A covariant model for the decuplet to octet Dalitz decays

In the last years it became possible to measure in HADES the dilepton decays of several baryons. The baryon dilepton decays provide information about the electromagnetic structure of the baryons in the timelike region. In the present work, we study the $B^\prime \to e^+ e^- B$ decays, where $B^\prime$ is a baryon decuplet member and $B$ is a baryon octet member. Our calculations are based on the covariant spectator quark model, where the contribution of the quark core is complemented with an $SU(3)$ contribution from the pion cloud. The pion cloud contribution prove to be relevant in the range of study. We present predictions for the $\Sigma^0 (1385) \to e^+ e^- \Lambda(1116)$ and $\Sigma^{+}(1385) \to e^+ e^- \Sigma^+(1193)$ decays, which may be tested at HADES in a near future. Predictions for the remaining decuplet baryon Dalitz decays are also presented. We conclude that different orders of magnitudes are expected for the baryon decuplet Dalitz decay widths, according to the quark content of the baryons. We also conclude that the dependence of the transition form factors on the square momentum transfer ($q^2$) is important for some transitions.


I. INTRODUCTION
In the last decades, there was a significant progress in the study of the electromagnetic structure of the nucleon (N ) and nucleon excitations (N * ) [1][2][3]. Most of the measured data were obtained through the scattering of electrons on nucleon targets (e − N → e − N * ), which probes the region where the square four-momentum transfer q 2 is negative (q 2 < 0), also known as the spacelike region. In the electron scattering experiments the analysis of the data is based on the γ * N → N * transition, where the spacelike virtual photon is produced by the incoming electron, and the γ * N → N * transition form factors are extracted from the experimental cross sections. Experiments based on electron-nucleon scattering have been performed in facilities such as Jefferson Lab, MIT-Bates, ELSA, MAMI among others, to probe the electromagnetic structure of N * states in the first three resonance regions [1][2][3][4][5][6].
The electron scattering technique can also be used to probe the electromagnetic structure of the hyperons (baryons with strange quarks) based on the γ * B → B ′ transitions, where B and B ′ are generic hyperons. In practice, however, the technique is almost exclusively limited to nucleon targets, since hyperons targets are difficult to produce due to their short lifetime, except in the limit q 2 = 0. In that limit, there are measurements of magnetic moments of a few hyperons and some magnetic transition moments [6][7][8][9][10][11][12][13]. Another limitation of the electron scattering technique is that it is restricted to the q 2 ≤ 0 region.
Another timelike sub-region, not discussed in the present work, is the region probed by e + e − and pp collisions at BaBar, BES-III, CLEO and PANDA/FAIR [31][32][33][34][35], which access the baryon (B) elastic form factors when q 2 ≥ 4M 2 B , where M B is the baryon mass [7,36]. HADES provides a unique opportunity to explore the electromagnetic structure of baryons based on the B ′ → γ * B transitions, where B ′ and B are generic baryons, through the dilepton decays (B ′ → e + e − B) [14,15,18,[20][21][22][23]. Different from the traditional electron-nucleon scattering, at HADES one can probe the electromagnetic structure of the hyperons in the kinematic region 4m 2 e ≤ q 2 ≤ (M B ′ − M B ) 2 , where M B ′ , M B are the baryon masses [23,30,37]. Measurements of strangeness production are possible due to the large acceptance and excellent particle identification, including dileptons in the final state [17]. In progress are feasibility studies on the Σ(1385), Λ(1404) and Λ(1520) Dalitz decays by the HADES collaboration [37][38][39]. Those studies suggest that those decays can be measured at GSI in the next few years and subsequently also at FAIR [18,20,37].
From the theoretical side there are not many models available for baryon electromagnetic transitions in the timelike region [19,[40][41][42][43][44]. An important constraint on those models is that the transition between the spacelike region and the timelike region (interval between q 2 = 0 and q 2 = 4m 2 e ) must be smooth [18,21,30]. There are a few theoretical issues, which need to be discussed: What happens in the transition between the spacelike region (q 2 ≤ 0), where the transition form factors are real functions and the region q 2 > 4m 2 π , where the transition form factors became complex functions (m π is the pion mass). How important are the physical poles associated with the meson resonances. How significant is the q 2 dependence of the form factors, and how are form factors modified near the pseudothreshold [15,24,[45][46][47][48].
From the analysis of the spacelike data, one can conclude that models based strictly on the quark degrees of freedom are insufficient to explain the measured transition form factors. The effects associated with the meson cloud dressing of the bare cores are crucial to describe the data in the region 0 ≥ q 2 > −2 GeV 2 , as demonstrated already for the ∆(1332) [1][2][3][4][49][50][51]. Our model for the ∆(1232) Dalitz decay [25], which describe the HADES data [15], corroborates also the importance of the of pion cloud for the γ * N → ∆(1232) transition in the timelike region, as in the spacelike region. There is therefore a great interest in studying the roles of the valence quark and meson cloud effects in the timelike region [3,[25][26][27]52].
Motivated by the experiments planned for HADES, in the present work we focus on the B ′ → γ * B transitions, where B ′ is a baryon decuplet member and B is a baryon octet member (decuplet baryon decays). We restrict for now our study to baryon systems that best fit an SU (3) quark model classification (baryon octet and baryon decuplet). Our calculations are based on the covariant spectator quark model [3,53,54] developed previously for the γ * B → B ′ transitions in the spacelike region [13].
The covariant spectator quark model provides an alternative to valence quark models which do not take into account meson cloud excitations of the bare cores, and simplified vector meson dominance (VMD) models [19,[42][43][44], which do not take into account the underlying quark substructure of the baryons. The formalism has been used in the study of the electromagnetic and the axial structure of the nucleon, several nucleon excitations, and hyperons [7-11, 25-27, 49-64].
The covariant spectator quark model of the γ * B → B ′ transition [13] is extended in the present work to the timelike region. Within the formalism, the octet baryon to decuplet baryon electromagnetic transitions are dominated by the magnetic transition form factor [3,53,54], which can be decomposed into valence quark and meson cloud contributions [12,13]. The meson cloud contribution is calculated from a microscopic pion-baryon model, calibrated by the γ * N → ∆(1232) transition, and extended to the octet baryon to decuplet baryon electromagnetic transitions [12,13,25].
We use our formalism to estimate the baryon decuplet Dalitz decay widths in terms of the square invariant mass of the dilepton pair q 2 , and the square invariant energy W 2 of the γ * B system [14,24]. We present, in particular, predictions for the Σ 0 (1385) → e + e − Λ(1116) and Σ + (1385) → e + e − Σ + (1193) decays, which may be tested by future HADES experiments [18,37]. As for the remaining decays, we estimate that the magnitudes of the Ξ 0 (1530) → e + e − Ξ 0 (1318) and Σ + (1385) → e + e − Σ + (1193) decay widths are comparable to the magnitude of the ∆(1232) → e + e − N decay width, as suggested by SU (3) and U -spin estimates [13,65]. We present also calculations for the radiative decay widths in terms of the invariant mass W , and compare our estimates with the available data. We conclude also that the radiative decays; Σ − (1385) → γ Σ − (1193) and Ξ 0 (1530) → γ Ξ 0 (1318), which are unknown at the moment, are close to the present experimental limits, and may be measured in a near future.
This article is organized as follows: In the next section, we review the formalism associated with the radiative and Dalitz decays of 3/2 + baryons into 1/2 + baryons. The covariant spectator quark model is discussed in Sec. III, where we present also numerical results for the transition form factors. Our results for the radiative and Dalitz decays of the decuplet baryons B ′ are presented and discussed in Sec. IV. The outlook and conclusions are presented in Sec. V.
We assume that B ′ is a member of the baryon decuplet (state 3 2 + ) and that B a is a member of the baryon octet (state 1 2 + ). Both baryons have positive parity. As before, M B ′ and M B represent the mass of B ′ and B, respectively.
The Dalitz decay of the baryon B ′ is determined by the function Γ γ * B (q, W ), where W is the energy of the resonance B ′ , q = q 2 and q 2 is the virtual photon (γ * ) square four-momentum. The baryon B ′ Dalitz decay is the consequence of the decay of the timelike virtual photon into a pair of electrons (γ * → e + e − ).
The photon decay width is defined by the limit q 2 = 0 [24,67] Γ γB (W ) = Γ γ * B (0, W ). (2.4) The Dalitz decay width Γ e + e − B (W ), it is determined by integrating according to In the previous equation the interval of integration 4m 2 e ≤ q 2 ≤ (W − M B ) 2 is the consequence of the threshold of the dilepton production and the maximum value of the photon square four-momentum allowed by the B ′ → γ * B decay: q 2 = (W − M B ) 2 . This is the value of q 2 obtained when the photon three-momentum vanishes |q| = 0 [45][46][47]52]. The function Γ ′ e + e − B (q, W ) can be determined [15,24,25,67] by The relations (2.4), (2.6) and (2.7) demonstrate that the decay widths Γ γB (W ) and Γ e + e − B (W ) are determined, once one has a model for the effective form factor |G T (q 2 , W )|. Note, however, that the model should be defined for arbitrary values of W (invariant energy of the γ * B system), since the measurements are performed for values of W which may differ from the decuplet baryon mass (M B ′ ). Our model for |G T (q 2 , W )| is described in the next section.
The baryon B ′ radiative decay (B ′ → γ B) measured in the experiments, correspond to the result from Eq. (2.1) in the limits W = M B ′ and q 2 = 0: (2.8)

III. COVARIANT SPECTATOR QUARK MODEL
In the present section, we describe the formalism associated with the covariant spectator quark model [3,53,54]. The covariant spectator quark model was derived from the covariant spectator theory [53,70]. In this framework a baryon is described as a three-constituent quark system, where a quark is free to interact with the electromagnetic fields. Integrating over the internal degrees of freedom of the non-interacting quarkpair, one reduces the three-quark system to a quarkdiquark system where the spectator quark-pair is represented by an on-mass-shell diquark with an effective mass m D [53,54,56]. In the electromagnetic interaction with the quarks, we take into account the structure associated with gluon and quark-antiquark dressing of the quarks. This structure is parametrized in terms of constituent quark electromagnetic form factors [53,54].

A. Formalism
In the covariant spectator quark model the baryon wave functions Ψ B (P, k) depend on the baryon (P ) and diquark (k) momenta, as well as the flavor and spin projection indices. Those wave functions are constructed conveniently by the symmetrized states of the diquark (12), and the off-mass-shell quark (3) [53,54,56]. The wave function for the octet and decuplet baryons are presented in Refs. [10,12,54].
The radial wave functions ψ B (P, k) can be parametrized in term of the variable The representation of ψ B (P, k) in terms of the single variable χ B is possible because the baryon B and the diquark are both on-mass-shell [53]. The γ * B → B ′ transition current in relativistic impulse approximation takes the form [53,54,56] where P + (P − ) is the final (initial) baryon momentum, k is the diquark momentum (on-mass-shell), and j µ q (q 2 ) is the quark current operator, depending on momentum transfer q = P + − P − [3,49,53]. The integration symbol represents the covariant integration in k, and the sum is over the diquark polarization states, including the scalar and vector components. The factor 3 takes into account the sum in the quarks based on the wave function symmetry.
The quark current j µ q , where q = u, d, s, includes the electromagnetic structure of the constituent quark (gluon and quark-antiquark dressing effects) [53,54]. The quark current operator is represented in the form [54] where j i (i = 1, 2) are the Dirac and Pauli flavor operators, acting on the third quark component of the wave function, and M N is the nucleon mass, as before.
The components of the quark current j i (i = 1, 2) can be decomposed as the sum of operators where are the flavor operators. These operators act on the quark wave function in flavor space, q = ( u d s ) T . The functions f i+ , f i− (i = 1, 2) represent the quark isoscalar and isovector form factors, respectively, based on the combinations of the quarks u and d. The functions f i0 (i = 1, 2) represent the structure associated with the strange quark.
The explicit form for the quark form factors is included in Appendix A. For the present discussion, the relevant part is that the quark form factors are represented in terms the vector meson mass poles associated with the mesons ρ, ω and φ depending of the type (l = ±, 0). The expressions of the quarks form factors are valid for the spacelike and timelike regions. In the timelike region, however, the vector mass poles are corrected with a finite decay width. The isovector transitions, like γ * N → ∆(1232) and γ * Λ(1116) → Σ 0 (1385), depend on the isovector form factors (meson ρ). Other transitions depend on a combination of isovector, isoscalar and strange quark form factors.
We consider here the covariant spectator quark model for the γ * B → B ′ transition from Refs. [10,12,13]. We assume in first approximation that the octet baryon (Ψ B ) and the decuplet baryon (Ψ B ′ ) wave functions are both described by the dominant S-wave quark-diquark configuration. The explicit expressions are presented in Refs. [9,10,12,54]. In the S-wave approximation, the transition is dominated by the magnetic dipole form factor, G M as in the γ * N → ∆(1232) transition When we take into account the pion cloud effects, one can decompose G M into two components [25,49,64]: where G B M represent the contribution from the threequark core (bare contribution) and G π M represent the contribution from the pion cloud. In the previous equation, we use q 2 = −Q 2 to convert the spacelike relations for G B M and G π M to the timelike region, and use W to generalize the dependence of the form factor on the resonance mass (M B ′ in the spacelike expressions). We omit the indices B and B ′ in the form factors for simplicity. In some octet baryon to decuplet baryon electromagnetic transitions, the contributions of the kaon cloud may be also considered. For a discussion of the magnitude of the kaon cloud contributions check Ref. [13].
It is worth noticing that the dominance of the magnetic dipole form factor is an approximation, and a consequence of the S-wave quark-diquark structure. In the case of the γ * N → ∆(1232) transition there is evidence that the quadrupole form factors G E and G C may have significant pion cloud contributions [46][47][48]. The contributions of those form factors to |G T (q 2 , W )| from Eq. (2.3) are, however, not significant, since G E is very small and G C is suppressed for small q 2 .
The valence quark contribution G B M and the pion cloud contribution G π M are discussed in the two next subsections. The numerical results for transition form factors are presented afterwards. We anticipate here that as in the case of the γ * N → ∆(1232) transition, the pion cloud/meson cloud contributions are relevant for the description of the γ * B → B ′ transitions.

B. Valence quark contributions
The contributions from the valence quarks to the octet baryon to decuplet baryon electromagnetic form factors (γ * B → B ′ ) were calculated in previous works. The expression for the magnetic form factor can be written 2) used to calculate the valence quark contributions for the transition form factors. The label γ * N → ∆ includes the γ * p → ∆ + and γ * n → ∆ 0 transitions (n is the neutron).

as [12]
where is the overlap integral of the octet baryon and decuplet baryon radial wave functions, and The functions j S i represent the projection of the flavor operators into the flavor components of the decuplet baryon and the symmetric component of the octet baryon flavor state [12]. The explicit expressions in terms of the quark form factors are presented in Table I.
In Table I and along the draft, we use the asterisk ( * ) to represent the excited states of Σ and Ξ, members of the baryon decuplet. The label γ * N → ∆ includes the γ * p → ∆ + and γ * n → ∆ 0 transitions (n is the neutron).
The overlap integral (3.8) is invariant and can be evaluated in any frame. For convinience we use the baryon The spacelike region, q 2 ≤ 0, is characterized by |q| ≥ |q| 0 and the timelike region, In the calculations, we use the experimental masses M N = 0.939 GeV, M Λ = 1.116 GeV, M Σ = 1.192 GeV and M Ξ = 1.318 GeV, for the baryon octet. As before, W represents the decuplet baryon masses. In the calculations associated with the physical decuplet baryon, we use the physical masses: M ∆ = 1.232 GeV, M Σ * = 1.385 GeV, and M Ξ * = 1.533 GeV.
The octet baryon radial wave functions take the form proposed on Refs. [9,10] for the study of the octet baryon electromagnetic form factors , (3.14) where N B are normalization constants and β i (i = 1, 2, 3, 4) are square momentum-range parameters in units M B m D . The parameters determined in Ref. [9], are β 1 = 0.0532, β 2 = 0.809, β 2 = 0.603 and β 2 = 0.381. This parametrization reflects the natural order for the size of the baryon cores β 2 > β 3 > β 4 . As for the decuplet baryon, we use the parametrization from Ref. [54] where N B ′ are normalization constants and α i (i = 1, 2) are square momentum-range parameters in units M B m D . In the present case the power associated with the factors in α 1 and α 2 is related with the number of strange quarks (0, 1 or 2). The radial wave function of the Ω − , unnecessary for the present study, can be found in Ref. [54]. In the calculations we use the values determined in the study of the decuplet baryon electromagnetic form factors α 1 = 0.3366 and α 2 = 0.1630 [54].
The normalization constants are determined by the conditions The parametrizations of the octet baryon and decuplet baryon radial wave functions were obtained from fits to the lattice QCD simulations of the electromagnetic form factors for pion masses larger than 400 MeV (small meson cloud contributions) [10,54,71,72] . The estimates of the valence quark contributions to the octet baryon and decuplet baryon elastic form factors are extrapolated to the physical regime using our extension of the model from the lattice to the physical case. Details of the procedure can be found in Refs. [9,10,51,54,64].
Our estimates for the γ * N → ∆(1232) transition form factors compare very well with the lattice QCD simulations with the corresponding pion masses [51]. Our results are also consistent with the bare core estimates from the EBAC model [25,51]. The EBAC model is a meson-baryon coupled-channel dynamical model where the meson-baryon couplings are calibrated by the pion electro-production data and photo-production data [4,73]. The contributions of the bare core are obtained when we set the meson-baryon coupling to zero [73].
Based on the results for the γ * N → ∆(1232) for the lattice QCD regime, where meson cloud effects are negligible, and on the comparison with the EBAC results at the physical point, one can conclude that the calibration of the valence quark degrees of freedom is under control [13]. Our parametrizations of the pion cloud contributions, discussed below, are inferred from the comparison between the extrapolation to the physical limit and the physical data [3,25,50].
A final note about the global normalization of the wave functions is in order. The wave functions associated to the baryon decuplet are normalized properly because the decuplet baryons are described by a model where we neglect the pion cloud contributions. As for the baryon octet, the normalization of the valence quark component is modified due to the inclusion of the pion cloud component. We note, however, that this correction only affects G B M and that, due to the magnitude of the normalization constant and the relative contribution from the valence quark contributions, the normalization effects can be estimated as 3% at most. One concludes, then, that in a first approximation, we can ignore the normalization correction due to the pion cloud dressing.

C. Pion cloud contributions
The pion cloud contribution to the γ * B ′ → B transition are estimated by the SU (3) extension of our pion cloud model for the γ * N → ∆(1232) transition [49][50][51]. We use, in particular the results of Ref. [13], where the meson cloud contributions of the diagrams of Fig. 1 are determined explicitly in the limit q 2 = 0. The calculations of the meson cloud loops are based on the cloudy bag model [74][75][76]. The explicit calculations use the meson-baryon couplings for the possible octet baryon and decuplet baryon intermediate states from Fig. 1. The connection with the quark microscopic properties between the covariant spectator quark model and cloudy bag model is performed matching the Dirac and Pauli couplings. In addition to the pion, we considered also the contributions of the kaon and the eta [13]. The eta contributions prove to be very small.
In the present work, we consider the simplest approximation, taking into account only the pion cloud contributions, and drooping the kaon cloud contributions, since the extrapolation of the pion cloud contributions to finite q 2 , based on the results of the γ * N → ∆(1232) transition is straightforward.
The generalization of the pion cloud contributions to the timelike region follows the lines of our work for the γ * N → ∆(1232) [25]. We represent then where G πa M (0) and G πb M (0) are the pion contributions for the diagrams (a) and (b), respectively, F π is the pion electromagnetic form factor, Λ 2 π = 1.53 GeV 2 , andG D is a generalization of the traditional dipole form factor. The coefficients G πa M (0) and G πb M (0) are presented in Table II. In Eq. (3.19) we omit the dependence on W , since the coefficients G πa M (0) and G πb M (0) are determined in the physical limit (W = M B ′ ).
The extension of the model with the inclusion of the kaon cloud will require the generalization of the two terms from Eq. (3.19) to the case of the kaon. This non trivial generalization is planed for a future work.
In the last column of Table II, we include for convenience the bare contribution G B M (0, M B ′ ) to the magnetic form factor at q 2 = 0. The relative magnitude of the pion cloud at q 2 = 0 can then be estimated by

D. Transition form factors
We now discuss the results for the transition form factor associated to Eqs. (3.6), (3.7) and (3.19). We present the results for |G M |, because although the form factor became complex in the timelike region (q 2 > 0), only the magnitude of G M is relevant for the radiative and Dalitz decays. In the spacelike region G M is real.
The numerical results for |G M |, for several values of W near the physical mass M B ′ are presented in Fig. 2 by the thick lines. In addition to |G M |, we include also the result of the valence quark contribution |G B M | (thin lines) and the absolute values of the pion cloud contribution (dotted line), according to Eq. (3.19). The line associated to the pion cloud corresponds, in fact, to the estimate associated with the largest value of W . The remaining cases have the same shape, except that the estimates are limited to q 2 ≤ (W − M B ) 2 .
The data included in the graph represent the magnitude of the experimental magnetic form factors for q 2 = 0, estimated from the radiative decay width data. The G M (0) data is discussed in more detail in the next section. The experimental values for |G M (0)| are important to infer the accuracy of the constant form factor model. The model associated with the constant form factor corresponds to a horizontal line with the magnitude of the experimental value for |G M (0)|.
One can notice that the model estimates for the ∆ → γ * N and Σ * 0 → γ * Λ decays have the magnitude comparable with the data. In the case of the Σ * + → γ * Σ + decay the model underestimate clearly the data. This underestimation is in part the consequence of neglecting the kaon cloud contributions. When those effects are taken into account one obtain G M (0) = 3.22, only 1.5 standard deviations from below the data [13].
In Fig 2, one can observe the dependence of the transition form factors on the variable W . In general, for a fixed value of q 2 the magnitude of G M decreases with W , as a consequence of our analytic expressions for G B M . This W -dependence was tested in our calculations in the lattice QCD regime, where the masses of the baryons and mesons are larger that the physical ones [3,51,64]. The W -dependence of our results is an important characteristic of our formalism, which has an impact on the calculation of the radiative and the Dalitz decay widths in terms of W , presented in the Sec. IV C.
In this aspect the present model is distinct of other models, like the constant form factor model and some VMD models [43]. The Iachello-Wan model [24,44] includes only a weak W -dependence on the transition form factors.
In the graphs, the spacelike results for G M are equivalent to the results presented in the graph for |G M |, in most cases, since G M (0) > 0. The exceptions are the Σ * − → γ * Σ − and Ξ * − → γ * Ξ − decays, where G M (0) < 0, according to the estimates from Ref. [13]. Our numerical values for G M (0) are presented in the next section (see Table III).
The results for the ∆ → γ * N form factors are almost identical to the results from Ref. [25], except that in the previous work we use the approximation G πa M (0) = G πb M (0) = 1 2 G π M (0) (pion cloud contributions equally divided between the two pion cloud processes from Fig. 1). The results for the ∆ → γ * N form factors are interesting because there is a deeper penetration in the timelike region due the large values of the upper limit (W − M B ) 2 , where B is the nucleon. For larger values of W one can notice that the valence quark contribution line became more flat. For larger values of W one can also observe the enhancement of |G M | for large q 2 , a direct consequence of the pion cloud contribution regulated by Eqs. (3.19) and (3.20), characterized by the peak of F π (q 2 ) near q 2 ≃ α ≈ m 2 ρ . The first detailed study of the ∆(1232) Dalitz decay at HADES suggests that the constant form factor model is insufficient to describe the data and that the signature of the form factor dependence on q 2 is present in the data [15].
In the graphs for the Σ * − → γ * Σ − and Ξ * 0 → γ * Ξ 0 transitions, one can notice nodes in both |G M | and |G B M | in the spacelike region. Those nodes are a consequence of zeros of G M due to a sign change. Since both form factors are negative near q 2 = 0, the nodes indicate the point where the functions became negative. The zero crossings are the consequence of the sign change of the valence quark contributions, according to our SU (3)  parametrization of the quark form factors (see Table I).
Similar results were also obtained in a previous study based on the covariant spectator quark model [12], with a not so general description of the pion cloud contributions. The Σ * − → γ * Σ − and Ξ * 0 → γ * Ξ 0 transitions are the transitions with smaller valence quark contributions. This result is also a consequence of our approximated SU (3) symmetry. In the exact SU (3) limit the form factors f i+ , f i− and f is are undistinguished and the valence quark contribution vanishes because j S i ≡ 0, according to Table I In the next section, we study the impact of our model for the transition form factors on the radiative and Dalitz decay widths.

IV. RADIATIVE AND DALITZ DECAY WIDTHS
We present here our estimates for the B ′ radiative and Dalitz decay widths. We start with the radiative decays at the pole: Γ γB (M B ′ ). Latter on, we discuss the functions d dq Γ e + e − B (q, W ), Γ e + e − B (W ) and Γ γB (W ).

A. Electromagnetic decay widths
Using the dominance of the magnetic dipole form factor, we can write [9,10,43] In Table III, we present the model estimates for G M (0) and Γ ≡ Γ γB in the second and fifth columns, and compare those estimates with the experimental data [6,[77][78][79][80][81], in the fourth and sixth columns. |G M (0)| exp is determined from Γ exp using Eq. (4.1). The numerical results were calculated in Ref. [13]. For the decays for which there are no data, we include the experimental estimate of the upper limit. The estimate of the third column, G M (0)| π , correspond to the calculation which exclude the kaon cloud contribution (only pion cloud), as in Sec. III D.
As discussed in the previous section, our estimate of G M (0), given by G M (0)| π , is consistent with the data for the ∆ → γN and Σ * + → γΛ decays, and underestimates the result for the Σ * + → γΣ + decay. A detailed comparison between model estimates and experimental data can be found in Refs. [12,13].
On Table III, one can notice that the experimental limits for the Σ * − and Ξ * 0 decays are close to our model estimates. One can conclude then that there is some hope that those decay widths can be measured in a near future.

B. Dalitz decay rates
The results for the Dalitz decay rates are presented in Fig. 3, for all the decuplet baryon decays, for several values of W . We include the labels B ′ → e + e − B in order to identify the decaying decuplet baryon. Recall that Σ 0 (1385) decay on Λ(1116) and on Σ 0 (1193).
The thick solid lines indicate the final result: the combination of valence quark and pion cloud contributions. The thin lines indicate the valence quark contributions (when we drop the pion cloud contributions).
The dashed lines indicate the result of the constant form factor model, obtained when we consider: G M ≡ G M (0), also known as QED estimate. To represent the QED estimate, we consider the following convention: • In cases where experimental data exist (∆ → γ N , Σ * 0 → γ Λ and Σ * + → γ Σ + ), we use the magnetic form factor determined by the electromagnetic decay width (see Table III • In the remaining cases, we use our best estimate given by the results from Table III, corresponding to the value of G M (0) which include the pion and kaon clouds (second column).
For the discussion of the q 2 -dependence of our model, we include also the model estimate of the Dalitz decay when we replace G M (q 2 ), by G M (0)| π . The results are represented by the dotted lines. In the case of the ∆ → e + e − N decay we ommit this estimate because it overlaps the estimate of the constant form factor model (dashed line).
The difference of magnitude between the valence quark contribution and the total is a consequence of the relative magnitude of those estimates for the transition form factors. We recall that based on the estimates from Ref. [13], also presented in Table II, the valence quark contributions to the transition form factors are about 55-70% of the total. One concludes, then, that when the pion cloud contribution are about 50% of the total, the bare estimates for d dq Γ e + e − B are about 1/4 of the total, since the decay widths are proportional to |G M | 2 . This rough estimate is valid for most decays. The main exceptions are the Σ * + → e + e − Σ + , Σ * 0 → e + e − Σ 0 and Ξ * 0 → e + e − Ξ 0 decays, where the relative contribution of the core is larger (smaller pion cloud contributions).
Concerning the comparison with the constant form factor model (dashed lines), one can conclude that the results are very close for the ∆ → e + e − N decay, for small values of W . This happens because our model is compatible with the experimental value for |G M (0)|, as discussed earlier. In the remaining cases, our result underestimates the constant form factor model. This underestimation is mainly a consequence of the non inclusion of the kaon cloud contribution in our q 2 -dependent estimates, in contrast with the constant form factor model. This underestimation was discussed in detail in Sec. III D for the Σ * + → γ * Σ + form factor.
The impact of the form factor dependence on q 2 can be inferred from the comparison between the exact estimate (thick solid line) and the dotted line. As anticipated in Sec. III D, the q 2 -dependence is more relevant for the Σ * 0 → e + e − Λ, Σ * − → e + e − Σ − and Ξ * − → e + e − Ξ − decays. The dominance of the exact result over the dotted line is clearly observed for large q 2 , particularly for large values of W .
In Fig. 4, we compare the magnitudes of the Σ * and Ξ * Dalitz decay rates at the mass poles. Note the similarity between the results for Σ * + /Ξ * 0 decays, as well as Σ * − /Ξ * − decays in the region of q 2 where they can be compared. These similarities are the consequence of the SU (3) symmetry structure of the covariant spectator quark model, combined with similar relative pion cloud contributions for the decays under discussion. The relations between the valence quark contributions, G B M , given by Eq. (3.7) are explained by their dependence on the functions j S i , which, according to Table I are identical in the cases Σ * + /Ξ * 0 and Σ * − /Ξ * − .
The similarities discussed above are also explained by the U -spin symmetry [13,65], which is valid to the valence quark component of the transition form factors. The U -spin symmetry, states that the decay transitions are similar when we replace a d-quark by a s-quark in the initial and final states [65].
The U -spin symmetry predicts also similar magnitudes for the ∆ → γ N and the Σ * 0 → γ Λ Dalitz decay rates [12,13]. This property, however, is not valid in the context of our model due to the difference of magnitudes of the pion cloud contributions (larger in the first case).
Our model is compatible with the U -spin symmetry, because it is based on an approximate SU (3) flavor symmetry. In the present case, the symmetry implies that the quark form factors associated with the u quark (combination of isovector and isoscalar components) and the s quark are similar at low q 2 . We recall, however, that the U -spin symmetry is valid only for the valence quark component of the transition. The covariant spectator quark model estimates provide then a more consistent description of the radiative and Dalitz decays.

C. Decay widths in terms of the invariant mass
The results for the radiative (B ′ → γB ) and Dalitz (B ′ → e + e − B) decay widths in terms of W are presented in Fig. 5, for all the decuplet baryon decays. The thick lines represent our estimates. The thin lines represent the estimates of the constant form factor model. We include also the data for Γ γB at the physical mass in the cases: ∆ → γ N , Σ * 0 → γ Λ and Σ * + → γΣ + , according to the results from Table III. In the cases Σ * 0 → γ Λ and Σ * + → γ Σ + , one can notice some underestimation of the data. This result is in part the consequence of including only the contribution of the pion cloud. The inclusion of the kaon cloud approaches the model estimate to the data, as can be inferred also from Table III (compare the second and the  third columns).
We choose to not include the kaon cloud contributions on the radiative decays, because our extension for finite q 2 is justified, at the moment, only for the pion cloud contribution. In general, the kaon cloud contributions are at most 20% of the pion cloud contributions (Table III), except for Ξ * 0 , where the effect of the kaon is about 25% of the pion cloud.
Concerning the comparison with the results of the   Regarding the inclusion of q 2 -dependent kaon cloud contributions, our expectation is that the slope associated with the kaon cloud contributions is larger than the slope associated with the pion cloud contributions near q 2 = 0, since the kaon cloud effects are more sup- pressed than the pion cloud effects in the spacelike region 2 . The consequence of this trend is that the transition form factors are expected to be enhanced in the timelike region with the inclusion of the kaon cloud contributions. We recall, however, that the kaon cloud contributions near q 2 = 0 are at most about 20% of the pion cloud contributions. There is then the possibility that the kaon cloud effects may not be very relevant in the region 0 < q 2 ≤ (W − M B ) 2 . Only more detailed calculations can determine how important may be the enhancement of the transition form factors due to the kaon cloud effects, in the timelike region.
Our estimates for the B ′ → e + e − B Dalitz decays underestimate, in general, the QED model. This tendency is a consequence of the results obtained for the Dalitz decay rates (Fig. 3), where the QED model overestimates, in general, the covariant spectator quark model. The exception is the ∆ → e + e − N decay, where oour model and the QED estimates are close.
It is worth noticing, that the only the ∆(1232) Dalitz decay was measured experimentally at the pole (W ≃ 1.232 GeV). Our estimate of the ∆(1232) Dalitz decay width is consistent with the result of HADES [15]. All estimates for the B ′ → e + e − B Dalitz decays at the physical decuplet baryon mass (M B ′ ) are presented in Table IV. Excluding the ∆(1232), the remaining estimates are predictions to be tested by future experiments. 2 This effect can be better understood assuming that the meson cloud contributions to the transition form factors can, near q 2 = 0, be simulated by a multipole function 1/(1 + Q 2 /Λ 2 ) n , where n > 2 is an integer (the exact value is not important for the discussion) and Λ 2 is a cutoff of Q 2 . One conclude, then, that the cutoff associated with the kaon cloud is smaller than the one associated to the pion, since the kaon cloud effects are suppressed more strongly than the pion cloud effects. The consequence of this relation between cutoffs is that the magnitude of the derivative of the kaon cloud multipole (∝ 1/Λ 2 ) at q 2 = 0 is larger than the magnitude of the derivative of the pion cloud multipole.

V. OUTLOOK AND CONCLUSIONS
The HADES facility provides a rare opportunity to study electromagnetic transitions between baryon states in the timelike region (q 2 > 0). Those experiments complement the information obtained from electro-production of baryon resonances in the spacelike region (q 2 ≤ 0). The recent and the up-coming results from HADES motivate the development of theoretical models for the γ * B → B ′ transition form factors in the timelike region, where B and B ′ are generic baryons.
Of particular interest are the Dalitz decays of baryons (B ′ → e + e − B), including hyperons. Measurements of the ∆(1232) Dalitz decays have been reported recently. The analysis of the Σ 0 (1385) → e + e − Λ(1116) decay is expected in a near future. Due to the capability of HADES to produce hyperons, other decuplet baryon Dalitz decays are expected to be measured in the following years. The next natural candidate, based on the estimated magnitude, is the Σ + (1385) → e + e − Σ + (1193) decay.
To complement the experimental activity at HADES, we present here model estimates for the Dalitz decay rates and Dalitz decay widths for all decuplet baryons. Our calculations are based on the covariant spectator quark model for the octet baryon to decuplet baryon electromagnetic transitions, extended in the present work to the timelike region. The model was previously calibrated by lattice QCD data for the baryon octet and baryon decuplet, and takes into account the pion cloud dressing of the baryon cores. The model is successful in the description of the radiative decays: ∆(1232) → γN , Σ 0 (1385) → γ Λ(1116) and Σ + (1385) → γ Σ + (1193). Under study is the extension of the present model with the inclusion of the kaon cloud contribution for finite q 2 , which may approach the model estimates to the data.
We conclude that, in general, the valence quark effects give the dominant contribution to the transition form factors and to the Dalitz decay widths, but that pion cloud contribution provides significant corrections, which improve the description of the data. In most cases, the pion cloud effects contribute with about 30-45% to the transition form factors near q 2 = 0. In some cases, those contributions are only about 20% (Σ + (1385), Σ 0 (1385) and Ξ 0 (1530) decays).
We also analyze the role of the q 2 -dependence of the form factors. We conclude that, in general, the QED approach (constant form factor model) is not a good approximation, as already observed in the case of the ∆(1232) Dalitz decay. The impact of the q 2 -dependence of the form factors is, however, less significant than in the case of the ∆(1232). The Σ 0 (1385) → γ * Λ(1116) transition form factors are enhanced in the timelike region due to the pion cloud effects. The q 2 -dependence is also relevant for the Σ − (1385) → e + e − Σ − (1193) and Ξ − (1530) → e + e − Ξ − (1318) decays.
The covariant spectator quark model proved also to be a useful framework to study Dalitz decays of nucleon excited states (N * ), more specifically in the cases of the ∆(1232) and N (1520) resonances [25,26]. Under study is the possibility of extending the formalism to other baryon systems, which may also be regarded as a combination of valence quark cores combined with meson cloud excitations of the baryon cores [27].
For the range of the calculation of the present work (W < 2 GeV) the regularization of the φ pole is not very relevant, since m 2 φ ≃ 1 GeV 2 ≫ q 2 . The singularities associated to the φ-meson appear, then only for W ≥ M B + m φ > 2.1 GeV. Nevertheless, we regularize the φ-propagator for consistence. We note also that even the calculations more dependent on the φ-pole, in particular the Dalitz decay widths Γ e + e − B (W ), are weakly dependent on the shape of Γ φ (q 2 ).