P H Y S I C A L R E V I E W D V O L U M E 13, N U M B E R 1 1 1 J U N E 1976

Magnetic moments of charmed baryons. I

A. L. Choudhury Department of Mathematics, Elizabeth City State University, Elizabeth City, North Carolina 27909

V. Joshi* Department of Physics, Old Dominion University, Norfolk, Virginia 23508

(Received 15 July 1975)

The magnetic moments of all baryons belonging to the totally symmetric 20representation, the mixed symmetry 20' representation, and the totally antisymmetric 4 representation have been compared under U(4) symmetry. For the 20' representation the usual results of SU(3) symmetry remain unchanged except the relation p(Ao) = - p(Z0) This result is not valid in U(4). The magnetic moments of all charmed particles have been expressed in terms of the moments of proton, neutron, and A particles in the case of U(4) symmetry.

I. INTRODUCTION

The recent discovery of the $ particles' has lead to renewed interest in introducing a new quark c with charmed quantum number 1, in addition to the three quarks: a pair u and d, an isotopic-spin doublet, and s, an isotopic-spin singlet with charm assignment zero. The z) particle i s as- sumed to be the quark-antiquark, cF combination and a vector meson with associated quantum num- bers, J P G = I--. The introduction of c as an ad- ditional quark, with isotopic spin I = 0, and hyper- charge Y = - $, has lead to renewed interest in extending the SU(3) symmetry classification to U(4) o r SU(4), a theory in which four quarks have to play the fundamental role in the reproduction of the other particles that appear in nature.

The SU(4) group2 for the classification of the fundamental particles has been suggested earl ier by Glashow, Iliopoulos, and Maiani, to eliminate the strangeness nonconserving neutral currents. In this model, a large number of new charmed pa r - ticles show up, in addition to the usual noncharmed particles. In strong interaction, the charm quan- tum number i s supposed to be strictly conserved. A s soon as the discovery of the $ particle was announced, a large number of explanations on the nature of the particle were p r o p o ~ e d . ~ Based on the SU(4) model Borchardt, Mathur, and Okubo5 proposed that the z) particle i s cE state and sug- gested an extension of the Gel1 -Mann-Nishijima formula as follows:

The q-~antum number C in (1) stands for the charm of the particles, which is assumed to be zero for all particles known until now. In the quark model, for the u, d, and s quarks we as - sume C = 0, and we assume that the C value of the fourth quark i s 1. The assignment of the other

quantum numbers is to be done according to the fractional scheme of Glashow, Iliopoulos, and T ~ T a i a n i . ~ ~ ~ We also assume here that all the bary- ons a re obtained by the combination of qqq, where q i s a quark.

A large number of experimental and theoretical works since then have been carried out. Assum- ing that the U(4) classification has a promising future, we have decided to calculate the magnetic moments of the baryons in light of ~ ( 4 ) symmetry.

The calculation of the magnetic moments for the SU(3) baryon octets has been done by Glashow and C01eman.~ Okubo8 has also shown that the same results can be reproduced from a very general consideration of the transformation property of the charge operator. Afterwards, Okubog indica- ted how the higher-order correction to the mo- ments, assuming a more complex magnetic -mo- ment operator, could be obtained. A more gener- a l fornlulation can also be found in the paper by Rosen, lo who gave a closed form of the magnetic - moment operator in terms of the U spin and the charge of the particles,

All these calculations have been very much im- proved by extending the symmetry group to SU(6), with the inclusion of intrinsic spin of the particles. Bgg, Lee, and Pais" assumed that the baryon oc- tets and the decouplets belong to the 56-dimension- a1 representation of SU(6) and the magnetic-mo- ment operator transforms like a (8 ,3) number of the 35 representation. Their r e s u h g a v e the ratio p ( p ) / ~ ( n ) = -2 . Thirring12 also obtained the same results straightforwardly from the quark model by vectorially adding the magnetic moments of the quarks which constitute the particle concerned.

In a sequence of two papers we would like f i rs t to extend the calculations of SU(3) to that of U(4). Then, enlarging the group to U(8) o r SU(8), we would like to find the magnetic moments of the charmed particles. In this paper, we restrict our-

3116 A . L . C H O U D H U R Y A N D V . J O S H 1 13 -

selves to the smal ler group U(4). In Sec. 11, we discuss the U(4) classification and introduce the conventional nomenclatures. We also write down the corresponding baryon s ta tes with the help of a tensor B,,,.13 In Sec. 111, we construct very general currents , l4 which appear in a magnetic - moment tensor. In Sec. IV, we obtain the mag- netic moments, assuming U(4) symmetry. In Sec. V, we discuss the general aspects of the results obtained.

11. U(4) CLASSIFICATION OF BARYONS

The baryons a r e to be obtained by the combina- tion of three quarks qqq. In t e rms of the irreduc- ible representation of U(4), we know

4 8 4 @ 4 = 20+2X 201+4, - - - - - (2)

where 20 i s the dimension of the completely sym- metricrepresentat ion, 20' i s the dimension of the representation of mixedsymmetry, and 4 i s the dimension of the totally antisymmetric representa-

tion. On the other hand, if we express the 20 multi-

plet in t e rms of SU(3)@ U, indices, we 6 d

20= (10,O) + (6, 1) + (3,2)+ (1, 3), - (3

where ( m , n) indicates the m -dimensional SU(3) multiplet with a charm quantum number n. Sim- ilarly for 201, we get -

20f= (8,O) + ( 6 , l ) + (?,I) + (3, 2) - (4) In the equation (4), ( 3 , l ) represents the contra- gredient triplet s tate with charm quantum number 1. The ~nult iplet 4 i s given by -

4=(3 , l )+ ( l ,O) . (5)

The baryon wave function for the 20 representation can be expressed in t e rms of the m 4 ) indices by a totally symmetric tensor B:~, which is normal- ized to the number of part icles in the multiplet. The tensor B:~, can be expressed in te rms of the SU(3) multiplets a s follows:

In the above expression d i j k represents the decouplets of SU(3) and the indices i, j, and k only run from 1 through 3, whereas p , v, and p run from 1 through 4, SS(') represents the sextuplet charm-1 baryons and i s totally symmetric in i and j. The states T:(~) a r e the SU(3) triplets, with charm quantum number 2. The state s : ( ~ ) i s the (1,3) s tate of the equation (3),15

Similarly, we find for 20'

In the above expression, whenever we have put two indices within the curly brackets the t e rms a r e antisymmetric with respect to those indices. Here again the Greek indices run from 1 through 4, whereas the Latin indices run from 1 through 3. The symbols lV[iklj, sf:), ~i( : ; ] , and Ti''' stand f o r a baryon octet, a sextuplet with charm 1, a con- tragredient triplet with charm 1, and a tr iplet with charm 2, respectively. The baryon octet can be expressed in t e rms of the well-known symbol Nj by the formula

where ~ [ i j j stands for the charm-1 contragredient triplet and So' i s the singlet with charm zero.

Here we would exclusively follow the identifica- tion of the particles a s introduced by Gaillard, Lee, and Rosner.' Thus, for the sextuplet belong- ing to the 20 representation, we define -

Here ei,, is the Levi-Civita totally antisymmetric st:') = T*O. tensor in three dimensions.

Finally, for the baryon - multi~let, we can write For the triplet members, (3, 2), of the 20 r ep re - - 1 sentation, we identify the particles a s

BtILvPl = ~(6:6;64p + 6;6/6: - 6:6;6:) TI:;; T * ( ~ ) = x * + + 77;(2)=~$+, and T,*(~)=x,*+. I u 9 (9b)

1 The new baryons belonging to the 20' represen- + ~ 6 : 6 ~ 6 ~ ~ , ~ ~ ~ ~ ' tation, the members of the sentupleGj:) [(6, I)],

13 - M A G N E T I C M O M E N T S O F C H A R M E D BARYONS. I 3117

a r e given by the equations (9a) just by dropping the asterisks. Similarly the multiplets (3,2) of the 2Q' representation are given by the equations (9b), Z e r e we also have to drop the asterisks. For the multiplet (3, I), we write

For (?,I) belonging to 4, we go over to the par- ticles by adding a prime t o the relations in the equations (9c).

111. MOST-GENERAL CURRENTS

In calculating the magnetic moment coming from the first-order electromagnetic interaction, we need to construct the most general current tensors J: (Ref. 14) with the help of the multiplet wave functions B given by the equations (6), (7), and (8). For the 20 representation, the most general cur- rent t h a t y e can form is

where po and g: a re two arbitrary constants and (B B),, stands for the trace of the tensor defined a s follows:

The most general current J(N): that can be con- structed with the tensors B and B belonging to the 20' representation is given by -

J(N): =DL + 6:Gz, (12) where

- - - D; = B L ; ; ~ ~ B ~ ~ ' ( K P I V +a2B20* ( P V I F ~ Z O ' ( K P I V + ~ ~ R Z O * I V U I P ~ Z O ~ {KPI,~

- + b l ~ 6 ~ ? 1 V ~ ~ 0 , ~ I , + b2~$"'BfOpllK + b3B~;~JP~q; ;1V

- (PV)UBZO' + E(YLL)PBZO'

+ ~ i ~ 6 ~ F ~ ~ ~ ? % ~ p + ~ 2 ~ 2 0 1 ( Y K ) P 3 20' ( V K ~ P

(13) and Gg is an expression obtained from D: by sett- ing p = K = (Y and replacing the constants a,, b,, and ci - s by new constants a;, b( , and c; - s.

By using the symmetry relations

we can simplify Eq. (12) into a very convenient form. We should notice that Eqs. (16) and (17) a r e the outcome of Eqs. (14) and (15). We finally get

Subsequently, we use the abbreviation

to express the magnetic moments of the baryons in terms of the constants p,, p,, g,, and g,.

Similarly, for the representation 4, which is totally antisymmetric, the current tensor i s

J(A): = p l B ~ u V P ) ~ t , V p , fgl6: (B B ) ~ , (20)

where again p, and g, a r e two arbitrary constants and

(B B), = B & u Y P 1 ~ ~ , V p ~ . (21)

In the next section we use the currents in Eqs. (lo), (18), and (20) to obtain the magnetic mo- ments.

IV. MAGNETIC MOMENTS IN U(4)

A s shown by Glashow and Coleman, the mag- netic moment for-baryons can be obtained by form- ing the trace of BBQ in all possible contractions, where Q is the charge operator, a 4 x 4 matrix, and can be written as

Q",=4&6K,, (22)

where q, = q, = $ and q, = q, = - i a r e the charges of the quarks. We follow the prescription of Okubo et ~ 1 . ~ for the charge operator as given by the equation (1).

We assume that for a low-momentum case the expectation value of the magnetic -moment opera- tor is proportional to

J(X):QK,, (23)

where X stands for either D, N, o r A. We have altogether suppressed the spin part of the magnet- ic moment as a first approximation.

A. 2_0 representation

For (10,O) members of the 20 representation, we find, using Eqs. (6), (lo), a n d (23), that

p ( ~ * + + ) = Q I J . ~ + $ ~ ; , p ( ~ * + ) = + p , + $ ~ ; ,

p(~i*O)= $ g ~ , p(N*-)= -Qpo+ $ g ~ , (24)

and

For the sextuplet (6, 1)

Similarly, for the triplet (3,2)

3118 A . L . C H O U D H U R Y A N D V. J O S H 1 13 -

p(X:++) = p(&*++), p(Xz')= ~ ( x Z ' ) = p(N*'). (29)

For the singlet (1, 3) (identifying s,*(~'=R*"), we find

B. 2_0' representation

If we now go to the 20' representation and use the equations (T), (18), and (23), the magnetic mo- ments of the baryon octet (8,O) come out to be

p ( p ) = P(c*) = 4 p, - 5 p y + +got ( 3 1 4

A n ) = w ( ~ O ) = + ~ , + d ~ ~ + g g ~ , (31b) 1 1 2 p(Z")= p (C- )= - 3 P , + ~ p y + 3 g o , ( 3 1 ~ )

p(C0)= -&P, -&py +$go, (31d)

! J ( A ~ ) = $ P , + & P , + ~ ~ ~ . (31dt)

The transition moments a r e given by

If we compare these results with the U(3) r e - sul ts a s quoted by Okubo,' we find the unchanged relations a r e

d p ) = p(C+), p(ZO)= A n ) , p(Z-)= IL(C-), (3lf)

P(P)+ 1 l ( Z - ) + 4 ~ ( n ) = 6i~.(A'), (31g)

and

p(C+) + p ( ~ - ) = 2p(C0). (3 1h)

However, we should notice that

p(z0) + - AAO) (31i)

in U(4) in contrast to the result p(zO) = ~ ( A O ) in u(3).

For the charmed sextuplet (6 , l )

p(Cl+)= +P, -+I,+ $go, (324

~ ( c ; ) = F(S')=%P, - & ~ ~ + + g ~ , (32b)

and

In te rms of p(p) , p(n) , and p(Ao), we can write

and P(C:)= F ( P ) + ~ P ( ~ ) -3dA0) .

For the charmed baryon triplet (3,2), we find

AX; +) = P@;+), ( 3 3 4

!J(Xd+) = P(XJ = AP). (33b)

If we now go over to the contragredient baryon triplet (s, I) , we find

Expressing (34a) and (34b) in t e rms of proton, neutron, and A particle magnetic moments, we find

~J.(cA) = P(P) + ~ ( n ) - d h O ) (344

and

p(AO) = - 3i~.(t1.) + 4p(A0).

Owing to the degeneracy of quantum numbers of some baryon sextuplet particles with the contra- gredient triplets, we find that transition magnetic moments appear between sextuplet and tr iplet par- ticles. The results a r e

C. 4 representation

F o r the particles belonging to the totally anti- symmetric representation 4, the magnetic mo- ments f o r the contragredient t r ip le ts can be found using Eqs. (8), (20), and (23). We get for the con- tragredient charmed tr iplets

and

For the singlet (1,O) we get

V. CONCLUDING REMARKS

We have calculated the magnetic moments in the low-frequency limit in the U(4) symmetry, assum- ing that the magnetic- moment operator i s propor- tional to the charge operator. The expectation val- ue of the magnetic moment could thus be written a s in Eq. (23). This equation finally has produced the results quoted in Sec. IV. Almost a l l the re- sul ts of U(3) have been reproduced in U(4). The only different result U(4) yields is

We have also derived the magnetic moments of all new charmed baryons in t e r m s of the magnetic

M A G N E T I C M O M E N T S O F C H A R M E D B A R Y O N S . I

moments of the proton, neutron, and A' particles in the representation and in t e rms of the mag- netic moments of ,V* - s in the 20 representation in U(4). Since Q is not an operator in SU(4), the results cannot be extended in SU(4) symmetry. Moreover, we would like to mention here that our results would be valid fo r both total magnetic mo- ments and anomalous magnetic moments for ex- actly the same reason that the results obtained by Coleman and Glashow7 would be valid for both,

In the following paper we will show how the r e - sul ts of ~ ( 4 ) can be extended to the ~ ( 8 ) symmetry,

ACKNOWLEDGMENT

The authors like to express their s incerest thanks to Professor L. C. Biedenharn, Jr. of Duke University, Durham, North Carolina, fo r en- couragement and some fruitful suggestions. The f i r s t author (A.L.C.) also expresses his gratitude to the Department of Physics, University of North Carolina at Chapel Hill f o r allowing him to use the facilities of the department during the summer months. He is particularly grateful to Professor J. W, Straley, Professor E. Merzbacher, and Pro- f e s so r H. van Dam fo r helpful cooperation.

*Work partially supported by NASA, Langley, Virginia. 'J. J. Aubert et al ., Phys. Rev. Lett. 33, 1404 (1974);

J.-E. Augustin et a l ., ibid. 33, 1406 (1974); C. Bacci et a1 . , ibid. 33, 1408 (1974).

2 ~ o r SU(4) review see M. K. Gaillard, B. W. Lee, and J . L. Rosner, Rev. Mod. Phys. 47, 227 (1975).

3 ~ . L. Glashow, J. niopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970).

4 ~ . Borchardt, V. S. Mathur, and S. Okubo, Phys. Rev. Lett. 34, 38 (1975); A. S. Goldhaber and M. Goldhaber, ibid. 34, 36 (1975); J. Schwinger, ibid. 34, 37 (1975); R. Michael Barnett, ibid. 34, 4 1 (1975); T. Applequist and H. D. Poli tzer, &id. 34, 43 (1975); A. De Rfijula and S. L. Glashow, ibid. 34, 46 (1975); H. T. Nieh, T. T. Wu, and C. N. Yang; ibid. 34, 49 (1975); C. G. Callan et a1 ., ibid. 34, 52 (1975); J. J. Sakurai, ibid. 34, 56 (1975). -

5 ~ . Okubo, V. S. Mathur, and S. Borchardt, Phys. Rev. Lett. 34, 236 (1975).

b t h e r possible choices of quantum-number assignments can be found. Fo r other possible combination of the quarks to construct a baryon see D. Amati, H. Bacry, J. Nuyts, and J. Prentki, Phys. Lett. 11, 190 (1964) ; Nuovo Cimento 34, 1732 (1964). See also J. W. Moffat, Phys. Rev. D 2, 288 (1975), and V. Rabl, G. Campbell,

and K. C. I a l i , J. Math. Phys. l6, 2494 (1975). IS. Coleman and S. L. Glashow, Phys. Rev. Lett. 5, 423

(1961). %. Okubo, Prog. Theor. Phys. 21, 949 (1962). 's. Okubo, Phys. Lett. 4, 14 (1963). "s. P. Rosen, Phys. Rev. Lett. 11, 100 (1963). "M. A. B. BBg, B. W. Lee, and A. Pais , Phys. Rev.

Lett. 2, 514 (1964). 12w. Thirring, in Quantum Electrodynamics, proceed-

ings of the IV Schladming Conference on Nuclear Phys- ics , edited by P. Urban (Springer, Berlin, 1965) [Acta Phys. Austriaca Suppl. 2 (1965)], p. 205.

13see also S. Okubo, Phys. Rev. D 11, 3211 (1975). 1 4 ~ . Sakita, Phys. Rev. 136, B1756 (1964); A. Salam,

R. Delbourgo, and J. Strathdee, Proc. R. Soc. London A284, 146 (1965). -

15~c tua l ly we should indicate here that there i s a certain amount of arbitrariness in choosing the constants in front of each state. If we write (with r ea l O L , P , y , 6 )

~ 2 , " ~ = ~ B : O ~ + B B ! ~ ~ +YB; , , ,~ + 6 ~ $ , ~

and normalize B iO to 10, B 6 t o 6, etc., then we get

20 = a210 + p26 + y23 +62.

We have made the choice a = p = y = 6 = 1.