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