Magnetic transitions and radiative decays of singly heavy baryons

A pion mean-field approach allows one to investigate light and singly heavy baryons on an equal footing. In the large $N_c$ limit, the light and singly heavy baryons are viewed respectively as $N_c$ and $N_c-1$ valence quarks bound by the pion mean fields created self-consistently, since a heavy quark can be regarded as a static color source in the limit of the infinitely heavy quark mass. The transition magnetic moments of the baryon sextet are determined entirely by using the parameters fixed in the light-baryon sector without any additional parameters introduced. Assuming that the transition $E2$ moments are small, we are able to compute the radiative decay rates of the baryon sextet. The numerical results are discussed, being compared with those from other approaches.


I. INTRODUCTION
In the large N c (the number of colors) limit, an ordinary baryon can be viewed as a state of N c valence quarks bound by the meson mean fields, which are self-consistently created by the presence of the N c valence quarks [1,2]. The chiral quark-soliton model (χQSM) [3] realizes effectively this picture (for a review, we refer to Refs. [4,5]). On an equal footing, a singly heavy baryon, which consists of one heavy quark and two light quarks, can be regarded as a state of N c − 1 valence quarks bound by the pion mean fields, if the heavy-quark mass is taken to be infinity (m Q → ∞) [6,7] (see also a recent review [8]). The masses of the lowest-lying singly heavy baryons in both the charmed and bottom flavor sectors were successfully described within the framework of this mean-field approach, and in particular the mass of the Ω b was predicted [7,9] with all parameters fixed in the light baryon sector. More recently, the five narrow Ω c resonances newly observed by the LHCb Collaboration [10] were studied within the same framework and two of them were advocated as exotic pentaquark baryons belonging to the anti-decapentaplet (15) [11]. This classification of the narrow Ω c resonances as the members of the baryon anti-decapentaplet has been further supported by computing strong decays of the newly found two narrow Ω c s [12]. The magnetic moments of singly heavy baryons have been also investigated without any new additional parameters [13]. Very recently, the electromagnetic form factors of the singly heavy baryons have been also computed within the self-consistent χQSM [14].
While there is no experimental information on radiative decays of heavy baryons, the CLEO Collaboration identified the two resonances of the singly heavy baryons, Ξ +′ c and Ξ 0′ c , using their radiative decays Ξ +′ c → Ξ + c γ and Ξ 0′ c → Ξ 0 c γ, since the masses of the Ξ ′ c isodoublet lie below threshold for their strong decays [15]. Theoretically, radiative decays of singly heavy baryons have been studied within various different approaches: The nonrelativistic quark model (NRQM) [16][17][18][19], the bag models [20,21], potential models [22,23], QCD sum rules [24][25][26][27], heavy-quark effective theories [28,29], chiral perturbation theories [30][31][32], relativistic quark models [33], the heavy quark symmetry with a diquark picture [34], the chiral constituent quark model [35,36], and lattice QCD [37,38]. In the present work, we want to investigate the transition magnetic moments and radiative decays of singly heavy baryons belonging to the baryon sextet with both spin 1/2 and 3/2. The main virtue of the present approach lies in the fact that all the dynamical parameters, which are required to compute the transition magnetic moments of the singly heavy baryons, have already been fixed in the light-baryon sector [13,39]. Thus, we can predict the transition magnetic moments and the radiative decay rates of the lowest-lying singly-heavy baryons in a robust manner.
The present work is organized as follows: In Section II, we explain a general formalism of the χQSM to compute the transition magnetic moments of the heavy baryons. In Section III, we explicitly calculate the transition magnetic moments within the present framework. In Section IV, we present the numerical results of the transition magnetic moments and radiative decays of the heavy baryons. The last Section is devoted to the summary and conclusions of the present work.

II. GENERAL FORMALISM
We first introduce the electromagnetic (EM) current that consists of both the light and heavy quark currents whereQ stands for the charge operator of the light quarks, defined aŝ Here, λ 3 and λ 8 represent the well-known flavor SU(3) Gell-Mann matrices. The second part of Eq. (1) is the heavyquark EM current with heavy-quark charge e Q (e c = 2/3 for the charm quark or as e b = −1/3 for the bottom quark). However, the heavy-quark EM current is not involved in the infinitely heavy-quark mass limit m Q → ∞, since the magnetic moment of a heavy quark is proportional to the inverse of the corresponding mass, i.e. µ ∼ (e Q /m Q )σ. We expect that the corrections from the next-to-leading order in the 1/m Q expansion should be rather small in comparison with the light-quark contributions to the transition magnetic moments. So, we consider only the first term of Eq. (1) when we calculate the transition magnetic moments of heavy baryons. Since we employ the limit of m Q → ∞, we obtain the same results for both the charmed and bottom baryons, which is the consequence of heavy flavor symmetry. Starting from the transition matrix element of the EM current sandwiched between the heavy-baryon states, we are able to derive the collective operator for the magnetic moments within the framework of the χQSM. In fact, they have been already derived in previous works [39][40][41][42][43]. Considering the rotational 1/N c and linear strange current-quark mass m s corrections, we obtain the collective operator for the magnetic moments aŝ whereμ (0) andμ (1) denote the leading and rotational 1/N c contributions, and the linear m s corrections respectively, which are expressed aŝ The indices of symmetric tensor d pq3 run over p = 4, · · · , 7.Ĵ 3 andĴ p correspond respectively to the third and the pth components of the spin operator acting on the soliton. D   The structure of the collective operatorμ in Eq. (5) is in fact very general and is considered as a model-independent one. It is deeply related to hedgehog symmetry and the embedding of the SU(2) soliton into SU(3) [2], which provides a minimal way of combining the ordinary space with the internal flavor space. Because of this embedding, the symmetry we have is SU(2) T × U(1) Y . Therefore, the collective operator for the magnetic moment is expressed in terms of the which yield the general expression for the collective operator given in Eq. (5). Hence, instead of computing w i in a specific model, we can determine w i by the experimental data on the magnetic moments of the baryon octet as done in Refs. [39,41,43].
To derive the transition magnetic moments of the heavy baryons, we need to compute the matrix elements of the collective operatorμ in Eq. (3) between heavy baryon states. Moreover, the presence of w 4,5,6 comes from the perturbative treatment of m s , so that the collective wavefunctions for the soliton consisting of the light-quark pair are not pure states but mixed ones with higher representations. In order to derive the collective wavefunctions, we need to diagonalize the collective Hamiltonian for flavor SU(3) symmetry breaking, expressed as where α, β, and γ are the dynamical coefficients for the lowest-lying singly heavy baryons, of which the explicit expressions can be found in Refs. [7,9]. Diagonalizing Eq. (8), we obtain the wavefunctions for the baryon anti-triplet (J = 0) and the sextet (J = 1) respectively [9] as with the mixing coefficients respectively, in the basis [Λ Q , Ξ Q ] for the anti-triplet and for the sextets. The expressions for the parameters p 15 , q 15 , and q 24 are also found in Refs. [9,13] and the corresponding numerical values are given as p 15 = −0.104 ± 0.003, q 15 = 0.238 ± 0.005, q 24 = −0.049 ± 0.002. (11) Note that the mixing coefficients are proportional to m s linearly. So, the effects of flavor SU(3) symmetry breaking arise also from the collective wavefunctions for the heavy baryons in addition to the collective operator for the magnetic moments, which contains w 4,5,6 . The complete wavefunction for a heavy baryon can be constructed by coupling the soliton wavefunction to the heavy quark such that the heavy baryon becomes a color singlet, which are expressed as where χ JQ3 stand for the Pauli spinors and C J,J3 JQ JQ3 denote the Clebsch-Gordan coefficients. J ′ and J ′ 3 represent the spin and its third component of the heavy baryons whereas T and T 3 are the corresponding isospin and its third component. The wavefunctions |B µ ; (J, J 3 )(T, T 3 ) with representation µ are given in Eq. (9). Calculating the matrix elements ofμ becomes just the simple D-function algebra. Thus, we can easily obtain the transition magnetic moments of the heavy baryons.
In Ref. [13], we discussed in detail how w 1 , w 4 , w 5 , and w 6 are modified for the singly heavy baryons that consist of N c − 1 light valence quarks. Accordingly, the first coefficient w 1 was revised as where σ is introduced to compensate the deviation arising from the nonrelativistic limit used in the course of deriving w 1 . The numerical value of σ was already determined in Ref. [12]: σ ∼ 0.85. The other two coefficients w 2 and w 3 are kept to be intact in the course of reducing the number of valence quarks from N c to N c − 1 whereas w 4,5,6 need to be modified as Using the numerical values of w i determined in Ref. [39] and Eqs. (13) and (14), we obtain the following values [13] 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, Thus, once we know the numerical values of these dynamical parameters, we can straightforwardly compute the transition magnetic moments of singly heavy baryons, which will be shown in the next Section.

III. TRANSITION MAGNETIC MOMENTS AND PARTIAL DECAY WIDTHS OF RADIATIVE DECAYS
The matrix elements of the EM current between the baryon sextet with spin 1/2 to the baryon antitriplet can be parametrized by the transition EM form factors F i (q 2 ) as follows: where q 2 is the square of the four-momentum transfer, and M B ′ 1/2 and M B 1/2 denote respectively the masses of the baryon antitriplet and sextet with spin 1/2. The u B(B ′ ) 1/2 is the Dirac spinor corresponding to the baryon B(B ′ ).
Similarly, the transition EM form factors from the baryon sextet with spin 3/2 to the sextet or antitriplet with spin 1/2, i.e. 6 3/2 → 3 1/2 or 6 3/2 → 6 1/2 , are defined as [48] where G * M , G * E , and G * C stand respectively for the transition magnetic dipole, electric quadrupole, and Coulomb form factors. The K M,E,C are the covariant tensors defined in Ref. [48]: u β B 3/2 represents the Rarita-Schwinger spinor for the baryon sextet with spin 3/2. At q 2 = 0, note that the transition magnetic dipole form factors F 2 (q 2 ) and G * M (q 2 ) are the same as the transition magnetic moments. Since the magnetic dipole transitions (M 1) are experimentally dominant over the electric quadrupole transitions (E2) in hyperon radiative decays, one can neglect the contribution from the E2 transitions to the radiative decays of heavy baryons. This is a plausible approximation, since the the size of the E2/M 1 ratio for the ∆ isobar is yielded in the range of (1 − 3) % experimentally [49][50][51][52][53][54]. Though there is no experimental data on the E2/M 1 ratio for the singly heavy baryons, we can safely assume that the size of the E2/M 1 ratio is very small. It indicates that the effects of the E2 are expected to be negligible on the radiative decay widths of the singly heavy baryons, because even the squared values of the E2 contribute to them. In fact, a recent lattice study has predicted the magnitude of the E2 form factor of Ω * c to be much smaller than those of the M 1 form factors [37]: M 1(Ω * c ) = −0.657 and E2(Ω * c ) = −0.012. A very recent calculation of the electromagnetic transition form factors of the singly heavy baryons within the self-consistent SU(3) χQSM has come to a similar conclusion [55].
So, we can express the radiative decay rates in terms of the transition magnetic moments. Using Eqs. (16,17) and neglecting the E2 transitions, we obtain the radiative decay rates for B 1/2 → B ′ 1/2 and for B 3/2 → B ′ 1/2 , respectively: respectively, where M BJ is the mass of baryon B with spin J and µ B ′ J ′ BJ is the transition magnetic moments for the radiative decay B J → B ′ J ′ γ. In Eq.(20,21) α EM denotes the fine structure constant and E γ the energy of the produced photon: Sandwiching the collective operator for the magnetic moment given in Eq. (3) between the heavy baryon states expressed by the collective wavefunctions (12), we finally arrive at the expressions of the transition magnetic moments for the baryon from the baryon sextet with spin 1/2 to the antitriplet, 6 1/2 → 3 1/2 , as where Q designates the electric charges of the corresponding heavy baryons. µ (0) [B c → B ′ c ] denotes the contribution from the leading order. µ (op) stands for the linear m s terms from the collective operator (5) and µ (wf) comes from the collective wavefunctions (9).
Similarly, we can derive the expressions of the transition magnetic moments for the 6 3/2 → 3 1/2 γ decays as and for 6 3/2 → 6 1/2 γ as As for the transition magnetic moments of the baryon sextet to the antitriplet, we find the following general relations These relations are satisfied even with the effects of flavor SU(3) symmetry breaking considered. Note that the NRQM [16][17][18] also satisfy Eq. (26). In the chiral limit, the U -spin symmetry yields various relations: The similar relations are obtained in Refs. [30,32], in which recombinations of Eq. (27) are found.

IV. RESULTS AND DISCUSSION
We now present the numerical results of the transition magnetic moments for the lowest-lying singly heavy baryons. As mentioned previously, all the dynamical parameters were already fixed by using the experimental data on the magnetic moments of the baryon octet. We want to emphasize that the present approach describes successfully the magnetic moments and transition magnetic moments for the baryon decuplet with those parameters [12] only. So, the present results for heavy baryons are obtained without any additional free parameters. In Table I, we list the numerical results of the transition magnetic moments for the charmed baryons. The third column lists those in the case of flavor SU(3) symmetry, whereas the fourth one does the total results with the linear m s corrections taken into account. Note that for the transition magnetic moments the relative signs are not important, because the wavefunctions for the heavy baryons can include a phase factor. Thus, the sign differences between different models do not matter. So, we will compare the magnitudes of the transition magnetic moments with those from other works.  [16][17][18] and Ref. [19] for Bc corresponding to 6 1/2 and 6 3/2 , respectively. The values from the modified bag model [21], light-cone QCD sum rules [24,25], chiral perturbation theory [32], the chiral constituent quark model [35], the Skyrme model (SM) [45], and the bound state approach with heavy quark symmetries [46] are also listed. In the last column, we list the results from a lattice QCD simulation [37,38].  As explained in the previous Section, the effects of flavor SU(3) symmetry breaking come from both the collective operators with w 4,5,6 and the collective wavefunctions. The total results listed in Table I contain both the contributions, though we do not show them separately. In general, the effects from the collective operators are dominant over those from the wavefunction corrections. However, the wavefunction corrections are not negligible specifically in the transitions Σ + c → Λ + c γ, Ξ ′0 c → Ξ 0 c γ, Σ * + c → Λ + c γ, and Ξ * 0 c → Ξ 0 c γ. As shown in Eqs. (23) and (24), the transition magnetic moments for the decays Ξ ′ c → Ξ c γ and Ξ * c → Ξ c γ are proportional to the corresponding charge Q. Thus,  [16][17][18] and Ref. [19] for Bc corresponding to 6 1/2 and 6 3/2 , respectively. The values from modified bag model [21], light-cone QCD sum rule [24,25], chiral perturbation theory [32], Skyrme model (SM) [45], and bound state approach [46] are also listed.
1.76 ± 0.08 1.69 ± 0.08 2.03 1.32 1.71 ± 0.60 1.06 --  Table II lists the results of the transition magnetic moments of the bottom baryons. In the present pion mean-field approach, we take the limit of the infinitely heavy quark mass (m Q → ∞). Thus, the results are in fact the same as those of the charmed baryons. Only the signs of the results are different, because charge of the bottom quark Q b = −1/3 is different from that of the charm quark Q c = 2/3. Nevertheless, the results are quite comparable with those of other works. In particular, we find that the present result of the Σ 0 b → Λ 0 b γ is almost the same as that from the Skyrme model [45]. III. Numerical results of the radiative decay rates for the charmed baryons in units of keV. The results are compared with those from the modified bag model [21], light-cone QCD sum rule [25], chiral perturbation theory [30,32], and lattice QCD [37,38]. It is straightforward to compute the radiative decay rates of the heavy baryons by using Eqs. (20) and (21), once we have known the corresponding transition magnetic moments. The results for the charmed baryons are presented in Table III. They are quite similar to those from the modified bag model [21] and light-cone sum rule [25], the whereas they are somewhat deviated from those of Ref. [32] even though the results of the transition magnetic moments are comparable each other. On the other hand, lattice QCD, χPT of Ref. [30] and the quark models used the same formulae as in the present work. If we recalculate the radiative decay rates, using the results of the transition magnetic moments from Ref. [32], we find that which are quite close to the present ones. Since Ref. [30] presented only the results of the radiative decay widths from χPT, we compare the present ones with them. Those of Ref. [30] are larger than those of all other models as well as the present results. In the last column of Table III, the results from the calculation of lattice QCD are listed [37,38], which are also comparable to the present ones.  [21], light-cone QCD sum rule [25], and chiral perturbation theory [30,32].   In Table IV, we list the results of the radiative decay rates for the bottom baryons. Again, they are comparable with those from the other works. By the same reason, the results from χPT [32] are different from the present ones but the recalculated results with Eqs. (20) and (20) are quite comparable with the present ones.

V. SUMMARY AND CONCLUSIONS
In the present work, we have investigated the transition magnetic moments of the lowest-lying heavy baryon sextets within a pion mean-field approach or a "model-independent chiral quark-soliton model. Since all the dynamical parameters for the magnetic moments of the singly-heavy baryons were fixed in the light baryon sector, we were able to compute the transition magnetic moments of the baryon sextet without any additional parameters introduced. We also derived the various relations among the transition magnetic moments as in the case of the magnetic moments of the heavy baryons. In addition, we found the relations that arise from the U -spin symmetry. The results of the transition magnetic moments for the charmed and bottom baryons were compared with those from other works. In particular, the present results are in good agreement with those from a simulation of lattice QCD. We obtained the radiative decay rates of the heavy baryons and compared the results with those from other works. While they are quite comparable with the results from the modified bag model, the light-cone QCD sum rule, and the simulation of lattice QCD, those from chiral perturbation theory seem different from the present ones. The reason arises from the fact that different formulae for the radiative decay rates were used. Using the same formulae, we found that the present results are also in good agreement with those from chiral perturbation theory.
Information on the transition magnetic moments may provide the vector meson coupling to the heavy baryons. Though the experimental data on them are still absent, theoretical investigations may shed lights on how the vector mesons can be coupled to the heavy baryons. In fact, this will provide essential information on the hadronic description of heavy hadron productions. The corresponding study is under way.