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Dirac mass spectra of Qq -like mesons in a power-law potential
Volume
122B. number
DIRAC
MASS
PHYSICS
2
SPECTRA
OF Q+LIKE
MESONS
LETTERS
3 March
1983
IN A
JENA and T. TRIPATI Department of Physics, Aska Science College, Aska, Ganjam, Orissa,India Received 31 August 1982 Revised manuscript received
4 November
1982
The mass spectra of Qq-like mesons are studied in the Dirac equation with an equally mixed 4-vector and scalar powerpotential can satisfactorily describe law potential of the form V(r) = Are.’ + V,J. It is found that this flavor-independent the mass levels of D, F and B mesons along with those of e and T families in a unified manner and that the quark masses in quarkonia and Qq-like mesons are very close to the current quark masses.
1. Introduction.
potential
Recently the empirical power-law of the form
V(r) = Ar” + V0 ,
(1)
with v = 0.1 and A > 0 was extensively discussed [l] in accordance with the very accurate data of the Tfamily and seems to be preferable to the logarithmic potential. These discussions are mostly based on the non-relativistic Schrbdinger type approach. It would be interesting to see how this potential works in the relativistic system inside a hadron. This work aims to study atomlike mesons Qq composed of a heavy quark Q and a light quark q in the Dirac equation with an equally mixed 4-vector and scalar power-law potential. In case of atomlike mesons where a light quark q is moving around an almost fured heavy quark Q with a high speed, the use of non-relativistic SchrBdinger treatment to study their bound states may not be quite appropriate. Therefore these mesons must be discussed in the framework of the relativistic theory and have not been studied systematically. In the present paper our chief interest is to describe the QQ and Qq systems in a unified way by taking same potential parameters for both systems. Here we discuss only the gross structure of the energy levels but not the problem including fine-hyperfme splittings. We assume the Lorentz structure of the confining potential to be an equal admixture of scalar and vector 0 031-9163/83/0000-OOOO/$
03.00 0 1983 North-Holland
parts in accordance with the phenomenology [2] of fine-hyperfme splittings of heavy mesons. The reason for this assumption is that an equally mixed scalar -vector potential in the Dirac equation can realize quark confinement to generate relativistic quarkantiquark bound states [3]. We solve the QQ systems with Q = c, b, t by the Schrbdinger equation and the Qq systems with Q = c, b, and q = u or d, s, or c by the Dirac equation. The availability of numerous and convincing experimental data in case of CCand bb systems makes it plausible to extract the static potential parameters of the model from the Schrbdinger formalism. Then assuming flavour-independence of the confining potential the same potential parameters are used in the Dirac equation to find out the bound state masses of Qq systems. Atomlike (Qq) mesons such as Do(&), D+(cd) and F+(cS) have already been observed experimentally in ground states only with their hyperfme mass splittings. Other such mesons in this category like B-(bti), Bo(bd), G”(bS) and H-(b?) are expected to be found in near future. 2. Bound state mass of 20 system. For the nonrelativistic QQ system, the reduced radial Schriidinger equation is
‘3 +(mQ(E - v(r)) -y)
U(r) = 0 .
(2) 181
Volume
122B, number
2
PHYSICS
Taking V(r) as given in eq. (1) and substituting p = (r/ru) with the scale factor chosen asr,, = (rr~~A)-~l(*~) eq. (1) reduces to the form (3) where .
frill = m&!? - Vo) (,($-2/(~+2)
= 2mQ + V,, t a(a/mQ)vl(v+2)en1
.
(5)
3. Bound state mass of Qe system. Without considering the recoil effects of the heavy quark Q, the Dirac equation for the Qq bound state massM(Qq) can be written as
= W(Q$ ~ mQ1
q(r)
>
(6)
where mQ and mq are masses of heavy and light quarks respectively. With the assumed Lorentz structure of I’(r) in the form of an equal admixture of scalar and vector parts [V,(r) = V,(r) = i V(r)] eq. (6) leads to an equation satisfied by the reduced radial part of the “large” component of the Dirac spinor q(r) in the form [3]
d2 U’(r) dr2 (7) where E’ = (M(Qq) - ma). Now taking V(r) as in eq. (1) and putting p = (r/r;) and A = ar’+l with the scale factor rb = [av+l(E’+
m9)] -ll(u+B
,
(8)
eq. (7) reduces to the SchrBdinger form
d2U’(P)+ E’
nl -p
O2 where 182
Cl
=
3 March
1983
(,v+l)-2/(v+2)
X (E’ + mq)v/(ve2)(E’-
mq - Vo) ,
(10)
Eq. (9) being identical to its non-relativistic counter part would yield the value of eAl = e,/ > 0 corresponding to the confined bound states of quarks which must be the same as that obtained from eq. (3). Using the substitutions,
(4)
For Y> 0, eq. (3) can be solved by an exact numerical method to yield positive definite values of en [. Now with A = u”+l eq. (4) would yield the formula for the bound state mass of the QQ system as M,/(QQ)
LETTERS
(9)
(E’-
mq - VO) = axnl ,
(11)
(2mq + VO) = aB ,
(12)
eq. (10) becomes
x$+~)“(x,, l +
(d2)lV. B) = enl
(13)
Then a unique positive root solution for xnl in eq. (13) gives the Qq bound state masses as Mn~(Q~)=mQ+mgtVo+ax,t.
(14)
4. Phenomenological results. In our phenomenological study the spin dependence is eliminated by using the spin-averaged level values as inputs. Since the two lowest S-wave CChyperfme splittings have now been experimentally measured [4], the spin-averaged levels for 1S and 2S states of cC system are directly calculated asMlg(E) = 3.067 GeVandMzs(cZ)=3.662 GeV respectively. In case of the bb system the 1S and 2S hyperfme splittings are not yet experimentally known. An estimation [2] for those values gives the spin-averaged masses of the two lowest S-wave states of bb system as M, s(bb) = 9.428 GeV and Mzs(bb) = 9.99 GeV. Now we would use the 1S and 2S spin-averaged states of CCand bb system in the Schradinger equation to determine the static potential parameters and the heavy quark masses which would then be used in the Dirac equation to find out the relativistic bound states of Qq-like mesons. First of all, we fm the parameter v = 0.1 in close agreement with the previous findings [ 11 and solve numerically eq. (3) to obtain e,/ values for different quark-antiquark bound states. Then taking the spinaveraged massesM1s(~C),M2s(c~),Mls(bb) and M2s(bb) as inputs in eq. (S), we determine the potential parameters a, Vu and the quark masses m, and mb as follows: (a, Vo) = (5.678, -7.4; GeV) ,
(15)
Table 1 Spinaveraged mass spectrum M,l(Q@ of cE. bG and t? families obtained from the SchrBdinger equation (2): ~... -__ n
I
EnI
1
s
1.2364
2 3 4 5 1 2 1
s s s s P P D
1.3341 1.3923 1.4335 1.4657 1.3071 1.3731 1.3544
Jfnl(Cq GeV)
Wzl(b@ GeV)
3.067 3.662 4.011 4.260 4.455 3.495 3.894 3.781
9.428 9.990 10.320 10.555 10.739 9.832 10.210 10.103
(mc,m,,)=(1.49,4.88;GeV).
mu =md=lOMeV,
GeV) 39.212 39.737 40.045 40.266 40.438 39.590 39.943 39.843 ~~ (16)
we obtain the spin-averaged ground state mass values of D, F and B mesons in the following manner
=
1.970 GeV ;
M, S(C”),~ =2.112GeV, Ml S(cs)exp
=2.113GeV;
Ml S(bE)exp
=
5.16-5.27
GeV .
(18)
Our calculated results are found to be in excellent agreement with the corresponding experimental spin-averaged mass values *i written below each result in (18). It is interesting to find that all the quarks except the s-quark have their masses comparable to the current quark masses [lo] rather than to the phenomenological constituent quark masses [ 111. Only the squark mass seems to lie between current and constituent quark masses. We are not going to interpret and discuss these results from theoretical point of view. However these points have already been discussed in the framework of Quantum-ChromoDynamics (QCD) by Kaburagi et al. [ 121. Finally with the same parameters fxed as in eqs. (15), (16) and (17), we predict some higher spinaveraged levels of the D, F and B families along with all such levels of G and H families. The results are displayed in table 2. All these predictions are shortly going to be tested in future experiments at CERN and FNAL. In fact our calculated results for charmed and b-flavored mesons compare quite well with the predictions of Kaburagi et al. [12,13]. These authors have taken in ref. [ 121 a purely scalar power-law potential in the Dirac equation to generate Qq bound states. However, if such a static power-law potential generating spin dependence in the usual manner with its Lorentz structure as a scalar, is used in a non-relativistic perturbative approach for CCsystem, it would be impossible in this case to explain the observed *’ For the D meson see ref. [7], for the F meson ref. [8], for the B meson ref. [9]. Table 2 Spin-averaged mass spectrum Mnl(Q$ of cii, cS, bIi, bS and bZ families obtained from the Dirac equation (6).
(17)
ms = 400 MeV ,
Ml S(cu)cal = 1.984 GeV ,
= 5.369 GeV ,
MIs(bii),,
J!fnl(ti)
Using these parameters we calculated the spin-averaged mass spectra for CCand bb systems from eq. (5). These results are presented in table 1. We can not make at this stage a definite quantitative comparison of our results with the experimental values unless we compute the fine-hyperfme splittings which we do not discuss in this paper. However looking at the mean mass values obtained here we can notice a very good qualitative agreement with the experimental mass spectra. We also predict the non-relativistic spinaveraged mass spectrum for the yet unobserved tt system by futing the quark mass m, arbitrarily at 20 GeV [S] . Such predictions are also enlisted in table 1. Although recent experiments at PETRA [6] have found no evidence for such heavy mesons upto an energy scale of 36.72 GeV, they are expected to be seen in future experiments. Now we shall find out the Dirac bound state masses of atomlike mesons Qq by using eq. (14). With the parameters taken to be the same as in eqs. (1 S) and (16) and the light quark mass parameters as
Ml S(cu)exp
3 March 1983
PHYSICS LETTERS
Volume 122B, number 2
n
1
M&CL3 M&3 GeV) GeV)
1
s
1.984
2 3 4 5 1 2 1
s s s s P P D
2.378 2.623 2.822 2.913 2.261 2.541 2.464
2.112 2.573 2.852 3.055 3.215 2.441 2.758 2.668 __
M&3 GeV)
Mnl(bs) GeV)
&z(bS) GeV)
5.369 5 .I64 6.018 6.207 6.358 5.647 5.932 5.849
5.497 5.958 6.238 6.440 6.600 5.826 6.144 6.053
6.221 6.142 7.044 7.260 7.429 6.597 6.943 6.845 183
Volume
122B, number
2
PHYSICS
fine-hyperfme splittings [2]. Therefore we believe that for a power-law potential model, a more consistent and unified approach in realizing fine-hyperfme spectra of light and heavy mesons in a non-relativistic formalism [ 141 which at the same time guarantees relativistic quark confinement in successfully generating Dirac bound states of Qq like mesons, is based mainly on the formal assumption of an equally mixed scalarvector Lorentz structure of the static potential. In the present work we find that with a power-law potential V(r) = A?’ + Vu which describes heavy mesons reasonably well, a relativistic description of Qqlike light mesons requires current quark masses. Now if we would take constituent quark masses mu = md = 0.336 GeV and ms = 0.540 GeV [l 1] instead of the light quark masses (17) which has been found to correspond to the current quark masses [lo] then with the same set of parameters as fned in eqs. (1.5) and (16) we would obtain the spin-averaged ground state masses of D and F mesons as MIS(cu) = 2.08 GeV ,
Mls(cS) = 2.18 GeV ,
3 March
1983
states of quarkonia have been obtained from the Schrbdmger equation and those of atomlike mesons from the Dirac equation. Flavour independence of the potential have been strictly maintained by taking same set of potential parameters and quark masses to generate the bound states of QQ and Qq-systems. We have found the remarkable result that the quark masses are close to the current quark masses rather than to the phenomenological constituent quark masses. In fact the former provides an excellent tit to the observed spin-averaged mass of D and F mesons. In conclusion we point out that the equally mixed vector-scalar Lorentz structure of a power-law potential which was phenomenologically observed in the Schrbdinger approach to explain the fme-hyperfme splittings of the meson spectra [2] can satisfactorily describe the Dirac mass spectra of Qq-like mesons. We would like to thank Dr. N. Barik for useful discussions. The computational help of the Computer Centre, Utkal University is gratefully acknowledged.
(19)
which are quite high as compared with their corresponding experimental values written in eq. (18). Thus from our phenomenological study it is evident that for the power-law potential the calculated spinaveraged masses with the current quark mass gives an excellent agreement with the experiment of D and F mesons whereas the ones with the constituent quark mass are too high. Hence we may conclude that for the study of relativistic bound states of Qq-like mesons the current quark mass is certainly preferable to the constituent mass. Such a conclusion was also reached in ref. [ 121 by Kaburagi et al. in their study of Dirac bound states of atomlike mesons Qq with different potential models such as (i) scalar power-law potential (ii) scalar logarithmic potential (in) Coulomb plus scalar linear potential. Therefore our conclusion that the relativistic description of Qq-like light mesons requires current quark masses is in fact general to any reasonable confining potential. 5. Conclusion. In this work, we have studied the spin-averaged mass levels of atom-like mesons Qq in the Dirac equation with an equally mixed scalar and vector potential of power-law type. The quarkonia and the atom-like mesons have been discussed in a unified manner by such a confining potential. Bound 184
LETTERS
[l]
A. Martin, Phys. Lett. 93B (1980) 338; N. Barik and S.N. Jena, Phys. Lett. 97B (1980) 261; A. Khare, Phys. Lett. 98B (1981) 385. [2] N. Barik and S.N. Jena, Phys. Lett. 97B (1980) 265. [3] E. Magyari, Phys. Lett. 95B (1980) 295; N. Barik and S.N. Jena, Phys. Rev. D, to be published. [4] D. Scharre, 1981 Intern. Symp. on Lepton and photon interactions at high energies (Bonn, August 1981); Intern. Conf. on Physics in collision (Blacksburg, VA, May 1981), SLAC-PUB-276 1; K. Berkelman, Proc. XXth Intern. Conf. on High energy physics (Madison, WI, July 1980), eds. L. Durand and LG. Pondron, AIP Conf. Proc., no. 68 (ATP, New York). [S] D.V. Nanopoulous, CERNPUN-TH-2866 (1980). [6] J.F. Grivaz, in: New flavors and hadron spectroscopy, Proc. of the XVI Rencontre du Mortond (Les Arcs, France, 1981) ed. J. TranThanh Van (Editions Frontieres, Dreux, France, 1981). [7] P.A. Rapidis et al., Phys. Rev. Lett. 39 (1977) 526. [ 81 R. Brandelik et al., Phys. Lett. 70B (1977) 132; 80B (1979) 412. [9] C. Bebek et al., Phys. Rev. Lett. 46 (1981) 84: K. Chadwick et al., Phys. Rev. Lett.46 (1981)88; G. Monti, SLAC preprint HEPSY MEMO 15-80 (1980). 1101 H. Fritzsch, Lecture at the 10th GIFT Seminar on Theor. Phys. (Jacca, Spain, 1979), CERN Preprint TH.2699. [Ill A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [I21 M. Kaburagi et al., Z. Phys. C-part. Fields 9 (1981) 213. [131 M. Kaburagi et al., Phys. Lett. 97B (1980) 143. S.N.Jena,Phys.Rev.D26 (1982) 618. [I41 N.Barikand
122B. number
DIRAC
MASS
PHYSICS
2
SPECTRA
OF Q+LIKE
MESONS
LETTERS
3 March
1983
IN A
JENA and T. TRIPATI Department of Physics, Aska Science College, Aska, Ganjam, Orissa,India Received 31 August 1982 Revised manuscript received
4 November
1982
The mass spectra of Qq-like mesons are studied in the Dirac equation with an equally mixed 4-vector and scalar powerpotential can satisfactorily describe law potential of the form V(r) = Are.’ + V,J. It is found that this flavor-independent the mass levels of D, F and B mesons along with those of e and T families in a unified manner and that the quark masses in quarkonia and Qq-like mesons are very close to the current quark masses.
1. Introduction.
potential
Recently the empirical power-law of the form
V(r) = Ar” + V0 ,
(1)
with v = 0.1 and A > 0 was extensively discussed [l] in accordance with the very accurate data of the Tfamily and seems to be preferable to the logarithmic potential. These discussions are mostly based on the non-relativistic Schrbdinger type approach. It would be interesting to see how this potential works in the relativistic system inside a hadron. This work aims to study atomlike mesons Qq composed of a heavy quark Q and a light quark q in the Dirac equation with an equally mixed 4-vector and scalar power-law potential. In case of atomlike mesons where a light quark q is moving around an almost fured heavy quark Q with a high speed, the use of non-relativistic SchrBdinger treatment to study their bound states may not be quite appropriate. Therefore these mesons must be discussed in the framework of the relativistic theory and have not been studied systematically. In the present paper our chief interest is to describe the QQ and Qq systems in a unified way by taking same potential parameters for both systems. Here we discuss only the gross structure of the energy levels but not the problem including fine-hyperfme splittings. We assume the Lorentz structure of the confining potential to be an equal admixture of scalar and vector 0 031-9163/83/0000-OOOO/$
03.00 0 1983 North-Holland
parts in accordance with the phenomenology [2] of fine-hyperfme splittings of heavy mesons. The reason for this assumption is that an equally mixed scalar -vector potential in the Dirac equation can realize quark confinement to generate relativistic quarkantiquark bound states [3]. We solve the QQ systems with Q = c, b, t by the Schrbdinger equation and the Qq systems with Q = c, b, and q = u or d, s, or c by the Dirac equation. The availability of numerous and convincing experimental data in case of CCand bb systems makes it plausible to extract the static potential parameters of the model from the Schrbdinger formalism. Then assuming flavour-independence of the confining potential the same potential parameters are used in the Dirac equation to find out the bound state masses of Qq systems. Atomlike (Qq) mesons such as Do(&), D+(cd) and F+(cS) have already been observed experimentally in ground states only with their hyperfme mass splittings. Other such mesons in this category like B-(bti), Bo(bd), G”(bS) and H-(b?) are expected to be found in near future. 2. Bound state mass of 20 system. For the nonrelativistic QQ system, the reduced radial Schriidinger equation is
‘3 +(mQ(E - v(r)) -y)
U(r) = 0 .
(2) 181
Volume
122B, number
2
PHYSICS
Taking V(r) as given in eq. (1) and substituting p = (r/ru) with the scale factor chosen asr,, = (rr~~A)-~l(*~) eq. (1) reduces to the form (3) where .
frill = m&!? - Vo) (,($-2/(~+2)
= 2mQ + V,, t a(a/mQ)vl(v+2)en1
.
(5)
3. Bound state mass of Qe system. Without considering the recoil effects of the heavy quark Q, the Dirac equation for the Qq bound state massM(Qq) can be written as
= W(Q$ ~ mQ1
q(r)
>
(6)
where mQ and mq are masses of heavy and light quarks respectively. With the assumed Lorentz structure of I’(r) in the form of an equal admixture of scalar and vector parts [V,(r) = V,(r) = i V(r)] eq. (6) leads to an equation satisfied by the reduced radial part of the “large” component of the Dirac spinor q(r) in the form [3]
d2 U’(r) dr2 (7) where E’ = (M(Qq) - ma). Now taking V(r) as in eq. (1) and putting p = (r/r;) and A = ar’+l with the scale factor rb = [av+l(E’+
m9)] -ll(u+B
,
(8)
eq. (7) reduces to the SchrBdinger form
d2U’(P)+ E’
nl -p
O2 where 182
Cl
=
3 March
1983
(,v+l)-2/(v+2)
X (E’ + mq)v/(ve2)(E’-
mq - Vo) ,
(10)
Eq. (9) being identical to its non-relativistic counter part would yield the value of eAl = e,/ > 0 corresponding to the confined bound states of quarks which must be the same as that obtained from eq. (3). Using the substitutions,
(4)
For Y> 0, eq. (3) can be solved by an exact numerical method to yield positive definite values of en [. Now with A = u”+l eq. (4) would yield the formula for the bound state mass of the QQ system as M,/(QQ)
LETTERS
(9)
(E’-
mq - VO) = axnl ,
(11)
(2mq + VO) = aB ,
(12)
eq. (10) becomes
x$+~)“(x,, l +
(d2)lV. B) = enl
(13)
Then a unique positive root solution for xnl in eq. (13) gives the Qq bound state masses as Mn~(Q~)=mQ+mgtVo+ax,t.
(14)
4. Phenomenological results. In our phenomenological study the spin dependence is eliminated by using the spin-averaged level values as inputs. Since the two lowest S-wave CChyperfme splittings have now been experimentally measured [4], the spin-averaged levels for 1S and 2S states of cC system are directly calculated asMlg(E) = 3.067 GeVandMzs(cZ)=3.662 GeV respectively. In case of the bb system the 1S and 2S hyperfme splittings are not yet experimentally known. An estimation [2] for those values gives the spin-averaged masses of the two lowest S-wave states of bb system as M, s(bb) = 9.428 GeV and Mzs(bb) = 9.99 GeV. Now we would use the 1S and 2S spin-averaged states of CCand bb system in the Schradinger equation to determine the static potential parameters and the heavy quark masses which would then be used in the Dirac equation to find out the relativistic bound states of Qq-like mesons. First of all, we fm the parameter v = 0.1 in close agreement with the previous findings [ 11 and solve numerically eq. (3) to obtain e,/ values for different quark-antiquark bound states. Then taking the spinaveraged massesM1s(~C),M2s(c~),Mls(bb) and M2s(bb) as inputs in eq. (S), we determine the potential parameters a, Vu and the quark masses m, and mb as follows: (a, Vo) = (5.678, -7.4; GeV) ,
(15)
Table 1 Spinaveraged mass spectrum M,l(Q@ of cE. bG and t? families obtained from the SchrBdinger equation (2): ~... -__ n
I
EnI
1
s
1.2364
2 3 4 5 1 2 1
s s s s P P D
1.3341 1.3923 1.4335 1.4657 1.3071 1.3731 1.3544
Jfnl(Cq GeV)
Wzl(b@ GeV)
3.067 3.662 4.011 4.260 4.455 3.495 3.894 3.781
9.428 9.990 10.320 10.555 10.739 9.832 10.210 10.103
(mc,m,,)=(1.49,4.88;GeV).
mu =md=lOMeV,
GeV) 39.212 39.737 40.045 40.266 40.438 39.590 39.943 39.843 ~~ (16)
we obtain the spin-averaged ground state mass values of D, F and B mesons in the following manner
=
1.970 GeV ;
M, S(C”),~ =2.112GeV, Ml S(cs)exp
=2.113GeV;
Ml S(bE)exp
=
5.16-5.27
GeV .
(18)
Our calculated results are found to be in excellent agreement with the corresponding experimental spin-averaged mass values *i written below each result in (18). It is interesting to find that all the quarks except the s-quark have their masses comparable to the current quark masses [lo] rather than to the phenomenological constituent quark masses [ 111. Only the squark mass seems to lie between current and constituent quark masses. We are not going to interpret and discuss these results from theoretical point of view. However these points have already been discussed in the framework of Quantum-ChromoDynamics (QCD) by Kaburagi et al. [ 121. Finally with the same parameters fxed as in eqs. (15), (16) and (17), we predict some higher spinaveraged levels of the D, F and B families along with all such levels of G and H families. The results are displayed in table 2. All these predictions are shortly going to be tested in future experiments at CERN and FNAL. In fact our calculated results for charmed and b-flavored mesons compare quite well with the predictions of Kaburagi et al. [12,13]. These authors have taken in ref. [ 121 a purely scalar power-law potential in the Dirac equation to generate Qq bound states. However, if such a static power-law potential generating spin dependence in the usual manner with its Lorentz structure as a scalar, is used in a non-relativistic perturbative approach for CCsystem, it would be impossible in this case to explain the observed *’ For the D meson see ref. [7], for the F meson ref. [8], for the B meson ref. [9]. Table 2 Spin-averaged mass spectrum Mnl(Q$ of cii, cS, bIi, bS and bZ families obtained from the Dirac equation (6).
(17)
ms = 400 MeV ,
Ml S(cu)cal = 1.984 GeV ,
= 5.369 GeV ,
MIs(bii),,
J!fnl(ti)
Using these parameters we calculated the spin-averaged mass spectra for CCand bb systems from eq. (5). These results are presented in table 1. We can not make at this stage a definite quantitative comparison of our results with the experimental values unless we compute the fine-hyperfme splittings which we do not discuss in this paper. However looking at the mean mass values obtained here we can notice a very good qualitative agreement with the experimental mass spectra. We also predict the non-relativistic spinaveraged mass spectrum for the yet unobserved tt system by futing the quark mass m, arbitrarily at 20 GeV [S] . Such predictions are also enlisted in table 1. Although recent experiments at PETRA [6] have found no evidence for such heavy mesons upto an energy scale of 36.72 GeV, they are expected to be seen in future experiments. Now we shall find out the Dirac bound state masses of atomlike mesons Qq by using eq. (14). With the parameters taken to be the same as in eqs. (1 S) and (16) and the light quark mass parameters as
Ml S(cu)exp
3 March 1983
PHYSICS LETTERS
Volume 122B, number 2
n
1
M&CL3 M&3 GeV) GeV)
1
s
1.984
2 3 4 5 1 2 1
s s s s P P D
2.378 2.623 2.822 2.913 2.261 2.541 2.464
2.112 2.573 2.852 3.055 3.215 2.441 2.758 2.668 __
M&3 GeV)
Mnl(bs) GeV)
&z(bS) GeV)
5.369 5 .I64 6.018 6.207 6.358 5.647 5.932 5.849
5.497 5.958 6.238 6.440 6.600 5.826 6.144 6.053
6.221 6.142 7.044 7.260 7.429 6.597 6.943 6.845 183
Volume
122B, number
2
PHYSICS
fine-hyperfme splittings [2]. Therefore we believe that for a power-law potential model, a more consistent and unified approach in realizing fine-hyperfme spectra of light and heavy mesons in a non-relativistic formalism [ 141 which at the same time guarantees relativistic quark confinement in successfully generating Dirac bound states of Qq like mesons, is based mainly on the formal assumption of an equally mixed scalarvector Lorentz structure of the static potential. In the present work we find that with a power-law potential V(r) = A?’ + Vu which describes heavy mesons reasonably well, a relativistic description of Qqlike light mesons requires current quark masses. Now if we would take constituent quark masses mu = md = 0.336 GeV and ms = 0.540 GeV [l 1] instead of the light quark masses (17) which has been found to correspond to the current quark masses [lo] then with the same set of parameters as fned in eqs. (1.5) and (16) we would obtain the spin-averaged ground state masses of D and F mesons as MIS(cu) = 2.08 GeV ,
Mls(cS) = 2.18 GeV ,
3 March
1983
states of quarkonia have been obtained from the Schrbdmger equation and those of atomlike mesons from the Dirac equation. Flavour independence of the potential have been strictly maintained by taking same set of potential parameters and quark masses to generate the bound states of QQ and Qq-systems. We have found the remarkable result that the quark masses are close to the current quark masses rather than to the phenomenological constituent quark masses. In fact the former provides an excellent tit to the observed spin-averaged mass of D and F mesons. In conclusion we point out that the equally mixed vector-scalar Lorentz structure of a power-law potential which was phenomenologically observed in the Schrbdinger approach to explain the fme-hyperfme splittings of the meson spectra [2] can satisfactorily describe the Dirac mass spectra of Qq-like mesons. We would like to thank Dr. N. Barik for useful discussions. The computational help of the Computer Centre, Utkal University is gratefully acknowledged.
(19)
which are quite high as compared with their corresponding experimental values written in eq. (18). Thus from our phenomenological study it is evident that for the power-law potential the calculated spinaveraged masses with the current quark mass gives an excellent agreement with the experiment of D and F mesons whereas the ones with the constituent quark mass are too high. Hence we may conclude that for the study of relativistic bound states of Qq-like mesons the current quark mass is certainly preferable to the constituent mass. Such a conclusion was also reached in ref. [ 121 by Kaburagi et al. in their study of Dirac bound states of atomlike mesons Qq with different potential models such as (i) scalar power-law potential (ii) scalar logarithmic potential (in) Coulomb plus scalar linear potential. Therefore our conclusion that the relativistic description of Qq-like light mesons requires current quark masses is in fact general to any reasonable confining potential. 5. Conclusion. In this work, we have studied the spin-averaged mass levels of atom-like mesons Qq in the Dirac equation with an equally mixed scalar and vector potential of power-law type. The quarkonia and the atom-like mesons have been discussed in a unified manner by such a confining potential. Bound 184
LETTERS
[l]
A. Martin, Phys. Lett. 93B (1980) 338; N. Barik and S.N. Jena, Phys. Lett. 97B (1980) 261; A. Khare, Phys. Lett. 98B (1981) 385. [2] N. Barik and S.N. Jena, Phys. Lett. 97B (1980) 265. [3] E. Magyari, Phys. Lett. 95B (1980) 295; N. Barik and S.N. Jena, Phys. Rev. D, to be published. [4] D. Scharre, 1981 Intern. Symp. on Lepton and photon interactions at high energies (Bonn, August 1981); Intern. Conf. on Physics in collision (Blacksburg, VA, May 1981), SLAC-PUB-276 1; K. Berkelman, Proc. XXth Intern. Conf. on High energy physics (Madison, WI, July 1980), eds. L. Durand and LG. Pondron, AIP Conf. Proc., no. 68 (ATP, New York). [S] D.V. Nanopoulous, CERNPUN-TH-2866 (1980). [6] J.F. Grivaz, in: New flavors and hadron spectroscopy, Proc. of the XVI Rencontre du Mortond (Les Arcs, France, 1981) ed. J. TranThanh Van (Editions Frontieres, Dreux, France, 1981). [7] P.A. Rapidis et al., Phys. Rev. Lett. 39 (1977) 526. [ 81 R. Brandelik et al., Phys. Lett. 70B (1977) 132; 80B (1979) 412. [9] C. Bebek et al., Phys. Rev. Lett. 46 (1981) 84: K. Chadwick et al., Phys. Rev. Lett.46 (1981)88; G. Monti, SLAC preprint HEPSY MEMO 15-80 (1980). 1101 H. Fritzsch, Lecture at the 10th GIFT Seminar on Theor. Phys. (Jacca, Spain, 1979), CERN Preprint TH.2699. [Ill A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [I21 M. Kaburagi et al., Z. Phys. C-part. Fields 9 (1981) 213. [131 M. Kaburagi et al., Phys. Lett. 97B (1980) 143. S.N.Jena,Phys.Rev.D26 (1982) 618. [I41 N.Barikand