L]~TTERE AL NUOVO CIMENTO VOL. 36, N. 10 5 ]Y[arzo 1983

Flavour Symmetry of Hadrons and Nonleptonic Decays of Charmed D-Mesons.

A. TODA

Department of Appl ied Physics, Tohoku Gakuin University - Taga]o 985, J a p a n

( r icevuto i l 23 Dicembre 1982)

PACS. 13.25. - Hadron ic decays of mesons.

A pecul iar , bu t in te res t ing mode l of t he f lavour s y m m e t r y of hadrons has recent ly been proposed by EBATA (~,2). Accord ing to h im, t he cons t i tuent quarks and the non- exot ic mesons mus t be classified into weak isospin 8U2~-mult iple ts , which consists of two double ts (ql, qz), four t r ip le ts (P, K, D, F ) and four s inglets (Po, Ko, Do, Fo)(z):

(1)

( ) ( c ) ql = q2 = d c o s 0 + s s i n 0 ' - - d s i n 0 + s c o s 0 '

P(O) = (~hxI'ql), K(O) = (~2xrql) , D(O) = (q~xl"q2),

Po(O) = (~tJ-'q~), Ko(O) --- (q2Fql), Do(O) = (qlFq2),

F(O) = (q~ xFq2) ,

Fo(o) = (~2_r'q2),

where the mesons are bound states of (qq) and F = ~5(Y~) for J P = 0- (1-)-mesons. The angle 0 is a free p a r a m e t e r in t he s t rong in te rac t ion and becomes p robab ly the Cabibbo 's 0 c by e lec t romagnet ic or igin (2). The weak isospin ( = Iv) invar iance w i t h f lavour conserva t ion then generates a 8 U, s y m m e t r y inc lud ing the usua l isospin I as a subgroup. T h e ass ignment (1) can wel l reproduce the A I = 1 rule of K-meson decays as wel l as m a n y o ther p h e n o m e n a at low energies (1,2). I n th is paper , we wi l l examine the non lep ton ic decays of D --~ KT~ and D --~ Kr:= wi th in t he above scheme.

The r e l evan t mode l {1,2) presupposes t h a t the effect ive weak H a m i l t o n i a n conserves t he weak isospin I v in S U2,, symmet ry , though i t v io la tes the f lavour conservat ion. F o r non lep ton ic decays, we pu t the angle 0 (_ ~ 0c) ~ 0, because the Cabibbo 's 0 c is small.

(1) T. EBATA: Phys. Lelt. B, 104, 55, 59 (1981), Tohoku University preprint TU/82/238 (1982). ('*) F. T2~KAGI and T. EB~TA: Tohoku University preprint TU/81/232 (1981); T. ]~BATA and F. TA KAGI: Tohoku University preprint TU/81/233 (1981).

316

FLAVOUR SYMMETRY OF HADRONS AND NO~LEPTONIC DECAYS ETC. 317

The meson mul t ip le t s (1) for 0 = 0 are expl ic i t ly g iven by (~)

(2)

P(O) = , Po(O) = ~-~ 7~ + ~ 7~,

1 1 K ( 0 ) = - - ~ ( K ~ ~ , Ko(0 ) = ~ ( K o + . D o ) ,

D -

1 o D o) Do(0) = 1 _ D ( 0 ) = - - ~ , ~ - ~ ( K ~ 1 7 6

( ) 1 1 1 1 1 1 F(o) = ~-~ 7o + ~-~ 7~ - - ~ 71 , ~o(0) = ~ 7o-- ~-~ 7~ + ~ 71,

where 7s = (u~ + d d - - 2s~)/~/6, 71 = (u~ + dd + s~)/~/3, 7o = (c~) and the o ther pseudo- scalar mesons have the conven t iona l conf igurat ion of (q~). Vec tor mesons (p, K*, D*, etc.) h a v e the same representa t ion except sp in-par i ty as eq. (2) (2).

F o r the dec~y D --~ Kn, we now obta in the effect ive H a m i l t o n i a n wi th CP- and I , - i n v a r i a n c e s (*),

(3) T[~ ~) = ig (DK o - KDo). P ,

where g is ~ coupl ing constant , and the a rgumen t 0 = 0 is abbrev ia t ed (e.g., P = P(0)) . This decay is t hen descr ibed only by one p a r a m e t e r g, so t h a t the re la t ive decay ampli- tudes are def ini t ly de te rmined as

A(D+ --+ K%:+) A(D~ K% ~ (4) - 1 , = - - V ~ .

A(DO-+ K-n+) A(DO-+ K-=+)

W e no te tha t t he resul t (4) satisfies the wel l -known re la t ion (a)

(5) A(D + -->- K%+) - - A ( D ~ K-n+) = ~/~A(DO--+ ~.%o) .

F o r the decay D -+ Knr:, we consider three channels i) D --~ ]s fol lowed by p --~ n=, ii) D --~ K*n fol lowed by f~* --~ ]~n and iii) the (( d i rect ~> coupl ing (4,5). The CP- and

(*) See, also the t h i rd p a p e r of ref. (1). (3) M. PESHKIN and J . L. ROSNER: Nucl. Phys. B, 122, 14~ (1977). ( ' ) S. 1 ) . ROSEN: Phys. Rev. Lett., 41, 3 (1978). (5) 2~. TODA: Phys. Rev. D, 22, 173 (1980).

3 1 8 A. ~oD.~

I ~ - i n v a r i a n t t t a m i l t o n i a n s for D --* ]~p, D -~ K*r~ a n d D -+ ~ K ~ : ~ are g iven b y

H ~ ~) = i/~(D ~ K o - K ~ D o ) �9 0, + / ~ ( D • ~ K ) - - ( ~ D • K)]" p~ ,

(6) H'(~*m = igi(DKo* ~ - - KDo* ). ~ # p + iga(DoKf ~ _ KoD*) . Oa P +

+ g3[(D x ~ ' P ) ' K ~ - - ( O ' P • K ) . D ~ ]

a n d

l=f ~r~=~l ~ h~(D.P)(K.P) + h~(D'K)(P.P) + hoDoKo(P.P ) w

respec t ive ly , whe re ]~, g~ a n d h~ are coup l ing cons t an t s . F r o m t h e s e t I a m i l t o n i a n s , we o b t a i n t h e r e l e v a n t a m p l i t u d e s , w h i c h g ive r ise to severa l r e l a t ions a m o n g t h e m as fol lows:

A(D+ ___> ~.o=+) _ A(D o_~ K*-T:+) = ~/gA(DO-- . ~ . % o ) ,

A(D + _ . ~op+) _ A(D o__. K-p+) = ~ / g A ( D o_~ ]~opo) = 0 , (7)

(1 /~ /2(Aa(D + --* K-r:+7: +) = - -Ad(D+- -> ~%+~o) : A,~(DO__~ K-=+uo) ,

Aa(D ~ -~ ~Orc+r~-) - - Aa(D o --~ K-~+r~o) = ~/~Aa(D ~ --,- 1~%%o) ,

w h e r e Ad(D -* K=r~) deno te s a n a m p l i t u d e of t h e ~ d i rec t ~> coupl ing. T h e f i rs t two r e l a t i ons a re c o n s i s t e n t w i t h t h e usua l i sosp in se lec t ion ru le (AI = 1, A I 3 : 1)(6).

L e t us n o w c o m p a r e ou r r e su l t s w i t h t h e p r e s en t d a t a (7,s). T h e b r a n c h i n g r a t i o s c o m i n g f r o m eq. (4) a re

B (D + __> KOr:+) B (D o __. ~%o) (8) ~ , = 2 ,

B(D~ K-n+) B ( D ~ K-T:+)

where r = F(D~ +) (F(D) is the to t a l decay ra te of D). The (~ best ~> es t imate Of 9 9 +0.9 t h e d a t a is r = ~.~-o.~ (8), b u t one e x p e r i m e n t a l g roup r ep o r t s a s ma l l e r v a l u e of

r = 0.9+~ (s,0). I f t h e l a t t e r v a l u e is cor rec t , t h e f irst r e l a t i o n of eq. (8) is cons i s t en t w i t h t h e d a t a 0.77 • 0.28 (7). T h e second one reflects a t e n d e n c y to a l a rge v a l u e of B(DO__~ ~o~o) ( = (2.2 • 1 .1)% (7)) b u t seems to be a l i t t l e too large. As for D - + K r ~ : , B(D~ ls176 ~ = 0 is a good resu l t , whose d a t a is o ~+o.6o/ . . . . o.i/o (7). W h e n we choose t h e

(6) See, for example, NI. K. GAILLARD and B. "vV. LEE: Rev. Mod. Phys., 47, 277 (1975). (7) ]~. H. SCIIINDLER, M. S. ALAM, A. M. BOYARSKI, M. BREIDENBACII, D. L. BURKE, J. DOREN- BOSCII, J . M. DORFAN, ~ . J . FELDMAN, ~ , E. B. FRANKLIN, G. HANSON, ]~[. G. HAYES, T. I-IIMEL, D. G. HITLIN, R . J . HOLLEBEEK, W. R . INNES, J . A. JAROS, P . JENNI, R . R . LARSEN, V. LOTII, M. L. PERL, B. ~ICIITER, A. ROUSSAIRE, D. L. SCIIARRE, R . F . SCIIWITTERS, J . L. SIEGRIST, H . TAUREG, M. TONUTTI, R . A. VIDAL, J . M. ~VEISS, t L ZACCONE, G. ~kBRAMS, C. fix. BLOCKER, A. BLONDEL, W. C. CARITHERS, %V. C]KINOWSKY, M. %V. VOLES, S. COOPER, W. E. DIETERLE, J. B. DILLION, M. ~V. EATON, G. GIDAL, G. GOLDHABER, A. D. JOHNSON, J . A. KA:DYK, A. J . LANKA~ORD, ]~. E. ~r 1VI. ]~. ]NVELSON, C. Y. PANG, J . F . PATRICK, J . STRAIT, G. H . TRILLING, E. N. YELL& a n d I . VIDEAU: Phys. Rev. D, 24, 78 (1981); G. H. TRILLING: Phys. Rep., 75, 57 (1981). (s) G. KALMUS: Talk at the X X I International Conference on High Energy Physics (Paris, 1982). (9) ~d~. ABE, T. C. BACON, J . BALLAM, L. BERNY, A. V. BEVAN, H . H . B1NGHAM, J . E. BRAU, D. B R I C I I ,

W . 2r BUGG, J . BUTLER, W . CAMERON, J . T. CARROLL, C. V. CAUTIS, J . S. CHIMA, H . O. COliN, D. C. VOLLEY, G. W. CONDO, S. ])ADO, R . DIAMOND, P. J . DORNAN, R . ERICKSON, T. FIEGUTH, ~c~. C. FIELD, L. FORTNEY, B. FRANEK, ~r. FUJIWARA, R . GEARIIART, J . GOLDBERG, G. P . GOPAL, A. T. GOSIIAW, E. S. HAFEN, V. HAGOPIAN, G. HALL, E. R . HANCOCK, T. HANDLER, H . ft. HARGIS, E. L. HART,

FLAVOUR SYMMETRY OF I tADEONS AND NONLEPTONIC DECAYS :ETC. 319

coupl ing g~ so as to get (*)

(9) A(D +--~ Is K*-~+) = 1/V/g,

the b ranch ing ra t ios of D -+ K . r : are g iven by (~)

(I0)

B(K-r:+~+) = c @ a / 3 ,

B(K~ ~ : c/2 -~ b 4- a / 6 ,

B(K~ -) = r-I(d -t- 2a/3) ,

where B(K~r~) s tands for B(D --~ ] ~ ) , and

a = I g ~ - g : [ : / [ ( V N - a):r(/)+)], (11)

I = lhil,/r(D+),

b = I/~-- 2]e[2/[2F(D+)],

d = lh~-- h~I2/[2F(D+)].

Unfor tuna te ly , there are too m a n y pa ramete r s to get a definite es t imate . We ten- t a t ive ly , however , pu t these pa ramete r s to be

a = 3.0, b : 4 .0 , c = 5 .3 , d = 1.4,

then our resul ts (%) are

(12)

B(K-~+r: +) -- 6.3 (6.3 j= 1.5),

B (K:=+~ ~ = 8.6 (8.5 • 3.2) ,

B(I~~176 ~ = 2 .3 ,

B(K~ ~ ~ 7.2 (12.9 ~ 8.4) ,

B(K%+r~ -) : 3.8 (3.8 • 1.2),

where we use r = 0.9, and the expe r imen ta l va lues (7) are in parenthesis . I t is possible to say t h a t t he present mode l contains a s t ruc ture consis tent wi th the data , t hough several coupl ing constants r emain st i l l to be ambiguous.

P. HARIDAS, K. HASEGAWA, T. HAYASHINO, D. Q. HUANG, ]%. I. HULSIZER, S. ISAA0SON, M. JOBES, G. E. KALMUS, D. P . KELSEY, J . I~ENT, T. KITAGAKI, P . LANG, J . LANNUTTI, A. LEVY, P. W. LUCAS, W. A. MANN, W. MARUYAI~L%, M. MACDERMOTT, R . 1V~ERENYI, l~. MILBURN, C. MILSTENE, K . C. MOFFEIT, Z. J . IV~URRAu A. INT~kPIER, S. NOGUCHI, F . OCHIAI, S. O. NEALE, A. P . T. PALOUNEK, I . A. PLESS, M. RABIN, P. RANKKN, %V. J . ROBERTSON, A. I-I. ROGERS, E. RON2~T, H . RUDNICKA, T. SATO, J . SCHNEPS, J . SHANK, ~k. M. SHAPIRO, C. SINCLAIR, I~. SUGAHARA, t . SUZUIKI, 1~. TAKAttASHI, K . TAMAI, S, TANAKA, S. TETHER, H . B. WALD, W. D. VV~ALKER, ~4I. WIDGOFF, C. G. WILKINS, S. WOLBERS, C. A. ~r Y. WU, _2k. YAMAGUCHI, R . K . YAMAMOTO, S. YAMASHITA, G. YEKUTIELI, Y. YOSHI- 2r G. P . YOST a n d ]=[. YUTA: P o s t e r e x i b i t i o n a t t h e X X I International C~nlerence on High Energy Physics (Par i s , 1982); see a lso , Phys. Rev. Lett., 48, 1526 (1982). (*) E q u a t i o n (9) p r a c t i c a l l y g ives a f a v o u r a b l e choice of g d see, A. TODA: Soryushiron Kenkyu, to be published.

3 2 0 A. TODA

I f we p u t 0 = 0 e i n s t ead of 0 ~ 0, our mode l genera tes , as a rule, too m a n y effect ive c o u p l i n g s - - f o r example , 13 coupl ings for D --~ Kp to first order of sin 0 c. In t h e above calcula t ion, therefore , we neg lec ted th is order of values even if we can exp la in t h e s m a l l v a l u e of B ( D ~ K0po). F ina l l y we r emark t h a t we can get f avourab lc resul ts , also for t h e case of r = 2.2 excep t t he first r e la t ion of eq. (8).

The au tho r is i n d e b t e d to F. TAKAGI for m a n y he lpfu l discussions. He has also benef i ted f rom conversa t ions w i t h T. EBAT~.

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