the
Wave AxialVector Mesons
In the Covariant Scheme
Abstract
We study the properties of axial vector mesons and as relativistic Swave states which are predicted in the scheme, through the analyses of their radiative and pionic decays. Specifically, partial widths of the strong processes, their wave amplitude ratios, and radiative transition widths of processes are calculated by using a simple decay interaction model, and made a comparison with the respective experimental values.
1 Introduction
In recent years we have proposed the
scheme[1, 2],
a relativistically covariant levelclassification
scheme of hadrons.
In this scheme, the ground state (GS) of
light meson system is assigned
as 
representation of the group
at their rest frame.
The group includes,
in addition to the conventional nonrelativistic
group, the new symmetry
^{1}^{1}1The new degree of
freedom corresponding to the
symmetry is called
the spin, after the wellknown
decomposition of
Dirac matrices. ,
which corresponds to the degree of freedom associated
with negative energy Dirac spinor solutions
of confined quarks inside hadrons.
By inclusion of this extra spin freedom,
it leads to the possible existence of some extra
multiples, called
chiral states,
which do not exist in the ordinary
nonrelativistic quark model (NRQM).
As an example, the light
scalar / meson,
a controversial particle for long time,
is identified as
wave chiral state
as well as meson,
and they play mutually the role of chiral partners
in the scheme.
As is well known, in conventional level classification
scheme based on NRQM,
lowest scalar meson is obliged to be
assigned as orbital wave excited state.
As an another example,
the meson, possibly
to be identified as
the wave axialvector mesons
in the scheme.
They form a linear representation of chiral symmetry
with the wave meson.
Here it is notable that these and
mesons are expected to have the light mass compared with
the conventional case of the wave states.
Furthermore, representation includes
the another axialvector
meson state with ,
to be identified with the meson.
In this work, we try to elucidate the properties of
our newtype wave axialvector
mesons, and ,
whose existence are predicted in
the scheme,
through the analyses of their radiative and
pionic decay.
In the actual analyses, we identify
our chiral wave and mesons as
the experimentally wellknown states, and
, respectively.
Then, by using a simple decay interaction, their
partial widths of the strong
() () decays
(with wave amplitude ratios, ) and
radiative transition widths of
()
processes are calculated
in comparison with
the respective experimental values.
2 Wave functions of the and mesons as wave chiral states
In this section we collect the concrete expressions
of meson wave function
(WF) in our scheme
necessary for the relevant applications
^{2}^{2}2In more detail,
see Ref. [1, 2, 4].
The basic framework of our levelclassification scheme is
what is called the boosted LScoupling (bLS) scheme.
In this scheme, the WF of
GS mesons are
given by the following (bilocal Klein Gordon)
field with one each upper and lower indices
^{3}^{3}3
For simplicity, the only positive frequency part of WF
is shown here.
,
(1) 
Where () denotes
Dirac spinor and flavor indices
respectively, () represents
the center of mass (CM) (relative) coordinate
of the composite meson. The
(, being the mass of meson;
, )
denotes 4momentum (4velicity) of the relevant mesons.
In the bLS scheme, respective
spin () and
spacetime
^{4}^{4}4
We have been adopted a
definite metric type 4dimensional oscillator
function as [7].
() parts of WF
are, separately,
made covariant by boosting
from the corresponding parts of NR ones.
Important feature of
scheme is that
the spin WF contains
extra spin degree of freedom,
called spin.
As expansion bases of spinor WF, we use
the Dirac spinor with hadron onshell 4velocity,
(2) 
Here, corresponds to conventional constituent quark degree of freedom, while is indispensable for covariant description of confined quarks ^{5}^{5}5They form the chiral partner in basic representation of the chiral group. . Accordingly, expansion basis of meson WF is given by direct product of the respective spinor WF corresponding to the relevant constituent quark. They consist of totally 16 members in space as,
(3) 
We show the specific form of spin WF for the respective members of wave mesons, appeared in the relevant applications, in Table 1. Here it should be noted that, in the actual application, being based on its success[6] with description for nonet, it seems that its WF should be taken as the form containing only positive states. This is made by taking the equalweight superposition of two spin WF which belongs to the different chiral representation, respectively.
Mesons  

[9]  
3 Radiative decays of the and mesons
At first, we will consider the radiative decays of and mesons. In this work, we focus on the radiative transitions among the GS mesons. Therefore we are able to adopt simply the effective spintype interaction,
(4) 
Here we introduced two independent coupling parameters and . The term contributes to only quark chirality conserving transitions, while the term does to chirality nonconserving ones. By applying the quarkphoton interaction (4), the effective meson current is given by the following formulas,
(5) 
Here, subscript () represents the coupling of the emitted single photon with the relevant meson system through constituent quark (antiquark). The specific form of the current is represented by
(6)  
(7) 
for the case of the chirality conserving transition; and similarly
(8)  
(9) 
for the case of the chirality nonconserving transition. Here denotes the 4momentum of emitted photon, is overlapping integral (OI) of spacetime oscillator function, which gives a Lorentz invariant transition form factor as
(10)  
(11) 
where we introduce the parameter corresponding to the
Regge slope inverse.
In our scheme the relativistic covariance
of the spin current, due to the inclusion
of Dirac spinor with
negative value,
plays an important role in some radiative
transition processes.
To clarify this point, we rewrite the spin
current vertex operator as
(12) 
In the cases of transition between both positive
(negative) Dirac spinors,
as is well known, the main contribution comes from
the magnetic interaction.
On the other hand, in the case of transitions between Dirac spinors
with positive and negative values,
the electric interaction, coming from the
term, becomes a dominant contribution.
As a results, this intrinsic electric dipole[5]
transition gives an important role
for the transition accompanied with their parity change,
such as processes.
In this work, we take the following values of parameters in our scheme.

(, )=(, ) from and

from
The masses of the respective mesons are taken from PDG[3], except for the one of the pion in the form factor with . The estimated widths are in comparison with experiment in Table 2. Results for this calculation are consistent with experiments.
Process  Our results  Experimental values 

68 (input)  687  
230 (input)  23060  
604  640246 
4 Pion emissions of and mesons
Next we consider the strong decays with one pion emission. We adopt simply the following two types of effective quarkpion interactions;
(13)  
(14) 
Note that here, ( and ) meson is treated as an external localfield. Resultant matrix elements are given as a sum of two terms;
(15)  
(16)  
(17) 
In the above case, the OI of the spacetime WF is given by
(18)  
(19) 
where , being the 4momentum of emitted pion. The relevant decay amplitude is
(20) 
The explicit forms of and are shown in Table 3.
It may be worthwhile to note that at least
two coupling types ( expressed and in the above )
are required to reproduce the experimental data on wave amplitude ratios.
Our decay interaction contains two
independent coupling parameters, and ,
which will be commonly applied to all quarkpion vertices
^{6}^{6}6
As an example, it is applied to the study of
‘extra’ meson[8].
.
These are determined from the experimental data of
wave amplitude ratio and total width of meson as,

from

from .
The masses of the relevant mesons are taken from PDG[3]. The numerical results are shown in Table 4.
Width (MeV)  

process  Our results  Experimental values  
0.277(input)  0.277 0.027  142(input)  1429  
0.344  0.108 0.016  191 
5 Concluding remarks
In this work, we investigate the decay properties of
wave and mesons
in the scheme,
by assigning them with
and mesons, respectively.
At first, it is shown that the radiative decay widths of
processes are
consistently reproduced by using the simple spintype
quarkphoton effective interaction
in the framework of the scheme.
Secondly, for the strong onepion emission decays,
assuming the  and type quarkpion effective
interactions, the wave amplitude ratios and partial widths of
decays
are evaluated.
As a results, by inputting the data for the meson,
the sign of wave amplitude ratio for the
decay agrees with the
experiments, but its absolute value is about
three time larger than experiment.
Partial width of is predicted
with .
The interaction adopted in this work
for the radiative/strong decays should be
tested by applying it to other various decay processes.
References
 [1] S. Ishida et al.,Prog. Theor. Phys. 104, 785 (2000).
 [2] S. Ishida and M. Ishida , Phys. Lett. B 539, 249 (2002).
 [3] W.M. Yao et al., Journal of Physics G 33, 1 (2006) .
 [4] S. Ishida et al., hepph/0408136 v3.

[5]
S Ishida et al., AIP Conf. Proc. 717, 716 (2004).
Also in ePrint Archive: hepph/0310061.  [6] M. Oda, M. Ishida, and S. Ishida, Prog. Theor. Phys. 101,1285 (1999).
 [7] S. Ishida and K. Yamada, Phys. Rev. D 35, 265 (1987).
 [8] K. Yamada, these proceedings.
 [9] I. Yamauchi, these proceedings.