Volume 97B, number 2 PHYSICS LETTERS 1 December 1980

FINE-HYPERFINE SPLITTINGS OF QUARKONIUM LEVELS

IN AN EFFECTIVE POWER-LAW POTENTIAL

N. BARIK and S.N. JENA 1 Department o f Physics, Utkal University, Vani Vihar, Bhubaneswar 751004, Orissa, India

Received 22 July 1980 Revised manuscript received 10 October 1980

We have shown that an effective non-coulombic power-law potential generating spin dependence through scalar and vec- tor exchanges in almost equal proportions along with a very small or zero quark anomalous moment can describe very satis- factorily the up-to-date data on the fine-hyperfine levels and the leptonic width ratios of the vector mesons in the cg and bb families in a flavour independent manner.

Implications of the power-law potential model have been investigated by many authors [1 ] particularly in the context of the phenomenological logarithmic po- tential model for the heaw quark-ant iquark bound systems. But in view of some recent experiments giv- ing very accurate data on the T system [2] and also some fresh information about the pseudoscalar part- ners of 6 and ~' [3], the power-law potential model has again come back into the game. Martin [4] has shown recently on the basis of semi-classical solutions that the most up-to-date data on the charmonium and upsilon spectra can be fitted by a power-law potential of the form V ( r ) = ar v, with v close to 0.1. We have corroborated Martin's idea in obtaining exact numeri- cal solutions to the Schr6dinger equation for obtain- ing the ce and bl~ bound states. We have found that a power-law potential in the form

V ( r ) = ar u + b , (1)

with the potential parameters and the quark-mass pa- rameters taking values as

(a, b, v) = (6.08 GeV, -6 .41 GeV, 0.106), (2)

(mc, rob) = (1.334 GeV, 4.72i GeV),

describes the gross features of the ce and bl~ spectra

1 Work supported in part by the University Grants Commis- sion, India under the Faculty Improvement Programme.

quite well [4]. Although this phenomenological po- tential gives quark confinement (a, v > 0) at long dis- tances, its short distance non-singular behaviour is in apparent contradiction with what one expects from quantum chromodynamics (QCD). One may argue that in the phenomenological fit, the short distance behaviour of the potential may not be reflected in the gross features of the spectrum and further this poten- tial may indeed simulate the actual one in a wide range of quark-antiquark separation distances ~> 0.2 fm appropriate to the average sizes of the hea W me- sons under consideration. However, the short distance part of the potential is believed to play an important role in the hyperfine splittings and therefore the f ine- hyperfine structures of the quarkonium levels may re- veal the inadequacy if any in this non-coulombic po- tential. With this motivation we discuss in this note the spin structure of this power-law potential and the fine-hyperfine splittings of the ce and bl~ systems.

The quantitative explanation of the fine-hyperfine levels depends on the spin structure of the quark antiquark potential. If we regard this power-law poten- tial V ( r ) = ar v + b to be an admixture of vector and scalar components with vector fraction r / then with a non-zero quark anomalous moment ~t, the spin de- pendent correction terms generated by this potential can be obtained in the usual manner as

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Volume 97B, number 2 PHYSICS LETTERS 1 December 1980

Vs pin (r) = A 1 (r) L" S + A 2 (r) S 1 • S 2

+ A3(r)S12 , (3)

Table 1 Spin-averaged mass MnL and <r v 2) L for the c~ and bro sys- tems.

where A 1 (r), A 2(r) and A 3 (r) are the radially depen- dent potential functions for spin orbit, spin-spin and tensor interactions, respectively. These functions are obtained through the standard reduction formulas. However, a word of caution is necessary. Although the effective potential [eq. (I)] may describe accu- rately the gross features of the data, its derivatives may be far remote from reality. In any case it will be found to be relatively reassuring to obtain in this meth- od the hyperfine splittings not so drastically different from those of Martin [4]. Then we write in explicit form the potential functions as

Al(r)=(av/2m~)(rV-2)[4r~(l +p)- 1], (4)

A2(r) = [2av(v + 1)/3m2](rV-2)[(1 +p)2~] , (5)

A3(r ) = [av(2 - v)/3m2q] (rV-2)[(1 + p)2r~]. (6)

Now with a perturbation approach to the spin-depen- dent correction term 6Vspin(r) one can obtain the fine hyperfine levels in the following manner with the mass formulas written conveniently in matrix form as

M(I S0) 1 -3/4I \(A2(r)) s '

IM(M(3P2)\ Qi 1 1/4 - l / l~ /Mnp \ M(3P1)~ = - I 1/4 1/2 ~[(Al(r))pl" M(3P0) j - 2 1 / 4 - 1 0 /l(A2(r))p] (8)

1P1) / 0 - 3 / 4 \(A3(r))p/ Here MnL are the spin-averaged masses for the L-orbi- tal state of the quarkonium obtained from the exact numerical solution to the Schr6dinger equation with the static potential V(r) in eq. (1) and (A 1 (r)}L, (A2(r)) L and (A 3(r))L are the corresponding expecta- tion values of the potential functions. In fact from eqs. (4) to (6) it is clear that these quantities depend on the expectation value (rV-2) L and also on the pa- rameters r/and p.

We have presented in table 1 the required expecta- tion values (rV-2} L for various bound states of the cg and bl~ systems along with the corresponding

nL c~ system bb system

MnL <rV-2>L MnL <rV 2~ L (GeV) (GeV)

1S 3.067 0.7026 9.428 2.1503 2S 3.672 0.3661 9.986 1.0284 3S 4.0251 0.2174 10.313 0.6697 4S 4.267 0 . 1 5 7 1 10.529 0.4952 IP 3.513 0.17115 9.831 0.48919

spin-averaged masses mnL. Now it is a question of choosing and fitting the parameters r~ and p. As a first choice if we assume p = 0, then a vector fraction = 0.57 describes the ground state hyperfine splitting and the 1P fine structures of the charmonium spec- trum in quite good agreement with experiment. Partic- ularly the pseudoscalar partner r/c of ~ comes out with a mass M~c = 2.9867 GeV which is almost the value obtained in a recent experiment at SLAC [3]. We also find the 1S 0 partner of ~' to be at a mass value 3.63 GeV. This result is quite good in the sense that the proximity of the predicted 1S 0 partners of ~ and ~ ' would make the Ml-transition rate quite small. The fine-hyperfine levels of the ce and bb systems calculated in this manner are presented in table 2. We find that not only we obtain a very good fit to the charmoni- um spectrum with correct 1P-level splittings, but also the vector meson masses of T, T' , T" and T " of the bl~ system come out in close agreement with recent experiments [2]. Now coming to the alternative choice of p ¢ 0, we find that a vector fraction ~/= 0.42 with p = 0.193 gives almost identical results as obtained with the previous choice. Except for the fact that in the latter case the 3P 0 level of charmonium comes out some 20 MeV higher, there is absolutely nothing to choose between the two alternatives. Therefore we may conclude that the non-coulombic power-law po- tential generating spin dependence through scalar and vector exchanges in almost equal parts along with a very small or zero quark anomalous moment can ex- plain very satisfactorily the f ine-hyperfine levels of the charmonium and upsilon spectra. Thus we find that there is absolutely no need to make the ad hoc assumption as done by Martin [4] that the hyperfine splittings are proportional to the I~ns(0)l 2 which is

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Volume 97B, number 2 PHYSICS LETTERS 1 December 1980

Table 2 Fine hyperfine structure of the e~ and bb systems.

q~ cE system state . . . . . . . . . . . .

predicted experimental mass (GeV) mass (GeV)

bb system

predicted mass (GeV)

experimental mass (GeV)

1 1S 0 2.9867 2.983 (?)

1 381 3.094 3.095 +- 0.003

2 1S o 3.63

2 3S1 3.686 3.684 +- 0.009

3 1S o 4.00 -

3 3S 1 4.034 4.040 + 0.010

4 1S o 4.249 - 4 3S 1 4.273 4.417 -+ 0.010

1 3P 0 3.418 3.413 -+ 0.005 a) 1 3P 1 3.491 3.508 +- 0.004 1 3P 2 3.557 3.554 -+ 0.005 1 1P 1 3.493 -

9.4084 9.4345

9.9766 9.9893

10.3068 10.315

10.5245 10.531

9.8092 9.8259 9.8410 9.8265

9.4345 +_ 0.0004

9.993 +- 0.001

10.3232 +_ 0.0007

10.546 +- 0.002

a) Ref. [71.

reminiscent of a short distance Coulomb-like potential due to one-gluon exchange. In that case it cannot be taken for granted that the charmonium and the upsi- lon spectra point absolutely towards a short distance Coulomb-like behaviour of the quark-ant iquark poten- tial predicted by QCD. A further test of the short dis- tance behaviour of the potential can be provided by the still heavier meson family of the tf system. Al- though recent experiments at PETRA [5] have found no evidence of such mesons within the energy range up to 35.8 GeV, future experiments may be able to detect them which may throw some light on the point under discussion.

Finally for completeness we calculate the leptonic decay widths of the vector mesons in the ~ and T families using the Van Royen-Weisskopf formula [6]

P(V -+ e+e - ) = 167r c~2e~Mv 2 ]~ns(0)[ 2 . (9)

The results are in reasonably good agreement with the experimental values as noted in table 3. However, if

we doubt the correctness of this formula, then the leptonic width ratios calculated from (9) as

P(Vns ~ e+e - ) _ (My(1S)'~ 2 I~ns(0)l 2 (10)

P(Vls -+e+e - ) \ ~ ] [~ls(0)l 2 '

can be a meaningful and reliable quantity to be com- pared with the corresponding experimental values. Table 3 lists these results to show in particular the re- markable agreement with recent experiments on the T family [2].

Thus we conclude that an effective non-coulombic power-law potential, generating spin dependence through scalar and vector exchanges in almost equal proportions along with a very small or zero quark anomalous magnetic moment can describe very satis- factorily the f ine-hyperf ine levels and the leptonic width ratios for the vector mesons in the c~ and bt~ families in a flavour independent manner. The short distance non-coulombic behaviour of this potential in apparent contradiction with the predictions of QCD does not pose any problem in the phenomenological description of the charmonium and upsilon systems.

We are thankful to Professor B.B. Deo for his con- stant inspirations and valuable suggestions. We also

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Volume 97B, number 2 PHYSICS LETTERS

Table 3 Leptonic decay widths and their ratios for the vector mesons in the c~ and bb families.

1 December 1980

Vector Mass hqJnL(0)l 2 re+ e - (keV) meson (GeV)

F(nS ~ e+e-)/r(1 S --* e+e -)

predicted experimental predicted experimental

3.094 0.042 5.21 4.8 -+ 0.6 q~' 3.686 0.027 2.43 2.1 +- 0.3 t~" 4.034 0.019 1.38 0.75 -+ 0.15 ¢"' 4.273 0.015 0.98 0.49 -+ 0.13

T 9.4345 0.32 1.07 1.2 -+0.2 q" 9.9893 0.159 0.47 0.51

T" 10.315 0.115 0.32 0.41

"r'" 10.531 0.091 0.24 0.26

I 1 0.466 0.45 -+ 0.09 0.265 0.1 88

1 1 0.439 0.44 +- 0.06

0.39 +- 0.06 0.299 0.35 -+ 0.04

0.32 -+ 0.04 0.234 0.20 +- 0.06

thank the Com pu te r Centre , Utkal Universi ty for its

t imely coopera t ion in the compu ta t iona l work.

References

[1] C. Quigg, Fermilab-Conf-79/74-THY (Sept. 1979); C. Quigg and J.L. Rosner, Phys. Lett. 71B (1977) 153; A. Khare, Phys. Lett. 73B (1977) 296.

[2] T. Bohringer et al., Phys. Rev. Lett. 44 (1980) 1111 ; D. Andrews et al., Phys. Rev. Lett. 44 (1980) 1108; Cor- nell preprint CLNS 80/452 (1980).

[3] E.D. Bloom, Invited talk 1979 Intern. Symp. on Lepton and photon interaction at high energy, SLAC-PUB 2425 (Nov. 1979); T.M. Himel et al., Phys. Rev. Lett. 44 (1980) 920.

[4] A. Martin, CERN preprint TH-2843 (1980), TH-2876 (198O); N. Barik and S.N. Jena, Phys. Lett. 97B (1980) 261.

[5 ] D.P. Barber et al., Phys. Rev. Lett. 44 (1980) 1722. [6] R. Van Royen and V.F. Weisskopf, Nuovo Cimento 50A

(1967) 617; R. Barbieri et al., Nucl. Phys. B105 (1976) 125.

[7] C. Bricman et al., Phys. Lett. 75B (1978) 1.

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