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共同研究者 : 山田賢治，石田晋（日大），織田益穂（国士舘大）

前田　知人　（日大短大）. 共同研究者 : 山田賢治，石田晋（日大），織田益穂（国士舘大）. Introduction Covariant Description of Composite Hadrons in the U ~ (12) SF ×O(3,1) L - Scheme Possible Assignments for Observed Mesons Electro-Magnetic and Pionic Interactions of Hadrons Summary. Contents. 1. Introduction.

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共同研究者 : 山田賢治，石田晋（日大），織田益穂（国士舘大）

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  1. 前田　知人　（日大短大） 共同研究者: 山田賢治，石田晋（日大），織田益穂（国士舘大） 少数粒子系物理の現状と今後の展望@RCNP

  2. Introduction Covariant Description of Composite Hadrons in the U~(12)SF×O(3,1)L - Scheme Possible Assignments for Observed Mesons Electro-Magnetic and Pionic Interactions of Hadrons Summary Contents 少数粒子系物理の現状と今後の展望@RCNP

  3. 1. Introduction • Non-relativistic Quark Model (NRQM) … has been used to study the properties of low-lying hadrons with remarkable success. (at least until recently ?) • Extension to Relativistic Quark Model (RQM) Note that, in this case, the word ``relativistic’’ has two different kinds of meaning. • For Center of Mass (CM) motion : • relevant to ; transition with large mass differences, large angle scattering, form factor in large q2 region, … etc. • For quark motion (in the case of large internal velocity): • Non-negligible even at the rest frame of hadrons • e.g. Godfrey-Isgur (1985), ``relativised Q.M.’’ 少数粒子系物理の現状と今後の展望@RCNP

  4. Relativistic Covariant Oscillator Quark Model (COQM) • (since ~1970) concerning the CM motion ! Feynman, Kislinger and Ravndal (1971), Y. S. Kim et al. (1973) , Namiki et al. (1970) , Ishida et al. (1971) Basic framework is ``boosted L-S coupling scheme’’. A remarkable point is that WFs of hadron are described as the direct product of spin part and space-time part. ( Covariant, but not fully relativised! ) 少数粒子系物理の現状と今後の展望@RCNP

  5. Purpose of this talk We emphasize the importance of covariant treatment of composite systems • It lead to some phenomenologically desirable properties Conserved EM current, Liner rising Regge Trajectory, …. etc. • Furthermore, it have been pointed out the possibility of the existence of new meson multiplets (called chiral states), in connection withrelativistic treatment of composite hadrons. 少数粒子系物理の現状と今後の展望@RCNP

  6. 2. Covariant Description of Composite Hadrons Boosted L-S ( U~(12) ×O(3,1) ) WF General WF of qqbar mesons are given by the following Klein-Gordon field with one each upper and lower indices. Definite Metric Type 4-Dim. Oscillator WF Bargman-Wigner Spinor WF Relative Coordinate Flavor WF C.M. Coordinate Spin Space-Time (Here etc. denotes Dirac spinor / flavor indices) A relativistic extension of conventional NRQM by separately boosting! 少数粒子系物理の現状と今後の展望@RCNP

  7. (1) Space-time part : 4-dimentional oscillator function • Basic equation of motion ( Potential ) pure conf. limit • CM and Relative coordinates • Plane Wave Expansion 2-nd quantized! 少数粒子系物理の現状と今後の展望@RCNP

  8. Definite type oscillator WF Note that boson type ; subsidiary condition; liner rising Regge trajectory Nomalizable! ( M2∝L) (Ground States) (Excite States) 少数粒子系物理の現状と今後の展望@RCNP

  9. (2) U(4) spin part The expansion basis of qqbar meson spin WF is given by direct product of the respective Dirac spinors corresponding to relevant constituent quarks and anti-quarks. They consist of totally 16 members of bi-Dirac space. Total 16 comp. Complete set of bi-Dirac spinor for describing the qqbar Ps ×2 V ×2 S ×2 A× 2 少数粒子系物理の現状と今後の展望@RCNP

  10. To fully utilize relativistic 4-components… Dirac spinors with on-shell 4-velocity of hadrons. ， ， ， ， Chirality : Parity : 少数粒子系物理の現状と今後の展望@RCNP

  11. The (u+, ,v+,)corresponds to conventional constituent quark degree of freedom. On the other hand, We suppose that the (u- ,v-) is also realized, independently of (u+, v+), as the physical degrees of freedom in composite hadrons. U~(12)SF – Scheme S. Ishida, M. Ishida, and T.M. PTP104 (2000) S. Ishida, M. Ishida, PLB539 (2002) M. Ishida, PLB627 (2005) The u- and v- with exotic quantum numbers (jp=(1/2)-) leads to a new type of `exotic’ states, called chiral states, which do not appear in the non-relativistic scheme. 少数粒子系物理の現状と今後の展望@RCNP

  12. Accordingly, a conventional non-relativistic symmetry, is extend into ρ- spin `at the rest frame of hadrons’. The remarkable point in this scheme is that it contains a new symmetry SU(2)ρ for “Confined Quarks”. 少数粒子系物理の現状と今後の展望@RCNP

  13. ：polarization vector of mesons, Expansion of Spin WF of qqbar meson 4 ×4* = 16 representation in U~(4)S Boost op. complete set of SU(2)σ×SU(2)ρ Boost op. 少数粒子系物理の現状と今後の展望@RCNP

  14. (Example) Wave functions of two ground-state vector mesons For the vector meson sector, there exist a “extra” vector-meson nonet in ground states in addition to ordinal rho(770) nonet, both with JPC = 1−−. Here it should be noted that, in the actual application, being based on the success of SU(6)-description for rho(770)-nonet, it seems that its WF should be taken as the form containing only positive rho3- and rho3bar-states. This corresponds to taking these spin WF as the irreducible representation of total rho-spin of qqbar. 少数粒子系物理の現状と今後の展望@RCNP

  15. identical to NRQM WF in the meson rest frame! Physical states are expected to be mixing states of them in equal weight. V V’ Candidates 少数粒子系物理の現状と今後の展望@RCNP

  16. 3. Possible Assignments for Observed Mesons in U~(12)SF ×O(3,1)L Scheme Here we try to assign some of the observed mesons to the predicted ground-state qqbar multiplets in the U~(12)SF classification scheme, resorting to their particle properties, and estimate the masses of missing members of the ground-state multiplets. 少数粒子系物理の現状と今後の展望@RCNP

  17. K. Yamada, arXiv: hep-ph/06012337 Experimental Candidates (Ground States) PDG. 少数粒子系物理の現状と今後の展望@RCNP

  18. K. Yamada, arXiv: hep-ph/06012337 Experimental Candidates (Excited States) 少数粒子系物理の現状と今後の展望@RCNP

  19. K. Yamada, arXiv: hep-ph/06012337 Experimental Candidates (Excited States) Cont’d 少数粒子系物理の現状と今後の展望@RCNP

  20. 4. Electro-Magnetic and Pionic Interactions of Hadrons By using the following method, we can obtain the decay interaction vertex, systematically. Notice There is a crucial difference for the ``small component’’ between of our BW spinors and of the usual constituent quark ones. i.e. Absence of relative motion of quarks only for the spinor part ! (Space-time part includes relative motion of quarks. ) Single BW spinor (P,E,M) ; Hadronic Variable Initial hadron at rest 少数粒子系物理の現状と今後の展望@RCNP

  21. (1) Electro Magnetic Interaction `Feynman Trick’ Minimal Subst. = Conserved E.M. Current (concerning the CM motion ) (See fordetail, S.Ishida K.Yamada and M. Oda, PRD40(1989)) 少数粒子系物理の現状と今後の展望@RCNP

  22. (2) Pionic Interaction (One Pseudo-scalar Emission) Here we suppose that emitted Ps-meson is local object. By the analogies to the case of E.M. interaction, similar ( but heuristic ) minimal substitution leads ; ( Feynman, Kislinger and Ravndal (1971)) V1 = + ( 1 ⇔ 2 ) Taking matrix element of V1 among u+(v) and u+bar(v), it yields On the other hand, in the case of u-(v) and u+bar(v), it gives no S-wave decay term. Therefore, we put the additional term, V2 = + ( 1 ⇔ 2 ) for u-(v) to u+bar(v), and ~0 for u+(v) and u+bar(v). 少数粒子系物理の現状と今後の展望@RCNP

  23. In the conventional chiral-quark model ; Matrix Elements 少数粒子系物理の現状と今後の展望@RCNP

  24. 5. Summary Characteristic qualities of the U(12)×O(3,1) Quark Model • It is covariant. • Excited states are on the linear Regge trajectory in terms of squared masses. • Electromagnetic current is conserved even for the transitions from excited states. • SU(2)ρ- symmetry leads to the possibility of the existence of the ``exotic’’ chiral-states. 少数粒子系物理の現状と今後の展望@RCNP

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