共同研究者 : 山田賢治，石田晋（日大），織田益穂（国士舘大）

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）

23/04/21 1少数粒子系物理の現状と今後の展望

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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


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1. Introduction

• Extension to Relativistic Quark Model (RQM) Note that, in this case, the word ``relativistic’’ has two different kinds of meaning.

1.1. For Center of Mass (CM) For Center of Mass (CM) motion ::

relevant to ; transition with large mass differences, large angle scattering, form factor in large q2 region, … etc.

2.2. For quark motion (in the case of large internal velocity):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 ?)


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• Relativistic CovariantCovariant 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’’.

A remarkable point is that WFs of hadron are described as the direct product of spin part and space-time part. direct product of spin part and space-time part.

( Covariant, but not fully relativised! )( Covariant, but not fully relativised! )


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Purpose of this talk

We emphasize the importance of covariant treatment of composite systems

• It lead to some phenomenologically desirable propertiesConserved EM current, Liner rising Regge Trajectory, …. etc.

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


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Boosted L-S ( U~(12) ×O(3,1) ) WF

Definite Metric Type 4-Dim. Oscillator WFC.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(Here etc. denotes Dirac spinor / flavor indices)

Relative Coordinate Bargman-Wigner Spinor WF

2. Covariant Description of Composite Hadrons


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(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!


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Definite type oscillator WF

Note that

boson type

;

subsidiary condition;

(Ground States)

(Excite States)

liner rising Regge trajectory( M2 ∝ L 　 )Nomalizable!


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Complete set of bi-Dirac spinor for describing the qqbar

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


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To fully utilize relativistic 4-components…

，

，，，

Chirality :Parity :

Dirac spinors with on-shell 4-velocity of hadrons.


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The (u+, ,v+,) corresponds to conventional constituent quark degree of freedom.

We suppose that the (uWe suppose that the (u-- ,v ,v--) is also realized, ) is also realized,

independently of (uindependently of (u++, v, v++), as the physical degrees ), as the physical degrees

of freedom in composite hadrons. of freedom in composite hadrons.

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.

On the other hand,

U~(12)12)SF SF – Scheme

S. Ishida, M. Ishida, and T.M. PTP104 (2000)S. Ishida, M. Ishida, PLB539 (2002)M. Ishida, PLB627 (2005)


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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’.


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complete set of SU(2)σ×SU(2)ρ

Expansion of Spin WF of qqbar meson

Boost op.

Boost op.4 ×4* = 16 representation in U~(4)S

： polarization vector of mesons,


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(Example) Wave functions of two ground-state vector mesons

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 the irreducible representation of total rho-spin of qqbarrho-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 JPC = 1−−.


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Physical states are expected to be mixing states of them in equal weight.Physical states are expected to be mixing states of them in equal weight.

V V’

identical to NRQM WF identical to NRQM WF in the meson rest frame!in the meson rest frame!

Candidates


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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


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K. Yamada, arXiv: hep-ph/06012337

Experimental Candidates (Ground States)

PDG.


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Experimental Candidates (Excited States)


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Experimental Candidates (Excited States) Cont’d



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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 !

NoticeNotice


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(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.


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Here we suppose that emitted Ps-meson is local object.

(2) Pionic Interaction (One Pseudo-scalar Emission)


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+ ( 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 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,

V1 =

V2 = + ( 1 ⇔ 2 )

for u-(v) to u+bar(v), and ~0 for u+(v) and u+

bar(v).


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In the conventional chiral-quark model ;

Matrix Elements

5. Summary

Characteristic qualities of the U(12)×O(3,1) Quark Model1. 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.


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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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