Strong Decays of Excited Heavy-Light Mesons in the U~(12)-Scheme

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Contents

1. Introduction
2. Covariant Description of Composite Hadrons in the
   U(12)SFO(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

• 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.

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• 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! )
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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
  with relativistic treatment of composite hadrons.

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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
Space-Time
Spin
(Here
etc. denotes Dirac spinor / flavor indices)
A relativistic extension of conventional NRQM by
separately boosting!
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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
liner rising Regge trajectory
( M2 ?L )
Nomalizable!
(Ground States)
(Excite States)
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(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
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To fully utilize relativistic 4-components
Dirac spinors with on-shell 4-velocity of
hadrons.
,
,
,
,
Chirality
Parity
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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.
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Accordingly, a conventional non-relativistic
symmetry,
is extend into
?- spin
at the rest frame of hadrons.
The remarkable point in this scheme is that it
contains a new symmetry SU(2)? for Confined
Quarks.
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Expansion of Spin WF of qqbar meson
4 4 16 representation in U(4)S
Boost op.
complete set of SU(2)sSU(2)?
Boost op.
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(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.
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Physical states are expected to be mixing states
of them in equal weight.
V
V
Candidates
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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.
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K. Yamada, arXiv hep-ph/06012337
Experimental Candidates (Ground States)
PDG.
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K. Yamada, arXiv hep-ph/06012337
Experimental Candidates (Excited States)
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K. Yamada, arXiv hep-ph/06012337
Experimental Candidates (Excited States) Contd
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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.
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
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(1) Electro Magnetic Interaction
Feynman Trick
Minimal Subst.

Conserved E.M. Current (concerning the CM motion
) (See for detail, S.Ishida K.Yamada and M. Oda,
PRD40(1989))
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(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))
( 1 ? 2 )
V1
Taking matrix element of V1 among u(v) and
ubar(v), it yields
On the other hand, in the case of u-(v) and
ubar(v), it gives no S-wave decay
term. Therefore, we put the additional term,
V2
( 1 ? 2 )
for u-(v) to ubar(v), and 0 for u(v) and
ubar(v).
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In the conventional chiral-quark model
Matrix Elements
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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)?- symmetry leads to the possibility of the
   existence of the exotic chiral-states.