Internat. J. Math. & Math. Scl.
(1987) 181-185
181
Vol. 10 No.
ANALYTIC APPROACH TO
q SYSTEMS IN POTENTIAL MODELS
LALI CHATTERJEE
Physics Department
Jadavpur University
700032, India
Calcutta
and
S. BHATTACHARYYA
Gokhale Memorial College
Calcutta
700020, India
(Received May 7, 1985 and in revised form January 21, 1986)
ABSTRACT.
solutions
Analytic
approximation
funnel
the
to
for
qq
from
obtained
systems
are
potential
bb
to
applied
a
and
cut-off
cc
type
systems.
Perturbative corrections to oscillator energy levels due to inclusion of short range
a/r
effect are also obtained.
KEY WDRDS AD PBRASES.
Potential Models, Schrodinger
1980 ANS SUBJECT CLASSIFICATION CODE.
I.
Equations.
81.
IlCrRODUCTION.
Bound state characteristics of heavy quark-antiquark pairs are generally well
The funnel potential
described in the framework of non-relativistic potential models.
(1.1)
a/r + gr + C
Vl(r)
which takes both confinement and asymptotic freedom into account by
J
2fm
f
fpe2JeJy Is gnel co_ns h
[2], [3], using
has been shown y inverse scattering methods
that
parameters
in
potentials
and
qualitative,
often in quantitative
V2(r)
oscillator potential
V2(r)
However,
radial wave
aroriate lJ.
Vl(r)
agreement with those
evaluated at
(I/r) part of
Vl(r)
V(r)
2
mm.
m
As
the origin are required in expressions for the
V2(r)
to
r
0
that dominates close to r=0.
Vl(r)
by a cut-off type potential
air
0
Kr2/2
r
r
r
r2
r
<
=0
K
from the
obtained
Kr2/2.
In this communication we approximate
with
are generally
completely ignores the short-range Coulombic type potential.
functions
It
charmonium and upsilon
decay width, it does not seem a good approximation to extrapolate
when it is
the linear and
are flavour-independent for the range 0.I fm
It has also been found that solutions obtained from
fm.
r
quarkonium
o$
reduced mass of the system.
r
r
2
(1.2)
L. CHATTERJEE and S. BHATTACHARYYA
182
We investigate the solution obtained for Quarkonium systems in such a potential,
Vl(r).
and compare the results with those using
not possible with
Vl(r)
of
permits us to obtain exact analytic wave functions which are
V(r)
The use of
and avoids the necessity of neglecting the Coulombic part
Vl(r)
V2(r)
that occurs if
We take a
alone is used.
from funnel
.27
potential parameters [3].
We calculate also the perturbation to the oscillator energy due to the inclusion
of
(-a/r) potential for
a
0
r
rl,
relying heavily on the assumption that the
colour interaction has a small coupling constant for small
r, thereby justifying use
of perturbation techniques.
2.
TE WAVE FNTIONS FOR EA RFION.
equation which is operative for a potential (-a/r) can be
The radial Schrodinger
reduced to the confluent form
x
--2x
) Y
8x
+ (C
ay
(2.1)
0
with
This yields general solutions
?(a,c,x)
F(a, c, x) and
As (a,c,x) is not regular
F(a,c,x) as the solution. So,
Ae
R
-r/2
FI(
+ 1- X
for integer c
2 + 2
which coincides
0) we reject this, and
0 (or r
at x
with our case.
retain
X
+
2 + 2, a
r, c
x
(2.2)
or)
where
2/2mE
We assume m
The
X
am/(2#E ),
IEI
E
mass of the constituent quarks of the heavy bound quarkonium system.
radial Schrodinger equation
for potential
with non-zero
V2(r
Kr2/2
can be reduced by suitable substitutions to the form
where
z
(r)
2
)
X
z}
+2+3
4
y
(2.3)
0
for the range
rl
r r 2.
R
This yields solutions for
2
B e
3
( +
+
8z 2
-(= r) /2
as
+ C e -(s
+ 2 + 3
IF1 (.X 4
r) 2/2
-2-2
(a r)
(’r)
IFI (X
2
3/2, (e r)
-2+I
4
+/-2
,
(= r)
2
(2.4)
q
ANALYTIC APPROACH TO
SYSTEMS IN POTENTIAL MODELS
183
where
4/--,
a
3.
%
--,
E
c
i.
ESTIMATE OF HEAVY QUAI MASSES.
At the outer boundary, which we expect to be the hadronic radius or confinement
radius
r
2
R
(3.1)
0
2
and at the inner boundary
R
For values of
R
3100 MeV [4],
E
the boundary conditions.
inverse pion mass
So
that
(3.2)
2
(M)’(
300 MeV [3]
o
We note, for m
2
r
23 and
GeV
I)
we have evaluated
and
( r I)
and
r
r
2
m, using
are of order of
36.
asymptotic expansion of the confluent hypergeometric functions at the
boundary may be considered valid.
the ratio of the wave function and
This yields
their derivatives at the boundary, as the equation
H
B
(I + k/= r)]
[-
r(x
+
3)
4
(3.3)
+ H + H3] + H4
2
[H
where
2
2
[1/2 (= r I) ],
[-= /2(= r I) ], H 2
H
,2
H
3
[=
rl)]/[2( rl)]
+ (X /2- 1/2)/(=
r
,2
H
4
C [(=
r
(( + l)/=r)
H
From (3.2) we have
B
P
p
[B/{2a
(X
r
r
I/2)/(e
+ ( / 2
I)
<<
rl)]/F(% 4+ I)
=/2 and [( /2
+
3)}]
4
+ [C/F
., +
I/2)/(= r I)]
<< =
r
1.
[(----)1
pc, where
’-F2"-2"
=
r
.06
2
(%
+4 3.)
(i/2)
F
for
m
(l’
(I/#(= r2 I/2
i"
+
(3.4)
F (1/2)
4
1GeV.
Equation (3.3) can be reduced to
p
/ 2mE
4#2
F(
+
4
F(X’4
+
I)
3)
+
rl ,r mm 0
(3.5)
184
L. CHATTERJEE and S. BHATTACHARYYA
Neglecting the Ist term on the
1.5m
3
and 87m
2
/,
(m~1 GeV),
/- /E
/
since it is of order
r.h.s.,
We
find
.0046 MeV
r
the
-i
rlm 0
a, when
and
and is not dependent much on r
m0’
when
1650 MeV
m
I/M
.0073 MeV
-1
Alternatively, if we wish to fix r
I/2 fermi
r
behaviour falls off and the linear confining effect sets in,
.0025 MeV
which is assumed generally to be the region where inverse
we get a much larger
5.5116 GeV.
m
Further, from
we can get straight away A
s
707.107 from Eq. (2.2).
B
Similarly
-r
If we assume the Upsilon
of
C
and
/5
x I0
may be found.
potential to be of the same form with the same value
5 GeV for the mass of
as obtained by level fitting from charmonium a value of
0
is obtained for
300 MeV.
0
r_2
at
value for
2
is small.
a
funnel potential prediction of mass
We can then choose
compared to
we estimate
Thus m is sensitive to the value of r
and on the magnitude of
.02m,
the beauty quark is obtained.
A value 372 MeV for
lower value of
obtained by level fitting of IS, 2S levels of
mO
b.
For the case of Upsilon,
9460 [4], we obtain by similar analysis m
using E
4967.416 for the beauty quark with Coulombic radius
MeV for m
The
0
yields a
m
level fitting yields
from
analytic
solutions
obtained
m
3230.647 MeV for
q
be
can
-i
0.0046 MeV
used
in
while a value 372
0.0046 MeV
r
problems
involving
-1
the
non-
relativistic potential models for heavy quarkonium systems assuming the potential to be
.
flavour independent.
CORRECTIONS TO ENEI LEVELS.
We now estimate the correction to the first oscillator energy level due to the
perturbation by the Coulombic potential for
AE
where
0
(V-V0) ?0
r
O, J
The perturbation energy for the L
between zero and
r
I.
0 level may be written
(4.1)
dr
the unperturbed wave function and the perturbation potential
(V-V
a/r
O)
0
V
0
V
0
0
for
for
r
for
r
2
r
r
r
r
r
2
(V-V
0)
is
ANALYTIC APPROACH TO
where
V
unperturbed potential.
0
Due to confinement
0 for
0
AE
f
I
where
rl
r e
2
dr and
n
I
.5 fm,
and
m0
f
12
r
4
12121
-K
e
-br
185
and we have eventually
r2,
(4.2)
2
We
dr
have
used
for
the
unperturbed
0
I/2 e-(
a
as given by
r
r
0
wavefunction
SYSTEMS IN POTENTIAL MODELS
12/ {-a I
4(a
_br
q
r12)
i 2nn!
2
H
n
(r)
(4.3)
[6].
12
are evaluated using the error function [6]. We get, for r
300 MeV,
mc
AE
Alternatively if we take
AE
0026
I. 17 GeV.
0.002 MeV
r
.0020, m
q
1.65 MeV,
retaining
m
0
300 MeV, we obtain
0.0035 MeV
The small magnitude of the perturbation energy shows that perturbation corrections
are justified.
However, if the short range coupling was greater than a
term.
(al
.27, the per-
turbation energy would increase correspondingly through
The first author wishes to thank U.G.C., India for financial support
A(NOWLKDGKMENT:
for the research.
REFERENCES
I.
NOVIKOV, V.A., et. al.
2.
QUIGG, C.
3.
4.
MOXHA, P., ROSNER, J. L. and QUIGG, C. Beyond Y: Heavier Quarkonia and Interquark
Force, Ph. ys. Rev. D23 1981, 2638-2646.
ERDKLEI, A. (Editor) Hisher Transcendental Functions I___., MaGraw Hill Book Co.
5.
FLUGGE, S.
Charmonium and Gluons,
Physics Repo. rts. 41C 1978, 1-135.
and ROSNKR, J. L. Further Evidence for Flavour Independence of the
Quark-antiquark Potential, Phys. Rev. D2__3 1981, 2625-2637.
1953.
Practical
0uantum Mechanics,
Springer Verlag i, 1974.
ABRADVITCH, I. and STEGON, I. A. Hand Book of Mathematical Functions, Dover PubPublications Inc., NY, 1965.