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） 2015/11/211

Contents 1.Introduction 2.Covariant Description of Composite Hadrons in the U ~ (12) SF ×O(3,1) L - Scheme 3.Possible Assignments for Observed Mesons 4.Electro-Magnetic and Pionic Interactions of Hadrons 5.Summary 2015/11/212

1. Introduction Extension to Relativistic Quark Model (RQM) Note that, in this case, the word ``relativistic’’ has two different kinds of meaning. 1.For Center of Mass (CM) : 1.For Center of Mass (CM) motion : relevant to ; transition with large mass differences, large angle scattering, form factor in large q 2 region, … etc. 2.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.’’ Non-relativistic Quark Model (NRQM) … has been used to study the properties of low-lying hadrons with remarkable success. (at least until recently ?) 2015/11/213

Covariant Relativistic Covariant Oscillator Quark Model (COQM) (since ~1970) Feynman, Kislinger and Ravndal (1971), Y. S. Kim et al. (1973), Namiki et al. (1970), Ishida et al. (1971) concerning the CM motion ! Basic framework is ``boosted L-S coupling scheme’’. direct product of spin part and space-time part. 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! ) 2015/11/214

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. existence of new meson multiplets (called chiral states) Furthermore, it have been pointed out the possibility of the existence of new meson multiplets (called chiral states), in connection with relativistic treatment of composite hadrons. 2015/11/215

Boosted L-S ( U ~ (12) ×O(3,1) ) WF Definite Metric Type 4-Dim. Oscillator WF C.M. Coordinate Flavor WF Space-Time A relativistic extension of conventional NRQM by separately boosting! General WF of qqbar mesons are given by the following Klein-Gordon field with one each upper and lower indices. Spin (Hereetc. denotes Dirac spinor / flavor indices) Relative Coordinate Bargman-Wigner Spinor WF 2. Covariant Description of Composite Hadrons 2015/11/216

(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! 2015/11/217

Definite type oscillator WF Note that boson type ; subsidiary condition; (Ground States) (Excite States) liner rising Regge trajectory ( M 2 ∝ L ) Nomalizable! 2015/11/218

Complete set of bi-Dirac spinor for describing the qq bar Ps ×2 V ×2 S ×2 A× 2 Total 16 comp. 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. (2) U(4) spin part 2015/11/219

To fully utilize relativistic 4-components… ， ，，， Chirality : Parity : Dirac spinors with on-shell 4-velocity of hadrons. 2015/11/2110

The (u +,, v +, ) corresponds to conventional constituent quark degree of freedom. We suppose that the (u -,v - ) is also realized, independently of (u +, v + ), as the physical degrees of freedom in composite hadrons. The u - and v - with exotic quantum numbers (j p =(1/2) - ) leads to a new type of `exotic’ states, called chiral states, which do not appear in the non- relativistic scheme. On the other hand, 12) SF U ~ (12) SF – Scheme S. Ishida, M. Ishida, and T.M. PTP104 (2000) S. Ishida, M. Ishida, PLB539 (2002) M. Ishida, PLB627 (2005) 2015/11/2111

Accordingly, a conventional non-relativistic symmetry, is extend into ρ- spin The remarkable point in this scheme is that it contains a new symmetry SU(2) ρ for “Confined Quarks”. `at the rest frame of hadrons’. 2015/11/2112

complete set of SU(2) σ ×SU(2) ρ Expansion of Spin WF of qqbar meson Boost op. 4 ×4* = 16 representation in U ~ (4) S ： polarization vector of mesons, 2015/11/2113

(Example) Wave functions of two ground-state vector mesons the irreducible representation of total rho-spin of qqbar 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 rho 3 - and rho 3 bar-states. This corresponds to taking these spin WF as the irreducible representation of total rho-spin of qqbar. For the vector meson sector, there exist a “extra” vector-meson nonet in ground states in addition to ordinal rho(770) nonet, both with J PC = 1 −−. 2015/11/2114

Physical states are expected to be mixing states of them in equal weight. VV’ identical to NRQM WF in the meson rest frame! Candidates 2015/11/2115

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. 3. Possible Assignments for Observed Mesons in U ~ (12) SF ×O(3,1) L Scheme 2015/11/2116

K. Yamada, arXiv: hep-ph/ Experimental Candidates (Ground States) PDG. 2015/11/2117

K. Yamada, arXiv: hep-ph/ Experimental Candidates (Excited States) 2015/11/2118

Experimental Candidates (Excited States) Cont’d K. Yamada, arXiv: hep-ph/ /11/2119

4. Electro-Magnetic and Pionic Interactions of Hadrons By using the following method, we can obtain the decay interaction vertex, systematically. 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 ! Notice 2015/11/2120 (Space-time part includes relative motion of quarks. ) (P,E,M) ; Hadronic Variable Initial hadron at rest Single BW spinor

(1) Electro Magnetic Interaction `Feynman Trick’ = Conserved E.M. Current (concerning the CM motion ) (See for detail, S.Ishida K.Yamada and M. Oda, PRD40(1989)) Minimal Subst. 2015/11/2121

Here we suppose that emitted Ps-meson is local object. (2) Pionic Interaction (One Pseudo-scalar Emission) 2015/11/ ( 1 ⇔ 2 ) By the analogies to the case of E.M. interaction, similar ( but heuristic ) minimal substitution leads ; ( Feynman, Kislinger and Ravndal (1971)) Taking matrix element of V 1 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, V 1 = V 2 = + ( 1 ⇔ 2 ) for u - (v) to u + bar (v), and ~0 for u + (v) and u + bar (v).

2015/11/2123 In the conventional chiral-quark model ; Matrix Elements

5. Summary Characteristic qualities of the U(12)×O(3,1) Quark Model 1.It is covariant. 2.Excited states are on the linear Regge trajectory in terms of squared masses. 3.Electromagnetic current is conserved even for the transitions from excited states. 4.SU(2) ρ - symmetry leads to the possibility of the existence of the ``exotic’’ chiral-states. 2015/11/2124