Mass and Magnetic Moments of the Heavy Flavored Baryons with $J=3/2$ in Light Cone QCD Sum Rules

Inspired by the results of recent experimental discoveries for charm and bottom baryons, the masses and magnetic moments of the heavy baryons with $J^P=3/2^+$ containing a single heavy quark are studied within light cone QCD sum rules method. Our results on the masses of heavy baryons are in good agreement with predictions of other approaches, as well as with the existing experimental values. Our predictions on the masses of the states, which are not yet discovered in the experiments, can be tested in the future experiments. A comparison of our results on the magnetic moments of these baryons and the hyper central model predictions is presented.


Introduction
Recently, considerable experimental progress has been made in the spectroscopy of baryons containing a single heavy quark. The CDF Collaboration has observed four bottom baryons Σ ± b and Σ * ± b [1]. The DO [2] and CDF [3] Collaborations have seen the Ξ b . The BaBar Collaboration discovered the Ω * c state [4].
The CDF sensitivity appears adequate to observe new heavy baryons.
Study of the electromagnetic properties of baryons can give noteworthy information on their internal structure. One of the main static electromagnetic parameters of the baryons is their magnetic moments. Magnetic moments of the heavy baryons in the framework of different approaches are widely discussed in the literature. In [5,6,7] the magnetic moments of heavy baryons containing c-quark have been calculated using the naive quark model. In [8,9,10],the magnetic moments of charm and bottom baryons are computed in quark model and in [11,12] heavy baryon magnetic moments are investigated within soliton type approaches. Calculation of the magnetic moments of heavy baryons in the framework of relativistic quark model is done in [13].
In [14] the magnetic moments of Σ c and Λ c baryons are estimated in QCD sum rules with external electromagnetic field. The magnetic moments of the Λ Q , (Q = c, b) and Σ Q Λ Q transition magnetic moments in light cone QCD sum rules are calculated in [15,16]. The magnetic moments of Ξ Q (Q = c, b) in the same framework is studied in [17] (a detailed description of this method can be found in [18] and references theirin).
In the present work, we study the magnetic moments and masses of the ground state baryons with total angular momentum 3/2 and containing one heavy quark within light cone QCD sum rules. The paper is organized as follows. In section 2, the light cone QCD sum rules for mass and magnetic moments of heavy baryons are calculated. Section 3 is devoted to the numerical analysis of the mass and magnetic moment sum rules and discussion. After these preliminary remarks let us start calculation of the magnetic moments of the heavy flavored hadrons. The basic objects in LCSR method is the correlation function where hadrons are represented by the interpolating quark currents. For this aim, we consider the correlator where η µ is the interpolating current of the heavy baryon and γ means the electromagnetic field. In QCD sum rules method, this correlation function is calculated in two different approaches: On the quark level, it describes a hadron as quarks and gluons interacting in QCD vacuum. In the phenomenological side, it is saturated by a tower of hadrons with the same flavor quantum numbers. The magnetic moments are determined by matching two different representations of the correlation function, i.e., theoretical and phenomenological forms, using the dispersion relations.
From Eq. (1), it follows that to calculate the correlation function from QCD side, we need the explicit expressions of the interpolating currents of heavy baryons with the angular momentum J P = 3/2 + . The main condition for constructing the interpolating currents from quark field is that they should have the same quantum numbers of the baryons under consideration. For the heavy baryons with J P = 3/2 + , the interpolating current is chosen in the following general form where C is the charge conjugation operator and a, b and c are color indices.
The value of A and quark fields q 1 and q 2 for each heavy baryon is given in Table 1.
The phenomenological part of the correlation function can be obtained by inserting the complete set of states between the interpolating currents in (1) with quantum numbers of heavy baryons.
where p 1 = p + q, p 2 = p and q is the photon momentum. The vacuum to baryon matrix element of the interpolating current is defined as where ε ρ is the photon polarization vector, f i and G i are the form factors and they are functions of q 2 = (p 1 − p 2 ) 2 . In order to obtain the explicit expressions of the correlation function, we perform summation over spins of the spin 3/2 particles using Using Eqs. (3)(4)(5)(6) in principle one can write down the phenomenological part of the correlator. But, the following two drawbacks appear: a) all Lorentz structures are not independent, b) not only spin 3/2, but spin 1/2 states also contribute to the correlation function. Indeed the matrix element of the current η µ between vacuum and spin 1/2 states is nonzero and is determined as where the condition γ µ η µ = 0 is imposed.
There are two different ways to remove the unwanted spin 1/2 contribution and retain only independent structures in the correlation function: 1) Introduce projection operators for the spin 3/2 states, which kill the spin 1/2 contribution, 2) Ordering Dirac matrices in a specific order and eliminate the structures that receive contributions from spin 1/2 states. In this work, we will follow the second method and choose the ordering for Dirac matrices as γ µ p ε qγ ν . With this ordering for the correlator, we get g µν p ε q g M 3 + other structures with γ µ at the beginning andγ ν at the end or which are proportional to p 2µ or p 1ν , where g M /3 = f 1 +f 2 and at q 2 = 0, g M is the magnetic moment of the baryon in units of its natural magneton, eℏ/2m B c. The factor 3 is due the fact that in the non-relativistic limit the interaction Hamiltonian with magnetic field On QCD side, the correlation function (1) can be evaluated using operator product expansion. After contracting out the quark pairs in Eq. (1) using the Wick's theorem, we get the following expression for the correlation function in terms of quark propagators where S ′ = CS T C and S Q (S q ) is the full heavy (light) quark propagator.
In calculation of the correlation function from QCD side, we take into account terms linear in m q and neglect quadratic terms.The correlator contains three different contributions: 1) Perturbative contributions, 2) Mixed contributions, i.e., the photon is radiated from freely propagating quarks at short distance and at least one of quark pairs interact with QCD vacuum non-perturbatively. The last interaction is parameterized in terms of quark condensates. 3) Non-perturbative contributions, i.e., when photon is radiated at long distances. In order to get expressions of the contributions when the photon is radiated at short distance, it is enough to replace the propagator of the quark that emits the photon by where the Fock-Schwinger gauge, x µ A µ (x) = 0 has been used. The expressions of the free light and heavy quark propagators in x representation are: where K i are Bessel functions. The non-perturbative contributions to the correlation function can be obtained from Eq. (9) by replacing one of the light quark propagators (the quark that emits the photon) by where Γ is the full set of Dirac matrices Γ j = 1, γ 5 , γ α , iγ 5 γ α , σ αβ / √ 2 and sum over index j is implied. Under this procedure, two remaining quark propagators are full propagators involving perturbative as well as nonperturbative contributions. Therefore, for the calculation of the correlation function from QCD side, the expressions of the heavy and light quark propagators in the presence of external field are needed.
The light cone expansion of the quark propagator in external field is done in [19]. The propagator receives contributions fromqGq,qGGq andqqqq, where G is the gluon field strength tensor. In present work, we neglect terms with two gluons as well as four quarks operators due to the fact that their contributions are small [20]. In this approximation the heavy and light quark propagators have the following expressions: Here, we would like to make the following remark about the parameter Λ. In order to achieve a factorization of large and small scales in the OPE, all infrared logarithms should be removed from coefficient functions and absorbed in the matrix elements of operators. In our case, this means that the lnΛ must be included in the condensates of different operator or distribution amplitude. A more detailed discussion on this point can be found in [21]. For this reason, we will choose the scale parameter Λ as a factorization scale, i.e., Λ = 1 GeV .
In order to calculate the contributions of the photon emission from large distances, the matrix elements of nonlocal operatorsqΓ i q between the photon and vacuum states are needed, γ(q) |qΓ i q | 0 . These matrix elements are determined in terms of the photon distribution amplitudes (DA's) as follows [22].
where χ is the magnetic susceptibility of the quarks, ϕ γ (u) is the leading twist are the twist 4 photon DA's, respectively. The photon DA's is calculated in [22] and for completeness their explicit expressions are presented in the numerical analysis section. The measure Dα i is defined as Note that, in definition of photon DA's with leading twist there are terms proportional to the strange quark mass. In our calculations, we neglect these terms. Using the expressions of the light and heavy full propagators and the photon DA's and separating the coefficient of the structure g µν p ε q, the expression of the correlation function from QCD side is obtained. Separating the coefficient of the same structure from phenomenological part and equating these representations of the correlator, sum rules for the magnetic moments of the J P = 3/2 + heavy baryons is obtained. In order to suppress the contribution of higher states and continuum, Borel transformation with respect to the variables p 2 2 = p 2 and p 2 1 = (p + q) 2 is applied. Before presenting the sum rules for the magnetic moments, few words about the relations between the correlation functions are in order. Note that, the coefficient of any structure in the correlation function can be written in the form where Π 1 corresponds to the emission of the photon from the quark q 1 , Π ′ 1 to emission from q 2 and Π 2 to emission from the heavy quark Q. Note that the functions Π i (q 1 , q 2 , q 3 ) (i = 1, 2) do not depend on the quark charges in the limit where we neglect electromagnetic corrections.
From the explicit form of the interpolating current it follows that it is symmetric under the exchange of the light quarks, i.e. q 1 ←→ q 2 , and for this reason Π 1 (q 1 , q 2 , Q) = Π ′ 1 (q 2 , q 1 , Q). Due to the symmetries between the interpolating currents, any interpolating current can be obtained from the interpolating current for Σ * 0(+) b(c) . This also leads to relations between their correlation functions. In terms of the defined Π 1 (q 1 , q 2 , Q) and Π 2 (q 1 , q 2 , Q), all the correlation functions can be written as: and the relations for the charmed baryons can simply be obtained from Eq.
where, for simplicity, we have used the short hand notation Π i (q 1 , q 2 , Q) ≡ and the corresponding expressions for the charmed baryons. Any deviation from these relations in the magnetic moments is a sign of SU(3) f violation.
Note that the first relation in Eq. (19) is a direct consequence of the isospin subgroup of SU(3) f . Since in this work, we set m u = m d and ūu = d d , i.e. we assume isospin symmetry, our results satisfy this relation exactly.
And hence, in the following, we will not show numerical results form the magnetic moments of Σ * 0(+) b(c) . Once the explicit forms of the functions Π i (q 1 , q 2 , Q)(i = 1, 2) are known, the sum rules for the magnetic moments can be obtained using Eq. (17) and The functions Π i (q 1 , q 2 , Q) can be written as: where + 4γ E (23 + ψ 02 + 2ψ 12 + ψ 22 ) + 4ψ 32 ]m q 1 − 6(−1 + 2ψ 12 + ψ 22 )m q 2 + 6(16 + 2ψ 02 − 18ψ 03 − 50ψ 10 + 9ψ 21 + 93ψ 23 − 4ψ 32 + 120ψ 33 + 45ψ 43 ) where Li 2 (x) is the dilogarithm function. The functions also entering Eqs. (22)(23)(24)(25) are given as The residue is determined from two point sum rules. For the interpolating current given in Eq. (2), we obtain the following result for λ 2 B : where The masses of the considered baryons can be determined from the sum rules. For this aim, one can get the derivative from both side of Eq. (27) with respect to −1/M 2 and divide the obtained result to the Eq. (27), i.e., 3 Numerical analysis  Table 2 in comparison with some theoretical predictions and experimental results. Note that the masses of the heavy flavored baryons are calculated in the framework of heavy quark effective theory (HQET) using the QCD sum rules method in [27].
After determination of the mass as well as residue of the heavy flavored  Table 2: Comparison of mass of the heavy flavored baryons in GeV from present work and other approaches and with experiment.
baryons our next task is the calculation of the numerical values of their magnetic moments. For this aim, from sum rules for the magnetic moments it follows that the photon DA's are needed [22] : , The constants appearing in the wave functions are given as [22]  The magnetic moments also are practically insensitive to the variation of Borel mass parameter when it varies in the working region. We should also stress that our results practically don't change considering three values of χ which we presented at the beginning of this section. Our final results on the magnetic moments of heavy flavored baryons are presented in Table 3. For comparison, the predictions of hyper central model [35] are also presented.
The quoted errors in Table 3 are due to the uncertainties in m 2 0 , variation of s 0 and M 2 as well as errors in the determination of the input parameters.