Radiative $\Omega_{Q}^{*}\rightarrow\Omega_{Q}\gamma$ and $\Xi_{Q}^{*}\rightarrow\Xi^{\prime}_{Q}\gamma$ transitions in light cone QCD

We calculate the magnetic dipole and electric quadrupole moments associated with the radiative $\Omega_{Q}^{*}\rightarrow\Omega_{Q}\gamma$ and $\Xi_{Q}^{*}\rightarrow\Xi^{\prime}_{Q}\gamma$ transitions with $Q=b$ or $c$ in the framework of light cone QCD sum rules. It is found that the corresponding quadrupole moments are negligibly small while the magnetic dipole moments are considerably large. A comparison of the results on the considered multipole moments as well as corresponding decay widths with the predictions of the vector dominance model is performed.


Introduction
In the recent years, there have been significant experimental progresses on hadron spectroscopy. Many new baryons containing heavy bottom and charm quarks as well as many new charmonium like states are observed. Now, all heavy baryons with single heavy quark have been discovered in the experiments except the Ω * b baryon with spin-3/2. In the case of doubly heavy baryons only the doubly charmed Ξ cc baryon has been discovered by SELEX Collaboration [1,2]  In the present work we calculate the electromagnetic form factors of the radiative Ω * Q → Ω Q γ and Ξ * Q → Ξ ′ Q γ transitions in the framework of the light cone QCD sum rules as one of the most applicable non-perturbative tools to hadron physics. Here, baryons with * correspond spin-3/2 while those without * are spin-1/2 baryons. Using the electromagnetic form factors at static limit (q 2 = 0), we obtain the magnetic dipole and electric quadrupole moments as well as decay widths of the considered radiative decays. We compare our results with the predictions of the vector meson dominance model (VDM) [3] which uses the values of the strong coupling constants between spin-3/2 and spin-1/2 heavy baryons with vector mesons [4] to calculate the magnetic dipole and electric quadrupole moments of the transitions under consideration. The electromagnetic multipole moments of heavy baryons can give valuable information on their internal structure as well as their geometric shapes. Note that other possible radiative transitions among heavy spin-3/2 and spin-1/2 baryons with single heavy quark, namely Σ * Q → Σ Q γ , Ξ * Q → Ξ Q γ and Σ * Q → Λ Q γ have been investigated in [5] using the same framework. Some of these radiative transitions have also been previously studied using chiral perturbation theory [6], heavy quark and chiral symmetries [7,8], relativistic quark model [9] and light cone QCD sum rules at leading order in HQET in [10].
The outline of the paper is as follows. In next section, QCD sum rules for the electromagnetic form factors of the transitions under consideration are calculated. In last section, we numerically analyze the obtained sum rules. This section also includes comparison of our results with the predictions of VDM on the multipole moments as well as the corresponding decay widths.

Theoretical framework
The aim of this section is to obtain light cone QCD sum rules (LCQSR) for the electromagnetic form factors defining the radiative Ω * Q → Ω Q γ and Ξ * Q → Ξ ′ Q γ transitions. For this goal we use the following two-point correlation function in the presence of an external photon field: where η and η µ are the interpolating currents of the heavy flavored baryons with spin 1/2 and 3/2, respectively. The main task in the following is to calculate this correlation function once in terms of hadronic parameters called the hadronic side and in terms of photon distribution amplitudes (DAs) with increasing twist by the help of operator product expansion (OPE). By equating the coefficients of appropriate structures from hadronic to OPE side, we obtain LCQSR for the transition form factors. To suppress the contribution of the higher states and continuum, we apply Borel transformations with respect to the momentum squared of the initial and final baryonic states. For further pushing down those contributions, we also apply continuum subtraction to both sides of the LCQSRs obtained.

Hadronic side
To obtain the hadronic representation, we insert complete sets of intermediate states having the same quantum numbers as the interpolating currents into the above correlation function.
As a result of which we get where the dots indicate the contributions of the higher states and continuum and q is the photon's momentum. In the above equation, 1(p + q, s)| and 2(p, s ′ )| denote the heavy spin 3/2 and 1/2 states and m 1 and m 2 are their masses, respectively. To proceed, we need to know the matrix elements of the interpolating currents between the vacuum and the baryonic states. They are defined in terms of spinors and residues as 1(p + q, s) |η µ (0) | 0 = λ 1ūµ (p + q, s), where u µ (p, s) is the Rarita-Schwinger spinor; and λ 1 and λ 2 are the residues of the heavy baryons with spin 3/2 and 1/2, respectively which are calculated in [5]. The matrix element 2(p, s ′ ) | 1(p + q, s) γ is also defined as [11,12] 2(p, where G i are electromagnetic form factors, ε µ is the photon's polarization vector and P = p+(p+q) 2 . In the above equation, the term proportional to G 3 is zero for the real photon which we consider in the present study. At q 2 = 0, the transition magnetic dipole moment G M and the electric quadrupole moment G E are defined in terms of the remaining electromagnetic form factors as Now, we use Eqs. (4) and (3) in Eq. (2) and perform summation over spins of the Dirac and Rarita-Schwinger spinors. In the case of spin 3/2 this summation is written as Using Eqs. (3)(4)(5)(6), in principle, one can straightforwardly calculate the hadronic side of the correlation function. But, here appear two unwanted problems: • there is pollution from spin-1/2 baryons, since the interpolating current η µ couples with spin-1/2 baryons also.
• All Lorentz structures are not independent.
Multiplying both sides of this equation by γ µ and using γ µ η µ = 0 as well as the Dirac from this expression it follows that contributions of spin-1/2 states are either proportional to the γ µ at the end or (p + q) µ . Taking into account this fact, from Eq. (6) it follows that only terms proportional to g µν contain contributions coming only from spin-3/2 states. This observation shows that how spin-1/2 states' contributions coupled to η µ can be removed.
The second problem can be solved if one orders the Dirac matrices in an appropriate way.
In this work, we choose the ordering ε q pγ µ . After some calculations, for the hadronic side of the correlation function, we get +other structures with γ µ at the end or which are proportional to (p + q) µ , where, we need two invariant structures to calculate the form factors G 1 and G 2 . In the present work, we select the structures ε pγ 5 q µ and q pγ 5 (εp)q µ for G 1 and G 2 , respectively.
The advantage of these structures is that these terms do not receive contributions from contact terms.

OPE Side
On OPE side, the aforementioned correlation function is calculated in terms of QCD degrees of freedom and photon DAs. For this aim, we substitute the explicit forms of the interpolating currents of the heavy baryons into the correlation function in Eq. (1) and use the Wick's theorem to obtain the correlation in terms of quark propagators.
The interpolating currents for spin 3/2 baryons are taken as where q 1 and q 2 stand for light quarks; a, b and c are color indices and C is the charge conjugation operator. The normalization factor A and light quark content of heavy spin 3/2 baryons are presented in table 1:

Heavy spin 3/2 baryons
2/3 s d Table 1: The normalization factor A and light quark content of heavy spin 3/2 baryons.
The general form of the interpolating currents for the heavy spin 1/2 baryons under consideration can be written as (see for instance [13]) where β is an arbitrary parameter and β = −1 corresponds to the Ioffe current. The constant B and quark fields q 1 and q 2 for the corresponding heavy spin 1/2 baryons are given in table 2. The correlation function in OPE side receives three different contributions: 1) perturbative contributions, 2) mixed contributions at which the photon is radiated from short distances and at least one of the quarks forms a condensate and 3) non-perturbative contributions where photon is radiated at long distances. The last contribution is parameterized by the matrix element γ(q) |q(x 1 )Γq(x 2 ) | 0 which is expanded in terms of photon DAs with definite twists. Here Γ is the full set of Dirac matrices Γ j = 1, γ 5 , γ α , iγ 5 γ α , σ αβ / √ 2 .

Heavy spin 3/2 baryons
The perturbative contribution at which the photon interacts with the quarks perturbatively, is obtained by replacing corresponding free quark propagator by where the free light and heavy quark propagators are given as with K i being the Bessel functions.
The non-perturbative contributions are obtained by replacing one of the light quark propagators that emits a photon by where sum over j is applied, and the remaining by full quark propagators involving the perturbative as well as the non-perturbative parts. The full heavy and light quark propagators which we use in the present work are (see [14,15]) where Λ is the scale parameter and we choose it at factorization scale Λ = (0.5 − 1) GeV [16].
In order to calculate the non-perturbative contributions, we need the matrix elements γ(q) |qΓ i q | 0 . These matrix elements are determined in terms of the photon DAs as [17] γ where ϕ γ (u) is the leading twist 2, ψ v (u), ψ a (u), A and V are the twist 3; and h γ (u), A and T i (i = 1, 2, 3, 4) are the twist 4 photon DAs [17]. Here χ is the magnetic susceptibility of the quarks.
The measure Dα i is defined as In order to obtain the sum rules for the form factors G 1 and G 2 , we equate the coefficients of the structures ε pγ 5 q µ and q pγ 5 (εp)q µ from both hadronic and OPE representations of the same correlation function. We apply the Borel transformations with respect to the variables p 2 and (p + q) 2 as well as continuum subtraction to suppress the contributions of the higher states and continuum. Finally, we obtain the following schematically written sum rules for the electromagnetic form factors G 1 and G 2 : where the functions Π i [Π ′ i ] can be written as where s 0 is the continuum threshold and we take M  [17]. The value of the magnetic susceptibility are calculated in [19,20,21]. Here we use the value χ(1 GeV ) = −4.4 GeV −2 [21] for this quantity. The LCQSR for the magnetic dipole and electric quadrupole moments also include the photon DAs [17] whose expressions are given as , where, the constants inside the DAs are given by . The sum rules for the electromagnetic form factors contain three more auxiliary parameters: Borel mass parameter M 2 , the continuum threshold s 0 and the arbitrary parameter β entering the expressions of the interpolating currents of the heavy spin 1/2 baryons. Any physical quantities, like the magnetic dipole and electric quadrupole moments, should be independent of these auxiliary parameters. Therefore, we try to find "working regions" for these auxiliary parameters such that in these regions the G M and G E are practically independent of these parameters. The upper and lower bands for M 2 are found requiring that not only the contributions of the higher states and continuum are less than the ground state contribution, but also the contributions of the higher twists are less compared to the leading twists. By these requirements, the working regions of Borel mass parameter are obtained as 15 GeV 2 ≤ M 2 ≤ 30 GeV 2 and 6 GeV 2 ≤ M 2 ≤ 12 GeV 2 for baryons containing b and c quarks, respectively. The continuum threshold s 0 is the energy square which characterizes the beginning of the continuum. In we denote the ground state mass by m, the quantity √ s 0 − m is the energy needed to excite the particle to its first excited state with the same quantum numbers. The         quadrupole moment G E correspond to the considered radiative transitions as presented in    → Ω − b γ channel at which our result is roughly one order of magnitude small compared to that of [3]. When we compare our results with those of [24,25], we see considerable differences in orders of magnitudes between two models predictions except for Ω * 0 c → Ω 0 c γ and Ξ * + c → Ξ ′+ c γ channels that our predictions are in the same orders of magnitude with those of [24,25]. The big differences among our results, [3] and [24,25] Table 4: Widths of the corresponding radiative transitions in KeV.
In summary, we have calculated the transition magnetic dipole moment G M and electric quadrupole moment G E as well as decay width for the radiative Ω * Q → Ω Q γ and Ξ * Q → Ξ ′ Q γ transitions within the light cone QCD sum rule approach and compared the results with the predictions of the VDM. Considering the recent progresses on the identification and spectroscopy of the heavy baryons, we hope it will be possible to study these radiative decay channels at the experiment in near future.

Acknowledgment
K. A. and H. S. would like to thank TUBITAK for their partial financial support through the project 114F018.