Strong decays of baryons and missing resonances
R. Bijker, J. Ferretti, G. Galatà, H. García-Tecocoatzi, E. Santopinto
aa r X i v : . [ h e p - ph ] O c t Strong decays of baryons and missing resonances
R. Bijker
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, AP 70-543, 04510 M´exico DF, M´exico
J. Ferretti
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, AP 70-543, 04510 M´exico DF, M´exico andDipartimento di Fisica and INFN, “Sapienza” Universit`a di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy ∗ G. Galat`a
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, 04510 M´exico DF, M´exico
H. Garc´ıa-Tecocoatzi
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, 04510 M´exico DF, M´exico andINFN, Sezione di Genova and Dipartimento di Fisica,Universit`a di Genova, via Dodecaneso 33, 16146 Genova, Italy
E. Santopinto † INFN, Sezione di Genova, via Dodecaneso 33, 16146 Genova, Italy
We provide results for the open-flavor strong decays of strange and non-strange baryons intoa baryon-vector/pseudoscalar meson pair. The decay amplitudes are computed in the P pair-creation model, where s ¯ s pair-creation suppression is included for the first time in the baryonsector, in combination with the U(7) and hypercentral models. The effects of this s ¯ s suppressionmechanism cannot be re-absorbed in a redefinition of the model parameters or in a different choiceof the P model vertex factor. Our results for the decay amplitudes are compared with the ex-isting experimental data and previous P and elementary meson emission model calculations. Inthis respect, we show that distinct quark models differ in the number of missing resonances theypredict and also in the quantum numbers of states. Therefore, future experimental results will beimportant in order to disentangle different models of baryon structure. Finally, in the appendices,we provide some details of our calculations, including the derivation of all relevant flavor couplingswith strangeness-suppression. This derivation may be helpful to calculate the open-flavor decayamplitudes starting from other models of baryons. PACS numbers: 12.39.-x, 13.30.Eg, 14.20.Gk, 14.20.Jn
I. INTRODUCTION
It is well known that the baryon spectrum is very richand much more complex than the meson spectrum. Nev-ertheless, at the moment, the number of known light-quark mesons is much larger than the number of knownbaryon resonances [1]. This problem has to do withthe difficulty of identifying those high-lying baryon res-onances that are only weakly coupled to the
N π chan-nel [2, 3] and thus cannot be seen in elastic
N π scatter-ing experiments. Since the experimental observations ofbaryon resonances mainly come from reactions in whichthe pion is present either in the incoming channel, such as
N π → N π , or in the outgoing one, such as
N γ → N π ,it would not be surprising if some baryon resonances,very weakly coupled to the single pion, were missing fromexperimental results. These baryons may decay mainlyinto two pion channels (
N ππ ) or into channels such as ∗ Present address † Corresponding author: [email protected]
N η , N η ′ , N ω and K + Λ, where the final-state mesonis different from the pion [3]. Although interesting re-sults were provided by CB-ELSA [4], TAPS [5], GRAAL[6], SAPHIR [7] and CLAS [8], theoretical calculationsof strong, electromagnetic and weak decays of baryonsmay still help experimentalists in their search for thoseresonances that are still unknown.The QCD mechanism behind the OZI-allowed strongdecays [9] is still not clear. For this reason, several phe-nomenological models have been developed in order tocarry out this type of study, including pair-creation mod-els [10–17], elementary meson emission models [18–23]and effective Lagrangian approaches (for example, seeRef. [24]). Attempts at modeling strong decays withinthe quark model (QM) formalism date from Micu’s sug-gestion [10] that hadron decays proceed through q ¯ q pairproduction with vacuum quantum numbers, i.e. J P C =0 ++ . Since the q ¯ q pair corresponds to a P quark-antiquark state, this model is known as the P pair-creation model [10, 11, 16]. A few years after its intro-duction, Le Yaouanc et al. used the P model to com-pute meson and baryon open-flavor strong decays [11]and also evaluated the strong decay widths of ψ (3770), ψ (4040) and ψ (4415) charmonium states [25]. The P model, extensively applied to the decays of light mesonsand baryons [26], has recently been applied to heavy me-son strong decays in the charmonium [27–29], bottomo-nium [30, 31], open-charm [32–35] and open-bottom [34]sectors. In the 90’s, Capstick and Roberts calculatedthe N π and the strange decays of non-strange baryons[3] by using relativized quark model wave functions forbaryons and mesons. The baryon and meson spectra werepredicted within the relativized QMs of Refs. [21, 36]. Itis also worthwhile to cite Ref. [37], where the authorscomputed the open-flavor strong decays of Ξ baryons upto N = 2 shell in a chiral quark model.In this paper, we present our results for the open-flavorstrong decay widths of light baryons (i.e. made up of u, d, s valence quarks) into a baryon-pseudoscalar me-son pair and a baryon-vector meson pair. The widthsare computed within a modified version of the P pair-creation model where, for the first time in the baryoncase, a strange quark pair suppression mechanism hasbeen taken into account, analogous to what was donein the meson sector to suppress unphysical heavy quarkpair-creation [28–31, 38]. The effects of this mechanism,which breaks the SU(3) symmetry, cannot be re-absorbedin a redefinition of the model parameters or in a differentchoice of the P model vertex factor.In the next section, we briefly mention the models forthe baryon spectrum and structure that we have used forthe calculation of the strong decays: the U (7) algebraicmodel [39, 40], by Bijker, Iachello and Leviatan, and thehypercentral model (hQM) [41], developed by Gianniniand Santopinto. In Section III, we review the P modelfor the two-body decay of a baryon into a baryon and ameson, including a discussion of the phase space factor.The results for the strong decays are presented in Sec-tion IV. Finally, we present a summary and conclusions.Some details of the calculation of P matrix elementsare presented in the appendices. Of particular interestis Appendix E, where the flavor couplings with SU (3)breaking induced by the presence of the strange pair sup-pression mechanism have been derived explicitly for thefirst time. II. STRANGE AND NON-STRANGE BARYONSPECTRAA. U (7) algebraic model The baryon spectrum is computed by means of alge-braic methods introduced by Bijker, Iachello and Levi-atan [39, 40]. The algebraic structure of the model con-sists of combining the symmetry of the internal spin-flavor-color part, SU sf (6) ⊗ SU c (3), with that of the spa-tial part, U (7) into U (7) ⊗ SU sf (6) ⊗ SU c (3) . (1) The U (7) model was introduced [39] to describe the rela-tive motion of the three constituent parts of the baryon.The general idea is to introduce a so-called spectrum gen-erating algebra U ( k + 1) for quantum systems character-ized by k degrees of freedom. For baryons, there are the k = 6 relevant degrees of freedom of the two relativeJacobi vectors ~ρ = 1 √ ~r − ~r ) ,~λ = 1 √ ~r + ~r − ~r ) , (2)and their canonically conjugate momenta, ~p ρ = ( ~p − ~p ) / √ ~p λ = ( ~p + ~p − ~p ) / √
6. The U (7) modelis based on a bosonic quantization which consists of in-troducing two vector boson operators, b † ρ and b † λ , asso-ciated to the Jacobi vectors, and an additional auxiliaryscalar boson, s † . The scalar boson does not representan independent degree of freedom, but is added underthe restriction that the total number of bosons N is con-served. The model space consists of harmonic oscillatorshells with n = 0 , , . . . , N .The baryon mass formula is written as the sum of threeterms ˆ M = M + ˆ M + ˆ M , (3)where M is a constant, ˆ M is a function of the spa-tial degrees of freedom and ˆ M depends on the internaldegrees of freedom. The spin-flavor part is treated in thesame way as in Ref. [40] in terms of a generalized G¨ursey-Radicati formula [42], which in turn is a generalizationof the Gell-Mann-Okubo mass formula [43, 44]ˆ M = a (cid:16) ˆ C ( SU sf (6)) − (cid:17) + b (cid:16) ˆ C ( SU f (3)) − (cid:17) + c (cid:18) ˆ C ( SU s (2)) − (cid:19) + d (cid:16) ˆ C ( U Y (1)) − (cid:17) + e (cid:16) ˆ C ( U Y (1)) − (cid:17) + f (cid:18) ˆ C ( SU I (2)) − (cid:19) . (4)The operators ˆ C ( G ) and ˆ C ( G ) correspond to the linearand quadratic Casimir invariants of the relevant groupsfor the internal degrees of freedom. The values of thecoefficients M , a , b , c , d , e and f are taken from [40].Since the space-spin-flavor wave function is symmetricunder permutation group S of the three identical con-stituents, the permutation symmetry of the spatial wavefunction has to be the same as that of the spin-flavorpart. Thus, the spatial part of the mass operator ˆ M has to be invariant under the S permutation symme-try. The dependence of the mass spectrum on the spatialdegrees of freedom is given by:ˆ M = ˆ M + ˆ M . (5)In Refs. [23, 40], strong decays of baryons were studiedin the collective string model, which is a special case of U (7) in which the radial excitations are interpreted asrotations and vibrations of an oblate top, in combinationwith the elementary emission model for the strong de-cays. Here, we do the same, but calculate the decays inthe P pair-creation model [10, 11, 16]. The rotationalpart of the operator (5) is written in the same form as in[40] ˆ M = α r ˆ L · ˆ L + 14 , (6)with eigenvalues M = α (cid:18) L + 12 (cid:19) . (7)In this way one gets linear Regge trajectories with a slope α , as required by the phenomenology [45]. The spectrumof the vibrational part is given by [40]ˆ M = M = κ v + κ v , (8)where v = n u and v = n v + n w are the vibrational quan-tum numbers, corresponding to the symmetric stretch-ing vibration along the direction of the strings (breath-ing mode), and two degenerate bending vibrations of thestrings. The spectrum consists of a series of vibrationalexcitations characterized by the labels ( v , v ), and atower of rotational excitations built on top of each vibra-tion. κ and κ are free parameters fitted to the data.The spectra calculated for the non-strange and strangebaryons are shown in Tables III–XVI, second/third col-umn. The baryon wave functions are denoted in the stan-dard form as (cid:12)(cid:12) S +1 dim { SU f (3) } J [dim { SU sf (6) } , L Pi ] (cid:11) , (9)where S and J are the spin and total angular momentum ~J = ~L + ~S . As an example, in this notation the nucleonand delta wave functions are given by (cid:12)(cid:12) / [56 , +1 ] (cid:11) and (cid:12)(cid:12) / [56 , +1 ] (cid:11) , respectively. B. Hypercentral model
The starting point of the hQM is the assumption thatquark interaction is hypercentral, i.e. it only depends onthe hyperradius x [41, 46], V q ( ~ρ, ~λ ) = V ( x ) , (10)with x = q ~ρ + ~λ [47]. Thus, the spatial part of thethree-quark wave function, ψ space , is factorized as ψ space = ψ q ( ~ρ, ~λ ) = ψ γν ( x ) Y [ γ ] l ρ l λ (Ω ρ , Ω λ , ξ ) , (11)where the hyperradial wave function, ψ γν ( x ), is labeledby the grand angular quantum number γ and the num-ber of nodes ν . Y [ γ ] l ρ l λ (Ω ρ , Ω λ , ξ ) are the hyperspherical harmonics, with angles Ω ρ = ( θ ρ , φ ρ ), Ω λ = ( θ λ , φ λ ) andhyperangle ξ = arctan ρλ [47]. The dynamics is containedin ψ γν ( x ), which is a solution of the hyperradial equation[ d dx + x ddx − γ ( γ +4) x ] ψ γν ( x )= − m [ E − V q ( x )] ψ γν ( x ) . (12)In the hQM, the quark interaction has the form [41, 46] V ( x ) = − τx + αx , (13)where τ and α are free parameters fitted to the reproduc-tion of the experimental data. Eq. (13) can be seen asthe hypercentral approximation of a Cornell-type quarkinteraction [12], whose form can be reproduced by Lat-tice QCD calculations [48]. Now, to introduce splittingswithin the SU(6) multiplets, an SU(6)-breaking termmust be added. In the case of the hQM, such viola-tion of the SU(6) symmetry is provided by the hyperfineinteraction [49, 50]. The complete hQM Hamiltonian isthen [41, 46] H hQM = 3 m + ~p ρ m + ~p λ m − τx + αx + H hyp , (14)where ~p ρ and ~p λ are the momenta conjugated to the Ja-cobi coordinates ~ρ and ~λ . In addition to τ and α , thereare two more free parameters in the hQM, the constituentquark mass, m , and the strength of the hyperfine interac-tion. The former is taken, as usual, as 1 / τ and α , is fitted in[41] to the reproduction of the *** and **** resonancesreported in the PDG [1]. The hQM has an approximate O (7) symmetry [41]. Extension to strange baryons
In Ref. [51], the hQM was extended to strangebaryons. Specifically, the authors considered a Hamil-tonian whose SU(6) invariant part is the same as in theHypercentral Model [41], while the SU(6) symmetry isbroken by a G¨ursey-Radicati-inspired interaction [42].The complete Hamiltonian is given by H = H + H GR , (15)where H = 3 m + p λ m + p ρ m + V ( x ) (16)and V ( x ) = − τx + αx . (17)The G¨ursey-Radicati-inspired interaction, H GR , whichdescribes the splittings within the SU(6) baryon multi-plets [51], is written in terms of Casimir operators as, H GR = A ˆ C ( SU sf (6)) + B ˆ C ( SU f (3))+ C ˆ C ( SU s (2)) + D ˆ C ( U Y (1))+ E (cid:18) ˆ C ( SU I (2)) −
14 ˆ C ( U Y (1)) (cid:19) , (18) FIG. 1: The P pair-creation model of hadron vertices; the q ¯ q pair (45) is created in a P flavor-color singlet. A is theinitial-state baryon; B and C are the final baryon and mesonstates, respectively. where A , B , C , D and E are free parameters fitted tothe data with the values reported in Ref. [51]. III. P PAIR-CREATION MODEL
The P pair-creation model is an effective model tocompute open-flavor strong decays [10, 11, 16]. Here, ahadron decay takes place in its rest frame and proceedsvia the creation of an additional q ¯ q pair. The quark-antiquark pair is created with the quantum numbers ofthe vacuum, i.e. J P C = 0 ++ (see Fig. 1), and the decayamplitude can be expressed as [10, 11, 17, 27, 28, 30, 31]Γ A → BC = Φ A → BC ( q ) X ℓ (cid:12)(cid:12) h BCq ℓJ | T † | A i (cid:12)(cid:12) . (19)In this paper, we focus on the open-flavor strong de-cays of light baryons (i.e. made up of u, d, s quarks)in the P model. We assume harmonic oscillator wavefunctions, depending on a single oscillator parameter α b for the baryons and α c for the mesons. The final stateis characterized by the relative orbital angular momen-tum ℓ between B and C and a total angular momentum ~J = ~J b + ~J c + ~ℓ . A. Phase space factor
The coefficient Φ A → BC ( q ) is the phase space factorfor the decay. We show three possible prescriptions. Thefirst in the non-relativistic expression,Φ A → BC ( q ) = 2 πq M b M c M a , (20) TABLE I: Effective meson and baryon masses, ˜ M [see Eq.(22)], from Refs. [3, 15].State ˜ M (GeV) N π ρ η ω depending on the relative momentum q between B and C and on the masses of the two intermediate-statehadrons, M b and M c . The second option is the standardrelativistic form,Φ A → BC ( q ) = 2 πq E b ( q ) E c ( q ) M a , (21)depending on q and on the energies of the twointermediate-state hadrons, E b = p M b + q and E c = p M c + q . The third possibility is to use an effectivephase space factor [3, 15],Φ A → BC ( q ) = 2 πq ˜ M b ˜ M c M a , (22)where ˜ M b and ˜ M c are the effective baryon and me-son masses, respectively, evaluated by means of a spin-independent interaction (see Table I). According to Ref.[15], this is valid in the weak-binding limit, where ρ and π are degenerate and ˜ m π = 5 . m π .In the case of heavy baryons and mesons, whose inter-nal dynamics is almost non-relativistic and the hyperfineinteractions are relatively small, the three types of phasespace factors provide almost the same results. B. Transition operator
The transition operator of the P model is given by[28, 30, 31]: T † = − γ Z d~p d~p δ ( ~p + ~p ) C F V ( ~p − ~p )[ χ × Y ( ~p − ~p )] (0)0 b † ( ~p ) d † ( ~p ) . (23)Here, γ is the pair-creation strength, and b † ( ~p ) and d † ( ~p ) are the creation operators for a quark and an anti-quark with momenta ~p and ~p , respectively. The q ¯ q pairis characterized by a color-singlet wave function C , aflavor-singlet wave function F , a spin-triplet wave func-tion χ with spin S = 1 and a solid spherical harmonic Y ( ~p − ~p ), since the quark and antiquark are in a relative P -wave. V ( ~p − ~p ) = e − α ( ~p − ~p ) / is the pair-creationvertex or quark form factor. If one does not consider thequark form factor, i.e. α d = 0, the vertex reduces to aconstant ( V = 1), see Appendix A.In our calculations, we introduce a strange quark-pairsuppression mechanism in the baryon sector analogouslyto what was done in charmonium to suppress unphysicalheavy quark pair-creation [28, 30, 31, 38]. The mecha-nism consists in introducing a reduction factor m n /m q inthe pair-creation operator which suppresses the creationof strange quark pairs ( q = s ) by a factor of m n /m s with respect to the nonstrange quark pairs ( q = u and q = d ). This particular choice for the pair-creationstrength breaks the SU (3) flavor symmetry and sup-presses the creation of s ¯ s pairs. Its effect cannot be re-absorbed in a redefinition of the model parameters or ina different choice of the P model vertex factor.In Appendix D we present the derivation of the transi-tion amplitudes in the P pair-creation model using theJacobi coordinates of Eq. (2) and including the effects ofa gaussian smearing of the pair-creation vertex. In Ap-pendix E we present a derivation of the flavor couplingcoefficients including the strangeness suppression mech-anism which is valid for both pseudoscalar and vectormesons. C. Mixing between N (1535) S and N (1650) S To better reproduce the experimental data, we intro-duce a mixing between N (1535) S and N (1650) S res-onances, | N (1535) S i = | / i cos θ + | / i sin θ , | N (1650) S i = −| / i sin θ + | / i cos θ , (24)where θ = 38 ◦ is the mixing angle. This was done inRefs. [40, 52], to correct the disagreement between ex-perimental and theoretical results for the helicity am-plitudes of the N (1535) S and N (1650) S resonances.In Ref. [53] this problem was solved with a mixture ofpseudoscalar meson-baryon and vector meson-baryon ina coupled-channels scheme. IV. OPEN-FLAVOR STRONG DECAYS.RESULTS AND DISCUSSION
In this section, we present the results of our calcu-lations of the open-flavor strong decays of non-strangebaryons and hyperons into baryon-pseudoscalar andbaryon-vector mesons, using the U(7) model (see Sec.IV A) and hQM model (see Sec. IV B). The decay am-plitudes are computed in the P model of Refs. [10, 11]and Sec. III. TABLE II: P model parameter values used in the calcula-tions, in combination with the relativistic phase space factorof Eq. (21) (column 2) and the effective phase space factorof Eq. (22) (column 3). The parameter values are obtainedin a fit to the experimental strong decay widths, see App. Aand Table XVIII, last column. The values of the constituentquark masses m n ( n = u, d ) and m s are used in the pair-creation operator of Eq. (D10). α b is the harmonic oscillatorparameter of baryons A and B , α c that of meson C and α d is the quark form factor parameter.Parameter Rel. PSF Eff. PSF γ α b − − α c − − α d − − m n m s A. Open-flavor strong decays calculated by usingthe U(7) model
Here, we show our results for the open-flavor decaysby using the U(7) algebraic model of Sec. II A and Refs.[39, 40].The strong decay widths are computed in the P model using Eqs. (19), (D7) and (D12), by consider-ing two possible choices for the phase space factor: thestandard relativistic form of Eq. (21) and the effectivephase space factor of Eq. (22). The results obtainedwith the relativistic phase space factor (RPSF) and themodel parameters of the second column of Table II arereported in Tables III–XVI; those obtained with the ef-fective phase space factor (EPSF) and the model parame-ters of the third column of Table II are reported in TablesIII–IV. The P model parameters of Table II are fittedto a sample of 9 transitions, as discussed in App. A. Inour calculations, whenever available we use the experi-mental values for the masses of the decaying resonancesfrom the PDG [1], otherwise the theoretical predictionsof Sec. II A. B. Open-flavor strong decays calculated by usingthe hQM
Below, we provide results for the open-flavor decaywidths of strange and non-strange baryons into lightbaryons plus pseudoscalar or vector mesons in the P model formalism of Sec. III, using the hQM results ofRefs. [41, 46, 51]. For the results, see Tables III-IX.The decays are calculated with the new values of the P model parameters of Table XVII, which we fitted toa sample of 9 transitions: ∆ → N π , N (1520) → N π , N (1535) → N π , N (1650) → N π , N (1680) → N π , N (1720) → N π , ∆(1905) → N π , ∆(1910) → N π and∆(1920) → N π . TABLE III: Strong decay widths of three- and four-star nucleon resonances (in MeV), calculated with the U (7) algebraic modeland the hypercentral QM. For the U (7) model the calculation is done for the relativistic and the effective phase space factorsof Eqs. (21) and (22), respectively, in combination with the parameters of Table II (RPSF and EPSF). For the hypercentralQM we present the results for the relativistic phase space factor (RPSF) in combination with the parameters of Table XVII.The experimental values are taken from Ref. [1]. Decay channels labeled by – are below threshold. The symbols ( S ) and ( D )stand for S - and D -wave decays, respectively.Model Resonance Status M [MeV] Nπ Nη Σ K Λ K ∆ π Nρ NωN (1440) P **** 1430-1470 110 −
338 0 − −
101 Exp.U(7) / [56 , +2 ] 1444 85 – – – 13 – – RPSFU(7) / [56 , +2 ] 1444 108 – 22 EPSFhQM / [56 , +2 ] 1550 105 – – – 12 – – RPSF N (1520) D **** 1515-1530 102 0 342 Exp.U(7) / [70 , − ] 1563 134 0 – – 207 – – RPSFU(7) / [70 , − ] 1563 102 0 342 EPSFhQM / [70 , − ] 1525 111 0 – – 206 – – RPSF N (1535) S **** 1520-1555 44 −
96 40 − < / [70 , − ] 1563 63 75 – – 16 – – RPSFU(7) / [70 , − ] 1563 106 86 14 EPSFhQM / [70 , − ] 1525 84 50 – – 6 – – RPSF N (1650) S **** 1640-1680 60 −
162 6 −
27 4 −
20 0 −
45 Exp.U(7) / [70 , − ] 1683 41 72 – 0 18 – – RPSFU(7) / [70 , − ] 1683 71 83 15 EPSFhQM / [70 , − ] 1574 51 29 – 0 4 – – RPSF N (1675) D **** 1670-1685 46 −
74 0 − < −
99 Exp.U(7) / [70 , − ] 1683 47 11 – 0 108 – – RPSFU(7) / [70 , − ] 1683 29 7 79 EPSFhQM / [70 , − ] 1579 41 9 – – 85 – – RPSF N (1680) F **** 1675-1690 78 −
98 0 − −
21 Exp.U(7) / [56 , +1 ] 1737 121 1 – 0 100 – – RPSFU(7) / [56 , +1 ] 1737 63 0 99 EPSFhQM / [56 , +1 ] 1798 91 0 0 0 92 – – RPSF N (1700) D *** 1650-1750 7 −
43 0 − < −
225 ( S ) Exp. <
50 ( D )U(7) / [70 , − ] 1683 9 3 – 0 561 – – RPSFU(7) / [70 , − ] 1683 5 2 657 EPSFhQM / [70 , − ] 1606 0 0 0 0 0 – – RPSF N (1710) P *** 1680-1740 3 −
50 5 −
75 3 −
63 8 −
100 3 −
63 Exp.U(7) / [70 , +1 ] 1683 5 9 0 3 56 – – RPSFU(7) / [70 , +1 ] 1683 11 9 58 EPSFhQM / [70 , +1 ] 1808 18 12 0 14.1 70 – – RPSF N (1720) P **** 1650-1750 12 −
56 5 −
20 2 −
60 90 −
360 105 −
340 Exp.U(7) / [56 , +1 ] 1737 111 7 0 14 36 5 0 RPSFU(7) / [56 , +1 ] 1737 123 7 28 EPSFhQM / [56 , +1 ] 1797 141 8 0 12 30 77 5 RPSF N (1875) D *** 1820-1920 3 −
70 0 −
22 0 − −
192 ( S ) 0 −
38 22 −
90 Exp.11 −
86 ( D ) Exp.U(7) / [70 , − ] 1975 0 0 0 0 0 0 0 RPSFU(7) / [70 , − ] 1975 0 0 0 EPSFhQM / [70 , − ] 1899 14 8 2 0 560 80 82 RPSF N (1900) P *** 1875-1935 20 −
37 24 −
44 6 −
26 0 −
37 75 −
120 Exp.U(7) / [70 , +1 ] 1874 11 12 1 13 63 64 24 RPSFU(7) / [70 , +1 ] 1874 17 11 33 EPSFhQM / [70 , +1 ] 1853 15 12 1 13 70 53 23 RPSF TABLE IV: As Table III, but for ∆ resonances. The symbols ( S ), ( P ), ( D ) and ( F ) stand for S -, P -, D - and F -wave decays,respectively. Model Resonance Status M [MeV] Nπ Σ K ∆ π ∆ η Σ ∗ K Nρ ∆(1232) P **** 1230-1234 114 −
120 Exp.U(7) / [56 , +1 ] 1246 71 – – – – – RPSFU(7) / [56 , +1 ] 1246 115 – – EPSFhQM / [56 , +1 ] 1240 63 – – – – – RPSF∆(1600) P *** 1550-1700 22 −
105 88 − <
88 Exp.U(7) / [56 , +2 ] 1660 17 – 65 – – – RPSFU(7) / [56 , +2 ] 1660 24 74 – EPSFhQM / [56 , +2 ] 1727 31 – 69 – – – RPSF∆(1620) S **** 1615-1675 26 −
45 39 −
90 9 −
38 Exp.U(7) / [70 , − ] 1649 5 – 76 – – – RPSFU(7) / [70 , − ] 1649 10 61 – EPSFhQM / [70 , − ] 1584 9 – 59 – – – RPSF∆(1700) D **** 1670-1770 20 −
80 50 −
200 ( S ) 6 −
28 48 −
165 Exp.10 −
60 ( D )U(7) / [70 , − ] 1649 46 – 311 – – – RPSFU(7) / [70 , − ] 1649 27 343 – EPSFhQM / [70 , − ] 1584 40 – 333 – – – RPSF∆(1905) F **** 1855-1910 24 − < >
162 Exp.U(7) / [56 , +1 ] 1921 31 1 188 19 0 99 RPSFU(7) / [56 , +1 ] 1921 14 139 14 EPSFhQM / [56 , +1 ] 1844 26 0 182 15 – 88 RPSF∆(1910) P **** 1860-1910 33 −
102 9 −
48 70 −
299 Exp.U(7) / [56 , +1 ] 1921 26 38 32 4 – 64 RPSFU(7) / [56 , +1 ] 1921 39 27 3 EPSFhQM / [56 , +1 ] 1871 49 38 34 4 – 60 RPSF∆(1920) P *** 1900-1970 9 −
60 3 − −
102 ( P ) 13 −
69 0 Exp.45 −
195 ( F )U(7) / [56 , +1 ] 1921 7 23 132 22 5 105 RPSFU(7) / [56 , +1 ] 1921 14 96 15 EPSFhQM / [56 , +1 ] 1856 17 22 137 20 – 102 RPSF∆(1930) D *** 1920-1970 11 −
75 Exp.U(7) / [70 , − ] 1946 0 0 0 0 0 0 RPSFU(7) / [70 , − ] 1946 0 0 0 EPSF∆(1950) F **** 1940-1960 82 −
151 1 − − <
34 Exp.U(7) / [56 , +1 ] 1921 172 5 92 1 0 22 RPSFU(7) / [56 , +1 ] 1921 72 40 1 EPSFhQM / [56 , +1 ] 1851 146 3 70 1 – 16 RPSF TABLE V: As Table III, but for Σ and Σ ∗ resonances.Model Baryon Status M [MeV] NK Σ π Λ π Σ η Ξ K ∆ K Σ ∗ π NK ∗ Σ ρ Λ ρ Σ ω Σ(1660) P *** 1630-1690 4 −
60 seen seen Exp. U (7) / [56 , +2 ] 1604 3 38 14 – – – 7 – – – – RPSFhQM / [56 , +2 ] 1704 4 45 18 – – – 6 – – – – RPSFΣ(1670) D **** 1665-1685 3 −
10 12 −
48 2 −
12 Exp. U (7) / [70 , − ] 1711 5 78 8 – – – 36 – – – – RPSFhQM / [70 , − ] 1799 4 62 7 – – – 29 – – – – RPSFΣ(1750) S *** 1730-1800 6 − <
13 seen 9 −
88 Exp. U (7) / [70 , − ] 1711 3 109 3 28 – – 9 – – – – RPSFhQM / [70 , − ] 1799 5 151 6 27 – – 7 – – – – RPSFΣ(1775) D **** 1770-1780 39 −
58 2 − −
27 8 −
16 Exp. U (7) / [70 , − ] 1822 101 17 38 0 – 4 11 – – – – RPSFhQM / [70 , − ] 1914 79 13 30 0 – 2 7 – – – – RPSFΣ(1915) F **** 1900-1935 4 −
24 seen seen < U (7) / [56 , +1 ] 1872 6 58 33 2 1 96 23 6 – 2 – RPSFhQM / [56 , +1 ] 1906 4 44 26 1 0 88 22 5 – 2 – RPSFΣ(1940) D *** 1900-1950 <
60 seen seen seen seen seen Exp. U (7) / [56 , − ] 1974 0 0 0 0 0 0 0 0 – 0 – RPSFhQM / [70 , − ] 1914 31 6 11 0 0 554 99 251 – 0 – RPSFΣ ∗ (1385) P **** 1383-1385 30 −
32 4 − U (7) / [56 , +1 ] 1382 – 3 27 – – – – – – – – RPSFhQM / [56 , +1 ] 1372 – 3 24 – – – – – – – – RPSFΣ ∗ (2030) F **** 2025-2040 26 −
46 8 −
20 26 − < −
40 8 −
30 Exp. U (7) / [56 , +1 ] 2012 54 37 75 8 1 30 37 7 0 4 0 RPSFhQM / [56 , +1 ] 2085 44 30 62 6 1 22 27 5 0 2 0 RPSF Although we use the same decay model as in Sec. IV A,there are some differences with respect from the previouscase: 1) As in Sec. IV A, for *** and **** states we usethe experimental values of the masses, but the quantumnumber assignments are provided by the hQM and do notalways coincide with those of the U(7) model, since thetwo models are different. For example, this happens forthe N (1650) S , N (1700) D and N (1875) D . Thus,in these cases, we expect to obtain quite different re-sults for the decay widths; 2) The hQM and U(7) modelspredict the existence of a few missing states below theenergy of 2.1 GeV. The masses, quantum numbers andalso quantity of missing states in the two previous modelsare different. Information on missing states is importantto the experimentalists in their search for new baryonresonances. C. Comparison with other QM calculations
The quality of our results is comparable to that of Refs.[3, 54]. Capstick and Roberts (CR) studied the strongdecays of non-strange baryons, nucleon and delta reso-nances, by using Capstick and Isgur’s relativized baryonmodel [36] to describe the masses of unknown resonances,harmonic oscillator wave functions and an effective phasespace [3, 54]; they did not calculate the open-flavor de-cays of strange baryons. Our study is more complete, asit includes many more decay channels (such as, for exam- ple, decays into Ξ + meson) and a detailed analysis of thedecays of strange baryons. Both calculations, ours andCapstick and Roberts’ [3, 54], are performed within the P model, though there are some differences. The mainone is in the pair-creation mechanism: in Refs. [3, 54]it does not depend on the flavor of the created qq pair,while in the present case the strange quark pair-creationis suppressed with respect to the nonstrange pairs. Theeffects of this strangeness-suppression mechanism cannotbe re-absorbed in a redefinition of the model parametersor in a different choice of the P model vertex factor.Whether s ¯ s pair-creation has to be suppressed or notmay be evaluated by comparing theoretical predictionsand experimental data for these particular decay chan-nels. Unfortunately, we think that the large uncertain-ties on experimental decay widths into channels due to s ¯ s pair-creation do not permit a strong definitive con-clusion to be drawn. Nevertheless, it is worthwhile toobserve that the introduction of a strangeness suppres-sion mechanism may be justified by experimental resultsconcerning the electro-production ratios of Λ K + , Σ ∗ K , pπ and nπ + baryon meson-states from N ∗ ’s [55].In our paper, we also compared results obtained withrelativistic or effective phase space factors. Our conclu-sion is that the quality of the results for the amplitudescalculated by using a phase space or the other are sim-ilar. Thus, we think that it is preferable to use a rel-ativistic phase space, in order to reduce the number ofunnecessary parametrizations. Finally, we can say that TABLE VI: As Table III, but for Λ and Λ ∗ resonances.Model Baryon Status M [MeV] NK Σ π Λ η Ξ K Σ ∗ π NK ∗ Σ ρ Λ ω Λ(1600) P *** 1560-1700 8 −
75 5 −
150 Exp. U (7) / [56 , +2 ] 1577 79 40 – – 19 – – – RPSFhQM / [56 , +2 ] 1627 93 46 – – 18 – – – RPSFΛ(1670) S **** 1660-1680 5 −
15 6 −
28 3 −
13 Exp. U (7) / [70 , − ] 1686 217 33 9 – 7 – – – RPSFhQM / [70 , − ] 1722 267 39 0 – 5 – – – RPSFΛ(1690) D **** 1685-1690 10 −
21 10 −
28 Exp. U (7) / [70 , − ] 1686 150 16 0 – 168 – – – RPSFhQM / [70 , − ] 1722 119 23 0 – 168 – – – RPSFΛ(1800) S *** 1720-1850 50 −
160 seen seen Exp. U (7) / [70 , − ] 1799 0 67 85 – 13 – – – RPSFhQM / [70 , − ] 1837 0 98 90 – 10 – – – RPSFΛ(1810) P *** 1750-1850 10 −
125 5 −
100 seen Exp. U (7) / [70 , +1 ] 1799 16 4 2 – 40 – – – RPSFhQM / [56 , +3 ] 1973 0 0 0 – 0 – – – RPSFΛ(1820) F **** 1815-1825 39 −
59 6 −
13 4 − U (7) / [56 , +1 ] 1849 78 31 1 0 73 – – – RPSFhQM / [56 , +1 ] 1829 57 22 0 0 66 – – – RPSFΛ(1830) D **** 1810-1830 2 −
11 21 − > U (7) / [70 , − ] 1799 0 99 9 0 82 – – – RPSFhQM / [70 , − ] 1837 0 84 7 0 64 – – – RPSFΛ(1890) P **** 1850-1910 12 −
70 2 −
20 seen Exp. U (7) / [56 , +1 ] 1849 96 69 31 2 30 28 – – RPSFhQM / [56 , +1 ] 1829 120 79 11 1 24 27 – – RPSFΛ(2110) F **** 2090-2140 8 −
63 15 −
100 seen Exp. U (7) / [70 , +1 ] 2074 0 14 3 1 136 0 38 16 RPSFhQM / [70 , +1 ] 1995 167 20 0 3 69 15 79 25 RPSFΛ ∗ (1405) S **** 1402-1410 48 −
52 Exp. U (7) / [70 , − ] 1641 – 230 – – – – – – RPSFhQM / [70 , − ] 1658 – 222 – – – – – – RPSFΛ ∗ (1520) D **** 1518-1520 6 − − U (7) / [70 , − ] 1641 10 17 – – – – – – RPSFhQM / [70 , − ] 1658 8 13 – – – – – – RPSFTABLE VII: As Table III, but for Ξ and Ξ ∗ resonances.Model Baryon Status M [MeV] Σ K Λ K Ξ π Ξ ∗ π Ξ(1690) S *** 1680-1700 Exp. U (7) / [70 , − ] 1828 58 85 14 0 RPSFhQM / [70 , − ] 1938 55 86 15 0 RPSFΞ(1820) D *** 1818-1828 2 −
18 3 −
12 0 − −
18 Exp. U (7) / [70 , − ] 1828 38 26 6 55 RPSFhQM / [70 , − ] 1938 29 21 5 55 RPSFΞ ∗ (1530) P **** 1531-1532 9 −
10 Exp. U (7) / [56 , +1 ] 1524 – – 11 – RPSFhQM / [56 , +1 ] 1511 – – 9 – RPSF TABLE VIII: Strong decay widths of missing nucleon resonances (in MeV) calculated in the U(7) Model of Sec. II A andRefs. [39, 40] (top) and the hypercentral QM of Sec. II B and Refs. [41, 46] (bottom). The calculations are carried out usingthe model parameters of Table II (second column) and Table XVII, respectively. in combination with the relativistic phasespace factor of Eq. (21). Tentative assignments of one and two star resonances are labeled by ‡ . N Mass
Nπ Nη Σ K Λ K ∆ π Σ ∗ K Nρ Nω Σ K ∗ Λ K ∗ ∆ ρU (7) Model J [20 , +1 ] 1713 0 0 0 0 0 – – – – – – / [70 , +1 ] 1796 0 3 5 0 65 – 7 7 – – – / [70 , +1 ] 1874 ‡
106 10 0 3 79 – 161 8 – – – J [70 , − ] 1874 0 0 0 0 0 – 0 0 – – – / [70 , +1 ] 1975 ‡ / [70 , +1 ] 1975 ‡ / [70 , +1 ] 1975 ‡ / [70 , +1 ] 1975 ‡
25 13 4 0 99 0 5 4 – – – J [70 , − ] 1975 ‡ / [56 , − ] 2094 5 1 1 5 3 2 48 6 2 2 14 / [56 , − ] 2094 ‡
27 0 0 1 23 1 53 11 0 2 13 / [70 , − ] 1829 ‡
42 7 0 1 38 – 0 0 – – – / [70 , − ] 1933 8 12 3 0 0 0 0 0 – – – / [70 , − ] 1933 0 0 3 0 0 0 0 0 – – – / [70 , − ] 1933 0 2 5 0 1 0 0 0 – – –hQM / [70 , +1 ] 1835 4 8 7 0 97 - 8 7 - - - / [20 , +1 ] 1836 0 0 0 0 0 - 0 0 - - - / [20 , +1 ] 1836 0 0 0 0 0 - 0 0 - - - / [70 , +1 ] 1839 ‡ / [70 , +1 ] 1840 ‡
12 4 0 0 25 - 0 0 - - - / [70 , +1 ] 1844 ‡ / [70 , +1 ] 1851 ‡ / [70 , +1 ] 1863 ‡ / [70 , − ] 1887 ‡ / [70 , − ] 1937 0 0 0 0 0 - 0 0 - - - / [70 , − ] 1942 ‡ / [56 , +3 ] 1943 ‡ / [70 , − ] 1969 0 0 0 0 0 0 0 0 - - - TABLE IX: As Table VIII, but for missing ∆ resonances.∆ Mass Nπ Σ K ∆ π ∆ η Σ ∗ K NρU (7) Model / [70 , +1 ] 1764 ‡ / [70 , − ] 1946 0 0 0 0 0 0 / [70 , +1 ] 1947 1 3 106 5 1 80 / [70 , +1 ] 1947 ‡
18 1 107 18 4 32 / [70 , − ] 1904 ‡ / [70 , − ] 1904 0 0 0 0 0 0hQM / [70 , +1 ] 1832 ‡ / [70 , +1 ] 1843 4 1 43 1 - 51 / [70 , − ] 1947 ‡ / [70 , − ] 1947 ‡ / [70 , +1 ] 1859 ‡
10 0 97 7 - 13 / [56 , +3 ] 2103 0 0 0 0 0 0 both studies are characterized by the same problems (afew results are far from data, like Γ N (1700) D → ∆ π ) andstrengths (a great number of data is fitted with a fewparameters); among other things, CR obtained for thedecay widths N (1535) S → N π , N (1700) D → ∆ π and N (1720) P → N ρ , the following theoretical results216 MeV, 778 MeV, and 11 MeV, respectively, which areto be compared with the experimental data 44 −
96 MeV,10 −
225 MeV and 105 −
340 MeV, respectively.We can also compare our results with those of Refs.[23, 40]. Bijker, Iachello and Leviatan (BIL) [23, 40] com-puted the open-flavor decay amplitudes within a modifiedversion of the elementary meson emission model (EME),with two parameters, and used the U(7) algebraic modelto calculate the baryon spectrum. The EME is an effec-tive model, in which the decay occurs by the emission of ameson from the decaying hadron. Even though the EMEand P models share some features, there are importantdifferences. For example, the internal quark dynamics isinvisible to the EME vertex, because it does not dependon the meson internal wave function, which brings a non-local character to the P matrix elements. Moreover, asin the CR case, BIL did not consider a suppression of s ¯ s pair-creation, though it has been shown that this mecha-nism can be beneficial in meson strong decays [28, 30, 31].It can also be shown that it is impossible to get a corre-spondence between EME and P models, unless the fac-tor in front of the recoil term in the EME is taken equal to k / m = 1. However, this factor has been taken as a freeparameter in BIL (and also in other EME studies, Refs.[20, 23, 40]) and by fit it came out to be equal 0.04, whichmeans that the recoil term is practically absent in BIL.This fact, in combination with the non local character ofthe P model, the quark form factor and flavor suppres- sion, explain the differences in the two model predictions.A more detailed explanation is contained in Appendix F.Finally, we can say that the general quality of our re-sults is comparable to those of Refs. [23, 40]. Both ofthem reproduce the general trend of the data, but, insome cases, they show a few large disagreement with ex-periments. For example, the results of Refs. [23, 40]do not agree with the data in the case of the decays N (1720) P → Λ K and N (1440) P → ∆ π , where theyget null amplitudes, while ours do agree. On the contrary,our results for other channels, like N (1520) D → ∆ π and Λ(1670) S → N ¯ K , do not agree with the data, be-ing much larger, while those of Refs. [23, 40] do agree orthey are closer.There was also an attempt to improve the study ofthe strong decays using covariant calculations with rel-ativistic constituent quark models (rCQM) by Melde etal. [56, 57]. The authors computed the transitions forall π , η , and K strong decay modes of several well-established non-strange and strange baryon states, usingthe so-called point-form spectator model (PFSM), whosenon relativistic limit is the classic EME; they did not cal-culate decay amplitudes into baryon-vector meson pairs.Unfortunately, these results still cannot provide a sat-isfactory explanation of the experimental decay widthsand, in general, underestimate the available experimentaldata. Finally, it is worthwhile to cite the results of a dy-namical coupled-channels study of πN → ππN reactionsof Ref. [58], where the authors provided a comprehensiveanalysis of world data of πN , γN and N ( e, e ′ ) reactions,including a coupled channel model for meson productionreactions considering all possible final states. D. Exotic states and alternative decay modes
As widely discussed in the literature, there are severalbaryons (mesons) whose nature may not be a pure three-quark (quark-antiquark) one. Some well-known examplesare the X (3872) [28, 29, 38, 59–62] and the D ∗ s (2317) and D s (2460) [63, 64] mesons. Some alternative hypotheses(hadron-hadron molecules, hybrids, tetra/penta-quarks,and so on) might explain why na¨ıve quark models failto reproduce some of the main properties of these reso-nances, including mass and decay modes.Let us focus on the Λ ∗ (1405). By combining U(7) and P models, we get a mass of 1641 MeV and a Σ π ampli-tude of 230 MeV; these numbers can be compared withexperimental results, namely 1402 − −
52 MeV (Σ π amplitude) [1]. Such a strong devi-ation between theoretical predictions and data might beexplained if we interpret the Λ ∗ (1405) as a baryon-mesonmolecular state (for example, see Ref. [65, 66]). Giventhis, our predictions would refer to the lowest-lying qqq state with the same quantum numbers as the Λ ∗ (1405).The ∆(1930) D represents another failure of our P /QM predictions, as we get a null N π width, whileit should be in the range 11 −
75 MeV [1]. Nevertheless,2
TABLE X: As Table VIII, but for missing Σ resonances.Σ Mass NK Σ π Λ π Σ η Ξ K ∆ K Σ ∗ π Σ ∗ η Ξ ∗ K NK ∗ Σ ρ Λ ρ Σ ω ∆ K ∗ U (7) Model / [70 , − ] 1822 24 11 6 20 10 5 4 – – – – – – – / [70 , − ] 1822 22 4 8 0 0 602 98 – – – – – – – / [70 , +1 ] 1822 ‡ J [20 , +1 ] 1849 ‡ / [56 , +1 ] 1872 4 62 19 23 12 17 6 – – 12 – – – – / [70 , +1 ] 1926 0 0 0 5 1 65 12 – – 25 – 3 – – / [70 , +1 ] 1999 1 31 1 7 14 28 8 1 – 26 11 6 1 – / [70 , +1 ] 1999 4 76 6 1 1 60 13 4 – 7 13 21 3 – / [70 , − ] 1999 0 0 0 0 0 0 0 0 – 0 0 0 0 – / [70 , +1 ] 2095 4 2 1 3 4 18 4 4 1 23 5 9 7 – / [70 , +1 ] 2095 2 1 1 2 2 84 18 13 2 39 7 14 11 – / [70 , +1 ] 2095 ‡
15 3 5 1 0 128 29 18 3 45 8 15 11 – / [70 , +1 ] 2095 69 13 24 2 1 54 15 1 0 17 0 3 1 – J [70 , − ] 2095 0 0 0 0 0 0 0 0 0 0 0 0 0 – / [70 , − ] 1957 ‡ / [70 , − ] 1957 0 0 0 0 0 0 0 0 – 0 – 0 – – / [70 , − ] 2055 0 0 0 0 0 0 0 0 – 0 0 0 0 – / [70 , − ] 2055 0 0 0 0 0 1 0 0 – 0 0 0 0 – / [70 , − ] 2055 0 0 0 0 0 0 0 0 – 0 0 0 0 –hQM / [56 , +1 ] 1906 4 69 22 19 20 20 7 – – 21 – – – – / [70 , − ] 1914 9 7 3 17 22 22 9 – – 100 – – – – / [56 , +3 ] 2050 † / [70 , +1 ] 2072 1 31 1 7 16 39 11 – – 29 – – – – / [70 , +1 ] 2072 5 89 7 3 3 66 15 – – 11 – – – – / [70 , − ] 2149 0 0 0 0 0 0 0 0 0 0 0 0 0 0 / [70 , +1 ] 2187 3 2 1 3 4 16 3 5 1 21 6 9 2 16 / [70 , +1 ] 2187 1 1 1 1 2 91 21 17 5 40 10 15 3 18 / [70 , +1 ] 2187 19 3 6 1 0 147 34 23 6 52 10 18 3 49 / [70 , +1 ] 2187 83 15 29 2 1 83 21 3 0 30 2 7 1 159 J [20 , +1 ] 2238 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J [70 , − ] 2263 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TABLE XI: As Table VIII, but for missing Σ ∗ resonances.Σ Mass NK Σ π Λ π Σ η Ξ K ∆ K Σ ∗ π Σ ∗ η Ξ ∗ K NK ∗ Σ ρ Λ ρ Σ ω ∆ K ∗ U (7) Model / [70 , − ] 1755 4 5 4 11 – 1 30 – – – – – – – / [70 , − ] 1755 9 6 14 0 – 181 165 – – – – – – – / [70 , +1 ] 1863 0 1 0 1 0 45 39 – – 5 – – – – / [56 , +1 ] 2012 12 18 16 35 14 21 17 0 – 24 6 28 76 – / [56 , +1 ] 2012 6 9 8 18 7 79 69 1 – 35 9 40 106 – / [56 , +1 ] 2012 11 7 15 1 0 112 101 1 – 37 9 41 106 – / [70 , +1 ] 2037 ‡ / [70 , +1 ] 2037 5 4 7 1 0 63 56 1 0 10 2 9 62 – J [70 , − ] 2037 0 0 0 0 0 0 0 0 0 0 0 0 0 – / [56 , +2 ] 1765 ‡ J [70 , − ] 1996 0 0 0 0 0 0 0 0 – 0 0 0 0 –hQM / [56 , +2 ] 1883 6 9 8 11 2 68 63 – – 14 – – – – / [56 , +1 ] 2085 6 9 9 18 8 86 76 2 – 39 25 53 19 – / [70 , +1 ] 2136 0 1 1 2 1 61 63 0 2 29 25 43 20 1 / [70 , +1 ] 2136 7 5 9 2 0 70 63 1 10 17 7 18 5 4 J [70 , − ] 2212 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TABLE XII: As Table VIII, but for missing Λ resonances.Λ Mass NK Σ π Λ η Ξ K Σ ∗ π Ξ ∗ K NK ∗ Σ ρ Λ ωU (7) Model / [70 , − ] 1799 0 15 1 – 447 – – – – J [20 , +1 ] 1826 0 0 0 0 0 – – – – / [70 , +1 ] 1904 0 3 4 2 54 – 0 – 0 / [70 , +1 ] 1978 27 6 4 10 31 – 56 1 4 / [70 , +1 ] 1978 109 12 2 1 58 – 123 3 16 J [70 , − ] 1978 0 0 0 0 0 – 0 0 0 J [70 , − ] 2074 0 0 0 0 0 0 0 0 0 / [70 , +1 ] 2075 0 13 11 12 17 1 0 20 9 / [70 , +1 ] 2075 0 6 6 6 82 4 0 28 13 / [70 , +1 ] 2075 0 51 10 2 57 0 0 1 2 J [70 , − ] 1936 0 0 0 0 0 – 0 – 0 / [70 , − ] 2034 0 0 0 0 0 0 0 0 0 / [70 , − ] 2034 0 0 0 0 1 0 0 0 0 / [70 , − ] 2034 0 0 0 0 0 0 0 0 0hQM / [70 , − ] 1837 0 15 2 – 477 – – – – / [70 , +1 ] 1995 38 8 0 10 29 – 55 2 4 / [70 , − ] 2072 0 0 0 0 0 – 0 – 0 / [70 , − ] 2072 0 0 0 0 0 0 0 0 0 / [70 , +1 ] 2110 0 0 1 4 35 11 0 41 8 / [70 , +1 ] 2110 0 18 13 12 35 2 0 23 9 / [70 , +1 ] 2110 0 10 6 2 87 7 0 33 14 / [70 , +1 ] 2110 0 50 10 2 19 0 0 2 2 J [20 , +1 ] 2160 0 0 0 0 0 0 0 0 0 / [70 , − ] 2186 0 0 0 0 0 0 0 0 0 / [70 , − ] 2186 0 0 0 0 1 0 0 0 0 / [70 , − ] 2186 0 0 0 0 0 0 0 0 0TABLE XIII: As Table VIII, but for missing Λ ∗ resonances.Λ Mass NK Σ π Λ η Ξ K Σ ∗ π Ξ ∗ K NK ∗ Σ ρ Λ ωU (7) Model / [70 , +1 ] 1756 29 44 14 – – – – J [20 , +1 ] 1891 0 0 0 0 0 – – / [70 , +1 ] 1939 35 66 36 17 39 – 6 / [70 , +1 ] 1939 88 85 10 0 94 – 15 J [70 , − ] 1939 0 0 0 0 0 – 0 J [70 , − ] 1896 0 0 0 0 0 – 0hQM / [70 , − ] 2008 0 1 0 0 – – 1 – 0 / [70 , − ] 2008 0 0 0 0 – – 0 – 0 unlike the Λ ∗ (1405) case, here we obtain a predictionfor the mass which is compatible with the experimen-tal data [1]. Possible explanations of this deviation fromthe data include the eventuality that the ∆(1930) D is a ∆ ρ bound state [67] or simply the fact that the N π decay may proceed in a different way. It would bethus worthwhile to investigate whether the inclusion ofbaryon-meson higher Fock components in ∆(1930) D ’swave function via the unquenched quark model (UQM)formalism [68] may help to solve this problem. Similarissues also occur in the Σ(1750) and Σ(1940) cases. SeeRefs. [69–71].Another interesting case of departure from experimen-tal data is that of the ∆(1700) D ∆ π width [1], forwhich we predict a larger value. On the contrary, in themolecular picture this decay mode is dynamically sup-pressed. It would be worthwhile to investigate this casein the UQM formalism and see if the introduction of con-tinuum components, also determining a renormalizationof the ∆(1700) wave function, may improve the qualityof our result. It would also be interesting to calculate thecouplings for the ∆ η virtual channel, which is relevant toseveral reactions. This coupling has been evaluated inthe molecular model in Refs. [72–74].In addition to a calculation of the decay amplitudeswithin the UQM, it would also be interesting to inves-tigate baryon-meson-meson decays. A possible way todo that is via the formalism of quasi-two-body decays,discussed in Refs. [3, 15, 29]. In quasi-two-body decays,the decay of a baryon A into a baryon B and mesons C and C proceeds as A → B ∗ C → BC C , where B ∗ isa baryon resonance. Alternately, one may also decide touse the coupled-channel approach. Of particular inter-est is the N (1710) D → N ππ decay mode, which is themain one of the N (1710) (40 − N π width, even if thelatter has more phase space for the decay. For example,see Refs. [75].
V. SUMMARY AND CONCLUSION
We computed the open-flavor strong decays of lightbaryons (i.e. made up of u , d , s valence quarks) intobaryon-pseudoscalar and baryon-vector mesons using amodified version of the P pair-creation model [10, 11],in which we considered a flavor-dependent pair-creationstrength to suppress the contributions from heavier q ¯ q pairs, like s ¯ s with respect to u ¯ u ( d ¯ d ).The baryon models, which we used in our study to getpredictions for missing or higher-lying states, were the U (7) [40] and hypercentral [41, 46] models. The possibil-ity of using two models to extract the baryon spectrummakes it possible to give two different points of view, es-pecially in the study of the energy region above 1 . − TABLE XIV: As Table VIII, but for missing Ξ resonances.Ξ Mass Σ K Λ K Ξ π Ξ η Σ ∗ K Ξ ∗ π Λ K ∗ Σ K ∗ Ξ ρ Ξ ωU (7) Model / [70 , +1 ] 1932 36 6 1 11 7 13 – – – – / [70 , − ] 1932 43 20 69 0 0 4 – – – – / [70 , − ] 1932 4 7 22 0 216 152 – – – – / [70 , − ] 1932 23 39 132 0 2 19 – – – – J [20 , +1 ] 1957 0 0 0 0 0 0 – – – – / [56 , +1 ] 1979 198 7 6 47 4 7 – – – – / [56 , +1 ] 1979 59 5 4 1 20 27 – – – – / [70 , +1 ] 2031 2 1 3 0 24 19 2 – – – / [56 , +2 ] 1727 26 4 3 – – 2 – – – –hQM / [56 , +2 ] 1843 125 6 5 – – 15 – – – – / [70 , − ] 2053 8 11 37 0 223 154 – – – – / [56 , +3 ] 2190 0 0 0 0 0 0 0 0 0 0 / [70 , − ] 2288 0 0 0 0 0 0 0 0 0 0 / [70 , − ] 2288 0 0 0 0 0 0 0 0 0 0 / [70 , +1 ] 2327 3 1 8 1 6 5 8 10 40 1 / [70 , +1 ] 2327 2 1 4 0 35 32 16 16 62 1 / [70 , +1 ] 2327 6 7 24 0 57 53 20 18 69 1 / [70 , +1 ] 2327 26 33 108 1 33 33 11 5 16 0 J [20 , +1 ] 2377 0 0 0 0 0 0 0 0 0 0 J [70 , − ] 2403 0 0 0 0 0 0 0 0 0 0TABLE XV: As Table VIII, but for missing Ξ ∗ resonances.Ξ Mass Σ K Λ K Ξ π Ξ η Σ ∗ K Ξ ∗ π Λ K ∗ Σ K ∗ Ξ ρ Ξ ωU (7) Model / [70 , − ] 1869 17 10 7 7 – 7 – – – – / [70 , − ] 1869 5 10 9 0 – 61 – – – – / [70 , +1 ] 1971 2 1 1 2 51 14 – – – – / [56 , +2 ] 1878 19 16 13 1 – 12 – – – –hQM / [56 , +2 ] 2022 19 12 13 12 – 24 – – – – / [56 , +1 ] 2225 33 19 23 35 33 7 40 34 31 16 / [56 , +1 ] 2225 17 10 12 17 133 29 61 50 44 23 / [56 , +1 ] 2225 14 19 16 3 194 44 66 51 45 24 / [56 , +1 ] 2225 64 87 70 13 56 16 13 3 2 1 / [70 , − ] 2352 0 0 0 0 0 0 0 0 0 0 / [70 , − ] 2352 0 0 0 0 1 0 0 0 0 0 / [56 , +3 ] 2369 0 0 0 0 0 0 0 0 0 0 TABLE XVI: As Table VIII, but for missing Ω resonances.Ω Mass Ξ K Ξ ∗ K Ω η Ξ K ∗ U (7) Model / [70 , − ] 1989 68 – – – / [70 , − ] 1989 20 – – – / [70 , +1 ] 2085 8 32 – – / [56 , +2 ] 1998 79 – – –hQM / [70 , − ] 2142 26 48 – – / [70 , − ] 2142 68 403 – – / [56 , +2 ] 2162 68 102 – – / [56 , +2 ] 2364 109 34 27 155 / [56 , +2 ] 2364 55 137 88 225 / [56 , +2 ] 2364 69 199 117 234 / [56 , +2 ] 2364 308 58 4 23 / [70 , − ] 2492 0 0 0 0 / [70 , − ] 2492 0 1 0 0 / [56 , +3 ] 2508 0 0 0 0TABLE XVII: Parameter values used in the calculations,in combination with the relativistic phase space factor ofEq. (21). The parameter values are fitted to a sampleof 9 transitions: ∆ → Nπ , N (1520) → Nπ , N (1535) → Nπ , N (1650) → Nπ , N (1680) → Nπ , N (1720) → Nπ ,∆(1905) → Nπ , ∆(1910) → Nπ and ∆(1920) → Nπ . Thequantum number assignments for the decaying states are nowtaken from the hQM results of Ref. [41, 46] and Table III.Parameter Value γ α b α c α d m n m s predict and also in the quantum number predictions forsome *** and **** states, like the N (1875) D . In a sub-sequent paper [78], the present results will be extendedup to to an energy region (2.5 GeV) which will be testedby forthcoming experiments at the JLab.It is worthwhile to enumerate some of the difficultiesand problems connected to this type of calculations. Oneproblem is related to the difficulty of assigning quantumnumbers to resonances within a QM. Sometimes, thiscan generate strong conflicts between theoretical resultsand experimental data. For example, this is the case ofthe N (1875) D , whose decay amplitudes change signifi-cantly whether we use the P model in combination withthe U(7) or hQM models. See Tables III.. Another prob-lem has to do with the large quantity of decay thresholds,sometimes lying at similar energies or almost overlapping with one another. Thus, we think that a more completestudy would require the introduction of continuum cou-pling effects (i.e. higher Fock components) in the baryonwave functions. In some cases, the presence of a thresh-old can deeply influence the quark structure of a hadron,close in energy, as in the well-known case of the X (3872)meson [28, 29, 60, 61]. In this respect, it is worthwhile tocite the results of the EBAC project, developed by Mat-suyama et al. [76]. This is a dynamical coupled-channelmodel for investigating the nucleon resonances in the me-son production reactions induced by pions and photons.See also the interesting coupled-channel model results ofthe Bonn-J¨ulich group of Refs. [77]. A similar formal-ism, which would make it possible to include meson cloudeffects in baryon and meson open- and hidden-flavor de-cays, will be the subject of a subsequent paper [78].Finally, one of the most important points is related tothe problem of the missing resonances. Is this a matter ofdegrees of freedom? In this case, the use of other typesof models, characterized by a smaller number of effec-tive degrees of freedoms, such as the quark-diquark one,could, at least partially, solve the problem [79]. Other-wise, does it have to do with the coupling of these missingstates with other types of decay channels, more difficultto observe? This is still an open question. Thus, wethink that it would be worthwhile to compare the resultsfor spectrum and decays of a three quark QM to thoseof other type of models, such as the quark-diquark one.Moreover, in the case of higher lying states, it would alsobe interesting to compare the predictions of three quarkmodels to those for hybrid baryons, where baryons are de-scribed as bound states of three constituent quarks anda constituent gluon [80]. Appendix A: Pair-creation vertex
Analogously to what is done in Ref. [81], we can studydifferent forms for the pair-creation vertex, to improvethe description of the experimental data. The determi-nation of the best vertex results from a χ -analysis basedon a sample of 9 transitions: ∆ → N π , N (1520) → N π , N (1535) → N π , N (1650) → N π , N (1680) → N π , N (1720) → N π , ∆(1905) → N π , ∆(1910) → N π and∆(1920) → N π . The different forms we consider aregiven by V (2 p ) = e − α d p / V (2 p ) = (1 + γ p ) e − α d p / V (2 p ) = 1 + γ e − α d p / V (2 p ) = 1 + ( γ + γ p ) e − α d p / (A1)where p = ( ~p − ~p ) /
4. As observed in Ref. [81],the forms containing a p parameter, such as V ∝ e − α d ( p − p ) or 1 / [( p − p ) + B ], present a bump around p − p , and thus do not show the expected decreasingbehavior. Thus, we do not include them in our analy-7 TABLE XVIII: Comparison of the results obtained with dif-ferent vertex functions, fitted to a selected number of experi-mental strong decays [1]. Columns 2 − V i ofEq. (A1) in combination with the effective phase space factorof Eq. (22).Channel V V V V Exp (MeV)∆(1232) → Nπ
115 118 116 120 114 − N (1520) → Nπ
102 98 101 98 55 − N (1535) → Nπ
106 108 102 107 44 − N (1650) → Nπ
71 72 68 72 60 − N (1680) → Nπ
63 55 60 50 78 − N (1720) → Nπ
123 114 114 118 12 − → Nπ
14 14 14 14 24 − → Nπ
39 42 38 43 33 − → Nπ
14 16 14 16 9 − sis. The quality of the description of the experimentaldata provided by the four vertices is equivalent (see Ta-ble XVIII). Thus, we choose the vertex with the smallestnumber of free parameters, the first vertex with V ( ~p − ~p ) = e − α d ( ~p − ~p ) / . (A2)This is the one used in the calculations of Sec. IV andfor the analytic derivation of the P amplitudes of App.D. The reader may object that we did not consider in ouranalysis the simplest choice for the pair-creation vertex,namely V = 1. Actually, this particular choice is a specialcase of V (2 p ), i.e. when α d = 0. Appendix B: Spin Wave Functions
In the following, we list the conventions used for thespin wave functions [20]: S = 1 / | χ ρ / i = √ ( | ↑↓↑i − | ↓↑↑i ) , | χ λ / i = √ (2 | ↑↑↓i − | ↑↓↑i−| ↓↑↑i ) ,S = 3 / | χ S3 / i = | ↑↑↑i . (B1)We only show the state with the largest component ofthe projection M S = S . The other states are obtainedby applying the lowering operator in spin space. Appendix C: Flavor Wave Fuctions
The meson and baryon states are written according tothe usual prescriptions. Below, we list the conventionsused for the flavor wave functions of mesons and baryons.
Mesons
Since the mixing angle θ ηη ′ between η and η ′ is small,we take θ ηη ′ = 0. Thus, we identify η = η and η ′ = η . • The octet mesons | π + i = −| u ¯ d i| π i = 1 √ | u ¯ u i − | d ¯ d i ] | π − i = | d ¯ u i| K + i = −| u ¯ s i| K − i = | s ¯ u i (C1) | K i = −| d ¯ s i| ¯ K i = −| s ¯ d i| η i = 1 √ | u ¯ u i + | d ¯ d i − | s ¯ s i ] • The singlet mesons | η ′ i = 1 √ | u ¯ u i + | d ¯ d i + | s ¯ s i ] (C2) Baryons
For the baryon flavor wave functions, | ( p, q ) , I, M I , Y i ,we adopt the convention of Ref. [82]. We only showthe highest charge state M I = I with Q = I + Y /
2. Theother charge states are obtained by applying the loweringoperator in isospin space. • The octet baryons | (1 , , , , i : φ ρ ( p ) = √ [ | udu i − | duu i ]: φ λ ( p ) = √ [2 | uud i − | udu i−| duu i ] (C3) | (1 , , , , i : φ ρ (Σ + ) = √ [ | suu i − | usu i ]: φ λ (Σ + ) = √ [ | suu i + | usu i− | uus i ] (C4) | (1 , , , , i : φ ρ (Λ) = √ [2 | uds i − | dus i−| dsu i + | sdu i−| sud i + | usd i ]: φ λ (Λ) = [ −| dsu i − | sdu i + | sud i + | usd i ] (C5) | (1 , , , , − i : φ ρ (Ξ ) = √ [ | sus i − | uss i ]: φ λ (Ξ ) = √ [2 | ssu i−| sus i − | uss i ] (C6)8 • The decuplet baryons | (3 , , , , i : φ S (∆ ++ ) = | uuu i (C7) | (3 , , , , i : φ S (Σ + ) = √ [ | suu i + | usu i + | uus i ] (C8) | (3 , , , , − i : φ S (Ξ ) = √ [ | ssu i + | sus i + | uss i ] (C9) | (3 , , , , − i : φ S (Ω − ) = | sss i (C10) • The singlet baryons | (0 , , , , i : φ A (Λ) = √ [ | uds i − | dus i + | dsu i − | sdu i + | sud i − | usd i ] (C11) Appendix D: P amplitudes: general expression The color, spin and spatial parts of the P ampli-tude M A → BC ( q ) = h BCq ℓJ | T † | A i , excluding the fla-vor couplings, were derived in a harmonic oscillator basisby Roberts and Silvestre-Brac (RSB) [16]. They did notconsider a quark form factor ( α d = 0) and used the fol-lowing momenta ~p ρ = 12 ( ~p − ~p ) ,~p λ = 13 ( ~p + ~p − ~p ) ,~q c = 12 ( ~p − ~p ) ,~q = 12 (cid:16) ~K b − ~K c (cid:17) ,~P cm = ~K b + ~K c , (D1)with conjugate coordinates ~ρ = ~r − ~r ,~λ = 12 ( ~r + ~r − ~r ) ,~r c = ~r − ~r ,~r = 13 ~R b − ~R c ,~R cm = 16 ( ~r + ~r + ~r ) + 14 ( ~r + ~r ) . (D2)On the contrary, here we use the standard Jacobi coordi-nates and conjugate momenta for the initial baryon, A , and the final state baryon B and meson C [83, 84], ~ρ = 1 √ ~r − ~r ) ,~λ = 1 √ ~r + ~r − ~r ) ,~r c = ~r − ~r ,~r = ~R b − ~R c ,~R cm = m b ~R b + m c ~R c m b + m c , (D3)where m i and p i are the mass and momentum of thequark i (see Fig. 1), m b = m + m + m and m c = m + m the masses of the hadrons B and C . Theircenter of mass coordinates are defined as ~R b = 13 ( ~r + ~r + ~r ) ,~R c = 12 ( ~r + ~r ) , (D4)The conjugate momenta are given by ~p ρ = 1 √ ~p − ~p ) ,~p λ = 1 √ ~p + ~p − ~p ) ,~q c = 12 ( ~p − ~p ) ,~q = m c ~K b − m b ~K c m b + m c ,~P cm = ~K b + ~K c , (D5)and ~K b = ~p + ~p + ~p ,~K c = ~p + ~p . (D6)The final result is [83, 84] M A → BC ( q ) = 6 γ θ A → BC ǫ ( l λ b , l c , L bc , l, l λ a , L, q ) , (D7)where the angular momenta of the baryons A and B , L a and L b , are the sum of the ρ and λ oscillator con-tributions, l ρ and l λ , and the term θ A → BC contains thedependence on the color-spin-flavor part θ A → BC = F A → BC √ − l + l λa X L bc S bc ( − S a − S bc + L bc J ρ S b
12 12 S c S a S bc S b l λ b J b S c l c J c S bc L bc J bc X L ˆ L (cid:26) S a l λ a J a L S bc (cid:27) (cid:26) S bc L bc J bc l J a L (cid:27) . (D8)The coefficient F A → BC contains the flavor couplings andis defined as: F A → BC = h φ B φ C | φ φ A i . (D9)9Here, φ denotes the flavor wave function of the createdquark-antiquark pair | φ i = 1 p m n /m s ) h | u ¯ u i + | d ¯ d i + m n m s | s ¯ s i i , (D10)which, in the limit of equal quark masses, reduces to theusual expression for a flavor singlet. The coefficients insquare brackets are proportional to 9-j coefficients a b cd e fg h i = ˆ c ˆ f ˆ g ˆ h ˆ i a b cd e fg h i , (D11) where ˆ ℓ = √ ℓ + 1,Finally, ǫ ( l λ b , l c , L bc , l, l λ a , L, q ) represents the spatialcontribution [83, 84] ǫ ( l λ b , l c , L bc , l, l λ a , L, q ) = J N n λa l λa ( α b ) N ∗ n λb l λb ( α b ) N ∗ n c l c ( α c )( − L bc exp( − F q )2 G l λa + l λb + l c +4 X l ,l ,l ,l C l λb l C l c l C l C l λa l x − r ! l − r x ! l − r ! l ′ r x − ! l r ! l ′ x l X l ,l ,l ,l ,l ( − l + l ˆ l ˆ L l l ′ l λ b l l ′ l c l l L bc l l ′ l l ′ l λ a l l L (cid:26) l l l l L L bc (cid:27) B l l l B l ll B l l ′ l ′ B l l l B l l ′ l ′ X λ,µ,ν D λµν ( x ) I ν ( l , l , l , l ; L ) (cid:18) l ′ + l ′ + l ′ + l ′ + 2 µ + ν + 12 (cid:19) ! q l + l + l + l +2 λ + ν G µ + ν − l − l − l − l , (D12)with C Ll = s π (2 L + 1)!(2 l + 1)![2( L − l ) + 1]! B ll ,l = ( − l √ π ˆ l ˆ l (cid:18) l l l (cid:19) . (D13)Here l ′ = l λ b − l , l ′ = l c − l , l ′ = 1 − l , l ′ = l λ a − l and ˆ l = √ l + 1. J = 1 / √ { ~p , ~p , ~p , ~p , ~p } → { ~p ρ , ~p λ , ~q c , ~q, ~P cm } , seeEqs. (D5,D6), N n λa l λa ( α b ), N n λb l λb ( α b ) and N n c l c ( α c ), N n,L ( α ) = q n !Γ( n + L +3 / α − L − , (D14)are the normalization coefficients of the harmonic oscil-lator wave functions of the baryons A and B ,Φ nLM ( ~p λ , ~p ρ ) = X m h l ρ , m ; l λ , M − m | LM iN n ρ ,l ρ ( α b ) L l ρ +1 / n ρ ( p ρ /α b ) e − p ρ / α b Y l ρ m ( ~p ρ ) N n λ ,l λ ( α b ) L l λ +1 / n λ ( p λ /α b ) e − p λ / α b Y l λ M − m ( ~p λ ) , (D15)and meson C ,Φ n c l c m c ( ~q c ) = N n c l c ( α c ) L l c +1 / n c ( q c /α c ) e − q c / α c Y l c m c ( ~q c ) , (D16)where n is the number of nodes in the harmonic oscil-lator wave function, L L +1 / n ( αp ) a Laguerre polynomial and Y LM ( ~p ) a solid spherical harmonic. The remaining0coefficients are given by: G = α b + 13 α + 13 α ,x = 2 α + α + 2 α √ G ,F = α (12 α + 5 α ) + α (20 α + 3 α )24 (3 α + α + α ) . (D17)The present results for the P amplitudes were obtainedin a consistent way using the same Jacobi coordinates forthe baryon wave functions and the P matrix elements.We observe that the coefficients of Eqs. (D17) also de-pend on the parameter α d of the Gaussian quark formfactor, V ( ~p − ~p ) = e − α ( ~p − ~p ) / . The case V = 1 is asubcase of the previous one.For the special case of ground-state baryons and pseu-doscalar mesons, the orbital angular momenta vanish l λ a = l λ b = l c = L bc = 0 and therefore J a = S a , J b = S b , J c = S c = 0 and J bc = S bc = J b . Due to the conservationof angular momentum and parity, the relative orbital an-gular momentum between the baryon B and the meson C is equal to l = 1 = L . As a result, the general expressionfor the P transition amplitude simplifies considerably.The color-spin-flavor part θ A → BC reduces to θ A → BC = − r J b + 12 ( − J ρ + J a − (D18) × (cid:26) J a J b J ρ (cid:27) F A → BC , (D19)where J ρ the total angular momentum of the first twoquarks and F A → BC the flavor matrix elements of the A → BC transition [16]. The spatial contribution sim-plifies to ǫ ( q ) = − (cid:18) α b α c π (cid:19) / (4 α b + α c ) q e − F q (3 α b + α c + α d ) / . (D20) Appendix E: Flavor couplings
In the following, we give the flavor coefficients F A → BC .These expressions are valid for both pseudoscalar andvector mesons. • A → B + C couplings. For octet baryons the flavor wave function has twocomponents, labeled by ρ and λ , both of which giverise to the flavor coupling coefficient F ρA → BC = h φ ρ ( B ) φ ( C ) | φ φ ρ ( A ) i , F λA → BC = h φ λ ( B ) φ ( C ) | φ φ λ ( A ) i , (E1)where we introduced the superscripts ρ and λ todistinguish the two contributions. In this case, the flavor couplings are given by N ΣΛΞ → N π N η Σ K Λ KN ¯ K Σ π Λ π Σ η Ξ KN ¯ K Σ π Λ η Ξ K Σ ¯ K Λ ¯ K Ξ π Ξ η = N √ √ − √ m n m s √ √ √ − √ m n m s − √ − mnms √ m n m s √ √ − √ m n m s (E2)for the ρ component and by N − √ √ √ m n m s √ √ − √ − mnms √ √ m n m s √ √ m n m s − √ √ √ √ − mnms )9 (E3)for the λ component. The normalization coefficientis given by N = vuut
32 + (cid:16) m n m s (cid:17) . (E4) • A → B + C couplings. The flavor coefficients for octet baryons in combi-nation with a singlet meson are given by N ΣΛΞ → N η Σ η Λ η Ξ η = N mnms m n m s (E5)for the ρ component and by N mnms mnms (E6)for the λ component.1 • A → B + C couplings. In this case the final baryon belongs to the decupletwhich only couples to the λ component of the initialoctet baryon F A → BC = h φ S ( B ) φ ( C ) | φ φ λ ( A ) i . (E7)The flavor coefficients are given by N ΣΛΞ → ∆ π Σ ∗ K ∆ ¯ K Σ ∗ π Σ ∗ η Ξ ∗ K Σ ∗ π Ξ ∗ K Σ ∗ ¯ K Ξ ∗ π Ξ ∗ η Ω K = N − m n m s − √ √ √ √ mnms − √ √ m n m s √ − √ m n m s −
13 13 1+2 mnms − √ m n m s (E8) • A → B + C couplings. As in the previous case, the decuplet baryon onlycouples to the λ component of the octet baryon F A → BC = h φ λ ( B ) φ ( C ) | φ φ S ( A ) i . (E9)The flavor coefficients are given by ∆Σ ∗ Ξ ∗ Ω → N π Σ KN ¯ K Σ π Λ π Σ η Ξ K Σ ¯ K Λ ¯ K Ξ π Ξ η Ξ ¯ K = N − √ √ m n m s − √ √ √ √ −
13 1+2 mnms √ √ m n m s −
13 13 1+2 mnms (E10) • A → B + C couplings. The flavor coefficients for decuplet baryons in com-bination with a octet meson F A → BC = h φ S ( B ) φ ( C ) | φ φ S ( A ) i . (E11) are given by ∆Σ ∗ Ξ ∗ Ω → ∆ π ∆ η Σ ∗ K ∆ ¯ K Σ ∗ π Σ ∗ η Ξ ∗ K Σ ∗ ¯ K Ξ ∗ π Ξ ∗ η Ω K Ξ ∗ ¯ K Ω η = N √ √ √ − m n m s √ √ − √ mnms − − √ m n m s √
23 13 √ − (4 mnms − √ − m n m s √ − √ m n m s (E12) • A → B + C couplings. The flavor coefficients for decuplet baryons in com-bination with a singlet meson are given by ∆Σ ∗ Ξ ∗ Ω → ∆ η Σ ∗ η Ξ ∗ η Ω η = N mnms mnms m n m s (E13) Appendix F: Comparison between ElementaryMeson Emission and P Models
In the following, we shall compare the ElementaryEmission (EME) and P Models. In the EME model,the decay proceeds via the emission of a meson in termsof an elementary quantum. On the contrary, in the P Model the decay process is described in terms of the cre-ation of an additional quark-antiquark pair.In order to do a comparison between EME and P models, we consider the simplest form for the EME tran-sition operator [86], H s = g (2 π ) / (2 k ) / X c h ( ~σ · ~k )e − i~k · ~r (F1)+ k m ~σ · ( ~p e − i~k · ~r + e − i~k · ~r ~p ) (cid:21) , (F2)where g = G cqq / m is the ratio between the meson-emission strength (emission of a meson C by a quark)and the quark mass, X c is the flavor operator related tothe emission of meson C . On the other hand, the P operator in matrix form is( h C | T | i ) i ′ i = X i Ψ ∗ Ci i P i ′ i = P C † , (F3)where T is the P operator T † = − R d~p d~p δ ( ~p + ~p ) C F [ χ × Y ( ~p − ~p )] (0)0 b † ( ~p ) d † ( ~p ) . (F4)2In Ref. [86], it is shown that equation (F3) can also bewritten as P C † = γ π ) / e − i~k c · ~r X c h ~σ · ( ~k c − ~p ) i ψ C (2 ~p − ~k c ) . (F5)If we compare Eqs. (F2) and (F5), we can observethat in the P model there are direct and recoil termswith equal weights; on the contrary, the recoil term inthe EME has a different weight factor k / m . To havea perfect correspondence between EME and P models,this factor in the recoil term has to be taken equal to k / m = 1. However, this factor has been taken as afree parameter in BIL [23, 40] (and also in several otherstudies, like [20, 23, 40]) and, by fitting, it turned outto be equal to 0.04, which means that the recoil term ispractically absent in BIL.It is worthwhile noting that in the calculation of open-flavor strong decays for L b = L c = 0, the final BC stateis characterized by a relative angular momentum ℓ = L a ±
1, where L a is the angular momentum of the initialstate, A . In Ref. [86], it is shown that in the EME, thepartial wave ℓ = L a + 1 is essentially given by the directterm with a small correction from the recoil one, while ℓ = L a − k / m inthe EME operator can give rise to different results withrespect to the P model.Another important difference is due to the wave func-tion factor ψ C (2 ~p − ~k c ) in Eq. (F5), related to the com-posite structure of the meson C , which gives rise to a nonlocal character of the operator. Acknowledgments
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