Re-study of Nucleon Pole Contribution in J/ψ N Nπ Decay

Commun. Theor. Phys. Beijing, China 46 26 pp. 57 53 c International Academic Publishers Vol. 46, No. 3, September 5, 26 Re-study of Nucleon Pole Contribution in J/ψ N Nπ Decay ZONG Yuan-Yuan,,2 SHEN Peng-Nian,,3,4 ZOU Bing-Song,,3,4 LIU Ji-Feng, 2 and LIANG Wei-Hong 2 Institute of High Energy Physics, the Chinese Academy of Sciences, P.O. Box 984, Beijing 49, China 2 Department of Physics, Guangxi Normal University, Guilin 544, China 3 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 73, China 4 Institute of Theoretical Physics, the Chinese Academy of Sciences, Beijing 8, China Received December 7, 25 Abstract We re-study nucleon pole contribution in J/ψ N Nπ decays by including the imaginary part for the propagator of the off-shell nucleon with energy above πn threshold. It is found that when including the imaginary part in the propagator, the branching ratio of the decay width will descend about % compared with the result without including the imaginary part, no matter whether including the off-shell form factors or not. It also leads to a phase of up to 25 for the off-shell nucleon propagator at invariant mass around 4 MeV. This effect needs to be considered for detailed partial wave analysis of N resonances around this mass region. PACS numbers: 3.2.Gd Key words: J/ψ decay, form factor Introduction Nucleons have been studied for decades. It is an elementary block of matter. Nucleons are composite systems with many internal degrees of freedom. The constituents are quarks and gluons. The nucleon spectrum investigation would provide us necessary information for revealing the structure of nucleon. ] So far, in virtue of a variety of sources, more and more information about the nucleon and its excited states have been accumulated. However, to date, we still have a limited knowledge of the internal quark-gluon structure of nucleons and the interaction between composite nucleons. On the theoretical side, with the establishment and development of Quantum Chromo Dynamics QCD, in principle the inner structure of nucleon and its properties should be studied from QCD theory, but in reality, due to its nonperturbative character, various QCD models have been proposed. In succession, many nucleon resonances N have been predicted. On the experimental side, up to now, many N states predicted by well accepted quark models 2] have not been found in πn experiments. Should we search for these missing resonances by other means? Or should the quark models be further modified? Or are there other causes? These puzzles impel us to further study the N spectrum on both theoretical and experimental sides. In recent years, a large number of experiments on N physics have been carried out at new facilities such as CEBAF at JLAB, ELSA at BONN, GRAAL at Grenoble, and Spring-8 at JASRI. Now 58 million events have been collected at Beijing Electron-Positron Collider BEPC. The two-step decay process J/ψ N N N NM, where M refers to meson, can be another excellent source for studying light baryon resonances with many advantages. 3] It is worth while mentioning that the nucleon-pole diagrams would also contribute as a background in the N study via J/ψ N NM decays. For light mesons, especially for pion, nucleon-pole contributions might be sizable and should not be ignored. In order to extract a more accurate and reliable conclusion from the J/ψ hadronic decay data, it is necessary to study the nucleon-pole contributions in those decay channels. By analyzing J/ψ N Nπ data, R. Sinha and S. Okubo 4] pointed out that in the J/ψ N Nπ decay, the nucleon-pole contribution dominates in soft pion limit, and the N -pole contribution became important at the large pion energy region. But they did not consider the off-shell effect of nucleon propagator as virtual particle. Liang et al. 5] studied the off-shell effect by including off-shell form factors for the calculation of the nucleonpole contributions. In this work, we consider the effect to the nucleon pole contribution by including the imaginary part in the propagator. And we further calculate the effect by including both imaginary part and off-shell form factors, and analyze the results. The paper is organized in the following way: in the next section, the nucleon-pole contributions in the J/ψ N Nπ decay are studied. In Sec. 3 the conclusion is given. 2 Nucleon Pole Contribution in J/ψ N Nπ Decay We analyze cautiously the off-shell effect through different N Nπ couplings and including the imaginary part The project partly supported by National Natural Science Foundation of China under Grant Nos. 47589, 4358, 4472, 4722, and 225525, the Physical Science Fund of Guangxi Province under Grant No. 9745, and the Knowledge Innovation Key Project of the Chinese Academy of Sciences under Grant No. KJCX2-SW-N2

58 ZONG Yuan-Yuan, SHEN Peng-Nian, ZOU Bing-Song, LIU Ji-Feng, and LIANG Wei-Hong Vol. 46 of the off-shell nucleon propagator in the J/ψ N Nπ channel. 2. Nucleon Pole Contribution in J/ψ N Nπ Decay by Using Different πn N Couplings The nucleon-pole diagrams for J/ψ N Nπ decay are shown in Fig.. Fig. Nucleon-pole diagrams for J/ψ N Nπ decay. Generally, the J/ψ-N- N interaction can be written as N F M γ µ + 2m F q p µ] Nɛ µ P ψ, where P ψ, p, and p are the four-momenta of J/ψ, N, and N, respectively, m is the mass of the nucleon, and ɛ µ P ψ denotes the polarization vector of J/ψ. Dimensionless real decay constants F M and F can be determined by the experimental data of the two-body decay J/ψ p p. As to the pion-nucleon interaction, one can choose either as pseudoscalar-pseudoscalar PS-PS form, H = ig πnn Nγ5 τn π, 2 or pseudoscalar-pseudovector PS-PV form, H = 2m g πnn Nγ 5 γ µ τn µ π, 3 where g πnn is the standard pion nucleon coupling constant with 4π g πnn 2 4.8, 4 and τ is the isospin Pauli matrix. As given in Ref. 5], the total decay amplitude of Fig. in the PS-PS coupling πn interaction is k/ɛ/ M PS = M PS,a + M PS,b = ig πnn ūp, sγ 5 F M 2p k + k 2 ɛ/k/ 2p k + k 2 + F m k/ p ɛ 2p k + k 2 p ɛ ] 2p k + k 2 vp, s, 5 while in the PS-PV coupling case, the amplitude has an additional terms, M PV = i 2m 2 g πnnf ūp, sp p µ ɛ µ γ 5 vp, s + M PS. 6 The differential decay rate can be formulated by summing over possible spin states of final nucleon and anti-nucleon, dγ PS J/ψ N Nπ = 2π4 M PS 2 dφ 3 P ψ ; p, p, k = 2π 4 2g 2 πnn F M 2 A PS, + F 2 A PS,2 2 + Re F F M A PS,3 ]dφ 3 P ψ ; p, p, k, 7 dγ PV J/ψ N Nπ = 2π4 2 M P V 2 dφ 3 P ψ ; p, p, k = 2π 4 2g 2 πnn F M 2 A PS, + F 2 A PS,2 + A PV,2 with being the mass of J/ψ and + Re F F M A PS,3 + A PV,3 ]dφ 3 P ψ ; p, p, k 8 dφ 3 P ψ ; p, p, k = δ 4 P ψ p p d 3 p d 3 p d 3 k k 2π 3 2p 2π 3 2p 2π 3, 9 2k the three-body phase space. From the measured angular distribution of J/ψ p p, the value of F / F M has been determined to be in the range of.2.4. 5] By taking F / F M =.2, which gives the smallest difference for two different πnn couplings, ΓJ/ψ N Nπ can be calculated and the branching ratios BRs of the decay widths for Fig.. in the PS-PS and PS-PV cases are Γ PS J/ψ N Nπ Γ PV J/ψ N Nπ =.563, =.529,

No. 3 Re-study of Nucleon Pole Contribution in J/ψ N Nπ Decay 59 respectively. Comparing with the empirical ratio, 6] ΓJ/ψ N Nπ one sees that the resultant BRs in Eqs. and are very close to the data. =.5 ±.4, 2 2.2 Off-shell Effect by Including the Imaginary Part in Nucleon Propagator for J/ψ N Nπ Decay The propagator with half-integer spin can be constructed with its projection operator and the corresponding Breit- Wigner factor. The propagator for a spin /2 resonance reads G /2 R = p /2 2M R p 2 R M 2 R + im RΓ R, 3 where /p 2 R M 2 R + im RΓ R is the standard form of the Breit Wigner factor, and M R, p R, and Γ R are the mass, four-vector momentum, and width of the resonance, respectively. And the corresponding spin /2 projection operator can be written as p /2 = upūp = p/ R + M R 2M R. 4 Thus, equation 3 can be written as G /2 p/ R = R + M R p 2 R MR 2 + im. 5 RΓ R For an off-shell nucleon with invariant mass above πn threshold, the imaginary width part should also be included in its propagator. Then the decay amplitude of Figs. a and b in the PS-PS coupling interaction can be written as M off PS,a q/ + m = ig πnn ūp, sγ 5 q 2 m 2 F M γ µ + + imγ 2m F q p µ] ɛ µ P ψ, λ ψ vp, s p/ + k/ + m = ig πnn ūp, sγ 5 p + k 2 m 2 F M ɛ/ F + imγ m p ɛ vp, s k/ = ig πnn ūp, sγ 5 k 2 F M ɛ/ F + 2k p + imγ m p ɛ vp, s, 6 M off PS,b = ig πnn ūp, s F M γ µ + 2m F p q µ] q/ m ɛ µ P ψ, λ ψ q 2 m 2 + imγ γ 5vp, s = ig πnn ūp, sγ 5 F M ɛ/ m F p/ + k/ + m p ɛ k 2 + 2k p + imγ vp, s = ig πnn ūp, sγ 5 F M ɛ/ + F m p ɛ k/ k 2 + 2k p + imγ vp, s. 7 The total amplitude for Fig. is then obtained by summing over Eqs. 6 and 7, M PS M off PS,a + M off PS,b k/ɛ/ = ig πnn ūp, sγ 5 F M 2p k + k 2 + imγ + F m k/ p ɛ 2p k + k 2 + imγ p ɛ 2p k + k 2 imγ while in the PS-PV coupling case, the amplitude is ɛ/k/ 2p k + k 2 + imγ ] vp, s, 8 M PV = ig πnn 2m 2 F ūp, sp p µ ɛ µ γ 5 vp s + M PS. 9 Fig. 2 Diagram for off-shell N Nπ.

5 ZONG Yuan-Yuan, SHEN Peng-Nian, ZOU Bing-Song, LIU Ji-Feng, and LIANG Wei-Hong Vol. 46 The decay width Γ of the off-shell nucleon can be calculated with the Feynman diagram shown in Fig. 2. The decay amplitude is M Noff Nπ = ig πnn ūpγ 5 uq. 2 After including a vertex form factor, the squared amplitude for N off Nπ decay can be written as M Noff Nπ 2 = 3g 2 πnn s m 2 m 2 π] Λ 2 p 2 + Λ 2, 2 where m and m π are the masses of proton and pion, respectively, p is the pion momentum in the rest frame of the off-shell nucleon, and s is the invariant mass of the nucleon propagator. Λ is the cut-off parameter for the vertex form factor and we take a commonly used one Λ =.5 as in Ref. 7]. The differential decay rate can be formulated by summing over possible spin states of final nucleon, dγn off Nπ = 32π 2 M N off Nπ 2 p dω, 22 s where dω = dφdcos θ is the solid angle of final proton, and p = s m + m π 2 s m m π 2 ] /2 /2 s. Integrating Eq. 2, we obtain Γ = 3 s m 2 m 2 π] p Λ 2 4π g2 πnn s p 2 + Λ 2. 23 The decay width versus s is shown in Fig. 3 for the case with the vertex form factor solid curve and without the form factor dashed curve. The inclusion of the vertex form factor is absolutely necessary. Fig. 3 The pπ decay width of the off-shell nucleon vs. its invariant mass with solid curve and without dashed curve including the vertex form factor. The influence of the width to the off-shell nucleon propagator G = /s m 2 + imγ is shown in Fig. 4. The largest influence happens at energies around.4 GeV where the magnitude is reduced by about 2% and the phase is caused for up to 25. For detailed partial wave analysis of N resonances around this energy region, these effects may need to be taken into account. Fig. 4 The magnitude left and phase right of the off-shell nucleon propagator vs. pπ invariant mass with Γ = dotted line, Γ of Eq. 23 with solid line and without dashed line form factor.

No. 3 Re-study of Nucleon Pole Contribution in J/ψ N Nπ Decay 5 Inserting the Eq. 23 into Eqs. 8 and 9 we can obtain the new amplitudes. Then by summing over possible spin states of final nucleon and anti-nucleon one can formulate new differential widths as dγ PS J/ψ N Nπ = 2π4 M PS 2M 2 dφ 3 P ψ ; p, p, k = 2π 4 2g 2 πnn F M 2 A PS, + F 2 A PS,2 ψ + Re F F M A PS,3]dΦ 3 P ψ ; p, p, k, 24 dγ PV J/ψ N Nπ = 2π4 M PV 2M 2 dφ 3 P ψ ; p, p, k = 2π 4 2g 2 πnn F M 2 A PS, + F 2 A PS,2 + A PV,2 ψ + Re F F M A PS,3 + A PV,3]dΦ 3 P ψ ; p, p, k. 25 The explicit expressions for A PS,i i =, 2, 3 and A PV,i i = 2, 3 are shown in Appendix. Taking F / F M =.2 to calculate the BRs of Γ PS J/ψ N Nπ/ again, we get and Γ PS J/ψ N Nπ Γ PV J/ψ N Nπ =.5, 26 =.47, 27 Γ PV J/ψ N Nπ =.94. 28 Γ PS J/ψ N Nπ Comparing with the corresponding results without including the imaginary part as given in Eqs. and, one can see that by including the imaginary part in the intermediate propagator, the BRs of the widths in both PS-PS and PS-PV coupling have been infected. They are reduced about %. 2.3 Off-shell Effect by Including Both Imaginary Part in Nucleon Propagator and Off-shell Form Factors in J/ψ N Nπ Decay Including both the imaginary part in the nucleon propagator and the off-shell form factors the amplitude of the J/ψ N Nπ decay can be rewritten as M k/ɛ/ PS = ig πnn ūp, sγ 5 F M 2p k + k 2 + imγ F 2 q 2 ɛ/k/ 2p k + k 2 + imγ F 2 q 2 + F m k/ p ɛ 2p k + k 2 + imγ F 2 q 2 p ɛ ] 2p k + k 2 imγ F 2 q 2 vp, s, 29 M PV = ig πnn 2m 2 ūp, s { F M F 2 q 2 F 2 q 2 ]ɛ/ + F m F 2 q 2 p ɛ F 2 q 2 p ɛ] where F q 2 is off-shell form factor. They can be chosen to be one of the following forms: 8 2] } γ 5 vp, s + M off PS, 3 F q 2 = Λ2 + m 2 Λ 2 + q 2, 3 F 2 q 2 = Λ4 + m 4 Λ 4 + q 4, 32 F 3 q 2 = e q2 m 2 /Λ 2, 33 F 4 q 2 = + q 2 m 2 2 /Λ 4, 34 where m and q are the mass and the four-momentum of the intermediate particle, respectively, and Λ is the so-called cut-off momentum that can be determined by fitting the empirical data. It should be mentioned that all the form factors mentioned above are normalized to unity when the intermediate nucleon is on its mass shell. We take Λ =., 2. GeV, and F / F M =.2 and calculate the BR of ΓJ/ψ N Nπ/ with various form factors shown above. We show our results in Table together with those from Ref. 5] as a comparison. From the Table, one sees that the suppression of the BRs is still about % compared with the corresponding results of only including off-shell form factors. 5] It indicates that the effect of the off-shell form factors to the BRs is more evident than the effect of the imaginary part. By adjusting the off-shell form factor, it may reproduce the suppression by the width effectively. However, the phase caused by the imaginary part cannot be reproduced by adjusting the off-shell form factor.

52 ZONG Yuan-Yuan, SHEN Peng-Nian, ZOU Bing-Song, LIU Ji-Feng, and LIANG Wei-Hong Vol. 46 Table ΓJ/ψ N Nπ/ %. F.F. πn Λ =. GeV Λ =. GeV Λ = 2. GeV Λ = 2. GeV coupling Ref. 5] present work Ref. 5] present work F PS 6.69 5.94 9.6 6.93 PV 4.92 4.38 5.7 3.89 F 2 PS.8.9 9.39 7.6 PV.72.67 5.25 3.46 F 3 PS.4.97 3.9.55 PV.62.58.3 8.88 F 4 PS 3.23 2.97 29.39 25.82 PV.96.84 23.99 2.4 3 Conclusion J/ψ N NM decay is an ideal process to study N spectrum. As intermediate states, nucleon and N can all contribute to the decay BR. In the J/ψ N NM decay data analysis, nucleon-pole contribution would play an important role as background for N production. Understanding this contribution would enable us to get a more accurate and more reliable information of N. In this paper, we study the nucleon-pole contribution by including the imaginary part in the propagator in the J/ψ N Nπ decay. We found that if including the imaginary part in propagator the nucleon-pole contribution will be suppressed by around % no matter whether the off-shell form factors are included or not. The nucleon-pole contribution still plays an important role and cannot be ignored. The influence on the branching ratio by including the imaginary width part in the propagator can be effectively reproduced by adjusting the off-shell form factors which have much larger effect. However, by including the imaginary width part in the propagator, a phase of up to 25 is produced in the nucleon pole amplitude for πn invariant mass around.4 GeV. This cannot be reproduced by adjusting off-shell form factors. The phase may have some effects on the interference between nucleon pole amplitude and the N pole amplitude. So for detailed partial wave analysis for N around this energy region, the effect may need to be taken into account. Appendix In this appendix, we give the explicit expressions of A PS,i i =, 2, 3 and A PV,i and 25, i = 2, 3 appearing in Eqs. 24 M PS 2 = 4g 2 N Nπ FM 2 A PS, + F 2 A PS,2 + Re F F M A PS,3], A M PV 2 = 4g 2 Nπ N FM 2 A PS, + F 2 A PS,2 + A PV,2 + Re F F M A PS,3 + A PV,3 ], A2 A PS, = { 2m 2 + p p Pψ k 2 k 2 3 f f + 2 f f 2 2 Mψ 2 P ψ kp ψ pp k P ψ kp ψ p p k] f f + + Mψ 2 2P ψ kp ψ p p k + 2P ψ kp ψ pp k 3Mψ 2 k 2 m 2 + p p + 2Mψ 2 k 2 p p + 2p kp k 2k 2 P ψ pp ψ p ] A PS,2 = 3m 2 m 2 p p k 2 + 2p kp k ]{ A PS,3 = 2 3 m 2 f f { + p p f f + 2 f f 2 f f Pψ p + + f f + } 2 f f 2, A3 + P ψ p f 2 f 2 P ψ pp ψ p + ] f f 2 f f f f 2 ]}, A4 } k pp ψ kp ψ p k 2 P ψ pp ψ p + k 2 P ψ p 2 ] k pk p k 2 p p + k 2 p 2 ] f f

No. 3 Re-study of Nucleon Pole Contribution in J/ψ N Nπ Decay 53 + 2 { } 3 Mψ 2 k p P ψ kp ψ p k 2 P ψ pp ψ p + k 2 P ψ p 2 ] k pk p k 2 p p + k 2 p 2 ] { 3 Mψ 2 2k p P ψ kp ψ p + 2k pp ψ kp ψ p k 2 P ψ p 2 k 2 P ψ p 2 + 2k 2 P ψ pp ψ p ] } 2p k 2 + 2p k 2 k 2 p 2 k 2 p 2 + 2k 2 p p ] f f + 2 f f 2, A5 A PV,2 = { 2m 4 4m 2 p k + p k + + f f 2 f + Pψ pp ψ p P ψ p 2 f 2 f A PV,3 = P ψ p 2 f 2 P ψ p 2 Mψ 2 f { 6m 2 Mψ 2 + f f P ψ p 2 f 2 ] +m 2 + p p p p + m 2 P ψ p P ψ p 2 p p 2]}, A6 m 2 + p p P ψ pp ψ k P ψ p P ψ k p k + p k] + + f f + + Pψ pp ψ p Mψp 2 p + m 2 Mψ 2 p k + p k P ψ p 2 p k P ψ p 2 p k ]}, A7 f = 2p k + k 2 + imγ, f = 2p k + k 2 imγ, f 2 = 2p k + k 2 + imγ, f 2 = 2p k + k 2 imγ. References ] V.D. Burkert, Phys. Lett. B 72 997 9. 2] S. Capstick and W. Roberts, Prog. Part. Nucl. Phys. 45 2 S24, and references therein. 3] B.S. Zou, Nucl. Phys. A 675 2 67c; B.S. Zou, Nucl. Phys. A 684 2 33; BES Collaboration J.Z. Bai, et al. Phys. Lett. B 5 2 75; BES Collaboration M. Ablikim, et al., hep-ex/453. 4] R. Sinha and Susumu Okubo, Phys. Rev. D 3 984 2333. 5] W.H. Liang, P.N. Shen, B.S. Zou, and A. Faessler, Euro. Phys. J A 2 24 487. 6] Particle Data Group, Euro. Phys. J. C 5 2. 7] K. Tsushima, A. Sibrtsev, and A.W. Thomas, Phys. Lett. B 39 997 29. 8] J. Kogut, Rev. Mod. Phys. 5 979 659; Rev. Mod. Phys. 55 983 775. 9] Q. Haider and L.C. Liu, J. Phys. G 22 996 87; L.C. Liu and W.X. Ma, J. Phys. G 26 2 L59. ] V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen, and J.J.de Swart, Phys. Rev. C 49 994 295. ] H. Haberzettl, C. Bennhold, T. Mart, and T. Feuster, Phys. Rev. C 58 998 R4. 2] Y. Oh, A.I. Titov, and T.-S.H. Lee, Phys. Rev. C 63 2 252.