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LETTERE AL NUOVO CIMF.I~/TO VOL. 17, ~. 15 11 Dicembre 1976

Masses of Charmed Mesons (*).

G. 5AXI~OW

D@artment de Physique UniversitO du Quebec ~ Montrdal - C . P . 8888, Montrdal, Que., Canada

C. S. KiLl, AN (**)

Physics Department, Indiana University - Bloomington, Ind . 47401, U.S .A

(ricevuto il 28 Settcmbre 1976)

Suppose that A~j, i , j = 1, 2, 3, 4 are the generations of S U 4. Then

(1) [Ai~, Akin ] = 6imAk~-- ~ksAi~, i , j , k, m = 1, 2, 3, 4 ,

r (2) ~ A ~ = o , i , j = 1, 2, 3, 4 .

k~l

If Bst and B~5 arc quark and ant/quark operators, respectively, then

(3) [ A t i , B5k ] = ~ t~Bs i , i, j , k = 1, 2, 3, 4 ,

(4) [A/f, B j = - - ~ jB i5 , i , j , b = 1, 2, 3, 4 .

A closed Lie Algebra of A's and B's can be obtained by imposing a commutation of the B's that correspond to some dynamical condition. In a strong-coupling model, the nonobservance of quarks outside the interior of (~ elementary ~) particles can be described by the condition

(5) [Bs~, Bi5 ] = O(~itA55-- A j~) , i, j = 1, 2, 3, 4 ,

where

0 = + 1

0 = - - 1

0 = 0

corresponds to the Lie Algebra of SUs ,

to that of SU1,4,

to that of T1o| S U 4 �9

(*) Work supported in par t by the Nat ional Research Council (Canada) and in pa r t by the U.S. Energy Research an4 Development Adminis t ra t ion. (**) Permanent address: Loyola Campus, Concordia University, Montreal, Canada H i B 1R6.

511

512 G. JAKIMOW and c. s. KAL~iIAN

This condi t ion l imi ts the rad ia l field var iab le to a f ixed (( rad ius ))(~). The model is t hen analogous to t he 5lIT bag model (2). The group T10 @ SU~ is ru led out because by i ts use we e,~nnot p red ic t t r ans i t ions be tween e l emen ta ry par t ic les . The group SU~,~ will be t r e a t ed e lsewhere and the group SU~ is ex ami n ed here.

Calculat ions arc pe r fo rmed using a Gelfand (3) basis :

(6) r

~I15 m25 ~Y~35 / 4 5 ~b55 \

~14 7Yb24 m34 m44

~13 ?Yt23 ~b33

T~12 ~}~22

i l l

To sat isfy the un imodu la r condi t ion m ~ = 0. The r ema in ing p a r a m e t e r s are pos i t ive in tegers sa t i s fy ing the condi t ions

(7) m~y~mi,~_l~mi+~j j = 2 , 3 , 4 , 5 , i = 1 , 2 , 3 , 4 , 5 .

Mesons ~re, as usual, ident i f ied wi th t he SU 4 15-dimensional r ep re sen ta t ion conta in- ing the S U s oc te t r ep resen ta t ion . Por such a r ep re sen ta t ion m~4 ~ me4 ~- 1 = m3a + 1 =m44 .4-2. By eq. 7 t h e n m 3 ~ = m ~ 4 - - 1 . This equa t ion also res t r ic t s t he possible values of m25 and m4~ to four possible cases:

( 8 a ) m 1 4 = m25 = ~ 4 5 ~ - 1 ,

(Sb) m14 ~ m2~ = m4~ ~- 2 ,

(8C) m14= m25-b 1 = m45~- 1 ,

(8d) ~t~14 = m25-= 1 = m45 + 2 .

The p a r a m e t e r s in the b o t t o m three rows of eq. (6) can all be wr i t t en in t e r m s of the p a r a m e t e r mla; specific values depend on choice of SUn represen ta t ion . There are thus only two free p a r a m e t e r s m14 and ~t~l~. Each case n o t e d in eq. (8) will be ex ami n ed in t e r m s of these pa rame te r s .

Fo r an octet r ep resen ta t ion , we m u s t also set m ~ = m~a = ~2a ~- 1 = m3a-}- 2. The last two rows of eq. (6) are then descr ibed as follows:

K 0, K*0-~ , K +, K * O . ,

77 (m14 ~}$14- 2 t ~0

A~- \ , m . - - 2 / A. ~ 2

(1 ?4_2) i 14 1.2) , ~+ --O-

\ m l a - - A + \ ml~

(1) Y. I)OTHA_N O~IlCl Y. JNTE'ESL~-.N~: AEC Research and D e v e l o p m e n t Report CALT-68-41 (1965). (2) 1~. C. JAFFE and J. KISKIS: P h y s . Rev. D, 13, 1355 (1976). (3) I. •. GELFAI':D aB.(l ~1". I. GR2kEV: Amer . Math . Soc. T r a n s L , Scr. 2, 64, 116 (1967).

MASSI~S OF CHARMED ~ESOI~S 5111

( m14-- 1 m14-- 2) K - , K * - - +

\ m14-- 2 ] ( mli-- 1 m1~-- 2)

K o, K*O~

\ m14-- 1 /

m l a - - 1 m l a - - 1) .

Vs'-> m14-- 1 ]

The cha rmed par t i c les are con ta ined in SUs t r ip l e t ( C ~ - - 1 ) and un t i t r ip le t (C ~ + 1) represen ta t ions . F o r the t r i p l e t r ep resen ta t ion m~a= m13 ~ m~a + 1 = man + 1 and

( nh4 m14-- 1) D - D * - - +

m l ~ - 1

]~o~,o..+ ( ml* m14-- 1) ,

\ mx4

m l a - - 1 m14-- 1)

m l a - - I

F o r the an t i t r ip l e t r ep resen ta t ion ml , = mla + 1 = m2a + 1 = man + 2 and

mla-- 1 mla-- l ) 1~+--> -

m l a - - i

DOD*O~

F ina l l y

( T~14-- 1 9Ybl4-- 2) ,

qYb14- 2

m14-- 1 r o l l - - 2) D +, D*+-+

*r~ld-- 1

@, @(.v), @(T)~

Ima4-- I mia-- I mi~-- i~

mla-- 1ml a - lm14 - 1 ] .

I n a Gel fand bas is A~j, i , j = 1, 2, genera te t he i sospin s~lbgroup of SU~ and A~j i , j = 1 , 2 , 3 genera te the SUa subgroup. Then B15 can be ident i f ied wi th the (~u, quark , B25 wi th the (( d 7) quark , Ba5 wi th the (~ s ~> quark and Ba5 wi th the q c * quark . In f i rs t -order t ime - independen t theory , the mass of each meson is g iven b y (4)

(9) 4

M(m) = Co + (m] ~ Ci~B~4Bajlm ) = i , ]= l

= Co+ Cl<mls~lm)+ C~{m[velm)+ Ca<mlu~lm)+ Ca{m[d'3lm).

(') A linear mass formula is used ilx this articlc, because the mass predictions of the quadratic formltla disagree with the mass of a recent discovery of a meson whosc production and decay characteristics are just those expected of a charmed meson (see ref. (~)).

514 G. JAKI~VIOW a n d C. S. KAL~/Alff

Since the masses of the u- and d-quarks are much smal ler than those of the s- and c- quarks, the las t two te rms are expected to be smal l and will be ignored in this paper .

Fo r uncha rmed mesons

( l O a )

is constant . Hence

(10b)

Thus

( l l a )

(~b)

( l l c )

(tld)

where

(12a)

(12b)

(~2c)

r = Co + C~<m[e~lm>

]~ (K, K*) = Co ~ - ( C ~ ) / 2 4 ,

M(K, K*) = Co ~ - (C~a)/60 § (C~fl)/24-- (C~?)/15,

M(~, ~, A~) : Co ~ - (C~a)/48 § (C~fl)/16,

M(V s) = C~ - - 23(C~a)/720 § (C~fl)/144 - - 2(C~?)/45,

= (~rh~ - - 2) (mr5 - - m 14 + 6) (m~5 - - m14 § 5) (m4~ - - m14 § 3 ) ,

fl = m,14(m1~ - - m l t -'.- 4)(m2~ - - m14 -- 3)(m45-- ml~ § 1 ) ,

= (m14 § 3)(mls-- mla § 1)(m25-- m 1 4 ) ( m 4 5 - - m l a - - 2) .

Now consider the four cases of eq. (8). For eq. (8a) fl = y = 0.

(13a) M ( K , K * ) = Co 1 - ( C t a ) / 2 4 ,

(lab) M(K, K*) = Co 1 - (C~a)/60,

(13c) M/=,, e, A~) = C~-- (C1~)/48,

(13d) M(Vs) = Co ~ - - 23(C~a)/720.

Hence

Bu t M(K, K*) = I~I(K, K*) requires C le = 0, and then the masses of all the octet mesons are ident ica l in contradic t ion to exper iment .

In general , M(K, K*) = M(K, K*) requires

(14) 3a § 5fl = 87 .

For eq. (8d), se t t ing n = mls--m14 and making use of eq. (12), eq. (14) simplifies to

n m 1 4 = - - 4 n - - 8 .

Such an equat ion , however , contradic ts eq. (7). Thus the eases represented by eqs. (Sa), (8d) are in cont rad ic t ion to exper iment . The cases corresponding to eqs. (Sb), (Be) y ie ld ident ica l resul ts and so for convenience the re la t ion described by eq. (8b) will be used in the res t of the paper . Fo r this case y = 0 and eq. (14) reduces to f l = ~ . Set t ing A = (C2~)/120 and B = (C1~)/360 the masses of the mesons are then d e -

MASSES o~ C H ~ D ~ESONS ~1~

scribed by

05a)

(15b)

(15c)

(~5d)

(15e)

(15/)

M ( K , K * ) ~ G o + 7 A - - 1 5 B ,

M(=, 9, A~) = Go-~ 7A - - 21B,

M(Vs) ~- Go + 7A - - 13B,

M(~) = Go-~ 4A - - 20B,

M(D,D*) = G o + 5 A - - 2 1 B ,

M(F, F*) = G0-~ 5 A - - 15B.

By using the experimental values (s) of the masses of K, K*, ~, p, A~, ~, the constants ~0, A and B can be evaluated and the masses of the charmed mesons can then be calculated. In this evaluation the mass of ~(P) is taken to be 2.8 GeV. The results of this calculation are shown in table I. Note (s) tha t the mass of the pseudosealar D is iden-

T A ~ L ~ I , - Predic ted charmed part ic le masses ( in MeV).

Par t ic le / J ~ 0- 1- 2 +

D 1873 2306 2789

F 2231 2429 2900

tical ~o the mass of a recently discovered meson whose production and decay characteristics are just those expected of a charmed meson. The crucial test is the mass of the vector-meson D. If the experimental indications (s) are confirmed, the calculated value is 300 MeV too high.

(~) T . G . TRIPFE, A. BARBARO-GALTIERI, 1~. L. KELLY, A. R1T:~ENBERG, A. ]~. ROSENFELD, G. P . ~OST, N. BARAsn-SoHM_IDT, C. BRICMAs R . J . HEMINGWAY, M. J . LOSTY, M. ]~OOS, V. CHAIOUPKA a n d B. ARI~ISTRONG: ReV. Mod. -Phys., 48, 51 (1976). (~) G. GOLDHABER, F. lW. PIERRE, G. S. ABRAMS, M. S. ALAI~I, A . M. BO~'ARSKI, M. BREIDENBACtt, W . C. CARITHEI~, W. Ct~II~OWSKI r, S. C. COOPER, R . G. DEVoE, J . ~/L DORFA_N, G. J . FELDMAIV, C. E. FRIEDBERG, ]) . FR~rI~ERi~]~R, G. HANSON, J . JAROS, A. D. JOHNSON, J . A . KADY~K, R . R . LARSEN, D. L~KE, V. L ~ H , H . L . LYNCH, R . $. ]V~ADARAS, C. C. MOREI~OUSE, H . ]~. NGWYEN, J . 1~. PATERSON, M. L . PERL, I . PERUZZI, ~ . PICCOLO, T. P . PUN, P . RAPIDIS, ]3. R I C H E R , B. ~ADOULET, R . H . SCHINDLER, R . 1 ~. SO~ITTERS, J . SIEGRY~T, ~V. TANENEAUM, G. ]~. TRILLING, ~ . VANNUCCI, J . S. ~VH1TAKER a]l~l. J . •. WISS: Phys. Rev. Leit., 37, 255 (1976); I . PERUZZI, M. PICCOLO, G. J'. FELDMAIW, t [ . K . NGUYEN, J . E. WISS, G. S. ~kBRAM~, M. S. ALA~, A. M. BOYARSKI, lYE BREIDENBACH, W. C. CkRITHERS, W . CHI- NOWSKY, R . G. DEVOE, J . ~r DORFAiV, G. E . I~ISC~ER, C. E. I~RIEDBERG, D. FRY~ERGER, G. GOLDH.~BER, G. HANSON, J . A. JAROS, A. h . JOHNSON, J . A. KADyK, R . R . LARSEN, D. LOKE, V. Lf2TH, H . J~. LYNCH, R . J . ~r C. C. MOREHOUSE, J . M. PATERSON, !Yl. L. PERL, F. M. PIERRE, T. P . PUi"~, P . ~APIDIS, B. RICttTER, R . H . SC~[NDLER, R . F . SCHWITTERS, J . SIEGRIST, W . TANENBAUM~ G. H . TRILI~ING, F . VANNUCCI, J . ~. WHITAKER: S t a n f o r 4 L i n e a r A c c e l e r a t o r C e n t r e R e p o r t S L A C - P U B - 1 7 7 6 (1976).