P., Suebka

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2nd ReadingNovember 16, 2005 10:45 WSPC/143-IJMPE 00374International Journal of Modern Physics EVol. 14, No. 7 (2005) 987–994c© World Scientific Publishing Companyππ REACTION IN NON-RELATIVISTIC QUARK MODEL∗P. SUEBKA, C. KOBDAJ and Y. YANSchool of Physics, Suranaree University of Technology,111 University Avenue, Nakhon Ratchasima 30000, ThailandReceived 31 May 2005Revised 6 July 2005The reaction π+π− → π+π− is studied in the non-relativistic quark model with the3P0 quark–antiquark dynamics. The cross section of the reaction π+π− → π+π− is wellreproduced even for rather high energies.Keywords: Pion–pion scattering; non-relativistic quark model.1. IntroductionThe study of the low and intermediate pion–pion scattering as well as other stronginteraction processes lies within the domain of non-perturbative QCD. Due to thelack of effective methods in obtaining solutions to QCD in the non-perturbativeconfinement region, we have to resort to the development of effective models. Meson-exchange models, non-relativistic quark models and chiral perturbation theories areamong the most successful approaches in studying the strong interaction at low andintermediate energies.The meson-exchange models have made tremendous successes in the inves-tigation of the nucleon–nucleon, meson–nucleon and meson–meson and nucleon-antinucleon interactions at low and intermediate energies,1–6 and even in the studyof the elastic nucleon-nucleon scattering at high energies.7,8 The models, however,have many free parameters involved, which is the unavoidable shortcoming of themeson-exchange models.The chiral perturbation theory, which is the effective field theory of the StandardModel below the scale of spontaneous chiral symmetry breaking, has become a well-established method for describing the low-energy interactions of the pseudoscalaroctet. Elastic pion–pion scattering at low energies is a good example of mesonicchiral perturbation theory. A complete analytical calculation of the reaction ππ →ππ at the two-loop order has been performed.9 However, it is difficult to use the∗This work was supported in part by the Suranaree University of Technology grants SUT 1-105-46-12-44 and SUT 1-105-48-36-12.9872nd ReadingNovember 16, 2005 10:45 WSPC/143-IJMPE 00374988 P. Suebka, C. Kobdaj & Y. Yanmethod to describe reactions with higher energies, for example, for the reactionππ → ππ at an energy around the f2(1270) threshold.In the non-relativistic constituent quark model, quarks and antiquarks are keptas the relevant degrees of freedom whereas the interaction between the quarks, par-ticularly the confinement, is described by effective, QCD inspired potentials. Theadvantage of the quark model over the meson exchange model is based on the factthat a large number of experimental observables can be understood qualitativelyand quantitatively by a low number of free parameters. An overview of the variousquark models with a detailed discussion can be found in Ref. 10. The processes ofmeson decays, baryon decays, meson–baryon reactions and baryon–antibaryon anni-hilations have been successfully described in the non-relativistic quark models11–17in the 3P0 quark–antiquark dynamics which has been proven to be the dominantQ̄Q dynamics in the non-relativistic quark models.The reaction ππ → ππ at the isospin I = 2 channel has been successfullystudied in the non-relativistic quark model,18 where the 3P0 quark diagrams haveno contribution. We will now study the ππ → ππ reaction in the non-relativisticquark model where the 3P0 quark diagram dominates.2. Reaction π+π− → π+π− in 3P0The success of the 3P0 quark–antiquark dynamics in studying the reactions e+e− →π+π− and e+e− → N̄N suggests that the reactions are completely dominated bythe intermediate vector mesons.19 We may also expect that in the 3P0 quark–antiquark dynamics, the processes shown in Fig. 1 would dominate the reactionπ+π− → π+π−, where a π+π− pair annihilates into a virtual time-like meson, thenthe virtual meson decays into a π+π− pair. The transition amplitude for the twostep process takes the formT = 〈ππ|V †67|Ψm〉1E − M 〈Ψm|V23|ππ〉 , (1)ρ, f2, f0π−π+p1p2p3p4p5p6p7p8π−π+Fig. 1. π+π− → π+π− in the 3P0 quark model.2nd ReadingNovember 16, 2005 10:45 WSPC/143-IJMPE 00374ππ Reaction in Non-Relativistic Quark Model 989where E is the center-of-mass energy of the two π system. Ψm and M are respec-tively the wave function and mass of the intermediate mesons. 〈ππ|V †67|Ψm〉 and〈Ψm|V23|ππ〉 are respectively the transition amplitude of the intermediate mesonannihilation into two pions and the one of two pions annihilation into a virtualtime-like meson. Vij is the quark–antiquark3P0 vertex defined asVij = λσij · (pi − pj)F̂ij Ĉijδ(pi + pj)= λ∑µ√4π3(−1)µσµijy1µ(pi − pj)F̂ij Ĉijδ(pi + pj) (2)where y1µ(q) = |q|Y1µ(q̂), σij = (σi + σj)/2, pi and pj are the momenta of quarkand antiquark created out of the vacuum. F̂ij and Ĉij are the flavor and coloroperators projecting a quark–antiquark pair to the respective vacuum quantumnumbers. The derivation and interpretation of the quark–antiquark 3P0 dynamicsmay be found in the literature.11,12The evaluation of the transition amplitudes of one meson to two mesons inthe quark–antiquark 3P0 dynamics is straightforward (see details in Appendix A).There are two free parameters, the size parameter of the mesons and the effectivestrength parameter λ in the quark–antiquark 3P0 vertex. The size parameter bmay be nailed down by the reaction ρ0 → e+e−, as done in Ref. 19, where we getb = 3.847 GeV−1.The effective strength parameter λ may be determined by the reaction ρ0 →π+π−. The decay width of the reaction takes the formΓ =π2Mρk(MπEπ)2|Tρ→π+π− |2 , (3)where Tρ→π+π− is the transition amplitude given in Eq. (A.5) in Appendix A. kis the momentum of the final pion mesons in the center-of-mass system. We considerthe final pions to be rather relativistic. We associate each pion with a “minimal rela-tivity” factor (Mπ/Eπ)1/2.2 With the size parameter b = 3.847 GeV−1, determinedfrom the reaction ρ0 → e+e− in Ref. 19, the experimental value Γ = 150 MeV forthe decay width of ρ0 → π+π− requires the effective strength parameter λ to takethe value λ = 2.73.The differential cross section for the reaction a + b → c + d takes, in the center-of-mass system, the form21dσdΩ=vfvi|M(p,k)|2 , (4)withM(p,k) = −(2π)2 EcEdEcmT (p,k) , (5)where vf ≡ Ef /dp and vi ≡ dEi/dk are the final and initial speeds of the pions,respectively. k and p are the incoming and outgoing momenta, respectively. The2nd ReadingNovember 16, 2005 10:45 WSPC/143-IJMPE 00374990 P. Suebka, C. Kobdaj & Y. Yantotal cross section for the reaction π+π− → π+π−, in terms of the partial wavetransition amplitudes, isσ = (2π)4E2cm16∑L|TL(k)|2 , (6)with the partial wave transition amplitudes TL(p, k) defined asTL(p, k) = 2π∫ π0T (p,k)√2l + 14πPL(cos θ) sin θ dθ , (7)where θ is the angle between the momenta k and p. For the reaction π+π− → π+π−,the partial wave transition amplitudes TL in Eq. (6) are linear combinations of theamplitudes in the isospin basis, that isT2n =23T2n(I = 0) +13T2n(I = 2) ,T2n+1 = T2n+1(I = 1) ,(8)where I is the isospin of the ππ system.Figure 2 shows the predictions for the cross section of the reaction π+π− →π+π− in the diagram in Fig. 1. The dashed line is the prediction for which onlythe ρ and f2(1270) mesons20 are involved as the intermediate states. There is nofree parameter in the calculation. The length parameter b of the ρ meson is fixed inthe reaction ρ → e+e− and for simplicity we assign the meson f2(1270) the samelength parameter. The effective strength parameter λ of the 3P0 quark–antiquark0.2 0.4 0.6 0.8 1 1.2 1.4Ecm(GeV)20406080100120140V(mb)Fig. 2. Predictions for the cross section of the reaction π+π− → π+π− in the 3P0 quark modelwith the ρ and f2(1270) mesons as the intermediate states (dashed line) and with the ρ, f2(1270)and f0(600) mesons as the intermediate states (solid line). Experimental data (solid circles) aretaken from Refs. 22 and 23.2nd ReadingNovember 16, 2005 10:45 WSPC/143-IJMPE 00374ππ Reaction in Non-Relativistic Quark Model 991vertex is fixed in the reaction ρ → ππ. It is found that the prediction is reasonableat the resonance region of the reaction π+π− → π+π−, that is the center-of-massenergies from 0.6 to 1.4 GeV. However, it is noticed that the prediction for the lowenergy region (lower than 0.6 GeV), with only the ρ and f2(1270) mesons involvedas the intermediate states, is much lower than the experimental data.22,23The solid line in Fig. 2 stands for the model prediction where the ρ, f2(1270) andf0(600) mesons20 are considered as the intermediate states in Fig. 1. In this work,all the three mesons are assigned the same length parameter b = 3.847 GeV−1, asdetermined in the process ρ → e+e−. We employ the effective strength parameterλ = 2.73 for the processes π+π− → ρ → π+π− and π+π− → f2(1270) → π+π−, andλ = 2.0 for the process π+π− → f0(600) → π+π−. It is noticed that the applicationof the same strength parameter λ = 2.73 to all the three processes leads to poorpredictions for the low energy region. Although the contribution of the f0(600)intermediate state is negligible for the higher energy region (over 0.6 GeV), theinvolvement of the f0 meson is very much necessary for understanding the reactionπ+π− → π+π− at the low energy region (lower than 0.6 GeV).3. Discussions and ConclusionsThe cross section of the reaction π+π− → π+π− is well reproduced in the 3P0 quarkmodel in which there is only one free parameter involved. The reaction π+π− →π+π− at higher energies is dominated by the processes π+π− → ρ → π+π− andπ+π− → f2(1270) → π+π−, while the process π+π− → f0(600) → π+π−, is thedominant one at lower energies.The parameters for the processes ρ0 → e+e− and ρ0 → π+π− work well withthe meson f2 but are not applicable to the meson f0(600). This may indicate thatρ and f2(1270) are mesons of the same kind while f0(600) is something else.Appendix A. One Meson Annihilation Into Two Mesons in the3P0 ModelWe study the reaction of one meson annihilation into two mesons shown in Fig. 3 inthe quark–antiquark 3P0 vertex of Eq. (2). The σij in the vertex can be understoodas an operator projecting a quark–antiquark pair onto a spin-1 state. It can be easilyproven that〈0, 0|σµij |[χ̄i ⊗ χj ]JM 〉 = (−1)M√2δJ,1δM,−µ . (A.1)Concerning SU(2) flavor a quark–antiquark pair which annihilates into the vac-uum must have zero isospin. So the operator F̂ij has the similar property〈0, 0|F̂ij |T, Tz〉 =√2δT,0δTz,0. For the color part, we simply have 〈0, 0|Ĉij |qiαq̄jβ〉 =δαβ , where α and β are color indices. The transition amplitude for a meson decayinto two mesons in the 3P0 model is defined as T = 〈Ψi|V †45|Ψf 〉, where |Ψi〉 and|Ψf 〉 are the initial and final states, respectively. The initial state is simply the one2nd ReadingNovember 16, 2005 10:45 WSPC/143-IJMPE 00374992 P. Suebka, C. Kobdaj & Y. Yanρ, f2, f0p1p2p3p4p5p6π−π+Fig. 3. A meson annihilation into a π+π− pair in the 3P0 quark model.meson wave function (WF) having the form|Ψi〉S = NS e−18b2(p1−p2)2[12(1)⊗ 12(2)]Si[12(1)⊗ 12(2)]Ti, (A.2)for the S-wave meson (for example, the ρ meson), and|Ψi〉P= NP e− 18b2(p1−p2)2[y1µ(p1 − p2) ⊗[12(1)⊗ 12(2)]S′]Si[12(1)⊗ 12(2)]Ti,(A.3)for the P -wave mesons (for example, the f2(1270) meson), where y1µ(q) = |q|Ylµ(q̂).We have spin Si = 1 and isospin Ti = 1 for the ρ meson (the isospin projectionTz = 0 for ρ0), spin Si = 2 and isospin Ti = 0 for the f2(1270) meson, and spinSi = 0 and isospin Ti = 0 for the f0(600) meson. Here we have employed theharmonic oscillator interaction between quark and antiquark. The final state |Ψf 〉is formed by coupling the WFs of the two final mesons. For two S-wave mesons wehave|Ψf 〉 = NsNse−18b2(p3−p4)2e−18b2(p5−p6)2[[12(3)⊗ 12(4)]S1⊗[12(5)⊗ 12(6)]S2]Sf ,Mf×[[12(3)⊗ 12(4)]T1⊗[12(5)⊗ 12(6)]T2]T,Tz. (A.4)The transition amplitude is derived as2nd ReadingNovember 16, 2005 10:45 WSPC/143-IJMPE 00374ππ Reaction in Non-Relativistic Quark Model 993Tρ→π+π− = λ2433√3π1/4b3/2ke−112b2k2(−1)mY1m(k̂) ,Tf2(1270)→π+π− = λ24√334√5π1/4b5/2k2e−112b2k2 (−1)mY2m(k̂) , (A.5)Tf0(600)→π+π− = λ2334π1/4b1/2e−112b2k2(2b2k2 − 9) ,where k is the momentum of the outgoing π mesons in the center-of-mass system.Note that we have, for simplicity, set the ρ, f2(1270), f0(600) and π mesons to havethe same size parameter b, that is NS = (b2/π)3/4 and NP = (2b5/3π1/2)1/2.References1. W. N. Cottingham and R. V. Mau, Phys. Rev. 130 (1963) 735; W. N. Cottingham,M. Locombe, B. Loiseau, J. M. Richard and R. V. Mau, Phys. Rev. B 8 (1973) 800;M. Lacombe et al., Phys. Rev. C 21 (1980) 861.2. R. Machleidt, K. Holinde and Ch. Elster, Phys. Rep. 149 (1987) 1.3. D. Lohse, J. W. Durso, K. Holinde and J. Speth, Nucl. Phys. A 516 (1990) 513.4. T. Hippchen J. Haidenbauer, K. Holinde and V. Mull, Phys. Rev. C 44 (1991) 1323;V. Mull, J. Haidenbauer, T. Hippchen and K. Holinde, ibid. 44 (1991) 1337; J. Haiden-bauer, T. Hippchen and R. Tegen, ibid. 44 (1991) 1812.5. J. Côte, M. Lacombe, B. Loiseau, B. Moussallam and R. V. Mau, Phys. Rev. Lett. 48(1982) 1319; M. Lacombe, B. Loiseau, B. Moussallam and R. V. Mau, Phys. Rev. C29 (1993) 1800.6. Y. Yan and R. Tegen, Phys. Rev. C 54 (1996) 1441.7. W. R. Gibbs and B. Loiseau, Phys. Rev. C 50 (1994) 2742.8. Y. Yan, L. Pumsa-ard, R. Tegen, T. Gutsche, V. E. Lyubovitskij and A. Faessler,IJMPE 12 (2003) 367.9. J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. E. Sainio, Phys. Lett. B 374(1996) 210.10. C. B. Dover, T. Gutsche, M. Maruyama and A. Faessler, Prog. Part. Nucl. Phys. 29(1992) 87.11. A. Le Yaouanc et al., Phys. Rev. D 8 (1973) 2223; ibid. 9 (1974) 1415; ibid. 11 (1975)1272.12. M. Maruyama, S. Furui and A. Faessler, Nucl. Phys. A 472 (1987) 643; M. Maruyama,S. Furui, A. Faessler and R. V. Mau, ibid. 473 (1987) 649; T. Gutsche, M. Maruyamaand A. Faessler, ibid. 503 (1989) 737.13. A. Muhn, T. Gutsche, R. Thierauf, Y. Yan and A. Faessler, Nucl. Phys. A 598 (1996)285.14. Y. Yan, R. Tegen, T. Gutsche and A. Faessler, Phys. Rev. C 56 (1997) 1596.15. C. B. Dover, T. Gutsche and A. Faessler, Phys. Rev. C 43 (1991) 379.16. T. Gutsche, R. D. Viollier and A. Faessler, Phys. Lett. B 331 (1994) 8.17. Y. Yan, S. W. Huang and A. Faessler, Phys. Lett. B 354 (1995) 24.18. T. Barnes and E. S. Swanson, Phys. Rev. D 46, 131 (1992); Phys. Rev. C 63 (2001)025204.19. Y. Yan, C. Kobdaj, P. Suebka, Y. M. Zheng, A. Faessler, Th. Gutsche and V. E.Lyubovitskij, Phys. Rev. C 71 (2005) 025204.20. S. Eidelman et al., Phys. Lett. B 592 (2004) 1.2nd ReadingNovember 16, 2005 10:45 WSPC/143-IJMPE 00374994 P. Suebka, C. Kobdaj & Y. Yan21. C. Joachain, Quantum Collision Theory (North-Holland Publishing Comp., 1975).22. S. D. Protopopescu et al., Phys. Rev. D 7 (1973) 1279.23. E. A. Alekseeva et al., ZETF 82 (1982) 1007.

Reaction in non-relativistic quark model

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Reaction in non-relativistic quark model

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