Spectroscopy of atomlike mesons Qq in the Dirac equation with logarithmic confining potential

Volume 97B, number 1

PHYSICS LETTF.RS

17 November 1980

SPECTROSCOPY OF ATOMLIKE MESONS Q~ IN THE DIRAC EQUATION WITH LOGARITHMIC CONFINING POTENTIAL Makoto KABURAGI, Masaaki KAWAGUCHI and Toshiyuki MORII College of Liberal Arts, Kobe University, Kobe 657, Japan and Tetsuro KITAZOE and Junya MORISHITA Departntent of Physics, Kobe UniversiO,, Kobe 657, Japan Received 20 June 1980

Atomlike mesons Q~ are investigated in the Dirac equation with a scalar logarithmic confining potential. It is found that the current quark masses give an excellent fit to experiment of D and F meson levels together with 4' and T families.

Many candidates for a confining potential have been proposed and investigated in the spectroscopy of the ~ and T families [ 1]. Usually discussions are confined to the nonrelativistic model. It would be interesting to see how the potential works in the relativistic system inside a hadron. In the present paper atomlike heavy mesons Q~ composed of one heavy quark Q and one light quark q are discussed in the Dirac equation with a scalar logarithmic potential. By atomlike meson we mean that a light quark q is running around an almost fixed heavy quark Q with a high speed. The logarithmic potential [2] is the most promising one which can provide the same level spacing for the ~ and T families in terms of common parameters. We apply the same potential to the atomlike meson Q~. The energy eigenvalues o f charmed mesons, D, F and b o t t o m e d mesons, B, G(bg-), and their higher levels are obtained by solving the Dirac equation. Our interest in the present paper is to understand how the logarithmic potential is able to describe the QQ and Qq system in a unified way. Consequently, we are interested only in the gross level structure, but not in the detailed fitting to data including hyperfine splitting. The spin of the heavy quark Q is neglected throughout the present paper. The hamiltonian in the relativistic Q~ system is

then given as H D = m Q +c~p+fi(rnq + S ) ,

(1)

and the one in the nonrelativistic QQ system is H S = 2mQ +p2/rnQ + S ,

(2)

where S is the potential S = a -1 ln(r/ro),

(3)

with constants a and r 0. We take the logarithmic potential of a scalar type, because a scalar potential is able to confine a quark in the Dirac equation [3], whereas a vector one cannot confine a quark in a finite region like the situation manifested in the Klein paradox [4]. We use the variation method to find energy eigenvalues of eqs. (1) and (2). A trial wave function for the radial part is chosen with several parameters in such a way that this trial function satisfies the asymptotic behavior at infinity and the proper boundary conditions at the origin, and an unknown factor of the wave function is assumed to be a polynomial. Several examinations have been made to justify the accuracy of the solution. First, the degree of the polynomial in the trial function is increased until the solution becomes stable. Second, it is confirmed that the 143

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wave functions obtained satisfy the virial theorem *~ , which must be satisfied in a bound state. Third, the solution is compared with the one obtained by the method of finite difference and it is found that both agree with each other very well. For the nonrelativistic system QQ, the characteristic features of the logarithmic potential are that every level spacing of QQ is determined only by a single parameter a and that level structures of c c a n d bb- are quite consistent with experiment. Hence the parameter a is determined by level spacing of J / 4 and 4'- Two remaining parameters r 0 and m b are to be determined by the experimental masses of J / 4 ( 3 0 9 5 ) and T(9435) as functions of m c as shown in fig. 1. The Dirac equation for the Q~ system is solved by using the same potential parameters. It is noted that the D and F meson levels we want to fit are average of D and D*, and F and F*, i.e. (D) = (3D* + D)/4 ~1 In the present calculation the virial theorem is satisfied in the order of magnitude I<[rp,H] >/(//)1 ~< 10 -4.

L0

mb

0.5

L.5

~

4~

L

L

1.1

1.2

._L

13

i

1

1.4

1.5

"l

1.6 GeV

rT1c

Fig. 1. Quark masses and potential parameters as functions of m o where a = 1.3589 GeV -1 is used. The input data are J/vp = 3.095 GeV, vp' = 3.686 GeV, Y = 9.435 GeV,
TQ =p2/ZmQ,

(4)

in addition to DI. After solving the Dirac equation we can determine the quark masses mu(md) and m s by using the observed masses of (D) and (F). Since r 0 in the potential is a function of mc, every physical quantity is expressed in terms of m c as shown in fig. 1. TQ gives a remarkable effect on the light quark mass but not on the total energy. The contribution to the latter is several percent to each energy level. From the solution of DII, which is expectedly preferable to DI, we see the remarkable fact that the u(d) and s quark masses are close to the current quark masses rather than to the phenomenological constituent quark masses. We are not going to discuss the theoretical relation between current and constituent quark masses in detail. We only emphasize here that for the scalar potential the confining energy inside a "bag" raises the b o t t o m of the potential, instead of giving the current quark a dynamical mass [5] It is interesting to take current quark masses to describe the relativistic Dirac system (DII). The mass values taken here are [6] rn u = 0.005 G e V ,

m d = 0.010 G e V ,

ms=0.180GeV ,

mc=l.500GeV,

m h = 4.887 G e V .

O.

1l)

and (F) = (3F* + F)/4, because we neglect the hyperfine splitting at present. In order to study the recoil effect of the heavy quark, we solve two kinds of equations. One is eq. (1) where the heavy quark is fixed at the origin, which we call equation DI. The other is called DII and includes the kinetic energy of the heavy quark,

mslDl)

m u (DI)

-0.3

17 November 1980

(5)

It is noted that the adjustable parameters are only two, r 0 and a, once the current quark masses are given. We determine r 0 and a by using J / 4 and 4 ' masses. All the levels other than J / 4 and 4 ' are predicted and the resulting mass levels are shown in figs. 2 and 3 with a remarkable agreement with experiment. The calculated result shows that the mass level for the Q~ system is approximately written as c o + Clln n, where n is the radial quantum number and the c's are appropriate constants. The mass difference of the current quark masses m u and m d produces the mass difference (D +) - (DO).

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D(c~)

M (GeV)

17 November 1980

F (c~)

M(GeV)

3.080

3.0

3.0

2.939 2.871 2.798 2.699

2.682

2.890 2770

2.4~0

2.633

"2.543

2.414

2.811 2,?17

2.710

2.583

2.5

3.009 2.938

2.5

2.568

3035 2.907 2.851 2,672 2.597

2528

2.459 2.382

2.255

2.0

1.~

2.0

(1.970) I

I

-1

-2

S

I

I

-3

2

>k

I

-2

-1

1

2

D

G (bg)

M (GeV)

B(bO}

-3

P

S

D

P

M (GeV)

L

>k

I

1

2.074 (2. t 10)

6.347 6,277

6.216 6.151 6.091

6.0

6.005

6.126

6.059

5.974 5.890

5.765

6,220

6.174

5.?23

6.0

6.004.

6.092 6.014

5.876

5.848 5.?66

6.308 6.185 6.134 5.96? 5.892

5.829 5.695

5.58O

5.5

5.5 5.411 5.319 I

1 5

i

l

-2

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~

~

-3

2 D

>k

I

l

-1

-2

S

P

i

J

i

1

3

2

)k

D

Fig. 2. Mass levels solved from the equation DII for cg, c~-, bg and bg, by using the current quark masses (5), a = 1.3589 GeV -1 and a -1 In r 0 = 0.6372 GeV. Here k = l f o r j = l - 1/2 and k = - l - 1 f o r / = l + 1/2. The values in parentheses are experimental data. W h e n t h e C o u l o m b i n t e r a c t i o n b e t w e e n the q u a r k s is s w i t c h e d on, we o b t a i n (D +) - (D 0) = 4.5 M e V w i t h t h e given q u a r k mass in eq. (5), wh~re t h e m a g n e t i c m o m e n t i n t e r a c t i o n is n e g l e c t e d . This result is comp a r e d w i t h D + - D O = 5.1 -+ 0.8 M e V a n d D *+ - D * 0 = 2.6 -+ 1.8 M e V + 2. It is safely said t h a t the c u r r e n t q u a r k masses a d o p t e d h e r e are able to r e p r o d u c e t h e e x p e r i m e n t a l mass d i f f e r e n c e .

4:2 For the experimental value of (D), see ref. [7], for {F), see ref. [81.

We c o r m n e n t o n some p r o b l e m s o f the Q ~ system. (1) If we use t h e c o n s t i t u e n t q u a r k masses rn u = rn d = 0 . 3 3 6 G e V a n d m s = 0 . 5 4 0 G e V [9] i n s t e a d o f the c u r r e n t q u a r k masses (5), (D) a n d (F) result in 2 . 1 7 3 G e V a n d 2 . 3 1 6 G e V , in d i s a g r e e m e n t w i t h experim e n t . (2) One m a y ask h o w relativistic t h e s y s t e m is. The a n s w e r is t h a t the relativistic e f f e c t is very large. To investigate this effect, the difference o f eigenvalues A £ = E s - E D for the Q ~ s y s t e m is c a l c u l a t e d , w h e r e E D a n d E s are eigenvalues in the Dirac ( D I I ) a n d S c h r 6 d i n g e r e q u a t i o n s , respectively. F o r rn O = 1.5 145

Volume 97B, number 1

PIIYSICS LETTERS

M (GeV)

¢J~ ((::'E)

17 November 1980

M (GeV}

Y(b5)

4.5' l

4.409

4.347 4.237

4.0

10,749

10.686

4.286

4,.160

10.5

10576

4.083

4.011 (4.04)

10.423 10.351 (10,323)

3,909

10.249 3.808

10.147 10.0 -

3.686 (3.686)

10,026 (9.993) 9.8?4

3.534 (3.527)

3.5

10.625

10.499

9.5 3"~ 'T I

(3.095)

) t

9.435 (9.435) i

i

I

I

L

0

1

2

0

1

2

S

P

D

S

P

D

>l

Fig. 3. Mass levels in the Schr6dinger equation ~2) for ~ and T families by using the same parameters as those used in fig. 2. The values in parentheses show experimental data 4- . The experimental value of the lowest P state of the ff fanily represents the center of gravity.

GeV, we obtain 2d£= 1.223 GeV for mq = 0.010 GeV and 2xE = 0.269 GeV for mq = 0.180 GeV. This result shows that the nonrelativistic t r e a t m e n t o f the light quark q is quite inadequate. In conclusion, we study the level structure of atomlike m e s o n Q ~ described by the Dirac e q u a t i o n in a scalar logarithmic potential and find the remarkable result that the quark masses are close to the current quark masses rather than the p h e n o m e n o l o g i c a l constituent quark masses. In fact it is f o u n d that the former gives an excellent fit to the experimental levels of (12)) and (F). We also predict higher levels o f the D and F families and those of B and G with no parameter. Our predictions will be e x a m i n e d in a c o m i n g e x p e r i m e n t at C E R N and F N A L . 4.3

For the ~ family see ref. [10], for the × states ref. [11] and for the T family ref. [12].

[ 1 ] E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane and T.M. Yan, Phys. Rev. D21 (1980) 203;D17 (1978) 3090; T. Appelquist, R.M. Barnett and K.D. Lane, Ann. Rev. Nucl. Part. Sci. 28 (1978) 387; J.D. Jackson, C. Quigg and J.L. Rosner, Proc. 19th Intern. Conf. on High energy physics (Tokyo, 1978) p. 391. 146

[2] C. Quigg and J.L. Rosner, Phys. Lett. 71B (1977) 153; M. Machacek and Y. Tomozawa, Ann. Phys. (NY) 110 (1978) 407; Yu.B. Rumer, Sov. Plays. JETP 11 (1960) 1365; Y. Muraki, Prog. Theor. Phys. 41 (1968) 473. [3] D.W. Rein, Nuovo Cimento 38A (1977) 19. [4] J.D. Bjorken and S.D. Drell, Relativistic quantunr mechanics (McGraw-Hill, New York, 1964). [5} H. Fritzsch, Acta Phys. Austriaca, Suppl. XIX (1978) 249; S. Weinberg, Trans. NY Acad. Sci. Set. II, 38 (1977) 185; P. Hasenfratz and J. Kuti, Phys. Rep. 40 (1979) 75, and references therein; see also M. Ida, Prog. Theor. Phys. 62 (1979) 522. [61 H. Fritzsch, Lecture at the 10th GIFT Seminar on Theoretical Physics (Jacca, Spain, 1979) CERN preprint TH. 2699. [7] P.A. Rapidis et al., Phys. Rev. Lett. 39 (1977) 526. [8] R. Brandelik et al., Phys. Lett. 70B (1977) 132; 80B (1979) 412. [9] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [10] J.E. Augustin et al., Phys. Rev. Lett. 33 (1974) 1406; G.S. Abrams et al., Phys. Rev. Lett. 33 (1974) 1453; A.M. Boyarski et ak, Phys. Rev. Lett. 34 (1975) 764; R. Brandelik et al., Phys. Lett. 76B (1978) 361; J. Siegrist et al., Phys. Rev. Lett. 36 (1976) 700. [11] C.J. Biddick et al., Phys. Rev. Lett. 38 (1977) 1324. [12] D. Andrews et al., Phys. Rev. Lett. 44 (1980) 1108; T. B6hringer et al., Phys. Rev. Lett. 44 (1980) 1111.