Masses of charmed mesons
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 SU 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~Bsi , i, j, k = 1, 2, 3, 4 , (4) [A/f, B j = - - ~jBi5, 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| SU 4 9 (*) Work supported in part by the National Research Council (Canada) and in part by the U.S. Energy Research an4 Development Administration. (**) Permanent address: Loyola Campus, Concordia University, Montreal, Canada H iB 1R6. 511 512 G. JAKIMOW and c. s. KAL~iIAN This condit ion l imits the radial field variable to a fixed (( radius ))(~). The model is then analogous to the 5lIT bag model (2). The group T10 @ SU~ is ruled out because by its use we e,~nnot predict transit ions between e lementary particles. The group SU~,~ will be t reated elsewhere and the group SU~ is examined here. Calculations arc performed using a Gelfand (3) basis: (6) r ~I15 m25 ~Y~35 /45 ~b55 \ ~14 7Yb24 m34 m44 ~13 ?Yt23 ~b33 T~12 ~}~22 i l l To satisfy the unimodular condit ion m~= 0. The remaining parameters are posit ive integers satisfying the condit ions (7) m~y~mi,~_l~mi+~j j=2 ,3 ,4 ,5 , i= 1 ,2 ,3 ,4 ,5 . Mesons ~re, as usual, identif ied with the SU 4 15-dimensional representat ion contain- ing the SU s octet representat ion. Por such a representat ion m~4 ~ me4 ~- 1 = m3a + 1 =m44 .4-2. By eq. 7 then m3~=m~4--1 . This equation also restr icts the possible values of m25 and m4~ to four possible cases: (8a) m14 = m25 = ~45 ~- 1 , (Sb) m14 ~ m2~ = m4~ ~- 2 , (8C) m14= m25-b 1 = m45~- 1 , (8d) ~t~14 = m25-= 1 = m45 + 2 . The parameters in the bot tom three rows of eq. (6) can all be wr i t ten in terms of the parameter mla; specific values depend on choice of SUn representat ion. There are thus only two free parameters m14 and ~t~l~. Each case noted in eq. (8) will be examined in terms of these parameters. For an octet representat ion, we must also set m~= m~a = ~2a ~- 1 = m3a-}- 2. The last two rows of eq. (6) are then described 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- \ mla- - A + \ ml~ (1) Y. I)OTHA_N O~IlCl Y. JNTE'ESL~-.N~: AEC Research and Deve lopment Report CALT-68-41 (1965). (2) 1~. C. JAFFE and J. KISKIS: Phys. Rev. D, 13, 1355 (1976). (3) I. . GELFAI':D aB.(l ~1". I. GR2kEV: Amer. Math. Soc. TransL, 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 / mla- - 1 mla - - 1) . Vs'-> m14-- 1 ] The charmed part ic les are contained in SUs tr ip let (C~- -1 ) and untitr iplet (C ~ + 1) representations. For the tr ip let representat ion m~a= m13 ~ m~a + 1 = man + 1 and ( nh4 m14-- 1) D-D* - -+ ml~- 1 ]~o~,o..+ ( ml* m14-- 1) , \ mx4 mla - - 1 m14-- 1) mla - - I For the ant i tr ip let representat ion ml, = mla + 1 = m2a + 1 = man + 2 and mla-- 1 mla-- l ) 1~+--> - mla- - i DOD*O~ Final ly ( T~14-- 1 9Ybl4-- 2) , qYb14- 2 m14-- 1 ro l l - - 2) D +, D*+-+ *r~ld-- 1 @, @(.v), @(T)~ Ima4-- I mia-- I mi~-- i~ mla-- 1ml a - lm14 - 1 ] . In a Gelfand basis A~j, i,j = 1, 2, generate the isospin s~lbgroup of SU~ and A~j i , j= 1 ,2 ,3 generate the SUa subgroup. Then B15 can be identif ied with the (~u, quark, B25 with the (( d 7) quark, Ba5 with the (~ s ~> quark and Ba5 with the q c * quark. In f irst-order t ime- independent theory, the mass of each meson is given by (4) (9) 4 M(m) = Co + (m] ~ Ci~B~4Bajlm ) = i , ]=l = Co+ Cl514 G. JAKI~VIOW and C. S. KAL~/Alff Since the masses of the u- and d-quarks are much smaller than those of the s- and c- quarks, the last two terms are expected to be small and will be ignored in this paper. For uncharmed mesons ( lOa) is constant. Hence (10b) Thus ( l la ) (~b) ( l lc) (tld) where (12a) (12b) (~2c) r = Co + C~ ]~(K, K*) = Co ~- (C~) /24 , 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~ - - ml t -'.- 4)(m2~ - - m14 -- 3)(m45-- ml~ 1 ), = (m14 3)(mls-- mla 1)(m25-- m14)(m45- - mla - - 2) . Now consider the four cases of eq. (8). For eq. (8a) fl =y = 0. (13a) M(K ,K* ) = Co 1 - (Cta ) /24 , (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 But M(K, K*) = I~I(K, K*) requires Cle = 0, and then the masses of all the octet mesons are identical in contradiction to experiment. In general, M(K, K*) = M(K, K*) requires (14) 3a 5fl = 87. For eq. (8d), setting n= mls--m14 and making use of eq. (12), eq. (14) simplifies to nm14 = - -4n- -8 . Such an equation, however, contradicts eq. (7). Thus the eases represented by eqs. (Sa), (8d) are in contradict ion to experiment. The cases corresponding to eqs. (Sb), (Be) yield identical results and so for convenience the relation described by eq. (8b) will be used in the rest of the paper. For this case y= 0 and eq. (14) reduces to f l=~. Setting A = (C2~)/120 and B= (C1~)/360 the masses of the mesons are then de- MASSES o~ CH~D ~ESONS ~1~ scribed by 05a) (15b) (15c) (~5d) (15e) (15/) M(K,K*) ~Go+7A- -15B, M(=, 9, A~) = Go-~ 7A - - 21B, M(Vs) ~- Go + 7A - - 13B, M(~) = Go-~ 4A - - 20B, M(D,D*) = Go+5A- -21B, M(F, F*) = G0-~ 5A- - 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 calcu- lated. 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) that the mass of the pseudosealar D is iden- TA~L~ I, - Predicted charmed particle masses ( in MeV). Part 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 character- istics 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 and 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 ICHER, 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 . 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