of the baryons $\Xi_{c}^{\ast}$ as functions of the
parameters $M^{2}$ and $s_{0}$ demonstrate behavior similar to ones of the
spin-$1/2$ particles, therefore we do not provide corresponding graphics, by
noting that systematic errors of calculations do not exceed limits accepted in
the sum rule method.

As is seen, the sum rule method employed in the present work to find masses of
the spin-$1/2$ and -$3/2$ baryons $\Xi_{c}^{\prime}$ and $\Xi_{c}^{\ast}$
leads for the first orbitally and radially excited states to the same
predictions. Therefore, relying only on this information, it is impossible to
make assignment for three new resonances observed by the LHCb collaboration.
To compare with LHCb experimental data, one needs to determine also widths of
these particles.

##  III  Decays of $\Xi_{c}^{\prime}(1/2^{-})$ and $\Xi_{c}^{\prime}(1/2^{+})$
to $\Lambda_{c}^{+}K^{-}$

In this section we study the vertices
$\Xi_{c}^{\prime}(1/2^{-})\Lambda_{c}^{+}K^{-}$ and
$\Xi_{c}^{\prime}(1/2^{+})\Lambda_{c}^{+}K^{-}$, and calculate corresponding
strong couplings, which are required to compute width of the decays
$\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}$ and
$\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}$, respectively.
There are different sum rule methods to extract numerical values of these
couplings. They can be calculated using both the QCD three-point and light-
cone sum rule methods. But LCSR method have some advantages compared to the
three-point sum rule approach in calculations of the strong couplings and form
factors. The reason is that in the three-point sum rules higher orders in OPE
are enhanced by powers of the heavy quark mass, and for sufficiently large
masses OPE breaks down. The LCSR method does not suffer from such problems,
because it is consistent with the heavy-quark limit and provides more
effective tools for investigations than alternative approaches Belyaev:1994zk
. Therefore, for analyses of the vertices
$\Xi_{c}^{\prime}(1/2^{-})\Lambda_{c}^{+}K^{-}$ and
$\Xi_{c}^{\prime}(1/2^{+})\Lambda_{c}^{+}K^{-}$, we use the QCD LCSR method.

To this end, we start from analysis of the correlation function

$\Pi(p,q)=i\int d^{4}xe^{ipx}\langle
K(q)|\mathcal{T}\\{\eta_{\Lambda}(x)\overline{\eta}(0)\\}|0\rangle,$ (33)

where $\eta_{\Lambda}(x)$ is the interpolating field for the $\Lambda_{c}$
baryon. The $\Lambda_{c}$ is the flavor-antitriplet spin-$1/2$ particle, and
its current is given by the expression

$\displaystyle\eta_{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\epsilon^{abc}\left\\{2\left(u_{a}^{T}Cd_{b}\right)\gamma_{5}c_{c}+2\widetilde{\beta}\left(u_{a}^{T}C\gamma_{5}d_{b}\right)c_{c}\right.$
(34)
$\displaystyle+\left(u_{a}^{T}Cc_{b}\right)\gamma_{5}d_{c}+\widetilde{\beta}\left(u_{a}^{T}C\gamma_{5}c_{b}\right)d_{c}$
$\displaystyle\left.+\left(c_{a}^{T}Cd_{b}\right)\gamma_{5}u_{c}+\widetilde{\beta}\left(c_{a}^{T}C\gamma_{5}d_{b}\right)u_{c}\right\\},$

where $\widetilde{\beta}$ is the arbitrary mixing parameter.

First, we write the correlation function $\Pi(p,q)$ in terms of involved
baryons’ parameters, and find by this way the physical or hadronic side of the
sum rule. As a result, we obtain

$\displaystyle\Pi^{\mathrm{Phys}}(p,q)=\frac{\langle
0|\eta_{\Lambda}|\Lambda_{c}^{+}(p,s)\rangle}{p^{2}-m_{\Lambda}^{2}}\langle
K(q)\Lambda_{c}^{+}(p,s)|\Xi_{c}^{\prime}(p^{\prime},s^{\prime})\rangle$
$\displaystyle\times\frac{\langle\Xi_{c}^{\prime}(p^{\prime},s^{\prime})|\overline{\eta}|0\rangle}{p^{\prime
2}-m^{2}}+\frac{\langle
0|\eta_{\Lambda}|\Lambda_{c}^{-}(p,s)\rangle}{p^{2}-\widetilde{m}_{\Lambda}^{2}}$
$\displaystyle\times\langle
K(q)\Lambda_{c}^{-}(p,s)|\Xi_{c}^{\prime}(p^{\prime},s^{\prime})\rangle\frac{\langle\Xi_{c}^{\prime}(p^{\prime},s^{\prime})|\overline{\eta}|0\rangle}{p^{\prime
2}-m^{2}}$ $\displaystyle+\frac{\langle
0|\eta_{\Lambda}|\Lambda_{c}^{+}(p,s)\rangle}{p^{2}-m_{\Lambda}^{2}}\langle
K(q)\Lambda_{c}^{+}(p,s)|\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})\rangle$
$\displaystyle\times\frac{\langle\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})|\overline{\eta}|0\rangle}{p^{\prime
2}-\widetilde{m}^{2}}+\frac{\langle
0|\eta_{\Lambda}|\Lambda_{c}^{-}(p,s)\rangle}{p^{2}-\widetilde{m}_{\Lambda}^{2}}$
$\displaystyle\times\langle
K(q)\Lambda_{c}^{-}(p,s)|\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})\rangle$
$\displaystyle\times\frac{\langle\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})|\overline{\eta}|0\rangle}{p^{\prime
2}-\widetilde{m}^{2}}+\cdots,$ (35)

where $p^{\prime}=p+q,\ p$ and $q$ are the momenta of the $\Xi_{c}^{\prime},$
$\Lambda_{c}$ baryons and $K$ meson, respectively. The $\Lambda_{c}^{+}$ and
$\Lambda_{c}^{-}$ are baryons with quantum numbers $(1S,1/2^{+})$ and
$(1P,1/2^{-})$, and masses $m_{\Lambda}$ and $\widetilde{m}_{\Lambda}$,
respectively. The dots in Eq. (35) stand for contributions of the higher
resonances and continuum states.

To continue, we introduce the matrix elements of the baryons $\Lambda_{c}$

$\displaystyle\langle 0|\eta_{\Lambda}|\Lambda_{c}^{+}(p,s)\rangle$
$\displaystyle=$ $\displaystyle\lambda_{\Lambda}u(p,s),$ $\displaystyle\langle
0|\eta_{\Lambda}|\Lambda_{c}^{-}(p,s)\rangle$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{\Lambda}\gamma_{5}u(p,s),$ (36)

and also parametrize remaining unknown matrix elements in terms of the strong
couplings Aliev:2010yx ; Aliev:2018ube

$\displaystyle\langle
K(q)\Lambda_{c}^{+}(p,s)|\Xi_{c}^{\prime}(p^{\prime},s^{\prime})\rangle=g_{0}\overline{u}(p,s)\gamma_{5}u(p^{\prime},s^{\prime}),$
$\displaystyle\langle
K(q)\Lambda_{c}^{-}(p,s)|\Xi_{c}^{\prime}(p^{\prime},s^{\prime})\rangle=\widetilde{g}_{0}\overline{u}(p,s)u(p^{\prime},s^{\prime}),$
$\displaystyle\langle
K(q)\Lambda_{c}^{+}(p,s)|\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})\rangle=g_{1}\overline{u}(p,s)u(p^{\prime},s^{\prime}),$
$\displaystyle{}\langle
K(q)\Lambda_{c}^{-}(p,s)|\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})\rangle=\widetilde{g}_{1}\overline{u}(p,s)\gamma_{5}u(p^{\prime},s^{\prime}),$
(37)

where $\lambda_{\Lambda}$ and $\widetilde{\lambda}_{\Lambda}$ are pole
residues of $\Lambda_{c}^{+}$ and $\Lambda_{c}^{-}$, respectively.

Then using the matrix elements of the particles $\Xi_{c}^{\prime}$ and
$\Xi_{c}^{\prime}(1/2^{-})$, carrying out the summations over the spins $s$
and $s^{\prime}$, and applying the double Borel transformation with respect
$p^{2}$ and $p^{\prime 2}$, for the phenomenological side of the sum rules, we
obtain

$\displaystyle\mathcal{B}\Pi^{\mathrm{Phys}}(p^{2},p^{\prime
2})=g_{0}\lambda\lambda_{\Lambda}e^{-m^{2}/M_{1}^{2}}e^{-m_{\Lambda}^{2}/M_{2}^{2}}(\not{p}+m_{\Lambda})$
$\displaystyle\times\gamma_{5}\left(\not{p}^{\prime}+m\right)-\widetilde{g}_{0}\lambda\widetilde{\lambda}_{\Lambda}e^{-m^{2}/M_{1}^{2}}e^{-\widetilde{m}_{\Lambda}^{2}/M_{2}^{2}}(\not{p}-\widetilde{m}_{\Lambda})$
$\displaystyle\times\gamma_{5}\left(\not{p}^{\prime}+m\right)+g_{1}\widetilde{\lambda}\lambda_{\Lambda}e^{-\widetilde{m}^{2}/M_{1}^{2}}e^{-m_{\Lambda}^{2}/M_{2}^{2}}(\not{p}+m_{\Lambda})$
$\displaystyle\times\gamma_{5}\left(\not{p}^{\prime}-\widetilde{m}\right)-\widetilde{g}_{1}\widetilde{\lambda}\widetilde{\lambda}_{\Lambda}e^{-\widetilde{m}^{2}/M_{1}^{2}}e^{-\widetilde{m}_{\Lambda}^{2}/M_{2}^{2}}$
$\displaystyle\times(\not{p}-\widetilde{m}_{\Lambda})\gamma_{5}\left(\not{p}^{\prime}-\widetilde{m}\right),$
(38)

where $M_{1}^{2}$ and $M_{2}^{2}$ are the Borel parameters.

As is seen, Eq. (38) contains different Lorentz structures. To extract sum
rules, it is convenient to reorganize these terms into structures proportional
to $\not{q}\not{p}\gamma_{5}$, $\not{p}\gamma_{5}$, $\not{q}\gamma_{5}$ and
$\gamma_{5}$, and employ corresponding invariant amplitudes. The same
structures appear in the QCD side of the sum rule equality, which has to be
calculated using the quark propagators. After performing the double Borel
transformation of $\Pi^{\mathrm{OPE}}(p,q)$, we get
$\mathcal{B}\Pi^{\mathrm{OPE}}(p^{2},p^{\prime
2})=\Pi^{\mathrm{OPE}}(M_{1}^{2},M_{2}^{2})$ which is a function of two Borel
parameters. To proceed, it is convenient to introduce $M^{2}$ through the
relation

$\frac{1}{M^{2}}=\frac{1}{M_{1}^{2}}+\frac{1}{M_{2}^{2}},$ (39)

and use $M_{1}^{2}=M_{2}^{2}=2M^{2}$ to go from the double-dispersion integral
to the single integral representation by performing one of the dispersion
integrals. We set $M_{1}^{2}=M_{2}^{2}$ as the masses of the $\Xi_{c}$ and
$\Lambda_{c}$ baryons are close, and uncertainties expected due to this choice
are small. As a result, we get a single integral representation for
$\Pi^{\mathrm{OPE}}\left(M^{2}\right)$, which considerably simplifies the
continuum subtraction. By equating now $\Pi^{\mathrm{OPE}}(M^{2})$ with the
expression Eq. (38) and performing the continuum subtraction, we find the sum
rule equality which depends on $\Pi^{\mathrm{OPE}}(M^{2},s_{0})$: After the
subtraction procedure the correlation function
$\Pi^{\mathrm{OPE}}(M^{2},s_{0})$ acquires dependence on the continuum
threshold parameter $s_{0}$. The formulas necessary to carry out subtractions
can be found in Appendix B of Ref. Agaev:2016srl .

By equating invariant amplitudes corresponding to aforementioned Lorentz
structures in both sides of the sum rule equality, one finds four equations
which should be solved to determine sum rules for the strong couplings. We
denote invariant amplitudes corresponding to the structures
$\not{q}\not{p}\gamma_{5}$, $\not{p}\gamma_{5}$, $\not{q}\gamma_{5}$ and
$\gamma_{5}$ by $\Pi_{i}^{\mathrm{OPE}}(M^{2},s_{0})$, where $i=1,2,3$ and
$4$, respectively.

The solution of these equations for the coupling of interest $g_{1}$ is

$\displaystyle
g_{1}=\frac{e^{\widetilde{m}^{2}/M_{1}^{2}}e^{m_{\Lambda}^{2}/M_{2}^{2}}}{\widetilde{\lambda}\lambda_{\Lambda}(m+\widetilde{m})(m_{\Lambda}+\widetilde{m}_{\Lambda})}\left\\{\Pi_{1}^{\mathrm{OPE}}\left[m_{K}^{2}\right.\right.$
$\displaystyle\left.+m(\widetilde{m}_{\Lambda}-\widetilde{m})\right]+\Pi_{2}^{\mathrm{OPE}}(\widetilde{m}-m-\widetilde{m}_{\Lambda})$
$\displaystyle\left.+\Pi_{3}^{\mathrm{OPE}}(\widetilde{m}_{\Lambda}-\widetilde{m})-\Pi_{4}^{\mathrm{OPE}}\right\\}.$
(40)

Here, $m_{K}=(493.677\pm 0.016)~{}\mathrm{MeV}$ is the mass of the $K$ meson.
The sum rules for the strong coupling $g_{2}$ corresponding to the vertex
$\Xi_{c}^{\prime}(1/2^{+})\Lambda_{c}^{+}K^{-}$ and responsible for the decay
$\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}$ can be determined
from Eq. (40) by replacements $\widetilde{m}\rightarrow-m^{\prime}$ and
$\widetilde{\lambda}\rightarrow\lambda^{\prime}$.

In order to activate Eq. (40), it is necessary to calculate the correlation
function $\Pi^{\mathrm{OPE}}(p,q)$ and find the invariant amplitudes
$\Pi_{i}^{\mathrm{OPE}}(M^{2},s_{0})$. After contracting the quarks fields and
inserting into the obtained formula quark propagators, we get the expression
which depend on the non-local matrix elements of operators
$\overline{s}^{a}u^{b}$ placed between the states $\langle K(q)|$ and
$|0\rangle$. We should express the correlation function
$\Pi^{\mathrm{OPE}}(p,q)$ using the distribution amplitudes (DAs) of $K$ meson
with different quark-gluon compositions and twists. To this end, we use the
expansion

$\overline{s}_{\alpha}^{a}u_{\beta}^{b}=\frac{1}{12}\Gamma_{\beta\alpha}^{i}\delta_{ab}(\overline{s}\Gamma^{i}u),$
(41)

where $\ \Gamma^{i}=1,\ \gamma_{5},\ \gamma_{\mu},\ i\gamma_{5}\gamma_{\mu},\
\sigma_{\mu\nu}/\sqrt{2}$ are the Dirac matrices. These terms placed between
the $K$ meson and vacuum states generate the two-particle DAs of the leading
and nonleading twists. They are defined by the expressions Ball:2006wn

$\displaystyle\langle
0|\overline{q}(x)\gamma_{\mu}\gamma_{5}s(-x)|K(q)\rangle=if_{K}q_{\mu}\int_{0}^{1}due^{i\xi
qx}\left[\phi_{2:K}(u)\right.$
$\displaystyle\left.+\frac{1}{4}x^{2}\phi_{4:K}(u)\right]+\frac{i}{2}f_{K}\frac{x_{\mu}}{qx}\int_{0}^{1}due^{i\xi
qx}\psi_{4:K}(u),$ (42) $\langle
0|\overline{q}(x)i\gamma_{5}s(-x)|K(q)\rangle=\frac{f_{K}m_{K}^{2}}{m_{s}+m_{q}}\int_{0}^{1}due^{i\xi
qx}\phi_{3:K}^{p}(u),$ (43)

and

$\displaystyle\langle
0|\overline{q}(x)\sigma_{\alpha\beta}\gamma_{5}s(-x)|K(q)\rangle=-\frac{i}{3}\frac{f_{K}m_{K}^{2}}{m_{s}+m_{q}}$
$\displaystyle\times(q_{\alpha}x_{\beta}-q_{\beta}x_{\alpha})\int_{0}^{1}due^{i\xi
qx}\phi_{3:K}^{\sigma}(u),$ (44)

where $f_{K}=(155.72\pm 0.51)~{}\mathrm{MeV}$ is the decay constant of the $K$
meson. In expressions above $\xi=2u-1$, with $u$ being the longitudinal
momentum fraction carrying the quark in the $K$ meson. The subscripts in DAs
label the twist of these functions.

Figure 4: The leading twist diagram contributing to $\Pi^{\mathrm{OPE}}(p,q)$.

There are also three-particle twist-3 and -4 DAs of the kaon, which appear due
to insertions into operators $\overline{s}\Gamma^{i}u$ of the gluon field
strength tensor $G_{\lambda\rho}$ coming from quark propagators. The
definitions of these DAs and their models are collected in Ref. Ball:2006wn .
The main contribution to $\Pi^{\mathrm{OPE}}(p,q)$ arises from the terms,
where all the propagators are replaced by their perturbative components. It is
known as the leading twist contribution: the corresponding Feynman diagram is
plotted in Fig. 4. Contributions of terms containing three-particle DA of the
$K$ meson generate only nonleading twist effects. In the present work, we take
into account contributions due to two- and three-particle DAs including
twist-4 corrections. An analytic expression for the double Borel transformed
and subtracted correlation function $\Pi^{\mathrm{OPE}}(M^{2},s_{0})$ is
rather cumbersome, therefore we do not write down it here. From derived
expression of $\Pi^{\mathrm{OPE}}(M^{2},s_{0})$ one can extract invariant
amplitudes required for our calculations.

The functions $\Pi_{i}^{\mathrm{OPE}}(M^{2},s_{0})$ depend on DAs of $K$
meson. In numerical computations for these DAs, we have utilized models and
parameters presented in Ref. Ball:2006wn . Apart from DAs, the sum rules for
the couplings $g_{1}$ and $g_{2}$ contain also masses of the ground state
$\Lambda_{c}^{+}$ and first orbitally excited $\Lambda_{c}^{-}$ baryons for
which we use their values from Ref. Tanabashi:2018oca

$\displaystyle m_{\Lambda}$ $\displaystyle=$ $\displaystyle(2286.46\pm 0.14)\
\mathrm{MeV},\ $ $\displaystyle\widetilde{m}_{\Lambda}$ $\displaystyle=$
$\displaystyle(2592.25\pm 0.28)\ \mathrm{MeV.}$ (45)

The pole residue of $\Lambda_{c}^{+}$ denoted in Eq. (40) by
$\lambda_{\Lambda}$ is borrowed from the work Aliev:2018ube

$\lambda_{\Lambda}=(3.8\pm 0.9)\times 10^{-2}\ \mathrm{GeV}^{3}.$ (46)

The Borel and continuum threshold parameters for the decay of the baryons
$\Xi_{c}^{\prime}(1/2^{+})$ and $\Xi_{c}^{\prime}(1/2^{+})$ are fixed exactly
as in computations of their masses. The parameters $\beta$ and
$\widetilde{\beta}$ in the interpolating currents of $\Xi_{c}^{\prime}$ and
$\Lambda_{c}$ are taken equal to each other and varied within the limits
presented in Eq. (29).

Numerical calculations lead to the following predictions

$g_{1}=0.41\pm 0.04,\ |g_{2}|=7.19\pm 0.65.$ (47)

The widths of the decays
$\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}$ and
$\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}$ can be obtained in
terms of the strong couplings $g_{1}$ and $g_{2}$, respectively. They are
determined by the formulas

$\displaystyle\Gamma\left(\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}\right)$
$\displaystyle=$
$\displaystyle\frac{g_{1}^{2}}{8\pi\widetilde{m}^{2}}\left[(\widetilde{m}+m_{\Lambda})^{2}-m_{K}^{2}\right]$
(48) $\displaystyle\times f(\widetilde{m},m_{\Lambda},m_{K}),$

and

$\displaystyle\Gamma\left(\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle\frac{g_{2}^{2}}{8\pi m^{\prime
2}}\left[(m^{\prime}-m_{\Lambda})^{2}-m_{K}^{2}\right]$ (49)
$\displaystyle\times f(m^{\prime},m_{\Lambda},m_{K}),$

where the function $f(x,y,z)$ is given by the expression

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

The predictions for the width of the decays
$\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}$ and
$\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}$ are equal to

$\displaystyle\Gamma\left(\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle(7.3\pm 1.4)~{}\mathrm{MeV},$
$\displaystyle\Gamma\left(\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}\right)$
$\displaystyle=$ $\displaystyle(14.2\pm 2.7)~{}\mathrm{MeV}.$ (51)

Theoretical ambiguities in Eq. (51) are generated by the strong couplings
$g_{1}^{2}$ and $g_{2}^{2}$, and by the masses $\widetilde{m}$ and
$m^{\prime}$of excited baryons $\Xi_{c}^{\prime}(1/2^{-})$ and
$\Xi_{c}^{\prime}(1/2^{+})$, which have been extracted in the present work.
For the masses $m_{\Lambda}$, $\widetilde{m}_{\Lambda}$ and $m_{K}$, we use
their experimental values, which are known with high precision: Relevant
experimental errors are very small and do not affect error estimates for decay
widths.

Predictions for partial widths of these two channels can be used for further
studies of the baryons $\Xi_{c}^{\prime}(1/2^{-})$ and
$\Xi_{c}^{\prime}(1/2^{+})$.

##  IV Processes $\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Xi_{c}^{\prime}\pi$ and
$\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Xi_{c}^{\prime}\pi$

Analysis of the decays
$\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Xi_{c}^{\prime}\pi$ and
$\Xi_{c}^{\prime}(1/2^{+})$ $\rightarrow\Xi_{c}^{\prime}\pi$ does not differ
from our studies carried out in the previous section. The correlation function
to investigate these processes is given by the expression

$\Pi_{\pi}(p,q)=i\int
d^{4}xe^{ipx}\langle\pi(q)|\mathcal{T}\\{\eta(x)\overline{\eta}(0)\\}|0\rangle,$
(52)

with the same interpolating current $\eta(x)$.

Let us consider the decay
$\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Xi_{c}^{\prime}\pi$ to outline methods
of analysis. For this process the physical side of the LCSR has the form

$\displaystyle\Pi_{\pi}^{\mathrm{Phys}}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle
0|\eta|\Xi_{c}^{\prime}(p,s)\rangle}{p^{2}-m^{2}}\langle\pi(q)\Xi_{c}^{\prime}(p,s)|\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})\rangle$
(53)
$\displaystyle\times\frac{\langle\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})|\overline{\eta}|0\rangle}{p^{\prime
2}-\widetilde{m}^{2}}+\cdots.$

Matrix elements of the ground-state and excited baryons $\Xi_{c}^{\prime}$ and
$\Xi_{c}^{\prime}(1/2^{-})$ are well known. Additionally, we model the vertex
matrix element by introducing the strong coupling $g_{3}$

$\langle\pi(q)\Xi_{c}^{\prime}(p,s)|\Xi_{c}^{\prime}(1/2^{-},p^{\prime},s^{\prime})\rangle=g_{3}\overline{u}(p,s)u(p^{\prime},s^{\prime}).$
(54)

Then, the double Borel transformation of the correlation function is
determined by the expression

$\displaystyle\mathcal{B}\Pi_{\pi}^{\mathrm{Phys}}(p^{2},p^{\prime
2})=g_{3}\widetilde{\lambda}\lambda
e^{-\widetilde{m}^{2}/M_{1}^{2}}e^{-m^{2}/M_{2}^{2}}$
$\displaystyle\times(\not{p}+m)\gamma_{5}\left(\not{p}^{\prime}-\widetilde{m}\right).$
(55)

The QCD side of the LCSR is determined by the correlator $\Pi_{\pi}(p,q)$
expressed in terms of the $c$ and $s$ quark propagators and distribution
amplitudes of the pion. The matrix elements of operators
$\overline{d}(x)\Gamma^{j}d(0)$ in $\Pi_{\pi}^{\mathrm{OPE}}(p,q)$ can be
expanded over $x^{2}$ and expressed by means of the pion’s two-particle DAs of
different twist Braun:1989iv ; Ball:1998je . As examples, in the case of
$\Gamma=\ i\gamma_{\mu}\gamma_{5}$ and $\gamma_{5}$ we get

$\displaystyle\sqrt{2}\langle\pi^{0}(q)|\overline{d}(x)i\gamma_{\mu}\gamma_{5}d(0)|0\rangle$
(56) $\displaystyle=$ $\displaystyle
f_{\pi}q_{\mu}\int_{0}^{1}due^{i\overline{u}qx}\left[\phi_{\pi}(u)+\frac{m_{\pi}^{2}x^{2}}{16}\mathbb{A}_{4}(u)\right]$
$\displaystyle+\frac{f_{\pi}m_{\pi}^{2}}{2}\frac{x_{\mu}}{qx}\int_{0}^{1}due^{i\overline{u}qx}\mathbb{B}_{4}(u),$

and

$\displaystyle
2\sqrt{2}m_{d}\langle\pi^{0}(q)|\overline{d}(x)i\gamma_{5}d(0)|0\rangle=$
$\displaystyle\times
f_{\pi}m_{\pi}^{2}\int_{0}^{1}due^{iuqx}\phi_{3;\pi}^{p}(u),$ (57)

where $m_{\pi}=135~{}\mathrm{MeV}$ and $f_{\pi}=131~{}\mathrm{MeV}$ are
parameters of the pion $\pi^{0}$. In Eq. (56) $\phi_{\pi}(u)$ is the twist-2
distribution amplitude, and $\mathbb{A}_{4}(u)$ and $\mathbb{B}_{4}(u)$ are
higher-twist functions that can be expressed in terms of the pion two-particle
twist-4 DAs. One of two-particle twist-3 distributions $\phi_{3;\pi}^{p}(u)$
determines the matrix element given by Eq. (57). Another twist-3 DA
$\phi_{3;\pi}^{\sigma}(u)$ corresponds to matrix element with
$\sigma_{\mu\nu}$ insertion. The non-local operators that appear due to a
gluon field strength tensor $G_{\mu\nu}(ux)$ insertions to
$\overline{d}(x)\Gamma^{j}d(0)$ generate the pion’s three-particle
distributions. Their definitions and further details were presented in Ref.
Ball:2006wn ; Braun:1988qv ; Braun:1989iv ; Ball:1998je .

The leading twist DA $\phi_{\pi}(u)$ is expressible in terms of the Gegenbauer
polynomials $C_{2n}^{3/2}(\varsigma)$

$\phi_{\pi}(u,\mu^{2})=6u\overline{u}\left[1+\sum_{n=1,2,3\ldots}a_{2n}(\mu^{2})C_{2n}^{3/2}(u-\overline{u})\right],$
(58)

where $\overline{u}=1-u$. The coefficients $a_{2n}(\mu^{2})$ depend on a scale
$\mu$, as a result, $\phi_{\pi}(u,\mu^{2})$ is a function of $\mu$. The
Gegenbauer moments $a_{2n}(\mu^{2})$ at some normalization point $\mu=\mu_{0}$
should be fixed from phenomenological analysis or computed by employing, for
example, lattice simulations.

We derive the sum rule for the coupling $g_{3}$ by employing invariant
amplitudes corresponding to the structure $\not{q}\not{p}\gamma_{5}$ both in
$\Pi_{\pi}^{\mathrm{Phys}}(p,q)$ and $\Pi_{\pi}^{\mathrm{OPE}}(p,q)$, and get

$g_{3}=\frac{e^{\widetilde{m}^{2}/M_{1}^{2}}e^{m^{2}/M_{2}^{2}}}{\widetilde{\lambda}\lambda}\Pi_{\pi}^{\mathrm{OPE}},$
(59)

where $\Pi_{\pi}^{\mathrm{OPE}}$ is the relevant amplitude. In computations
the Borel and continuum threshold parameters $M^{2}$ and $s_{0}$ are chosen as
in Table 1. Parameters of the pion’s DAs are borrowed from Refs. Braun:1988qv
; Braun:1989iv . Numerical calculations of $g_{3}$ and the partial width of
the decay $\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Xi_{c}^{\prime}\pi$ lead to
results

$g_{3}=0.14\pm 0.02,$ (60)

and

$\Gamma\left(\Xi_{c}^{\prime}(1/2^{-})\rightarrow\Xi_{c}^{\prime}\pi\right)=(1.1\pm
0.2)~{}\mathrm{MeV}.$ (61)

The similar analysis can be done in the case of the second process
$\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Xi_{c}^{\prime}\pi$. The strong coupling
$g_{4}$ can be obtained from Eq. (59) by replacements
$\widetilde{m}\rightarrow m^{\prime}$ and
$\widetilde{\lambda}\rightarrow\lambda^{\prime}$. Computations give following
predictions

$\displaystyle g_{4}=1.04\pm 0.16,$
$\displaystyle\Gamma\left(\Xi_{c}^{\prime}(1/2^{+})\rightarrow\Xi_{c}^{\prime}\pi\right)=(0.39\pm
0.09)~{}\mathrm{MeV}.$ (62)

Information obtained here will be used to evaluate widths of the baryons
$\Xi_{c}^{\prime}(1/2^{-})$ and $\Xi_{c}^{\prime}(1/2^{+})$.

##  V  Decay channels
$\Xi_{c}^{\ast}(3/2^{-})\rightarrow\Lambda_{c}^{+}K^{-},\Xi_{c}^{\prime}\pi$
and
$\Xi_{c}^{{}^{\ast}}(3/2^{+})\rightarrow\Lambda_{c}^{+}K^{-},\Xi_{c}^{\prime}\pi$

The decays of the spin-$3/2$ baryons $\Xi_{c}^{\ast}(3/2^{-})$ and
$\Xi_{c}^{{}^{\ast}}(3/2^{+})$ to the final state $\Lambda_{c}^{+}\,K^{-}$ can
be explored by a manner described above for the spin-$1/2$ particles. To this
end, we begin from calculation of the correlation function

$\Pi_{\mu}(p,q)=i\int d^{4}xe^{ipx}\langle
K(q)|\mathcal{T}\\{\eta_{\Lambda}(x)\overline{\eta}_{\mu}(0)\\}|0\rangle,$
(63)

where $\eta_{\mu}$ is the interpolating current for spin-$3/2$ baryons
$\Xi_{c}^{\ast}$ given by Eq. (10).

To calculate the phenomenological side of the sum rules
$\Pi_{\mu}^{\mathrm{Phys}}(p,q)$, we write down it in the form similar to one
presented in Eq. (35) with simple modifications. We also define the strong
couplings $G_{0(1)}$ and $\widetilde{G}_{0(1)}$ using the matrix elements

$\displaystyle\langle
K(q)\Lambda_{c}^{+}(p,s)|\Xi_{c}^{\ast}(p^{\prime},s^{\prime})\rangle=G_{0}\overline{u}(p,s)u_{\alpha}(p^{\prime},s^{\prime})q^{\alpha},$
$\displaystyle\langle
K(q)\Lambda_{c}^{-}(p,s)|\Xi_{c}^{\ast}(p^{\prime},s^{\prime})\rangle=\widetilde{G}_{0}\overline{u}(p,s)\gamma_{5}u_{\alpha}(p^{\prime},s^{\prime})q^{\alpha},$
$\displaystyle\langle
K(q)\Lambda_{c}^{+}(p,s)|\Xi_{c}^{\ast}(3/2^{-},p^{\prime},s^{\prime})\rangle=G_{1}\overline{u}(p,s)\gamma_{5}u_{\alpha}(p^{\prime},s^{\prime})q^{\alpha},$
$\displaystyle{}\langle
K(q)\Lambda_{c}^{-}(p,s)|\Xi_{c}^{\ast}(3/2^{-},p^{\prime},s^{\prime})\rangle=\widetilde{G}_{1}\overline{u}(p,s)u_{\alpha}(p^{\prime},s^{\prime})q^{\alpha}.$
(64)

After some manipulations, for the Borel transformation of
$\Pi_{\mu}^{\mathrm{Phys}}(p,q)$, we obtain the following expression

$\displaystyle\mathcal{B}\Pi_{\mu}^{\mathrm{Phys}}(p^{2},p^{\prime
2})=G_{0}\lambda^{\ast}\lambda_{\Lambda}e^{-m^{\ast
2}/M_{1}^{2}}e^{-m_{\Lambda}^{2}/M_{2}^{2}}(\not{p}+m_{\Lambda})$
$\displaystyle\times\left(\not{p}^{\prime}+m^{\ast}\right)F_{\alpha\mu}(m^{\ast},p^{\prime})q^{\alpha}-\widetilde{G}_{0}\lambda^{\ast}\widetilde{\lambda}_{\Lambda}e^{-m^{\ast
2}/M_{1}^{2}}$ $\displaystyle\times
e^{-\widetilde{m}_{\Lambda}^{2}/M_{2}^{2}}(\not{p}-\widetilde{m}_{\Lambda})\left(\not{p}^{\prime}+m^{\ast}\right)F_{\alpha\mu}(m^{\ast},p^{\prime})q^{\alpha}$
$\displaystyle+G_{1}\widetilde{\lambda}^{\ast}\lambda_{\Lambda}e^{-\widetilde{m}^{\ast
2}/M_{1}^{2}}e^{-m_{\Lambda}^{2}/M_{2}^{2}}(\not{p}+m_{\Lambda})\left(\not{p}^{\prime}-\widetilde{m}^{\ast}\right)\gamma_{5}$
$\displaystyle\times
F_{\alpha\mu}(\widetilde{m}^{\ast},p^{\prime})\gamma_{5}q^{\alpha}-\widetilde{G}_{1}\widetilde{\lambda}^{\ast}\widetilde{\lambda}_{\Lambda}e^{-\widetilde{m}^{\ast
2}/M_{1}^{2}}e^{-\widetilde{m}_{\Lambda}^{2}/M_{2}^{2}}$
$\displaystyle\times(\not{p}-\widetilde{m}_{\Lambda})\left(\not{p}^{\prime}-\widetilde{m}^{\ast}\right)\gamma_{5}F_{\alpha\mu}(\widetilde{m}^{\ast},p^{\prime})\gamma_{5}q^{\alpha}.$
(65)

To derive the sum rules, we use available structures in Eq. (65). The same
terms are fixed in $\mathcal{B}\Pi_{\mu}^{\mathrm{QCD}}(p^{2},p^{\prime 2})$
and matched with ones from
$\mathcal{B}\Pi_{\mu}^{\mathrm{Phys}}(p^{2},p^{\prime 2})$. The final
expressions of the strong couplings are rather lengthy, therefore we do not
write down them here.

The strong coupling required to compute the width of the decay
$\Xi_{c}^{\ast}(3/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}$ is $G_{1}$. The
coupling $G_{2}$ necessary to find the width of the process
$\Xi_{c}^{\ast}(3/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}$ can be obtained from
the relevant sum rule after simple replacements. In numerical computations the
parameters $M^{2}$, $s_{0}$ are chosen as in the corresponding mass
calculations. For $G_{1}$ and $G_{2}$ our analysis leads to the following
predictions (in units of $\mathrm{GeV}^{-1}$)

$G_{1}=22.0\pm 2.6,\ \ |G_{2}|=3.6\pm 0.4.$ (66)

The information gained from these studies is enough to determine the widths of
the corresponding decay channels. In fact, the width of the decay
$\Xi_{c}^{\ast}(3/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}$ can be found using
the expression

$\displaystyle\Gamma(\Xi_{c}^{\ast}(3/2^{-})$ $\displaystyle\rightarrow$
$\displaystyle\Lambda_{c}^{+}K^{-})=\frac{G_{1}^{2}}{24\pi\widetilde{m}^{\ast
2}}\left[(\widetilde{m}^{\ast}-m_{\Lambda})^{2}-m_{K}^{2}\right]$ (67)
$\displaystyle\times f^{3}(\widetilde{m}^{\ast},m_{\Lambda},m_{K}),$

whereas for $\Gamma(\Xi_{c}^{\ast}(3/2^{+})\rightarrow\Lambda_{c}^{+}K^{-})$,
we employ

$\displaystyle\Gamma(\Xi_{c}^{\ast}(3/2^{+})$ $\displaystyle\rightarrow$
$\displaystyle\Lambda_{c}^{+}K^{-})=\frac{G_{2}^{2}}{24\pi
m^{\ast\prime^{2}}}\left[(m^{\ast\prime}+m_{\Lambda})^{2}-m_{K}^{2}\right]$
(68) $\displaystyle\times f^{3}(m^{\ast\prime},m_{\Lambda},m_{K}).$

Numerical computations lead to predictions

$\displaystyle\Gamma\left(\Xi_{c}^{\ast}(3/2^{-})\rightarrow\Lambda_{c}^{+}K^{-}\right)=(10.5\pm
2.1)~{}\mathrm{MeV},$
$\displaystyle\Gamma\left(\Xi_{c}^{\ast}(3/2^{+})\rightarrow\Lambda_{c}^{+}K^{-}\right)=(35.4\pm
7.2)~{}\mathrm{MeV}.$ (69)

Exploration of the second pair of processes
$\Xi_{c}^{\ast}(3/2^{-})\rightarrow\Xi_{c}^{\prime}\pi$ and
$\Xi_{c}^{\ast}(3/2^{+})\rightarrow\Xi_{c}^{\prime}\pi$ is performed using the
correlation function

$\Pi_{\mu}^{\pi}(p,q)=i\int
d^{4}xe^{ipx}\langle\pi(q)|\mathcal{T}\\{\eta(x)\overline{\eta}_{\mu}(0)\\}|0\rangle.$
(70)

The sum rules for $G_{3}$ and $G_{4}$ are obtained by means of the structures
$\not{q}\not{p}q_{\mu}$ in the physical and QCD representation of the
correlator $\Pi_{\mu}^{\pi}(p,q)$. Our analysis for strong couplings and
partial widths yields:

$\displaystyle G_{3}=4.3\pm 0.6,$
$\displaystyle\Gamma\left(\Xi_{c}^{\ast}(3/2^{-})\rightarrow\Xi_{c}^{\prime}\pi\right)=(4.0\pm
0.8)\times 10^{-2}~{}\mathrm{MeV},$ (71)

and

$\displaystyle G_{4}=0.45\pm 0.06,$
$\displaystyle\Gamma\left(\Xi_{c}^{\ast}(3/2^{+})\rightarrow\Xi_{c}^{\prime}\pi\right)=(1.7\pm
0.4)\times 10^{-1}~{}\mathrm{MeV}.$ (72)

Partial widths of processes considered here allow us to evaluate the full
widths of the baryons $\Xi_{c}^{\ast}(3/2^{-})$ and $\Xi_{c}^{\ast}(3/2^{+})$
saturated by two dominant decay modes. It is clear that effect of the process
$\Xi_{c}^{\ast}(3/2^{-})\rightarrow\Xi_{c}^{\prime}\pi$ on the full width of
$\Xi_{c}^{\ast}(3/2^{-})$ is negligible, and can be safely ignored. For width
of the baryon $\Xi_{c}^{\ast}(3/2^{+})$, we get

$\Gamma^{\ast\prime}=(35.6\pm 2.1)~{}\mathrm{MeV}.$ (73)

This estimate for $\Gamma^{\ast\prime}$, and prediction
$\widetilde{\Gamma}^{\ast}=(10.5\pm 2.1)~{}\mathrm{MeV}$ should be compared
with the LHCb data.

##  VI Analysis and Concluding notes

We have calculated the masses and pole residues of the ground state spin-$1/2$
and -$3/2$ baryons $\Xi_{c}^{\prime}$ and $\Xi_{c}^{\ast}$

$m_{\mathrm{th}}=(2576\pm 150)~{}\mathrm{MeV},$ (74)

and

$m_{\mathrm{th}}^{\ast}=(2655\pm 102)~{}\mathrm{MeV}.$ (75)

Comparing obtained theoretical predictions for masses of these particles with
experimental data (2) and (3), we see nice agreements between them:
Theoretical errors of $m_{\mathrm{th}}$ and $m_{\mathrm{th}}^{\ast}$ are
typical for this method and do not exceed allowed limits.

In the present work we have also computed the masses and widths of the
spin-$1/2$ flavor-sextet baryons $\Xi_{c}^{\prime}$, and spin-$3/2$ baryons
$\Xi_{c}^{\ast}$ in order to compare obtained information with results of the
LHCb collaboration. The masses of these particles have been extracted from
two-point sum rules, whereas to calculate their widths, we have used the QCD
light-cone sum rule approach.

The sum rule method is a powerful nonperturbative tool to explore features of
conventional and exotic hadrons. It relies on first principles on the QCD by
employing quark-gluon structure of particles under analysis, and universal
vacuum expectations values of various local quark, gluon, and mixed operators.
Predictions obtained in this context depend on a few auxiliary parameters of
computations, which limit theoretical accuracy of investigations. Main part of
uncertainties is generated by a choice of the Borel parameter $M^{2}$: its
variation within allowed working region leads to ambiguities in values of
extracted parameters. In this sense the mass of a hadron is most protected
physical quantity the reason being in a functional form of a relevant sum
rule. In fact, sum rules for the masses of hadrons are given as a ratio of
correlation functions (see, for instance Eq. (27)), which reduces
uncertainties and stabilize a final result.

In the present article ambiguities in the masses of the excited spin-$1/2$ and
-$3/2$ baryons $\Xi_{c}^{\prime}$ and $\Xi_{c}^{\ast}$ amount to
$\pm(2.2-3.8)\%$ of central values, which is nice accuracy for sum rule
computations. In other words, the masses of the baryons may be chosen from
values spanning approximately $(120-220)~{}\mathrm{MeV}$ region. Because,
resonances discovered by LHCb have very close masses and cover narrow range of
$\sim 40~{}\mathrm{MeV}$, the sum rule method could not resolve such fine
structure: its predictions are compatible with all of these resonances.
Performed analysis also does not ”see” resonances $\Xi_{c}(2790)$ and
$\Xi_{c}^{\ast}(2815)$, because central values for masses of the excited
states extracted in the present work are higher than masses of these
particles. Therefore, classification of the spin-$1/2$ and -$3/2$ excited
baryons $\Xi_{c}^{\prime}$ and $\Xi_{c}^{\ast}$, and their possible
interpretation as resonances $\Xi_{c}(2923)^{0}$, $\Xi_{c}(2939)^{0}$, and
$\Xi_{c}(2965)^{0}$ should be done using widths of these particles, which
differ from each other and have been evaluated with accuracy enough for such
differentiation.

Let us note that parameters of the flavor-antitriplet spin-$1/2$ states $csd$
were calculated in Ref. Aliev:2018ube . In that paper the authors considered
the baryon $\Xi_{c}(1/2^{-})$ with parameters $\widetilde{m}=(2922\pm
83)~{}\mathrm{MeV}$, and $\widetilde{\Gamma}=(19.4\pm 3.3)~{}\mathrm{MeV}$ as
the resonance $\Xi_{c}(2930)^{0}$. The radial excitation of the spin-$1/2$
antitriplet baryon $\Xi_{c}(1/2^{+})$ has the same mass but lower width

$\displaystyle m^{\prime}$ $\displaystyle=$ $\displaystyle(2922\pm
83)~{}\mathrm{MeV},$ $\displaystyle\Gamma^{\prime}$ $\displaystyle=$
$\displaystyle(13.6\pm 2.3)~{}\mathrm{MeV}.$ (76)

This particle should be taken into account in our present analysis.

Before confronting with experimental data the full widths of the flavor-sextet
baryons should be found using results of the sections III and IV. Simple
analysis leads to the following estimates

$\widetilde{\Gamma}=(8.4\pm 1.4)\mathrm{~{}MeV},\ \Gamma^{\prime}=(14.6\pm
2.7)\mathrm{~{}MeV.}$ (77)

Then, it is not difficult to see that sextet baryon
$\Xi_{c}^{\prime}(1/2^{-})$, which has the mass and width

$\displaystyle\widetilde{m}$ $\displaystyle=$ $\displaystyle(2925\pm
115)~{}\mathrm{MeV},$ $\displaystyle\widetilde{\Gamma}$ $\displaystyle=$
$\displaystyle(8.4\pm 1.4)~{}\mathrm{MeV},\ $ (78)

can be interpreted as the resonance $\Xi_{c}(2923)^{0}$ with parameters (5).

The second resonance $\Xi_{c}(2939)^{0}$ may be considered as the excited
spin-$3/2$ baryon $\Xi_{c}^{\ast}(3/2^{-})$

$\displaystyle\widetilde{m}^{\ast}$ $\displaystyle=$ $\displaystyle(2960\pm
67)~{}\mathrm{MeV},$ $\displaystyle\widetilde{\Gamma}^{\ast}$ $\displaystyle=$
$\displaystyle(10.5\pm 2.1)~{}\mathrm{MeV}.$ (79)

The interpretation of $\Xi_{c}(2965)^{0}$ with parameters (7) is twofold: it
may be considered as the spin-$1/2$ antitriplet baryon $\Xi_{c}(1/2^{+})$ with
$\Gamma^{\prime}=(13.6\pm 2.3)~{}\mathrm{MeV}$. But one can identify it also
with radially excited sextet particle $\Xi_{c}^{\prime}(1/2^{+})$ with the
width $(14.2\pm 2.7)~{}\mathrm{MeV}$. Let us note that masses of these
particles within theoretical errors are compatible with the LHCb data.

Because the radially excited spin-$3/2$ particle $\Xi_{c}^{\ast}(3/2^{+})$ has
the width $(35.6\pm 2.1)~{}\mathrm{MeV}$, it cannot be considered as a
candidate to new resonances. It is worth noting that parameters of
$\Xi_{c}^{\ast}(3/2^{+})$ are close to the mass and width of the baryon
$\Xi_{c}(2970)^{0}$ Yelton:2016fqw

$\displaystyle m$ $\displaystyle=$ $\displaystyle(2970.8\pm 0.7\pm
0.2)~{}\mathrm{MeV}$ $\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle
30.3\pm 2.3_{-4.0}^{+1.0}~{}\mathrm{MeV},$ (80)

Recently the Belle collaboration determined the spin-parity of $\Xi_{c}(2970)$
as $J^{\mathrm{P}}=1/2^{+\text{ }}$Moon:2020gsg . Conclusion about radially
excited spin-$1/2$ nature of the baryon $\Xi_{c}(2970)$ was made also in Ref.
Arifi:2020yfp . In the light of these circumstances $\Xi_{c}^{\ast}(3/2^{+})$
cannot be interpreted as $\Xi_{c}(2970)$. Identification of two resonances
$\Xi_{c}(2965)^{0}$ and $\Xi_{c}(2970)^{0}$ seems also to be problematic,
because they have significantly different decay widths.

We have noted above that new LHCb resonances were explored in Refs.
Yang:2020zjl ; Wang:2020gkn ; Lu:2020ivo ; Zhu:2020jke  using various
suggestions on their structure and employing different computational schemes.
Thus, in Ref. Yang:2020zjl  these states were investigated in the context of
the HQET. In this theory a $P$-wave baryon consists of one charm quark and
light diquark, and contains one orbital excitation between light quarks
($\rho$-mode), or between a charm and light diquark ($\lambda$-mode). In this
paper, the resonances $\Xi_{c}(2923)^{0}$, $\Xi_{c}(2939)^{0}$, and
$\Xi_{c}(2965)^{0}/\Xi_{c}(2970)^{0}$ were interpreted as $P$-wave baryons
$J^{\mathrm{P}}=1/2^{-}$, $J^{\mathrm{P}}=3/2^{-}$ and
$J^{\mathrm{P}}=3/2^{-}$containing one $\lambda$-mode: A difference in
organization of two last resonances with $J^{\mathrm{P}}=3/2^{-}$ was
explained in Ref. Yang:2020zjl . Our interpretation of $\Xi_{c}(2923)^{0}$,
and $\Xi_{c}(2939)^{0}$ is consistent with this picture. But in our analysis
the last resonance from this list $\Xi_{c}(2965)^{0}$ is either $2S$
antitriplet or sextet spin-$1/2$ particle.

The new resonances were considered also in Ref. Wang:2020gkn , in which first
two states were interpreted as $\lambda$-mode baryons with
$J^{\mathrm{P}}=3/2^{-}$, and the last state as $J^{\mathrm{P}}=5/2^{-}$.
Interpretation of only the resonance $\Xi_{c}(2939)^{0}$ in this scheme
coincides with our prediction. In Ref. Lu:2020ivo  the last particle
$\Xi_{c}(2965)^{0}$ was regarded as spin-$1/2$ flavor-sextet $2S$ baryon,
which is in accord with our assignment.

New measurements by LHCb provided information on parameters of three
resonances which can be considered as charm-strange baryons. In the present
work we have calculated the masses and widths of four excited $csd$ baryons
with different spins. Theoretical investigations of first orbitally and
radially excited spin-$1/2$ sextet and spin-$3/2$ baryons, as well as existing
results for spin-$1/2$ flavor-antitriplet states fix parameters of six
particles. We have used new resonances $\Xi_{c}(2923)^{0}$,
$\Xi_{c}(2939)^{0}$, and $\Xi_{c}(2965)^{0}$, and known ones
$\Xi_{c}(2930)^{0}$ and $\Xi_{c}(2970)^{0}$ to confront their parameters with
our predictions: Obtained results have been discussed above. From brief
analysis of theoretical articles it is evident that interpretations of excited
$csd$ baryons are controversial. Evidently, for comprehensive analysis of this
sector of hadron spectroscopy more detailed experimental information and
further investigations are required.

## ACKNOWLEDGEMENTS

S. S. A. is grateful to Prof. V. M. Braun for enlightening comments.

*

##  Appendix A  The invariant amplitudes $\Pi_{1}^{\mathrm{OPE}}(M^{2},s_{0})$
and $\Pi_{2}^{\mathrm{OPE}}(M^{2},s_{0})$

The invariant amplitudes $\Pi_{1(2)}^{\mathrm{OPE}}(p^{2})$ used to calculate
the mass and pole residue of the flavor-sextet spin-$1/2$ baryons after the
Borel transformation and subtraction prescriptions take the following form

$\Pi_{1(2)}^{\mathrm{OPE}}(M^{2},s_{0})=\int_{\mathcal{M}^{2}}^{s_{0}}ds\rho_{1(2)}^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi_{1(2)}(M^{2}),$
(A.81)

where $\mathcal{M}=m_{c}+m_{s}$. The spectral densities
$\rho_{1(2)}^{\mathrm{OPE}}(s)$ in Eq. (A.81) are found from the imaginary
part of the correlation function and encompass essential piece of
$\Pi^{\mathrm{OPE}}(p)$. The Borel transformations of remaining terms in
$\Pi^{\mathrm{OPE}}(p)$ are included into $\Pi_{1(2)}(M^{2})$, and have been
calculated directly from the expression of $\Pi^{\mathrm{OPE}}(p)$.

The functions $\rho_{1(2)}^{\mathrm{OPE}}(s)$ and $\Pi_{1(2)}(M^{2})$ contain
components of different dimensions and have the structure

$\rho_{1(2)}^{\mathrm{OPE}}(s)=\rho_{1(2)}^{\mathrm{pert.}}(s)+\sum\rho_{1(2)}^{\mathrm{DimN}}(s),\
\ \Pi_{1(2)}(M^{2})=\sum\Pi_{1(2)}^{\mathrm{DimN}}(M^{2}).$ (A.82)

The $\Pi_{1(2)}^{\mathrm{OPE}}(M^{2},s_{0})$ have been computed by setting
$m_{q}=0$ and $m_{s}\neq 0$, and used to perform numerical analyses. The
amplitudes $\Pi_{1(2)}^{\mathrm{OPE}}(M^{2},s_{0})$, in general, contain a few
hundred terms and exist as Mathematica files. Their explicit expressions are
cumbersome, therefore we provide below simplified formulas in which $m_{s}=0$.

The perturbative contribution and nonperturbative terms with dimensions $3$,
$4$, $5$ and $7$ in the case of the spectral density
$\rho_{1}^{\mathrm{OPE}}(s)$ are given by the expressions:

$\rho_{1}^{\mathrm{pert.}}(s)=\frac{(5+2\beta+5\beta^{2})}{2048\pi^{4}s^{2}}\left[m_{c}^{6}(8s-m_{c}^{2})+s^{3}(s-8m_{c}^{2})+12m_{c}^{4}s^{2}\ln\left(\frac{s}{m_{c}^{2}}\right)\right],$
$\rho_{1}^{\mathrm{Dim3}}(s)=\frac{\langle\overline{s}s\rangle+\langle\overline{d}d\rangle}{192\pi^{2}s^{2}}m_{c}\left(m_{c}^{2}-s\right)^{2}\left(1+4\beta-5\beta^{2}\right),$
$\rho_{1}^{\mathrm{Dim4}}(s)=\frac{\langle
g_{s}^{2}G^{2}\rangle}{3072\pi^{4}s^{2}}\left(s-m_{c}^{2}\right)\left[8m_{c}^{2}\left(1+\beta+\beta^{2}\right)+s\left(5+2\beta+5\beta^{2}\right)\right],$
$\rho_{1}^{\mathrm{Dim5}}(s)=\frac{\langle\overline{d}g_{s}\sigma
Gd\rangle+\langle\overline{s}g_{s}\sigma
Gs\rangle}{768\pi^{2}s^{2}}m_{c}(\beta-1)\left[m_{c}^{2}(7+11\beta)-6s(1+\beta)\right],$
$\rho_{1}^{\mathrm{Dim7}}(s)=\frac{\langle
g_{s}^{2}G^{2}\rangle\left[\langle\overline{s}s\rangle+\langle\overline{d}d\rangle\right]}{384\pi^{2}s^{2}}m_{c}(1-\beta^{2}).$

The function $\Pi_{1}(M^{2})$ is composed of the following components:

$\Pi_{1}^{\mathrm{Dim6}}(M^{2},s_{0})=\frac{\langle\overline{s}s\rangle\langle\overline{d}d\rangle}{72}\left(11\beta^{2}+2\beta-13\right)e^{-m_{c}^{2}/M^{2}},$
$\Pi_{1}^{\mathrm{Dim7}}(M^{2},s_{0})=\frac{\langle
g_{s}^{2}G^{2}\rangle\left[\langle\overline{s}s\rangle+\langle\overline{d}d\rangle\right]}{864\pi^{2}m_{c}}\left(\beta^{2}+\beta-2\right)e^{-m_{c}^{2}/M^{2}},$
$\displaystyle\Pi_{1}^{\mathrm{Dim8}}(M^{2},s_{0})=-\frac{\langle
g_{s}^{2}G^{2}\rangle^{2}}{27\cdot
2^{13}\pi^{4}M^{2}}\left(13\beta^{2}+10\beta+13\right)e^{-m_{c}^{2}/M^{2}}+\frac{\langle\overline{s}g_{s}\sigma
Gs\rangle\langle\overline{d}d\rangle}{288M^{4}}(1-\beta)$
$\displaystyle\times\left[m_{c}^{2}(26+22\beta)+M^{2}\left(25+23\beta\right)\right]e^{-m_{c}^{2}/M^{2}},$
$\displaystyle\Pi_{1}^{\mathrm{Dim9}}(M^{2},s_{0})=\frac{\langle\overline{s}g_{s}\sigma
Gs\rangle\langle g_{s}^{2}G^{2}\rangle}{27\cdot
2^{11}\pi^{2}m_{c}M^{4}}(1-\beta)\left[m_{c}^{2}(31+11\beta)-2M^{2}(1+\beta)\right]e^{-m_{c}^{2}/M^{2}},$
$\displaystyle\Pi_{1}^{\mathrm{Dim10}}(M^{2},s_{0})=0.$

For the spectral density $\rho_{2}^{\mathrm{OPE}}(s)$, we get

$\rho_{2}^{\mathrm{pert.}}(s)=\frac{m_{c}(13-2\beta-11\beta^{2})}{1536\pi^{4}s}\left[m_{c}^{6}+9sm_{c}^{4}-9m_{c}^{2}s^{2}-s^{3}+6m_{c}^{2}s(s+m_{c}^{2})\ln\left(\frac{s}{m_{c}^{2}}\right)\right],$
$\rho_{2}^{\mathrm{Dim3}}(s)=\frac{\langle\overline{s}s\rangle+\langle\overline{d}d\rangle}{192\pi^{2}s}\left(m_{c}^{2}-s\right)^{2}\left(1+4\beta-5\beta^{2}\right),\
$ $\displaystyle\rho_{2}^{\mathrm{Dim4}}(s)=\frac{\langle
g_{s}^{2}G^{2}\rangle}{9216\pi^{4}m_{c}s}\left(1-\beta\right)\left[(m_{c}^{2}-s)\left(s(13+11\beta)+m_{c}^{2}(53+67\beta)\right)\right.$
$\displaystyle\left.+3m_{c}^{2}s(11+13\beta)\ln\left(\frac{s}{m_{c}^{2}}\right)\right],$
$\rho_{2}^{\mathrm{Dim5}}(s)=\frac{\langle\overline{d}g_{s}\sigma
Gd\rangle+\langle\overline{s}g_{s}\sigma
Gs\rangle}{768\pi^{2}s}(1-\beta)\left[m_{c}^{2}(5+\beta)-6s(1+\beta)\right].$

The function $\Pi_{2}(M^{2})$ is determined by the components

$\Pi_{2}^{\mathrm{Dim6}}(M^{2},s_{0})=\frac{\langle\overline{s}s\rangle\langle\overline{d}d\rangle}{24}m_{c}\left(5\beta^{2}+2\beta+5\right)e^{-m_{c}^{2}/M^{2}},$
$\Pi_{2}^{\mathrm{Dim7}}(M^{2},s_{0})=\frac{\langle
g_{s}^{2}G^{2}\rangle\left[\langle\overline{s}s\rangle+\langle\overline{d}d\rangle\right]}{3456\pi^{2}}\left(\beta^{2}-8\beta+7\right)e^{-m_{c}^{2}/M^{2}},$
$\displaystyle\Pi_{2}^{\mathrm{Dim8}}(M^{2},s_{0})=\frac{\langle
g_{s}^{2}G^{2}\rangle^{2}}{27\cdot
2^{13}\pi^{4}m_{c}M^{2}}(m_{c}^{2}-2M^{2})\left(11+2\beta-13\beta^{2}\right)e^{-m_{c}^{2}/M^{2}}+\frac{\langle\overline{s}g_{s}\sigma
Gs\rangle\langle\overline{d}d\rangle}{144M^{4}}m_{c}$
$\displaystyle\times\left[M^{2}\left(\beta-1\right)^{2}-3m_{c}^{2}(5+2\beta+5\beta^{2})\right]e^{-m_{c}^{2}/M^{2}},$
$\displaystyle\Pi_{2}^{\mathrm{Dim9}}(M^{2},s_{0})=\frac{\langle\overline{s}g_{s}\sigma
Gs\rangle\langle g_{s}^{2}G^{2}\rangle}{27\cdot
2^{11}\pi^{2}M^{4}}m_{c}^{2}(\beta^{2}+28\beta-29)e^{-m_{c}^{2}/M^{2}},$
$\displaystyle\Pi_{2}^{\mathrm{Dim10}}(M^{2},s_{0})=0.$

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