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PHYSICAL REVIEW D VOLIJME 17, NUMBER 11 1 JUNE 1978 Electromagnetic mass differences of the charmed baryons A. C. David Wright* Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2Jl (Received 6 December 1977) Born-term contributions to the electromagnetic mass splittings of the charmed and uncharmed baryons are ' estimated using form factors constructed with the help of vector-meson dominance and SU(4) symmetry. The remaining contribution to the mass splittings is assumed to be a tadpole transforming like the third component of isospin. Our results are compared with previous estimates based on the quark model. In previous papers,'*' we have obtained esti- mates for the electromagnetic mass splittings of the charmed pseudoscalar and vector mesons D and D*. In this approach, Born-term and first- intermediate-state contributions to mass split- tings are calculated using pole-dominated form factors constructed with the help of SU(4) sym- metry. The remaining contribution is assumed to be due to a tadpole3 transforming like the third component of iosopin. The predicted D mass splitting, m,+ - m,o = 5.8 MeV, is in good agree- ment with the experimental value4 of 5.1 + 0.8 MeV. This success, and the fact3 that the tadpole approach works better for the SU(3) baryons than for the SU(3) mesons, suggests that it would be where The form factors Fl(q2) and F2(q2) are normalized at q2 = 0 in terms of the charge Q and magnetic moment y of the baryon: Assuming pole dominance, a form factor can be written as worthwhile to apply the method to the problem of the charmed-baryon electromagnetic mass differ- F(q2)= C ctFi(q2), ences. Furthermore, results already obtained5-" i=p, W, 0, + in the context of the quark model vary widely, so that it would be interesting to compare the results of our approach with those of Refs. 5-11. Because of the uncertainties involved in the B,~,+B,~,+Y and B312+B312+y couplings, we shall re- strict our attention in the following to the g multi- plet. This means that we shall neglect inelastic intermediate-state contributions, and assume that mass splittings are given reasonably accurately by the sum of the Born and tadpole terms. The Born-term contribution to the electromag- where Fi($) is the contribution due to vector mes- on i. To determine the coefficients c, in (5), we impose the Okubo-Zweig-Iizuka (OZI) rule13 and SU(4) invariance for the VBB couplings. We as- sume pure F-type coupling for P,(O) and the SU(6) F/D ratio f o r G,(O) to obtain the relative p, w, @, and 4 couplings listed in Table I. If the Drell-Yan threshold relation14 is to hold, the large-$ behavior of the form factors is re- quired to be netic self-mass of a 2 baryon can be written as1' F,(qZ) = O((qZY2), (6) LY 6M= -2. 1 qdq2{[(1 + P2)11' - P]3fl(-q2) F ~ ( ~ ~ ) = . One way15 of satisfying (6) within the pole-domin- -.[(I + ~ ~ ) ~ / ~ ( i - 2p2) + 2p3] ance picture is to write F,(q2) (i=p, w, $, $) as a ~f2(-q')), (1) product of pole factors: where p= q/2M and M is the mass of the baryon. The functions fl(q2) and f2(q2) are given in terms of the electric and magnetic form factors (2) The appropriate Regge slopes are of= 1.0 for the uncharmed mesons, and, taking the $' as the first recurrence of the $, af=0.25 for the $ and its 17 - 3130 O 1978 The American Physical Society

Electromagnetic mass differences of the charmed baryons

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Page 1: Electromagnetic mass differences of the charmed baryons

P H Y S I C A L R E V I E W D V O L I J M E 1 7 , N U M B E R 1 1 1 J U N E 1 9 7 8

Electromagnetic mass differences of the charmed baryons

A. C. David Wright* Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2Jl

(Received 6 December 1977)

Born-term contributions to the electromagnetic mass splittings of the charmed and uncharmed baryons are ' estimated using form factors constructed with the help of vector-meson dominance and SU(4) symmetry. The

remaining contribution to the mass splittings is assumed to be a tadpole transforming like the third component of isospin. Our results are compared with previous estimates based on the quark model.

In previous papers, '* ' we have obtained es t i - m a t e s for the electromagnetic m a s s splittings of the charmed pseudoscalar and vector mesons D and D*. In th i s approach, Born-term and f i r s t - intermediate-s tate contributions t o m a s s split- t ings a r e calculated using pole-dominated form fac tors constructed with the help of SU(4) sym- metry. The remaining contribution i s assumed to b e due to a tadpole3 t ransforming like the third component of iosopin. The predicted D m a s s splitting, m,+ - m,o = 5.8 MeV, i s in good agree- ment with the experimental value4 of 5.1 + 0.8 MeV. T h i s success , and the fact3 that the tadpole approach works be t te r fo r the SU(3) baryons than f o r the SU(3) mesons, suggests that it would b e

where

The f o r m f a c t o r s F l (q2) and F2(q2) a r e normalized a t q2 = 0 in t e r m s of the charge Q and magnetic moment y of the baryon:

Assuming pole dominance, a f o r m factor can b e wri t ten a s

worthwhile to apply the method to the problem of the charmed-baryon electromagnet ic m a s s differ- F(q2)= C c t F i ( q 2 ) , ences. Fur thermore , r e s u l t s a lready obtained5-" i = p , W, 0, + in the context of the quark model vary widely, s o that it would be interest ing to compare the r e s u l t s of our approach with those of Refs. 5-11.

Because of the uncertaint ies involved in the B ,~ ,+B,~ ,+Y and B312+B312+y couplings, we shall r e - s t r i c t our attention in the following t o the g multi- plet. T h i s means that we shall neglect inelastic intermediate-s tate contributions, and assume that m a s s splittings a r e given reasonably accurately by the sum of the Born and tadpole t e r m s .

The Born-term contribution to the electromag-

where Fi($) is the contribution due to vector mes- on i. To determine the coefficients c , in (5), we impose the Okubo-Zweig-Iizuka (OZI) rule13 and SU(4) invariance f o r the VBB couplings. We a s - sume pure F-type coupling f o r P,(O) and the SU(6) F/D rat io f o r G,(O) to obtain the relat ive p, w , @, and 4 couplings l is ted in Table I.

If the Drell-Yan threshold relation14 is to hold, the large-$ behavior of the f o r m fac tors is re - quired to be

netic se l f -mass of a 2 baryon can b e written as1' F,(qZ) = O((qZY2), (6)

LY 6 M = -2. 1 qdq2{[(1 + P2)11' - P]3fl(-q2) F ~ ( ~ ~ ) = .

One way15 of satisfying (6) within the pole-domin- -.[(I + ~ ~ ) ~ / ~ ( i - 2p2) + 2p3] ance picture is to wr i te F,(q2) ( i = p , w , $, $) as a

~f2( -q ' ) ) , (1) product of pole factors:

where p = q/2M and M i s the m a s s of the baryon. T h e functions fl(q2) and f2(q2) a r e given in t e r m s of the e lec t r ic and magnetic fo rm fac tors

(2) The appropriate Regge s lopes a r e o f = 1.0 f o r the uncharmed mesons, and, taking the $' a s the f i r s t r e c u r r e n c e of the $, a f = 0 . 2 5 f o r the $ and i t s

17 - 3130 O 1978 The American Physical Society

Page 2: Electromagnetic mass differences of the charmed baryons

17 - E L E C T R O M A G R E T I C M A S S D I F F E R E N C E S O F T H E C H A R M E D . . . 3131

TABLE I. Contributions of p , w, @, and # to the form factors F 1 and Gy.

= I G,,, = ( F ~ + ~ M F ~ ) a

P W O i l , P W @ +

P + + o o : b o o n -A 1 0 0 5 - 1 0 0

2 2 4 4

z+ 1 - 0 + # + o 20 0 - 0 o + + o 2- -1 4 -g 0 2 - 4 - 5 0

?O * f $ 2 0 1 1 2 - - - 4 1 2 3 0

-- - -1 - 0 -1 1 4 0 2 6 3

Cf+ l j O + 1 4 0 -4 : 0 4 0 2 0 0 0 0

'2 - 1 3 0 8 1 - 1 - 0

S+ 3 4 - 8 - ; 0 0 0 0

s O -1 1 -1 L + - L L 2 6 3 3 6 3 4

A+ 3 ' - L " 6 3 3 0 ° 0 1

AO 1 -- 4 -g 2 2 0 0 0 1

x; + - ; - t; 0 + - - - - : A 0 $ x: -- - 4 1 - -- 8 ; & o r 6 1'8 O ?

a In units of the magnetic moment of each particle.

recurrences. These product-type form factors have the advantage of providing a well defined prescription for the calculation of any fo rm factor. On the other hand, the couplings of the recur- rences of the $ can be l a rge r than those of the + itself, which i s unphysical. Furthermore, pro- duct-type form factors predict charmed-baryon pa i r production c ros s sections in e'e' annihilation which a r e much too la rge near threshold16 due to the proximity of the recurrences of the +. How- ever, the dominant contribution17 to the integral in (1) comes f rom the spacelike region -1 Gev2

q2 6 0, where the $ contribution does not vary greatly from i t s normalized value a t q2=0 . There- fore, Born-term contributions to self-masses a r e principally determined by the SU(4) structure of the form factors, rather than their behavior a t la rge q2.

Using the pole-dominated form factors of (7), we find the Born-term contributions to baryon electromagnetic mass splittings gived in Table 11.' For the charmed-baryon masses, we have taken M,, = 2.41 GeV, M, = 2.56 GeV, M A = 2.47 GeV, and M , = 3.65 GeV, a s i s suggested by the quark model.'' Our results a r e not sensitive to variations in the masses. Also, we have used the SU(4) values of the baryon magnetic moments.

TABLE 11. Baryon electromagnetic mass splittings A M in MeV, with Born-term (AiLIB) and tadpole ( A M t a d ) contributions. The experimental values ( A M e x p t ) a r e taken from Ref. 19.

A M B AMtad A N *Mexpt

The value1' ( p - n), = 0.79 MeV given in Table I1 i s based on the dipole f i t to the form factors, with a dipole (mass)' of 0.71 GeV2. The product-type form factors (7) yield 0.70 MeV for this contribu- tion. In general, Born- te rm contributions to mass splittings a r e grea ter for the charmed baryons than for the uncharmed baryons; the analogous situation has been foundlv2 to hold for the pseudo- scalar and vector mesons.

Assuming that the tadpole te rm transforms like the third component of isospin, we find for the tadpole contributions

(A' - AO),,= (1 -+Y)X,

where (1 - a)/a i s the F/D ratio of the tadpole

TABLE 111. A comparison of our resul t for the CTf - C! splitting in MeV with values obtained in various ~ u a r k models (Refs. 5-11).

Reference Ct+ - Cy

This calculation -1.4 Itoh et al. 6.5 Ono 6.1 Lane and Weinberg -6 Deshpande et a l . -3 to -18 C han 0.4 Lichtenberg 3.4 Kalnlan and Jakimow -2.7

Page 3: Electromagnetic mass differences of the charmed baryons

3132 A . C . D A V I D W R I G H T - 17

couplings. Taking X= -2.08 MeV and a = -0.75 to yield agreement with the uncharmed-baryon mass splittings,lg we obtain the results given in Table 11.

It is interesting to compare our results with those previously obtained in the quark Table III l ists the various values predicted for the C;'- C: mass difference, which is the one most likely to be measured in the not-too-distant future. Our negative value, resulting from the

dominance of the tadpole over the Born term, corresponds to quark models in which the d-u mass difference dominates the Coulomb and mag- netic interactions of the quarks. Unless inelastic contributions a re sizable, this general feature of our approach would appeai- to be inescapable.

I am grateful to the Killam Foundation for fi- nancial support. This work was supported in part by the National Research Council of Canada.

*Present address: Atomic Energy of Canada Limited, Sheridan Park, Mississauga, Ontario, Canada L5K 1B2.

ID. H. Boa1 and A. C. D. Wright, Phys. Rev. D 16, 1505 (1977).

'A. C. D. Wright, Phys. Rev. D s , 2265 (1977). 3 ~ . Coleman and S L. Glashow, Phys. Rev. 134, B671

(1964); R. H. Socolow, ibid. 137, B1221 (1965). *G. J. Feldman, SLAC Report No. SLAC-PUB-2000,

1977 (unpublished). 5 ~ . Itoh, T . WIinamikawa, K. WIiura, and T. Watanabe,

Prog. Theor. Phys. 54, 908 (1975). %. Ono, Phys. Rev. D 15, 3492 (1977). 'K. Lane and S. Weinberg, Phys. Rev. Lett. 2, 717

(1976). 'N. Deshpande, D. Dicus, K. Johnson, and V . Teplitz,

Phys. Rev. D 15, 1885 (1977). 'L.-H. Cban, Phys. Rev. D g , 2478 (1977). 'OD. B. Lichtenberg, Phys. Rev. D16 , 231 (1977). "C S. Kalman and G , J ak~mow Lett. Nuovo C~men to 19, 403 (1977).

"M. Elitzur and H. Harari , Ann. Phys. (N.Y.) 56, 81 (1970).

I3s. Okubo, Phys. Lett. 2, 165 (1963); G. Zweig, CERN Report No. CERN-TH402, 1964 (unpublished); J. Iizuka, Prog. Theor. Phys, Suppl. 37-38, 21 (1966).

14s. D. Drell and T.-11. Yan, Phys. Rev. Lett. 2, 181 (1970).

1 5 ~ . C. E . Devenish, T . S. Eisenschitz, and J. G. K6r- ner , Phys. Rev. D f i , 3063 (1976); J. G. Korner and M. Kuroda, ibid. 2, 2165 (1977).

16A calculation of charmed-baryon pair production c ros s sections employing pole-dominated form factors modi- fied to include continuum contributions i s given in A. C. D. Wright, Phys. Lett. m, 425 (1977).

"Using product-type form factors, the Born-term con- tributions e) to baryon electromagnetic mass splittings a r e all given within +20% by truncating the integral a t q2=1 G ~ v ' .

"A. De RBjula, H. Georgi, and S. L. Glashow, Phys. Rev. D 12, 147 (1975). We have taken the charmed quark mass 50 MeV greater than have these authors to obtain a C; mass of 2.25 GeV, in agreement with the experimental value of B. Knapp et n l . , Phys. Rev. Lett. 2, 882 (1976).

article Data Group, Rev. Mod. Phys. 48, S1 (1976).