Electromagnetic transition form factors, $E2/M1$ and $C2/M1$ ratios of the baryon decuplet

We investigate the electromagnetic transition form factors of the baryon decuplet to the baryon octet, based on the self-consistent SU(3) chiral quark-soliton model, taking into account the effects of explicit breaking of flavor SU(3) symmetry. We emphasize the $Q^2$ dependence of the electromagnetic $N\to \Delta$ transition form factors and the ratios of $E2/M1$ and $C2/M1$ in comparison with the experimental and empirical data. In order to compare the present results of the electromagnetic transition form factors of the $N\to \Delta$ with those from lattice QCD, we evaluate the form factors with the pion mass deviated from its physical value. The results of the $E2/M1$ and $C2/M1$ ratios are in good agreement with the lattice data. We also present the results of the electromagnetic transition form factors for the decuplet to the octet transitions.


I. INTRODUCTION
Understanding how a baryon is shaped electromagnetically has been one of the most important issues in hadronic physics. The EM structure of the baryon decuplet, which contains the first excitations of the nucleon and hyperons, is far from complete understanding. The reason is that it is very difficult to get access to their structures experimentally on account of their ephemeral nature. On the other hand, the EM transition N γ * → ∆ can be examined experimentally by using the electroproduction of the pion [1 -10]. Moreover, the EM transitions from the baryon octet to the decuplet are experimentally accessible [11][12][13][14]. Thus, the structure of the baryon decuplet can be investigated by using the EM transition from the baryon octet to the decuplet. Theoretically, the EM transitions of the N γ * → ∆ have been extensively studied within various frameworks over decades: for example, the linear sigma model [15], the Skyrme models [16][17][18][19], the chiral bag model [20], the Dyson-Schwinger approaches [21][22][23], constituent quark models [24], relativistic quark models [25], QCD sum rules [26][27][28][29], a πN dynamical model [30], lattice QCD [31][32][33], AdS/QCD [34], chiral perturbation theory [35], and so on.
In the present work, we want to investigate the EM transition form factors and related observables from the baryon octet to the decuplet within the framework of the self-consistent SU(3) chiral quark-soliton model (χQSM). The EM transition N γ → ∆ was already studied in the χQSM [36][37][38][39][40]. In Ref. [40] the EM transition form factors for the N γ * → ∆ were computed without the effects of the flavor SU(3) symmetry. Moreover, the large N c argument was used to improve quantitatively the magnetic and quadrupole transition moments. However, it is known that the absolute magnitudes of the helicity amplitudes for the N γ → ∆ excitation based on the χQSM are underestimated, compared with the experimental and empirical data. The lattice data are also known to be quite smaller than the experimental data. Thus, in the present work, we emphasize the dependence of the form factors on the momentum transfer and the ratios of E2/M 1 and C2/M 1.
In the present work, we will compute all possible EM transitions from the baryon octet to the decuplet with the effects of flavor SU(3) symmetry breaking taken into account. Since radiative decays of the negative decuplet baryons vanish in the exact flavor SU(3) symmetric case due to the U -spin symmetry, it is of great importance to consider explicit breaking of flavor SU(3) symmetry. We first examine the N γ * → ∆ transition, because the lattice data as well as the experimental data exist. The existing lattice calculation [32] still uses unphysical pion mass, so that we employ the corresponding values of the pion mass to compare the numerical results with the lattice data. We then present the numerical results of all possible EM transition form factors from the baryon octet to the decuplet, focusing on the explicit breaking of flavor SU(3) symmetry. The results of the E2/M 1 and C2/M 1 ratios are compared with the experimental and lattice data.
The present work is organized as follows: In Section II, we briefly explain the definition of the EM transition form factors and helicity amplitudes. In Section III, we describe shortly how the expressions of the EM transition form factors are obtained. In Section IV, the numerical results are presented and are compared with the experimental data. We also discuss them in comparison with the lattice data. In the final Section, summary and conclusions are given.

II. EM TRANSITION FORM FACTORS FOR RADIATIVE EXCITATIONS B8γ * → B10
The radiative excitation from an octet baryon to a decuplet baryon, B 8 γ * → B 10 , is schematically shown in Fig. 1 in the rest frame of a member of the baryon decuplet. In this rest frame, p 10 , p 8 , and q, which are the momenta of a B 10 (p 10 ) FIG. 1. Schematic diagram for the γ * B8 → B10 transition decuplet baryon and an octet one, and the momentum transfer carried by the photon, are expressed respectively as where q and ω q denote the three-momentum and energy of the virtual photon. In the rest frame of the decuplet baryon, the energy-momentum relations are given as E 2 8 = M 2 8 + |q| 2 and E 2 10 = M 2 10 . Thus, the momentum and energy of the virtual photon are written as where Q 2 = −q 2 > 0.
To explain the radiative excitation B 8 γ * → B 10 , we need to evaluate the matrix element of the EM current V µ between B 10 and B 8 as follows as [41] B 10 (p 10 where the EM current V µ is defined as with the charge operator given byQ = diag(2/3, −1/3, −1/3). u β (p, s) and u(p, s) stand for Rarita-Schwinger and Dirac spinors, respectively. Γ βµ in Eq. (3) can be expressed in terms of the multipole form factors [41,42] where G * M 1 , G * E2 , and G * C2 are known respectively as the magnetic dipole (M 1) transition form factor, the electric quadrupole (E2) one, and the Coulomb quadrupole (C2) one. The corresponding Lorentz tensors K βµ M 1 are written as The Lorentz tensors are required to satisfy the identities q µ K βµ M 1,E2,C2 = 0 by conservation of the EM current. In describing the radiative excitations of the baryon decuplet, it is convenient to define two different thresholds, i.e. the physical threshold and the pseudothreshold, which are defined by q 2 = −Q 2 t ≡ (M 10 + M 8 ) 2 and q 2 = −Q 2 pt ≡ (M 10 − M 8 ) 2 , respectively. At these two thresholds, the electric quadrupole form factor can be identified as the Coulomb one [41] The transition magnetic moment 1 and the transition electric quadrupole moment are defined respectively by [43,44]: where µ N denotes the nuclear magneton defined by µ N = e/2M N . It is also of great interest to examine the helicity amplitudes, since they can be extracted from experimental data. The transverse and Coulomb helicity amplitudes are defined respectively in terms of the spatial and temporal components of the EM current 1 Note that the definition of the magnetic transition moment in the present work is different from that in Refs. [40], where the following approximation was used M 10 /M 8 ≈ 1 + O(N −2 c ). In the present work, we strictly follow the definition used in experiments.
The explicit expressions for the densities Q i , X , and M i can be found in Appendix A. The results on the matrix elements of collective operators B 10 |...|B 8 are given in Appendix B. I i and K i stand for the moments and anomalous moments of inertia [47]. m 1 and m 8 are defined as with the average mass of the up and down current quark, m, and the mass of the strange current quark, m s .
The expression of the electric quadrupole form factor is given as with the corresponding density G B8→B10 E2 (r) The explicit expressions of I 1E2 (r) and K 1E2 (r) can be found in Appendix A. The Coulomb quadrupole form factor G B8→B10 * C2 is written as where G B8→B10

C2
(r) is simply the same as G B8→B10 E2 (r). It is more convenient to decompose the densities into three different terms The first term represents the SU(3)-symmetric ones including both the leading and rotational 1/N c terms, the second one denotes the linear m s corrections arising from the current-quark mass term of the effective chiral action. The last term is originated from the collective wave functions. If the effects of the flavor SU(3) symmetry breaking are considered, a collective baryon wave function is not any longer in a pure state but a state mixed with higher representations, as shown in Eq. (B2). Thus, there are two different terms that provide the effects of flavor SU(3) symmetry breaking. The explicit expressions of these three terms are given for the magnetic dipole form factor Similarly, the densities for the electric quadupole form factors are written by The densities for the Coulomb quadrupole form factors are identical to those for the E2 form factors, i.e. G B8→B10 (r). While the leading-order term in the 1/N c expansion, which is expressed by Q 0 , contributes to the M 1 transition form factor, it vanishes for the E2 transition form factor because of the headgehog ansatz in the present approach. It indicates that the rotational 1/N c corrections take a role of the leading-order contribution. Moreover, we have only the single rotational 1/N c term, which contains the density I 1E2 (r). The corresponding expression can be found in Eq .A2 in Appendix A. Similarly, the C2 form factors does not get any contribution from the leading-order term.

IV. RESULTS AND DISCUSSION
The parameters in the χQSM except for the dynamical quark mass were already fixed by reproducing properties of the pion. Since the contributions of the sea quarks need to be regularized, the cutoff mass Λ should be introduced. This is fixed by reproducing the pion decay constant, f π = 93 MeV. the average mass of the up and down current quarks m is determined by the physical pion mass m π = 139 MeV. While the dynamical quark mass M can be considered as a free parameter, it is also fixed by describing the electric form factor of the proton. Once we fix all these parameters, we compute various observables including both the lowest-lying light and singly heavy baryons. Therefore, we do not have any room to fit the parameters in the present calculation.
In is already known from previous investigations [37,38,40] that the magnitudes of the EM transition form factors of the ∆ are rather underestimated, compared with the experimental data while the E2/M 1 and C2/M 1 ratios are well described. There are several reasons why it is so. In fact, the pion-loop effects come into essential play in explaining the nature of the ∆ isobar, since it decays strongly into the πN . Moreover, the ∆ isobar has a rather broad width, so that the corresponding wavefunction should contain such information arising from this broad width. In Ref. [48] the strong decay widths of the baryon decuplet were scrutinized based on the χQSM in a model-independent approach, where all the dynamical parameters were fixed by experimental data without calculating them self-consistently. While the strong decay width of the ∆ is still underestimated, the widths of all the other members of the baryon decuplet are in good agreement with the experimental data. It implies that one should go beyond the pion mean-field approaximation to describe the properties of the ∆ baryon. Since it is rather difficult to take into account the pion-loop corrections beyond the pion mean-field approximation in the present framework, we will consider the Q 2 dependence of the form factors and the ratios of E2/M 1 and C2/M 1. These effects beyond the mean fields may be cancelled in the calculation of these ratios. Note that the lattice calculations also have similar problems in reproducing the experimental data.
We first discuss the results of the N → ∆ EM transition form factors, focusing on the Q 2 dependence of the form factors. In order to compare the Q 2 dependence of the present results with the experimental and empirical data, we normalize the values of the form factors at Q 2 = 0.06 GeV 2 , using the experimental data on the helicity amplitudes by the A1 Collaboration [3]. We explicitly multiply the present values of the M 1, E2, and C2 form factors by 1.82, 3.13, and 3.18, respectively. The upper left panel of Fig. 2 draws the result of the magnetic dipole transition form factor as a function of Q 2 with the strange current quark mass taken to be m s = 180 MeV. We take the experimental and empirical data taken from Refs. [1, [3][4][5][6]. The present result seems to fall off slightly more slowly than those of the empirical and experimental data, as Q 2 increases. However, the general tendency of the result is in agreement with the data. In the upper right panel of Fig. 2 we show the result of the electric quadrupole form factor for the N → ∆ EM transition as a function of Q 2 . The result exhibits Q 2 dependence, which is different from that of the M 1 form factor. It falls off faster than the empirical and experimental data. In fact, this is a well-known problem. As with the strange current quark mass ms taken to be 180 MeV. In the upper left and right panels, we draw the results of the magnetic dipole and electric quarupole transition form factors respectively, whereas in the lower panel, we depict those of the Coulomb qudrupole form factor. They are compared with the experimental and empirical data. We normalize the numerical results by the experimental data at Q 2 = 0.06 GeV 2 , i.e. the M 1 form factors by 1.82, the E2 ones by 3.13, and C2 ones by 3.18. The solid curve illustrates the result of the EM transition form factors with ms = 180 MeV. The result is normalized by the data from Ref. [3], i.e. by the factor of 1.82. The red triangle denotes the data taken from Ref. [6], the black circles from Ref. [3], the brown square from Ref. [4], the blue squares from Ref.
we will discuss explicitly later, the present results of the E2/M 1 ratio deviates from the experimental data because of this Q 2 dependence of the E2 form factor. Actually, one can understand this Q 2 behavior of the present results.
Since the E2 transition form factor is proportional to ω q within this model expression, it is strongly suppressed when ω q (Q 2 ) = 0, which corresponds approximately to Q 2 0.6 GeV 2 for the N γ * → ∆ excitation. That explains why the E2 form factor decreases drastically as Q 2 increases. On the other hand, The result of the Coulomb form factor describes relatively well the experimental data, as shown in the lower panel of Fig. 2.
In order to compare the present results with the lattice data [33], we have to compute the EM transition form factors, employing the values of the unphysical pion mass, which were used by Ref. [33]. To do that, we have to derive the solutions of the pion mean fields with specific values of the unphysical pion mass, i.e. m π = 297 MeV and 353 MeV. Then we can compute the EM transition form factors, using these solutions with the values of m π . In fact, Goeke et al. [49] showed that the stable mean-field soliton still exists in the wide range of the pion mass 0 ≤ m π ≤ 1500 MeV. It indicates that the results from the χQSM can be directly compared with those from lattice QCD, the pion mass used in it being employed. They indeed described remarkably the mass of the nucleon in comparison with the lattice data. The same method was extended to the description of the energy-momentum form factors of the nucleon [50] [52][53][54][55]. While the general Q 2 dependence of results of the M 1 transition form factor is similar to those of the lattice calculation, the present results decrease still faster than the lattice ones as Q 2 increases. In the upper left panel of Fig. 3 we represent the results of the E2 transition form factors with the pion mass varied as in the case of the M 1 form factor. The sizes of the E2 form factor are drastically diminished by increasing the value of the pion mass. This tendency was already seen in the E2 form factors of the ∆ and Ω − in Ref. [46]. As shown in the upper left panel of Fig. 3, the present results with larger pion masses are in better agreement with the lattice data in the lower Q 2 region (Q 2 0.5 GeV 2 ). In the lower panel of Fig. 3, we depict the results of the C2 transition form factor. As the pion mass increases, the sizes of the C2 one decreases as in the case of the E2 form factor. However, the results get underestimated compared with the lattice data in the lower Q 2 region. The left and right panels of Fig. 4 show respectively the results of the E2/M 1 and C2/M 1 ratios for the N γ * ∆ excitation as functions of Q 2 , being compared with the experimental and empirical data [1, 4, 6-10] as well as those of the lattice calculation [33]. The results of E2/M 1 ratios are in qualitatively good agreement with the experimental data near Q 2 ≈ 0, the present ones fall off faster than the experimental and empirical data. This arises from the Q 2 dependence of the E2 transition form factors, of which the results decrease much faster than those of the M 1 form factors. On the other hand, the results of the C2/M 1 form factors are more or less in agreement with the data. Note that the lattice data on the C2/M 1 are underestimated in comparison with the experimental data.  [6], Skyrme model [19], Linear sigma model(LSM) [15], non-relativistic quark model(NQM) [24], QCD sum rule(QCDSR) [29], chiral constituent quark model(χCQM) [57] and chiral perturbation theory(χPT) [35].
The ratios R EM and R SM ratios for all the members of the baryon decuplet have not been much investigated. The R EM is an only known ratio experimentally [6,56]. Moreover, there are no experimental data and are few theoretical results on the C2/M 1 ratios for the whole baryon decuplet. In Table I, we list the numerical results of the E2/M 1 (R EM ) and C2/M 1 (R SM ) ratios at Q 2 = 0 in comparison with those from other models. The second and fourth columns list the results of the R EM without and with the effects of flavor SU(3) symmetry breaking. Comparing the results in these two columns with each other, we find that the contributions of the m s corrections seem to be not at all small. However, one has to keep in mind that the effects of flavor SU(3) symmetry breaking to the M 1 form factors are smaller than to the E2 and C2 form factors, as we will show later explicitly. Thus, the effects of flavor SU(3) symmetry breaking apparently look amplified. We want to mention that while the M 1, E2, and C2 form factors for the EM Σ − → Σ * − and Ξ − → Ξ * − transitions vanish in exact flavor SU(3) symmetry due to the U -spin symmetry, the ratios R EM and R CM do not vanish. The reason can be easily understood by examining Eqs. (20) and (23), which are the SU(3) symmetric leading contributions to the M 1 and E2 form factors, respectively. The matrix elements of the collective operators for both the M 1 and E2 form factors have basically the same structures, so that the ratios of these form factors are proportional to the ratios of the densities given in terms of Q 0 and so on. Therefore, even though form factors for the Σ − → Σ * − and Ξ − → Ξ * − photo-transitions vanish in exact flavor SU(3) symmetry, the ratios R EM and R CM turn out finite.
The present value of R EM for the N γ → ∆ is understimated by about 20 % in comparion with the experimental data. This discrepancy may be overcome by going beyond the pion mean-field approximation, as was hinted by the results of χPT. The results of R EM for all decuplet hyperons are comparable with those of chiral perturbation theory (χPT) [35] except for those of R EM for the Σ − γ → Σ * − and Ξ − γ → Ξ * − excitations. Interestingly, both channels are forbidden by the U -spin symmetry. On the other hand, the results from the chiral constituent quark model [57] are overall larger than the present ones. The results for the Σ − γ → Σ * − and Ξ − γ → Ξ * − transitions from the QCD sum rules [29] are very large, compared with those of the present work.
The EM transition form factors should comply with the U -spin symmetry in the exact flavor-SU(3) symmetric case. The U -spin symmetry is inherited in Eqs. (20) and (23) as it should be. The U -spin relations for the magnetic transition moments were given in Refs. [39,58]. In particular, the magnetic transition form factors for the negative charged decuplet baryons should vanish in exact flavor SU(3) symmetry, which one can easily see from Eqs. (20). Some years ago, the SELEX Collaboration measured the upper limit of the partial width for the radiative decay of Σ * − , which is given as Γ(Σ * − → Σ − γ) < 9.5 keV. It indicates that the corresponding magnetic transition moment should satisfy the upper limit |µ Σ * − Σ − | < 0.82 µ N [39]. Thus, the experimental data can provide a clue as to how much the U -spin symmetry is broken in the case of the EM transitions for the baryon decuplet. Figure 5 draws the results of the magnetic dipole transition form factors without and with the effects of flavor SU(3) symmetric breaking. The solid curves depict those with the linear m s corrections whereas the dashed ones exhibit those in the exact SU(3) symmetric case. One can see that the effects of flavor SU(3) symmetry breaking contribute to the M 1 form factors in general below 10 %, which is in agreement with the quark-model prediction [59]. Note that they have almost a negligible contribution to the Ξγ * → Ξ * transition. However, when it comes to the EM transitions for the negatively charged decuplet hyperons, the linear m s terms take a leading role, since the flavor-SU(3) symmetric contributions vanish because of the U -spin symmetry. The magnitudes of these forbidden transition form factors lie below the upper limit imposed by the SELEX experiment. In Figs. 6 and 7, we draw respectively the E2 and C2 transition form factors of all the hyperons of the baryon decuplet. Since the densities for the E2 and C2 form factors are in fact the same each other, the general behaviors of these form factors are very similar, as shown in Figs. 6 and 7. Except for the N γ * → ∆ transition, the effects of flavor SU(3) symmetry breaking are noticeable. In particular, when it comes to the Σ 0 γ → Σ * 0 transition, the contributions of flavor SU(3) symmetry breaking are of almost the same order as the SU(3)-symmetric term. However, as we mentioned already in the previous Section, the E2 and C2 transition form factors do not have any leading-order contributions. It means that the rotational 1/N c correction plays a role of the leading-order contribution. While the linear m s corrections should be usually smaller than the leading-order contributions as in the case of the M 1 transition form factor, they become rather important when the leading-order contributions vanish. A typical example can be found in the calculation of the singlet axialvector charge [60] for which the leading-order contribution disappears too. Thus, the linear m s corrections come into significant play, when the E2 and C2 transition form factors are discussed.
As already mentioned previously, the magnitudes of the radiative decay rates for the baryon decuplet based on the χQSM are quite underestimated. However, their ratios are still interesting. For example, the results of the ratios for some decay widths are given as Thus, the present results for these ratios are in qualitative agreement with the data.

V. SUMMARY AND CONCLUSIONS
In the present work, we aimed at investigating the electromagnetic transition form factors of the baryon decuplet within the framework of the self-consistent SU (3)  form factors fall off faster than the data. In order to compare the present results of the form factors with the lattice data, we employed the values of the unphysical pion mass, i.e. m π = 297 MeV and 353 MeV, so that we are directly able to compare the results with the lattice data. In addition, we normalized the present results with the lattice data at Q 2 = 0.06 GeV 2 such that we can see how the Q 2 dependences are different from those of the lattice calculation. The form factors fall off more slowly as the pion mass increases. Moreover, the magnitudes of the E2 and C2 form factors are much reduced by using the unphysical pion masses. We then computed the E2/M 1 and C2/M 1 ratios as functions of Q 2 . The E2/M 1 ratios fall off faster than the experimental data. On the other hand, the results of the the Q 2 dependence of the C2/M 1 ratios are in good agreement with the experimental data.
We then presented the results of the E2/M 1 and C2/M 1 ratios at Q 2 = 0. There exists an experimental data only on the E2/M 1 ratio for the EM N → ∆ transition. The comparison of the present result with the data shows around 20 % deviation from it. We then examined the effects of flavor SU(3) symmetry breaking on the electromagnetic transition form factors of the decuplet hyperons. While they are rather marginal on the magnetic dipole transition form factors, they play an important role in describing the E2 and C2 transition form factors. The reason is that the leading-order contributions vanish for the E2 and C2 transtion form factors. Thus, the rotational 1/N c and linear m s corrections are equally important.
While the present results of the EM transition form factors for the baryon decuplet describe well the Q 2 denpendence, the magnitudes of the N γ * → ∆ form factors are still underestimated, compared with the experimental data. This is already a well-known feature of the χQSM. There is in fact a way of improving the present work. One can combine the Q 2 behavior of the form factors obtained by the present work with the magnetic transition moments evaluated in a model independent approach [39]. In principle, the quadrupole transition moments can be determined in a similar way. Actually, it is of great importance to describe the eletromagnetic transition form factors of the baryon decuplet, since all the determined dynamical parameters can be employed when we compute the strangeness-changing transitions. While there is no experimental information on semileptonic decays of the baryon decuplet except for the Ω − , they are still very important in determining the strong vector and tensor coupling constants for the baryon decuplet and the octet to the vector meson vertices through the Goldberger-Treiman relations. The corresponding work is under way.
The regularization functions in Eqs. (A1) and (A2) are defined by where |val and |n denote the states of the valence and sea quarks with the corresponding eigenenergies E val and E n of the single-quark Hamiltonian h(U c ), respectively [47].

Appendix B: Matrix elements of the SU(3) Wigner D function
The collective wavefunction of a baryon with flavor F = (Y, T, T 3 ) and spin S = (Y = −N c /3, J, J 3 ) in the representation ν is expressed in terms of a tensor with two indices, i.e. ψ (ν; F ),(ν; S) , one running over the states F in the representation ν and the other one over the states S in the representation ν. Here, ν denotes the complex conjugate of the ν, and the complex conjugate of S is written by S = (N c /3, J, J 3 ). Thus, the collective wavefunction is expressed as where dim(ν) stands for the dimension of the representation ν and Q S a charge corresponding to the baryon state S, i.e. Q S = J 3 + Y /2.
where α and γ are the parameters appearing in the collective Hamiltonian, which are written by Here, Σ πN is the well-known πN sigma term. We list the results of the matrix elements of the relevant collective operators for the EM transition form factors in Tables II, III