Available via license: CC BY 4.0
Content may be subject to copyright.
arXiv:1906.08353v1 [hep-ph] 19 Jun 2019
Magnetic dipole moments of the spin- 3
2doubly heavy baryons
Ula¸s ¨
Ozdem1, ∗
1Department of Physics, Dogus University, Acibadem-Kadikoy, 34722 Istanbul, Turkey
The magnetic dipole moments of the spin- 3
2doubly charmed, bottom and charmed-bottom
baryons are obtained by means of the light-cone QCD sum rule. The magnetic dipole mo-
ments of these baryons encode important information of their internal structure and shape de-
formations. The numerical results are given as, µΞ∗++
cc = 2.94 ±0.95, µΞ∗+
cc =−0.67 ±0.11,
µΩ∗+
cc =−0.52 ±0.07, µΞ∗0
bb = 2.30 ±0.55, µΞ∗−
bb =−1.39 ±0.32, µΩ∗−
bb =−1.56 ±0.33,
µΞ∗+
bc
= 2.12 ±0.68, µΞ∗0
bc =−0.96 ±0.32 and µΩ∗+
bc
=−1.11 ±0.33, respectively. A compari-
son of our results on the magnetic dipole moments of the spin- 3
2doubly heavy baryons with the
predictions of different approaches is presented. The consistency of the predictions with some (but
not all) theoretical predictions is good.
Keywords: Electromagnetic form factors, Magnetic moment, Doubly heavy baryons, Light-cone QCD sum
rules
I. MOTIVATION
The doubly heavy baryons presumably contain two heavy quark and one light quark. One of them was first reported
by the SELEX Collaboration in the decay mode Ξ+
cc →Λ+
cK−π+with the mass MΞ+
cc = 3519 ±1M eV [1]. However,
neither Belle [2], nor FOCUS [3], nor BABAR [4] could confirm the doubly heavy baryons in e−e+annihilations. It
is worth pointing out that the analysis of the SELEX experiment with other experimental groups is achieved through
different production mechanisms. Therefore, the results of the SELEX Collaboration cannot be ruled out. In 2017,
LHCb Collaboration observed another doubly heavy baryon Ξ++
cc in the mass spectrum of Λ+
cK−π+π+with the
mass MΞ++
cc = 3621.40 ±0.72 ±0.27 ±0.14 M eV [5]. The investigation for the doubly heavy baryons may provide
with valuable knowledge for comprehension of the nonperturbative QCD effects. One of the several point of views
which makes the physics of doubly heavy baryons charming is that the binding of two charm quarks and a light quark
provides a unique perspective for dynamics of confinement. The research of the properties of doubly heavy baryons is
one of the active and interesting branches of particle physics. The dynamic (e.g. strong, radiative and weak decays)
and static (e.g. masses and magnetic dipole moments) properties of the spin- 1
2and spin- 3
2doubly heavy baryons have
been studied extensively in literature [6–81].
One of the main characteristic parameters of the doubly heavy baryons is their electromagnetic properties. As the
electromagnetic properties characterize necessary aspects of the internal structure of baryons, it is very important
to investigate the baryon electromagnetic form factors, especially the magnetic dipole moments. The magnitude
and sign of the dipole magnetic moment ensure crucial information on size, structure and shape deformations of
baryons. Apparently, determining the magnetic dipole moment is an important step in our comprehension of the
baryon properties with regards to quark-gluon degrees of freedom. The magnetic dipole moments of the spin- 3
2
doubly heavy baryons were first extracted by Lichtenberg [75] in the framework of naive quark model. Besides naive
quark model the magnetic dipole moments of the spin- 3
2doubly heavy baryons have also been calculated in the
MIT bag model [71–73], the effective quark mass and screened charge scheme [63,64], nonrelativistic quark model
(NRQM) [22], the relativistic harmonic confinement model (RHM) [74], skyrmion model [76], hypercentral constituent
quark model (HCQM) [38–40], chiral constituent quark model (χCQM) [67] and chiral perturbation theory [59].
In this study, we are going to concentrate on the doubly heavy baryons (hereafter we will denote these baryons
as B∗
QQ ) with spin-parity JP=3
2
+, and calculate their magnetic dipole moments by the help of light-cone QCD
sum rule (LCSR) approach, which is one of the powerful nonperturbative methods in hadron physics providing us to
calculate properties of the particles and processes. In LCSR, the hadronic properties are expressed with regards to
the vacuum condensates and the light-cone distribution amplitudes (DAs) of the on-shell particles [for details, see
for instance [82–84]]. This method is quite successful in determining properties of the doubly heavy baryons (see e.g.
[24,31,32,48–50,79]).
∗uozdem@dogus.edu.tr
2
The outline of the paper is as follows. In section II, the details of the magnetic dipole moments calculations for the
doubly heavy baryons with spin- 3
2are presented. In section III, we numerically analyze the sum rules obtained for
the magnetic dipole moments. Section IV is reserved for discussion and concluding remarks.
II. FORMALISM
To obtain the magnetic dipole moment of the doubly heavy baryons by using the LCSR approach, we begin with
the subsequent correlation function,
Πµνα (p, q) =i2Zd4xZd4y eip·x+iq·yh0|T {JB∗
QQ
µ(x)Jα(y)JB∗†
QQ
ν(0)}|0i.(1)
Here, Jµ(ν)is the interpolating current of the B∗
QQ baryons and the electromagnetic current Jαis given as,
Jα=X
q=u,d,s,c,b
eq¯qγαq, (2)
where eqis the electric charge of the corresponding quark.
From a technical point of view, the correlation function can be rewritten in a more appropriate form by the help of
external background electromagnetic (EBGEM) field,
Πµν (p, q) = iZd4x eip·xh0|T {JB∗
QQ
µ(x)JB∗†
QQ
ν(0)}|0iF,(3)
where F is the EBGEM field and Fαβ =i(εαqβ−εβqα) with εβand qαbeing the polarization and four-momentum
of the EBGEM field, respectively. Since the EBGEM field can be made arbitrarily small, the correlation function in
Eq. (3) can be acquired by expanding in powers of the EBGEM field,
Πµν (p, q) = Π(0)
µν (p, q) + Π(1)
µν (p, q) + ...., (4)
and keeping only terms Π(1)
µν (p, q), which corresponds to the single photon emission [85] (the technical details about
the EBGEM field method can be found in [86]). The main advantage of using the EBGEM field approach relies on
the fact that it separates the soft and hard photon emissions in an explicitly gauge invariant way [85]. The Π(0)
µν (p, q)
is the correlation function in the absence of the EBGEM field, and gives rise to the mass sum rules of the hadrons,
which is not relevant for our case.
After these general comments, we can now move on deriving the LCSR for the magnetic dipole moment of the doubly
heavy baryons. The correlation function given in Eq. (3) can be calculated with regards to hadronic properties, known
as hadronic representation. In addition to this it can be obtained with regards to the quark-gluon properties in the
deep Euclidean region, known as QCD representation. By matching the results of these representations using the
dispersion relation and quarkhadron duality ansatz, one can acquire the corresponding sum rules.
We start to calculate the correlation function with respect to hadronic degrees of freedom including the physical
properties of the particles under investigation. To this end, we insert an intermediate set of B∗
QQ baryons into the
correlation function. As a consequence, we obtain
ΠHad
µν (p, q) = h0|JB∗
QQ
µ|B∗
QQ(p)i
[p2−m2
B∗
QQ ]hB∗
QQ(p)|B∗
QQ(p+q)iFhB∗
QQ(p+q)|¯
JB∗
QQ
ν|0i
[(p+q)2−m2
B∗
QQ ]+..., (5)
where the dots stand for contributions of higher states and the continuum. The matrix elements in Eq. (5) are defined
as [87,88],
h0|Jµ(0) |B∗
QQ(p, s)i=λB∗
QQ uµ(p, s),(6)
hB∗
QQ(p)|B∗
QQ(p+q)iF=−e¯uµ(p)(F1(q2)gµν ε/ −1
2mB∗
QQ hF2(q2)gµν +F4(q2)qµqν
(2mB∗
QQ )2iε/q/ +F3(q2)
(2mB∗
QQ )2
×qµqνε/)uν(p+q),(7)
3
where λB∗
QQ is the residue of B∗
QQ baryon and uµ(p, s) is the Rarita-Schwinger spinor. Summation over spins of B∗
QQ
baryon is carried out as:
X
s
uµ(p, s)¯uν(p, s) = −p/ +mB∗
QQ hgµν −1
3γµγν−2pµpν
3m2
B∗
QQ
+pµγν−pνγµ
3mB∗
QQ i.(8)
Substituting Eqs. (5)-(8) into Eq. (3) for hadronic side we obtain
ΠHad
µν (p, q) = −
λ2
B∗
QQ p/ +mB∗
QQ
[(p+q)2−m2
B∗
QQ
][p2−m2
B∗
QQ
]hgµν −1
3γµγν−2pµpν
3m2
B∗
QQ
+pµγν−pνγµ
3mB∗
QQ i
×(F1(q2)gµν ε/ −1
2mB∗
QQ hF2(q2)gµν +F4(q2)qµqν
(2mB∗
QQ )2iε/q/ +F3(q2)
(2mB∗
QQ )2qµqνε/).(9)
In principle, make use of the above equations, we can get the final expression of the hadronic representation of the
correlator, however we encounter two problems. One of them is related to the fact that not all Lorentz structures
appearing in Eq. (9) are independent. The second problem is the correlator can also receive contributions from
spin-1/2 particles, which should be removed. To eliminate the spin-1/2 contributions and obtain only independent
structures in the correlator, we perform the ordering for Dirac matrices as γµp/ε/q/γνand remove terms with γµat the
beginning, γνat the end and all those proportional to pµand pν[89]. As a result, for hadronic side we get,
ΠHad
µν (p, q) = −
λ2
B∗
QQ
[(p+q)2−m2
B∗
QQ
][p2−m2
B∗
QQ
]"−gµν p/ε/q/ F1(q2) + mB∗
QQ gµν ε/q/ F2(q2)
+ other independent structures#.(10)
The magnetic dipole moment form factor, GM(q2), is defined with respect to the form factors Fi(q2) in the following
manner [87,88]:
GM(q2) = F1(q2) + F2(q2)(1 + 4
5τ)−2
5F3(q2) + F4(q2)τ(1 + τ),(11)
where τ=−q2
4m2
B∗
QQ
. At q2= 0, the magnetic dipole form factors are obtained with respect tothe functions Fi(0) as:
GM(0) = F1(0) + F2(0).(12)
The magnetic dipole moment (µB∗
QQ ), is defined in the following way:
µB∗
QQ =e
2mB∗
QQ
GM(0).(13)
In this work we derive sum rules for the form factors Fi(q2) at first, then in numerical analyses we will use the
above relations to obtain the values of the magnetic dipole moments using the QCD sum rules for the form factors.
The final form of the hadronic representation with respect to the chosen structures in momentum space is:
ΠHad
µν (p, q) = ΠH ad
1gµν p/ε/q/ + ΠHad
2gµν ε/q/ +..., (14)
where ΠHad
iare functions of the form factors Fi(q2) and other hadronic parameters; and ... represents other indepen-
dent structures.
To obtain the expression of the correlation function with respect to the quark-gluon parameters, the explicit form
for the interpolating current of the B∗
QQ baryons needs to be chosen. In this work, we consider the B∗
QQ baryons with
the quantum numbers JP=3
2
+. The interpolating current is given as [32],
JB∗
QQ
µ(x) = 1
√3ǫabcn(qaT CγµQb)Q′c+ (qaT CγµQ′b)Qc+ (QaT CγµQ′b)qco,(15)
where qis the light; and Qand Q′are the two heavy quarks, respectively. We give the quark content of the spin-3/2
doubly heavy baryons in Table I.
After contracting pairs of quark fields and using the Wick’s theorem, the correlation function becomes:
4
Baryon q Q Q′
Ξ∗
QQ uor d b or c b or c
Ξ∗
QQ′uor d b c
Ω∗
QQ s b or c b or c
Ω∗
QQ′s b c
TABLE I: The quark content of the spin-3/2 doubly heavy baryons.
ΠQCD
µν (p) = −i
3εabcεa′b′c′Zd4xeip·xh0|(Scc′
Q′TrhSba′
Qγνe
Sab′
qγµi+Scc′
qTrhSba′
Q′γνe
Sab′
Qγµi+Scc′
QTrhSba′
qγνe
Sab′
Q′γµi
+Scb′
Qγνe
Saa′
Q′γµSbc′
q+Sca′
Qγνe
Sbb′
qγµSac′
Q′+Sca′
Q′γνe
Sbb′
QγµSac′
q+Scb′
Q′γνe
Saa′
qγµSbc′
Q+Sca′
qγνe
Sbb′
Q′γµSac′
Q
+Scb′
qγνe
Saa′
QγµSbc′
Q′)|0iF,(16)
where ˜
Sij
Q(q)(x) = CS ij T
Q(q)(x)Cand, Sij
q(x) and Sij
Q(x) are the light and heavy quark propagators, respectively. The
light and heavy quark propagators are given as [90,91],
Sq(x) = Sfree
q−¯qq
121−imqx/
4−¯qσ.Gq
192 x21−imqx/
6−igs
32π2x2Gµν (x)/xσµν +σµν /x,
SQ(x) = Sfree
Q−gsmQ
16π2Z1
0
dv Gµν (vx)σµν x/ +x/σµν K1(mQ√−x2)
√−x2+ 2σµν K0(mQp−x2),(17)
where
Sfree
q=1
2π2x2ix/
x2−mq
2,
Sfree
Q=m2
Q
4π2K1(mQ√−x2)
√−x2+ix/ K2(mQ√−x2)
(√−x2)2,(18)
with Gµν is the gluon field strength tensor, Kiare Bessel functions of the second kind, mqand mQare the light and
heavy quark mass respectively.
The correlator in Eq. (16) includes different contributions: the photon can be emitted both perturbatively or
nonperturbatively. When the photon is emitted perturbatively, one of the propagators in Eq. (16) is replaced by
Sfree(x)→Zd4y Sf r ee(x−y) /A(y)Sfree (y),(19)
and the remaining two propagators are replaced with the full quark propagators including the free (perturbative) part
as well as the interacting parts (with gluon or QCD vacuum) as nonperturbative contributions. The total perturbative
photon emission is acquired by performing the replacement mentioned above for the perturbatively interacting quark
propagator with the photon and making use of the replacement of the remaining propagators by their free parts.
In case of nonperturbative photon emission, the light quark propagator in Eq. (16) is replaced by
Sab
αβ → −1
4(¯qaΓiqb)(Γi)αβ ,(20)
where Γirepresent the full set of Dirac matrices. Under this approach, two remaining quark propagators are taken
as the full propagators including perturbative as well as nonperturbative contributions. Once Eq. (20) is inserted
into Eq. (16), there seem matrix elements such as hγ(q)|¯q(x)Γiq(0)|0iand hγ(q)|¯q(x)ΓiGαβq(0)|0i, representing the
nonperturbative contributions. Furthermore, nonlocal operators such as ¯qq ¯qq and ¯qG2qare anticipated to appear.
In this study, we consider operators with only one gluon field and contributions coming from three particle nonlocal
5
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
(a)
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q
(b)
FIG. 1: Feynman diagrams for the magnetic dipole moments of the spin-3/2 doubly heavy baryons. The thick, thin, wavy and
curly lines represent the heavy quark, light quark, photon and gluon propagators, respectively. Diagrams (a) corresponding to
the perturbative photon vertex and, diagrams (b) represent the contributions coming from the DAs of the photon.
operators and ignore terms with four quarks ¯qq¯qq, and two gluons ¯qG2q. In order to calculate the nonperturbative
contributions, we need the matrix elements of the nonlocal operators between the photon states and the vacuum and
these matrix elements are described with respect to the photon DAs with definite twists. Up to twist-4 the explicit
expressions of the photon DAs are given in [85]. Using these expressions for the propagators and DAs for the photon,
the correlation functions from the QCD side can be computed.
The QCD and hadronic sides of the correlation function are then matched using dispersion relation. The next step
in deriving the sum rules for the magnetic dipole moments of the spin-3
2doubly heavy baryons is applying double
Borel transformations (B) over the p2and (p+q)2on the both sides of the correlation function in order to suppress
the contributions of higher states and continuum. As a result, we obtain
BΠHad
µν (p, q) = BΠQC D
µν (p, q),(21)
which leads to
BΠHad
1=BΠQCD
1,BΠHad
2=BΠQCD
2,(22)
corresponding to the structures gµν p/ε/q/ and gµν ε/q/. In this manner we extract the QCD sum rules for the form factors
F1and F2. They are very lengthy functions, therefore we do not give their explicit expressions here.
6
III. NUMERICAL ANALYSIS
In the present section, we achieve numerical analysis for the spin- 3
2doubly heavy baryons. We use mu=md= 0,
ms= 96+8
−4MeV, mc= 1.67±0.07 GeV, mb= 4.78 ±0.0 6 GeV, f3γ=−0.0039 GeV2[85], h¯qqi= (−0.24 ±0.01)3GeV3
[92], m2
0= 0.8±0.1 GeV2,hg2
sG2i= 0.88 GeV 4and χ=−2.85 ±0.5 GeV−2[93]. The masses of the Ξ∗
QQ, Ξ∗
QQ′,
Ω∗
QQ and Ω∗
QQ′baryons are borrowed from Ref. [32], in which the mass sum rules have been used to compute
them. These masses are used to have the following values: MΞ∗
cc = 3.72 ±0.18 GeV, MΩ∗
cc = 3.78 ±0.16 GeV,
MΞ∗
bc = 7.25 ±0.20 GeV, MΩ∗
bc = 7.30 ±0.20 GeV, MΞ∗
bb = 10.40 ±1.00 GeV and MΩ∗
cc = 10.50 ±0.20 GeV. In
order to specify the magnetic dipole moments of doubly heavy baryons, the value of the residues are needed. The
residues of the doubly heavy baryons are computed in Ref. [32]. These residues are calculated to have the following
values: λΞ∗
cc = 0.12 ±0.01 GeV3,λΩ∗
cc = 0.14 ±0.02 GeV3,λΞ∗
bc = 0.15 ±0.01 GeV3,λΩ∗
bc = 0.18 ±0.02 GeV3,
λΞ∗
bb = 0.22 ±0.03 GeV3and λΩ∗
bb = 0.25 ±0.03 GeV3. The parameters used in the photon DAs are given in Ref. [85].
The QCD sum rule for the magnetic dipole moments of the doubly heavy baryons, besides the above mentioned
input parameters, include also two more extra parameters. These parameters are the continuum threshold s0and
the Borel mass parameter M2. According to the QCD sum rules philosophy we need to find the working regions of
these parameters, where the magnetic dipole moments of the doubly heavy baryons be insensitive to the variation
of these parameters in their working regions. The continuum threshold is not totally arbitrary, it is chosen as the
point at which the excited states and continuum begin to contribute to the computations. To designate the working
region of the s0, we enforce the conditions of operator product expansion (OPE) convergence and pole dominance.
In this respect, we choose the value of the continuum threshold within the interval s0= (16 −20) GeV2for Ξ∗
cc,
s0= (58 −62) GeV2for Ξ∗
bc,s0= (116 −120) GeV2for Ξ∗
bb,s0= (18 −22) GeV2for Ω∗
cc,s0= (60 −64) GeV2for Ω∗
bc
and s0= (118 −122) GeV2for Ω∗
bb baryons. The working window for M2is acquired by requiring that the series of
OPE in QCD side is convergent and the contribution of higher states and continuum is adequately suppressed. Our
numerical analysis shows that these conditions are fulfilled when M2changes in the regions: 4 GeV2≤M2≤6 GeV2
for Ξ∗
cc, 7 GeV2≤M2≤9 GeV2for Ξ∗
bc, 10 GeV2≤M2≤14 GeV2for Ξ∗
bb, 5 GeV2≤M2≤7 GeV2for Ω∗
cc,
8 GeV2≤M2≤10 GeV2for Ω∗
bc and 11 GeV2≤M2≤15 GeV2for Ω∗
bb baryons. As an example in Fig. 2, we
present the dependencies of the magnetic dipole moments of doubly charmed baryons on M2at several fixed values
of the continuum threshold s0. As is seen from the figure, though being not completely insensitive, the magnetic
dipole moments exhibit moderate dependency on the auxiliary parameters, continuum threshold s0and the Borel
mass parameter M2which is reasonable in the error limits of the QCD sum rule formalism.
Our final results on the magnetic dipole moments for the spin- 3
2doubly heavy baryons are
µΞ∗++
cc = 2.94 ±0.95,
µΞ∗+
cc =−0.67 ±0.11,
µΩ∗+
cc =−0.52 ±0.07,
µΞ∗0
bb = 2.30 ±0.55.
µΞ∗−
bb =−1.39 ±0.32,
µΩ∗−
bb =−1.56 ±0.33,
µΞ∗+
bc = 2.12 ±0.68,
µΞ∗0
bc =−0.96 ±0.32,
µΩ∗+
bc =−1.11 ±0.33,(23)
where the quoted errors in the results are due to the uncertainties in the values of the input parameters and the photon
DAs, as well as the variations in the calculations of the working regions M2and s0. We also need to emphasize that
the main source of uncertainties is the variations with respect to s0and the results weakly depend on the choices of
the Borel mass parameter.
Table II shows a comparison of our results magnetic dipole moments with those from various other models such as,
quark model (QM) [75], the relativistic harmonic confinement model (RHM) [74], MIT bag model [71–73], Skyrmion
model [76], nonrelativistic quark model (NRQM) [22], hyper central constituent model (HCQM) [38–40], effective mass
and screened charge scheme [63,64], chiral constituent quark model (χCQM) [67] and heavy baryon chiral perturbation
theory (HBChBT) [59]. From a comparison of our results with the predictions of other models we observe from this
table that for the the Ξ∗++
cc baryon, almost all approaches give, more or less, similar predictions. For the Ξ∗+
cc and
Ω∗+
cc baryons, there are large discrepancy among results not only the magnitude but also by the sign. For the the
Ξ∗0
bb baryon, our estimation is consistent within the errors with Refs. [22,59,74] and unlike other approaches. For
the the Ξ∗−
bb baryon, almost all approaches give, more or less, similar predictions except the results of Refs. [71–73],
7
Ξ∗++
cc Ξ∗+
cc Ω∗+
cc Ξ∗0
bb Ξ∗−
bb Ω∗−
bb Ξ∗+
bc Ξ∗0
bc Ω∗0
bc
[75] 2.60 -0.19 0.17 - - - - - -
[76] 3.16 -0.98 -0.20 - - - - - -
[67] 2.66 -0.47 0.14 - - - - - -
[38] 2.75 -0.17 0.12 - - - - - -
[39,40] 2.22 0.07 0.29 1.61 -1.74 -1.24 1.56 -0.38 -0.18
[63,64] 2.41 -0.11 0.16 1.60 -0.98 -0.70 2.01 -0.55 -0.28
[63,64] 2.52 0.04 0.21 1.50 -1.02 -0.80 2.02 -0.50 -0.30
[22] 2.67 -0.31 0.14 1.87 -1.11 -0.66 2.27 -0.71 -0.26
[71] 2.54 0.20 0.39 1.37 -0.95 -1.28 2.03 -0.39 -0.23
[72] 2.00 0.16 0.33 0.92 -0.65 -0.52 1.41 -0.25 -0.11
[73] 2.35 -0.18 -0.05 1.40 -0.88 -0.70 1.88 -0.54 -0.33
[74] 2.72 -0.23 0.16 2.30 -1.32 -0.86 2.68 -0.76 -0.32
[59] 3.51 -0.27 -0.64 2.83 -1.33 -1.54 3.22 -0.84 -1.09
[59] 3.63 -0.37 -0.65 2.87 -1.38 -1.55 3.27 -0.89 -1.10
This work 2.94 -0.67 -0.52 2.30 -1.39 -1.56 2.12 -0.96 -1.11
TABLE II: Magnetic dipole moments of the spin- 3
2doubly heavy baryons (in nuclear magnetons µN).
which are small. For the Ω∗−
bb baryon, our estimation is consistent within the errors with Refs. [39,40,59,71] and
different from other results. For the the Ξ∗+
bc baryon, we see that within errors our predictions in good agreement
with the Refs [22,59,63,64,71,74]. For the Ξ∗0
bc baryon, while the sign of the magnetic dipole moment is correctly
determined, there is a large discrepancy among results. For the Ω∗0
bc baryon, almost all approaches give, more or
less, similar predictions except the results of Ref. [59] and this work, which are quite large. As can be seen from the
magnetic dipole moment results of the doubly heavy baryons given in this table, the results obtained using different
models lead to rather different estimations, which can be used to distinguish these models. Obviously, more studies
are needed to understand the current situation.
IV. DISCUSSION AND CONCLUDING REMARKS
In the presented paper we have calculated the magnetic dipole moments of the spin- 3
2doubly heavy baryons by
means of the light-cone QCD sum rule. The magnetic dipole moments of the doubly heavy baryons encodes important
information of their internal structure and shape deformations. Measurement of the magnetic dipole moments of the
spin- 3
2doubly heavy baryons in future experiments can be very helpful understanding the internal structure of these
baryons. However, the direct measurement of the magnetic dipole moments of the spin- 3
2doubly heavy baryons are
unlikely in the near future. Therefore, any unstraightforward estimations of the magnetic dipole moments of the
spin- 3
2doubly heavy baryons could be very helpful. Comparison of our results with the estimation of other theoretical
models is presented.
V. ACKNOWLEDGEMENTS
We are grateful to V. S. Zamiralov for useful discussions, comments and remarks.
8
44.5 5 5.5 6
M2[GeV2]
0
1.5
3
4.5
6
7.5
9
µ Ξ*++[ µN]
s0 = 16 GeV2
s0 = 18 GeV2
s0 = 20 GeV2
44.5 5 5.5 6
M2[GeV2]
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
µ Ξ*+[ µN]
s0 = 16 GeV2
s0 = 18 GeV2
s0 = 20 GeV2
5 5.5 66.5 7
M2[GeV2]
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
µ Ω*+[ µN]
s0 = 18 GeV2
s0 = 20 GeV2
s0 = 22 GeV2
FIG. 2: The depend ence of the magnetic dipole moments for the spin- 3
2doubly charmed baryons on the Borel parameter
squared M2at different fixed values of the continuum threshold.
[1] M. Mattson et al. (SELEX), Phys. Rev. Lett. 89, 112001 (2002).
[2] R. Chistov et al. (Belle), Phys. Rev. Lett. 97, 162001 (2006).
[3] S. P. Ratti, Nucl. Phys. Proc. Suppl. 115, 33 (2003).
[4] B. Aubert et al. (BaBar), Phys. Rev. D74, 011103 (2006).
[5] R. Aaij et al. (LHCb), Phys. Rev. Lett. 119, 112001 (2017).
[6] E. Bagan, M. Chabab, and S. Narison, Phys. Lett. B306, 350 (1993).
[7] R. Roncaglia, D. B. Lichtenberg, and E. Predazzi, Phys. Rev. D52, 1722 (1995).
[8] D. Ebert, R. N. Faustov, V. O. Galkin, A. P. Martynenko, and V. A. Saleev, Z. Phys. C76, 111 (1997).
9
[9] S.-P. Tong, Y.-B. Ding, X.-H. Guo, H.-Y. Jin, X.-Q. Li, P.-N. Shen, and R. Zhang, Phys. Rev. D62, 054024 (2000).
[10] C. Itoh, T. Minamikawa, K. Miura, and T. Watanabe, Phys. Rev. D61, 057502 (2000).
[11] S. S. Gershtein, V. V. Kiselev, A. K. Likhoded, and A. I. Onishchenko, Phys. Rev. D62, 054021 (2000).
[12] V. V. Kiselev and A. K. Likhoded, Phys. Usp. 45, 455 (2002).
[13] V. V. Kiselev, A. K. Likhoded, O. N. Pakhomova, and V. A. Saleev, Phys. Rev. D66, 034030 (2002).
[14] I. M. Narodetskii and M. A. Trusov, Phys. Atom. Nucl. 65, 917 (2002), [Yad. Fiz.65,949(2002)].
[15] R. Lewis, N. Mathur, and R. M. Woloshyn, Phys. Rev. D64, 094509 (2001).
[16] D. Ebert, R. N. Faustov, V. O. Galkin, and A. P. Martynenko, Phys. Rev. D66, 014008 (2002).
[17] N. Mathur, R. Lewis, and R. M. Woloshyn, Phys. Rev. D66, 014502 (2002).
[18] J. M. Flynn, F. Mescia, and A. S. B. Tariq (UKQCD), JHEP 07, 066 (2003).
[19] J. Vijande, H. Garcilazo, A. Valcarce, and F. Fernandez, Phys. Rev. D70, 054022 (2004).
[20] T.-W. Chiu and T.-H. Hsieh, Nucl. Phys. A755, 471 (2005).
[21] S. Migura, D. Merten, B. Metsch, and H.-R. Petry, Eur. Phys. J. A28, 41 (2006).
[22] C. Albertus, E. Hernandez, J. Nieves, and J. M. Verde-Velasco, Eur. Phys. J. A32, 183 (2007), [Erratum: Eur. Phys.
J.A36,119(2008)].
[23] A. P. Martynenko, Phys. Lett. B663, 317 (2008).
[24] L. Tang, X.-H. Yuan, C.-F. Qiao, and X.-Q. Li, Commun. Theor. Phys. 57, 435 (2012).
[25] X. Liu, H.-X. Chen, Y.-R. Liu, A. Hosaka, and S.-L. Zhu, Phys. Rev. D77, 014031 (2008).
[26] W. Roberts and M. Pervin, Int. J. Mod. Phys. A23, 2817 (2008).
[27] H.-S. Li, L. Meng, Z.-W. Liu, and S.-L. Zhu, Phys. Lett. B777, 169 (2018).
[28] A. Valcarce, H. Garcilazo, and J. Vijande, Eur. Phys. J. A37, 217 (2008).
[29] L. Liu, H.-W. Lin, K. Orginos, and A. Walker-Loud, Phys. Rev. D81, 094505 (2010).
[30] C. Alexandrou, J. Carbonell, D. Christaras, V. Drach, M. Gravina, and M. Papinutto, Phys. Rev. D86, 114501 (2012).
[31] T. M. Aliev, K. Azizi, and M. Savci, Nucl. Phys. A895, 59 (2012).
[32] T. M. Aliev, K. Azizi, and M. Savci, J. Phys. G40, 065003 (2013).
[33] Y. Namekawa et al. (PACS-CS), Phys. Rev. D87, 094512 (2013).
[34] M. Karliner and J. L. Rosner, Phys. Rev. D90, 094007 (2014).
[35] Z.-F. Sun, Z.-W. Liu, X. Liu, and S.-L. Zhu, Phys. Rev. D91, 094030 (2015).
[36] H.-X. Chen, W. Chen, Q. Mao, A. Hosaka, X. Liu, and S.-L. Zhu, Phys. Rev. D91, 054034 (2015).
[37] Z.-F. Sun and M. J. Vicente Vacas, Phys. Rev. , 094002 (2016).
[38] B. Patel, A. K. Rai, and P. C. Vinodkumar (2008) arXiv:0803.0221 [hep-ph] .
[39] Z. Shah, K. Thakkar, and A. K. Rai, Eur. Phys. J. C76, 530 (2016).
[40] Z. Shah and A. K. Rai, Eur. Phys. J. C77, 129 (2017).
[41] V. V. Kiselev, A. V. Berezhnoy, and A. K. Likhoded, Phys. Atom. Nucl. 81, 369 (2018), [Yad. Fiz.81, no.3, 356(2018)].
[42] H.-X. Chen, Q. Mao, W. Chen, X. Liu, and S.-L. Zhu, Phys. Rev. D96, 031501 (2017), [Erratum: Phys.
Rev.D96,no.11,119902(2017)].
[43] J. Hu and T. Mehen, Phys. Rev. D73, 054003 (2006).
[44] F.-S. Yu, H.-Y. Jiang, R.-H. Li, C.-D. L¨u, W. Wang, and Z.-X. Zhao, Chin. Phys. C42, 051001 (2018).
[45] R.-H. Li, C.-D. L¨u, W. Wang, F.-S. Yu, and Z.-T. Zou, Phys. Lett. B767, 232 (2017).
[46] L. Meng, N. Li, and S.-L. Zhu, Phys. Rev. D95, 114019 (2017).
[47] S. Narison and R. Albuquerque, Phys. Lett. B694, 217 (2011).
[48] J.-R. Zhang and M.-Q. Huang, Phys. Rev. D78, 094007 (2008).
[49] Z.-G. Wang, Eur. Phys. J. A45, 267 (2010).
[50] Z.-G. Wang, Eur. Phys. J. C68, 459 (2010).
[51] Z.-H. Guo, Phys. Rev. D96, 074004 (2017).
[52] W. Wang, F.-S. Yu, and Z.-X. Zhao, Eur. Phys. J. C77, 781 (2017).
[53] W. Wang, Z.-P. Xing, and J. Xu, Eur. Phys. J. C77, 800 (2017).
[54] Y.-J. Shi, W. Wang, Y. Xing, and J. Xu, Eur. Phys. J. C78, 56 (2018).
[55] Q.-F. L¨u, K.-L. Wang, L.-Y. Xiao, and X.-H. Zhong, Phys. Rev. D96, 114006 (2017).
[56] L.-Y. Xiao, K.-L. Wang, Q.-f. Lu, X.-H. Zhong, and S.-L. Zhu, Phys. Rev. D96, 094005 (2017).
[57] X.-Z. Weng, X.-L. Chen, and W.-Z. Deng, Phys. Rev. D97, 054008 (2018).
[58] E.-L. Cui, H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Phys. Rev. D97, 034018 (2018).
[59] L. Meng, H.-S. Li, Z.-W. Liu, and S.-L. Zhu, Eur. Phys. J. C77, 869 (2017).
[60] B. Silvestre-Brac, Few Body Syst. 20, 1 (1996).
[61] B. Julia-Diaz and D. O. Riska, Nucl. Phys. A739, 69 (2004).
[62] A. Faessler, T. Gutsche, M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij, D. Nicmorus, and K. Pumsa-ard, Phys. Rev.
D73, 094013 (2006).
[63] R. Dhir and R. C. Verma, Eur. Phys. J. A42, 243 (2009).
[64] R. Dhir, C. S. Kim, and R. C. Verma, Phys. Rev. D88, 094002 (2013).
[65] B. Patel, A. K. Rai, and P. C. Vinodkumar, J. Phys. G35, 065001 (2008).
[66] T. Branz, A. Faessler, T. Gutsche, M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij, and B. Oexl, Phys. Rev. D81, 114036
(2010).
[67] N. Sharma, H. Dahiya, P. K. Chatley, and M. Gupta, Phys. Rev. D81, 073001 (2010).
[68] K. U. Can, G. Erkol, B. Isildak, M. Oka, and T. T. Takahashi, Phys. Lett. B726, 703 (2013).
10
[69] K. U. Can, G. Erkol, B. Isildak, M. Oka, and T. T. Takahashi, JHEP 05, 125 (2014).
[70] H.-S. Li, L. Meng, Z.-W. Liu, and S.-L. Zhu, Phys. Rev. D96, 076011 (2017).
[71] S. K. Bose and L. P. Singh, Phys. Rev. D22, 773 (1980).
[72] A. Bernotas and V. Simonis, (2012), arXiv:1209.2900 [hep-ph] .
[73] V. Simonis, (2018), arXiv:1803.01809 [hep-ph] .
[74] A. N. Gadaria, N. R. Soni, and J. N. Pandya, DAE Symp. Nucl. Phys. 61, 698 (2016).
[75] D. B. Lichtenberg, Phys. Rev. D15, 345 (1977).
[76] Y.-s. Oh, D.-P. Min, M. Rho, and N. N. Scoccola, Nucl. Phys. A534, 493 (1991).
[77] M.-Z. Liu, Y. Xiao, and L.-S. Geng, Phys. Rev. D98, 014040 (2018).
[78] A. N. Hiller Blin, Z.-F. Sun, and M. J. Vicente Vacas, Phys. Rev. D98, 054025 (2018).
[79] U. ¨
Ozdem, J. Phys. G46, 035003 (2019).
[80] Y.-J. Shi, W. Wang, and Z.-X. Zhao, (2019), arXiv:1902.01092 [hep-ph] .
[81] Y.-J. Shi, Y. Xing, and Z.-X. Zhao, (2019), arXiv:1903.03921 [hep-ph] .
[82] V. L. Chernyak and I. R. Zhitnitsky, Nucl. Phys. B345, 137 (1990).
[83] V. M. Braun and I. E. Filyanov, Z. Phys. C44, 157 (1989), [Yad. Fiz.50,818(1989)].
[84] I. I. Balitsky, V. M. Braun, and A. V. Kolesnichenko, Nucl. Phys. B312, 509 (1989).
[85] P. Ball, V. M. Braun, and N. Kivel, Nucl. Phys. B649, 263 (2003).
[86] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Fortsch. Phys. 32, 585 (1984).
[87] V. Pascalutsa, M. Vanderhaeghen, and S. N. Yang, Phys. Rept. 437, 125 (2007).
[88] G. Ramalho, M. T. Pena, and F. Gross, Phys. Lett. B678, 355 (2009).
[89] V. M. Belyaev and B. L. Ioffe, Sov. Phys. JETP 57, 716 (1983), [Zh. Eksp. Teor. Fiz.84,1236(1983)].
[90] K.-C. Yang, W. Y. P. Hwang, E. M. Henley, and L. S. Kisslinger, Phys. Rev. D47, 3001 (1993).
[91] V. M. Belyaev and B. Yu. Blok, Z. Phys. C30, 151 (1986).
[92] B. L. Ioffe, Prog. Part. Nucl. Phys. 56, 232 (2006).
[93] J. Rohrwild, JHEP 09, 073 (2007).
