Transition magnetic moments of $J^P=\frac{3}{2}^+$ decuplet to $J^P=\frac{1}{2}^+$ octet baryons in the chiral constituent quark model

In light of the developments of the chiral constituent quark model ($\chi$CQM) in studying low energy hadronic matrix elements of the ground-state baryons, we have extended this model to investigate their transition properties. The magnetic moments of transitions from the $J^P=\frac{3}{2}^+$ decuplet to $J^P=\frac{1}{2}^+$ octet baryons have been calculated with explicit valence quark spin, sea quark spin and sea quark orbital angular momentum contributions. Since the experimental data is available for only a few transitions, we have compared our results with the results of other available models. The implications of other complicated effects such as chiral symmetry breaking and SU(3) symmetry breaking arising due to confinement of quarks have also been discussed.


I. INTRODUCTION
Understanding the internal structure of hadrons within the nonperturbative regime of quantum chromodynamics (QCD) is one of the most challenging area in theory as well as experiment. The electromagnetic properties, obtained from the measurements of electromagnetic Dirac and Pauli form factors, are further related to the static low-energy observables like masses, charge radii, magnetic moments, etc.. It constitutes one of the most promising area and can provide valuable insight into the underlying dynamics and the nonperturbative aspects of QCD. At present, electromagnetic form factors have been precisely obtained for the case of nucleons [1][2][3][4][5][6] whereas, for other baryons, the experimental data are available only for magnetic moments.
Magnetic moment of baryons is one of the most important quantity in scrutinizing the structure and the properties of light baryons. Continuous theoretical efforts are being made to investigate the magnetic moments and the calculations have benefited a lot from the information being made available through the experiments. At present, the magnetic moments of the J P = 1 2 + octet baryons (except Σ 0 ) have been accurately measured experimentally [6]. Our information about the J P = 3 2 + decuplet baryons is however limited to only ∆ + and Ω − because of the difficulty in measuring their properties experimentally on account of their short lifetimes.
Further, the low-lying baryon decuplet to octet electromagnetic transitions play a very important role in probing the internal spin structure as well as the deformation of the octet and decuplet baryons. The ∆(1232) resonance is the lowest-lying excited state of the nucleon in which the search for transition amplitudes from the spin-parity selection rules has been carried out. The ∆ + → pγ transition amplitude contains the magnetic dipole moment (G M 1 ), the electric quadrupole moment (G E2 ), and the Coulumb quadrupole moment (G C2 ).
One of the important model which finds application in the nonperturbative regime of QCD is the chiral constituent quark model (χCQM) [39][40][41] where chiral symmetry breaking and its spontaneous breaking is implemented. The χCQM uses the effective interaction Lagrangian approach of the strong interactions, where, the important phenomenon of quarkantiquark excitations is included. This results in the presence of the meson cloud at low energies where the effective degrees of freedom are the valence quarks and the internal Goldstone bosons (GBs), which are coupled to the valence quarks [41][42][43][44][45]. This perspective is in common with the modern effective field theory approaches. The χCQM has successfully been applied to calculate the spin and flavor distribution functions including the strangeness content of the nucleon [43,44], weak vector and axial-vector form factors [46], nucleon structure functions and longitudinal spin asymmetries [47], electromagnetic and axial-vector form factors of the quarks and nucleon [48], charge radii and quadrupole moment [49]. The magnetic moments of octet baryons, the transition within the octet baryons Σ → Λ and the Coleman-Glashow sum rule have already been calculated [50]. The work was further extended to the calculations of the magnetic moments of decuplet baryons [51], the magnetic moments baryon resonances [52], magnetic moments of Λ resonances [53] etc..
Considering the above developments of the χCQM in studying low energy hadronic matrix elements of the ground-state baryons, it becomes desirable to extend this model to investigate their transition properties. We will calculate the magnetic moments of transitions from the J P = 3 2 + decuplet to J P = 1 2 + octet baryons. Taking benefit from the earlier studies of J P = 3 2 + decuplet and J P = 1 2 + octet baryons [50], the explicit contributions coming from the valence quarks, quark sea polarization, and its orbital angular momentum have been calculated. The implications of other complicated effects such as chiral symmetry breaking and SU(3) symmetry breaking arising due to confinement of quarks have also been discussed.

II. TRANSITION MAGNETIC MOMENTS
In this section, we calculate the transition magnetic moments for the radiative decays transitions. In the present calculations we have considered only the S z = 1 2 spin projection for the 3 2 + decuplet as the matrix elements for other spin projections will come out to be zero.
The transition magnetic moment can be calculated from the matrix element where µ z corresponds to the magnetic moment operator, B 1 2 + and B 3 2 + correspond to the spin-flavor wavefunctions of the octet and decuplet baryons respectively expressed as The spin wavefunctions (χ s for the case of decuplet baryons and χ ′ and χ ′′ for the case of octet baryons) are expressed as The flavor wavefunctions φ s for the decuplet baryons of the types B 3 2 whereas the flavor wavefunctions φ ′ and φ ′′ for the octet baryons of the type B 1 2 where Q 1 , Q 2 , and Q 3 correspond to any of the u, d, and s quarks. For the case of Λ(uds) and Σ 0 (uds), the wavefunctions are given as The details of the spatial wave functions (ψ s , ψ ′ , ψ ′′ ) can be found in Ref. [54].
The magnetic moment of a given baryon in the χCQM receives contribution from the valence quark spin, sea quark spin and sea quark orbital angular momentum. The total magnetic moment is expressed as to as the orbital angular momentum contribution of the quark sea [41].
where µ q = eq 2Mq (q = u, d, s) is the quark magnetic moment in the units of µ N (nuclear magneton), ∆q The spin structure of a decuplet to octet transition matrix element is defined as This can be used to calculate the quark spin polarizations (for q = u, d, s) for a given The valence quarks spin polarizations ∆q 3 2 + → 1 2 + V for a given baryon transition can be calculated using the SU (6) spin-flavor wave functions defined in Eqs. (2) and (3). Using these, the magnetic moment contribution coming from the valence quarks can be calculated from Eq. (9) and have been summarized in Table I for all the decuplet to octet transitions.
For the calculation of the sea quarks spin polarizations ∆q 3 2 + → 1 2 + S for a given baryon transition, we will use the basic idea of the chiral constituent quark model (χCQM) [39] where the set of internal GBs couple directly to the valence quarks in the interior of hadron and we have  Here the transition probability of the emission of a GB from any of the q quark P (q, GB) and the transition probability of the q ↑↓ quark P (q ↑↓ , GB) can be calculated from the effective Lagrangian describing interaction between quarks and GBs expressed as where c 8 is the coupling constant for the octet GB. The GB field Φ ′ can be expressed in terms of the GBs and their transition probabilities as The fluctuation process describing the effective Lagrangian is where qq ′ + q ′ constitute the sea quarks. In Eq. (17), the chiral fluctuations u(d) → d(u) + π +(−) , u(d) → s + K +(0) , u(d, s) → u(d, s) + η, and u(d, s) → u(d, s) + η ′ are given in terms of the transition probabilities P π , P K , P η and P η ′ respectively [41,42,44].
From Eq. (15) the transition probability of the emission of a GB from any of the q quark are expressed in terms of the transition probabilities P π , P K , P η and P η ′ as whereas the transition probability of the q ↑↓ quark can be expressed as Using the sea spin polarizations, the magnetic moment contributions coming from the sea quarks can be calculated from Eq. (10) and have been summarized in Table II for all the decuplet to octet transitions.
The magnetic moment contribution of the angular momentum of a given sea quark can be expressed in terms of the orbital angular momenta of quarks and GB L q , L GB which are further related to the masses of quarks and GB (M q , M GB ) as The magnetic moment arising from all the possible transitions of a given valence quark to the GBs is obtained by multiplying the orbital moment of each process to the probability for such a process to take place. The general orbital moment for any quark q is given as The orbital moments of u, d and s quarks after including the transition probabilities P π , P K , P η and P η ′ as well as the masses of GBs M π , M K and M η can be expressed as The orbital contribution to the magnetic moment of the decuplet to octet transition for the baryon the type B(Q 1 Q 2 Q 3 ) is given as

III. RESULTS AND DISCUSSION
The transition probabilities P π , P K , P η and P η ′ as well as the masses of GBs M π , M K and M η are the input parameters needed for the numeric calculations of the baryon transition magnetic moments µ B 3 2 + → B 1 2 + in the χCQM. The hierarchy followed by the transition probabilities P π , P K , P η and P η ′ which represent respectively the probabilities of fluctuations of a constituent quark into pions, K, η and η ′ is given as This order is because of the fact that probability of emission of a particular GB is dependent on its masses implying that the probability of emission a heavier meson from a lighter quark is much smaller than that of the lighter mesons. The transition probabilities are usually fixed by the experimentally known spin and flavor distribution functions measured from the DIS experiments [3,4,6,44]. A detailed analysis leads to the following probabilities: On the other hand, the orbital angular momentum contributions are characterized by the masses of quarks and GBs (M q and M GB ). The on mass shell mass values can be used in accordance with several other similar calculations [41,50].
The inputs discussed above are used to calculate the explicit valence µ B  Table III. The limited experimental data available for the 3 2 + → 1 2 + transitions has also been presented in the table.
It can immediately seen that the contributions coming from valence and orbital contributions have same signs whereas the sea contributions have opposite signs. All of ultimately add up to give the total magnetic moment. It is also observed that in some cases the orbital part dominates over the sea part making the total magnetic moments even higher than the valence part. This is for the case of ∆ → p, and Ξ * 0 → Ξ 0 transitions. One can generalize this as follows, whenever there the number of u, d or s quarks are more, there is dominance of the orbital part, whereas when the u, d and s quarks are in equal numbers, there is some variation from this behavior. For example, in the case of ∆ → p and Σ * + → Σ + the baryon quark content is uud and clearly    For the sake of comparison with other models, we have presented the results of the available phenomenological and theoretical models in Table IV. We have presented the results from chiral perturbation theory (χPT) [25], light cone QCD sum rules and light cone QCD (LCQCDSR) [26], large-N c chiral perturbation theory (Large N C PT) [27,28], relativistic quark model (Rel-QM) [29], QCD sum rules (QCDSR) [32,33], lattice QCD [34], chiral quark model (χQM) [35] effective mass quark model (EMQM) [36], meson cloud model (MCM) [37] , U-spin [38]. The experimental values from the PDG [6] have also been listed.
Since the experimental data is available for ∆ → p, Σ * + → Σ + , and Σ * 0 → Λ transitions, we can compare these with the χCQM results as well as with the results of other available models. It is evident from the table that a good agreement corresponding to the case of ∆ → p is obtained. The magnetic moment of ∆ → p+γ transition is a long standing problem and most of the approaches in literature underestimate it. The empirical estimate for the magnetic moment of the ∆ → p + γ transition can be made from the helicity amplitudes [6], A1 2 = − 0.135 ± 0.005 GeV − 1 2 and A3 2 = − 0.250 ± 0.008 GeV − 1 2 [6] as inputs in the decay rate. The extracted magnetic moment comes out to be µ ∆→p = 3.46 ± 0.03 µ N . Our predicted value of 3.87 µ N is very close to the experimental results. The difference in sign with some of the other models may be due to the different model predictions and signs of the wavefunctions. In the case of Σ * + → Σ + , even though our results are almost half of the experimental value, except for a very few models all other models predict a value close to our results. In the case of and Σ * 0 → Λ transitions our results are more or less in good agreement with the results of other models as well as with the experimental data.
To summarize, the chiral constituent quark model (χCQM) is able to describe the transition properties of the low lying baryons. In a very interesting manner, the χCQM) is able to phenomenologically estimate the explicit contributions coming from the valence quarks, Other models  decuplet to J P = 1 2 + octet baryons. The results immediately suggest that the contributions coming from various sources with same and opposite signs ultimately add up to give the total magnetic moment. Whenever the number of u, d or s quarks are more, there is a dominance of the orbital part, whereas when the u, d and s quarks are in equal numbers, there is some variation from this behavior. These observations endorse that the sea quarks and the orbital angular momentum of the sea quarks perhaps provide the dominant dynamics of the constituents in the low-energy regime of QCD. The qualitative and quantitative description of the results confirms that constituent quarks and weakly interacting Goldstone bosons provide the appropriate degree of freedom in the nonperturbative regime of QCD. A further precise measurement of these magnetic moments, therefore, would have important implications for the χCQM.