LETTERE AL NUOV0 ClMV.NTO VOL. 18, ~'. 17 23 Apr i le 1977

Effect o f u and d Quark Differences

o n the Masses of Charmed Mesons (*).

G. JAKIMOW

D~partment de Physique, Universitd du Quebec a Montrdal . C.P. 8888, Montrdal, Quebec, Canada

C. S. K&LMAI~ (**)

Physics Depazment, Ind iana University . Bloomington, Ind. 47401, U.S .A .

( r icevuto il 25 Gennaio 1977)

In prev ious calculat ions (1,~) of the masses of cha rmed mesons, t he mass of each meson was g iven by

(1) M(m) = O0 + C1~ml8~1 m) + C~(mle~lm) + C30nl~lm) + C, id~lm),

where Co, Cx, C2, and C 3 are all constants . I t was then assumed t h a t since the masses of the u and d quarks are much smal ler t han those of the s and c quarks, t h e l a s t two t e rms are expec ted to be smal l and could be ignored. In this paper these t e rms are expl ic i t ly calculated. The effect is to renormal izc the masses of all the mesons wi thou t changing the resul ts of the prev ious calculatons and to spli t the K and D isoplets.

I f A i j , i , j = 1, 2, 3, 4 are the genera tors of SU~, then

(2) [A~, A~,,] = ~ , ~ A ~ j - - ~ A ~ , , , i, j , k, m = 1, 2, 3, 4 ,

4

(3) ~ : a ~ = 0 , i, j = 1, 2, 3, 4 . k--I

I f qj and qk are quark and an t iqua rk generators , respect ively , then

(4) [ A ~ , qk] ~ 5~q~ , i, j , k -= 1, 2, 3, 4 ,

(5) [Ai~, qk] = - - 5kjq~, i , j , k = 1, 2, 3, 4 .

(*) Work supported in part by the National Research Council (Canada) and in part by the U.S. Energy Research an4 Development Administration. (**) Permanent address: Ooyola Campus, Goncordia University, Montreal, Canada H4B 1R6. (t) G. JAKI~OV an4 C. S. KxL~,w: Concordia Universlt6 du Quebec Report CUQ/EPP-76/15 (1976); L e t L Nuovo Cimenlo, to be published. (J) C. S. K_~AN: Indiana University Report IUHET-I1 (1976).

544

EFFECT OF U AND d QUARK DIFFERENCES ON THE MASSES, ETC. 545

The algebra of q's a n d A ' s is closed if

(6) [q,, qj] = O(($i~A~-- A.) , i , j = 1, 2, 3, 4 ,

where 0 = + 1 corresponds to the Lie algebra of S U 5,

0 = - 1 corresponds to t ha t of SU1, 4,

0 = 0 corresponds to tha t of T10 | S U,

and As~ is a (~ d i a g o n a l , genera tor in addi t ion to A~j, i = 1, 2, 3, 4, needed to complete the first two of the above Lie algebras.

This condi t ion (3) l imi ts the quark radial-field var iable to a fixed , r a d i u s , and thus corresponds to the nonobservance of free quarks. The group Tie | S U 4 is no t considered because b y i ts use one canno t predic t t rans i t ions be tween e lementa ry par- ticles. Since the same values of the masses (1,2) are ob ta ined us ing ei ther S U 5 or SU1,4,

for convenience S U s will be used in this paper . Calculat ions are per formed us ing a Ge l ' l and (4) basis :

(4) I"~) =

t~'/bl5 ~b25 m35 7n'45 ~b55 ~

'D'bl4 ~b24 7Yb34 9~b44

mla m2a ross

'D?/12 'm.22

nhl

To sat isfy the un imodu la r condi t ion m55 = 0. The remain ing parameters are posit ive integers sat isfying the condi t ions

(5) m i 1 ~ m t j _ l ~ , ~ b t + l j , j = 2 , 3 , 4 , 5 , i = 1 , 2 , 3 , 4 , 5 .

Mesons are as usua l identif ied wi th the S U 4 15-dimensional representa t ion m~, = m u + l = m 3 4 + 1 = m 4 4 + 2. By eq. (7) then m 3 s = m 1 4 - - 1 . There values of m~

and m45 are restr ic ted (1) to two possible cases

(6a)

(6b) 'mlt=m~5+ 1 =m,l~+ 1.

Since bo th cases yield ident ica l results, for convenience the re la t ion described b y eq. (6a) will be used in the rest of the paper . The values of the parameters in the bo t tom three rows of eq. (4) are described in the previous paper (1). Three cases were, however, mis- labelled there and are corrected below:

( rex4 ~14- -1 / , ( mz4 ~14--1 / Y~I, - 1 ~1t--2 K+' K*+-> K~ K*~ ' K ~ 1 7 6 / "

9/b14 / \ m14-- 1 / \ ml~-- 1 /

(a) Y. DOTHAI~" and Y. NE'EM&~: AEC Research and Development Report CALT-68-41 (1965). (') I. 1~. (]EL'FAND and M. I. GRAEV: .~'fser. Math. See. Transl., 64, 116 (1967).

54~ O. JAKIMOW and c. s. KALMAN

In eq. (1), se t t ing A = (G,~)/120, B----(01a)/36o, D,------ Ca~/720, D ~ - - - - - C4~/720, (7 ---- 01 + O2, where g = (m14 - - 2)(mls - - ~h4 + 6)(m25 - - m14 -[- 5)(m4s - - m14 -{- 3), one obta ins

(7) ~ ( ~ ) = 0o +

(8) M ( K o ) = M ( K o) = C o +

(9) ~ ( K ~) = Oo +

(10) M(D ~) = Go +

(11) M(D o ) - - M(D ~ ---- G o +

(12) M(Ps) = (7 0 +

(13) M(~v) = O 0 +

(14) M ( F :~) = O o +

7 A - - 21B + 30C,

7 A - - 15B + 36C + 6D 1 - 6D~,

7 A - - l S B + 3 6 G - - 6 D 1 + 6D 2,

5 A - - 2 1 B + 3 6 C + 6 D 1 - - 6 D 2 ,

5A - - 21B + 36G - - 6D 1 -{- 6D 2 ,

7 A - - 13B + 38C,

4A - - 20B + 40G,

5 A - - 15B + 42G.

Iden t i ca l equa t ions are ob ta ined if one replaces K by K*, 7: by p or A 2 etc. Aside f rom the usual Ge l l -Mann-0kubo mass relat ion, these equat ions imply t h a t

(15)

(16)

9M(D) = 6M(r + a M ( g ) - - M ( K ) ,

J / ( F ) - - M(D) = ~ ( K ) - - ~ / (~ ) .

There are ident ica l (~) to the sum rules ob ta ined prev ious ly wi th C8 and C4 of eq. (1) set to zero. The only addi t iona l resul t is the spl i t t ing of the K and D isoplets. F r o m eqs. (8-11) we get

(17) M(D +) - - M(D ~ = M ( K ~ - - M ( K +) .

I t should be no ted tha t eq. (17) does no t include effects due to Coulomb and magnet ic- m o m e n t in terac t ions be tween the quarks, and is thus consis tent wi th the usual calcu- lat ions of e lec t romagnet ic spl i t t ings of cha rmed mesons (5).

(6) D. B. LIOBTEI~BERO: Phys. Rev. D, 42, 3760 (1975).