Spectroscopy of charmed baryons

Z. Phys. C - Particies and Fields 17, 219 227 (1983) Partkz S for Physik C

and Fields �9 Springer-Verlag 1983

Spectroscopy of Charmed Baryons

T. Morii* Department of Physics, University of Wuppertal, D-5600 Wuppertal 1, Federal Republic of Germany

M. Kaburagi, M. Kawaguchi College of Liberal Arts, Kobe University, Kobe 657, Japan

T. Kitazoe, M. Tomita Department of Physics, Kobe University, Kobe 657, Japan

J. Morishita, M. Oka Graduate School of Science and Technology, Kobe University, Kobe 657, Japan

Received 30 June 1982

Abstract. An atomlike baryon composed of one heavy quark and two light quarks is studied in a similar way to the Breit theory of a helium atom. Spectroscopy of charmed baryons is given for the vector Coulomb plus scalar linear confining potential.

1. Introduction

Since the discovery of charmonium the nonrelativistic treatments have been made to explain the spectroscopy and decay properties of these particles and charmed baryons [1-3]. All the charmed, bottomed and topped hadrons which contain one or two light quarks, however, must be investigated in the framework of the relativistic theory. In our recent paper, spectroscopy of an atomlike meson composed of a heavy quark Q(c, b, t,...) and a light quark c](fi, d, g) was discussed in the Dirac equation with a scalar confining potential [4,5]. By atomlike meson we mean that a light antiquark is orbitting an almost fixed heavy quark Q, similar to a hydrogen atom.

The aim of the present paper is to investigate an atomlike baryon Qqq composed of a heavy quark Q fixed at an origin and of two light quarks qq, similar to a helium atom. Our fundamental equation is the Breit equation with an appropriate quark potential [6, 7]. As for the potential we choose one, the Coulomb

* Research fellow of the Alexander yon Humboldt Foundation. On leave of absence from College of Liberal Arts, Kobe University

plus linear potential, which has a capability of inter- preting spectroscopy and decay properties of heavy quarkonia Q(~ [2] and atomlike mesons Qc] [5].

We have two alternatives to consider the quark potential inside a baryon. One is that the q-q potential is obtained from q-g/ potential by a charge conjugation. It is noticed that an attractive color de- pendent scalar potential for the quark-antiquark system gives rise to a repulsive force for the quark- quark system. It must be emphasized in the relativistic theory that the Qqq bound state exists irrespective of the sign of scalar confining potential. The bound state with a "repulsive" scalar potential, however, does not have a nonrelativistic limit, so that we do not have any intuitive interpretation of this bound state. In addition, this bound state has a strange nature, namely the heavier the light quark is, the baryon is not necessarily the heavier. When we choose an attractive vector confinement potential for the quark-antiquark system, the quark-quark potential is also attractive. It is however known that a vector potential cannot confine a quark in a finite region [8].

The other choice of the potential in a baryon is suggested from the string picture of hadrons [9, 10]. Since the string is closely related to the vacuum structure inside a hadron, there is no strict relation between q-q and q-~. We assume that the string potential is linear and the strength is an adjustable parameter. We further assume that the string is attractive and spans along three sides of a triangle

220 T. Morii eta[. : Spectroscopy of Charmed Baryons

Qqq. The latter assumption comes from technical need to solve the potential problem since it is difficult to treat the three body interactions. We think that the essential feature in the usual string model is incor- porated by this assumption.

In Sect. 2, a method of calculating the energy eigenvalues and wave functions of an atomlike baryon Qqq is developed. In our formulation, the heavy quark Q is fixed at an origin and gives a central potential to two light quarks, which are moving around Q and interacting with each other. As we treat the light quark q relativistically, the qq wave function has 16(= 4 x 4) components. By separating the spin and the orbital angular momentum, we finally get a set of equations of 8 components.

In Sect. 3, we study masses of charmed baryons, A~(= cud :I = 0), s cuu :I = 1) and f2~(= css). In order to obtain energy eigenvalues and wave functions of Qqq system, we use a variation method. Cal- culations are carried out for both repulsive and attractive confining potentials. In both cases the calculated masses of A~ and Z~ show a reasonable fit to the average experimental mass, although A~ and Sc are almost degenerate. The reason is discussed there why the hyperfine tuning is difficult.

In Sect. 4, we present discussion about the numerical results and some properties of the Dirac equation with "repulsive" potential. It is discussed that there is an abnormal relation in some of attractive and repulsive potentials that the baryon mass decreases as the light quark becomes heavy. Consequently, there is a possibility that (2r becomes light, comparable to or even lighter than Ao or 12~. Conclusion is given in Sect. 5.

2. Equation of Motion in Qqq System

The Hamiltonian of the Qqq system is given in the same way as the Breit Hamiltonian of a helium atom [6, 7],

H = H 1 + H 2 +H12 , (2.1)

H i =~i.p~ +fli(m,+S(r~))+ V(r,), ( i= 1,2) (2.2)

H,2 = fix fi2 S(q2) + (1 - a s "%) V(q2). (2.3)

H~ is the Dirac Hamiltonian of the l{ght quark q~ interacting with the heavy quark Q, where m~, pi, and r~ are the mass, momentum and coordinate of % as shown in Fig. 1. H~2 is the interaction Hamiltonian between ql and q2. rlz is the distance between q~ and q2. S and V are scalar and vector potentials, and three-body interaction is not taken into considera- tion. Then the equation of motion is given by

H T (1,2) = E T(1,2). (2.4)

The angular momentum of the Q q q system is treated in the following way. The total angular momentum of the system is the sum of spin sQ of the heavy quark

Q(M) 1'2 q2(rr12,-P2.-$2)

Fig. 1. Notations for Qqq system

and total angular momentum of qq system j,

J = j + sQ, (2.5)

where j is decomposed either by the j j coupling or by the LS coupling,

J =Jl +J2 = L + S, (2.6)

with Ji, S and L being defined as

Jl = sl + li,

S = s 1 + s2,

L =1:+12.

Here si, l i and Ji are the spin of the quark % orbital and total angular momenta of the Qqi subsystem, respectively. Since the interaction between sQ and j is proportional to 1/m 0 and much smaller than the other interactions, we do not take it into account in this paper. Consequently, E c and St* states are degenerate, for instance, and total angular momentum states are classified in terms of j hereafter. The jj coupling is convenient to characterize the eigenstates of H a + H 2 and the LS coupling is useful to analyse the interaction H~2. We consider the j j coupling first and then calculate H12 in terms of the LS scheme.

The wave function for two light quarks having the angular momenta j l and J2 is expressed as

T ( 1 , 2 ) = a _ 0 - - + a _ + 0 -+

+a+_~0 +- + a + + O ++, (2.7)

where the signs, say, - + denote l 1 = j ~ - � 8 9 and 12 =J2 + �89 a+ _+ is a 2 x 2 matrix in the p space and its elements are functions of r 1 and r 2. ~1 is a function of Q~ and Q2, defined by [11]

-+ -+ -= -+ o2)

= 2 (j~rn,j2mzljm)r (2.8) m l m 2

Urn) being the total angular momentum of two quark system. Here Ojm(O) is the spin harmonics given by,

(of re(f2) =-- (~(s = �89 = j + �89 jm, (2)

= ~ (s # 1 v lJ In) Zu Y7 (f2). (2.9) #v

where Z, denotes the spin wave function and Y~ is the spherical harmonics.

T. Morii et al. : Spectroscopy of Charmed Baryons

We discuss the parity of the radial wave functions a's. The wave function 7~(r,, r2) is transformed into fll ~ ( - q , - r2)f12 by inversion. The transformation property of the spherical functions qS,m tells that the

- - + + " " , + j 2 ~ l parity of ~ and ~, ~s ( - I) a and the one oft~ +- and ~ - + i s ( - 1)h+aL It is also found that the diagonal element of a's remains unchanged under the parity transformation whereas the off-diagonal element changes its sign. Consequently the radial wave functions are written as

o I o ] a _ _ = 0 - b _ _ ' a + + = 0 - b + +

I ~ I -~ a + = ib_+ 0 , a + _ = _ i b + _ 0 '(2.10a)

- v + + v ~]+) ~V ) o ~(~+) - v - + v o &+) ~( ; ' o M - + v J i +)

o ~+~ ~ + ) - M + + v

for the parity ( - l) jz +jz- , state,

where we define

M e = (m 1 + S ( q ) ) +_ (m: + S(r2) ),

V = V ( q ) + V(r2),

FII+) O = - - + (1 _+ IJ~ + 1/2[) 1 , 6 3 r i ri

221

Lo [o - + ] ib+

a _ _ = ib"_ 0 , a + + = ib'++ 0 J"

a _ + = 0 -b_+~ , a + _ = - b ' + _ '

(2.10b)

for the parity ( - 1) h +J~ state, where b's are real functions of r, and r z. We are interested in the ground states and will confine ourselves to the case of natural parity P = ( - 1 )d, +J~ - ~ hereafter.

In the case of H,2 = 0, the equation of motion of the Qqq system (2.4) is simply expressed in terms of b,

v + + v &-) &-) o

~ - ) 0 M - + V 3(,-) 0 & - ) 3~-~ - V + + v _

(2.12)

(2.13)

(2.14a)

(2.14b)

Equation (2.11) is separated into two parts without the interaction H~E. The upper half of b's corresponds to the state P ~ = ( - ) ~ - a / 2 , P z = ( - ) J 2 - ~ / E and the lower one to P1 = (-)i~ + 1/2, Pz = (-)/~ + 1/a where Pi is the parity of the i-th quark.

The term from Ha2 which contributes to (2.11) is written as,

0 _ _ I

O-+ I

0 . _ I

0++ I

0 + + I

0+_ I

O- . t

o__ I

(2.15)

rS__ 0 0 V. S+_ 0 0 V_ 0 SD - V~ 0 0 - S~ Vo 0

o - v t s~ o o v ~ - - & o

V+_ 0 0 S++ V++ 0 0 S+_

S+_ 0 0 V++ S++ 0 0 V+_

o - sE v~ o o s ~ - v ~ o

0 VD - Se 0 0 - V~ So 0 _V__ 0 0 S+_ V+_ 0 0 S__

Here we define

I ~ b

lb -+ [b+_

]b++

~+ +

b+_

5_+1 Lb'__

i

-----E'

I

- b _ _ -

b_+

b+_

b++

b++

b + _

b_+

b _

(2.11t

s _ _ = ( O - - I s + r i o - - ) ,

s++ =<0++lS+Vl~ ,++> , s + _ - - < 0 + + l s + v l • - - > ,

sD = ( 0 - + 1 - s + v l 0 - + ) , & = ( g , - + l - s + v l o + - > ,

and

v_ _ = ( ~ - -te~ "~ vl~,- - ), v+ + = ( O + + I0",'('2 r i o + + ) ,

v+_ = < O+ + I0"a'0"2 Vl~ , - - ) , V D = ( ~ l - - + [0' 1 "tflr 2 V[[//- + >,

VE = ( 0 - + I0"1 "('2 v i e + - ).

(2.16)

(2.17)

The LS coupling scheme is employed to calculate the matrix elements (2.16) and (2.17), using the formulae given in Appendix. When the interaction H 12 is switched on, not only the two parts previously separated are mixed up but also Jl and J2 cannot be good quantum numbers, j = Jl +J2 is the only good quantum number of the total Qqq system so that all the combinations o f j l , j 2 to compose j can contribute to the eigenstate.

3. M a s s e s o f C h a r m e d B a r y o n s

3.1. H o w to Solve the Equation

We are going to solve the equation of motion by the variation method. A set of trial functions are chosen

222

in a form of the Gaussian expansion,

b+_ _+ (or/~+ _+) = ~ cijrlx ~ rl2 2 e x p ( - ~i#/1)~2-1 -- ~(12)r2~j 2,, (3.1) i j

where c~]s and ~ 's are variation parameters. By applying this set of trial functions to (2.11) and (2.15), we reduce the Breit equation into simultaneous equations linear in the parameters c~. In the present calculation, we take 9 c~j's (i,j = 1,2, 3) for each of 8 b's and 3 e's for each orbital angular momentum. Consequently we get a 72 dimensional eigenvalue equation for c's.

The solutions are chosen by the following criterions. (i) They must satisfy the virial theorem,

( V i r ) = ( 3 H ( p / 2 , 2 q ) / O 2 ) ~ . = 1 / < H ( p , q ) ) < - 1 0 -3. (3.2)

(ii) When the interaction 9H12 is gradually switched off like g = 1 ~ g = 0, the energy eigenvalue E must be continuously connected to a sum of two positive single particle energies such that E = E ~ + E 2 with El, E 2 > 0. (iii) When two eigenvalues cross over in the inter- mediate 9(0 < g < i), we have examined the continuity of several physical quantities such as (e p >, ( V i r ) and wave functions. (iv) The ground states we are mostly interested in have the following symmetries under the exchange o f q and r 2 ,

bo(rl, r2) = bij(r2, rl) with ij = - - and + + ,

bij(rl ,r2)=bij(r2, rl) with i j = - - and + + ,

b + _ ( q , r 2 ) = b + (r2, r l ) ,

~+ _ ( q , r=) = ~_ + (r~, q ) . (3.3)

The ground states, which we want to solve in the present paper, are A~ with j = 0 and Z~(= 2;*), s with j = 1, where j = j l + j 2 . The states with ( j l , j2)=

1 1 1 3 3 1 ~-),. have contribution. (>9, (> .. (>~), a

We assume that ground states are dominated by (1/2,1/2) state which consists of (Sa/z,S~/2) and (P~/>Pl/2). Orbital excited states can also be studied in the above formulation, although the higher (Jl ,J2) configurations have to be taken into account with an appropriate symmetric or antisymmetric condition instead of (3.3). In addition, much more variation parameters are needed in order to get accurate solutions.

In order to test the variation calculation, we have applied our program to the real helium atom and have obtained a result consistent with the other calculations [7].

3.2. Potent ia ls

Among various models of quarkonium the Coulomb plus linear potential gives successful description of energy levels and decay processes [2]. We have shown that the averaged masses of atomlike mesons D, F and B are well described in terms of the Coulomb plus


linear potential, which gives a good fit to quarkonia. The current quark masses were found there to give a better fit to the observed masses of D, F and B than the constituent quark masses [5].

When one considers the confining potential inside a baryon, there are two alternatives. One is the potential of ordinary type where the q - q force is related to the q-c~ force by the charge conjugation. We call this type of force as "repulsive" since

S q q = - - Sqq/2, (3.4)

for qq of color 3*. The other originates from the interactions between vacuum and quark. Several con- siderations in this respect have been made so far. It is usually believed that the interactions manifest them- selves as an electric flux confined along a string or in a bagged region [9, 10]. Here we want to make a two body potential simulate the string interactions and assume

S ~q = eSq,, (3.5)

where e is left as a positive parameter. We call it as "attractive". A typical value of e is 1/2 [10].

As is well known, the q - q Coulomb potential is

Vqq = V qo/2, (3.6)

for both cases. For the qc7 potential, we employ a potential, which

gives the best fit to quarkonia [2].

S q~(r) = r/a 2 + b, (3.7)

and

V q~(r) = - (4/3)%/r, (3.8)

with the parameters

a = 2.34 G e V - ~, b = - 0.84 GeV, a s = 0.39,

m~ = 1.84 GeV.

For light quarks the current masses are chosen as well,

mu, e = 0.01 GeV and m s = 0.20 GeV. (3.9)

3.3. Resul t s o f Numer ica l Calculat ions

Calculated masses of charmed baryons obtained finally in the "repulsive" case are

A t = 2.36 GeV (Exp. 2.260 GeV) [ 12],

<Zc) = 2.31 GeV (Exp. 2.420 GeV for Xc)[12],

f2 c = 2.31 GeV. (3.10)

< S c ) = ( Z c +22;*)/3 means the averaged mass of Z~ and 27*, which are degenerate in the present study. It is remarked that the rough values of A~ and ( S c) are obtained, while Ac, (Z~) and f2 c are almost degenerate. The degeneracy can be explained as follows. As shown in Table 1, the absolute value of < fll ) ( = </72)) decreases so much by introducing the interaction H12, while 1(/71/32 > [ increases. Consequently we get a


Table 1. Root mean square radii, expectation values of e.p, fl> fi~fl2 and the Hamiltonian for 9 = 0 and g = 1, 9 being the parameter multiplied to H~2

(GeV -~) (GeV) (f i , ) (fllfl2) (GeV)

9 = 0 6.03 0.22 - 0.353 0.119 0.36 A~

9= I 5.45 0.25 -0.060 -0.373 0.52

9 =0 6.04 0.22 -0.363 0.127 0.35

9 = 1 5.85 0.24 - 0.081 0.421 0.47

g =0 7.53 0.16 -0,.190 0.020 0.25 Q~

g = 1 5.97 0.21 0,.094 -0.452 0.47

spec t rum which is a lmos t independen t of the quark mass, because the mass term (f l~ml +f12m2) is u n i m p o r t a n t relat ive to the interact ion.

In the case of the "a t t rac t ive" confining potent ia l , we ob ta in

A c = 2.29 GeV,

( s ) = 2.28 GeV,

f2~ = 2.37 GeV, (3.1 1)

for e = 1/2. F igure 2 shows e dependence of those mass spectra. As e increases, masses of A~ and Zc increase sharp ly while f2r decreases slowly. I t is no tab le tha t the mass of ~2~ becomes smal la r than A~ and S~ in some region of e. It is, however, difficult to adjust the hyperf ine spl i t t ing of A~ and Sc a l though their average mass has a r easonab le value. De ta i l ed discussions on this account will be given in Sec. 4.

The spl i t t ing of A~ and 22 c comes from the off- d iagona l terms of the vec tor C o u l o m b potent ia l , V+ _

j=0 J=l

t 2.6 , , , %,~-0.4

~2.4

~2.3

/ m S~ /);0, 2.1 I I i ~ O.P9 0.1 0.5 0 9 0.1 0.5

E E

Fig. 2. e dependence of masses of the baryons A~, X~ and Q~ with m,.~ = 0.01 GeV, ms = 0.2 GeV. Results for other choices of quark mass are also shown. A sharp decrease of f2~ curve in the small e (< 0.1) region as e--. 0 shows that the confining potential is switched off and the system becomes under the control of pure Coulomb potential just like a helium atom

223

and Ve in (2.15). This can simply be es t imated as

A(Z,c - Ac)~ - - ~ , l ! ) (az -- aAc), (3.12)

where a's are p robab i l i t y to cause hyperf ine spl i t t ing and calcula ted from the re levant combina t i on of b's. By using a rough es t imat ion ( 1 / r ) ~ 1 /5 .2GeV -1, azo = 0.22 and aAc = 0.17 for the a t t rac t ive poten t ia l (~ = 0.5), we get

A (S c - Ac) ~_ - 0.005 a e V , (3.13)

which is consis tent with the s t ra igh t forward calcula- t ion (3.11). The similar es t imat ions can be done for the spli t t ings of D* - D and J/t~ - t l c as,

A (D* - D) ~ - 5c% 2 -~2r 2 x 2 ( _ 4)aD ~ 0.025 GeV

(3.14)

where M = 1.84 GeV is the mass of the cha rm quark. a D ~ 0.96 and ( 1 / r 2 ) ~ (1/4.2) 2 are es t imated from a solut ion of the Di rac equa t ion of D and D* [5]. We also have

_ 4 1 A(J/O r/c) ~ - g ~ S / 4 M 2 r 3 / ' 2 x ( - 4)

0.057 GeV. (3.15)

where we use ( 1 / r 3) ~ (1/2.0) 3 [-5]. These results are qual i ta t ively insensitive to the choice of parameters . The p rob lem of small spl i t t ing of A~ and S~ is o r ig ina ted from how to tune the other hyperf ine and fine spli t t ings in quarkonia .

4. Discussions

This sect ion is devoted to discuss some general p ro- pert ies character is t ic of the relat ivist ic theory of qua rk in teract ions and the Qqq system.

4.1. Properties of One-body Dirac Equation

Let us consider the one body Di rac equa t ion with mass m in scalar and vector potent ia ls , S(r) and V(r),

(E - m - S(r) - V(r) ) f = r (E + m + S(r) - V(r))g = ~ - p f , (4.1)

where p and E are the m o m e n t u m and energy of the qua rk q. The confining potent ia ls S(r) and V(r) are considered to be infinity at infinite distance, so tha t equa t ion (4.1) is wri t ten as

[-(E - V) 2 - (m + S ) 2 - pZ] f

= [ , i ~ . v ( s - V ) ] ( E + m + S - V ) - ~ . p f . (4.2)

The r ight hand side of(4.2) is negligible at r -~ 0o. Then p is real if ]V(r)[ > IS(r)[, which means that a quark can exist a t infinity. O n the other hand, [ V(r)[ < [S(r)[ gives imaginary p, which satisfies a necessary condi t ion of quark confinement. In the case of pure scalar

224

potential, confinement is possible irrespective of the sign of the potential. Pure vector potential, however, cannot confine quarks in a finite region.

We consider the relation between attractive and repulsive potentials. The Dirac equation of a massless particle with an "attractive" scalar potential is,

(iT"p + S(r))t) = 0, (4.3)

and that with a "repulsive" one is,

(i~'p - S(r))O = 0. (4.4)

By applying 78 transformation to (4.4), we have

( - i ~ . p - s ( r ) ) ~ = o , ( 4 .5 )

with tk =YsO. This means that (4.4) has the same energy eigenvalues as (4.3).

For a quark with mass, S(r) in (4.3) is replaced by (m + S(r)) and those in (4.4) and (4.5) by ( - m + S(r)). Therefore the energy eigenvalues of the repulsive potential are the same with those of attractive potential with negative quark mass. Since it is impossible to construct the nonrelativistic kinetic energy with negative mass, the Dirac equation with the "repulsive" potential does not have any intuitive nonrelativistic limit.

By separating the angular variables, we get the radial part of the Dirac equation with a scalar and a vector potential, as

(E - V(r))w = - P2 idw/dr

+ &(k/r)w + p3(m + S(r))w (4.6)

w th

Here the angular momentum parameter k is defined as

k= - (1+ 1)= -0 '+ �89 f o r / = j - � 8 9 k = l = j + 1 f o r l = j + � 8 9 (4.7)

The following symmetry property is obtained by multiplying Pz to (4.6),

E(k, m + S(r), V(r)) = E ( - k, - (m + S(r)), V(r)) (4.8)

where energy eigenvalue E is considered as a function of k,m + S(r) and V(r). We conclude from (4.8) that when bound states exist for an attractive scalar potential S(r) there exist the corresponding positive energy bound states for the repulsive scalar potential - S(r). It should also be noted that if s~/2 is the ground state

T a b l e 2. Q u a r k mass dependence of var ious expectat ion values

T. Morii et al. : Spectroscopy of C h a r m e d Baryons

and Pa/2 is the first excited state for S(r),p~/2 is the ground state and s~/2 is the first excited state for - S(r). Consequently Pl/2 state is found to become important in the repulsive scalar potential.

4.2. Properties of Qqq System

We study parameter dependence of several physical quantities and try to understand the qualitative fea- tures of Qqq system.

Consider the repulsive confining potential first. Figure 3 shows quark mass dependence of energy eigenvalues. A remarkable thing is that the eigenvalues of A c and Z c (Oc) initially decrease and then turn to increase when the light quark mass increases. This abnormal phenomenon is explained as follows. When the quark is light, the system is extremely relativistic and the repulsive force mixes the Pl/2 state with a great deal as stated in Sect. 4.1. This means that ( f l ) happens to be negative and the mass term ( t im) becomes a decreasing function of m. (See Table 2.) Anyway it is remarkable, in the repulsive scalar potential, to notice that there is a good chance that f2c is almost degenerate to or lighter than S~.

2.4 > (D

:E 2 .3 .

~2.2/ 0.0

~ 1 1 1 1 1 ,

~c~c

i I i I i [ i 0.1 0.2 0.3 0.4 m (GeV)

Fig. 3. Q u a r k mass dependence of the masses of A c (j = 0) and -rc/~? ~ (j = 1) for the ' repulsive ' conf inement potent ia l

for the "repulsive" confining potential

j m ~ <~1"P1> </~1> <&&> </4> <s1> <vl> <&2> <v12> (GeV) (GeV- 1) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV)

0 0.01 5.45 0.256 - 0.060 - 0.374 0.517 - 0.025 - 0.069 0.209 - 0.014 0.3 7.57 0.196 0.199 - 0.447 0.560 - 0.033 - 0.061 0.274 - 0.038

1 0.01 5.85 0.255 - 0 . 0 8 7 - 0 . 4 1 7 0.466 - 0 . 0 2 7 - 0 . 0 6 1 0.164 - 0 . 0 3 0 0.3 6.33 0.179 0.159 - 0.433 0.492 - 0.014 - 0.059 0.210 - 0.025

T. Mori i et al. : Spect roscopy of C h a r m e d Baryons

/A 2.5 / !

Eclat//!/

2.4

m

23 =_.,_r /

E = 0 . 9

- - - - - - E = 0 . 5

2 .2 ~ I ~ I ~ I I 0.0 0.1 0.2 0.3 0.4

m ( G e V )

Fig. 4. Q u a r k mass dependence of the masses of A~ and Z j r 2 c for the 'a t t ract ive ' conf inement potent ia l . The solid line is for e = 0.9 and the dashed line for e = 0.5

In the case of the attractive potential, the quark mass dependence is shown in Fig. 4. We see the curves similar to those of repulsive case and again have an abnormal relation between quark mass and eigenvalue. The analysis of Table 3, however, shows that the mechanism underlying the attractive potential case is entirely different from those of the repulsive case. When the light quark mass is increased in the strong attractive potential, the relativistic system moves

225

toward the nonrelativistic system. In the extreme relativistic system, the heavier the quark mass, the more the quark orbit is crushed. We have a decrease of (fllf12S12) stronger than increases of the other terms in the Hamiltonian. (See Fig. 5.) Consequently,

0.6

0.5

A

>~ o.,~ r

o.3 : 3

= 0.2 o

~ 0 . 1 x

l.U

0.0

- 0 . I

. J

Vl

6

( .9

to 5

" t3

m 4 :E I :E

T , I , I , T 0.0 0.1 0.2 0.3

m (GeV)

Fig. 5. Dependence of var ious expec ta t ion values on the q u a r k mass for Z~/f2 c with the 'a t t ract ive ' conf inement potent ia l , e = 0.9

Table 3. Dependence of var ious expec ta t ion values on the q u a r k mass m and the coup t ing p a r a m e t e r E for the "a t t rac t ive" conf ining potent ia l

j m e ~ (cq p~) (]~1) (/~t f ie ) ( H ) ( S ~ ) ( V 1 ) ( $ 1 2 ) ( V 1 ; ) (GeV) (GeV- ~) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV)

0.01 0.1 8.53 0.152 0.390 0.476 0.291 0.014 - 0 . 0 4 2 0.059 - 0 . 0 2 3 0.5 5.72 0.214 - 0 , 1 1 7 0.520 0.453 - 0 . 0 2 6 - 0 . 0 6 1 0.218 - 0,015 0.9 5.56 0.193 - 0.389 0.660 0.597 - 0.042 - 0.060 0,433 - 0.0l 1

0.3 0.1 5.37 0,185 0.758 0,603 0.664 0,000 - 0.070 0.016 - 0.038 0.5 3.95 0.330 0.471 0.315 0.596 - 0 . 0 5 9 - 0 . 0 9 1 0.014 - 0 . 0 6 1 0.9 3.97 0.345 0.195 0.264 0.542 - 0.063 - 0.087 0.097 - 0.062

0.01 0.1 8,57 0.153 0.436 0.454 0.297 0.016 - 0.042 0.058 - 0.026 0.5 5.62 0.242 0.025 0,459 0.440 0.025 - 0.061 0.163 - 0,036 0.9 5.48 0.218 - 0.283 0.609 0.537 - 0.027 - 0,064 0.321 - 0.033

0.3 0.1 5.43 0.5 3.97 0.9 3.89

0.188 0.752 0.583 0,669 0,000 - 0.069 0.023 - 0.044 0,332 0.481 0.302 0.620 - 0.061 - 0,088 0.022 - 0.055 0.357 0.271 0.212 0.553 - 0.074 - 0.089 0.055 - 0.053

226

3 . 0 [-[ , , , , i , , ,

2.2 , l , , I , i ,

0.0 0,5

o 6

2

1.0

4c I I I I I I I [

0.5 1.0

- b ( G e V )

Fig. 6. b dependence of masses of A~ and Z~ for the 'attractive' confinement potential, e = 0.5. Root mean square radii are also shown

we have a decrease of the eigenvalue until the system becomes nonrelativistic.

As for the hyperfine splitting of A~ and Z~, any change of s and quark masses cannot explain the experimental values�9 (See Fig. 2) The change of b shown in Fig. 6 cannot give a successful result. We think that it is important to make the Coulomb interaction stronger. It is desirable to obtain the best potential parameters by taking account of the fine and hyperfine splittings in the quarkonia [13], while the parameters in the present calculation gives the best fit to the average levels of quarkonia.

The recoil effect of the heavy quark is neglected in the present investigation, which may not be justified in (css) system and in the case of baryons with some other combination of flavor. We, however, expect that it does not change the above results qualitatively because the root-mean-square radii are not so small.

5. Conc lud ing R e m a r k s

In this paper we have discussed atomlike charmed baryons in a unified way together with charmonium and atomlike charmed mesons in terms of the Coulomb plus linear potential. Our fundamental equation for atomlike baryons is essentially the Breit equation of helium atom. It must be noted that a Qqq bound state exists independent of the sign of scalar confining potential in the relativistic theory. Equation of motion is solved by the variation method, and masses of A~, Z~ and D~ are calculated.

In conclusion, we would like to emphasize that (1) We have developed a method to deal with atomlike baryons in the relativistic theory together with atomlike mesons and nonrelativistic quarkonia.


(2) The Coulomb plus scalar linear potential with the parameters introduced by Eichten et al. [2] is capable to give reasonable values of charmed baryon masses. (3) The calculated masses of Qqq show no essential difference between the repulsive and the attractive confining potential. (4) We cannot obtain the hyperfine splitting large enough to interpret the experimental data. (5) There is a possibility that (2 c be light, with mass comparable to or lighter than A c or Zc. The attractive potential does not exclude this possibility.

A p p e n d i x

We employ the LS coupling scheme to calculate the matrix elements of (2.16) and (2.17), that is,

j = S + L

with

S : s I --~s 2

and

L = I 1 + l 2.

The angular part ~b • + defined in (2.8) is expressed in terms of the LS coupling base ](s 1 =!sz, 2=�89 S, (ll, I2)L:j) as

~+m +- =- ~l jm(lljl, Izj2)

I s1 11 J l ] = ~ sl 12 J2 I(sl@S,(l l lz)L:J}, (a l )

S,LLS L J

where the coefficient is written in terms of the S U(2) 9j symbol as [11]

l~ 1 = , J ( z A + 1)(2j2 + 1)(2s + lt(2L + l) L

�9 12 J2

L j

Using the expansion (A1), we obtain

( ~j,~(/'lJ~,/'2J2)1 -t- S + VI ~jm(Iljl,/2J2)}

tljl = 2 �89 12 Jz 1'2 ( ( l i O L l + S + V l ( l a l 2 ) L }

L j L

for the matrix elements in (2.16) and

( tPim(l'ljl, l'2j2)[0" 1 "O" 2 V lt~jm(lajl, 12J2)}

~- ll Jl ~ ll Jl = 2 21 12 Ii =

s,L L L

(A3)


�9 (4S - 3) (( l ] l'2)L I Vl(l 112)L ) (A4)

for those in (2.17). In the actual calculations of Sect. 3, we restrict

ourselves to Jl =J2 = �89 and can explicitly write down from (AI),

r - = I s = 0, (t~ = 0, l~ = 0 )L = 0 1 ,

r = - I S = 1,( l i = l , G = 0 ) L = 1 ) ,

~ - + = I S = 1,(/1 : O , l 2 = l ) L = 1 ) ,

r ~/~ = I s = 1,(lt = 1,12 = 1 ) C = l )

-'~ IS = 0 , ( l l = 1, l 2 : I ) L = 0 1 (A5)

for j = 0 and

r = I S = 1,(11 =O, lz =O)L=O),

r ; = [ S = O , ( l l = l , 1 2 = O ) L = l )

+ I S = 1,(ll = 1,12 = 0 ) L : : 11,

~-+ ,j~ = _ I S = O , ( l l = O , 1 2 = l ) L = l )

+ I S = l , ( l l = O , 1 2 = l ) L = l ) ,

= - ] S = 1,(1~ = 1,1 e = 1)L = 0 1

+ [S = O,(l t = 1,12 = 1 ) L = 1)

+ [ S = 1,(l t = 1,l 2 = 1 ) L = 2 1 (A6)

for j = 1 states. One can calculate the orbi tal matr ix elements

((l'll'2)LlOJ(lll:)L) in (A3) and (A4) by introducing the relative coordinate r = r 1 - r 2 and the center of mass coordinate R = (r~ + r21(2. Using (2.8), (2.9) and (3. l ), we have

b_+ + I(11 I2)L ) = Z clsr~' r~ ~e ~?~4-~2~r~ i j

"[ Yh (;1) x Yl2 (;2)3 tL), (A7)

where [ ](LI denotes a tensor p roduc t with the rank L. A well known formula,

rll r,12 ,~ "tq(L)

/2lt ~1(212 = a~2(211 + 1)(212 + 1 ) ~ 2 / 1 a ] \ 2 2 2 ]

227

.(_l)Z~ z~ z~ ~ + < R 1 "~'1--'~2

�9 Y~ 0t - & 0 ;~0rd~0) (2~01~ - 2 2 0 1 4 0 ) did2

2 2 l z - 2 2 [Yd,(/~) • Yd~(t~')] ~L) (A8)

dl d 2

is useful and especially in the case of 11, l 2 = 0 or 1 we have

q l l t = l , l z = O : L = l ) = R + , (A9)

qr21/1 = 1,l 2 = I : L )

3 = ~ - n [R x R - �88 x r + 2X(r x R - R x r)](L). (A10)

After a s t ra ightforward calculation, the matr ix elements ((l' 1 l;)LlO[(l112)L) can be given analytically�9

Acknowledgements. One of us (T.M.) thanks Professor K. Schilling and the members of the university of Wuppertal for their kind hospitality and many fruitful discussions. He also thank Prof. P. Kroll for critical reading of the manuscript. Numerical calculation has been done at the computer centers of Osaka University and Kobe University, which is financially supported in part by the Institute for Nuclear Study, University of Tokyo.

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Spectroscopy of charmed baryons