Radiative decays of dynamically generated charmed baryons

In this work we study the radiative decay of dynamically generated $J^P=\oh^-$ charm baryons into the ground state $J^P=\oh^+$ baryons. Since different theoretical interpretations of these baryonic resonances, and in particular of the $\Lambda_c(2595)$, give different predictions, a precise experimental measurement of these decays would be an important step for understanding their nature.

The decay modes of a resonance may provide a way of learning about the nature of the state: whether it fits in the conventional qqq-baryon or qq-meson picture or it has a more exotic interpretation. In particular, the radiative decays into lower lying states may represent a significant part of the decay width when the hadronic modes are suppressed by phase-space restrictions or/and small values of the coupling constants. Note that electromagnetic transitions are also useful in the determination of the quantum numbers of states decaying into a final hadron with well established quantum numbers.
Many of the former approaches also have been applied to obtain the radiative decays of excited heavy flavored baryons, such as the Λ c (2595). A first qualitative estimation was given in Ref. [60], where the HHCPT formalism was extended to include the lowest lying ex-cited baryon doublet, Λ c (2595) (J P = 1 2 − ) and Λ c (2625) The results of Ref. [54] were obtained by exploiting the alternative light diquark SU(2N f )×O(3) symmetry. Radiative decays of excited charmed baryons were also calculated within the relativistic quark model of Refs. [55,56], as well as using light-cone QCD sum-rules in Ref. [61]. All these works consider the excited heavy baryon as being an orbital excitation of the three-quark Qqq system with a unit of angular momentum inserted between the heavy quark Q and the light diquark qq. A different perspective is provided by the model of Ref. [62], which considers the excited Λ c (2595) and Λ c (2625) as being D * N bound systems N in the first excited state of a harmonic potential adjusted phenomenologically to reproduce the spin-averaged excitation energy, or by the model of Ref. [63] which considers the radiative decay of the Λ c (2940) in a hadronic D * N molecular picture.
In the present work we study the radiative decays of excited baryons in the C = 1 sector. We focus on the resonances Λ c (2595), Σ c (2800), Ξ c (2790) and Ξ c (2970), all having J P = 1 2 − , which have been generated as meson-baryon molecular states on several models based on coupled-channel dynamics [40][41][42][43][44][45][46][47]. The radiative transitions to baryons of the J P = 1 2 + ground state multiplet proceeds, as we will see, through the coupling of the photon to the various meson-baryon components of the resonance, as determined by the coupled-channel dynamical model of Ref. [47]. This is essentially different from the quark models for which, in the heavy quark limit, the photon only couples to the light diquark system. Therefore, a precise measurement of the radiative decays of excited heavy flavored baryons would help in understanding their internal structure.
The radiative decays of dynamically generated charm mesons has already been addressed recently. In [64][65][66], the radiative decay of the D * s0 (2317) meson has been studied from the point of view that it is generated dynamically mainly from the interaction of the D meson with a kaon. Also radiative decays of the puzzling X(3872) has been calculated from the point of view that this state is a D * D + c.c. molecule [67]. More recently, many different radiative decays of the controversial X, Y and Z states have been analyzed assuming that their structure is determined by the interactions of two vector mesons [68][69][70] This work is organized as follows: in the next sec-tion we explain the model for generating dynamically the J P = 1 2 − baryon resonances from the interaction of ground state baryons with pseudo-scalar mesons. In Sec. III we present the framework for evaluating the diagrams needed in the calculation of the radiative decays of the resonances. The results of the calculation are presented and discussed in Sec. IV and in Sec. V we present a brief overview and our conclusions.

II. THE DYNAMICALLY GENERATED STATES
In this section we will review the coupled-channel approach employed in [47] to obtain dynamically generated open-charm baryon resonances.
The scattering amplitudes T , which describe the scattering of the pseudo-scalar meson fields off the groundstate baryon fields can be obtained by solving the wellknown Lipmann-Schwinger equation, which schematically reads The loop function J is the product of the meson and baryon single-particle propagators, and the scattering kernel V describes the interaction between the pseudoscalar mesons and the ground-state baryons. Following the original work of Hofmann and Lutz [43], we identify a t-channel exchange of vector mesons as the driving force for the s-wave scattering between pseudo-scalar mesons in 16-plet and baryons in 20-plet representations. The scattering kernel takes the form (see [43] for details) where the sum runs over all vector mesons of the SU(4) 16-plet, (ρ, K * ,K * , ω, φ, D * , D * s ,D * ,D * s , J/Ψ), m V is the mass of the exchanged vector meson, g is the universal vector meson coupling constant, p i , q i (p j , q j ) are the four momenta of the incoming (outgoing) baryon and meson, and the coefficients C (I,S,C) ij;V denote the strength of the interaction in the different isospin (I), strangeness (S), charm (C) sectors, and meson-baryon channels (i, j). For the coupling constant g we use the value 6.6, which reproduces the decay width of the ρ meson [71]. The s-wave projection of the scattering kernel is easily obtained and, in the center-of-mass (c.m.) frame, it takes the analytical form with a, b, α and β being where k i , k j are the initial and final relative momenta, m i , m j , M i , M j are the masses of the incoming and outgoing mesons and baryons, their corresponding energies, which have been taken to be their on-shell values, and Ω(| k|) stands for the total energy ω(| k|) Once the scattering kernel has been constructed, it can be inserted in the s-wave projected Lipmann-Schwinger equation, where the loop function J explicitly reads and a dipole-type form factor F (| k|), have been introduced in order to regularize the integral. This form is typically adopted in studies of hadronhadron interactions within the scheme of Lipmann-Schwinger-type equations in the light flavor sector [72]. The value of the cut-off Λ is a free parameter of our model that is fitted to the existing data. In Table I we collect, from the resonances generated dynamically in Ref. [47] in various C = 1 sectors, those states that, using a cutoff value of around 1 GeV, can be readily identified with observed resonances [71].
In order to associate a given enhancement of the scattering amplitude to a resonance, we look for a characteristic pole, z R , in the unphysical sheet of the complex energy plane. Our prescription of unphysical sheet is such that, whenever the real part of the complex energy crosses a meson-baryon threshold cut, the sign of the onshell momentum is changed for this channel and for the already opened ones, as described in detail in Ref. [73]. If the pole z R lies not too far from the real axis, its value determines the Breit-Wigner mass (M = Re z R ) and width (Γ = 2 Im z R ) of the resonance as seen from real energies. The couplings of the resonance into its various mesonbaryon components are obtained from the residues of the scattering amplitude since, close to the pole, it can be parameterized in the form: Note that the value of the coupling constants of Eq.  Tables II to IV.

III. RADIATIVE DECAY CALCULATION
The reaction we study is given by: where P , K and Q are four-momenta which are related by P = K +Q, ǫ is the photon polarization and χ i and χ f are the polarization of the initial and final spin 1 2 baryons B * and B, respectively.  For the evaluation of these radiative decays, we follow a gauge invariant scheme already used for non-charmed resonances [74][75][76][77]. As shown there, the mechanisms for the decay of the resonance B * into γB are given by the diagrams of Fig. 1, where the photon couples to the constituent mesons and baryons. The amplitude for the radiative decay has the structure: with σ µ = (0, σ), where σ i are the usual Pauli matrices. For the tensor T µν , there are five possible independent Lorentz structures that one can construct with the two independent four-momenta: This expression is simplified by noting that, due to the Lorentz condition ǫ ν K ν = 0, the terms with the coefficients c and e in Eq. (11) will not contribute to the radiative decay amplitude of Eq. (10). Moreover, gauge invariance imposes T µν K ν = 0 and we obtain: from where we conclude that b = 0, while a and d are related through a = −d(P.K). Finally, working in the rest frame of the decaying baryon ( P = 0) and taking the Coulomb gauge (ǫ 0 = 0), only the a term contributes to the amplitude. The a coefficient is more easily calculated through the d coefficient, using the above mentioned relation. This is so because, as we will show, the integral for evaluating the d coefficient converges and it receives contributions from only the (b) and (c) diagrams of Fig. 1. The amplitude for diagram 1(b) is given by: where g RBM is the coupling of the resonance to its meson and baryon constituents, g BBM is the coupling of the baryonic current to a meson, g γMM is the coupling of the meson to a photon, and m M and m B are masses of the meson and baryon in the loop, respectively. Using the Feynman parameterization and the identity together with the relation between the a and d terms of the amplitude, one finds the following contribution of diagram 1(b) to the a term: Analogously, one can calculate the contribution coming from the diagram in Fig. 1 (c) as: With respect to the couplings appearing in Eq. (16) and Eq. (18), the values of g RBM , relating the resonance to its constituents, are listed in Tables II to IV of previous section, where the channels appearing there are given in the isospin basis. The couplings of the mesons to the photon, g γMM , and those of the baryons to the photon, g γBB , are eQ where Q is the electric charge (−1, 0, 1 or 2) of the meson or the baryon which emits the photon. Finally, the coupling of the mesons to the ground state baryons is given by: where D + F =1.26, D − F =0.33 [78]. We have taken f = 1.15f M with f M = f π = 93 MeV for light mesons and f M = f D = 165 MeV for heavy mesons. The coefficients α i and β i , which depend on the SU(4) flavor structure of the interaction, have been obtained from the Lagrangian for the interaction of the ground state spin 1 2 + baryons and the pseudo-scalar mesons, which is constructed in the following.
Our prescription of field definitions for the baryons and mesons is that of Ref. [43]. The baryons are represented by a three index tensor B ijk , antisymmetric in the first two indices, where i, j and k run from 1 to 4: The pseudo-scalar meson field is represented by a matrix: The Lagrangian for the BBM interaction reads: Note that the physical η, η ′ and η c fields are related to the mathematical η 8 and η 15 fields, belonging to the pseudo-scalar meson fifteen-plet, and the singlet η 1 field. The mixture is given by: Therefore, in order to obtain a Lagrangian for the physical fields, we we need to add to Eq. (23) an interaction of the baryon current with the singlet field η 1 : where the coefficients δ D = − 1 √ 2 and δ F = 3 √ 2 are obtained by consistently imposing that processes where the η c meson couples to light baryons (N , Λ, Σ or Ξ) should vanish.
The complete Lagrangian L ′ BBM of Eq. (25) allows us to determine all coupling constants g BBM , which we write in the form of Eq. (20). The specific values of the α i and β i coefficients for all the transitions needed in the different radiative decays are collected in Tables VI to X of the appendix.
With all the required ingredients having been established, one can finally obtain the the radiative decay width from: where E γ is the photon energy, M a i = M a (b) + M a (c) , and the sum runs over all contributing intermediate M B channels.

IV. RESULTS
We evaluate the following radiative decays: for Λ c (2595), Σ c (2792), Ξ c (2790) and Ξ c (2970), which are the resonances that, according to the models of Ref. [40][41][42][43][44][45][46][47], can plausibly be interpreted as dynamically generated meson-baryon molecules. We start showing the results for the Λ c (2595), which is the analog in the charm sector of the Λ(1405) in the strange sector and, therefore, is a good candidate for being a meson-baryon molecular state. The radiative decay widths are: These values are also collected in Table V, where the contribution to the width coming from each intermediate channel is also shown, together with the sign, (+) or (−), of the corresponding amplitude. This allows one to analyze the constructive or destructive character of the interferences between the various channels. First of all, we note the tremendous difference in size, of two orders of magnitude, between the radiative decay rate of the Λ c (2595) into Λ c γ and Σ + c γ states. To understand the origin of this difference we focus on the most important contribution, which corresponds to the D + n intermediate state in both cases. First of all, the ratio between the couplings D + n → Λ c and D + n → Σ + c is 3.8, as can be inferred from the α and β coefficients of Table VI. Moreover, the different kinematical variables of the two processes produce a D + n loop which is, for Λ c γ decay, a factor 2 larger than for Σ + c γ decay. The square of this two factors, together with the ratio of photon energies, which are E γ = 290 MeV for Λ c γ and E γ = 138 MeV for Σ + c γ, explain the factor 100 difference between the D + n contributions to both decays. We also note that there are constructive interferences between the most important contributions (D 0 p, D + n, D + s Λ) to Λ c γ decay, which enhance even further this rate compared to the Σ + c γ one.  In Table V we also compare our results to those obtained by other calculations performed within the relativistic quark model [55], using light-cone QCD sum-rules [61], or adopting a bound D * N picture for the Λ c (2595) [62]. We observe a large diversity of results. Note also that the two orders of magnitude difference between the radiative decays into Λ c γ or Σ c γ states found in the present work is not obtained by any of the other models displayed in Table V, nor by the HHCPT results of Ref. [60], which estimated partial widths of the same order of magnitude. Obviously, the tremendous differences between models calls for a measurement of these decay modes which could bring essential information about the nature of the Λ c (2595). It would also be interesting to see whether our predictions, based in a molecular picture for these resonances, are also very different from those obtained in quarkmodel theoretical approaches.

V. CONCLUSIONS AND OVERVIEW
We have studied the radiative decay width of charmed baryon resonances, which have been obtained as poles in the T -matrix of a coupled channels dynamical model and, therefore, they are interpreted as composite mesonbaryon molecular states.
We have obtained a sizable value for the one-photon radiative width of the Λ c (2595). This resonance decays radiatively mostly into Λ c γ states, with a partial rate of Γ Λc(2595)→Λcγ = 278 KeV, while only a tiny amount of the width, Γ Λc(2595)→Σ + c γ = 2 KeV, is due to the decay into Σ c γ final states. Our results are very different, in size and distribution among decay channels, to what is found by other approaches in the literature.
We have also presented predictions for the radiative decay of other excited charmed baryons. The radiative decay widths of the Σ c (2800) and Ξ c (2970) resonances are found to be relatively small, of the order of a few tenths of KeV. However, the transitions Ξ c (2790) + → Ξ + c γ and Ξ c (2790) 0 → Ξ 0 c γ are substantially larger, worth exploring experimentally.
The sizable value of some widths, especially those of the Λ(2595) and the Ξ c (2790) resonances, makes the study of these radiative reactions a very useful tool to obtain information about the characteristics of these charmed baryons.