Magnetic moments of the lowest-lying singly heavy baryons

A light baryon is viewed as $N_c$ valence quarks bound by meson mean fields in the large $N_c$ limit. In much the same way a singly heavy baryon is regarded as $N_c-1$ valence quarks bound by the same mean fields, which makes it possible to use the properties of light baryons to investigate those of the heavy baryons. A heavy quark being regarded as a static color source in the limit of the infinitely heavy quark mass, the magnetic moments of the heavy baryon are determined entirely by the chiral soliton consisting of a light-quark pair. The magnetic moments of the baryon sextet are obtained by using the parameters fixed in the light-baryon sector. In this mean-field approach, the numerical results of the magnetic moments of the baryon sextet with spin $3/2$ are just 3/2 larger than those with spin $1/2$. The magnetic moments of the bottom baryons are the same as those of the corresponding charmed baryons.


I. INTRODUCTION
Very recently, Ref. [1] showed that when the number of colors (N c ) goes to infinity singly heavy baryons can be described as N c − 1 valence quarks bound by the meson mean fields that also have portrayed light baryons as N c valence quarks bound by the same mean fields [2,3], being motivated by Diakonov [4]. The masses of the lowest-lying singly heavy baryons were well reproduced in both the charmed and bottom sectors, and the mass of the Ω b was predicted within this framework. Using the method developed in Ref. [1], we were able to interpret two narrow Ω c resonances as exotic baryons belonging to the anti-decapentaplet (15) [5] among five Ω c s found by the LHCb Collaboration [6]. This mean-field approach is called the chiral quark-soliton model (χQSM) [7] (for a review, see Refs. [8,9] and references therein). Very recently, the model has also described successfully strong decays of heavy baryons [10] including those of the newly found two narrow Ω c s.
The magnetic moments of the heavy baryons have been already investigated within various different approaches such as quark models [11][12][13], the MIT bag model [14], the quark potential model [15,16], the Skyrme models in bound-state approaches [17,18], a relativistic quark model [19], lattice QCD [20][21][22], heavy-baryon chiral perturbation theories [23,24], and QCD sum rules [26,27], and so on. Since there are no experimental data available yet, it is of great interest to compare the results with those from several different approaches. Since the baryon anti-triplet (3) consists of the light-quark pair with the total light-quark spin J = 0, the corresponding magnetic moment vanishes in the present mean-field approach with the infinitely heavy-quark mass limit considered. Thus, in the present work, we want to employ the χQSM to compute the magnetic moments of the lowest-lying singly heavy baryons, in particular, the baryon sextet (6) with both spin J ′ = 1/2 and J ′ = 3/2. The magnetic moments of the light baryons were already studied within the χQSM [28,29]. A merit of this approach is that we can deal with light and heavy baryons on the same footing. All the dynamical parameters required for the present analysis were determined in Ref. [30] based on the experimental data on the magnetic moments of the baryon octet, we have no additional free parameter to handle for those of the heavy baryons. We obtain the results for the magnetic moments of the baryon sextet and compare them with those from other models and lattice QCD. The results turn out to be consistent with those from the other works, in particular, with those from Ref. [19]. Compared with the results from the lattice QCD [20][21][22], the present ones are consistently larger than them except for the Σ ++ c magnetic moment. The structure of the present work is sketched as follows: In Section II, we briefly review the general formalism of the χQSM in order to compute the magnetic moments of the heavy baryons. In Section III, we show how to carry out the calculation of the magnetic moments within the present framework, using the dynamical parameters fixed in the light baryon sector. In Section IV, we present the numerical results of the magnetic moments of the heavy baryons, examining the effects of flavor SU(3) f breaking. We summarize the present work in the last Section.

II. GENERAL FORMALISM
In the mean-field approach, a heavy baryon can be expressed by the correlation function of the N c − 1 lightquark operators, while a heavy quark inside it is regarded as a static color source in the limit of the infinitely heavy quark mass (m Q → ∞). The heavy quark is required only to makes the heavy baryon a color singlet state. The electromagnetic current we now consider consists of both the light and heavy quark currents whereQ denotes the charge operator of the light quarks in flavor SU(3) space, defined bŷ Here, λ 3 and λ 8 are the well-known flavor SU(3) Gell-Mann matrices. The e Q in the second part of the electromagnetic current in Eq. (1) stands for the heavy-quark charge, which is given as e c = 2/3 for the charm quark or as e b = −1/3 for the bottom quark. The magnetic moment of a heavy quark is proportional to the inverse of the corresponding mass, i.e. µ ∼ (e Q /m Q )σ, so that it should be very small in comparison with the light-quark contributions. It plays an essential role only in describing the baryon anti-triplet, which is understandable, because the light-quark pair constitutes a spin-zero state. However, its effect is rather small when it comes to the baryon sextet. In the lattice QCD [20][21][22], it is known that the contribution of the heavy quark to the magnetic moments of the baryon sextet is approximately one order smaller than the light-quark contributions. Ref. [19] also examined the heavy-quark contribution separately and found that its effect is in general tiny on the magnetic moments of the baryon sextet.
In principle, one could consider the heavy-quark effects on the magnetic moments as done in the quark models. It would give an overall constant contribution to the magnetic moments of the baryon sextet such as −e Q /6m Q [19], which is parametrically very small. However, if one wants to consider the heavy-quark contribution within the present formalism consistently, one should go beyond the mean-field approximation. This is yet a difficult task, since we do not know proper nonperturbative interactions between the light and heavy quarks. Thus, we want to restrict ourselves to the light-quark contribution from the mean-field approximation, so we will ignore in the present work that from the heavy quark current in the limit of m Q → ∞.
Hence, we will deal with the first term of Eq. (1) when we compute the magnetic moments of heavy baryons by considering the following baryon matrix elements: Since we have ignored the heavy-quark contributions, we obtain the same results for both the charmed and bottom baryons. So, we will mainly focus on the magnetic moments of the charmed baryon sextet in the present work. The general expressions for the magnetic moments of light baryons have been constructed already in previous works [28][29][30][31]. We will extend the formalism for those of heavy baryons in this work. Taking into account the rotational 1/N c and linear m s corrections, we are able to write the collective operator for the magnetic moments aŝ whereμ (0) andμ (1) represent the leading and rotational 1/N c contributions, and the linear m s corrections respectivelŷ The indices of symmetric tensor d pq3 run over p = 4, · · · , 7.Ĵ 3 andĴ p denote the third and the pth components of the spin operator acting on the soliton. D which is obtained from the SU(3) rotation of the electromagnetic octet current. The coefficients w i in Eq. (5) encode a concrete dynamics of the chiral soliton and are independent of baryons involved. In fact, w 1 includes the leading-order contribution, a part of the rotational 1/N c corrections, and linear m s corrections, whereas w 2 and w 3 represent the rest of the rotational 1/N c corrections. The m s dependent term in w 1 is not explicitly involved in the breaking of flavor SU(3) symmetry. Thus, we will treat w 1 as if it had contained the SU(3) symmetric part, when the magnetic moments are computed. On the other hand, w 4 , w 5 , and w 6 are indeed the SU(3) symmetry breaking terms. Yet another m s corrections will come from the collective wave functions, which we will discuss soon. In principle, w i can be computed within a specific chiral solitonic model such as the χQSM [28,31]. We want to emphasize that the structure of Eq. (5) is rather model-independent and is deeply rooted in the hedgehog Ansatz or hedgehog symmetry. Since we consider the embedding of the SU(2) soliton into SU(3) [3], which keeps the hedgehog symmetry preserved, we have SU(2) T × U(1) Y symmetry. So, the structure of the collective operator is determined by the SU (2) In this respect, we will determine w i by using the experimental data on the magnetic moments of the baryon octet as done in Refs. [29,30,32], instead of relying on a specific model. We will briefly show how to fix w i , using the experimental data in the next Section. To obtain the magnetic moments of the heavy baryons, the operatorμ in Eq. (4) needs to be sandwiched between heavy baryon states. Since we consider the linear m s corrections perturbatively, the collective wave functions for the soliton consisting of the light-quark pair are no longer pure states. The collective Hamiltonian for flavor SU(3) symmetry breaking [1], which is expressed as brings about the mixing of the baryon wave functions with those in higher SU (3) representations. The parameters α, β, and γ for heavy baryons are written as Note that the three parameters α, β, and γ are expressed in terms of the moments of inertia I 1, 2 and K 1, 2 . The valence parts of them are different from those in the light baryon sector by the color factor N c − 1 in place of N c . The expression of Σ πN is similar to the πN sigma term again except for the N c factor: As mentioned previously, a singly heavy baryon consists of N c − 1 light valence quarks, so the constraint imposed on the right hypercharge should be changed from Then the wave functions for the baryon anti-triplet (J = 0) and the sextet (J = 1) are obtained respectively as [33] with the mixing coefficients respectively, in the basis [Λ Q , Ξ Q ] for the anti-triplet and for the sextets. The parameters p 15 , q 15 , and q 24 are given by where To carry out actual computation, we need to know the explicit expression of the wave functions formally given in Eq. (10). The wave function of a state with flavor F = (Y, T, T 3 ) and spin S = (Y ′ = −2/3, J, J 3 ) in the representation ν is obtained in terms of a tensor with two indices, i.e. ψ (ν; F ),(ν; S) , one running over the states F in the representation ν and the other one over the states S in the representation ν. Here, ν stands for the complex conjugate of the representation ν, and the complex conjugate of S is written as S = (2/3, J, J 3 ). N c being taken to be 3, the spin with hypercharge is given by S = (2/3, J, J 3 ). Thus, the collective wave function for the soliton with a light-quark pair is expressed as where dim(ν) denotes the dimension of the representation ν and Q S a charge corresponding to the baryon state S, i.e. Q S = J 3 + Y ′ /2.
To construct the complete wave function for a heavy baryon, we need to couple the soliton wave function to the heavy quark such that the heavy baryon becomes a color singlet. Thus, the wave function for the heavy baryon should be written as where χ JQ3 denote the Pauli spinors and C J ′ J ′ 3 J,J3 JQ JQ3 the Clebsch-Gordan coefficients. Using the wave functions given in Eq. (14), we can derive the magnetic moments of the heavy baryons where µ arises fromμ (1) in Eq. (4), which contain w 4 , w 5 , and w 6 . µ

III. MAGNETIC MOMENTS OF THE BARYON SEXTET
In the present approach, it is trivial to compute the magnetic moments of the baryon anti-triplet that consists of the soliton with spin J = 0. Since any scalar particle does not carry the magnetic moment, the magnetic moments of the baryon anti-triplet all turn out to vanish [19,23]. This implies that the magnetic moments of Λ + c and Ξ c should be tiny. So, we concentrate in this work on the magnetic moments of the baryon sextet.
We start with the magnetic moments in the chiral limit, for which we need to consider µ (0) B in Eq. (15). As mentioned already, w 1 contains both the leading-order and a part of the 1/N c rotational corrections. Since we will use the experimental data to fit w 1 , it is difficult to decompose these two different terms as already discussed in Ref. [10]. Thus, we will follow the argument of Ref. [10] to fit the parameters required for the magnetic moments of the baryon sextet. Using the explicit expression of the magnetic moments given in Refs. [31,34], we can write w 1 , w 2 , and w 3 as where the explicit forms of M 0 , M can be found in Refs. [31,34]. I denote the moments of inertia with the notation of Ref. [34] taken. It was shown in Ref. [34] that in the limit of the small soliton size the parameters in Eq. (16) can be expressed as These results with the small soliton size lead to the expressions of the magnetic moments in the nonrelativistic (NR) quark model. Indeed, Eq. (16) reproduces the correct ratio of the proton and magnetic moments µ p /µ n = −3/2. So, the limit of the small soliton size is identical to the NR limit [34]. In the NR limit, we also obtain the relation M To carry on the computation, we have to assume that this relation is also valid in the case of the realistic soliton size. Then, we are able to express the leading-order contribution M 0 in terms of w 1 and w 3 Since a heavy baryon consists of N c − 1 valence quarks, the original M 0 should be modified by introducing (N c − 1)/N c as done similarly in Ref. [10]. The denominator N c will cancel the same N c factor in the original expressions of w 1 and w 3 (for explicit expressions of w 1 and w 3 , we refer to Ref. [35]) so that they have the proper prefactor N c − 1 arising from the presence of the N c − 1 valence quarks inside a heavy baryon. Theoretically, only the valence part of M 0 requires this scaling factor. However, as far as we fix the values of w i using the experimental data, it is not possible to fix separately the valence and sea parts. Thus, it is plausible to definew 1 as where σ is introduced to compensate possible deviations arising from the relation M assumed to be valid in the realistic soliton case, and from the scaling factor (N c − 1)/N c introduced to M 0 without separation of the valence and sea parts. We want to mention that σ has been already determined in Ref. [10]: σ ∼ 0.85.
While w 2 and w 3 are kept to be the same as in the case of light baryons, w 4,5,6 are required to be modified by introducing the same factor (N c − 1) /N c as done for M 0 . So, we redefine w 4, 5, 6 as Employing the numerical values provided in Ref. [30] and Eq. (20), we obtain the following values w 1 = −10.08 ± 0.24, w 2 = 4.15 ± 0.93, w 3 = 8.54 ± 0.86, w 4 = −2.53 ± 0.14, w 5 = −3.29 ± 0.57, Using these numerical values, we can now derive the magnetic moments of the baryon sextet. So, we want to emphasize that there is no additional free parameter to fit. The explicit expressions of the magnetic moments for the baryon sextet with J ′ = 1/2 in Eq. (15) are derived as in the basis of [Σ c , Ξ ′ c , Ω c ]. Q denotes the electric charge of the corresponding baryon. For the baryon sextet with spin J ′ = 3/2, we obtain in the basis of [Σ * c , Ξ * c , Ω * c ] for the charmed sextet. Before we present the numerical results of the magnetic moments of the heavy baryons, we first discuss general relations we find in Eqs. (26) and (27). In the present mean-field approach, there is no difference between charmed and bottom baryons, since there is no contribution from the heavy quark in the limit of m Q → ∞. Thus, even though the electric charges of the charm and bottom baryons are different each other, we have exactly the same numerical values of the magnetic moments for both the charm and bottom baryon belonging to the same representation, which can be written as Furthermore, we find a general interesting relations between the magnetic moments of the baryon sextet with J ′ = 1/2 and those with J ′ = 3/2. The difference is just fact 3/2 given as In fact, this relation was already found in both the bound-state approach of the Skyrme model [17] and the SU(3) quark models [11][12][13]. Coleman and Glashow found various relations between the magnetic moments of the baryon octet [36], which arise from the isospin invariance. Similar relations have been also obtained in the case of the baryon decuplet [29,37]. We find here the generalized Coleman-Glashow relations for the spin-1/2 baryon sextet as Similar relations were also discussed in Ref. [24]. Though the usual Coleman-Glashow relations are satisfied in the chiral limit, the relations in Eq. (30) are the robust ones even when the effects of SU(3) flavor symmetry breaking are taken into account. In the chiral limit, we obtain the relation according to the U -spin symmetry Another interesting relation in the chiral limit is the sum rule given as We know that Eq. (32) is very similar to the sum rule for the magnetic moments of the baryon decuplet. In the case of the decuplet, the sum of all the magnetic moments is the same as that of all the electric charges of the corresponding baryons [29]. On the other hand, Eq. (32) is identical to the sum of 2Q − 1 for all the members of the baryon sextet as shown in Eq. (25), which yields also the null result as given in Eq. (32). We can derive the same relations from Eqs. (30)(31)(32) for the J ′ = 3/2 sextet baryon and also for the bottom baryons.
Concerning the magnetic moments of the baryon anti-triplet, we already mentioned that those of all members turn out to be zero, since the spin of the soliton with a light-quark pair is zero. Lichtenberg derived a relation based on the quark model, which shows that all the magnetic moments of the baryon anti-triplet are the same [13] µ This relation is trivially satisfied because all of them vanish in the present work. In Table I and II, we list the results of the magnetic moments for the charmed baryon sextet with J ′ = 1/2 and J ′ = 3/2, respectively. In the second columns, the the numerical values of the magnetic moments in the chiral limit are listed whereas in the third columns the total results are given. The effects of flavor SU(3) symmetry breaking contribute to the magnetic moments of the baryon sextet found to be around (5 − 10) % except for those of Σ 0 c and Ω 0 c , for which the effects are found to be around (15 − 20) %.

IV. NUMERICAL RESULTS
In Table I, we compare the results for the baryon sextet with J ′ = 1/2 with those from the Skyrme model in the bound-state approach [17,18], the relativistic quark model [19], and the lattice QCD [20][21][22]. While the original Skyrme model in the bound-state approach [17] is constructed, based on the light pseudoscalar meson fields together with the heavy pseudoscalar meson fields, Ref. [18] introduced the light and heavy vector mesons in addition. We find that in general the magnitudes of the present results are slightly larger than those of Ref. [17] except for µ(Ξ + c ) and µ(Ξ ′0 c ). The results of µ(Σ ++ c ) and µ(Σ 0 c ) from Ref. [18] are, respectively, around 15 % and 60 % larger than the present results in magnitude. The results are qualitatively similar to those of Refs. [19], though Ref. [19] used a rather different model, i.e. the relativistic quark model. Comparing the present results with those from the lattice QCD [20][21][22], we find that the results are systematically larger than those from the lattice QCD except for the Σ ++ c magnetic moment.
Since the results of the magnetic moments of the bottom baryons are exactly the same as those of the charmed baryons, it seems redundant to present the corresponding results. The present results for the magnetic moments of the bottom baryon sextet are even in better agreement with those from Ref. [19].

V. SUMMARY AND CONCLUSION
In the present work, we have studied the magnetic moments of the lowest-lying singly heavy baryons, based on the chiral quark-soliton model. All the dynamical parameters were fixed in the light baryon sector. The magnetic moments of the baryon anti-triplet vanish in the present mean-field approach, because the spin of the soliton for the anti-triplet is J = 0. Since the first term in the expression of the heavy baryon magnetic moments consists of the leading-order and the 1/N c rotational corrections, we had to decompose them, using the limit of the small soliton size. Having properly considered the scaling factor, we were able to compute the magnetic moments of the baryon sextet with both spins J ′ = 1/2 and J ′ = 3/2. The results were compared with those from other models such as the Skyrme model in bound-state approaches, the relativistic quark model, and the lattice QCD. They are in particular consistent with those from the relativistic quark model. We also compared the present results with those from the lattice QCD. Except for the Σ ++ c magnetic moment, we obtained systematically larger values of the magnetic moments than those from the lattice QCD.
The same method can be applied to compute the transition magnetic moments of the singly heavy baryons, which provide essential information on radiative decays of them. The corresponding investigation is under way. The magnetic moments of doubly heavy baryons can be also studied within the present mean-field approach, assuming that N c − 2 valence quarks can produce the modified pion mean fields. The corresponding study is under investigation.