Volume 97B, number 2 PHYSICS LETTERS 1 December 1980

A NON-COULOMBIC EFFECTIVE POWER-LAW POTENTIAL FOR THE HEAVY QUARKONIUMS

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

Received 22 July 1980 Revised manuscript received 10 October 1980

An effective power-law potential of. the form V(r) = 6.08 r 0 ' 1 0 6 - - 6.41 is found to describe satisfactorily the gross fea- tures of the mass spectra and the leptonic width ratios of the cg and bl~ systems in a flavour-independent manner.

Non-relativistic potential models for bound heavy quark-an t iquark systems like c~ and bl~ representing respectively the heavy meson families of ~ and T have proved to be quite successful. The interquark force believed to be flavour independent is generally described by the tradit ional Coulomb plus linear po- tential [1 ]. Although one expects quantum chromody- namics (QCD) to be the theory underlying this descrip- t ion, one cannot as yet derive such a potential com- pletely from first principles. In such circumstances one often turns towards phenomenology for further understanding of the dynamics of bound qua rk -an t i - quark systems. Then with no theoretical prejudice one can take a simple power law potential like V(r) = ar v, to describe the quarkonium spectroscopy. In fact im- plications of such a potential have been investigated in the past by Quigg, Rosner and many other authors [2]. But the available data being consistent with a small positive v very close to zero, at tent ion was shift- ed in a limited sense towards a better looking loga- rithmic potential [3]. However the logarithmic poten- tial leads to exact equality of the mass differences

(AMs)qff = [M2s(qgt) -Mls(qgt ) ] for all qC t systems independent of the constituent quark mass mq, which is strictly not true according to more accurate experi- ments. Apart from this, some recent experiments have come up with very accurate data on the T-family [4] and also some other ones with new information re- garding the 1S 0 partners of ff and 4 ' [5]. In view of

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

this, it is worthwhile to have a fresh look at an effec- tive power-law potential model. Martin [6] recently revived interest in such a potential model by pointing out on the basis of semiclassical solutions that with v close to 0.1, it is possible to describe very well the main features of the charmonium and upsilon spectra. Our purpose here is to corroborate the same view by deriving the potential parameters and the quark masses in a different manner from WKB solutions before ob- taining detailed numerical fits for the charmonium and upsilon systems.

We take the power-law potential in the form,

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

where a and v are assumed to be positive. Then semi- classical solutions to the SchrSdinger equation for the qgt-bound states with this potential are possible which would give the binding energy and the absolute square of the S-state wave functions at the origin in the fol- lowing form [7]:

EnL = b + (a/m~/2) 2/(v+ 2)

1 1 2v/(v+2) × [ A ( v ) ( n + - ~ L - z ) ] , (2)

i ~ns(0)l 2 = (1/27r 2) [v/(v+2)] {arnq [A (v)] v} 3/(0+ 2)

X (n - -~1 )2(v-1)/(v+2), (3)

where

A ( v ) = 2x/~I ' (3 /2 + 1/v) /F(1 + 1/v). (4)

Now taking some experimental inputs from the c~ and

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bt3 systems such as (AMs)bl S and (AMs)c6 , the spin- averaged mass M ls(c6) and finally [ ff 1 s(0)12e and Iffls(0)12t~, we can obtain from eqs. (2) and (3) an esti- mate of the potential parameters a, b, v and the con- stituent quark masses m c and rn b in the following manner. If we denote the ratio (Ams)cU(AMs)b5 as D O and I ffls(0)121JI ~ls(0)12~ as W 0 then we get

v = 3 ln(Do)lln(Wo), (5)

a = (AMs)q{ [3~ Fqq(V)]vl3/Q(v ) [A(v)lv, (6)

mq = Q(v) [3 Fqq(O)]2/3 /(AMs)qq, (7)

b = M 1 s(C~) - 2m c - (a/m ~t/2)2/(v + 2) [3 A (o)] 2o/(o+2),(8)

where

Q(o) = [(7[3) 2v/(v+ 2) - 1], (9)

P ( V l s - e+e - ) -

Fqq-(V) = [Tr2(v+2)l 4qs(0)lZq/v]. (10)

Since with the static potential we can only get the spin- averaged mass spectrum of the q~t-bound states, the in- puts should be taken in an average sense only. There- fore (AMs)bf ~ and (AMs)c~ are chosen around the cor- responding experimental values referring to the vector meson masses. We take (AMs)c~ = 0.605 GeV and (AMs)b6 = 0.568 GeV. According to the recent SLAC experiment [5 ], if we consider M% = 2.983 GeV, we get the spin-averaged mass Mls(C¢ ) = 3.069 GeV. Final-

obtain the IV ls(0)12q values from the ly we experimen- tal leptonic widths expressed in terms of the some- what corrected Van Royen-Weisskopf formula [8] in the form

Table 1

1 December 1980

16rro~2e2q [1 8 %(qgl!] 2

M2s(qq ) I~ls(0)12 ~ J •

(11)

If we take the quark-gluon coupling constants as(C~ ) = 0.41 and as(bl~ ) = 0.24 in conformity with the deep- inelastic scattering data and the requirements of asymptotic freedom, we obtain IV ls(0)l 2 = 0.094 GeV 3 and I~b ls(0)121~ = 0.5678 GeV3. With these val- ues, we obtain the potential parameters and the quark masses from eqs. (5) to (10) in the following manner:

(a, b, v) = (6.08 GeV, -6 .41 GeV, 0.106), (12)

(mc,mb) = (1.346 GeV, 4.759 GeV). (13)

Now with these parameters we calculate the mean mass spectra for the c~ and bb systems using the semi- classical formula for the binding energy given in eq. (2). However this formula is an approximate one which be- comes worse for the excited states particularly with L 4= 0. For example, eq. (2) would predict a degenerate 2S and 1D level of the @pbound systems which is not true at least in the case of the charmonium spectrum. Therefore, for exactness we go for the numerical solu- tions to the Schr6dinger equation with the potential parameters fixed as given in eq. (12). We vary the quark masses m e and m b around the corresponding es- timated values given in eq. (13) so as to obtain fits to the spin-averaged ground state masses Mls(C~ ) = 3.067 GeV and Mls(bl3 ) = 9.428 GeV, respectively. We must point out here that we have decided to take the Cornell scale as the absolute energy scale in order to be consistent with the recent experimental data on the

The spin averaged mass spectrum MnL(q~l) of the cE, bl~ and tt families obtained from (i) numerical and (ii) WKB calculations

Bound state MnL (cc--) MnL (bl~) MnL (tt) (GeV) (GeV) (GeV)

( i ) ( i i ) ( i ) ( i i ) ( i ) ( i i )

mq (GeV) 1.334 1.346 4.721 4.759 20.0 20.0

1S 3.067 3.069 9.428 9.475 39.567 39.512 2S 3.672 3.674 9.986 10.043 40.067 40.04 3S 4.025 4.019 10.313 10.366 40.359 40.346 4S 4.267 4.264 10.529 10.597 40.571 40.562 1P 3.513 3.428 9.831 9.812 39.841 39.83 2P 3.897 3.864 10.195 10.221 40.234 40.21 1D 3.799 3.674 10.088 10.043 - -

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T system [4] although at present most experimentalists seem to prefer the DESY energy scale. We obtain m c = 1.334GeV and m b = 4.721 GeV which are very close to our earlier estimates. Then with these quark masses we obtain the spin-averaged mass values for a few other excited states of the c~ and b13 systems. The re- sults are given in table 1 along with the values obtained from the WKB calculations for comparison. We find the agreement bet ter particularly in S-states but worse in P- and D-states which justifies our apprehension. We cannot make at this stage a definite quantitative comparison of our results with the experimental val- ues unless we discuss the f ine-hyper f ine splittings which will be reported in a subsequent work [9]. How- ever, looking at the mean mass values obtained here we can notice a very good qualitative agreement with the experimental mass spectra.

For completeness we also obtain the average sizes (r), the kinetic energy (T) = (½ r d V/dr), the velocity parameter/32 = (T)/mq and finally the I~ns(0)l 2 = rnq(dV/dr)/47r for the S-states of the charmonium and upsilon systems which are presented in table 2. We observe that the values of the average kinetic ener- gies and the corresponding velocity parameters so ob- tained for all these states definitely just ify the non- relativistic approach. Finally using the computed val-

ues of I ffns(0)l 2, we obtain the leptonic width ratios for the c~ and bb systems from the following relation:

F ( n S - , e+e - ) ( M l s ] 2 [~ns(0) 12 (14)

F (1S-+ e+e - ) =\Mns.ns ! Iffls(0)l 2 '

Table 2 also fists these values which are found to be in excellent agreement with the corresponding experi- mental quantities.

Thus we find that this non-coulombic power-law potential describes quite well the gross features of the mass spectra and the leptonic width ratios for the ce and bb systems in a flavour-independent manner. This potential , unlike the Coulomb plus linear or the loga- rithmic one is non-singular at the origin having a finite bo t tom of the order of ~6 .4 GeV. Therefore its short range behaviour is in contradiction with what one ex- pects from QCD. However it is probably not possible to probe the short range behaviour of the qCq potential with the systems under consideration, since the aver- age sizes of all the mesons in the ff and T family as given in table 2 are found to be greater than or of the order of 0.2 fro. The effect of the short range part of the potential may be felt in the hyperfine splittings where it is supposed to play an important role. We will discuss this aspect in a separate communication [9] to show that a suitable Lorentz structure of this non- coulombic power-law potential can adequately describe the f ine-hyper f ine splittings of the ce and bl~ systems. In that case one would have serious doubts in saying that the charmonium and the upsilon spectra suggest a short range Coulomb-like behaviour of the q u a r k - antiquark potential well in accordance with QCD pre- dictions. Another way to probe the short distance be- haviour of the potential is perhaps through the stxll heavier quarkonium ft. With the consti tuent quark mass m t > 18 GeV, the average size of such mesons

Table 2 Results for <r), (T),/32, IqJ s(0)l 2 and P (nS ~ e+e-)/l'(1S ~ e+e -) of some S-wave levels of the c~, bfo and t{- families.

(T) /32 [~ s(0)12 r (nS ~ e+e -) (GeV) (GeV 3) I" (1S ~ e+e -)

calc. exp.

2.033 0.344 0.258 0.042 1 1 ¢' 4.146 0.37 0.277 0.027 0.448 0.45 -+ 0.09

" 6.649 0.389 0.292 0.019 0.263 0.156 q/" 9.063 0.402 0.301 0.015 0.185 0.102

T 1.125 0.323 0.068 0.32 1 1 T' 2.409 0.35 0.074 0.159 0.443 0.44 -+ 0.06 T" 3.685 0.365 0.077 0.115 0.3 0.32 +- 0.04 T"' 4.945 0.377 0.08 0.091 0.228 0.20 -+ 0.06

0.502 0.296 0.015 2.41 1 1 ~' 1.176 0.324 0.016 1.29 0.52 -

qfi. {r) (GeV -1 )

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would be less than or equal to 0.1 fm. Therefore it may be possible to detect any inadequate behaviour of this non-coulombic power-law potential when its pre- dictions are compared with the experimental mass spectrum of the ff system. With this in view we have obtained the mean mass spectrum of this system with a quark mass m t = 20 GeV [10]. The predictions are listed in tables 1 and 2. Although recent experiments at PETRA [11 ] have found no evidence for these heavy mesons up to an energy scale of 35.8 GeV, future experiments may be able to settle this issue.

We are thankful to Professor B.B. Deo for his con- stant inspiration and valuable suggestions. We also thank the Computer Centre, Utkal University Ibr its timely cooperation in the computational work.

R eferen ces

[1] E. Eitchen et al., Phys. Rev. Lett. 34 (1975) 369.

[2] C. Quigg and J.L. Rosner, Phys. Lett. 71B (1977) 153; C. Quigg, FERMILAB-Conf-79/74-THY (Sept. 1979); C. Qu~gg and J.L. Rosner, Phys. Rep. 56C (1979) 167; A. Khare, Phys. Lett. 73B (1977) 296.

[3] C. Quigg and J.L. Rosner, Phys. Lett. 71B (1977) 153. [4] T. Bohringer et aL, Phys. Rev. Lett. 44 (1980) 1111;

D. Andrews et al., Phys. Rev. Lett. 44 (1980) 1108; Cornell preprint CLNS-80/452 (1980).

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

[6] A. Martin, CERN preprints TH-2876 (1980), TH-2843 (1980).

[7] C. Quigg and J.L. Rosner, Phys. Rep. 56C (1979) 167. [8] R. Van Royen and V.F. Weisskopf, Nuovo Cimento 50A

(1967) 617; R. Barbieri et al., Nucl. Phys. B105 (1976) 125; W. Celmaster, SLAC-PUB-2151 (1978); E.C. Poggio and H.J. Schnitzer, Phys. Rev. D20 (1979) 1179.

[9] N. Batik and S.N. Jena, Phys. Lett. 97B (1980) 265. [10] D.V. Nanopoulous, CERN-PUB-TH-2866 (1980). [11] D.P. Barber et al., Phys. Rev. Lett. 44 (1980) 1722.

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