J/ψ→p p ¯ ϕ decay in the isobar resonance model
aa r X i v : . [ nu c l - t h ] O c t J/ψ → p ¯ pφ decay in the isobar resonance model Jian-Ping Dai , , ∗ Peng-Nian Shen , , , , † Ju-Jun Xie , , ‡ and Bing-Song Zou , , , § Based on the effective Lagrangian approach, the
J/ψ → p ¯ pφ decay is studiedin an isobar resonance model with the assumption that the φ -meson is producedfrom intermediate nucleon resonances. The contributions from the N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100) states are considered. In terms of thecoupling constants g φNN ∗ and g φNN ∗ extracted from the data of the partial decaywidths of the N ∗ s to the N π channel, the reaction cross section of the π − p → nφ process and the partial decay widths of the J/ψ → p ¯ pη and J/ψ → p ¯ nπ − processes,respectively, the invariant mass spectrum and the Dalitz plot for J/ψ → p ¯ pφ arepredicted. It is shown that there are two types of results. In the type I case, a largepeak structure around 2.09GeV implies that a considerable mount of N φ or qqqs ¯ s component may exist in the narrow-width N ∗ / − (2090) state, but for the wide-width N ∗ / + (2100) state, it has little qqqs ¯ s component. In the type II case, a small peakaround 2.11GeV may only indicate the existence of a certain mount of pφ or qqqs ¯ s component in the narrow-width N ∗ / + (2100) state, but no information for the wide-width N ∗ / − (2090) state. Further BESIII data with high statistics would help us todistinguish the strange structures of these N ∗ s. PACS numbers: 13.75.-n, 13.75.Cs, 14.20.Gk ∗ Electronic address: [email protected] † Electronic address: [email protected]
I. INTRODUCTION
In past decades, many excited states of nucleon were observed and their properties, suchas the mass, width, decay modes, decay branching fractions and etc., were more or lessaccurately measured. Most of these states and their properties can be well-explained byquark models, but some of them cannot be fitted into the nucleon spectrum predicted bythe three-valence-quark model. To explain the discrepancy, except that the data is lackof higher accuracy and statistics due to the limited experimental technique and method,one speculated that these states may contain some constituents other than three u and d valance quarks, especially the s and ¯ s quarks, and suggested to check this conjecture throughexperiments. Later, in the high energy physics and nuclear physics experiments, throughthe data analysis, one found that some excited states of nucleon ( N ∗ ) couple strongly withstrange particles. For instance, in either J/ψ → ¯ pK + Λ decay, or pp → p Λ K + reactionnear the kaon-production threshold[1, 2], or γp → K + Λ kaon-photoproduction process[3–6], N ∗ (1535) has a significant strength of coupling to the K Λ channel. This indicates that the N ∗ (1535) state may contain a considerable mount of s ¯ s component, which is consistent witha very large branching fraction of 45 ∼
60% for the N ∗ (1535) → N η decay.On the other hand, the φ -meson is mainly composed of s ¯ s . According to the Okubo-Zweig-Iizuka (OZI) rule [7], the production rate of φ -meson in the nuclear process would besuppressed if the initial interacting particles do not contain a constituent with s and ¯ s quarks.On the contrary, if a N ∗ contains strange constituents, its coupling with a channel involvinga φ -meson might be relatively strong. In fact, it is found that the pp → ppφ and π − p → nφ reaction data can be well-explained as long as the coupling constant of φN N ∗ (1535) issufficiently large, which implies that such a significant coupling is closely related to a factthat a considerable mount of s ¯ s component is involved in the wave function of N ∗ (1535) [8].Therefore, in the charm physics experiment at BESIII, say the measurements of J/ψ hadronicdecays, the
N φ decay channel of
J/ψ would also be a good place to check whether some N ∗ s , as the intermediate states in the decay process, have strange components, althoughthe branching fractions of such decays are not large. ‡ Electronic address: [email protected] § Electronic address: [email protected]
Similar to N ∗ (1535), some nucleon resonances which have not yet been well-establishedand cannot be fitted into the nucleon spectrum from theoretical models have remarkablebranching fractions in some decay channels involving strange particles, say N η , Λ K and etc.For instance, the branching fractions for N ∗ (2090) → N η , N ∗ (2100) → N η and N ∗ (1900) → N η are about 41%, 61% and 14%, respectively [19]. This implies that these states mighthave sizable strange constituents, and the effect of such ingredients should show up in the
J/ψ → p ¯ pφ decay.In fact, the branching fraction of J/ψ → p ¯ pφ was measured by the DM2 Collaboration in1988 [9]. However, due to the insufficient statistics, no resonance information was extracted.Recently, the luminosity of BEPCII has reached over 3 × cm − s − around J/ψ peak,a huge amount of
J/ψ events, say 3 × , will be collected at BESIII in one year. Thenew data set would offer an opportunity to study the possible strange ingredient or evenpentaquark in the nucleon resonance.Based on the mostly accepted assertion that the J/ψ → p ¯ pφ decay is dominated bya process with intermediate nucleon resonances, so-called resonance model, we study thepossibility of strange ingredients in the mentioned resonances through this decay in aneffective Lagrangian approach. It is our hope that the information of the strange structurein nucleon resonances, especially those which are not well-established, can be deduced, anda reference for coming BESIII data analysis can be provided.The paper is organized in the following way. In Section II, the theoretical model andformalism are briefly introduced. The results are presented and discussed in Section III.And in Section IV, a concluding remark is given. II. MODEL AND FORMALISM
In the resonance model, the
J/ψ → p ¯ pφ decay undergoes a two-step process, namely J/ψ firstly decays into an intermediate ¯ pN ∗ ( p ¯ N ∗ ) state, and then N ∗ ( ¯ N ∗ ) successivelydecays to φ and p (¯ p ). Corresponding Feynman diagrams are drawn in Fig.1, where k , p , p , p and q ( q ′ ) are the four-momenta of J/ψ , p , ¯ p , φ and N ∗ ( ¯ N ∗ ), respectively. The embed-ded intermediate N ∗ state should have following characters. Its mass, in principle, shouldrange from m p + m φ to m J/ψ − m ¯ p , namely from 1.96GeV to 2.15GeV due to the phasespace restriction. However, some N ∗ states whose mass is smaller than the pφ threshold J/ψ ( k ) φ ( p )¯ p ( p ) p ( p ) J/ψ ( k ) p ( p ) ¯ p ( p ) φ ( p ) N ∗ ( q ) ¯ N ∗ ( q ′ ) FIG. 1: Feynman diagrams for the
J/ψ → p ¯ pφ decay in the resonance model. may also contribute, because of their relatively larger branching fractions for some decaychannels involving strange particles and their off-shell effect as well. The spin of the in-termediate N ∗ can be any half-integer due to a relative angular momentum between N ∗ and ¯ p . For minimizing our calculation without affecting our qualitative conclusion, in thepresent approach, we only consider those intermediate N ∗ states whose contributions maydominate the decay width. Thus, the selected N ∗ state should have following features:It should have a relatively large branching fraction for a decay in which strange particlesare involved. As a consequence, it might have, according to the OZI rule, a configurationwith strangeness, so that it would be easier decaying into N φ and relatively important inthe
J/ψ → p ¯ pφ decay. It would also be reasonable to take the embedded N ∗ state withits spin up to 5 / N ∗ state with higher spin wouldencounter a power suppression due to a large relative angular momentum. In the practi-cal calculation, in the mass region above the pφ threshold, we only take N ∗ / − (2090) S and N ∗ / + (2100) P with the N η decay branching fractions of about 0 .
41 and 0 .
61, respec-tively, although the contribution from the later one would subject to a p − wave suppression,and ignore N ∗ / − (2080) D and N ∗ / + (2000) F because of their tiny N η branching frac-tions (about 0 . N ∗ / − (1535) S and N ∗ / + (1900) P into account due to their large N η branchingfractions of about 0 . − .
60 and 0 .
14, respectively. The reason for disregarding other sub-threshold resonances like N ∗ / − (1650) S , N ∗ / + (1710) P , N ∗ / + (1720) P , N ∗ / + (1680) F , N ∗ / − (1675) D , N ∗ / − (1520) D , N ∗ / + (1440) P , and N ∗ / + (939) P is as follows. For N ∗ / − (1650) S , its branching fractions to N η and Λ K are about 0.03 ∼ ∼ N ∗ / − (1535) S . Moreover, one arguedthat due to the weak coupling of N ∗ / − (1650) S to N ρ from SU(3) symmetry, the couplingbetween N ∗ / − (1650) S to N φ might also be weak [8]. In fact, if both N ∗ / − (1535) S and N ∗ / − (1650) S are used to fit the π − p → nφ data, the later one would give an inappropriatecontribution at the higher energies and the fitted result shows an almost zero contributionfrom N ∗ / − (1650) S [8]. For N ∗ / + (1720) P , its branching fractions to N η and Λ K are lessthan 0.04 and about 0.01 ∼ ∼ N ∗ / + (1900) P state. For N ∗ / + (1710) P , its branching fraction to N η is less than 0.06,which is almost 10 times less than that for N ∗ / + (2100) P , and its branching fraction to Λ K is about 0.05 ∼ N ∗ / + (2100) P . Addi-tionally considering the factor that N ∗ / + (1720) P and N ∗ / + (1710) P are the states belowthe N φ threshold, their contributions would be smaller than that from N ∗ / + (1900) P andmuch smaller than that from N ∗ / + (2100) P , respectively. In fact, their contributions areevidently small near the N φ threshold region in the π − p → nφ reaction. For N ∗ / + (939) P , N ∗ / + (1440) P , N ∗ / − (1520) D , N ∗ / − (1675) D , and N ∗ / + (1680) F , their branching frac-tions to the N η channel are almost zero, and the contributions from the D-wave and F-wavestates even suffer from the high-partial-wave suppression [8]. Based on such a discussion andthe result given by the partial wave analysis (PWA) in Ref. [11, 12], we can safely assumethat if N ∗ / − (1535) S , N ∗ / + (1900) P , N ∗ / − (2090) S and N ∗ / + (2100) P can give a con-tribution about 85% to 90% of the total, taking these four N ∗ states to study a system withstrange particles, for instance the J/ψ → p ¯ pφ process, would be meaningful. For simplicity,we omit the spectroscopic symbol in the notation of the N ∗ state hereafter.To reveal the decay property of the J/ψ → p ¯ pφ process, the coupling constants g φNN ∗ and g ψNN ∗ should be fixed at the beginning. A. Determination of g φNN ∗ As mentioned in Ref. [8], the φ -meson production near the threshold in the π − p → nφ reaction is dominated by the intermediate nucleon resonances in the s -channel, and the u -channel N ∗ exchange and the t -channel ρ -meson exchange between pion and proton are foundto be negligible, although in some references the t -channel ρ -meson exchange and/or nucleonpole contributions were assumed to be important [8, 13]. Based on this argument, thecoupling constant g φNN ∗ can be extracted by fitting the cross section data of the π − p → nφ reaction [8, 10]. The s -channel Feynman diagram for such a process is shown in Fig. 2, where p , p , p , p , and q denote the four-momenta of the incoming π − and proton, outgoing φ andneutron, and intermediate N ∗ , respectively. In this diagram, the coupling constant g πNN ∗ p p p φp π − p nN ∗ FIG. 2: s -channel Feynman diagram for the π − p → nφ reaction in the resonance model. ( g ηNN ∗ ) can be determined in terms of a commonly used effective Lagrangian [8, 14, 15]. Fora nucleon resonance with J PN ∗ = − where J and P denote its spin and parity, respectively,the effective Lagrangian can be written as [8, 14, 15] L πNN ∗ = g πNN ∗ ¯ N ∗ ~τ · ~πN + h.c., (1)with J PN ∗ =
12 + L πNN ∗ = ig πNN ∗ ¯ N ∗ γ ~τ · ~πN + h.c., (2)and with J PN ∗ =
32 + , say N ∗ (1900), [14] L πNN ∗ = i g πNN ∗ M N ∗ ¯ N ∗ µ ~τ · ∂ µ ~πN + h.c., (3)where g πNN ∗ , N ∗ µ , N , ~π and ~τ denote the coupling constant of a pion to a nucleon anda N ∗ , the Rarita-Schwinger field of N ∗ with its spin of 3/2 and mass of M N ∗ , the field ofnucleon, the field of ~π and the isospin matrices, respectively. And the effective Lagrangianfor the ηN N ∗ (1535) coupling can be expressed by: L ηNN ∗ = g ηNN ∗ ¯ N ∗ ηN + h.c.. (4)With these Lagrangians, the partial decay widths of the N ∗ states can easily be derivedby evaluating the transition from the initial N ∗ state to the final N π ( N η ) stateΓ N ∗ (1535) → Nπ = 3 g πNN ∗ ( m N + E N ) p cmπ πM N ∗ , (5)Γ N ∗ (1900) → Nπ = g πNN ∗ ( m N + E N )( p cmπ ) πM N ∗ , (6)Γ N ∗ (2090) → Nπ = 3 g πNN ∗ ( m N + E N ) p cmπ πM N ∗ , (7)Γ N ∗ (2100) → Nπ = 3 g πNN ∗ ( E N − m N ) p cmπ πM N ∗ , (8)Γ N ∗ (1535) → Nη = g ηNN ∗ ( m N + E N ) p cmη πM N ∗ (9)with p cmπ ( η ) = s ( M N ∗ − ( m N + m π ( η ) ) )( M N ∗ − ( m N − m π ( η ) ) )4 M N ∗ , (10)and E N = q ( p cmπ ( η ) ) + m N . (11)By re-producing the mass, the width and the πN ( ηN ) channel branching fraction of the N ∗ state [19] measured in the experiment, the phenomenological coupling constant g πNN ∗ ( g ηNN ∗ ) can be extracted.Furthermore, the effective Lagrangians of the φN N ∗ interaction for various N ∗ states areadopted as follows: For a N ∗ with J PN ∗ = − , say N ∗ (1535) or N ∗ (2090), [8] L φNN ∗ = g φNN ∗ ¯ N ∗ γ ( γ µ − q µ qq )Φ µ N + h.c. (12)with q being the four-momentum of N ∗ and Φ µ being the field of φ meson. For a N ∗ with J PN ∗ =
12 + , say N ∗ (2100), L φNN ∗ = g φNN ∗ ¯ N ∗ γ µ Φ µφ N + h.c., (13)and for a N ∗ with J PN ∗ =
32 + , say N ∗ (1900), [14] L φNN ∗ = ig φNN ∗ ¯ N ∗ µ γ Φ µφ N + h.c.. (14)To reckon for the off-shell effect of N ∗ , a form factor F N ∗ ( q ) = Λ Λ + ( q − M N ∗ ) (15)with Λ being the cut-off parameter is introduced in the M N N ∗ vertex [16, 17].The propagator G J P ( q ) of a N ∗ J P with the quantum number J P and momentum q can bewritten in a Breit-Wigner form [18]. For the J N ∗ = 1 / G ± ( q ) = i ( ± 6 q + M N ∗ ) q − M N ∗ + iM N ∗ Γ N ∗ , (16)where Γ N ∗ denotes the total decay width of the N ∗ state, and the + and − signs on the leftof q are the signs for the positive and negative parity states, respectively. For the J N ∗ = 3 / G µν ± ( q ) = i ( ± 6 q + M N ∗ ) q − M N ∗ + iM N ∗ Γ N ∗ ( − g µν + 13 γ µ γ ν ∓ M N ∗ ( γ µ q ν − γ ν q µ ) + 23 M N ∗ q µ q ν ) . (17)In terms of the effective Lagrangian, form factor and N ∗ J P propagator mentioned above,we use Feynman rules to write the invariant amplitude contributed by a N ∗ in the s − channel π − p → nφ reaction as M π − pN ∗ ∝ √ g πNN ∗ g φNN ∗ F N ∗ ( q )¯ u ( p n , s n )Γ φNN ∗ ϕ φ ( p φ , s φ ) G N ∗ ( q ) ϕ π ( p π , s π )Γ πNN ∗ u ( p p , s p ) , (18)where u , ϕ φ and ϕ π denote the fields of the nucleon, φ − meson and π − meson, respectively, p n , p p , p φ and p π represent the momenta of the proton, neutron, φ − meson and π − meson,respectively, s n , s p , s φ and s π describe the spins of the proton, neutron, φ − meson and π − meson, respectively, and Γ φNN ∗ and Γ πNN ∗ stand for the vertex functions of φN N ∗ and πN N ∗ , respectively. The formulae of the invariant amplitude for various N ∗ s are givenin Appendix A. Summing up all amplitudes for the N ∗ s considered, we obtain the totalinvariant amplitude M π − p → nφ = X N ∗ M π − pN ∗ , (19)where N ∗ runs over all the considered states. Consequently, we can calculate the total crosssection of the π − p → nφ reaction by using the following equation σ = Z d Φ ( P , p p , p π , p n , p φ ) (2 π ) q ( p p · p π ) − m p m π |M π − p → nφ | , (20)with d Φ being an element of the two-body phase space, and P being the total momentumof the system. By adjusting the coupling constants g φNN ∗ to fit the total cross section ofthe π − p → nφ reaction, we can extract a set of phenomenological g φNN ∗ . We would furthermention that the contributions from the u − channel and meson-exchange channel will notbe included in the calculation, because they are negligibly small [8]. B. Determination of g ψNN ∗ The coupling constant g ψNN ∗ can be extracted from the BESII data for the J/ψ → p ¯ nπ − and J/ψ → p ¯ pη decays [11, 12]. The Feynman diagrams for these decays are the same asthose in Fig.1 except that the φ -meson is replaced with the π - and η -mesons, respectively.The effective Lagrangian for the J/ψN N ∗ interaction can be chosen in the following form:For a N ∗ with J PN ∗ = − , say N ∗ (1535) or N ∗ (2090), [1] L ψNN ∗ = ig ψNN ∗ ¯ N ∗ γ σ µν p νψ ǫ µ ( ~p ψ , s ψ ) N + h.c. (21)with p ψ and ε ( p ψ ) being the four-momentum and the polarization vector of J/ψ , respectively,for a N ∗ with J PN ∗ =
12 + , say N ∗ (2100), L ψNN ∗ = g ψNN ∗ ¯ N ∗ γ µ ǫ µ ( ~p ψ , s ψ ) N + h.c., (22)and for a N ∗ with J PN ∗ =
32 + , say N ∗ (1900), [14] L ψNN ∗ = ig ψNN ∗ ¯ N ∗ µ γ ǫ µ ( ~p ψ , s ψ ) N + h.c.. (23)It should be mentioned that J/ψ meson produced in BEPCII is transversely polarized,namely s = ±
1. The completeness condition of polarization vector obeys X s = ± ǫ µ ( ~p, s ) ǫ ∗ ν ( ~p, s ) = δ µν ( δ µ + δ µ ) , (24)where ǫ µ ( ~p, s ), ~p and s denote the polarization vector, the momentum, and the polarizationdirection of J/ψ , respectively, and δ is a Kronecker Delta symbol.Then, the invariant decay amplitude contributed by a specific N ∗ in the J/ψ → p ¯ nπ − ( J/ψ → p ¯ pη ) decay can easily be written as M J/ψ decayN ∗ ∝ ξg π ( η ) NN ∗ g ψNN ∗ F N ∗ ( q )¯ u ( p p , s p )Γ π ( η ) NN ∗ ϕ π ( η ) ( p π ( η ) , s π ( η ) ) G N ∗ ( q ) ϕ ψ ( p ψ , s ψ )Γ ψNN ∗ v ( p ¯ p , s ¯ p ) (25)with u ( v ) being the field of proton (anti-proton), ϕ ψ and ϕ π ( η ) being the fields of ψ and π ( η ),respectively, p p , p ¯ p , p π ( η ) and p ψ being the momenta of the proton, anti-proton, ψ -meson and π - ( η -)meson, respectively, s p , s ¯ p , s π ( η ) and s ψ being the spins of the proton, anti-proton, ψ -meson and π - ( η -)meson, respectively, and Γ π ( η ) NN ∗ and Γ ψNN ∗ being the vertex functionsof π ( η ) N N ∗ and ψN N ∗ , respectively. The coefficient ξ is taken to be √ p ¯ nπ − p ¯ pη reaction. The formulae of the invariant amplitude for various N ∗ s are given in Appendix B. Consequently, the total invariant amplitude can be obtainedby summing over all possible N ∗ states M J/ψ decay = 12 X N ∗ M J/ψ decayN ∗ , (26)where the factor of 1/2 comes from the average over the J/ψ spin. The partial decay widthof
J/ψ → p ¯ nπ − ( J/ψ → p ¯ pη ) can be calculated by d Γ = 12 12 M ψ p p d p p m N p p d p ¯ p m N d p π ( η ) p π ( η ) X s ψ X s p ,s ¯ p ,s π ( η ) |M J/ψ decay | (2 π ) − δ ( p ψ − p p − p ¯ p − p π ( η ) ) . (27)By fitting the branching fractions of (2 . ± . × − for J/ψ → p ¯ nπ − and (2 . ± . × − for J/ψ → p ¯ pη [19], respectively, the magnitude of g ψNN ∗ can be extracted. C. J/ψ → p ¯ pφ decay Using the effective Lagrangians mentioned above, the invariant amplitude of the
J/ψ → p ¯ pφ decay can easily be derived. Its form is the same as that in Eq.(25) except that π issubstituted with φ M J/ψ → p ¯ pφN ∗ ∝ g φNN ∗ g ψNN ∗ F N ∗ ( q )¯ u ( p p , s p )Γ φNN ∗ ϕ φ ( p φ , s φ ) G N ∗ ( q ) ϕ ψ ( p ψ , s ψ )Γ ψNN ∗ v ( p ¯ p , s ¯ p ) . (28)The formulae of the invariant amplitude contributed by various N ∗ s are given in AppendixC. Then, the total invariant amplitude can be obtained by summing over the contributionsfrom all possible N ∗ states M J/ψ → p ¯ pφ = 12 X N ∗ M J/ψ → p ¯ pφN ∗ , (29)where the factor of 1/2 is due to the average over the J/ψ spin as usual. The invariant massspectrum of pφ in the J/ψ → p ¯ pφ decay can be expressed as [19] d Γ d Ω ∗ p d Ω ¯ p = 12 1(2 π ) (2 m p ) M ψ |M J/ψ → p ¯ pφ | | p ∗ p || ¯ p | dm pφ , (30)1where ( | p ∗ p | , Ω ∗ p ) is the momentum of proton in the rest frame of p and φ , and d Ω ¯ p is theangle of anti-proton in the rest frame of the decaying J/ψ . Integrating over all the anglesin the rest frame of
J/ψ , the Dalitz plot can be derived in the following form [19] d Γ = 12 1(2 π ) (2 m p ) M ψ |M J/ψ → p ¯ pφ | dm pφ dm pφ , (31)with m pφ = p pφ = ( p ψ − p ¯ p ) , m pφ = p pφ = ( p ψ − p p ) . (32) III. RESULTS AND DISCUSSION
Based on the discussion in the last section, only N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090)and N ∗ (2100) P are adopted as the intermediate state in the practical calculation. Thecoupling constant g π ( η ) NN ∗ for these N ∗ s are extracted by using the decay width formulasfor N ∗ → N π ( η ) shown in the last section, where the masses of N , N ∗ , π and η are taken fromPDG [19], namely m N =0.938GeV, m π =0.139GeV and m η =0.547GeV, M N ∗ (1535) =1.535GeV, M N ∗ (1900) =1.900GeV, M N ∗ (2090) =2.090GeV, and M N ∗ (2100) =2.100GeV, respectively [19].It should be noted that the total width and the branching fractions of N ∗ S (1535) (orthe partial decay widths) for the N π and
N η channels have more or less accurately beenmeasured, thus g πNN ∗ (1535) and g ηNN ∗ (1535) can be estimated by using the averaged valuesof branching fractions given in PDG [19]. However, for the two-star state N ∗ / + (1900) andone-star states N ∗ / − (2090) and N ∗ / + (2100), their partial decay widths for the N π channelhave not precisely been confirmed yet. The extracted g πNN ∗ would be allowed to change in arange due to the mentioned large uncertainty. The range can roughly be estimated by usingthe maximal and minimal values [19] of the total width and the N π branching fraction forthe corresponding N ∗ . The extracted g πNN ∗ and g ηNN ∗ for each N ∗ are tabulated in Table I.From this table, it is clearly shown that the result is reasonable, namely it consists with thefact that the larger the partial decay width is, the stronger the N ∗ couples to the decayedparticles.Then the coupling constants g φNN ∗ for various N ∗ s can be extracted by fitting the totalcross section data for the π − p → nφ reaction. To consider the off-shell effect of N ∗ , a formfactor in Eq.(15) with a cut-off parameter Λ being 1.8GeV for N ∗ / − (1535) and 2.3GeV for N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100) is employed.2 TABLE I: Coupling constants g πNN ∗ and g ηNN ∗ for various N ∗ states. N total width decay mode branching fraction partial width g πNN ∗ ( g ηNN ∗ )(GeV)[19] [19] (GeV) N ∗ / − (1535) 0.150 N π
45% 0.675 × − N η
53% 0.795 × − × N ∗ / + (1900) 0.180 N π .
5% 0.990 × − × N π
26% 0.129 0.147 × N ∗ / − (2090) 0.095 N π .
0% 0.855 × − × − N π
18% 0.630 × − N π
10% 0.414 × − N ∗ / + (2100) 0.113 N π
15% 0.170 × − N π
10% 0.200 × − N π
12% 0.312 × − × Because the magnitudes of g πNN ∗ for the N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100) statescan respectively vary in a rather large range of their own, we combine possible g πNN ∗ valuesfor these N ∗ s into various cases to fit the reaction data. The fitted result shows that only twotypes of combinations can give the best fit. In type I, one can only take a smaller total widthfor N ∗ / − (2090) and a larger total width for N ∗ / + (2100), and in type II, it is the other wayround. To be specific, the restricted regions of the total widths for N ∗ / + (1900), N ∗ / − (2090)and N ∗ / + (2100) are bound by the following combined cases: Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) =180MeV/95MeV/200MeV, 180MeV/95MeV/260MeV, 498MeV/95MeV/200MeV,and 498MeV/95MeV/260MeV in type I, and 180MeV/350MeV/113MeV,180MeV/414MeV/113MeV, 498MeV/350MeV/113MeV and 498MeV/414MeV/113MeVin type II. The typical fitted curves of these two types are plotted in Figs. 3 (a) and (b)with χ being 3 .
62 and 3 .
05, respectively. In these figures, the dashed, dotted, dash-dottedand dash-double-dotted curves denote the contributions from individual N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100) states, respectively. The contributions frominterference terms in these cases are shown in Figs. 3 (c) and (d), respectively. In thesefigures, we only plot the the contributions from the interference term between N ∗ / − (1535)and N ∗ / − (2090) and from the sum of the rest terms, shown as the solid and dashed curves,3respectively, because the former is much larger than the later. The fitted cures are plottedby solid curves in Figs. 3 (a) and (b), respectively. They are obtained by summing over thecontributions from all the N ∗ states coherently. ( b ) s (a)Type I(Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) =498MeV/95MeV/260MeV) ( b ) s (b)Type II(Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) =180MeV/350MeV/113MeV) Sum of all other interference terms Interference term between N*(1535) and N*(2090) ( b ) s (c)Type I(Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) =498MeV/95MeV/260MeV) All other interference term Interference term between N*(1535) and N*(2090) ( b ) s (d)Type II(Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) =180MeV/350MeV/113MeV) FIG. 3: Total cross section of the π − p → nφ reaction. From Fig. 3, we find that N ∗ / − (1535) provides a major contribution in the whole energyrange considered, especially near the N φ threshold. The contribution from N ∗ / + (1900) isrelatively flat in the high energy region. The cross section from N ∗ / − (2090) or N ∗ / + (2100)has large uncertainty. Its shape depends on the total width of the state, Γ N ∗ . If Γ N ∗ issmall, the cross section curve would show a relatively narrow peak around the mass of the4 N ∗ , otherwise it presents a broad structure. This is simply because that a Breit-Wigner formfor the N ∗ propagator is adopted in the calculation. The P wave N ∗ / + (2100) state can onlyplay a minor role although it has a large branching fraction to N η , since its contributionnear the
N φ threshold is too small. Furthermore, the contribution from N ∗ (2090) cannotbe large because of a counter-contribution from the interference term between N ∗ (2090) and N ∗ (1535) in a region close to the N φ threshold, namely a larger contribution from N ∗ (2090)makes the fit worse. In conclusion, although some higher resonances are introduced, thedominate contribution in the π − p → nφ cross section still comes from the N ∗ / − (1535)state, which is consistent with discussion in Ref. [8]. It should further be mentioned thatthe contribution from all the interference terms is about 2% only. Thus, the assumptionthat the contribution from ignored N ∗ s including their interference terms is about 10 ∼ N ∗ sin a range of 85 ∼
90% of the total will not affect our qualitative conclusion.Based on the best fit, namely a small enough χ and a reasonable overall fit, we canextract the coupling constant g N ∗ Nφ for all adopted N ∗ s in all the mentioned cases. Theresultant g N ∗ Nφ for these N ∗ s are tabulated in Table II. The fractions of the individualcontributions from all the considered N ∗ s are given in the table as well. From this table, TABLE II: The extracted coupling constant g N ∗ Nφ and the corresponding fraction of contributionin the π − p → nφ reaction.Γ N ∗ (1900) / Γ N ∗ (2090) / Γ N ∗ (2100) g φNN ∗ (10 − )/(fraction of contribution(%)) N ∗ / − (1535) N ∗ / + (1900) N ∗ / − (2090) N ∗ / + (2100)type I 180MeV/95MeV/200MeV 140/67.1 7 . . one has following observations: (1) N ∗ / − (1535) is a dominant resonance in the π − p → nφ N φ strongly, and the coupling constant g N ∗ (1535) Nφ ranges from 1.1 to 1.4. The contribution andthe coupling to N φ in type I is larger than those in type II. (2) The N ∗ / + (2100) state isthe second largest contributor, which offers about 10% to 26% of the contribution. It alsoshows that N ∗ / + (2100) may couple to N φ remarkably. The value of the coupling constant g N ∗ (2100) Nφ stretches from 0.09 to 0.19. And the contribution and the coupling to N φ in typeII is larger than those in type I. (3) The contribution from N ∗ / − (2090) is about 3% to 6%,and g N ∗ (2090) Nφ spans a range from 0.007 to 0.014. The contribution and the coupling to N φ in type I is slightly larger than those in type II. (4) The contribution from N ∗ / + (1900) is evensmaller, about 0.4 % to 1.0%, and g N ∗ (1900) Nφ spreads in a range of 0.006 to 0.079. The effectfrom the total width uncertainty of this state is quite small. These observations are clearlyconsistent with the information from the curves shown in Fig. 3. Namely, N ∗ / + (2100) givesa contribution comparable to that from N ∗ / − (1535), especially in the higher energy region, N ∗ / − (2090) offers a visible contribution around the energy about its mass, and N ∗ / + (1900)only provides a very small contribution in the whole energy region.Next, we determine g ψNN ∗ in terms of the partial decay widths of the J/ψ → p ¯ pη and J/ψ → p ¯ nπ − processes, respectively. The partial wave analysis of the J/ψ → p ¯ pη datacollected at BESII shows that the partial decay width contributed by the intermediate N ∗ / − (1535) state is about (56 ± g ψNN ∗ (1535) . Again, a form factor with Λ being 1.8GeV in Eq.(15) is adopted in thecalculation to describe the off-shell effect of N ∗ / − (1535), and the extracted g ψNN ∗ (1535) istabulated in Table III. On the other hand, one notices that in analyzing the J/ψ → p ¯ nπ − data of BESII, assuming the contributions from N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100)to be about 5 ∼ . ± . stat. )) × − [12]. Therefore, we also approximately takethe contributions from N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100) to be 56%,10%, 10% and 10%, respectively, in the calculation of the J/ψ → p ¯ nπ − decay. Using theseassumptions, the extracted g πNN ∗ (1535) values and the form factor in Eq.(15) with Λ being1.8GeV for N ∗ / − (1535) and 2.3GeV for either N ∗ / + (1900), or N ∗ / − (2090) or N ∗ / + (2100),we can extract g ψNN ∗ for later three N ∗ s from the the branching fraction of the J/ψ → p ¯ nπ − decay. The resultant g ψNN ∗ and corresponding Λ are tabulated in Table III. From this table,we find that g ψNN ∗ (1535) is in the order of 10 − . Based on the ranges of the measured total6 TABLE III: g ψNN ∗ and Λ for N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100). N ∗ Total Width(GeV) g ψNN ∗ Λ=1.8GeV Λ=2.3GeV N ∗ / − (1535) 0.150 1.319 × − —— N ∗ / + (1900) 0.180 —— 2.422 × − × − N ∗ / − (2090) 0.095 —— 4.726 × − × − × − N ∗ / + (2100) 0.113 —— 1.362 × − × − × − width and the obtained g πNN ∗ for the N ∗ / + (1900), N ∗ / − (2090), and N ∗ / + (2100) states,the extracted values of g ψNN ∗ (1900) , g ψNN ∗ (2090) , and g ψNN ∗ (2100) could vary in the ranges of(0 . ∼ . × − , (1 . ∼ . × − , and (1 . ∼ . × − , respectively. It seems thatthe couplings of J/ψ to N and different N ∗ are about the same. This is understandable,because that J/ψ is merely composed of charmed quarks, N is consist of upper and downquarks only, and N ∗ is made up of upper, down and even strange quarks, thus the couplingmechanisms for different N ∗ s would be the same.In terms of the extracted the values of g ψNN ∗ and g φNN ∗ , we are in the stage of calculatingphysics observables in the J/ψ → p ¯ pφ decay with adopted intermediate states N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100). The resultant invariant mass spectra of pφ areplotted in Fig. 4. In sub-figures (a) and (b), the dashed and solid curves represent the upperand lower limits of the total invariant mass spectrum, which are caused by the uncertaintiesof the widths of the N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100) states. And in sub-figures(c) and (d), the dashed, dotted, dash-dotted and dash-double-dotted curves describe thesub-contributions from the N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100) states,respectively. The fractions of the contributions for these N ∗ s in the decay are tabulated inTable IV.From the numerical values in Table IV and the pφ invariant mass curves in Fig. 4, we have7 (a)Type I (Γ N ∗ (2090) =95MeV). (b)Type II (Γ N ∗ (2100) =113MeV).(c)Type I (Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) =498MeV/95MeV/260MeV). (d)Type II (Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) = 180MeV/350MeV/113MeV). FIG. 4: Invariant mass spectra of pφ in type I (Γ N ∗ (2100) =113MeV) and type II (Γ N ∗ (2090) =95MeV)with two curves covering the range of 4 sets of parameters. In (c) and (d), the solid-line is from N ∗ (2090)’s contribution; the dash-line is from N ∗ (2100)’s; the dash-dotted line is from N ∗ (1535)’s;the dash-dotted-dotted line is from N ∗ (1900)’s. following observations. From Fig. 4(a) and Table IV, one sees that in type I the contributionfrom N ∗ / − (2090) is about (46.3 ∼ N ∗ s shown in Fig. 4(c) tell us that this structure ismainly contributed by N ∗ / − (2090) due to its relatively narrow width, namely a strongercoupling between N and φ . This implies that there may exists a large N φ or qqqs ¯ s compo-8 TABLE IV: Fractions of contributions from N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100) in the J/ψ → p ¯ pφ decay.Γ N ∗ (1900) / Γ N ∗ (2090) / Γ N ∗ (2100) fraction(%) N ∗ (1535) N ∗ (1900) N ∗ (2090) N ∗ (2100)type I 180MeV/95MeV/200MeV 2.01 0.44 48.22 32.80498MeV/95MeV/200MeV 1.96 0.01 61.24 25.23180MeV/95MeV/260MeV 1.83 0.20 70.51 15.12498MeV/95MeV/260MeV 1.81 0.01 65.36 16.38type II 180MeV/350MeV/113MeV 1.66 0.36 1.42 76.26498MeV/350MeV/113MeV 1.65 0.01 1.83 80.03180MeV/414MeV/113MeV 1.69 0.18 2.72 82.38498MeV/414MeV/113MeV 1.63 0.01 2.53 87.56 nent in N ∗ / − (2090). Meanwhile the N ∗ / + (2100) state also provides a sizable contributionof about (15.1 ∼ N ∗ / − (2090),and the shape of the contribution is flatter due to a large width of the state. Therefore, thispiece of contribution would not affect the shape of the total contribution qualitatively. Thecontributions from N ∗ / − (1535) and N ∗ / + (1900) are negligibly small, their contributionsare about (1.81 ∼ ∼ N ∗ / − (1535) and N ∗ / + (1900) states and the interference terms will not affect the conclu-sion qualitatively. From Fig. 4(b) and Table IV, one finds that in type II the contributionfrom N ∗ / + (2100) is about (76.3 ∼ N ∗ / + (2100), because of its dominant contribution andrelatively narrow width. This also implies that its coupling to N φ could be remarkable, asignificant
N φ or qqqs ¯ s component may exist in N ∗ / + (2100). Meanwhile the contributionsfrom N ∗ / − (1535), N ∗ / + (1900) and N ∗ / − (2090) are negligibly small, their contributions areabout (1.63 ∼ ∼ ∼ N ∗ / − (1535) and N ∗ / + (1900) in the type I case and for9 (a)Type I(Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) =498MeV/95MeV/260MeV.) (b)Type II(Γ N ∗ (1900) /Γ N ∗ (2090) /Γ N ∗ (2100) =180MeV/350MeV/95MeV.) FIG. 5: Dalitz plots. N ∗ / − (1535), N ∗ / + (1900) and N ∗ / − (2090) in the type II case in this decay process, becausetheir informations are deeply submerged in the signals of the N ∗ / − (2090) and N ∗ / + (2100)states and the N ∗ / + (2100) state, respectively.Furthermore, the Dalitz plots for type I and type II are plotted in Fig.5. It is shown thatthe Dalitz plots of types I and II have distinguishable features. In the type I case, there areone vertical belt and one horizontal belt at 4.37( GeV /c ) and an enhancement in the upperright corner. But in the type II case, there is only two enhancements at the upper left andlower right corners. These patterns agree with the findings from the invariant mass curves.Finally, we need to mention that the value of the cut-off parameter in a certain rangedoes not qualitatively affect our conclusion. IV. SUMMARY
In this paper, the
J/ψ → p ¯ pφ decay is studied in the isobar resonance model with effectiveLagrangians. In such a model, the nucleon resonances are adopted as the intermediate states.Because of the s ¯ s structure of the φ -meson and the OZI rule, the major decay width of thisprocess will be contributed by the resonances who contain strange content. Therefore, thisdecay process could be used to study the possible strange structure of the nucleon resonances.0Based on a careful analysis, four N ∗ states, N ∗ / − (1535), N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100), are adopted in the calculation, so that the qualitative conclusion would notbe affected. The coupling constants g πNN ∗ for these N ∗ s and g ηNN ∗ for N ∗ / − (1535) are ex-tracted from the branching fractions of the N ∗ s to the N π channel and of N ∗ / − (1535) to the N η channel, respectively, in the first step. With determined g πNN ∗ , coupling constant g φNN ∗ for N ∗ s are obtained by fitting the cross section of the π − p → nφ reaction. Because the un-certainties of the partial width for N ∗ / + (1900), N ∗ / − (2090) and N ∗ / + (2100), the resultant g φNN ∗ s are allowed to change in certain regions. It is found that in the best fit, except thedominant contribution from N ∗ / − (1535) and negligible contribution from N ∗ / + (1900), thecontributions from N ∗ / − (2090) and N ∗ / + (2100) are visible and even remarkable in somecases, and the total widths of these two states cannot be large simultaneously. Therefore,there are two types of fits. In the first type, type I, N ∗ / − (2090) and N ∗ / + (2100) have asmaller total width and a larger total width, respectively, and in the second type, type II, itis the other way round. In the second step, the coupling constant g ψNN ∗ (1535) and g ψNN ∗ forother three N ∗ s are extracted by fitting the partial decay widths of the J/ψ → p ¯ pη processand the J/ψ → p ¯ nπ − process, respectively.Finally, we can calculate the physical observables in the J/ψ → p ¯ pφ decay by usingobtained g πNN ∗ s and g ψNN ∗ s in the type I and type II cases. The invariant mass spectrumof pφ in the type I case shows that there is a peak structure around 2.09GeV due to themajor contribution from the narrower N ∗ / − (2090) state. This means that its coupling to N φ is relatively strong, and a large
N φ or qqqs ¯ s component may exist in N ∗ / − (2090).Meanwhile the contribution from N ∗ / + (2100) is flatter and smaller, which implies that eventhere is a strange ingredient in this state, its coupling to N φ would be weaker. In thetype II case, the curve of the invariant mass spectrum of pφ has a small peak structurearound 2.11GeV, because of the dominant contribution from the narrow N ∗ / + (2100) stateand negligible contributions from other states. It suggests that its coupling to N φ is strong,a significant
N φ or qqqs ¯ s component might exist in the N ∗ / + (2100). However, one wouldnot be able to reveal the strange structure in N ∗ / − (2090), because its information is deeplysubmerged in the signal of the N ∗ / + (2100) state. For the same reason, no matter in whichcases, one cannot figure out strange structures of N ∗ / − (1535) and N ∗ / + (1900) from thisprocess.In summary, in the J/ψ → p ¯ pφ decay, the widths of N ∗ / − (2090) and N ∗ / + (2100)1cannot be large simultaneously. The proposed study of this channel with the high statisticsBESIII data [21] will tell us how the pφ invariant mass curve goes. If the shape of thecurve likes that of type I, the width of the N ∗ / − (2090) state is narrower and there wouldbe a considerable mount of pφ or qqqs ¯ s component in the state, while the width of the N ∗ / + (2100) state would be wider. If the shape of the curve is similar to that of type II,only the width of the N ∗ / + (2100) state is narrower and there would be a certain mountof pφ or qqqs ¯ s component in the state. Of course, the real data of high statistics on the J/ψ → p ¯ pφ decay may reveal more knowledge on all possible N ∗ s than our predictionsbased on the information from πN → φN . It will definitely provide us useful informationon the N ∗ resonances with large qqqs ¯ s component. And the pp → ppφ reaction should alsobe studied to confirm our prediction. Acknowledgments
This work is partly supported by the National Natural Science Foundation of China undergrants Nos. 10875133, 10847159, 10975038, 11035006, 11165005, and the Key-project bythe Chinese Academy of Sciences under project No. KJCX2-EW-N01, and the Ministry ofScience and Technology of China (2009CB825200). [1] B.C. Liu, B.S. Zou, Phys. Rev. Lett. , 042002 (2006);B.C. Liu, B.S. Zou,Phys. Rev. Lett. , 039102 (2007).[2] B.C. Liu, B.S. Zou, Commun. Theor. Phys. , 501 (2006).[3] G. Penner and U. Mosel, Phys. Rev. C66 , 055211 (2002); ibid.
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Appendix A
The invariant amplitudes of π − p → nφ reaction with N ∗ (1535), N ∗ (1900), N ∗ (2090) and N ∗ (2100) being intermediate states are as follows:For N ∗ (1535) M N ∗ (1535) = √ g πNN ∗ g φNN ∗ F N ∗ ( q )¯ u ( p n , s n ) γ ( γ ν − q ν qq ) ǫ ν ( p φ , s φ ) G N ∗ ( q ) u ( p p , s p ) , (A1)for N ∗ (1900) M N ∗ (1900) = i √ M N ∗ g πNN ∗ g φNN ∗ F N ∗ ( q )¯ u ( p n , s n ) γ ǫ µ ( p φ , s φ ) G µνN ∗ ( q ) p πν u ( p p , s p ) , (A2)For N ∗ (2090) M N ∗ (2090) = √ g πNN ∗ g φNN ∗ F N ∗ ( q )¯ u ( p n , s n ) γ ( γ ν − q ν qq ) ǫ ν ( p φ , s φ ) G N ∗ ( q ) u ( p p , s p ) , (A3)and for N ∗ (2100) M N ∗ (2100) = i √ g πNN ∗ g φNN ∗ F N ∗ ( q )¯ u ( p n , s n ) γ ν ǫ ν ( p φ , s φ ) G N ∗ ( q ) γ u ( p p , s p ) , (A4)where p π is the four momentum of π − meson, and ǫ ( p φ ) is the polarization vector of φ meson. Appendix B
The invariant amplitude of
J/ψ → p ¯ pη decay with N ∗ (1535) being the intermediate stateis written as: M N ∗ (1535) = ig ηNN ∗ g ψNN ∗ ¯ u ( p , s )[ G N ∗ ( q ) F N ( q ) γ σ µρ p ρψ ǫ µ ( k, s ψ )+ γ σ µρ p ρψ ǫ µ ( k, s ψ ) G ¯ N ∗ ( q ′ ) F N ( q ′ )] v ( p , s ) , (B1)with p ψ and ǫ ( p ψ ) being the four momentum and the polarization vector of J/ψ .The invariant amplitudes of
J/ψ → p ¯ nπ − reaction with N ∗ (1900), N ∗ (2090) and N ∗ (2100) being intermediate states are as follows:For N ∗ (1900), M N ∗ (1900) = i √ M N ∗ g πNN ∗ g ψNN ∗ ¯ u ( p , s ) γ ǫ µ ( k, s ψ ) G µν ¯ N ∗ ( q ′ ) F N ( q ′ ) p πν v ( p , s ) . (B2)4For N ∗ (2090), M N ∗ (2090) = i √ g πNN ∗ g ψNN ∗ ¯ u ( p , s ) γ σ µρ p ρψ ǫ µ ( k, s ψ ) G ¯ N ∗ ( q ′ ) F N ( q ′ ) v ( p , s ) . (B3)For N ∗ (2100), M N ∗ (2100) = i √ g πNN ∗ g ψNN ∗ ¯ u ( p , s ) γ µ ǫ µ ( k, s ψ ) G ¯ N ∗ ( q ′ ) F N ( q ′ ) γ v ( p , s ) . (B4) Appendix C
The invariant amplitudes of
J/ψ → p ¯ pφ decay with N ∗ (1535), N ∗ (1900), N ∗ (2090) and N ∗ (2100) being intermediate states are as follows:For N ∗ (1535), M N ∗ (1535) = ig φNN ∗ g ψNN ∗ ¯ u ( p , s )[ γ ( γ ν − q ν qq ) ǫ ν ( p , s φ ) G N ∗ ( q ) F N ( q ) γ σ µρ p ρψ ǫ µ ( k, s ψ )+ γ σ µρ p ρψ ǫ µ ( k, s ψ ) G ¯ N ∗ ( q ′ ) F N ( q ′ ) γ ( γ ν − q ′ ν q ′ q ′ ) ǫ ν ( p , s φ )] v ( p , s ) . (C1)For N ∗ (1900), M N ∗ (1900) = − g φNN ∗ g ψNN ∗ ¯ u ( p , s )[ γ ǫ ν ( p , s φ ) G νµN ∗ ( q ) F N ( q ) γ ǫ µ ( k, s ψ )+ γ ǫ µ ( k, s ψ ) G µν ¯ N ∗ ( q ′ ) F N ( q ′ ) γ ǫ ν ( p , s φ )] v ( p , s ) . (C2)For N ∗ (2090), M N ∗ (2090) = ig φNN ∗ g ψNN ∗ ¯ u ( p , s )[ γ ( γ ν − q ν qq ) ǫ ν ( p , s φ ) G N ∗ ( q ) F N ( q ) γ σ µρ p ρψ ǫ µ ( k, s ψ )+ γ σ µρ p ρψ ǫ µ ( k, s ψ ) G ¯ N ∗ ( q ′ ) F N ( q ′ ) γ ( γ ν − q ′ ν q ′ q ′ ) ǫ ν ( p , s φ )] v ( p , s ) . (C3)For N ∗ (2100), M N ∗ (2100) = g φNN ∗ g ψNN ∗ ¯ u ( p , s )[ γ ν ǫ ν ( p , s φ ) G N ∗ ( q ) F N ( q ) γ µ ǫ µ ( k, s ψ )+ γ µ ǫ µ ( k, s ψ ) G ¯ N ∗ ( q ′ ) F N ( q ′ ) γ ν ǫ ν ( p , s φ )] v ( p , s ) ..