Vol.2, No.11, 1292-1297 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.211156
Light D wave meson spectrum in a relativistic harmonic
model with instanton induced interaction
Antony Prakash Monteiro, Kanti Basavarajappa Vijaya Kumar*
Department of Physics, Mangalore University, Mangalagangothri, Mangalore, India; *Corresponding author: kbvijaya kumar@
yahoo.com
Received 20 July 2010; revised 25 August 2010; accepted 28 August 2010.
ABSTRACT
The mass spectrum of the D wave mesons is
considered in the frame work of relativistic
harmonic model (RHM). The full Hamiltonian
used in the investigation has the Lorentz scalar
plus a vector harmonic-oscillator potential, the
confined-one-gluon-exchange potential (COGEP)
and the instanton-induced quark-antiquark in-
teraction (III). A good agreement between calcu-
lated D wave meson masses with experimental
D wave meson masses is obtained. The respec-
tive role of III and COGEP in the D wave meson
spectrum is discussed.
Keywords: Quark Model; Confined-One-Gluon-
Exchange Potential; Instanton Induced Interaction;
D Wave Meson Spectra
1. INTRODUCTION
The Quantum Chromodynamics (QCD), the theory of
strong interactions, is not exactly solvable in the
non-perturbative regime which is required to obtain the
physical properties of the hadrons. Hence various ap-
proximation methods like lattice gauge theories are em-
ployed to solve QCD in the non-perturbative regime. In
the constituent quark model, conventional mesons are
bound states of a spin ½ quark and spin ½ antiquark
bound by a phenomenological potential. The phenome-
nological models developed to explain the observed
properties of mesons are either non-relativistic quark
models (NRQM) with suitably chosen potential or rela-
tivistic models where the interaction is treated pertur-
batively [1-3]. In most of the works that use NRQM, it is
assumed that the quark interaction is dominated by a
linear or quadratic confinement potential and is supple-
mented by a short range potential stemming from the
one-gluon exchange mechanism. The Hamiltonian of
these quark models usually contains three main ingredi-
ents: the kinetic energy, the confinement potential and a
hyperfine interaction term, which has often been taken
as an effective one-gluon-exchange potential (OGEP) [4].
Other types of hyperfine interaction like Instanton-
Induced Interaction (III) deduced by a non-relativistic
reduction of the ‘t Hooft interaction [5-12] have also
been introduced in the literature.
The success of the NRQM in describing the hadron
spectrum is somewhat paradoxical, as light quarks
should in principle not obey a non-relativistic dynamics.
This paradox has been avoided in many works based on
the constituent quark models by using for the kinetic
energy term of the Hamiltonian a semi-relativistic or
relativistic expression [9-12]. Even in the existing rela-
tivistic models though the effect of confinement of
quarks has been taken into account, the effect of con-
finement of gluons has not been taken into account
[13-15]. Therefore in our present work, we have inves-
tigated the effect of exchange of confinement of gluons
on the masses of light D wave mesons and their radially
excited states in the frame work of RHM with III
[6,11-12,16]. The essential new ingredient in our inves-
tigation of the light D wave mesonic states is to take into
account the confinement of gluons in addition to the
confinement of quarks. In the existing quark models,
Fermi-Breit interaction which gives rise to π- and N-
splitting is treated as perturbation. The OGEP being at-
tractive for π, and for a nucleon a naïve perturbative
treatment of one gluon hyperfine interaction is incorrect
and hence one obtains a high value for the pion mass. This
leads to further renormalization of strength of interaction
for a better fit. Also, the most prominent flaw of NRQM is
the neglect of relativistic effects and gluon dynamics. In
our present work, for the confinement of quarks we are
making use of the RHM which has been successful in
explaining the properties of light hadrons. For the con-
finement of gluons, we have made use of the current con-
finement model (CCM) which was developed in the spirit
of the RHM [13-15]. The CCM has been quite successful
in describing the glue-ball spectra. The confined gluon

A. P. Monteiro et al. / Natural Science 2 (2010) 1292-1297 1293
propagators (CGP) are derived in CCM. Using CGP we
have obtained confined one gluon exchange potential
(COGEP). The full Hamiltonian used in the investigation
has Lorentz scalar plus a vector harmonic-oscillator po-
tential, in addition to two-body COGEP and III. In our
present work we are extending the model to study D wave
light meson spectra. In our present work, the total mass of
the meson is obtained by calculating the energy eigen
values of the Hamiltonian in the harmonic oscillator basis
spanned over a space extending upto the radial quantum
number nmax = 4. The masses of D wave mesons are ob-
tained after diagonalising for various values of nma x .
In Section 2, we review the RHM and CCM models
and give a brief description of the CGP, COGEP and III.
We also discuss the parameters involved in our model.
The results of the calculations are presented in Section 3.
Conclusions are given in Section 4.
2. THE MODEL
In RHM, quarks in a hadron are confined through the
action of a Lorentz scalar plus a vector harmonic-oscilla-
tor potential [13-15].


22
0
11
2
conf
Vr ArM   

(1)
where 0

is the Dirac matrix:
0
10
01





, (2)
M is the quark mass and A2 is the confinement strength.
They have a different value for each quark flavour. In
RHM, the confined single quark wave function (
 ) is
given by:
N
EM








 
σP (3)
with the normalization

2
3
EM
NEM

; (4)
where E is an eigenvalue of the single particle Dirac
equation with the interaction potential given in (1). The
lower component is eliminated by performing the simi-
larity transformation,
U



(5)
where U is given by,

2
2
1
1
EM
NEM
EM









 
 
 

σP
1
σP
P1
(6)
formation operator. With this transformation, the upper
component  satisfies the harmonic oscillator wave
equation.

2
22
ArE M
EM







P (7)
which is like the three dimensional harmonic oscillator
equation with an energy-dependent parameter2
n

:

12
nn
AE M (8)
The eigenvalue of (7) is given by,


22 2
21 .
nn
EM n

 (9)
Note that Eq.7 can also be derived by eliminating the
lower component of the wave function using the
Foldy-Wouthuysen transformation as it has been done in
[13].
Adding the individual contributions of the quarks, we
obtain the total mass of the hadron. The spurious centre
of mass (CM) is corrected by using intrinsic operators for
the 2
i
ir

and 2
i
i


terms appearing in the Hamilto-
nian. This amounts to just subtracting the CM motion
zero point contribution from the 2
E expression. It
should be noted that this method is exact for the 0S-state
quarks as the CM motion is also in the 0S state.
The two body quark-antiquark potential is the sum of
COGEP and III potential.




ijij ij
q COGEPIII
VrVrVr


(10)
COGEP is obtained from the scattering amplitude
[3,13-15],

2
,
42 2
b
a
j
ab si
f
ii ij j
g
MDq





 

 (11)
where, 0



, /ij

are the wave functions of the
quarks in the RHM, ab
ab
DD


 are the CCM gluon
propagators in momentum representation, 24
s
g

(=
s

)
is the quark-gluon coupling constant and i

is the color
(3)c SU generator of the th iquark. The details can be
found in references [3,13]. Below, we list the expressions
for the central, tensor and spin-orbit part of the COGEP.
The central part of COGEP is [3,13-15],

 
4
342
01
2
4
4123
()
s
ij
cent ij
COGEPi j
ijij ij
N
DcrD
EM
Vr
rrr



 












σσ
λλ


(12)
To calculate the matrix elements (ME) of COGEP, we
have fitted the exact expressions of 0()Dr

and 1()Dr

by Gaussian functions. It is to be noted that the 0()Dr

and 1()Dr

are different from the usual Coulombic
propagators. However, in the asymptotic limit (0r

)
they are similar to Columbic propagators and in the in-
frared limit ()r

they fall like Gaussian. In the

A. P. Monteiro et al. / Natural Science 2 (2010) 1292-1297 1294
above expression the c (fm-1) gives the range of propaga-
tion of gluons and is fitted in CCM [13,14] to obtain the
glue ball spectra. The 0()Dr

and 1()Dr

are given by,

0
Dr

= r






exp
22
0
2
rc




; 1()Dr

= r

exp
22
0
2
rc




where 1

= 1.035994, 2

= 2.016150 fm-1, 0 c =
(3.001453)1/2 fm-1,

= 0.8639336.
And 2
c= (4.367436)1/2 fm-1.
Tensor part of COGEP is,
()
TENij
COGEP
Vr

=- 4
s

ij
λλ 2
()EM



11
() ()
33
ij ij Dr Dr
r







ij
ˆ
S (13)
where ij
ˆˆˆ
S[3(.)(.).]
ijiji j
rrσσ σσ
where ˆˆˆ
ij
rrr is the unit vector in the direction of r

.
In the above expression primes and double primes corre-
sponds to first and second derivatives of 1()Dr

. The de-
rivatives of 1()Dr

were fitted to Gaussian functions.
1()
ij
Dr

=1
r exp
22
3
2
rc




- 2
1
r

exp [
22
2
2
rc]
''
1()
ij
Dr

=

22 22
'' 3 2
132
22
exp exp
22
ij
rc rc
Dr rr 
  


 
 

+
22
24
1exp 2
rc
r
r





= -1.176029 fm-1,  =5.118019 fm-4 2 c=
(4.367436)1/2 fm-1 and 3 c= (2.117112)1/2 fm-1. The
spin-orbit part of COGEP is,

LS ij
COGEP
Vr

=- 4
s

ij

λλ

2
EM



1
2r



01
()2() ijij ij
ij
ij
rppDr Dr



 



σσ
 
(14)
where
0()
ij
Dr

=
 
22 22
0 1
2
11
exp exp
22
rc rc
rr
rr
 
  

 

 

2

1ij
Dr

= 2 (1
rexp
22
3
2
rc




- 2
1
r

exp
22
2
2
rc




)
where 1

=2.680358 fm-1, 2

=-7.598860 fm-2 and 1 c=
(2.373588)1/2.
It should be noted that in the limit c  0, the central,
tensor and spin-orbit part of the COGEP goes over to the
corresponding potentials of the OGEP [3].
The central part of III potential is given by [8-12,16],




,0 ,0
,0 ,0
,0 ,0
8,1,
8,1/2,
2
8,0
20
ij SL
ij SL
III
ijS L
gr forI
gr forI
V
gg
rforI
g




 
















(15)
The symbols S, L and I are respectively the spin, the
relative angular momentum and the iso-spin of the sys-
tem. The
g
and
g

are the coupling constants of the
interaction. The Dirac delta-function appearing has been
regularized and replaced by a Gaussian- like function:

2
32
1exp ij
ij
r








(16)
where

is the size parameter.
The non-central part of III has contributions from both
spin- orbit and tensor terms. The spin-orbit contribution
comes from relativistic corrections to the central poten-
tial of III. It is given by [8-12],




SO ij ijij
III LSL
Vr VrLSVrL


 


(17)
The first term in Eq.17 is the traditional symmetric
spin-orbit term proportional to the operatorLS



. The
other term is the anti-symmetric spin-orbit term propor-
tional to L




where

12
1
2

  


. The radial
functions of Eq.17 are expressed as [8-12],





22
2ij
22 3
=1
22
4ij 2
3
=3 -2
exp- r
11
π
exp-r
1
π
k
ij
LS k
k
ij k
k
k
k
ij k
Vr MM
MM




















(18)
and



22
6ij -4
ij 22 3
=5 -4
exp- r
11
-
π
k
Lk
k
ij k
Vr MM










(19)
The term ()
LS
Vr

is responsible for the spitting of the
3
J
L states with 1, L, L+1. JL

 With such a
term Lis still good quantum numbers but S is not. The
term
L
V

(r

) which couples states 1
J
L
L and 3
J
L
L

.
Due to the mass dependence in Eq.19, it is clear that this
term is inoperative when the quarks are identical. In
practice the antisymmetric spin obit term is important

A. P. Monteiro et al. / Natural Science 2 (2010) 1292-1297 1295
only in the K-sector. The terms i and i are free pa-
rameters in the theory [12]. Mi corresponds to the mass
of the strange quark (s) and Mj corresponds to mass of
(u/d) quark. This term accounts for the splitting between
11D2 and 13D2 states in the K sector.
The tensor interaction of III is [12],




22
8ij -4
ij 3
=7 -4
ˆexp- r
π
k
ij
TEN
III k
k
ij k
S
VrMM




 (20)
with the tensor interaction, L is no longer a good quan-
tum number since this term couples the states 3
1
J
L
L


and 3
1
(2)
J
L
L .
The parameters of the RHM are the masses of the
quarks, Mu=Md and Ms, the respective confinement
strengths, Au
2=Ad 2, As 2, and the oscillator size parameter
bn (=1/ n
). They have been chosen to reproduce various
nucleon properties: the root mean square charge radius,
the magnetic moment and the ratio of the axial coupling
to the vector coupling [13]. The confinement strength
Au,d is fixed by the stability condition for the nucleon
mass against the variation of the size parameter bn
0
n
NH N
b

 (21)
The parameters associated with the strange quark Ms
and As
2 have been fitted in order to reproduce the mag-
netic moments of the strange baryons, according to the
procedure described in [16]. The
s

of COGEP is
fixed from S wave meson spectroscopy [16]. The value
of
s

turns out to be 0.2 for D wave mesons, which is
compatible with the perturbative treatment. Among the
non central parts of the potential, the hyperfine terms of
III has 12 additional strength and size parameters

’s
and

’s ( in Eqns. 18-20) respectively. We note that the
 values can have both positive and negative values
[8-12]. The values of the III parameters  ’s and

’s
are fixed from S and P wave meson spectroscopy [12,16]
and are listed in Table 1 and 2.
Table 1. The parameters for the D wave mesons.
b 0.62 fm
Mu,d 380 MeV
Ms 560 MeV
s

0.2
1

0.2 fm
2

0.29 fm
3

1.4 fm
4

1.3 fm
1

1.8
2

1.7
3

1.9
4

2.1
5

-22.0
6

-24.5
Table 2. Values of 7

and 8

parameters.
Meson 7 
8

(1650)

28.0 36.0
K*(1680) 49.0 52.0
K2(1820) 36.0 45.0
3

(1670) -1.6 -6.5
K*(1780) 1.0 -2.5
3

(1850) -4.0 -8.0
3. RESULTS AND DISCUSSIONS
To calculate the meson masses, the product of
quark-antiquark oscillator wave functions is expressed in
terms of oscillator wave functions corresponding to the
relative and CM coordinates. The oscillator quantum
number for the CM wave functions is restricted to NCM =
0. The Hilbert space of relative wave functions is trun-
cated at radial quantum number nmax = 4. The Hamilto-
nian matrix is constructed for each meson separately in
the basis states of 21 0,0; S
CM CMJ
NL L

 and di-
agonalised.
We have obtained the D wave meson spectra using the
model described above. The masses of the singlet and
triplet D wave mesons after diagonalisation in harmonic
oscillator basis with nmax= 4 are listed in Tables 3 and 4
respectively. The results show that III potential along
with COGEP is necessary to obtain the meson mass
spectra. If COGEP is taken as the only source of hyper-
fine interaction, the value of αs necessary to reproduce
the hadrons spectrum is generally much larger than one;
this leads to a large spin-orbit interaction, which de-
stroys the overall fit to the spectrum. The inclusion of III
diminishes the relative importance of COGEP for the
hyperfine splitting. The important role played by III in
obtaining the masses of these mesons can be well under-
stood by examining the Table 6. In Table 6, we have
given the calculated masses of triplet D wave mesons
Table 3. Masses of the singlet mesons (in MeV ).
N2S+1LJ Meson Experimental
Mass
Calculated
Mass
2

(1670)1670±20 1673.8
11D2
K2 (1770) 1773±8 1768.8
Table 4. Masses of the triplet mesons ( in MeV ).
N2S+1LJMeson Experimental
Mass Calculated Mass
(1650)

1649 ± 24 1649.6
13D1 K*(1680)1717 ± 27 1718.9
13D2 K
2(1820)1816 ±13 1818.6
3

(1670)1667 ± 4 1667.8
K*(1780)1776 ± 7 1778.3
13D3
3

(1850) 1854 ± 7 1855.9

A. P. Monteiro et al. / Natural Science 2 (2010) 1292-1297 1296
Table 5. The diagonal contributions to the masses of mesons by Vconf, color-electric (CE), color-magnetic
(CM), spin-orbit, tensor terms of COGEP and spin-orbit, tensor terms of III (in MeV).
Meson Vconf CE
COGEP
V CM
COGEP
VLS
COGEP
V TEN
COGEP
V LS
III
V TEN
III
V
2

(1670) 1675.26 -2.83 1.43 ... ... ... ...
K2 (1770) 1770.79 -3.13 1.22 ... ... ... ...
(1650)

1675.26 -2.83 -0.48 2.12 0.44 -41.83 -257.09
K*(1680) 1770.79 -3.13 -0.41 1.81 -0.38 -29.74 -273.60
K2(1820) 1770.79 -3.13 -0.41 0.60 0.38 -9.91 220.59
3

(1670) 1675.26 -2.83 -0.48 -1.41 -0.13 27.89 9.59
K*(1780) 1770.79 -3.13 -0.41 -1.21 -0.11 19.83 1.34
3

(1850) 1866.32 -3.42 -0.35 -1.05 -0.09 12.84 6.43
Table 6. Masses of triplet mesons ( in MeV ) without III.
Meson Experimental Mass
Calculated Mass
without III
(1650)

1649 ± 24 1676.3
K*(1680) 1717 ± 27 1770.9
K2(1820) 1816 ±13 1768.2
3

(1670) 1667 ± 4 1670.4
K*(1780) 1776 ± 7 1765.9
3

(1850) 1854 ± 7 1861.4
without the inclusion of III potential. The role of III is
crucial in explaining the mass differences of D wave K
mesons. The Table 5 gives the diagonal contributions to
the masses of D wave mesons by the confinement poten-
tial, colour-electric (CE), colour-magnetic (CM), spin-
orbit, tensor terms of COGEP and spin-orbit, tensor
terms of III. In case of singlet D wave mesons, tensor
and spin-orbit terms of COGEP and III do not contribute
to the masses. In case of these singlet mesons the CE
part of COGEP is attractive, whereas the CM part of
COGEP is repulsive. The dominant contribution to the
calculated masses comes from the confinement potential.
In case of triplet D wave mesons the contribution of III
potential is very significant. From Table 5, we note that
the significant contribution to the masses of 13D1 mesons
arises from the tensor term of III which is attractive. The
tensor term involves the parameters k7 and k8. It was
necessary to tune k7 and k8 parameters to get a reasona-
bly good agreement with the experimental masses.
Hence in our model, we have only two free parameters
k7 and k8. For 13D1 mesons along with the tensor contri-
bution of III, the spin orbit contribution of III is also
significant and is attractive. The contribution of tensor
term of III in case of 13D2 is repulsive that significantly
increases the value of calculated mass. It is to be noted
that the anti-symmetric spin orbit potential of III con-
tributes substantially to the mass difference between the
11D2 and 13D2 mesons in the K meson sector. The mass
difference between K*(1680) and K2(1820) mesons is
due to the large difference in tensor part of III potential.
The tensor III potential is attractive for K*(1680) but is
repulsive for K2(1820) [17]. In case of 13D3 mesons the
contribution due to spin-orbit part of III potential is
dominant compared to that of tensor part which is repul-
sive. From Tables 3 and 4, it is clear that the calculated
meson masses are in good agreement with the experi-
mental masses [18].
4. CONCLUSIONS
We have investigated the effect of the III on the
masses of the D wave mesons in the frame work of
RHM. We have shown that the computation of the
masses using only COGEP is inadequate. The contribu-
tion of the III is found to be significant. To obtain the
masses of D wave mesons, 5x5 Hamiltonian matrix was
diagonalised. The contribution from the tensor and
spin-orbit part of the III is found to be significant in case
of triplet D wave mesons. To obtain the physical masses
of the mesons in the K sector it is necessary to include
the anti-symmetric part of III. There is a good agreement
between the calculated and experimental masses of D
wave mesons. To enhance the performance of the model,
there is a need to make a global chi-square fit of the pa-
rameters and also to test different wave functions which
would ultimately give better result. Also, the model has
to be tested to calculate the leptonic and radiative decay
widths. Work in this direction is in progress.
5. ACKNOWLEDGEMENTS
One of the authors (APM) is grateful to the DST, India, for granting
the JRF. The other author (KBV) acknowledges the DST for funding
the project (Sanction No. SR/S2/HEP-14/2006) and the fruitful discus-
sion he had with Y.L. Ma of IHEP, Beijing.
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