Magnetic Moments of 70-Plet Baryons in Quark Model and QCD Sum Rules

Magnetic moments of the positive parity 70-plet baryons are estimated in the framework of the nonrelativistic quark model and QCD sum rules method. It is found that the magnetic moments of the 70-plet baryons can be expressed in terms of the F and D couplings and exhibit unitary symmetry.TheQCD sum rules for themagneticmoments of the 70-plet octet baryons are formulated. A comparison of our results on magnetic moments of 56-plet and 70-plet baryons predicted from QCD sum rules is presented.


Introduction
Study of the electromagnetic properties of hadrons represents very important source of information about their internal structure and can provide valuable insight into understanding the mechanism of strong interactions at low energies, that is, about nonperturbative aspects of QCD.Particular interest deserves magnetic moments of baryons as a subject of permanent study due to growing experimental information [1].
Although the magnetic moments of the positive baryons are widely studied experimentally and theoretically, there is very limited information about their negative parity counterparts.Magnetic moments of these states can be extracted through the bremsstrahlung processes in photo and electroproduction reactions.
The positive octet and decuplet baryons usually enter the 56-plet and negative parity baryons lie in the 70-plet representation of the SU (6), which is the group-theoretical basis of the NRQM.In the unitary SU(3) model coupling constants of photons (and vector mesons) with baryons are expressed in terms of the  and  constants.In chiral model [19] mixing of 56-plet and 70-plet is studied for 1/2 + and 3/2 + baryons.
The basic element of QCD sum rules calculation is the interpolating current of the corresponding hadron which can be reduced to 56-plet wave function in nonrelativistic limit.Here we note that in studying the properties of the 1/2 − baryons the interpolating current for the positive parity baryons has been used although it has no usual NRQM limit for the 70-plet baryons.Despite this difficulty we will try to

𝐹 and 𝐷 Couplings in Quark-Diquark Model
Although the NRQM results of magnetic moments cannot be expressed in terms of  and  couplings, in [23], it was shown that they can be achieved in framework of diquarkquark model.The characteristic property of this model is that the photon or vector boson fields are to interact with diquark and single quark in a different way.
Let us discuss the magnetic moments of 56-plet baryons in NRQM in framework of diquark-quark model.As an example let us consider a proton from 56-plet.Its wave function can be written as where subindices 1 and 2 mean quark spin-up and spindown, respectively.The wave functions of other members of octet baryons can be obtained from proton wave function by appropriate replacements of quark fields.The results of the magnetic moments of the 56-plet baryons in terms of  and  couplings can be obtained by using their wave functions and introducing the following four different matrix elements: Using these matrix elements with   ŵ    as the modified magnetic moment operator, the magnetic moments of Σ 0 and Λ baryons in [23] are obtained: where (2/3) 11 = ,  12 = , and (2V 11 − V 12 ) = 3( − ) and also V 11 =  and V 12 = 3 − .These results show that the nonrelativistic diquark-quark model reveals the unitary symmetry pattern and the magnetic moments of baryons can be formulated in terms of the  and  couplings [23,24].
Performing similar analysis for the magnetic moments of Σ 0 * and Λ * baryons we obtain Transition moment  Σ 0 * Λ * 0 is obtained immediately from group-theoretical relation [23,24] where  is appropriate operator and reads and in the NRQM yields zero.
Advances in High Energy Physics 3 Magnetic moments of other members of octet baryons can be found with the help of appropriate replacements of quark fields.In the limit  = (2/3),  = 1 and replace   →   ; we get the NRQM result of the magnetic moments of 70-plet baryons.When, in (7), (8) take the relevant quark charges we get unitary symmetry results for the octet baryons in the 70-plet, similar to the relations for the 56-plet baryons (see ( 4)): Finally, magnetic moments of 70-plet octet baryons can be obtained from results of 56-plet baryons with the help of simple replacements by comparing ( 3), (7), and (8).For example, the magnetic moment of Σ 0 in 70-plet can be achieved from 56-plet one by replacing coefficients of  and  terms: These relations constitute one of the main results in the present work.

QCD Sum Rules for Magnetic Moments of 56-Plet and 70-Plet Baryons
These results of the magnetic moments of octet baryons can also be obtained from the QCD sum rules method which exhibits unitary symmetry patterns of a sense that they can be represented in terms of only two independent  and  type functions.Note that the conclusion is true not only for photon but for any vector field too (see, e.g., [26]).The key object of the QCD sum rules method is the interpolating currents.In the nonrelativistic limit the interpolating current of baryons can be reduced to their wave functions [27].
The magnetic moments of the octet baryons in framework of traditional and light-cone versions of QCD sum rules are calculated in various works (see, e.g., [6,8] and references therein).
The QCD sum rules method is based on the correlation function: where  is the time ordering operator,  means external electromagnetic field, and   is the interpolating current carrying the same quantum numbers as the corresponding baryon .As an example we present the interpolating current for Σ 0 baryon: where , ,  are color indices,  is the charge conjugation operator, and  is the arbitrary parameter ( = −1 corresponds to the so called Ioffe current).
In order to construct QCD sum rules for magnetic moments of the octet baryons the correlation function is calculated in terms of hadrons and quark-gluon degrees of freedom.By matching these two representations the QCD sum rules for the octet baryons magnetic moments are obtained [7].The magnetic moments of octet baryons in general form can be written as where   and   are the residue and magnetic moment of corresponding baryon, respectively, and Π  are the invariant functions in the coefficient of ¡  ¡  ¡  Lorentz structure and their expressions can be found in [6,8].Using the SU(2) symmetry (15) can be written as Using the relation obtained in [28] which connected Λ and Σ 0 interpolating currents one can easily find the expression for the magnetic moment of Λ-hyperon: We can now predict the magnetic moments of the octet in 70-plet with positive parity if we assume that the transformations obtained in NRQM (see (3), (8), and ( 12)) hold in the case of QCD sum rules, that is, at the level of correlation functions Π's.
In this case even the explicit expressions of interpolating currents of octet baryons belonging to the 70-plet representation are not known; one can predict the magnetic moments of these baryons.Using ( 8) and ( 12) as well as ( 16) and ( 17) for the magnetic moments of Σ and Λ baryons in 70-plet we get the following sum rules: Comparing these equations with the sum rules for the Σ and Λ baryons from 56-plet we see that they have changed drastically.
In order to get the idea about the magnitude of the magnetic moments of 70-plet baryons, in the obtained sum rules, we put their experimentally measured mass and for residues we put their values for 56-plet baryons.Under this assumption, the magnetic moments of the positive parity 70plet baryons are calculated and their values are presented in Table 1 (the 2nd column).For completeness we also present the magnetic moments of 56-plet baryons in this table (the 1st column).

Conclusion
It is shown that octet baryons in the 70-plet can be analyzed in the way similar to those of 56-plet.In particular, magnetic moments are written in terms of the  and  quantities characteristics for octet coupling.Moreover the main formulas for the magnetic moments are written in such a way as to obtain the NRQM results as well as unitary symmetry ones.The QCD sum rules for the magnetic moment of 70-plet octet baryons are constructed.A comparison of the magnetic moments for (1/2) + 56-plet and 70-plet baryons predicted from QCD sum rules is presented.