ANALYTIC APPROACH TO q SYSTEMS IN POTENTIAL MODELS

Analytic solutions for qq systems obtained from a cut-off type approximation to the funnel potential are applied to bb and cc systems. Perturbative corrections to oscillator energy levels due to inclusion of short range a/r effect are also obtained.


Vl(r)
a/r + gr + C (1.1) which takes both confinement and asymptotic freedom into account by the linear and J f 2fm fpe2JeJy Is gnel co_ns h o$ aroriate lJ.It has been shown y inverse scattering methods [2], [3], using charmonium and upsilon parameters that quarkonium potentials are flavour-independent for the range 0.I fm r fm.
It has also been found that solutions obtained from Vl(r) are generally in qualitative, and often in quantitative agreement with those obtained from the oscillator potential V2(r) Kr2/2.However, V2(r) completely ignores the short-range Coulombic type potential.As radial wave functions evaluated at the origin are required in expressions for the decay width, it does not seem a good approximation to extrapolate V2(r) to r 0 when it is (I/r) part of Vl(r) that dominates close to r=0.In this communication we approximate Vl(r) by a cut-off type potential V(r) We investigate the solution obtained for Quarkonium systems in such a potential, and compare the results with those using Vl(r).
The use of V(r) permits us to obtain exact analytic wave functions which are not possible with Vl(r) and avoids the necessity of neglecting the Coulombic part of Vl(r) that occurs if V2(r) alone is used.We take a .27from funnel potential parameters [3].
We calculate also the perturbation to the oscillator energy due to the inclusion of a (-a/r) potential for 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.

TE WAVE FNTIONS FOR EA RFION.
The radial Schrodinger equation which is operative for a potential (-a/r) can be reduced to the confluent form This yields general solutions F(a, c, x) and ?(a,c,x) for integer c which coincides with our case.
As (a,c,x) is not regular at x 0 (or r 0) we reject this, and retain F(a,c,x) as the solution.So, -r/2 FI( + 1-X 2 + 2 or)

IEI
We assume m mass of the constituent quarks of the heavy bound quarkonium system.
The radial Schrodinger equation with non-zero for potential V2(r Kr2/2 can be reduced by suitable substitutions to the form 3 X +2+3 2 For values of E 3100 MeV [4], 300 MeV [3] we have evaluated m, using o the boundary conditions.We note, for m GeV and r and r 2 are of order of 2 inverse pion mass (M)'( r I) 23 and ( r I) 36.
So that asymptotic expansion of the confluent hypergeometric functions at the boundary may be considered valid.This yields the ratio of the wave function and their derivatives at the boundary, as the equation where  Neglecting the Ist term on the r.h.s., since it is of order .02m,compared to 1.5m 3 and 87m 2 /, (m~1 GeV), we estimate /-/E / rlm0 Thus m is sensitive to the value of r and m0' and is not dependent much on r 2 and on the magnitude of a, when a is small.
We find the funnel potential prediction of mass m 1650 MeV is obtained for -i r .0046MeV when 0 300 MeV. -1 We can then choose r_2 I/M .0073MeV Alternatively, if we wish to fix r at I/2 fermi .0025MeV which is assumed generally to be the region where inverse r behaviour falls off and the linear confining effect sets in, we get a much larger value for m 5.5116 GeV.
Further, from we can get straight away A /5 x I0 s 707.107 from Eq. (2.2).Similarly B and C may be found.
If we assume the Upsilon -r potential to be of the same form with the same value of 0 as obtained by level fitting from charmonium a value of 5 GeV for the mass of the beauty quark is obtained.
A value 372 MeV for mO obtained by level fitting of IS, 2S levels of yields a lower value of m b.
For the case of Upsilon, using E 9460 [4], we obtain by similar analysis m -i 4967.416 for the beauty quark with Coulombic radius 0.0046 MeV while a value 372 -1 MeV for m from level fitting yields m 3230.647MeV for r 0.0046 MeV 0 q The analytic solutions obtained can be used in problems involving the non- relativistic potential models for heavy quarkonium systems assuming the potential to be flavour independent.
We now estimate the correction to the first oscillator energy level due to the perturbation by the Coulombic potential for r between zero and r I.
The perturbation energy for the L O, J 0 level may be written AE (V-V0) ?0 dr (4.1) where 0 the unperturbed wave function and the perturbation potential (V-V 0) is (V-V O) a/r V 0 for 0 r r 0 for r r r 2 V 0 for r 2 r where V 0 unperturbed potential.
Due to confinement 0 0 for r r2, and we have eventually AE 4(a 12/ {-a I -K 12121 I and 12 are evaluated using the error function [6].We get, for r 0026 .5 fm, m0 300 MeV, mc I. 17 .0020,m 1.65 MeV, retaining m 300 MeV, we obtain q 0 AE 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 .27, the per- turbation energy would increase correspondingly through (al term. A(NOWLKDGKMENT: The first author wishes to thank U.G.C., India for financial support for the research.
reduced mass of the system.