Magnetic dipole moments of the spin-$\frac{3}{2}$ doubly heavy baryons

TL;DR

This paper calculates the magnetic dipole moments of spin-3/2 doubly heavy baryons using light-cone QCD sum rules, providing insights into their internal structure and shape deformations.

Contribution

It presents the first detailed theoretical predictions of magnetic dipole moments for various doubly heavy baryons using light-cone QCD sum rules.

Findings

Magnetic moments for specific baryons are numerically estimated.

Results indicate significant variations in magnetic moments among different baryons.

Provides theoretical benchmarks for future experimental measurements.

Abstract

The magnetic dipole moments of the spin-$\frac{3}{2}$ doubly charmed, bottom and charmed-bottom baryons are obtained by means of the light-cone QCD sum rule. The magnetic dipole moments of these baryons encode essential knowledge of their inner structure and shape deformations. The numerical results are given as, $\mu_{\Xi_{cc}^{*++}} = 2.94 \pm 0.95$, $\mu_{\Xi_{cc}^{*+}} = - 0.67 \pm 0.11$, $\mu_{\Omega_{cc}^{*+}} =- 0.52 \pm 0.07$, $\mu_{\Xi_{bb}^{*0}} = 2.30 \pm 0.55$, $\mu_{\Xi_{bb}^{*-}} = -1.39 \pm 0.32$, $\mu_{\Omega_{bb}^{*-}} = -1.56 \pm 0.33$, $\mu_{\Xi_{bc}^{*+}} = 2.63 \pm 0.82$, $\mu_{\Xi_{bc}^{*0}} = - 0.96 \pm 0.32$ and $\mu_{\Omega_{bc}^{*+}} =- 1.11 \pm 0.33$, respectively.

Tables

Table 1. Table 1: The quark content of the spin-3/2 DHBs.

Table 2. Table 2: MDMs of the spin- 3 2 3 2 \frac{3}{2} DHBs (in nuclear magnetons μ N subscript 𝜇 𝑁 \mu_{N} ).

Approaches	${\Xi }_{cc}^{\ast ++}$	${\Xi }_{cc}^{\ast +}$	${\Omega }_{cc}^{\ast +}$	${\Xi }_{bb}^{\ast 0}$	${\Xi }_{bb}^{\ast -}$	${\Omega }_{bb}^{\ast -}$	${\Xi }_{bc}^{\ast +}$	${\Xi }_{bc}^{\ast 0}$	${\Omega }_{bc}^{\ast 0}$
NRQM  Albertus:2006ya	2.67	-0.31	0.14	1.87	-1.11	-0.66	2.27	-0.71	-0.26
HCQM  Shah:2016vmd ; Shah:2017liu	2.22	0.07	0.29	1.61	-1.74	-1.24	1.56	-0.38	-0.18
HBChBT-I  Meng:2017dni	3.51	-0.27	-0.64	2.83	-1.33	-1.54	3.22	-0.84	-1.09
HBChBT-II  Meng:2017dni	3.63	-0.37	-0.65	2.87	-1.38	-1.55	3.27	-0.89	-1.10
EMS  Dhir:2009ax ; Dhir:2013nka	2.41	-0.11	0.16	1.60	-0.98	-0.70	2.01	-0.55	-0.28
SCS  Dhir:2009ax ; Dhir:2013nka	2.52	0.04	0.21	1.50	-1.02	-0.80	2.02	-0.50	-0.30
MIT Bag model-I  Bernotas:2012nz	2.00	0.16	0.33	0.92	-0.65	-0.52	1.41	-0.25	-0.11
MIT Bag model-II  Simonis:2018rld	2.35	-0.18	-0.05	1.40	-0.88	-0.70	1.88	-0.54	-0.33
RHM  Gadaria:2016omw	2.72	-0.23	0.16	2.30	-1.32	-0.86	2.68	-0.76	-0.32
This work	2.94	-0.67	-0.52	2.30	-1.39	-1.56	2.63	-0.96	-1.11

Table 3. Table 3: Comparision of the first splitting.

Approaches	$\Delta {\mu }_u$	$\Delta {\mu }_d$	$\Delta {\mu }_s$
NRQM Albertus:2006ya	0.00	0.00	0.00
HCQM Shah:2016vmd ; Shah:2017liu	0.71	-0.91	-0.59
HBChBT-I Meng:2017dni	-0.10	0.08	0.00
HBChBT-II Meng:2017dni	-0.04	0.03	0.00
EMS Dhir:2009ax ; Dhir:2013nka	-0.01	0.01	0.02
SCS Dhir:2009ax ; Dhir:2013nka	-0.02	0.02	0.01
MIT Bag model-I Bernotas:2012nz	0.10	0.01	0.03
MIT Bag model-II Simonis:2018rld	-0.01	0.02	-0.09
RHM Gadaria:2016omw	-0.34	-0.03	-0.06
This work	-0.02	-0.14	0.14

Table 4. Table 4: Comparision of the second and third splittings.

Approaches	$\Delta {\mu }_{u-d}^{cc}$	$\Delta {\mu }_{u-d}^{bb}$	$\Delta {\mu }_{u-d}^{bc}$	$\Delta {\mu }_{d-s}^{cc}$	$\Delta {\mu }_{d-s}^{bb}$	$\Delta {\mu }_{d-s}^{bc}$
NRQM Albertus:2006ya	2.98	2.98	2.98	-0.45	-0.45	-0.45
HCQM  Shah:2016vmd ; Shah:2017liu	2.15	3.35	1.94	-0.22	-0.50	-0.20
HBChBT-I Meng:2017dni	3.78	4.16	4.06	0.37	0.21	0.25
HBChBT-II  Meng:2017dni	4.00	4.25	4.16	0.28	0.17	0.21
EMS Dhir:2009ax ; Dhir:2013nka	2.52	2.58	2.56	-0.27	-0.28	-0.27
SCS Dhir:2009ax ; Dhir:2013nka	2.48	2.52	2.52	-0.17	-0.22	-0.20
MIT Bag model-I Bernotas:2012nz	1.84	1.57	1.66	-0.17	-0.13	-0.14
MIT Bag model-II Simonis:2018rld	2.53	2.28	2.42	-0.13	-0.18	-0.21
RHM Gadaria:2016omw	2.95	3.62	3.44	-0.39	-0.46	-0.44
This work	3.61	3.69	3.59	-0.15	0.17	0.15

Equations

$${\Pi }_{\mu \nu \alpha }(p,q)$$

$$J_{\alpha }=\sum _{q=u,d,s,c,b}{e}_q\bar{q}{\gamma }_{\alpha }q,$$

$${\Pi }_{\mu \nu }(p,q)=i\int d^{4}x{e}^{ip\cdot x}⟨0|\mathcal{T}\{J_{\mu }^{B_{QQ}^{\ast }}(x)J_{\nu }^{B_{QQ}^{\ast †}}(0)\}|0{⟩}_F,$$

$${\Pi }_{\mu \nu }(p,q)={\Pi }_{\mu \nu }^{(0)}(p,q)+{\Pi }_{\mu \nu }^{(1)}(p,q)+....,$$

$${\Pi }_{\mu \nu }^{Had}(p,q)$$

$$⟨B_{QQ}^{\ast }(p)\mid B_{QQ}^{\ast }(p+q){⟩}_F$$

$$\frac{⟨B_{QQ}^{\ast }(p+q)\mid {\bar{J}}_{\nu }^{B_{QQ}^{\ast }}\mid 0⟩}{[(p+q{)}^2-m_{B_{QQ}^{\ast }}^2]}+...,$$

$$⟨0\mid J_{\mu }(0)\mid B_{QQ}^{\ast }(p,s)⟩$$

$$⟨B_{QQ}^{\ast }(p)\mid B_{QQ}^{\ast }(p+q){⟩}_F$$

$$\displaystyle-\frac{1}{2m_{{B^{*}_{QQ}}}}\Big{[}F_{2}(q^{2})g_{\mu\nu}+F_{4}(q^{2})$$

$$\displaystyle\times\frac{q_{\mu}q_{\nu}}{(2m_{{B^{*}_{QQ}}})^{2}}\Big{]}\varepsilon\!\!\!/q\!\!\!/+\frac{F_{3}(q^{2})}{(2m_{{B^{*}_{QQ}}})^{2}}$$

$$\displaystyle\times q_{\mu}q_{\nu}\varepsilon\!\!\!/\Bigg{\}}u_{\nu}(p+q),$$

$$\sum _s{u}_{\mu }(p,s){\bar{u}}_{\nu }(p,s)$$

$${\Pi }_{\mu \nu }^{Had}(p,q)$$

$$\displaystyle\Big{[}g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2\,p_{\mu}p_{\nu}}{3\,m^{2}_{{B^{*}_{QQ}}}}+\frac{p_{\mu}\gamma_{\nu}-p_{\nu}\gamma_{\mu}}{3\,m_{{B^{*}_{QQ}}}}\Big{]}$$

$$\displaystyle\times\Bigg{\{}F_{1}(q^{2})g_{\mu\nu}\varepsilon\!\!\!/-\frac{1}{2m_{{B^{*}_{QQ}}}}\Big{[}F_{2}(q^{2})g_{\mu\nu}$$

$$\displaystyle+F_{4}(q^{2})\frac{q_{\mu}q_{\nu}}{(2m_{{B^{*}_{QQ}}})^{2}}\Big{]}\varepsilon\!\!\!/q\!\!\!/+\frac{F_{3}(q^{2})}{(2m_{{B^{*}_{QQ}}})^{2}}q_{\mu}q_{\nu}\varepsilon\!\!\!/\Bigg{\}}.$$

$${\Pi }_{\mu \nu }^{Had}(p,q)=$$

$$\displaystyle\times\Bigg{[}-g_{\mu\nu}p\!\!\!/\varepsilon\!\!\!/q\!\!\!/\,F_{1}(q^{2})+m_{{B^{*}_{QQ}}}g_{\mu\nu}\varepsilon\!\!\!/q\!\!\!/\,F_{2}(q^{2})$$

$$\displaystyle+~{}\mbox{other independent structures}\Bigg{]}.$$

$$G_M(q^2)$$

$$G_M(0)$$

$${\mu }_{B_{QQ}^{\ast }}$$

$${\Pi }_{\mu \nu }^{Had}(p,q)$$

$$J_{\mu }^{B_{QQ}^{\ast }}(x)$$

$$\displaystyle+(Q^{aT}C\gamma_{\mu}Q^{\prime b})q^{c}\Big{\}}~{},$$

$${\Pi }_{\mu \nu }^{QCD}(p)$$

$$S_q(x)$$

$$\displaystyle-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\bigg{[}\hbox to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox to0.0pt{/\hss}{x}\bigg{]},$$

$$S_Q(x)$$

$$\displaystyle\times\frac{K_{1}(m_{Q}\sqrt{-x^{2}})}{\sqrt{-x^{2}}}+2\sigma^{\mu\nu}K_{0}(m_{Q}\sqrt{-x^{2}})\bigg{]},$$

$$\displaystyle S_{q}^{free}=\frac{1}{2\pi^{2}x^{2}}\Big{(}i\frac{{x\!\!\!/}}{x^{2}}-\frac{m_{q}}{2}\Big{)},$$

$$\displaystyle S_{Q}^{free}=\frac{m_{Q}^{2}}{4\pi^{2}}\bigg{[}\frac{K_{1}(m_{Q}\sqrt{-x^{2}})}{\sqrt{-x^{2}}}+i\frac{{x\!\!\!/}~{}K_{2}(m_{Q}\sqrt{-x^{2}})}{(\sqrt{-x^{2}})^{2}}\bigg{]},$$

$$S^{free}(x)\to \int d^{4}y{S}^{free}(x-y)to0.0pt/\text{hss}A(y)S^{free}(y),$$

$$S_{\alpha \beta }^{ab}\to -\frac{1}{4}({\bar{q}}^a{\Gamma }_i{q}^b)({\Gamma }_i{)}_{\alpha \beta },$$

$$\mathcal{B}{\Pi }_{\mu \nu }^{Had}(p,q)=\mathcal{B}{\Pi }_{\mu \nu }^{QCD}(p,q),$$

$$\mathcal{B}{\Pi }_1^{Had}=\mathcal{B}{\Pi }_1^{QCD},\mathcal{B}{\Pi }_2^{Had}=\mathcal{B}{\Pi }_2^{QCD},$$

$${\mu }_{{\Xi }_{cc}^{\ast ++}}$$

Full text

Electromagnetic form factors, Magnetic dipole moment, Doubly heavy baryons, Light-cone QCD sum rules

I Motivation

The doubly heavy baryons (DHBs) presumably contain two heavy quark and one light quark. One of them was firstly announced by the SELEX Collaboration in the decay mode $\Xi_{cc}^{+}\rightarrow\Lambda_{c}^{+}K^{-}\pi^{+}$ with the mass $M_{\Xi^{+}_{cc}}=3519\pm 1$ MeV Mattson:2002vu . However, neither Belle Chistov:2006zj , nor FOCUS Ratti:2003ez , nor BABAR Aubert:2006qw could confirm the DHBs 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 $\Xi_{cc}^{++}$ in the mass spectrum of $\Lambda_{c}^{+}\,K^{-}\pi^{+}\,\pi^{+}$ with the mass $M_{\Xi_{cc}^{++}}=3621.40\pm 0.72\pm 0.27\pm 0.14$ MeV Aaij:2017ueg . Recently, the LHCb Collaboration reconstructed their analysis via decay modes $\Xi_{cc}^{++}\rightarrow\Lambda_{c}^{+}\,K^{-}\pi^{+}\,\pi^{+}$ and $\Xi_{cc}^{++}\rightarrow\Xi_{c}^{+}\,\pi^{+}$ and they reported their mass value as $M_{\Xi_{cc}^{++}}=3621.55\pm 0.23\pm 0.30$ MeV Aaij:2019uaz . The investigation for the DHBs may provide with valuable knowledge for comprehension of the nonperturbative QCD effects. One of the several point of views which makes the physics of DHBs 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 DHBs is one of the active and interesting branches of particle physics. Therefore, the weak Albertus:2006ya ; Li:2017ndo ; Wang:2017mqp ; Wang:2017azm ; Shi:2017dto ; Shi:2019hbf ; Shi:2019fph , strong Hu:2005gf ; Xiao:2017udy and radiative decays Li:2017pxa ; Yu:2017zst ; Lu:2017meb ; Cui:2017udv , the magnetic dipole moments (MDMs) Can:2013zpa ; Branz:2010pq ; Bose:1980vy ; Patel:2008xs ; SilvestreBrac:1996bg ; Patel:2007gx ; Gadaria:2016omw ; JuliaDiaz:2004vh ; Faessler:2006ft ; Can:2013tna ; Li:2017cfz ; Bernotas:2012nz ; Lichtenberg:1976fi ; Oh:1991ws ; Simonis:2018rld ; Liu:2018euh ; Blin:2018pmj ; Meng:2017dni ; Dhir:2009ax ; Ozdem:2018uue and masses Bagan:1992za ; Roncaglia:1995az ; Ebert:1996ec ; Tong:1999qs ; Itoh:2000um ; Gershtein:2000nx ; Kiselev:2001fw ; Kiselev:2002iy ; Narodetskii:2001bq ; Lewis:2001iz ; Ebert:2002ig ; Mathur:2002ce ; Flynn:2003vz ; Vijande:2004at ; Chiu:2005zc ; Migura:2006ep ; Albertus:2006ya ; Martynenko:2007je ; Tang:2011fv ; Liu:2007fg ; Roberts:2007ni ; Valcarce:2008dr ; Liu:2009jc ; Alexandrou:2012xk ; Aliev:2012ru ; Aliev:2012iv ; Namekawa:2013vu ; Karliner:2014gca ; Sun:2014aya ; Chen:2015kpa ; Sun:2016wzh ; Shah:2016vmd ; Kiselev:2017eic ; Chen:2017sbg ; Hu:2005gf ; Meng:2017fwb ; Narison:2010py ; Zhang:2008rt ; Guo:2017vcf ; Lu:2017meb ; Xiao:2017udy ; Weng:2018mmf ; Can:2013zpa ; Branz:2010pq ; Bose:1980vy ; Patel:2008xs ; SilvestreBrac:1996bg ; Patel:2007gx ; Gadaria:2016omw of the DHBs have been examined broadly in literature by the help of the quark models, potential models, lattice QCD, Feynman-Hellmann theorem, extended chromomagnetic model, chiral perturbation theory, heavy quark effective theory, QCD sum rules, perturbative QCD, Faddeev approach, SU(3) flavor symmetry, nonperturbative string approach, local diquark approach, light-cone QCD sum rules, light front approach and extended on-mass-shell renormalization scheme.

Electromagnetic properties one of the major and meaningful parameters of the DHBs. As the electromagnetic properties symbolise necessary viewpoints of the intrinsic properties of hadrons, it is quite significant to analyze the baryon electromagnetic form factors, particularly the MDMs. The magnitude and sign of the MDM ensure crucial data on size, structure and shape deformations of baryons. Apparently, determining the MDM is an important step in our comprehension of the baryon features with regards to quark-gluon degrees of freedom. In this study, we are going to concentrate on the DHBs (from now on we will represent these particles as $B^{*}_{QQ}$ ) with spin-parity $J^{P}=\frac{3}{2}^{+}$, and calculate their MDMs by the help of the 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 of the on-shell particles Chernyak:1990ag ; Braun:1988qv ; Balitsky:1989ry . Since the MDMs are quantities in terms of the properties of the vacuum and distribution amplitudes of the particles, any uncertainties in these parameters are reflected to the uncertainties of the predictions of the MDMs. The first extraction of the MDMs of the spin-$\frac{3}{2}$ doubly heavy baryons were obtained by Lichtenberg Lichtenberg:1976fi by means of the naive quark model. Then, many scientists have used different theoretical models to compute the spin-$\frac{3}{2}$ doubly heavy baryon MDMs Albertus:2006ya ; Shah:2016vmd ; Shah:2017liu ; Meng:2017dni ; Dhir:2009ax ; Dhir:2013nka ; Bernotas:2012nz ; Simonis:2018rld ; Gadaria:2016omw . In Ref. Albertus:2006ya , they have been calculated the static properties and semileptonic decays of the DHBs with the help of nonrelativistic quark model. To examination the dependence of their results on inter-quark interaction they use various quark potential that contain hyperfine terms coming from one gluon exchange and Coulomb term as well. In Refs. Shah:2016vmd ; Shah:2017liu , the masses of the ground, orbital and radial states of the DHBs are evaluated in the framework of hypercentral constituent quark model with Coulomb plus linear potential. The MDMs of the ground state DHBs are also extracted. In Ref. Meng:2017dni , the MDMs of the spin-$\frac{3}{2}$ doubly charmed baryons are investigated up to NNLO by means of the heavy baryon chiral perturbation theory. As a by product, they obtained the numerical values of the doubly bottom and charmed-bottom baryons up to NLO. In Refs. Dhir:2009ax ; Dhir:2013nka , they are calculated the MDMs of singly and DHBs with spin-$\frac{3}{2}$ and spin-$\frac{1}{2}$ in the framework of effective quark mass and screened charge of quark. In Ref. Bernotas:2012nz , the MDMs of the DHBs are investigated in the MIT bag model with center of mass motion corrections. In Ref. Simonis:2018rld , the decay widths, MDMs and M1 transitions of the all ground states heavy baryons are evaluated by means of the extended MIT bag model. In Ref. Gadaria:2016omw , they are utilized the extended relativistic harmonic confinement model to acquire masses and MDMs of the heavy flavored ground state baryons.

The plan of the manuscript is as follows. In section II, the details of the MDMs computations for the DHBs with spin-$\frac{3}{2}$ are presented. Numerical analysis of the LCSR for the MDMs are given in section III. Section IV is reserved for discussion and concluding remarks.

II Formalism

To obtain the MDM of the DHBs by using the LCSR approach, we begin with the subsequent correlation function,

[TABLE]

Here, $J_{\mu(\nu)}$ is the interpolating current of the $B^{*}_{QQ}$ baryons and the electromagnetic current $J_{\alpha}$ is given as,

[TABLE]

where $e_{q}$ is the electric charge of the corresponding quark.

From a technical point of view, the Eq. (1) can be reconsidered in a more appropriate form by the help of external background electromagnetic (EBGEM) field,

[TABLE]

where F is the EBGEM field and $F_{\alpha\beta}=i(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})$ with $\varepsilon_{\beta}$ and $q_{\alpha}$ being 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,

[TABLE]

and keeping only terms $\Pi_{\mu\nu}^{(1)}(p,q)$, which corresponds to the single photon emission Ball:2002ps . The primary 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 Ball:2002ps . The $\Pi_{\mu\nu}^{(0)}(p,q)$ is the correlation function in the lack of the EBGEM field, and leads to the two point 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 MDM of the DHBs. 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 quark–hadron duality ansatz, one can acquire the corresponding sum rules.

We begin to evaluate the correlation function with respect to hadronic degrees of freedom comprising the physical properties of the particles under investigation. For that purpose, we embed a complete set of $B^{*}_{QQ}$ baryons into the correlation function. So, we get

[TABLE]

where the dots stand for contributions of higher states and the continuum. The matrix elements in Eq. (II) are defined as Pascalutsa:2006up ; Ramalho:2009vc ,

[TABLE]

where $\lambda_{{B^{*}_{QQ}}}$ is the residue of ${B^{*}_{QQ}}$ baryon and $u_{\mu}(p,s)$ is the Rarita-Schwinger spinor. Summation over spins of ${B^{*}_{QQ}}$ baryon is carried out as:

[TABLE]

Substituting Eqs. (II)-(8) into Eq. (3) for hadronic side we obtain

[TABLE]

As a principle, we can achieve the last form of the hadronic representation of the correlator using the Eq. (II), however we face with two problems. One of them is related to the fact that not all Lorentz structures appearing in Eq. (II) are independent. The second problem is the correlator can also get contributions from spin-1/2 baryons, which should be removed. To eliminate the spin one half contributions and acquire just independent structures in the correlator, we perform the ordering for Dirac matrices as $\gamma_{\mu}p\!\!\!/\varepsilon\!\!\!/q\!\!\!/\gamma_{\nu}$ and remove terms with $\gamma_{\mu}$ at the beginning, $\gamma_{\nu}$ at the end and all those proportional to $p_{\mu}$ and $p_{\nu}$ Belyaev:1982cd . As a result, for hadronic side we get,

[TABLE]

The MDM form factor, $G_{M}(q^{2})$, is defined with respect to the form factors $F_{i}(q^{2})$ in the following manner Pascalutsa:2006up ; Ramalho:2009vc :

[TABLE]

where $\tau=-\frac{q^{2}}{4m^{2}_{{B^{*}_{QQ}}}}$. At $q^{2}=0$, the magnetic dipole form factors are obtained with respect tothe functions $F_{i}(0)$ as:

[TABLE]

The MDM ($\mu_{{B^{*}_{QQ}}}$), is defined in the following way:

[TABLE]

In this study we achieve QCD sum rules for the form factors $F_{i}(q^{2})$ at first, after that in numerical calculations we will make use of the above equations to obtain the values of the MDMs 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:

[TABLE]

where $\Pi_{i}^{Had}$ are functions of the form factors $F_{i}(q^{2})$ and other hadronic parameters; and $...$ represents other independent 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 $J^{P}=\frac{3}{2}^{+}$. The interpolating current is given as Aliev:2012iv ,

[TABLE]

where $q$ is the light; and $Q$ and $Q^{\prime}$ are the two heavy quarks, respectively. We give the quark content of the spin-3/2 DHBs in Table 1.

After contracting pairs of quark fields and using the Wick’s theorem, the correlation function becomes:

[TABLE]

where $\tilde{S}_{Q(q)}^{ij}(x)=CS_{Q(q)}^{{ij}^{T}}(x)C$ and, $S_{q}^{ij}(x)$ and $S_{Q}^{ij}(x)$ are the light and heavy quark propagators, respectively. The light and heavy quark propagators are given as Yang:1993bp ; Belyaev:1985wza ,

[TABLE]

where

[TABLE]

with $G^{\mu\nu}$ is the gluon field strength tensor, $K_{i}$ are second kind of the Bessel functions, $m_{q}$ and $m_{Q}$ are the light and heavy quark mass respectively.

The correlator in Eq. (16) contains various contributions: the photon can be emitted both perturbatively or nonperturbatively. When the photon is emitted perturbatively, one of the propagators in Eq. (16) is substituted by

[TABLE]

and the surviving two propagators are substituted 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 achieved by carrying out the substitution mentioned above for the perturbatively interacting quark propagator with the photon and employing the substitution of the surviving propagators by their free parts.

In case of nonperturbative photon emission, the light quark propagator in Eq. (16) is substituted by

[TABLE]

where $\Gamma_{i}$ represent the full set of Dirac matrices. Under this approach, two surviving quark propagators are taken as the full propagators comprising perturbative as well as nonperturbative contributions. Once Eq. (20) is inserted into Eq. (16), there seem matrix elements such as $\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and $\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\alpha\beta}q(0)\right|0\rangle$, representing the nonperturbative contributions. Furthermore, nonlocal operators such as $\bar{q}q\bar{q}q$ and $\bar{q}G^{2}q$ are anticipated to appear. In this study, we take into account operators includes only one gluon field and three particle nonlocal operators and to disregard terms with four quarks $\bar{q}q\bar{q}q$, and two gluons $\bar{q}G^{2}q$. 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 distribution amplitudes with definite twists. Up to twist-4 the explicit expressions of the photon distribution amplitudes are given in Ball:2002ps . Using these expressions for the propagators and distribution amplitudes for the photon, the correlation functions from the QCD side can be computed.

The QCD and hadronic representations of the correlation function are then matched using dispersion relation. The next step in deriving the sum rules for the MDMs of the spin-$\frac{3}{2}$ DHBs is applying double Borel transformations (${\cal B}$) over the $p^{2}$ and $(p+q)^{2}$ on the both representations of the correlation function so as to suppress the contributions of higher states and continuum. As a result, we obtain

[TABLE]

which leads to

[TABLE]

corresponding to the structures $g_{\mu\nu}p\!\!\!/\varepsilon\!\!\!/q\!\!\!/$ and $g_{\mu\nu}\varepsilon\!\!\!/q\!\!\!/$. In this manner we extract the QCD sum rules for the form factors $F_{1}$ and $F_{2}$. They are very lengthy functions, therefore we do not give their explicit expressions here. The interested readers can find details of the calculations such as Borel transformations and continuum subtraction in Refs. Agaev:2016srl ; Azizi:2018duk

III Numerical analysis

In this section, we achieve numerical computations for the spin-$\frac{3}{2}$ DHBs. We use $m_{u}$ = $m_{d}$ = 0, $m_{s}$ = $96^{+8}_{-4}$ MeV, $m_{c}$ = $1.67\pm 0.07$ GeV, $m_{b}$ = $4.78\pm 0.06$ GeV, $f_{3\gamma}$ = $-0.0039~{}\text{GeV}^{2}$ Ball:2002ps , $\langle\bar{q}q\rangle$ = $(-0.24\pm 0.01)^{3}\,\text{GeV}^{3}$ Ioffe:2005ym , $m_{0}^{2}$ = $0.8\pm 0.1~{}\text{GeV}^{2}$, $\langle g_{s}^{2}G^{2}\rangle=0.88~{}GeV^{4}$ and $\chi$ = $-2.85\pm 0.5~{}\text{GeV}^{-2}$ Rohrwild:2007yt . The masses of the $\Xi^{*}_{QQ}$, $\Xi^{*}_{QQ^{\prime}}$, $\Omega^{*}_{QQ}$ and $\Omega^{*}_{QQ^{\prime}}$ baryons are borrowed from Ref. Aliev:2012iv , in which the mass sum rules have been used to compute them. These masses are used to have the following values: $M_{\Xi^{*}_{cc}}=3.72\pm 0.18~{}\text{GeV}$, $M_{\Omega^{*}_{cc}}=3.78\pm 0.16~{}\text{GeV}$, $M_{\Xi^{*}_{bc}}=7.25\pm 0.20~{}\text{GeV}$, $M_{\Omega^{*}_{bc}}=7.30\pm 0.20~{}\text{GeV}$, $M_{\Xi^{*}_{bb}}=10.40\pm 1.00~{}\text{GeV}$ and $M_{\Omega^{*}_{bb}}=10.50\pm 0.20~{}\text{GeV}$. In order to specify the MDMs of DHBs, the value of the residues are needed. The residues of the DHBs are computed in Ref. Aliev:2012iv . These residues are calculated to have the following values: $\lambda_{\Xi^{*}_{cc}}=0.12\pm 0.01~{}\text{GeV}^{3}$, $\lambda_{\Omega^{*}_{cc}}=0.14\pm 0.02~{}\text{GeV}^{3}$, $\lambda_{\Xi^{*}_{bc}}=0.15\pm 0.01~{}\text{GeV}^{3}$, $\lambda_{\Omega^{*}_{bc}}=0.18\pm 0.02~{}\text{GeV}^{3}$, $\lambda_{\Xi^{*}_{bb}}=0.22\pm 0.03~{}\text{GeV}^{3}$ and $\lambda_{\Omega^{*}_{bb}}=0.25\pm 0.03~{}\text{GeV}^{3}$. The parameters used in the photon distribution amplitudes are given in Ref. Ball:2002ps .

The QCD sum rule for the MDMs of the DHBs, besides the above mentioned input parameters, include also two more extra parameters. These parameters are the continuum threshold $s_{0}$ and the Borel mass parameter $M^{2}$. According to the QCD sum rules philosophy we need to find the working regions of these parameters, where the MDMs of the DHBs be insensitive to the variation of these parameters in their working regions. The $s_{0}$ is not entirely optional parameter, it is preferred as the point at which the continuum and excited states begin to contribute to the calculations. To decide the working interval of the $s_{0}$, we enforce the conditions of operator product expansion (OPE) convergence and pole dominance. Therefore, it is expected that $s_{0}$ varies in the interval $(M_{B^{*}_{QQ}}+0.3)^{2}$ GeV${}^{2}\leq s_{0}\leq(M_{B^{*}_{QQ}}+0.7)^{2}$ GeV2. From this point of view, we prefer the value of the $s_{0}$ within the interval $s_{0}=(16-20)~{}\text{GeV}^{2}$ for $\Xi^{*}_{cc}$, $s_{0}=(58-62)~{}\text{GeV}^{2}$ for $\Xi^{*}_{bc}$, $s_{0}=(116-120)~{}\text{GeV}^{2}$ for $\Xi^{*}_{bb}$, $s_{0}=(18-22)~{}\text{GeV}^{2}$ for $\Omega^{*}_{cc}$, $s_{0}=(60-64)~{}\text{GeV}^{2}$ for $\Omega^{*}_{bc}$ and $s_{0}=(118-122)~{}\text{GeV}^{2}$ for $\Omega^{*}_{bb}$ baryons. The working region for $M^{2}$ is achieved by requiring that the series of OPE in QCD representation is convergent and the contribution of higher states and continuum is efficiently suppressed. In technique language, the upper limit on $M^{2}$ is found demanding the maximum pole contribution. The lower limit is obtained demanding that the contribution of the perturbative part exceeds the nonperturbative one and series of the operator product expansion in the obtained sum rules converge. Our numerical calculations indicates that both conditions are satisfied when $M^{2}$ changes in the regions: $4~{}\text{GeV}^{2}\leq M^{2}\leq 6~{}\text{GeV}^{2}$ for $\Xi^{*}_{cc}$, $7~{}\text{GeV}^{2}\leq M^{2}\leq 9~{}\text{GeV}^{2}$ for $\Xi^{*}_{bc}$, $10~{}\text{GeV}^{2}\leq M^{2}\leq 14~{}\text{GeV}^{2}$ for $\Xi^{*}_{bb}$, $5~{}\text{GeV}^{2}\leq M^{2}\leq 7~{}\text{GeV}^{2}$ for $\Omega^{*}_{cc}$, $8~{}\text{GeV}^{2}\leq M^{2}\leq 10~{}\text{GeV}^{2}$ for $\Omega^{*}_{bc}$ and $11~{}\text{GeV}^{2}\leq M^{2}\leq 15~{}\text{GeV}^{2}$ for $\Omega^{*}_{bb}$ baryons. As an example in Fig. 2, we present the dependencies of the MDMs of doubly charmed baryons on $M^{2}$ at several fixed values of the $s_{0}$. As is seen from the figure, although being not entirely insensitive, the MDMs exhibit acceptable dependency on the extra parameters, $s_{0}$ and $M^{2}$ which is reasonable in the error limits of the QCD sum rule formalism.

Our final results on the MDMs for the spin-$\frac{3}{2}$ DHBs are

[TABLE]

where the quoted errors in the results are in connection with the uncertainties in the values of the input parameters and the photon distribution amplitudes, as well as the variations in the computations of the working windows $M^{2}$ and $s_{0}$. We also need to emphasize that the main source of uncertainties is the variations with respect to $s_{0}$ and the results weakly depend on the choices of the $M^{2}$.

In Table 2, we present the our numerical results for the MDMs and comparison of the with various other models such as, the relativistic harmonic confinement model (RHM) Gadaria:2016omw , MIT bag model Bernotas:2012nz ; Simonis:2018rld , nonrelativistic quark model (NRQM) Albertus:2006ya , hyper central constituent model (HCQM) Shah:2016vmd ; Shah:2017liu , effective mass (EMS) and screened charge scheme (SCS) Dhir:2009ax ; Dhir:2013nka and heavy baryon chiral perturbation theory (HBChBT) Meng:2017dni . From a comparison of our values with the predictions of other models we observe from this table that for the the $\Xi_{cc}^{*++}$ baryon, practically all methods give, approximately, similar predictions. For the $\Xi_{cc}^{*+}$ and $\Omega^{*+}_{cc}$ baryons, there are large discrepancy among results not only the magnitude but also by the sign. The reason for such inconsistencies is relatively easy to understand. The sign of the MDM depends on what is stronger -two heavy quarks and one light quarks. In our analysis, the light quark overcome two heavy quarks and give the dominant contributions. For the the $\Xi_{bb}^{*0}$ baryon, our estimation is consistent within the errors with Refs. Albertus:2006ya ; Gadaria:2016omw ; Meng:2017dni and unlike other approaches. For the the $\Xi_{bb}^{*-}$ baryon, nearly all models give, approximately, similar predictions except the values of Refs. Bernotas:2012nz ; Simonis:2018rld , which are small. For the $\Omega^{*-}_{bb}$ baryon, our estimation is consistent within the errors with Refs. Shah:2016vmd ; Shah:2017liu ; Meng:2017dni and different from other results. For the the $\Xi_{bc}^{*+}$ baryon, we see that within errors our predictions in good agreement with the Refs Albertus:2006ya ; Gadaria:2016omw ; Dhir:2009ax ; Dhir:2013nka ; Meng:2017dni . For the $\Xi^{*0}_{bc}$ baryon, while the sign of the MDM is correctly determined, there is a large discrepancy among results. For the $\Omega^{*0}_{bc}$ baryon, nearly all methods give, moderately, similar approximations except the results of Ref. Meng:2017dni and this work, which are quite large.

There is no experimental data for the MDMs of the DHBs. But, one can define some useful splittings rules for the checking and comparing the results. In the heavy quark limit, $\Xi_{QQq}^{*}$, $\Xi_{QQ^{\prime}q}^{*}$, $\Omega_{QQq}^{*}$ and $\Omega_{QQ^{\prime}q}^{*}$ baryons have the same MDMs. Because they have the same light degrees of freedom and the contribution of heavy quarks will vanished in this limit. If one keep the heavy quark mass, the MDMs are not equal anymore. However one can expect that they may satisfy following relations as the $\Xi_{QQ^{\prime}q}^{*}$ and $\Omega_{QQ^{\prime}q}^{*}$ are intermediate states:

[TABLE]

We observe from Table 3 that all baryons almost satisfy that condition except the results are obtained in Refs. Shah:2016vmd ; Shah:2017liu .

As we mentioned beginning of this section, we work on $m_{u}=m_{d}=0$ limit. The second splittings can be defined

[TABLE]

which gives the difference occur as a result of the different charge of $u$ and $d$ quark. The third splittings can also be defined,

[TABLE]

which gives the difference occur as a result of the different quark mass of $d$ and $s$. In the second and third splittings, the heavy quark contribution has been canceled out. Therefore, $\Delta\mu_{u-d/d-s}^{cc}=\Delta\mu_{u-d/d-s}^{bb}=\Delta\mu_{u-d/d-s}^{bc}$ is expected. We can see most results in the Table 4 satisfy this relation, except the $\Delta\mu_{d-s}^{cc}$ of this work.

IV Discussion and concluding remarks

In the presented paper we have evaluated the MDMs of the spin-$\frac{3}{2}$ DHBs by means of the light-cone QCD sum rule. The MDMs of the DHBs encodes key knowledge of their internal structure and shape deformations. Measurement of the MDMs of the spin-$\frac{3}{2}$ DHBs in future experiments can be very helpful understanding the internal structure of these baryons. However, the direct measurement of the MDMs of the spin-$\frac{3}{2}$ DHBs are unlikely in the near future. Therefore, any unstraightforward approximation of the MDMs of the spin-$\frac{3}{2}$ DHBs could be very helpful. Comparison of our results with the estimation of other theoretical models is presented. As can be seen from the MDM results of the DHBs given in Table 2, 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.

V Acknowledgements

We are grateful to V. S. Zamiralov for useful discussions, comments and remarks.