The radiative decays of the singly heavy baryons in chiral perturbation theory

In the framework of the heavy baryon chiral perturbation theory (HBChPT), we calculate the radiative decay amplitudes of the singly heavy baryons up to the next-to-next-to-leading order (NNLO). In the numerical analysis, we adopt the heavy quark symmetry to relate some low energy constants (LECs) with those LECs in the calculation of the magnetic moments. We use the results from the lattice QCD simulation as input. With a set of unified LECs, we obtain the numerical (transition) magnetic moments and radiative decay widths. We give the numerical results for the spin-$1\over 2$ sextet to the spin-$1\over 2$ antitriplet up to the next-to-leading order (NLO). The nonvanishing $\Gamma(\Xi_c^{'0} \rightarrow \Xi^0_c\gamma)$ and $\Gamma(\Xi_c^{*0} \rightarrow \Xi^0_c \gamma)$ solely arise from the U-spin symmetry breaking, and do not depend on the lattice QCD inputs up to NLO. We also systematically give the numerical analysis of the magnetic moments of the spin-$1\over 2$, spin-$3\over 2$ sextet and their radiative decay widths up to NNLO. In the heavy quark limit, the radiative decays between the sextet states happen through the magnetic dipole (M1) transitions, while the electric quadrupole (E2) transition does not contribute. We also extend the same analysis to the single bottom baryons.

In the framework of the heavy baryon chiral perturbation theory (HBChPT), we calculate the radiative decay amplitudes of the singly heavy baryons up to the next-to-next-to-leading order (NNLO). In the numerical analysis, we adopt the heavy quark symmetry to relate some low energy constants (LECs) with those LECs in the calculation of the magnetic moments. We use the results from the lattice QCD simulation as input. With a set of unified LECs, we obtain the numerical (transition) magnetic moments and radiative decay widths. We give the numerical results for the spin- 1 2 sextet to the spin-1 2 antitriplet up to the next-to-leading order (NLO). The nonvanishing Γ(Ξ ′ 0 c → Ξ 0 c γ) and Γ(Ξ * 0 c → Ξ 0 c γ) solely arise from the U-spin symmetry breaking, and do not depend on the lattice QCD inputs up to NLO. We also systematically give the numerical analysis of the magnetic moments of the spin- 1 2 , spin-3 2 sextet and their radiative decay widths up to NNLO. In the heavy quark limit, the radiative decays between the sextet states happen through the magnetic dipole (M1) transitions, while the electric quadrupole (E2) transition does not contribute. We also extend the same analysis to the single bottom baryons.

I. INTRODUCTION
A heavy baryon contains two light quarks and a heavy quark. In the SU(3) flavor symmetry, the two light quarks form a diquark in the antisymmetric3 f or the symmetric 6 f representation. Constrained by the Fermi-Dirac statistics, the J P of the diquark is 0 + or 1 + , respectively. Then the diquark and the heavy quark are combined to form the heavy baryon. For the ground antitriplet, the J P is 1 2 + . For the ground sextet, the J P is 1 2 + or 3 2 + . In the following, we use ψ3, ψ 6 , and ψ µ 6 * to denote the spin-1 2 antitriplet, spin-1 2 , and spin-3 2 sextet, respectively. For the transitions ψ 6 → ψ3 and ψ µ 6 * → ψ 6 , the radiative decays are quite important, since some strong decay channels are forbidden by the phase space. So far, the BaBar and Belle Collaborations have observed three radiative decay processes: Ω * c → Ω c γ [1,2], Ξ ′ + c → Ξ + c γ and Ξ ′ 0 c → Ξ 0 c γ [3][4][5]. More observations are expected at the BESIII, Belle II, LHCb and other collaborations in the future.
The radiative decay processes are good platforms for studying the electromagnetic properties, which are important to reveal the inner structures of the heavy baryons. In literature, theorists used many different models to study the radiative decays. In Refs. [6,7], the authors studied the decay widths and electromagnetic form factors of the processes Ω * c → Ω c γ and Ξ ′ c → Ξ c γ using the lattice QCD simulation. In Ref. [8], the authors constructed the chiral Lagrangains for the heavy baryons incorporating the heavy quark symmetry and studied the radiative decays of the heavy baryons and mesons. Later, the authors in Refs. [9][10][11][12] investigated the electromagnetic properties of the heavy baryons in the heavy hadron chiral perturbation theory. In Ref. [13], Jiang et al. calculated the electromagnetic decay widths of the heavy baryons up to the next-to-leading order (NLO) in the heavy baryon chiral perturbation theory (HBChPT). They found that the neutral radiative decay channels, e.g. Ξ ′ 0 c → Ξ 0 c γ and Ξ * 0 c → Ξ ′ 0 c γ, are suppressed due to the U-spin symmetry. Besides the lattice QCD and the effective field theory, theorists also studied the radiative decays with other phenomenological models: the heavy quark symmetry [14], the light cone QCD sum rule formalism [15][16][17][18][19], the bag model [20,21], the nonrelativistic quark model [22], the relativistic three quark model [23] and other various quark models [24][25][26][27][28][29][30].
The chiral perturbation theory is firstly used to study the properties of the pseudoscalar mesons [31][32][33][34]. It has a self-consistent power counting law which is in terms of the small momentum (mass) of the pseudoscalar mesons. When it is extended to the baryons, the mass of a baryon, which is at the same order as the chiral symmetry breaking scale in the chiral limit, breaks the consistent power counting [35]. To solve this problem, the heavy baryon chiral perturbation theory (HBChPT) is developed [36][37][38][39]. In this scheme, the baryon field is decomposed into the light and heavy components. The heavy component can be integrated out in the low energy region and the large mass of the baryon is eliminated. Now, the power counting law recovers and the expansion is in terms of the residue momentum of the baryons and the momentum (mass) of the pseudoscalar mesons.
So far, the amplitudes of the radiative decays are calculated up to NLO using the effective theory [8][9][10][11][12][13]. In this work, we systematically derive the radiative decay amplitudes up to the next-to-next-to-leading order (NNLO) in HBChPT. Many low energy coefficients (LECs) are involved in the analytical expressions. Some of them also appear in the magnetic moments up to NNLO. In this work, we will use the data of the magnetic moments and the radiative decay widths from the lattice QCD simulations as input to obtain the numerical results. In the numerical analysis, we adopt the heavy quark symmetry to reduce the number of the LECs [40,41]. We give the final results of (transition) magnetic moments, radiative decay widths in a group of unified LECs.
The paper is arranged as follows. In Section II, we derive the expressions of the decay widths using the form factors from the electromagnetic multipole expansion. In Section III, we present the effective Lagrangians that contribute to the radiative decays up to NNLO. In Section IV, we derive the analytical expressions of the decay amplitudes up to NNLO. In Section V, we construct the Lagrangians in the heavy quark limit and reduce the number of the LECs using the heavy quark symmetry. In Section VI, we use the data from the lattice QCD simulation as input to calculate the LECs. Then, we obtain the numerical results of the (transition) magnetic moments, the M1 transition form factors and the decay widths of the charmed baryons up to NNLO. In Section VII, we extend the calculations to the bottom baryons. Finally, we compare our results with those from other models and give a brief summary in VIII. In Appendix A, we give the magnetic moments of the spin-1 2 and spin-3 2 sextet as by-product. In Appendix B, we give some quark model results. In Appendix C, we list the details of the loop integrals.

II. THE RADIATIVE DECAY WIDTH
In the SU(3) flavor symmetry, the explicit matrix forms of the spin-1 2 antitriplet, spin-1 2 , and spin-3 2 sextet fields are with The radiative decay width reads,

III. THE LAGRANGIAN
We list the tree and loop diagrams that contribute to the radiative decay amplitudes up to O(p 4 ) in Fig. 1 and Fig. 2, respectively. The diagram with chiral dimension D χ contributes to the O(p Dχ ) radiative decay amplitude and O(p Dχ−1 ) transition magnetic moment. In the following, we list the Lagrangians involved in this work.
The leading-order Lagrangian for the pseudoscalar meson interaction reads with where A µ is the photon field, Q is the charge matrix of the light quarks. We use the m φ and F φ to denote the masses and the decay constants of the mesons, respectively. Their values are [32] m π = 137 MeV, m K = 496 MeV, m η = 548 MeV, The leading-order meson-baryon Lagrangian L Bφ reads with where Q B is the charge matrix of the heavy baryon. It is related to the charge matrix of the heavy quarkQ and that of the light quark Q through the relation Q B = Q + 1 2Q . For the charmed baryons, one hasQ c = diag( 2 3 , 2 3 , 2 3 ) and Q B = diag(1, 0, 0), respectively. M3 ,6,6 * denote the average masses of the antitriplet, spin-1 2 sextet, spin-3 2 sextet states, respectively.
The O(p 2 ) Lagrangians constructed from other building blocks do not contribute to the O(p 2 ) radiative decays. For instance, the following χ ± is O(p 2 ), qq is a parameter related to the quark condensate, m u,d,s denotes the current quark mass. At the leading order, the m u,d is ignored and 2B 0 m s are absorbed into the coupling constant. We obtain χ + = diag(0, 0, 1). We can construct the O(p 2 ) Lagrangians with χ + , for instance, Tr(B3χ + B 6 ). However, they do not contribute to the radiative decay amplitude at O(p 2 ).
The O(p 2 ) Bφφ vertex arising from L Bφφ contributes to the O(p 4 ) decay amplitude, The u µ and u ν form the flavor representations 8 ⊗ 8 = 1 ⊕ 8 1 ⊕ 8 2 ⊕ 10 ⊕ 10 ⊕ 27 as illustrated in Table I Fig. 2, the vertices Bφφ arising from the last three Lagrangian terms are symmetric for the opposite charged pseudoscalar mesons, while the φφγ vertex is antisymmetric. Then the loops with the opposite charged intermediate pseudoscalar mesons cancel out. Therefore, the above three Lagrangian terms do not contribute to the radiative decay.
The decay | 1 2 → | 1 2 γ is the M1 transition. The transition | 3 2 → | 1 2 γ may happen through the M1 and E2 transitions. The Lagrangian at O(p 3 ) contributes, represents that the scripts b(a) and the j(i) are symmetrized. S123 is the symmetrization operator for the subscripts 1, 2, and 3. ǫ ijk is the Levi-Civita symbol.
1 81 82 10 10 27 where n, n 2 , m 2 , andm 2 terms contribute to the G 1 . They cancel the divergences of the O(p 3 ) loop diagrams. The finite terms have the same structures as those in the O(p 2 ) tree diagrams when the same meson decay constants are adopted F π = F K = F η . Then they can be absorbed into the lower order f 2−4 andf 3 terms in Eq. (26). The n 1 , m 1 andm 1 terms contribute to G 2 , which contributes to the lowest-order E2 transition. The nonrelativistic form of L At O(p 4 ), the Lagrangian that contributes at the tree level is, As illustrated in Tables II and III, there are five and six independent Lagrangian terms for the transitions B 6 /B µ 6 * → B3γ and B µ 6 * → B 6 γ, respectively. The leading order expansion of the operator [χ + ,f µν ] vanishes, since they are diagonal matrices. Many terms are absorbed by the other Lagrangians.
In the nonrelativistic limit, L Bγ is written as  Group representation At the leading order, the O(p 2 ) tree diagram in Fig. 1 stems from the L Bγ and contributes to O(p 2 ) decay amplitude and O(p) transition magnetic moment, The superscript denotes the chiral order. The transition magnetic moment is in the unit of nuclear magneton. At NLO, the results from the tree diagrams are with ℓ = v · q. At NLO, the chiral corrections come from the loop diagrams (a), (b) and (i)-(l). After the integration, the amplitudes of the diagrams (i)-(l) vanish due to S · v = 0. The (a) and (b) diagrams contribute to the O(p 3 ) decay amplitude and O(p 2 ) transition magnetic moment: where n II 1 is the finite part of the loop integral and its explicit form is given in the Appendix. d is the dimension.
The O(p 3 ) magnetic moments from the loop diagrams read where J 2 , J ′ 2 and Λ 2 are the finite parts of the loop integrals. δ φ , γ φ 1,2 , a φ 66 * and other coefficients for the loops are listed in Table IV.
For the radiative transition B µ 6 * → B 3 γ, we give the explicit forms of the G 1 and G 2 . At the leading order, the L Bγ contributes to the form factor G 1 , where the superscript denotes that the value of G 1 comes from the O(p 2 ) tree diagram in Fig. 1. G 2 vanishes at this order. At NLO, both the M1 and E2 transitions contribute to the radiative decay. The L Bγ contributes at the tree level and the from factors read, The loop diagrams (a) and (b) also contribute at this order, At O(p 4 ), the analytical expressions of form factors coming from the loop diagrams are The spin-3 2 sextet decay into the spin-1 2 sextet through the M1 and E2 transitions. In this section, we show the analytical expressions of the form factors G 1 and G 2 . Then, one can obtain the analytical expressions of the decay amplitudes and the transition magnetic moments using Eqs. (12)- (17).
At the leading order, the transition amplitude arises from the L Bγ . The G 2 vanishes and the G 1 is where the C are the coefficients listed in Table V. At the next-to-leading order, both the tree and the loop diagrams contribute to the chiral corrections. The O(p 3 ) tree diagram arises from the L The analytical expressions of the O(p 3 ) loop diagrams (a), (b) in Fig. 2 are, where β φ , h φ and the following δ φ , θ φ and so on are the coefficients as listed in Table V. At NNLO, the form factors from the O(p 4 ) loop diagrams and the tree diagram are,

D. The U-spin symmetry in the analytical expressions
For the transitions B 6 → B3γ and B µ 6 * → B3γ, the form factors and the transition magnetic moments of the heavy baryons completely come from the dynamics of the two inner light quarks. The contributions from the two light quarks are destructive, which is clearer in the quark model as listed in Appendix B.
In the neutral decays Ξ    VI: The masses of the heavy baryons in the unit of MeV. The masses without special notations are from Ref. [46]. The † represents that the mass of the corresponding state is still absent. Then we estimate it with the average mass of the other states in the same isospin multiplet.   Another manifestation of the U-spin symmetry is the relations among the coefficients of the charged radiative decays. If we exchange the s quark and d quark, the heavy baryons transform as Σ + where X denotes the coefficients C (3) , δ φ and so on for the diagrams (a)-(d).
For the radiative decays B µ 6 * → B 6 γ, there are similar relations between the coefficients as Eq. (82), where X denotes the coefficients in Table V for the diagrams (a)-(d).
We also find some relations between the form factors in Table V. Up to NLO, one obtains similar relations as those in Ref. [11], The G E2 also satisfies the same relationships. Up to NNLO, Eq. (84) still holds. Eq. (85) is destroyed by the O(p 4 ) loop diagrams. In the calculation of the transition magnetic moments and the amplitudes, we use the baryon masses as listed in Table VI.

V. THE INDEPENDENT LECS IN THE HEAVY QUARK LIMIT
In previous works, we have calculated the magnetic moments of the spin-1 2 and spin-3 2 heavy baryons up to NNLO [40,41]. There are many common LECs for the magnetic moments and the radiative decay amplitudes. Thus, we perform the numerical analysis for the radiative decay widths together with the magnetic moments up to NNLO.
At the leading order, there are ten LECs: the d 2   [6,7,50,51]. The value of GM1(Ω * 0 c → Ω 0 c γ) is derived from Ref. [7]. The magnetic moment is in the unit of the nuclear magneton. The superscript ‡ denotes that the corresponding data is treated as input. LECs should have been estimated with the experiment data as input. So far, there are no experiment data. As a compromise, we use the data from the lattice QCD simulation as input, which is listed in Table VII. One notices that the number of the lattice QCD data is still smaller than that of the LECs. In the following section, we use the heavy quark symmetry to reduce the number of the LECs.

A. The heavy quark symmetry
Besides the Lagrangians in Section III, the magnetic moments up to NNLO involve the following Lagrangians, In the heavy quark limit,the spin-1 2 and spin-3 2 sextets are in the same multiplet. They can be described by a superfield [52], With the superfield, we construct the Lagrangians, the κ 1−8 terms, to reduce the number of the LECs.
The O(p 2 ) Lagrangians that contribute to the radiative decays read L (2) where the subscript "QB" represents the breaking effect of the heavy quark spin symmetry. Combining the two equations with Eq. (26), we reduce the seven LECs, d 5,6,8,9 , f 2,3 andf 3 , to three independent LECs, κ 1,2,3 , The Lagrangian that introduces the vertex Bφφ at O(p 2 ) is The transition magnetic moment and the decay width for the radiative transition B6/B µ 6 * → B3γ in the charmed baryon sector. µ is in the unit of nuclear magneton. The superscript ‡ denotes that the corresponding data is used as input.

VI. NUMERICAL RESULTS AND DISCUSSIONS
A. The radiative decays from the sextet to the antitriplet charmed baryons For the radiative transitions B 6 → B3γ and B µ 6 * → B3γ, we calculate the numerical results up to NLO. The numerical results are listed in Table VIII. Their analytical expressions contain three unknown coefficients f 2 , f 4 , and n 1 . The f 2 is estimated using µ(Ξ ′ + c → Ξ + c γ) from lattice QCD simulation and f 4 is related to f 2 through κ 2 in the heavy quark limit. The n 1 contributes to the G 2 form factor, which are important for the G E2 and has little influence on G M1 . The radiative decay width mainly arises from the M1 transition. Then we calculate the decay width without the G 2 contribution.
The radiative decay amplitudes of Ξ ′ 0 c → Ξ 0 c γ and Ξ * 0 c → Ξ 0 c γ completely come from the loops (a) and (b) up to NLO as illustrated in Section IV D. The amplitudes of the two loops only involve g 1−6 . Their values are [13,47,48] g 1 = 0.98, g 2 = − 3 8 g 1 = −0.60, g 3 = √ 3 2 g 1 = 0.85, The magnetic dipole form factor, transition magnetic moment and the decay width for the radiative transition from the spin-3 2 to the spin-1 2 sextet. The second to the forth columns represent contributions from the light quarks order by oder. The "Light" and "Heavy" represent the contributions from the light and heavy quarks, respectively. The sum of them are the total GM1 from factor. "..." denotes that there is no corresponding data in the lattice QCD simulations. where g 2,4 are calculated through the strong decay widths of the charmed baryons and others are obtained through the quark model. In Table VIII, one obtains The above results are independent of the inputs from the lattice QCD simulations. For the neutral decay channel Ξ * 0 c → Ξ 0 c γ, the E2 transition decay width is only 1.6 eV. The E2 transition is very strongly suppressed compared with the M1 transition.
B. The radiative decay width from the spin-3 2 sextet to the spin-1 2 sextet In the heavy quark limit, the average mass differences are The mass difference between the antitriplet and sextet does not vanish in the heavy quark symmetry limit. This will impact the convergence of the numerical results [40]. Thus, we do not consider the contributions of the intermediate antitriplet states in the loops in the numerical analysis. Since M − = δ 3 vanishes in the heavy quark limit, the G 2 does not contribute to the G M1 . The G E2 vanishes according to Eq. (12). Then the m 1 andm 1 do not appear in the analytical expressions. The LECs are reduced to κ 1 , κ 3 , κ 4 , κ 7 and κ 8 . In Refs. [40,41], we decomposed the magnetic moments of the heavy baryons into the contributions of the light and heavy quarks. We selected the average value µ c = 0.21µ N from the lattice QCD simulation as the magnetic moment of the charm quark. The heavy quark contributions to the magnetic moments of the antitriplet, the spin-  Table IX. The chiral expansion works well. The chiral corrections at NLO and NNLO to the (transition) magnetic moments cancel with each other in most channels. This helps to guarantee that the total results are mainly from the leading order.

VII. THE RESULTS FOR THE BOTTOM BARYONS
In this section, we extend the calculations to the singly bottom baryons. The charge matrices of the bottom quark and bottom baryons areQ The (transition) magnetic moments and the radiative decay amplitudes of the singly heavy baryons can be divided as where the superscripts "q" and "Q" denote the contributions from the light and heavy quarks, respectively. The Lagrangians and the LECs of the light quark sector are the same for the bottom and charmed baryons. For the heavy quark sector, one obtains the Lagrangians for the bottom baryons by replacing theQ c withQ b in the Tr(f + µν ). In the heavy quark limit, the mass differences for the bottom baryon states are The Γ(Ξ * − b → Ξ − b γ) is also mainly from the M1 transition, and the E2 decay width is only 0.20 eV. For the radiative decays B µ 6 * → B 6 γ, we use the predictions from the quark model to estimate the contributions from the bottom quarks [41]. The transition magnetic moments and the radiative decay widths are listed in Table XII.

VIII. SUMMARY
In this work, we calculate the radiative decay amplitudes and the transition magnetic moments for the singly heavy baryons. We derive their analytical expressions up to the next-to-next-to-leading order in the framework of the HBChPT. The expressions contain many LECs. Most of them also contributed to the magnetic moments. Thus, we perform the numerical analysis for the magnetic moments and the decay amplitudes of the singly heavy baryons simultaneously with a set of unified LECs. The heavy baryons have the heavy quark symmetry in the heavy quark limit. This helps to reduce the number of the independent LECs. For the decays B 6 → B3γ and B µ 6 * → B3γ, we calculate the numerical results up to the next-to-leading order. Due to the U-spin symmetry, the tree diagrams do not contribute to the transitions Ξ ′ 0 c → Ξ 0 c γ and Ξ * 0 c → Ξ 0 c γ. Their decay widths totally arise from the chiral corrections, which does not involve unknown LECs up to NLO. For Ξ * 0 c → Ξ 0 c γ, the E2 transition is suppressed. The above conclusions also hold for the radiative decays Ξ For the radiative decays B µ 6 * → B 6 γ, we calculate numerical results of the decay widths up to the next-to-next-toleading order. In the process, we do not include the antitriplet states as the intermediate states in the loops. We use the magnetic moments of the charmed baryons from the lattice QCD simulations are treated as input and predict the transition magnetic moments and the decay widths.
We extend the calculations to the bottom baryons. The light quark contributions are the same as those in the charmed baryon sector. The heavy quark contributions are estimated using the quark model.
In Tables XIII and XIV, we list our numerical results for the radiative decay widths in the charmed and bottom baryon sectors, respectively. We compare them with the results calculated using the lattice QCD [6,7], the extent bag model [20], the light cone QCD sum rule [53][54][55], the heavy hadron chiral perturbation theory (HHChPT) [11,56], the HBChPT [13] and the quark model [23]. For the radiative decays B 6 → B3γ and B µ 6 * → B3γ, our numerical results are consistent with those from other frameworks. For the radiative decay B µ 6 * → B6γ, we have estimated the LECs by adopting four magnetic moments from the lattice QCD simulations as input, which are smaller than those of other models [40,41]. Since the decay width is proportional to the square of the multipole form factor, the inputs from the lattice QCD may lead to smaller decay widths.
In the future, with more data from the experiment and the lattice QCD, we can update our numerical results using the analytical expressions. We expect the analytical expressions may be helpful for the extroplation of the lattice QCD simulation. Hopefully, our numerical results will be helpful to the experimental search of the radiative decays of the heavy baryons at LHCb, Belle II and BESIII. The decay widths of the charmed baryon transitions from different frameworks, the lattice QCD [6,7], the extent bag model [20], the light cone QCD sum rule [53][54][55], the heavy hadron chiral perturbation theory (HHChPT) [11,56], the HBChPT [13] and the quark model [23].