L:ETTERE AL NUOVO CIMElqTO VOL. 20, N. 14 3 Dicembre 1977

1+ Baryons Magnetic Moment Sum Rule of 2

in SUa for u-, d- and c-Quark Sector.

S. IwAo

Department o] Physics, College o] Liberal Arts, Kallazawa Utziversity - Kaltazawa 920

(ricevuto il 26 Settembre 1977)

Recently, the present author(~) has pointed out that the SUa symmetry in u-, d- and c-quark sector will be useful to be investigated, tie has shown that the mass formulae for nonet vector mesons for linear ones and the quark-line rule of ()Ku~o (~), Zw~m (~) and IIZUKA (4) are satisfied approximately for the J/~ and ~ production ratio observed at 150 and 225 GeV/c incident proton and pions on nuclear targets (5).

The purpose of this paper is to discuss the magnetic form factors of �89 baryons in u-, d- and c-quark sector. In what follows the magnetic moments of �89 baryons in that sector are expressed in terms of those of proton and neutron.

In our assumed SU~ symmetry in u-, d- and c-quark sector, we have a charm quantum number in place of strangeness. Thus the formulation of this theory proceeds completely parallel to that of the octet version of Gell-Mann and Ne'eman. We can use exactly the same generators of the conventional SUa group in our formulation of Lie algebra.

The electric charge in our S U 3 sector may be given by

1 1 1 (1) Q = ~ - ~ + ~

in units of proton charge, ttere 23 and ).s are familiar diagonal matrices of the con- ventional S U s group and ~ is a unit matrix multiplied by a constant factor, viz., ), = Vr-~l.

(') S. IWAO: Mass ]ormula ]or mesons and quark-line rule in u-, d- and e-quark sector, I IP ICK-042 , p r e p r l n t (Augus t 1977). (i) S. OKUBO: Phys. Le~t., 5, 165 (1963); .A review of quark-~i~te rule, UR-616, COO-3065-167, p r e p r i n t ( June 1977). ( ') G. ZwE[a : C E R N Repor t TH-401 a n d TH-40% unpub l i shed (1964). (4) J . IIZITKA: Prog. Theor. Phys. Suppl., 37-38, 21 (1966). (J) K . J . ANDERSON, G. G. HENRY, K. T. McDONALD, J. E. PILOHER, E. I . ROSENDERO, J. G. BRANSO.'.', G. H. SA~'DERS, A. J. S. S.~lI~ and J. J . THALER: Phys. Rev. Lett., 36, 237 (1976); 37, 799 (1976); J . G. BRANSON, G. H. SANDERS, A. J. S. SMITII, J . J . THALER, K. J. ANI)EP~ON, G. G. HENRY, K. T. McDoNALD, J. E. PILCHER and R. J. ROSENBERG: Phys. Rev. I~.tt., 38, 1334 (1977).

522

MAGNETIC MOMENT SUM I~ULE OF �89 BARYONS ETC. 523

The wave funct ions of octet ba ryons in the no ta t ions of GAILLhRD, :LEE and R0SSER (s) m a y be represented as

(2) B = 1 1 +

c", ~ ~ + ~ -~Co ~ .

2 _ x . + x . ++ ~ co +

The magnet ic par ts of pho ton -ba ryon in teract ions under considerat ion are defined by (7)

(3) iF2(t ) Tr (Bau~qv[Q, B]) + iD2(t ) Tr (BauvqV{Q, B}) ,

where F2(t ) and D2(t ) are form factors for S U 3 an t i symmet r i c and symmetr ic parts , respectively, auv = (i/2)[7u , yv], qv is the fou r -momen tum t ransfer and t = qL The calculat ion proceeds s t ra ightforwardly. One may finds the magnet ic moments of baryons in terms of these form factors. The results are

4 (4.1) /%(t) = ] D2(t) ,

1 (4.2) /~n(t) = - - F~(t) + -~ D2(t ) ,

4.3) /~c+(t) = D2(t) ,

1 (4.4) #c++(t) = F2(t ) - ] -~D2( t ) ,

1 (4.5) #c+(t) = ~D2(t ) ,

1 (4.6) /~c~(t) = - - F2(t ) + -~ D~(t) ,

(4.7) #c,+oo +(t) = ~3_ D2(t),

1 (4.8) #x++(t) = F~(t) § -~D~(t)

and

4 (4.9) ~x~ (t) = -zD2(t ) .

(6) ~I. K. GAILLARD, B. W. LEE and J. L. ]:tOSNER: Rev. Mod. Phys., 47, 277 (1975). (7) S. GASIOROWICZ: Elementary Particle Physics (New York, N.Y., 1966).

524 s. :wAo

Here, eq. (4.7) represents C~-C o t r ans ien t moment . The Marshak-0kubo-Sudarshan- like sum rule in SU2 (8) is satisfied by the above der ivat ion as i t should be, viz.,

(5) 2~,ct(t) = t%+ 4t) + t,r

The magnet ic momen t s of charmed baryons are expressed immedia te ly in te rms of those of pro ton and neut ron .

3 (6.1) /%o+(t) = -~/~p(t),

1 (6.2) t',~l +(t) -- ~ t,o(t) - - t , .(t),

1 (6.3) t%+(t) =-~ t tv( t ) ,

(6.4) t~r = ~ . ( t ) ,

(6.5) t,o~o+(t) = -2-- t, d t ) ,

1 (6.6) t,~++(t) = ~ t, d t ) - t~,,(t)

and

(6.6) /.tx~(t) =/.tD(t).

The largo mass difference between nonst rango and charmed baryons will require the large symmet ry breaking and the sum rules discussed above may largely be violated. We th ink , however, tha t these relat ions should be tested exper imental ly , a l though it is difficult to perform the exper iment due to the smallness of the l ifetime of charmed baryons , (10 - a 2 - 10 -a3) s.

(~) R..-'~IAI~IIAK, S. OKUBO and E. C. G. SUDARSHAN': Phys. Rev., 106,599 (1957); S. Iwxo: Prog. Theor. Phys., 55, 943 (1976).