††thanks: Corresponding author

# Properties of doubly heavy spin-$\frac{1}{2}$ baryons: the ground and
excited states

M. Shekari Tousia 0009-0007-7195-0838<EMAIL_ADDRESS>K. Azizia,b
0000-0003-3741-2167<EMAIL_ADDRESS>aDepartment of Physics, University of
Tehran, North Karegar Avenue, Tehran 14395-547, Iran
bDepartment of Physics, Doǧuş University, Dudullu-Ümraniye, 34775 Istanbul,
Turkey

###### Abstract

We determine the masses and residues of the ground and excited
spin-$\frac{1}{2}$ baryons consist of two heavy b or c quark utilizing the QCD
sum rule formalism. In the calculations, we consider the non-perturbative
operators up to ten mass dimensions in order to increase the accuracy compared
to the previous calculations. We report the obtained results for both the
symmetric and anti-symmetric currents defining the doubly heavy baryons of the
ground state (1S), first orbitally excited state (1P) and first radially
excited state (2S). We compare our results with the predictions of other non-
perturbative approaches as well as existing experimental data which is
available only for the ground state of $\Xi_{cc}$ channel. These predictions
can help the experimental groups in their searches for all members of the
doubly heavy baryons in their ground and exited states.

##  I Introduction

Investigation of the properties of the doubly heavy baryons is an important
and useful area in particle physics. The quark model makes predictions about
the hadronic states containing single, double and triple heavy quarks
GellMann:1964nj . All of the hadronic states with a single heavy quark
predicted by the quark model have already been observed in experiments. For a
while, scientists were unable to observe baryons consisting of two heavy
quarks, posing a longstanding puzzle within the quark model. However, a
breakthrough was made in 2002 by the SELEX collaboration with the discovery of
$\Xi^{+}_{cc}(3520)$ in the $pD^{+}K^{-}$ decay channel SELEX:2002wqn . This
discovery was later confirmed by the same experiment in 2005 SELEX:2004lln .
In 2012, a calculation showed that the mass of $\Xi_{cc}$ state may be greater
than the one measured by SELEX collaboration Aliev:2012ru . In 2017, the LHCb
collaboration announced the discovery of $\Xi^{++}_{cc}(3621)$ through the
$\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}K^{-}\pi^{+}\pi^{+}$decay channel
LHCb:2017iph , which was later confirmed by the same collaboration in 2018
with a statistical significance of 5.9 $\sigma$ in the decay channel
$\Xi^{++}_{cc}\rightarrow\Xi^{+}_{c}\pi^{+}$ LHCb:2018pcs . As it is seen, the
mass of $\Xi^{++}_{cc}$ measured by LHCb is about 100 $\mathrm{MeV}$ greater
than the measured mass for the $\Xi^{+}_{cc}$ by SELEX, representing a puzzle
since a unit charge difference can not produce such a large mass difference.
Hence this subject became one of the hot topics in hadron physics both from
the experimental and theoretical views. Experimental groups have since been
conducted to discover these types of baryons, but so far it has not led to the
discovery of another new doubly heavy baryons (for instance see Ref.
LHCb:2021xba ). Numerous theoretical researches have been conducted based on
these discoveries to determine the properties of the doubly heavy baryons such
as mass, lifetime, residue, decay widths, etc. within various methods
Ebert:2002ig ; Zhang:2008rt ; Wang:2010hs ; Alrebdi:2020rev ; Rostami:2020euc
; Aliev:2020aon ; Aliev:2021hqq ; Azizi:2020zin ; Olamaei:2020bvw ; Yu:2017zst
; Luchinsky:2020fdf ; Gerasimov:2019jwp ; Wang:2017mqp ; Meng:2017udf ;
Gutsche:2017hux ; Xiao:2017udy ; Xiao:2017udy ; Lu:2017meb ; Xiao:2017dly ;
Zhao:2018mrg ; Xing:2018lre ; Jiang:2018oak ; Gutsche:2018msz ; Dhir:2018twm ;
Gutsche:2019wgu ; Yu:2019yfr ; Gutsche:2019iac ; Ke:2019lcf ; Cheng:2020wmk ;
Hu:2020mxk ; Li:2020qrh ; Han:2021gkl ; Wang:2017azm ; Shi:2017dto ;
Zhang:2018llc ; Ivanov:2020xmw ; Rahmani:2020pol ; Li:2017pxa ;
Berezhnoy:2018bde ; Guo:2017vcf ; Ma:2017nik ; Yao:2018zze ; Yao:2018ifh ;
Meng:2018zbl ; Shi:2020qde ; Qiu:2020omj ; Olamaei:2021hjd ; Qin:2021dqo ;
Hu:2017dzi ; Li:2018epz ; Shi:2019hbf ; Aliyev:2022rrf ; Shi:2019fph ;
Alrebdi:2020rev ; Aliev:2020aon ; Aliev:2021hqq ; Aliev:2012iv ; Aliev:2019lvd
; Aliev:2012ru ; Aliev:2012nn ; Ozdem:2018uue ; Ozdem:2019zis ;
Padmanath:2019ybu ; Brown:2014ena ; Giannuzzi:2009gh ; Shah:2017liu ;
Shah:2016vmd ; Yoshida:2015tia ; Valcarce:2008dr ; Wang:2010it ; Ortiz-
Pacheco:2023kjn . Nevertheless, triply heavy baryons are still waiting to be
found in the future experiments.

In this study, we calculate the masses and residues of the doubly heavy
spin-$\frac{1}{2}$ baryons in their ground and exited states. For calculation
of these parameters, some non-perturbative approaches are needed. One of these
methods is the QCD sum rule formalism which was introduced in 1979 by Shifman,
Vainshtein and Zakharov based on the fundamental QCD Lagrangian, considering
the correlation function including different interpolating currents
Shifman:1978bx ; Shifman:1978by . This method is a successful approach in
hadron physics and has many applications in the study of hadrons Aliev:2010uy
; Aliev:2009jt ; Aliev:2012ru ; Agaev:2016dev ; Azizi:2016dhy  confirmed by
different experiments. In various approaches, like lattice QCD
Padmanath:2019ybu ; Brown:2014ena , relativistic quark model Ebert:2002ig ,
quark model Yoshida:2015tia , hypercentral constituent quark model
Shah:2016vmd ; Shah:2017liu , Faddeev method Valcarce:2008dr , Regge
phenomenology Oudichhya:2023pkg , Hamiltonian model Yoshida:2015tia , QCD sum
rules Aliev:2019lvd ; Zhang:2008rt ; Wang:2010it ; Wang:2010hs ; Aliev:2012ru
; Aliev:2012iv , Bathe-Salpeter equations Li:2022ywz , Salpeter model
Giannuzzi:2009gh  and hyper-central constituent quark model Shah:2016vmd ;
Shah:2017liu , researchers studied the spectroscopic parameters of the doubly
heavy baryons. To investigate any type of interaction/decay of the doubly
heavy baryons, we need their exact value of the mass and residue. For this
purpose, we compute the spectroscopic parameters with the aim of increasing
the accuracy in the calculations by applying the higher dimensional non-
perturbative effects in the ground, first orbitally and radially excited
states.

Calculations of the mass of the doubly heavy baryons in the ground state have
been done in different approaches such as relativistic quark model
Ebert:2002ig , double ratios of sum rules (DRSR) Narison:2010py  and QCD sum
rule method Lichtenberg:1995kg ; Zhang:2008rt ; Aliev:2012ru ; Wang:2010hs .
With the aim of reducing the error and increasing the accuracy by including
the higher-dimensional non-perturbative operators, we perform these
calculations up to 10 dimensions. In the previous work done by sum rules
Aliev:2012ru , the calculations were done up to 5 mass dimensions and the
reported error values are relatively high.

Calculations of the mass of the doubly heavy baryons for $1P$ excited state
have also been done in various methods such as relativistic quark model
Ebert:2002ig , quark model Roberts:2007ni ; Yoshida:2015tia , Faddeev method
Valcarce:2008dr  and QCD sum rules Aliev:2019lvd ; Wang:2010it . These
calculations for $2S$ excited state have also been done in different
approaches such as relativistic quark model Ebert:2002ig , quark model
Yoshida:2015tia , Salpeter model Giannuzzi:2009gh , hyper-central constituent
quark model Shah:2016vmd ; Shah:2017liu , Faddeev method Valcarce:2008dr  and
QCD sum rule formalism Aliev:2019lvd . For the $1P$ and $2S$ excited states,
we calculate the mass of all the members more accurately by including the
higher dimensional non-perturbative operators as discussed above.

The calculation of the residues of these baryons has been performed for the
ground states and their first radial and orbital excitations in fewer
references. For the ground states, researchers studied this parameter using
QCD sum rule method Zhang:2008rt ; Aliev:2012ru ; Wang:2010hs . These studies
for the $1P$ excited state have been done in QCD sum rules Aliev:2019lvd ;
Wang:2010hs . For the $2S$ excited state the residue calculation has also been
done in Ref. Aliev:2019lvd . We report the results of our calculations for the
residue of these baryons in the ground, $1P$ and $2S$ states with the higher
accuracy in the operator product expansion (OPE) as well. The obtained results
can be used in the future experiments of particle laboratories such as LHC, to
research for these types of baryons.

This work is organized as follows: Section II presents the derivation of sum
rules for masses and residues of the doubly heavy baryons, while Section III
reports the results of numerical analysis of the sum rules and comparison of
our findings with the existing literature. Section IV includes summary and
conclusion and we move some expressions to the appendix.

##  II Calculation of the mass and residue of the doubly heavy baryons

The QCD sum rule method follows a standard prescription, where a correlation
function is evaluated through two distinct approaches. The first approach
involves the use of hadronic degrees of freedom, referred to as the physical
or hadronic side. This approach yields results that contain physical
quantities like the mass and residue of the considered states. The second
approach involves QCD degrees of freedom, such as quark-gluon condensates, QCD
coupling constant and quark masses. This approach gives the QCD side of the
calculations. QCD sum rules for the physical quantities are obtained by
matching the results of both sides and considering the coefficients of the
same Lorentz structures. To start the calculation, we need to consider the
two-point correlation function as following form:

$\Pi(q)=i\int d^{4}xe^{iq\cdot x}\langle
0|\mathcal{T}\\{\eta(x)\bar{\eta}(0)\\}|0\rangle,$ (1)

where $\eta(x)$ represents the interpolating current of the doubly heavy
baryons and $\mathcal{T}$ is the time ordering operator. The interpolating
current of symmetric and anti-symmetric (concerning the heavy quark exchange)
of these baryons are:

$\displaystyle\eta^{S}=\frac{1}{\sqrt{2}}\epsilon_{abc}\\{(Q^{a^{T}}Cq^{b})\gamma_{5}Q^{\prime
c}+(Q^{\prime
a^{T}}Cq^{b})\gamma_{5}Q^{c}+t(Q^{a^{T}}C\gamma_{5}q^{b})Q^{\prime
c}+t(Q^{\prime a^{T}}C\gamma_{5}q^{b})Q^{c}\\},$ (2)

and

$\displaystyle\eta^{A}$
$\displaystyle=\frac{1}{\sqrt{6}}\epsilon_{abc}\\{2(Q^{a^{T}}CQ^{\prime
b})\gamma_{5}q^{c}+(Q^{a^{T}}Cq^{b})\gamma_{5}Q^{\prime c}-(Q^{\prime
a^{T}}Cq^{b})\gamma_{5}Q^{c}+2t(Q^{a^{T}}C\gamma_{5}Q^{\prime
b})q^{c}+t(Q^{a^{T}}C\gamma_{5}q^{b})Q^{\prime c}$ $\displaystyle-t(Q^{\prime
a^{T}}C\gamma_{5}q^{b})Q^{c}\\},$ (3)

where $Q$ and $Q^{\prime}$ represents heavy quark field ($b$ or $c$); $a$, $b$
and $c$ are color indices, $C$ is the charge conjugation operator and $t$ is
an arbitrary mixing parameter to be fixed from the analysis. $t=-1$
corresponds to the Ioffe current. The above interpolating currents are written
considering all the quantum numbers of the states under study. It should be
noted that the doubly heavy baryons including $\Xi_{cc}$, $\Xi_{bc}$,
$\Xi_{bb}$, $\Omega_{cc}$, $\Omega_{bc}$ and $\Omega_{bb}$ are symmetric and
$\Xi^{\prime}_{bc}$ and $\Omega^{\prime}_{bc}$ are anti-symmetric with respect
to the exchange of the two heavy quarks.

For the physical side, the correlator is transformed into its final form by
inserting complete sets of hadronic states in the relevant places:

$\displaystyle\Pi^{\mathrm{Phys}}(q)$ $\displaystyle=\frac{\langle
0|\eta|B_{QQ^{\prime}}(q,s)\rangle\langle
B_{QQ^{\prime}}(q,s)|\bar{\eta}|0\rangle}{m^{2}-q^{2}}+\frac{\langle
0|\eta|\tilde{B}_{QQ^{\prime}}(q,s)\rangle\langle\tilde{B}_{QQ^{\prime}}(q,s)|\bar{\eta}|0\rangle}{\tilde{m}^{2}-q^{2}}+\frac{\langle
0|\eta|B_{QQ^{\prime}}^{\prime}(q,s)\rangle\langle
B_{QQ^{\prime}}^{\prime}(q,s)|\bar{\eta}|0\rangle}{m^{\prime}{}^{2}-q^{2}}$
$\displaystyle+\cdots.$ (4)

The $|B_{QQ^{\prime}}(q,s)\rangle$, $|\tilde{B}_{QQ^{\prime}}(q,s)\rangle$ and
$|B_{QQ^{\prime}}^{\prime}(q,s)\rangle$ are the one-particle states of the
ground (positive parity), first orbital excitation $1P$ (negative parity) and
first radial excitation $2S$ (positive parity) states of baryon, respectively.
Here, $m$, $\tilde{m}$ and $m^{\prime}$ are their corresponding masses and
$\cdots$ represents the contributions of the higher states and continuum. The
matrix elements in Eq. (4) are defined as follows:

$\displaystyle\langle 0|\eta|B_{QQ^{\prime}}(q,s)\rangle$ $\displaystyle=$
$\displaystyle\lambda u(q,s),$ $\displaystyle\langle
0|\eta|\tilde{B}_{QQ^{\prime}}(q,s)\rangle$ $\displaystyle=$
$\displaystyle\tilde{\lambda}\gamma_{5}u(q,s),$ $\displaystyle\langle
0|\eta|B_{QQ^{\prime}}^{\prime}(q,s)\rangle$ $\displaystyle=$
$\displaystyle\lambda^{\prime}u(q,s),$ (5)

where $\lambda$, $\tilde{\lambda}$ and $\lambda^{\prime}$ are the
corresponding residues and $u(q,s)$ is the Dirac spinor. These matrix elements
are used in Eq. (4) and summation over spins of Dirac spinors, which is given
as

$\displaystyle\sum_{s}u(q,s)\bar{u}(q,s)=(\not\\!q+m),$ (6)

is performed. So, the physical side becomes

$\displaystyle\Pi^{\mathrm{Phys}}(q)=\frac{\lambda^{2}(\not\\!q+m)}{m^{2}-q^{2}}+\frac{\tilde{\lambda}^{2}(\not\\!q-\tilde{m})}{\tilde{m}^{2}-q^{2}}+\frac{\lambda^{\prime
2}(\not\\!q+m^{\prime})}{m^{\prime}{}^{2}-q^{2}}+\cdots.$ (7)

After the Borel transformation, the final result for the physical side becomes

$\displaystyle\tilde{\Pi}^{\mathrm{Phys}}(q)=\lambda^{2}(\not\\!q+m)e^{-\frac{m^{2}}{M^{2}}}+\tilde{\lambda}^{2}(\not\\!q-\tilde{m})e^{-\frac{\tilde{m}^{2}}{M^{2}}}+\lambda^{\prime
2}(\not\\!q+m^{\prime})e^{-\frac{m^{\prime}{}^{2}}{M^{2}}}+\cdots,$ (8)

where $\tilde{\Pi}^{\mathrm{Phys}}(q)$ denotes the correlation function after
the Borel transformation.

To analyze the QCD side, we use Eq. (1) and substitute the interpolating
currents from Eqs. (2) and (II) into it. The calculations involve conducting
possible contractions among the quark fields utilizing Wick’s theorem. The
contracted quark fields are substituted with heavy and light quarks
propagators expressed in coordinate space, which have explicit forms as
follows:

$\displaystyle
S_{q}^{ab}(x)=i\delta_{ab}\frac{\not{x}}{2\pi^{2}x^{4}}-\delta_{ab}\frac{m_{q}}{4\pi^{2}x^{2}}-\delta_{ab}\frac{\langle\overline{q}q\rangle}{12}+i\delta_{ab}\frac{\not{x}m_{q}\langle\overline{q}q\rangle}{48}-\delta_{ab}\frac{x^{2}}{192}\langle\overline{q}g_{s}\sigma
Gq\rangle$
$\displaystyle+i\delta_{ab}\frac{x^{2}\not{x}m_{q}}{1152}\langle\overline{q}g_{s}\sigma
Gq\rangle-i\frac{g_{s}G_{ab}^{\alpha\beta}}{32\pi^{2}x^{2}}\left[\not{x}{\sigma_{\alpha\beta}+\sigma_{\alpha\beta}}\not{x}\right]-i\delta_{ab}\frac{x^{2}\not{x}g_{s}^{2}\langle\overline{q}q\rangle^{2}}{7776}$
$\displaystyle-\delta_{ab}\frac{x^{4}\langle\overline{q}q\rangle\langle
g^{2}_{s}G^{2}\rangle}{27648}+\cdots,$ (9)

and

$\displaystyle
S_{Q}^{ab}(x)=i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\Bigg{\\{}\frac{\delta_{ab}\left({\not{k}}+m_{Q}\right)}{k^{2}-m_{Q}^{2}}-\frac{g_{s}G_{ab}^{\alpha\beta}}{4}\frac{\sigma_{\alpha\beta}\left({\not{k}}+m_{Q}\right)+\left({\not{k}}+m_{Q}\right)\sigma_{\alpha\beta}}{(k^{2}-m_{Q}^{2})^{2}}$
$\displaystyle+\frac{g_{s}^{2}G^{2}}{12}\delta_{ab}m_{Q}\frac{k^{2}+m_{Q}{\not{k}}}{(k^{2}-m_{Q}^{2})^{4}}+\frac{g_{s}^{3}G^{3}}{48}\delta_{ab}\frac{\left({\not{k}}+m_{Q}\right)}{(k^{2}-m_{Q}^{2})^{6}}\left[{\not{k}}\left(k^{2}-3m_{Q}^{2}\right)+2m_{Q}\left(2k^{2}-m_{Q}^{2}\right)\right]\left({\not{k}}+m_{Q}\right)+\cdots\Bigg{\\}},$

where $G_{\mu\nu}$ is the gluon field strength tensor,
$G_{ab}^{\alpha\beta}=G_{A}^{\alpha\beta}t^{A}_{ab}$, $t^{A}=\lambda^{A}/2$,
$G^{2}=G_{\alpha\beta}^{A}G_{A}^{\alpha\beta}$ and
$G^{3}=f^{ABC}G_{\alpha\beta}^{A}G^{B\beta\delta}G_{\delta}^{C\alpha}$.
$\lambda^{A}$ are the Gell-Mann matrices and $f^{ABC}$ are the color structure
constants of the $SU_{c}(3)$ group where the indices A, B and C take the
values from 1 to 8. In the QCD side, the correlation function is evaluated in
the deep Euclidean region by using the OPE. After applying the Wick’s theorem
and conducting all contractions of the quark fields for the symmetric part, we
obtain the subsequent expression in relation to the heavy and light quarks
propagators:

$\displaystyle\Pi_{QCD}^{S}(q)$
$\displaystyle=iA\epsilon_{abc}\epsilon_{a^{\prime}b^{\prime}c^{\prime}}\int
d^{4}xe^{iqx}\Bigg{\\{}-\gamma_{5}S_{Q}^{cb^{\prime}}S_{q}^{\prime
ba^{\prime}}S_{Q^{\prime}}^{ac^{\prime}}\gamma_{5}-\gamma_{5}S_{Q^{\prime}}^{cb^{\prime}}S_{q}^{\prime
ba^{\prime}}S_{Q}^{ac^{\prime}}\gamma_{5}+\gamma_{5}S_{Q^{\prime}}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q}^{ab^{\prime}}S_{q}^{\prime
ba^{\prime}}\Big{]}$
$\displaystyle+\gamma_{5}S_{Q}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q^{\prime}}^{ab^{\prime}}S_{q}^{\prime
ba^{\prime}}\Big{]}+t\Bigg{(}-\gamma_{5}S_{Q}^{cb^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}S_{Q^{\prime}}^{ac^{\prime}}-\gamma_{5}S_{Q^{\prime}}^{cb^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}S_{Q}^{ac^{\prime}}-S_{Q}^{cb^{\prime}}S_{q}^{\prime
ba^{\prime}}\gamma_{5}S_{Q^{\prime}}^{ac^{\prime}}\gamma_{5}$ $\displaystyle-
S_{Q^{\prime}}^{cb^{\prime}}S_{q}^{\prime
ba^{\prime}}\gamma_{5}S_{Q}^{ac^{\prime}}\gamma_{5}+\gamma_{5}S_{Q^{\prime}}^{cc^{\prime}}Tr\Big{[}S_{Q}^{ab^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}\Big{]}+S_{Q^{\prime}}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q}^{ab^{\prime}}S_{q}^{\prime
ba^{\prime}}\gamma_{5}\Big{]}+\gamma_{5}S_{Q}^{cc^{\prime}}Tr\Big{[}S_{Q^{\prime}}^{ab^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}\Big{]}$
$\displaystyle+S_{Q}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q^{\prime}}^{ab^{\prime}}S_{q}^{\prime
ba^{\prime}}\gamma_{5}\Big{]}\Bigg{)}+t^{2}\Bigg{(}-S_{Q}^{cb^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}\gamma_{5}S_{Q^{\prime}}^{ac^{\prime}}-S_{Q^{\prime}}^{cb^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}\gamma_{5}S_{Q}^{ac^{\prime}}+S_{Q^{\prime}}^{cc^{\prime}}Tr\Big{[}S_{q}^{ba^{\prime}}\gamma_{5}S_{Q}^{\prime
ab^{\prime}}\gamma_{5}\Big{]}$
$\displaystyle+S_{Q}^{cc^{\prime}}Tr\Big{[}S_{q}^{ba^{\prime}}\gamma_{5}S_{Q^{\prime}}^{\prime
ab^{\prime}}\gamma_{5}\Big{]}\Bigg{)}\Bigg{\\}},$ (11)

where $S^{\prime}=CS^{T}C$. If $Q^{\prime}=Q$ , the value of constant A
becomes 1 and when we have $Q^{\prime}\neq Q$, this value becomes
$\frac{1}{2}$. We also have the anti-symmetric part as

$\displaystyle\Pi_{QCD}^{A}(q)$
$\displaystyle=\frac{i}{6}\epsilon_{abc}\epsilon_{a^{\prime}b^{\prime}c^{\prime}}\int
d^{4}xe^{iqx}\Bigg{\\{}2\gamma_{5}S_{Q}^{cb^{\prime}}S_{Q^{\prime}}^{\prime
aa^{\prime}}S_{q}^{bc^{\prime}}\gamma_{5}+\gamma_{5}S_{Q}^{cb^{\prime}}S_{q}^{\prime
ba^{\prime}}S_{Q^{\prime}}^{ac^{\prime}}\gamma_{5}-2\gamma_{5}S_{Q^{\prime}}^{ca^{\prime}}S_{Q}^{\prime
ab^{\prime}}S_{q}^{bc^{\prime}}\gamma_{5}+\gamma_{5}S_{Q^{\prime}}^{cb^{\prime}}S_{q}^{\prime
ba^{\prime}}S_{Q}^{ac^{\prime}}\gamma_{5}$
$\displaystyle-2\gamma_{5}S_{q}^{ca^{\prime}}S_{Q}^{\prime
ab^{\prime}}S_{Q^{\prime}}^{bc^{\prime}}\gamma_{5}+2\gamma_{5}S_{q}^{ca^{\prime}}S_{Q^{\prime}}^{\prime
bb^{\prime}}S_{Q}^{ac^{\prime}}\gamma_{5}+4\gamma_{5}S_{q}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q}^{ab^{\prime}}S_{Q^{\prime}}^{ba^{\prime}}\Big{]}+\gamma_{5}S_{Q^{\prime}}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q}^{ab^{\prime}}S_{q}^{\prime
ba^{\prime}}\Big{]}+$
$\displaystyle\gamma_{5}S_{Q}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q^{\prime}}^{ab^{\prime}}S_{q}^{\prime
ba^{\prime}}\Big{]}+t\Bigg{(}2\gamma_{5}S_{Q}^{cb^{\prime}}\gamma_{5}S_{Q^{\prime}}^{\prime
aa^{\prime}}S_{q}^{bc^{\prime}}+\gamma_{5}S_{Q}^{cb^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}S_{Q^{\prime}}^{ac^{\prime}}-2\gamma_{5}S_{Q^{\prime}}^{ca^{\prime}}\gamma_{5}S_{Q}^{\prime
ab^{\prime}}S_{q}^{bc^{\prime}}+\gamma_{5}S_{Q^{\prime}}^{cb^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}S_{Q}^{ac^{\prime}}$
$\displaystyle-2\gamma_{5}S_{q}^{ca^{\prime}}\gamma_{5}S_{Q}^{\prime
ab^{\prime}}S_{Q^{\prime}}^{bc^{\prime}}+2\gamma_{5}S_{q}^{ca^{\prime}}\gamma_{5}S_{Q^{\prime}}b^{b^{\prime}}S_{Q}^{ac^{\prime}}+2S_{Q}^{cb^{\prime}}S_{Q^{\prime}}^{\prime
aa^{\prime}}\gamma_{5}S_{q}^{bc^{\prime}}\gamma_{5}+S_{Q}^{cb^{\prime}}S_{q}^{\prime
ba^{\prime}}\gamma_{5}S_{Q^{\prime}}^{ac^{\prime}}\gamma_{5}-2S_{Q^{\prime}}^{ca^{\prime}}S_{Q}^{\prime
ab^{\prime}}\gamma_{5}S_{q}^{bc^{\prime}}\gamma_{5}$
$\displaystyle+S_{Q^{\prime}}^{cb^{\prime}}S_{q}^{\prime
ba^{\prime}}\gamma_{5}S_{Q}^{ac^{\prime}}\gamma_{5}-2S_{q}^{ca^{\prime}}S_{Q}^{\prime
ab^{\prime}}\gamma_{5}S_{Q^{\prime}}^{bc^{\prime}}\gamma_{5}+2S_{q}^{ca^{\prime}}S_{Q^{\prime}}^{\prime
bb^{\prime}}\gamma_{5}S_{Q}^{ac^{\prime}}\gamma_{5}+4\gamma_{5}S_{q}^{cc^{\prime}}Tr\Big{[}S_{Q}^{ab^{\prime}}\gamma_{5}S_{Q^{\prime}}^{\prime
ba^{\prime}}\Big{]}$
$\displaystyle+4S_{q}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q}^{ab^{\prime}}S_{Q^{\prime}}^{\prime
ba^{\prime}}\gamma_{5}\Big{]}+\gamma_{5}S_{Q^{\prime}}^{cc^{\prime}}Tr\Big{[}S_{Q}^{ab^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}\Big{]}+S_{Q^{\prime}}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q}^{ab^{\prime}}S_{q}^{\prime
ba^{\prime}}\gamma_{5}\Big{]}+\gamma_{5}S_{Q}^{cc^{\prime}}Tr\Big{[}S_{Q^{\prime}}^{ab^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}\Big{]}$
$\displaystyle+S_{Q}^{cc^{\prime}}\gamma_{5}Tr\Big{[}S_{Q^{\prime}}^{ab^{\prime}}S_{q}^{\prime
ba^{\prime}}\gamma_{5}\Big{]}\Bigg{)}+t^{2}\Bigg{(}2S_{Q}^{cb^{\prime}}\gamma_{5}S_{Q^{\prime}}^{aaa^{\prime}}\gamma_{5}S_{q}^{bc^{\prime}}+S_{Q}^{cb^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}\gamma_{5}S_{Q^{\prime}}^{ac^{\prime}}-2S_{Q^{\prime}}^{ca^{\prime}}\gamma_{5}S_{Q}^{\prime
ab^{\prime}}\gamma_{5}S_{q}^{bc^{\prime}}$
$\displaystyle+S_{Q^{\prime}}^{cb^{\prime}}\gamma_{5}S_{q}^{\prime
ba^{\prime}}\gamma_{5}S_{Q}^{ac^{\prime}}-2S_{q}^{ca^{\prime}}\gamma_{5}S_{Q}^{\prime
ab^{\prime}}\gamma_{5}S_{Q^{\prime}}^{bc^{\prime}}+2S_{q}^{ca^{\prime}}\gamma_{5}S_{Q^{\prime}}^{\prime
bb^{\prime}}\gamma_{5}S_{Q}^{ac^{\prime}}+4S_{q}^{cc^{\prime}}Tr\Big{[}S_{Q^{\prime}}^{ba^{\prime}}\gamma_{5}S_{Q}^{\prime
ab^{\prime}}\gamma_{5}\Big{]}$
$\displaystyle+S_{Q^{\prime}}^{cc^{\prime}}Tr\Big{[}S_{q}^{ba^{\prime}}\gamma_{5}S_{Q}^{\prime
ab^{\prime}}\gamma_{5}\Big{]}+S_{Q}^{cc^{\prime}}Tr\Big{[}S_{q}^{ba^{\prime}}\gamma_{5}S_{Q^{\prime}}^{\prime
ab^{\prime}}\gamma_{5}\Big{]}\Bigg{)}\Bigg{\\}}.$ (12)

In order to obtain the result for the QCD side of the sum rule, the above
mentioned propagators are first used and the calculations are followed by
Fourier and Borel transformations. Finally, the process of continuum
subtraction is carried out, assisted by the quark-hadron duality assumption.
The result of the QCD side is presented in the form

$\tilde{\Pi}_{1(2)}^{S(A)}(q)=\int_{(m_{Q}+m_{Q\prime})^{2}}^{s_{0}}dse^{-s/M^{2}}\rho_{1(2)}^{S(A)}(s)+\Gamma_{1(2)}^{S(A)}(M^{2}),$
(13)

where subindex 1(2) corresponds to the coefficient of the Lorentz structure
$\not\\!q(I)$, $s_{0}$ demonstrates the continuum threshold and
$\rho_{1(2)}^{S(A)}(s)$ is the spectral density that is achieved by taking the
imaginary part of the correlation function,
$\rho_{1(2)}^{S(A)}(s)=\frac{1}{\pi}\mathrm{Im}[\tilde{\Pi}_{1(2)}^{S(A)}(q)]$.
In the appendix, we present the expressions of the spectral densities obtained
after calculations up to 5 non-perturbative operators.
$\Gamma_{1(2)}^{S(A)}(M^{2})$ is a function that presents the calculations of
the non-perturbative part from the mass dimension 6 to 10. The
$\Gamma_{1(2)}^{S(A)}(M^{2})$ is a very lengthy function, so we don’t present
its explicit form in the present study.

The results obtained from performing computations on both sides are compared
using dispersion relations that take into account the coefficients of the same
Lorentz structures that are $\not\\!q$ and $I$. This yields the QCD sum rules
for the relevant quantities, which can be expressed as:

$\displaystyle\lambda^{2}e^{-\frac{m^{2}}{M^{2}}}+\tilde{\lambda}^{2}e^{-\frac{\tilde{m}^{2}}{M^{2}}}+\lambda^{\prime
2}e^{-\frac{m^{\prime}{}^{2}}{M^{2}}}=\tilde{\Pi}^{S(A)}_{\not\\!q}(s_{0},M^{2}),$
(14)

and

$\displaystyle\lambda^{2}me^{-\frac{m^{2}}{M^{2}}}-\tilde{\lambda}^{2}\tilde{m}e^{-\frac{\tilde{m}^{2}}{M^{2}}}+\lambda^{\prime
2}m^{\prime}e^{-\frac{m^{\prime}{}^{2}}{M^{2}}}=\tilde{\Pi}^{S(A)}_{I}(s_{0},M^{2}).$
(15)

In the following, we derive QCD sum rules for the mass and residue of the
ground state for the $\not\\!q$ structure. To achieve this, we follow the
ground state+continuum approach, in which we treat the second and third terms
of Eq. (14) on the left-hand-side as components of the continuum. As a result,
we obtain an equation for the mass of the ground state as follows:

$\displaystyle
m^{2}=\frac{\frac{d}{d(-\frac{1}{M^{2}})}\tilde{\Pi}^{S(A)}_{\not\\!q}(s_{0},M^{2})}{\tilde{\Pi}^{S(A)}_{\not\\!q}(s_{0},M^{2})}.$
(16)

The residue is obtained as

$\displaystyle\lambda^{2}=e^{\frac{m^{2}}{M^{2}}}\tilde{\Pi}^{S(A)}_{\not\\!q}(s_{0},M^{2}).$
(17)

After that, we consider the first two terms on the left-hand side of Eq. (14)
and raise the threshold for the first two terms, while treating the third term
in the continuum. With the help of the ground state’s results as inputs, we
can determine the mass and residue for the first orbitally excited state,
denoted by $1P$. Subsequently, we utilize the results obtained from the ground
state and the $1P$ state in a similar mode and follow the ground state + first
orbitally excited state + first radially excited state + continuum scheme to
achieve the mass and residue of the radially excited, $2S$, state.

##  III Numerical Analysis

In this section, we present the numerical analysis of our results on the mass
and residues of the doubly heavy baryons in their ground, first orbitally and
first radially excited states. We consider some inputs that are shown in Table
1. In the calculations, the masses of $u$ and $d$ quarks are assumed as zero,
but we keep the $s$ quark mass.

Parameters | Values
---|---
$m_{c}$ |  $1.27\pm 0.02~{}\mathrm{GeV}$ Zyla:2020zbs
$m_{b}$ |  $4.18^{+0.03}_{-0.02}~{}\mathrm{GeV}$ Zyla:2020zbs
$m_{s}$ |  $93^{+11}_{-5}~{}\mathrm{MeV}$ Zyla:2020zbs
$\langle\bar{q}q\rangle$ |  $(-0.24\pm 0.01)^{3}$ $\mathrm{GeV}^{3}$ 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
$\langle\overline{q}g_{s}\sigma Gq\rangle$ | $m_{0}^{2}\langle\bar{q}q\rangle$
$\langle g_{s}^{2}G^{2}\rangle$ |  $4\pi^{2}(0.012\pm 0.004)$ $~{}\mathrm{GeV}^{4}$Belyaev:1982cd
$\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle$ |  $(0.012\pm 0.004)$ $~{}\mathrm{GeV}^{4}$Belyaev:1982cd
$\langle g_{s}^{3}G^{3}\rangle$ |  $(0.57\pm 0.29)$ $~{}\mathrm{GeV}^{6}$Narison:2015nxh
Table 1: Some input parameters entering the calculations.

In addition to the parameters listed in Table 1, there are three extra
parameters, namely Borel parameter $M^{2}$, threshold parameter $s_{0}$ and
arbitrary mixing parameter $t$. They are obtained from the analysis of results
based on the standard criteria of the QCD sum rule method. These criteria
include weak dependence of the results on auxiliary parameters, pole dominance
and convergence of the OPE. We determine the working regions of $t$ from the
analysis by considering a parametric plot of the results as functions of
$\cos\theta$, where $t=\tan\theta$. The QCD side of calculations in terms of
$\cos\theta$ is shown in Figure 1, which is valid for all states of the doubly
heavy baryons.

Figure 1:  $\tilde{\Pi}^{\mathrm{QCD}}_{\not\\!q}$ as a function of
$\cos\theta$ at the central values of $M^{2}$ and $s_{0}$.

We select the regions that show least variations with respect to the changes
in $\cos\theta$, which read

$\displaystyle-1\leq\cos\theta\leq-0.5~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}0.5\leq\cos\theta\leq
1.$ (18)

Additionally, we determine the working region of Borel mass through the
following criteria. The upper bound for $M^{2}$, is set by ensuring that the
pole contribution should be larger than the continuum and higher states
contributions. To meet this requirement, we stipulate that the ratio $PC$,

$\displaystyle PC=\frac{\Pi(s_{0},M^{2},t)}{\Pi(\infty,M^{2},t)},$ (19)

should surpass $0.50$. In Figure 2, we can see the satisfaction of this
condition for pole contribution of $\Xi_{cc}$ as a function of the Borel
parameter $M^{2}$ at fixed values of $s_{0}$.

Figure 2:  Pole contribution of $\Xi_{cc}$ as a function of the Borel
parameter $M^{2}$ at fixed values of $s_{0}$.

The lower bound for $M^{2}$ is set based on the condition that the OPE series
must be convergent, which means that the perturbative contribution should be
greater than the non-perturbative one and the higher the dimension of the non-
perturbative operator, the lower its contribution. For this purpose, we
introduce the ratio:

$R(M^{2})=\frac{\Pi^{\mathrm{DimN}}(s_{0},M^{2},t)}{\Pi(s_{0},M^{2},t)},$ (20)

Where $\mathrm{DimN}$ refers to the last three dimensions, i.e.
$\mathrm{DimN=Dim(8+9+10)}$. To determine the minimum of the Borel parameter,
the ratio should not exceed $0.05$ of the total contribution. As the final
parameter, the continuum threshold values $s_{0}$, which also depend on the
energy of the next first excited state, are chosen in such a way that the
integrals do not receive a contribution from the excited states during the
ground state calculations.

Based on our analysis, we have observed that there is only a weak dependence
between the physical quantities and the auxiliary parameters in the given
windows for $M^{2}$ and $s_{0}$. To portray how the masses are affected by the
auxiliary parameters, we include Figures 3, 4 and 5. We draw these figures at
the average value of the arbitrary mixing parameter, i.e. $t=0.75$. The
results demonstrate a high level of stability in relation to the Borel
parameter and continuum threshold within their respective working regions. It
should be noted that our results for all baryons show least dependence on
auxiliary parameters. For this reason, we include only the graphs for the
$\Xi_{cc}$ state. The principal sources of uncertainties in our numerical
outcomes are due to the uncertainties with respect to the auxiliary parameters
and errors of other input parameters.

Figure 3: Left: The variation of the mass of $\Xi_{cc}$ at $1S$ state as a
function of the Borel parameter $M^{2}$ and at the various amounts of the
parameter $s_{0}$. Right: The variation of the mass of $\Xi_{cc}$ at $1S$
state as a function of the threshold parameter $s_{0}$ and at the various
amounts of the parameter $M^{2}$.

Figure 4: Left: The variation of the mass of $\Xi_{cc}$ at $1P$ state as a
function of the Borel parameter $M^{2}$ and at the various amounts of the
parameter $s_{0}$. Right: The variation of the mass of $\Xi_{cc}$ at $1P$
state as a function of the threshold parameter $s_{0}$ and at the various
amounts of the parameter $M^{2}$.

Figure 5: Left: The variation of the mass of $\Xi_{cc}$ at $2S$ state as a
function of the Borel parameter $M^{2}$ and at the various amounts of the
parameter $s_{0}$. Right: The variation of the mass of $\Xi_{cc}$ at $2S$
state as a function of the threshold parameter $s_{0}$ and at the various
amounts of the parameter $M^{2}$.

After determination of the range of auxiliary parameters, we present the mass
and residue of the doubly heavy baryons with spin-$\frac{1}{2}$, obtained
through numerical analyses in Table 2. This table also presents the working
intervals for the Borel parameter $M^{2}$ and threshold parameter $s_{0}$ for
all the channels.

Baryon | State | $M^{2}~{}(\mathrm{GeV^{2}})$ | $s_{0}~{}(\mathrm{GeV^{2}})$ | $\mathrm{Mass}~{}(\mathrm{GeV})$ | $\mathrm{Residue}~{}(\mathrm{GeV^{3}})$
---|---|---|---|---|---
| $\Xi_{cc}(\frac{1}{2}^{+})(1S)$ | $3.00-5.50$ | $3.87^{2}-4.24^{2}$ | $3.69\pm 0.10$ | $0.16\pm 0.04$
$\Xi_{cc}$ | $\Xi_{cc}(\frac{1}{2}^{-})(1P)$ | $3.00-5.50$ | $4.37^{2}-4.74^{2}$ | $3.91_{-0.11}^{+0.09}$ | $0.18\pm 0.03$
| $\Xi_{cc}(\frac{1}{2}^{+})(2S)$ | $3.00-5.50$ | $4.87^{2}-5.24^{2}$ | $4.04\pm 0.08$ | $0.19\pm 0.03$
| $\Xi_{bc}(\frac{1}{2}^{+})(1S)$ | $6.00-9.00$ | $6.92^{2}-7.21^{2}$ | $6.73_{-0.13}^{+0.14}$ | $0.29\pm 0.06$
$\Xi_{bc}$ | $\Xi_{bc}(\frac{1}{2}^{-})(1P)$ | $6.00-9.00$ | $7.42^{2}-7.71^{2}$ | $6.94\pm 0.13$ | $0.32_{-0.04}^{+0.06}$
| $\Xi_{bc}(\frac{1}{2}^{+})(2S)$ | $6.00-9.00$ | $7.92^{2}-8.21^{2}$ | $7.12\pm 0.14$ | $0.33\pm 0.06$
| $\Xi_{bb}(\frac{1}{2}^{+})(1S)$ | $10.00-15.00$ | $10.58^{2}-10.86^{2}$ | $9.97\pm 0.19$ | $0.45_{-0.08}^{+0.09}$
$\Xi_{bb}$ | $\Xi_{bb}(\frac{1}{2}^{-})(1P)$ | $10.00-15.00$ | $11.08^{2}-11.36^{2}$ | $10.25\pm 0.18$ | $0.60_{-0.08}^{+0.09}$
| $\Xi_{bb}(\frac{1}{2}^{+})(2S)$ | $10.00-15.00$ | $11.58^{2}-11.86^{2}$ | $10.33_{-0.19}^{+0.18}$ | $0.70_{-0.10}^{+0.09}$
| $\Omega_{cc}(\frac{1}{2}^{+})(1S)$ | $3.00-5.50$ | $3.89^{2}-4.26^{2}$ | $3.70\pm 0.09$ | $0.17\pm 0.04$
$\Omega_{cc}$ | $\Omega_{cc}(\frac{1}{2}^{-})(1P)$ | $3.00-5.50$ | $4.39^{2}-4.76^{2}$ | $3.93_{-0.09}^{+010}$ | $0.19_{-0.04}^{+0.03}$
| $\Omega_{cc}(\frac{1}{2}^{+})(2S)$ | $3.00-5.50$ | $4.89^{2}-5.26^{2}$ | $4.07_{-0.09}^{+0.08}$ | $0.20_{-0.03}^{+0.04}$
| $\Omega_{bc}(\frac{1}{2}^{+})(1S)$ | $6.00-9.00$ | $6.92^{2}-7.21^{2}$ | $6.77_{-0.12}^{+0.13}$ | $0.30\pm 0.05$
$\Omega_{bc}$ | $\Omega_{bc}(\frac{1}{2}^{-})(1P)$ | $6.00-9.00$ | $7.42^{2}-7.71^{2}$ | $7.07\pm 0.12$ | $0.33_{-0.06}^{+0.05}$
| $\Omega_{bc}(\frac{1}{2}^{+})(2S)$ | $6.00-9.00$ | $7.92^{2}-8.21^{2}$ | $7.20\pm 0.13$ | $0.35\pm 0.06$
| $\Omega_{bb}(\frac{1}{2}^{+})(1S)$ | $10.00-15.00$ | $10.58^{2}-10.86^{2}$ | $9.98\pm 0.18$ | $0.46_{-0.08}^{+0.09}$
$\Omega_{bb}$ | $\Omega_{bb}(\frac{1}{2}^{-})(1P)$ | $10.00-15.00$ | $11.08^{2}-11.36^{2}$ | $10.31\pm 0.19$ | $0.63\pm 0.09$
| $\Omega_{bb}(\frac{1}{2}^{+})(2S)$ | $10.00-15.00$ | $11.58^{2}-11.86^{2}$ | $10.45\pm 0.18$ | $0.75_{-0.09}^{+0.08}$
| $\Xi^{\prime}_{bc}(\frac{1}{2}^{+})(1S)$ | $6.00-9.00$ | $6.92^{2}-7.21^{2}$ | $6.81\pm 0.11$ | $0.31\pm 0.05$
$\Xi^{\prime}_{bc}$ | $\Xi^{\prime}_{bc}(\frac{1}{2}^{-})(1P)$ | $6.00-9.00$ | $7.42^{2}-7.71^{2}$ | $6.99_{-0.12}^{+0.13}$ | $0.34_{-0.05}^{+0.04}$
| $\Xi^{\prime}_{bc}(\frac{1}{2}^{+})(2S)$ | $6.00-9.00$ | $7.92^{2}-8.21^{2}$ | $7.15_{-0.13}^{+0.14}$ | $0.36\pm 0.06$
| $\Omega^{\prime}_{bc}(\frac{1}{2}^{+})(1S)$ | $6.00-9.00$ | $6.92^{2}-7.21^{2}$ | $6.82\pm 0.12$ | $0.32\pm 0.06$
$\Omega^{\prime}_{bc}$ | $\Omega^{\prime}_{bc}(\frac{1}{2}^{-})(1P)$ | $6.00-9.00$ | $7.42^{2}-7.71^{2}$ | $7.03_{-0.12}^{+0.13}$ | $0.35\pm 0.06$
| $\Omega^{\prime}_{bc}(\frac{1}{2}^{+})(2S)$ | $6.00-9.00$ | $7.92^{2}-8.21^{2}$ | $7.24_{-0.13}^{+0.14}$ | $0.38_{-0.07}^{+0.06}$
Table 2: The auxiliary parameters and the outcomes of the masses and residues
for the ground, first orbitally excited and first radially excited states.

Our approach involves analyzing the ground state, first orbitally excited
state, first radially excited state and continuum, using Eq. (14) to progress
step-by-step. As previously said, firstly, we derive the mass and residue of
the doubly heavy baryons in the ground state using proper threshold parameters
based on the ground state+continuum approach. We determine the appropriate
range for $s_{0}$, also listed in Table 2. Secondly, we apply the same method
to determine the mass and residue of baryons for the ground state, first
orbitally excited state and continuum scheme, using suitable threshold
parameters. The outcomes of this stage are likewise provided in Table 2.
Finally, we include the radially excited $2S$ state using the ground state,
first orbitally excited state, first radially excited state and continuum
scheme and specify the appropriate threshold parameter. The results for the
$2S$ states can be found in Table 2 as well. It is necessary to mention that
the results of two Lorentz structures have roughly the same values. Therefore,
we only report the results of $\not\\!q$ structure.

As we can see in Table 2, for $\Xi_{cc}$, the masses of the excited states
$1P$ and $2S$ are about 0.22 and 0.35 $\mathrm{GeV}$ more than the mass of the
ground state, respectively. These values for other members are as follows:
$\Xi_{bc}$ (0.21 and 0.39 $\mathrm{GeV}$), $\Xi_{bb}$ (0.28 and 0.36
$\mathrm{GeV}$), $\Omega_{cc}$ (0.23 and 0.37 $\mathrm{GeV}$), $\Omega_{bc}$
(0.30 and 0.43 $\mathrm{GeV}$), $\Omega_{bb}$ (0.33 and 0.47 $\mathrm{GeV}$),
$\Xi^{\prime}_{bc}$ (0.18 and 0.34 $\mathrm{GeV}$) and $\Omega^{\prime}_{bc}$
(0.21 and 0.42 $\mathrm{GeV}$).

Calculations of the mass of the doubly heavy baryons in the ground state have
been done in different approaches Zhang:2008rt ; Aliev:2012ru ; Wang:2010hs ;
Ebert:2002ig ; Narison:2010py . To provide a comprehensive comparison, Table 3
includes predictions for the mass of the doubly heavy baryons obtained from
various approaches. We perform these calculations by including the non-
perturbative operators of the mass dimensions up to 10 to reduce the errors
and increase the accuracy. In the previous work done by QCD sum rule method
Aliev:2012ru , the calculations were done up to 5 mass dimensions. Our results
for the ground states, except for some channels, are overall in good
agreements with the predictions of different methods such as relativistic
quark model Ebert:2002ig  and QCD sum rules Zhang:2008rt ; Aliev:2012ru ;
Wang:2010hs  within the presented uncertainties. Our result on the mass of the
$\Xi_{cc}$ state is consistent within the errors with the experimental datumn
of LHCb collaboration available only for this channel.

The study of the mass of the doubly heavy baryons for $1P$ and $2S$ excited
states have also been done in various methods Ebert:2002ig ; Giannuzzi:2009gh
; Aliev:2019lvd ; Shah:2016vmd ; Shah:2017liu ; Wang:2010it ; Valcarce:2008dr
. For these excited states, our calculations are reported with higher
accuracies as well. As can be seen from Tables 5 and 7, our results for the
$1P$ and $2S$ excited states, except for some channels, are overall in good
consistencies with the predictions of various approaches such as relativistic
quark model Ebert:2002ig , quark model Yoshida:2015tia , Salpeter
modelGiannuzzi:2009gh , hyper-central constituent quark model Shah:2016vmd ;
Shah:2017liu , Faddeev method Valcarce:2008dr  and QCD sum rules Aliev:2019lvd
; Wang:2010it  within the presented errors as well.

Furthermore, we studied the residues of the ground, first orbitally and first
radially excited states of the doubly heavy baryons and reported our outcomes
in Tables 4, 6 and 8. For the ground states, using QCD sum rules Zhang:2008rt
; Aliev:2012ru ; Wang:2010hs , researchers have also studied the residue of
the doubly heavy baryons. These studies for the $1P$ excited state have been
done in QCD sum rules Aliev:2019lvd ; Wang:2010hs . For the $2S$ excited
state, the residue calculation has also been done in Ref. Aliev:2019lvd . As
mentioned before, our results are more accurate for the residue calculations
compared to the previous predictions, since contributions of the non-
perturbative operators up to dimension ten are taken into account in the OPE.
Although, we see overall good consistencies (there are some differences on
some channels) of the results for the masses among different approaches and
predictions within the presented uncertainties, overall we see considerable
differences in the values of the presented residues for the ground and excited
baryons among different studies (there are some consistencies among the
results in some channels). The presented uncertainties for the residues are
high compared to the masses. This is well understood from the fact that the
mass is ratio of two sum rules that leads to killing of the uncertainties in
ratio, while the residue is obtained from one sum rule. Our results on the
residues can be used to investigate different decay channels of the doubly
heavy baryons under study.

Baryon | This work | Ref. Aliev:2012ru  | Ref. LHCb:2018pcs  | Ref. Ebert:2002ig  | Ref. Zhang:2008rt  | Ref. Wang:2010hs
---|---|---|---|---|---|---
$\Xi_{cc}$ | $3.69\pm 0.10$ | $3.72\pm 0.20$ | $3.62\pm 0.0015$ | $3.620$ | $4.26\pm 0.19$ | $3.57\pm 0.14$
$\Xi_{bc}$ | $6.73_{-0.13}^{+0.14}$ | $6.72\pm 0.20$ | $-$ | $6.933$ | $6.75\pm 0.05$ | -
$\Xi_{bb}$ | $9.97\pm 0.19$ | $9.96\pm 0.90$ | $-$ | $10.202$ | $9.78\pm 0.07$ | $10.17\pm 0.14$
$\Omega_{cc}$ | $3.70\pm 0.09$ | $3.73\pm 0.20$ | $-$ | $3.778$ | $4.25\pm 0.20$ | $3.71\pm 0.14$
$\Omega_{bc}$ | $6.77_{-0.12}^{+0.13}$ | $6.75\pm 0.30$ | $-$ | $7.088$ | $7.02\pm 0.08$ | -
$\Omega_{bb}$ | $9.98\pm 0.18$ | $9.97\pm 0.90$ | $-$ | $10.359$ | $9.85\pm 0.07$ | $10.32\pm 0.14$
$\Xi^{\prime}_{bc}$ | $6.81\pm 0.11$ | $6.79\pm 0.20$ | $-$ | $6.963$ | $6.95\pm 0.08$ | -
$\Omega^{\prime}_{bc}$ | $6.82\pm 0.12$ | $6.80\pm 0.30$ | $-$ | $7.116$ | $7.02\pm 0.08$ | -
Table 3: The masses of the ground state $1S$ of the doubly heavy baryons in $\mathrm{GeV}$ and comparison of the results with other predictions. Baryon | This work | Ref. Aliev:2012ru  | Ref. Zhang:2008rt  | Ref. Wang:2010hs
---|---|---|---|---
$\Xi_{cc}$ | $0.16\pm 0.04$ | $0.16\pm 0.02$ | $0.042\pm 0.026$ | $0.115\pm 0.027$
$\Xi_{bc}$ | $0.29\pm 0.06$ | $0.28\pm 0.05$ | $0.046\pm 0.021$ | -
$\Xi_{bb}$ | $0.45_{-0.08}^{+0.09}$ | $0.44\pm 0.08$ | $0.067\pm 0.057$ | $0.252\pm 0.064$
$\Omega_{cc}$ | $0.17\pm 0.04$ | $0.18\pm 0.03$ | - | $0.138\pm 0.030$
$\Omega_{bc}$ | $0.30\pm 0.05$ | $0.29\pm 0.05$ | - | -
$\Omega_{bb}$ | $0.46_{-0.08}^{+0.09}$ | $0.45\pm 0.08$ | - | $0.311\pm 0.077$
$\Xi^{\prime}_{bc}$ | $0.31\pm 0.05$ | $0.30\pm 0.05$ | - | -
$\Omega^{\prime}_{bc}$ | $0.32\pm 0.06$ | $0.31\pm 0.06$ | - | -
Table 4: The residues of the ground state $1S$ of the doubly heavy baryons in $\mathrm{GeV^{3}}$ and comparison of the results with other predictions.  Baryon | This work | Ref. Aliev:2019lvd  | Ref. Ebert:2002ig  | Ref. Valcarce:2008dr  | Ref. Wang:2010it  | Ref. Yoshida:2015tia
---|---|---|---|---|---|---
$\Xi_{cc}$ | $3.91_{-0.11}^{+0.09}$ | $4.03\pm 0.20$ | $3.838$ | $3.880$ | $3.77\pm 0.18$ | $3.947$
$\Xi_{bc}$ | $6.94\pm 0.13$ | $7.14\pm 0.11$ | $-$ | $-$ | $-$ | $-$
$\Xi_{bb}$ | $10.25\pm 0.18$ | $10.32\pm 0.10$ | $10.368$ | $10.406$ | $10.38\pm 0.15$ | $10.476$
$\Omega_{cc}$ | $3.93_{-0.09}^{+010}$ | $4.16\pm 0.14$ | $4.002$ | $-$ | $3.91\pm 0.14$ | $4.086$
$\Omega_{bc}$ | $7.07\pm 0.12$ | $7.20\pm 0.11$ | $-$ | $-$ | $-$ | $-$
$\Omega_{bb}$ | $10.31\pm 0.19$ | $10.37\pm 0.10$ | $10.532$ | $-$ | $10.38\pm 0.15$ | $10.607$
$\Xi^{\prime}_{bc}$ | $6.99_{-0.12}^{+0.13}$ | $7.02\pm 0.07$ | $-$ | $-$ | $-$ | $-$
$\Omega^{\prime}_{bc}$ | $7.03_{-0.12}^{+0.13}$ | $7.09\pm 0.07$ | $-$ | $-$ | $-$ | $-$
Table 5:  The masses of $1P$ excitation of the doubly heavy baryons in $\mathrm{GeV}$ and comparison of the results with other predictions. Baryon | This work | Ref. Aliev:2019lvd  | Ref. Wang:2010it
---|---|---|---
$\Xi_{cc}$ | $0.18\pm 0.03$ | $0.099\pm 0.013$ | $0.159\pm 0.037$
$\Xi_{bc}$ | $0.32_{-0.04}^{+0.06}$ | $0.242\pm 0.020$ | $-$
$\Xi_{bb}$ | $0.60_{-0.08}^{+0.09}$ | $0.576\pm 0.046$ | $0.365\pm 0.089$
$\Omega_{cc}$ | $0.19_{-0.04}^{+0.03}$ | $0.125\pm 0.015$ | $0.192\pm 0.041$
$\Omega_{bc}$ | $0.33_{-0.06}^{+0.05}$ | $0.241\pm 0.021$ | $-$
$\Omega_{bb}$ | $0.63\pm 0.09$ | $0.620\pm 0.060$ | $0.444\pm 0.101$
$\Xi^{\prime}_{bc}$ | $0.34_{-0.05}^{+0.04}$ | $0.176\pm 0.084$ | $-$
$\Omega^{\prime}_{bc}$ | $0.35\pm 0.06$ | $0.174\pm 0.079$ | $-$
Table 6:  The residues of $1P$ excitation of the doubly heavy baryons in $\mathrm{GeV^{3}}$ and comparison of the results with other predictions.  Baryon | This work | Ref. Aliev:2019lvd  | Ref. Ebert:2002ig  | Ref. Shah:2017liu  | Ref. Giannuzzi:2009gh  | Ref. Shah:2016vmd  | Ref. Yoshida:2015tia
---|---|---|---|---|---|---|---
$\Xi_{cc}$ | $4.04\pm 0.08$ | $4.03\pm 0.20$ | $3.910$ | $3.920$ | $4.183$ | $-$ | $4.079$
$\Xi_{bc}$ | $7.12\pm 0.14$ | $7.14\pm 0.11$ | $-$ | $7.263$ | $7.495$ | $-$ | $-$
$\Xi_{bb}$ | $10.33_{-0.19}^{+0.18}$ | $10.32\pm 0.10$ | $10.441$ | $10.609$ | $10.751$ | $-$ | $10.571$
$\Omega_{cc}$ | $4.07_{-0.09}^{+0.08}$ | $4.16\pm 0.14$ | $4.075$ | $-$ | $4.268$ | $4.041$ | $4.227$
$\Omega_{bc}$ | $7.20\pm 0.13$ | $7.20\pm 0.11$ | $-$ | $-$ | $7.559$ | $7.480$ | $-$
$\Omega_{bb}$ | $10.45\pm 0.18$ | $10.37\pm 0.10$ | $10.610$ | $-$ | $10.830$ | $10.736$ | $10.707$
$\Xi^{\prime}_{bc}$ | $7.15_{-0.13}^{+0.14}$ | $7.02\pm 0.07$ | $-$ | $-$ | $-$ | $-$ | $-$
$\Omega^{\prime}_{bc}$ | $7.24_{-0.13}^{+0.14}$ | $7.09\pm 0.07$ | $-$ | $-$ | $-$ | $-$ | $-$
Table 7:  The masses of $2S$ excitation of the doubly heavy baryons in $\mathrm{GeV}$ and comparison of the results with other predictions. Baryon | This work | Ref. Aliev:2019lvd
---|---|---
$\Xi_{cc}$ | $0.19\pm 0.03$ | $0.144\pm 0.064$
$\Xi_{bc}$ | $0.33\pm 0.06$ | $0.306\pm 0.090$
$\Xi_{bb}$ | $0.70_{-0.10}^{+0.09}$ | $0.764\pm 0.114$
$\Omega_{cc}$ | $0.20_{-0.03}^{+0.04}$ | $0.205\pm 0.085$
$\Omega_{bc}$ | $0.35\pm 0.06$ | $0.361\pm 0.112$
$\Omega_{bb}$ | $0.75_{-0.09}^{+0.08}$ | $0.850\pm 0.160$
$\Xi^{\prime}_{bc}$ | $0.36\pm 0.06$ | $0.187\pm 0.05$
$\Omega^{\prime}_{bc}$ | $0.38_{-0.07}^{+0.06}$ | $0.255\pm 0.125$
Table 8: The residues of $2S$ excitation of the doubly heavy baryons in
$\mathrm{GeV^{3}}$ and comparison of the results with other predictions.

##  IV Summary and conclusion

The quark model makes predictions about the hadronic states containing single,
double and triple heavy quarks. The investigation of the properties of the
doubly heavy baryons represents a promising area in particle physics. To study
any type of interactions/decays of the doubly heavy baryons, we need the exact
values of the masses and residues of these baryons. The values of the
spectroscopic parameters calculated from theory can also guide experimental
groups to search for the unseen baryon members predicted by the quark model.
For this purpose, we determined the spectroscopic parameters of the doubly
heavy baryons with a higher accuracy in the ground, first orbitally and first
radially excited states. Among the non-perturbative approaches, we used the
QCD sum rule formalism as one of the most powerful and predictive methods
based on the QCD Lagrangian. In order to increase the accuracy, we included
into the analyses the non-perturbative operators up to dimension 10. After
fixing the auxiliary parameters, we extracted the masses and residues of the
ground, first orbitally and first radially excited states of the doubly heavy
baryons as shown in Table 2 and compared our results with the predictions of
other approaches and the existing datumn for the ground $\Xi_{cc}$ state at
different tables. Our result on the mass of the ground $\Xi_{cc}$ state is in
a good consistency with the experimental datumn of the LHCb collaboration. Our
results on the masses are overall consistent with the predictions of other
studies except for some channels. Regarding the residue, we have inverse
situation: Except for some channels, we see considerable discrepancies of our
results with the previous predictions.

Our predictions may help experimental groups in the course of their search for
the unseen members of the doubly heavy baryons. They may also be checked by
other non-perturbative approaches in future. As said, in the calculations of
any physical quantities related to the interactions/decays of these baryons,
the values of the residues are immediately required. Our results that have
been provided with higher accuracy can be served as inputs to study the decay
properties of the considered doubly heavy baryons in their ground and excited
states. Such investigations are required to estimate the width and lifetime of
the doubly heavy baryons.

## ACKNOWLEDGEMENTS

The authors thank H. R. Moshfegh for his useful discussions. K. Azizi is
thankful to Iran National Science Foundation (INSF) for the partial financial
support provided under the elites Grant No. 4025036.

## APPENDIX: EXPRESSIONS OF SPECTRAL DENSITIES

In this appendix, we present the explicit expressions of the spectral
densities obtained from the calculations:

$\displaystyle\rho^{S(pert)}_{1}(s)$
$\displaystyle=\frac{A}{128\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\frac{-3\,D_{1}\,\Theta(D_{1})}{Z_{1}^{4}\,Z_{3}^{2}}$
$\displaystyle\Bigg{\\{}\Bigg{(}-2\,m_{Q}\,u\,Z1^{2}\,(m_{Q^{\prime}}\,v-3\,m_{q}\,Z_{2})\,Z_{3}-v\,Z_{2}\,\Big{(}-6\,m_{Q^{\prime}}\,m_{q}\,Z_{1}^{2}\,Z_{3}+5\,u\,(3\,D_{1}\,Z_{1}^{2}+2\,s\,u\,v\,Z_{2}\,Z_{3})\Big{)}\Bigg{)}$
$\displaystyle+2\,t\,u\,v\,\Bigg{(}-3\,D_{1}\,Z_{1}^{2}\,Z_{2}-2(-m_{Q}\,m_{Q^{\prime}}\,Z_{1}^{2}+s\,u\,v\,Z_{2}^{2})Z_{3}\Bigg{)}$
$\displaystyle-t^{2}\Bigg{(}-2\,m_{Q}\,u\,Z_{1}^{2}\,(-m_{Q^{\prime}}\,v-3\,m_{q}\,Z_{2})\,Z_{3}-v\,Z_{2}\,\Big{(}-6\,m_{Q^{\prime}}\,m_{q}\,Z_{1}^{2}\,Z_{3}-5\,u(3\,D_{1}\,Z_{1}^{2}+2\,s\,u\,v\,Z_{2}\,Z_{3})\Big{)}\Bigg{)}\Bigg{\\}},$
(21)

$\displaystyle\rho^{S(pert)}_{2}(s)$
$\displaystyle=\frac{A}{64\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\frac{3\,D_{1}\,\Theta(D_{1})}{Z_{1}^{3}\,Z_{3}^{2}}$
$\displaystyle\Bigg{\\{}\Bigg{(}D_{1}\,Z_{1}^{2}\,\Big{(}3(m_{Q}\,u+m_{Q^{\prime}}\,v)-m_{q}\,Z_{2}\Big{)}-\Big{(}-s\,u\,v\,Z_{2}\,(3\,m_{Q}\,u-m_{q}\,Z_{2})+m_{Q^{\prime}}\,(5\,m_{Q}\,m_{q}\,Z_{1}^{2}-3\,s\,u\,v^{2}\,Z_{2})\Big{)}\,Z_{3}\Bigg{)}$
$\displaystyle-2\,t\,m_{q}\Bigg{(}-D_{1}\,Z_{1}^{2}\,Z_{2}-(-m_{Q^{\prime}}\,m_{Q}\,Z_{1}^{2}+s\,u\,v\,Z_{2}^{2})\,Z_{3}\Bigg{)}$
$\displaystyle+t^{2}\,\Bigg{(}D_{1}\,Z_{1}^{2}\,\Big{(}-3\,(m_{Q}\,u+m_{Q^{\prime}}\,v)-m_{q}\,Z_{2}\Big{)}-\Big{(}-s\,u\,v\,Z_{2}\,(-3\,m_{Q}\,u-m_{q}\,Z_{2})+m_{Q^{\prime}}\,(5\,m_{Q}\,m_{q}\,Z_{1}^{2}$
$\displaystyle+3\,s\,u\,v^{2}\,Z_{2})\Big{)}\,Z_{3}\Bigg{)}\Bigg{\\}},$ (22)

$\displaystyle\rho^{A(pert)}_{1}(s)$
$\displaystyle=\frac{1}{256\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\frac{D_{1}\,\Theta(D_{1})}{Z_{1}^{4}\,Z_{3}^{2}}$
$\displaystyle\Bigg{\\{}\Bigg{(}-2\,m_{Q}\,u\,Z_{1}^{2}\,(-13\,m_{Q^{\prime}}\,v-m_{q}\,Z_{2})\,Z_{3}-v\,Z_{2}\Big{(}-2\,m_{Q^{\prime}}\,m_{q}\,Z_{1}^{2}\,Z_{3}+15\,u(3\,D_{1}\,Z_{1}^{2}+2\,s\,u\,v\,Z_{2}\,Z_{3})\Big{)}\Bigg{)}$
$\displaystyle-2\,t\,\Bigg{(}-2\,m_{Q}\,u\,Z_{1}^{2}\,(m_{Q^{\prime}}\,v-2\,m_{q}\,Z_{2})\,Z_{3}-v\,Z_{2}(9\,D_{1}\,u\,Z_{1}^{2}-4\,m_{Q^{\prime}}\,m_{q}\,Z_{1}^{2}\,Z_{3}+6\,s\,u^{2}\,v\,Z_{2}\,Z_{3})\Bigg{)}$
$\displaystyle+t^{2}\,\Bigg{(}-2\,m_{Q}\,u\,Z_{1}^{2}(-11\,m_{Q^{\prime}}\,v-5\,m_{q}\,Z_{2})\,Z_{3}-5\,v\,Z_{2}\Big{(}-2\,m_{Q^{\prime}}\,m_{q}\,Z_{1}^{2}\,Z_{3}-3\,u(3\,D_{1}\,Z_{1}^{2}+2\,s\,u\,v\,Z_{2}\,Z_{3})\Big{)}\Bigg{)}\Bigg{\\}},$
(23)

$\displaystyle\rho^{A(pert)}_{2}(s)$
$\displaystyle=\frac{1}{128\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\frac{D_{1}\,\Theta(D_{1})}{Z_{1}^{3}\,Z_{3}^{2}}$
$\displaystyle\Bigg{\\{}\Bigg{(}-D_{1}\,Z_{1}^{2}\,(-m_{Q}\,u-m_{Q^{\prime}}\,v-13\,m_{q}\,Z_{2})-\Big{(}s\,u\,v\,Z_{2}(-m_{Q}\,u-13\,m_{q}\,Z_{2})+m_{Q^{\prime}}\,(15\,m_{Q}\,m_{q}\,Z_{1}^{2}-s\,u\,v^{2}\,Z_{2})\Big{)}Z_{3}\Bigg{)}$
$\displaystyle+2\,t\Bigg{(}D_{1}\,Z_{1}^{2}\Big{(}2\,(m_{Q}\,u+m_{Q^{\prime}}\,v)-m_{q}\,Z_{2}\Big{)}-\Big{(}-s\,u\,v\,Z_{2}\,(2\,m_{Q}\,u-m_{q}\,Z_{2})+m_{Q^{\prime}}(3\,m_{Q}\,m_{q}\,Z_{1}^{2}-2\,s\,u\,v^{2}\,Z_{2})\Big{)}\,Z_{3}\Bigg{)}$
$\displaystyle+t^{2}\,\Bigg{(}D_{1}\,Z_{1}^{2}\Big{(}-5(m_{Q}\,u+m_{Q^{\prime}}\,v)-11\,m_{q}\,Z_{2}\Big{)}-\Big{(}-s\,u\,v\,Z_{2}(-5\,m_{Q}\,u-11\,m_{q}\,Z_{2})+5\,m_{Q^{\prime}}\,(3\,m_{Q}\,m_{q}\,Z_{1}^{2}$
$\displaystyle+s\,u\,v^{2}\,Z_{2})\Big{)}Z_{3}\Bigg{)}\Bigg{\\}},$ (24)

$\displaystyle\rho^{S(3D)}_{1}(s)$
$\displaystyle=\frac{A}{32\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{(}3\,\langle\overline{q}q\rangle\,\Theta(D_{2})\Big{)}\Bigg{\\{}2\,m_{Q}\,u+(2\,m_{Q^{\prime}}+5\,m_{q}\,u)\,Z_{4}+2\,t\,(m_{q}uZ_{4})+t^{2}\,\Big{(}-2\,m_{Q}\,u$
$\displaystyle-(2\,m_{Q^{\prime}}-5\,m_{q}\,u)Z_{4}\Big{)}\Bigg{\\}},$ (25)

$\displaystyle\rho^{S(3D)}_{2}(s)$
$\displaystyle=\frac{A}{16\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{(}\langle\overline{q}q\rangle\,\Theta(D_{2})\Big{)}\Bigg{\\{}-\Big{(}2\,D_{2}+5\,m_{Q^{\prime}}\,m_{Q}+u\,(3m_{Q}\,m_{q}+s-s\,u)+3\,m_{Q^{\prime}}\,m_{q}\,Z_{4}\Big{)}$
$\displaystyle+2\,t\,(2\,D_{2}-m_{Q^{\prime}}\,m_{Q}+s\,u\,Z_{4})+t^{2}\,\Big{(}-2\,D_{2}-5\,m_{Q^{\prime}}\,m_{Q}+3\,m_{Q}\,m_{q}\,u+(3m_{Q^{\prime}}\,m_{q}-s\,u)Z_{4}\Big{)}\Bigg{\\}},$
(26)

$\displaystyle\rho^{A(3D)}_{1}(s)$
$\displaystyle=\frac{1}{192\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{(}\langle\overline{q}q\rangle\,\Theta(D_{2})\Big{)}\Bigg{\\{}\Big{(}2\,m_{Q}\,u+(2\,m_{Q^{\prime}}+45\,m_{q}\,u)Z_{4}\Big{)}$
$\displaystyle+2\,t\,\Big{(}4\,m_{Q}\,u+(4\,m_{Q^{\prime}}+9\,m_{q}\,u)Z_{4}\Big{)}+5\,t^{2}\,\Big{(}-2\,m_{Q}\,u-(2\,m_{Q^{\prime}}-9\,m_{q}\,u)Z_{4}\Big{)}\Bigg{\\}},$
(27)

$\displaystyle\rho^{A(3D)}_{2}(s)$
$\displaystyle=\frac{1}{96\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{(}\langle\overline{q}q\rangle\,\Theta(D_{2})\Big{)}\Bigg{\\{}\Big{(}26\,D_{2}-m_{Q^{\prime}}(15\,m_{Q}+m_{q}-m_{q}\,u)-u(m_{Q}\,m_{q}-13\,s\,Z_{4})\Big{)}$
$\displaystyle-2\,t\,\Big{(}2\,D_{2}+3\,m_{Q^{\prime}}\,m_{Q}+u\,(2m_{Q}\,m_{q}+s-s\,u)+2\,m_{Q^{\prime}}\,m_{q}\,Z_{4}\Big{)}$
$\displaystyle+t^{2}\,\Big{(}-22\,D_{2}+5\,m_{Q}\,m_{q}\,u+5\,m_{Q^{\prime}}(-3\,m_{Q}+m_{q}-m_{q}\,u)-11\,s_{u}\,Z_{4}\Big{)}\Bigg{\\}},$
(28)

$\displaystyle\rho^{S(4D)}_{1}(s)$
$\displaystyle=\frac{A}{256\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{\langle}\frac{\alpha_{s}GG}{\pi}\Big{\rangle}\
\,\frac{-3\,u\,v\,Z_{2}\,\Theta(D_{3})}{Z_{1}^{4}}$
$\displaystyle\Bigg{\\{}\Big{(}3u^{2}+u(-3+7\,v)-3\,v\,Z_{3}\Big{)}+2\,t\Big{(}u^{2}+u(-1+3\,v)-v\,Z_{3}\Big{)}+t^{2}\,\Big{(}3u^{2}+u(-3+7\,v)-3\,v\,Z_{3}\Big{)}\Bigg{\\}},$
(29)

$\displaystyle\rho^{S(4D)}_{2}(s)$
$\displaystyle=\frac{A}{128\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{\langle}\frac{\alpha_{s}GG}{\pi}\Big{\rangle}\
\,\frac{\Theta(D_{3})}{Z_{1}^{4}}$
$\displaystyle\Bigg{\\{}-\Bigg{[}m_{Q}\,u\,Z_{1}\Big{(}u^{2}+u(-1+5\,v)-3\,v\,Z_{3}\Big{)}+v\Big{[}\Big{(}m_{q}\,u+m_{Q^{\prime}}\,(2+8\,u)\Big{)}v^{3}-3\,m_{Q^{\prime}}\,v^{4}$
$\displaystyle-\Bigg{(}(4\,m_{Q^{\prime}}\,+m_{q})u(-1+2\,u)v+\Big{(}2\,m_{q}\,u+m_{Q^{\prime}}\,(-1+11\,u)\Big{)}v^{2}\Bigg{)}Z_{4}+(3\,m_{Q^{\prime}}\,+m_{q})u^{2}\,Z_{4}^{2}\Big{]}\Bigg{]}$
$\displaystyle-2\,t\Big{(}m_{q}\,u\,v\,Z_{1}\,Z_{2}\Big{)}+t^{2}\,\Bigg{[}m_{Q}uZ_{1}\Big{(}u^{2}+u(-1+5v)-3vZ_{3}\Big{)}+v\Bigg{(}m_{q}uZ_{1}Z_{2}+m_{Q^{\prime}}\Big{(}2(1+4\,u)v^{3}$
$\displaystyle-3\,v^{4}-\Big{(}4\,u(-1+2\,u)v+(-1+11\,u)v^{2}\Big{)}\,Z_{4}+3\,u^{2}\,Z_{4}^{2}\Big{)}\Bigg{)}\Bigg{]}\Bigg{\\}},$
(30)

$\displaystyle\rho^{A(4D)}_{1}(s)$
$\displaystyle=\frac{1}{512\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{\langle}\frac{\alpha_{s}GG}{\pi}\Big{\rangle}\
\,\frac{\,u\,v\,Z_{2}\,\Theta(D_{3})}{Z_{1}^{4}}$
$\displaystyle\Bigg{\\{}-\Big{(}u^{2}-u(1+11\,v)-v\,Z_{3}\Big{)}+2\,t\Big{(}u^{2}+u(-1+7\,v)-v\,Z_{3}\Big{)}-t^{2}\,\Big{(}u^{2}-u(1+11\,v)-v\,Z_{3}\Big{)}\Bigg{\\}},$
(31)

$\displaystyle\rho^{A(4D)}_{2}(s)$
$\displaystyle=\frac{1}{2304\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{\langle}\frac{\alpha_{s}GG}{\pi}\Big{\rangle}\
\,\frac{\Theta(D_{3})}{Z_{1}^{4}}$
$\displaystyle\Bigg{\\{}\Bigg{[}m_{Q}\,u(19\,u^{4}+2\,u^{3}(-17+8\,v)-u^{2}(-15+v)Z_{3}+12\,u\,v^{2}\,Z_{3}-15\,v^{2}\,Z_{3}^{2})+v\Bigg{(}33\,m_{q}\,u\,Z_{1}\,Z_{2}$
$\displaystyle+m_{Q^{\prime}}\Big{(}2(-17+8\,u)v^{3}+19\,v^{4}+12\,u^{2}\,v\,Z_{4}-(-15+u)v^{2}\,Z_{4}-15\,u^{2}\,Z_{4}^{2}\Big{)}\Bigg{)}\Bigg{]}$
$\displaystyle+2\,t\Bigg{[}2\,m_{Q}\,u(u^{4}+2\,u^{3}\,Z_{3}-6\,u\,v^{2}\,Z_{3}-u^{2}(3+v)Z_{3}+3\,v^{2}\,Z_{3}^{2})+v\Bigg{(}3\,m_{q}\,u\,Z_{1}\,Z_{2}+2m_{Q^{\prime}}\Big{(}v^{4}-6\,u^{2}\,vZ_{4}$
$\displaystyle-(3+u)v^{2}\,Z_{4}+2\,v^{3}\,Z_{4}+3\,u^{2}\,Z_{4}^{2}\Big{)}\Bigg{)}\Bigg{]}+t^{2}\,\Bigg{[}m_{Q}\,u(-23\,u^{4}+u^{3}(26-8\,v)+u^{2}(-3+(8-5\,v)v)$
$\displaystyle+12\,u\,v^{2}\,Z_{3}+3\,v^{2}\,Z_{3}^{2})+v\Bigg{(}-39\,m_{q}\,u\,Z_{1}\,Z_{2}+m_{Q^{\prime}}\Big{(}2(13-4\,u)v^{3}-23\,v^{4}+12\,u^{2}\,v\,Z_{4}+(-3+5\,u)v^{2}\,Z_{4}$
$\displaystyle+3\,u^{2}\,Z4^{2}\Big{)}\Bigg{)}\Bigg{]}\Bigg{\\}},$ (32)

$\displaystyle\rho^{S(5D)}_{1}(s)=0,$ (33)

$\displaystyle\rho^{S(5D)}_{2}(s)$
$\displaystyle=\frac{A}{64\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{(}5\,m_{0}^{2}\langle\overline{q}q\rangle\,u\,Z_{4}\,\Theta(D_{4})\Big{)}\Big{(}1-2\,t+t^{2}\Big{)},$
(34)

$\displaystyle\rho^{A(5D)}_{1}(s)=0,$ (35)

$\displaystyle\rho^{A(5D)}_{2}(s)$
$\displaystyle=\frac{1}{384\pi^{4}}\int_{0}^{1}\,du\int_{0}^{1-u}dv\,\Big{(}5\,m_{0}^{2}\langle\overline{q}q\rangle\,u\,Z_{4}\,\Theta(D_{4})\Big{)}\Big{(}-13+2\,t+11\,t^{2}\Big{)},$
(36)

where

$\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle-\frac{Z_{3}}{Z_{1}^{2}}\Big{(}(m_{Q}^{2}\,u+m_{Q^{\prime}}^{2}\,v)Z_{1}+s\,u\,v\,Z_{2}\Big{)},$
$\displaystyle D_{2}$ $\displaystyle=$ $\displaystyle-
m_{Q}^{2}u-(m_{Q^{\prime}}^{2}-su)Z_{4},$ $\displaystyle D_{3}$
$\displaystyle=$
$\displaystyle\frac{Z_{3}}{Z_{1}^{2}}\Big{(}(m_{Q}^{2}\,u+m_{Q^{\prime}}^{2}\,v)Z_{1}+s\,u\,v\,Z_{2}\Big{)},$
$\displaystyle D_{4}$ $\displaystyle=$ $\displaystyle-
m_{Q}^{2}\,u-(m_{Q^{\prime}}^{2}-s\,u)Z_{4},$ (37)

and we have define

$\displaystyle Z_{1}$ $\displaystyle=$ $\displaystyle
u^{2}+u(-1+v)+(-1+v)\,v,$ $\displaystyle Z_{2}$ $\displaystyle=$
$\displaystyle 1-u-v,$ $\displaystyle Z_{3}$ $\displaystyle=$ $\displaystyle
1-v,$ $\displaystyle Z_{4}$ $\displaystyle=$ $\displaystyle 1-u.$ (38)

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