P H Y S I C A L R E V I E W D V O L U M E 1 5 , N U M B E R 11 1 J 1 J N E 1 9 7 7

Electromagnetic mass differences of charmed baryons in the charmed-quark model

Seiji Ono*' III. Physikalisches Instltut, Technlsche Hochschule .4achen, Auchen, West-Germarry

(Rece~vzd 30 August 1976)

Electromagnetic mass differences of charmed baryons are calculated using the harmonic-osciallator charmed- quark model.

After the discovery of charmed mesons D and D*,lv2 charmed baryons a r e expectei to be found. In this note we study electromagnetic m a s s differ- ences of charmed baryons in the charmed-quark model. In our previous it w a s shown that the harmonic-oscillator quark model with R2 (wave- function radius) =2.75 G e T 2 can be used consis- tently and with remarkable s u c c e s s t o explain elec- tromagnetic m a s s differences of baryons, ampli- tudes of the p r o c e s s e s yiV- N* - Nn, and other p roper t i es of baryons. In a n analogous way we stu- died proper t i es sf uncharmed mesons4 alld charmed meson^.^

Following Refs. 3-5 we make the following a s - sumptions in studying charmed baryons:

(a) The electromagnetic m a s s differences of charmed baryons a r e caused ( i ) by the m a s s dif- ference (Am,) between the p quark (up quark) and the ~z quark (down quark), (ii) by the Coulomb force among quarks, and (iii) by the magnetic hyperfine interaction.

(b) The value of the gyromagnetic rat io of a l l quarks i s unity, i.e., the magnetic moment

= charge/2 x mass . F r o m magnetic moments of un- charmed baryons one gets pq = pn = 2.793 e/2 n z , and m, ( m a s s of uncharmed quarks) -336 MeV.

(c) F r o m the m a s s s p e c t r u ~ n of mesons ( i j l , Do, p,

T , etc.) we guess MZ, =4mq-582,. We u s e harmonic-oscillator wave functions f o r

s - s t a t e cqq sys tems and f o r s - s ta te ccq sys tems (Y i s the uncharmed quark):

In Eq. (1) [ ~ q . (2)] and F2 a r e coordinates of two uncharmed (charmed) quarks and F3 i s a coor- dinate of a charmed (uncharmed) quark. Since we know that SU(3)-symmetric wave functions work very ~ e 1 1 , ~ - ~ we a s s u m e the s a m e R, ( o r Y,) f o r a l l uricharmed quarks.

Taking into account the l a r g e m a s s difference be- tween m, and m, one ge t s f o r the cqq system.

In the s a m e way we get f o r the Ycc sys tem

E L E C T R O M A G N E T I C M A S S D I F F E R E N C E S O F C H A R M E D . . . 3493

We take

R: = 2.75 G~v- ' , Y, = +R, = $R,, = $Y,,,

Ame=-1.83 MeV. (10)

The value Ame = -1.83 MeV for baryons i s different f r o m that f o r b o ~ o n s . " ~ However, we have shown in Ref. 3 that t h i s choice of p a r a m e t e r s Am, = -1.83 MeV and R2 = 2.75 G e T 2 i s a very successful one in that it can explain the proper t i es of baryons

(electromagnetic m a s s difference, photoproduction and electroproduction processes) .

Now we must construct an SU(8) wave function f o r charmed baryons. We a s s u m e that each wave function must be totally symmetr ic under a n ex- change of any two quarks. However, t h e spat ial s - s t a t e wave function of t h e charmed baryon i s not symmetr ic because of the l a r g e m a s s difference between mQ and m, and because of the difference be- tween R, and R, ( o r Y, and y,). Therefore, in o r d e r to construct t h e symmetr ic total wave function we must u s e the broken SU(8) wave function. A s a n example we show the symmetr ic s - s t a t e wave func- tion f o r C y (we u s e the notation introduced by Gaillard, Lee, and Rosner7 f o r charmed baryons):

where +(I , 3, 2) can b e obtained by the permutation 1/2 e2

of coordinates Y,, Y,, r, in Eq. (1). Wave functions Am=Amea+(+) -c

RO of other s ta tes a r e constructed in a s imi la r way.

T h e symmetr ic s - s t a t e wave function belongs to - ! - T P ~ 1 1 (2T)3/2 vH9 (13) the representat ion 120 =(20, 4) +(20', 2) in SU(8),

which corresponds to 56 = (10,4) +(a, 2) in SU(6). where We l is t our resu l t s of electromagnetic m a s s differ- ences of charmed baryons in Table I, where we de- Ro = R,, f o r the cqq system,

fine Ro = Y,, f o r the ccq system,

a = Rca/R4, b = yc,/yc, bD = 2.793 e/2 nzp . d = (Rcq/Rq)3, f = ( ~ , , / y ~ ) ~ , (12) F r o m Table I we can derive the following sum

ru les : Y =m,/m, -$ .

c:+ - c:o=s*+ -S*O, The electromagnetic m a s s difference can be cal-

culated according t o t h e following formula: C : - C y = S + -So,

TABLE I. Electromagnetic mass differences of charmed baryons. We follow the notation introduced by Gaillard et al . , Ref. 7 , for charmed baryons. Parameters a , d , and v a r e de- fined in Eq. (12 ) .

(sc (4), 2S+ I) Mass difference

01 C H (MeV)

3494 S E I J I 0 Y 0 15 -.

In Eq. (13) the l as t t e r m which comes f rom the magnetic hyperfine interaction i s not negligibly small but gives a (10-20)% contribution. H , in Eq. (14) c o m e s only f r o m the magnetic hyperfine inter- action and we can determine the s t rength of the magnetic hyperfine interaction by measuring the m a s s differences in Eq. (14).

F r o m Table I we notice that a s a rule in the s a m e

isospin multiplet the l a r g e r the charmed baryon's charge, the heavier it becomes. These resu l t s a>re opposite t o that of the uncharmed baryons, in which the m a s s difference between the n quark (down quark) and the p quark (up quark) plays a n impor- tant role . This opposite resul t in the charmed bar- yon i s caused by the la rge Coulomb interaction be- tween the charmed quark with the r a t h e r l a r g e pos- itive charge and uncharmed quarks. T h i s resu l t i s a l s o dependent on the smal l value of Ihnz, 1. If we use Am, = -7.06 MeV a s i s used f o r mesons, the conclusion is reversed.

The author is grateful t o Dr. D. G. Sutherland f o r discussions. He would l ike to thank the Alex- ander von Humboldt Foundation f o r financial sup- port.

*Work supported in par t by the Alexander von Humbolt Foundation.

?Present address: Department of Physics, University of California, Davis, California 95616.

'v. Liith, worB presented at Neutrino Conference, Aachen, W. Germany, 1976 (unpublished).

'A. De Rfijula, work presented at Neutrino Conference, Aachen, W. Germany, 1976 (unpublished).

3 ~ . Ono, Nucl. Phys. w, 522 (1976). %. Ono (unpublished). ' s . Ono, Phys. Rev. Lett. 37, 655 (1976). 9. Minamikawa e t a1 ., Prog. Theor. Phys. Suppl.

37-35, 56 (1976). - 7 ~ . K . Gaillard, B. W. Lee, and J. L. Rosner, Rev.

Mod. Phys. 47, 227 (1975).