L E T T E R E AL NUOVO CIMENTO VOL. 16, N. 12 1 7 Luglio 1 9 7 6

N o n l e p t o n i c D e c a y s o f C h a r m e d P a r t i c l e s .

M. KATUYA and Y. KOH)~

Laboratory o/ Physics, Shizuoka Women's University - Yada 409, Shizuoka, Japan 422

(ricevuto il 10 Maggio 1976)

The weak-interaction scheme proposed by GLASHOW, ILIOPOULOS and MAIANI (GIM) (i) promises with respect to both understanding the suppressions of the strangeness-changing neutral current in semileptonic decays and interprets the ~-particles (~) as hidden-charm states. In the GIM theory, the charged weak hadronic current which is a member of the 1_55 representation of SU4 is

(1) J~ = ~7.(1 + 75)(cos 0 d ~- sin 0s) ?- ~ ( 1 ~- y s ) ( c o s O s - - s i n O d ) ,

where 0 is the Cabibbo angle and u, d, s and c represent quark fields. The bilinear form of this current contains two parts belonging to a 20 dimensional and an 84 dimensional representation. In the nonleptonic decays of ordinary hadrons, there exists the selec- tion rule, A1 = 1, which could be generalized as the 8 dominance in S U 3 and as the 15 and/or 20 dominance in S U 4. Therefore, it is natural in the GIM scheme, to assume

(~) S. L. GLASHOW, 3. ILIOPOULOS a n d L. MAIANI: Phys. Rev. D, 2, 1285 (1970), hereaf ter referred to as GIM. (3) J . J . AUBERT, W. BECKER, P. J . BIGGS, J . BURGER, M. CREN, G. EVERHART, P . GOLDHAGEN, J . LEONG, T. MOCORRISTON, T. P~I'[OADES, M. ROHDE, S. C. C. TING, S. L. TvVu a n d Y. Y. LEE: Phys. Rev. Left., 33, 140~ (1974); J . - E . AUGUSTIN, A. M. BOYARSKI, M. BREIDENBACIt, F. BULOS, J . T. DAKIN, G. J . FELDMAN, G-, E. FISCHER, D. FRYBERGER, G. HANSON, B. JEAN-MARIE, R . R . LARSEN, V. LOTH, H . L. LYNCH, D. LYON, C. C. MOREHOUSE, J . M. PATERSON, M. L. PERL, B. ~ICHTER, r . RAPIDIS, R . f . SCttWIT- TERS, W . M. TANENBAUM, F. VANNUCCI, G. S. ABRA~IS, D. BRIGOS, W. CHINOWSKY, C. S . PRIEDRERG, G. GOLDHABEI~, ]~. J . HOLLEBEEK, J . A . KADYK, ]~. LULU, F. PIERRE, G. H . TRILLING, J . S. WHITAKER, 5. WISS a n d J . E. ZIPSE: Phys. Rev. Left . ,33, 1406 (1974); C. BACCI, R . BALDINI CELIO, M. BERNARDINI, G. CAPON, a . DEL FABBRO, M. GRILLI, E. IAROCCI, L. JONES, M. LOCCI, C. MENOUCCINI, G. P . MURTAS, G. PENSO, G. SALVINI, M. SPANO, M. SPINETTI, B. STELLA, V. VALENTE, B. BARTOLI, D. BISELLO, B. ESPO o SITO, F. FELIOETTI, P . MONACELLI, M. NIGRO, L. PAOLUZZI, I . PERUZZI, G. PIANO MORTARI, M. PICCOLO, F . RONGA, •. SEBASTIANI, L. TRASATT], F . VANOLI, G. BARBARIN0, G. BARRIELLINI, C. BEMPORAD, ~=~. BIANCASTELLI, M. CALVETTI, M. CASTELLANO, ~. CEVENINI, f . COSTANTINI, P . LARICCIA, P. PARASCAN- DALO, E . SASS], C. SPENCER, L. TORTORA, U. TROYA a a 4 S. VITALE: Phys. Rev. Le~t., 33, 1408 (1974); G. S. ABRAI~IS, e . BRI@GS, ~V. ()HINOWSKY, C. E. FRIEDBERG, G. GOLDHABER, R . J . HOLLEBEEK, J . A . KADYK, A. LITKE, B. LULU, P. PIERRE, B. SADOULET, G. H . TRILLING, J . S. WHITAKER, J . WISS, 5. E. ZIPSE, 5.-]~. AUGUSTIN, A. M. BOYARSKI, M. BREIDENBACH, F. BULOS, G. J . FELDMAN, G. E. FISCHER, D. FRYBERGER, G. HANSON, B. JEAN-MARIE, R . ~:~. LARSEN, V. LUTH, H . L . LYNCH, D. LYON, C. C. ~OREHOUSE, J. M. PATERSON, M. L. PERL, B. ~ICHTER, P. ~APIDIS, ~:~. F . SCHWITrERS, W . TANEBAUM and F. VANNUCCI: Phys. Rev. Left. , 33, 1453 (1974).

357

3 5 8 M. KATUYA a n d Y. K O I D E

t h a t the 20 representa t ion is domina ted (a). Along the line of this thought , several au thors have a l ready discussed the AC = 0 and AC = 1 nonleptonic decays (~).

The purposes of this le t ter arc i) to give supplementa l discussion on the nonleptonic decays in the SUa symmet ry , ii) to derive some new sum rules of nonleptonic-decay ampl i tudes and iii) to predic t the decay rates of charmed hadrons.

Denote the 20 representa t ion in the GIM theory by traceless tensor as follows:

(2) T r~ b) ~ COS 30T[~ [~ 31 _}_ sin 0 cos O(T[~ [~ ~) -- Tr~ ~ a]) § h.c. [o d]

Here and hereaf te r we neglect the t e r m propor t iona l to sin ~ 0. The effective Hami l - ton ian for 20(�89 +) ~ 20(�89 +) + 15(0-) which t ransforms as 20 m a y be wr i t ten as

(3)

~ c l ~ [ i d ] p i t ~ r ] :~[ i i ]pd A_ ~ i ]t~[icJpd l ~ i l:~[cd] P i l

Here we h a v e dropped the reference to the spinor s t ructure . I f t he effective Hami l ton ian (3) is assumed to be of the nonder iva t ive type ,

B(]s + ]~75)BP, the CP- invar iance implies e = ] = g = h = a+ = b+ = c+ = d+ = 0 for the par i ty -v io la t ing par t and a_ = b_ = v_ = d_ = 0 for the par i ty -conserv ing par t .

IWASAKI and ALTARELLI et al. (3) h a v e poin ted out the re la t ion A_:X+: ~ _ - = = I : - - V ~ : - - 2 for the s-wave ampl i tudes . W e th ink t h a t t he agreement of the rela- t ion wi th t he exper iment is not sat isfactory. Note t h a t t he exper imenta l re la t ion S(A_):S(Z+):S(E --) = 1 . 0 : ( - - 1 . 0 • -4-0.1), while the A I = �89 rule and the Lee-Sugawara re la t ion hold precisely in the s-wave decay ampli tudes .

We will discuss in this le t ter another possibi l i ty t ha t the effective Hami l t on i an is assumed to be of the der iva t ive type, BTt,(]v -4- ]~,5)B~'P. I t is in teres t ing t h a t this choice gives I/~(z-)l--- IL(Z$)l-~ I/I, phenomenologicaUy (5), where ] is a coupling con- s tan t for 0 - ~ 0 - + 0-. (See eq. (5).) The CP- invar iance for par i ty -v io la t ing and -conserving in terac t ion Hami l ton ians implies a_ = b_ = c_ = d_ = 0. A l though twe lve paramete rs seem to be invo lved in the effective Hami l ton ian (3), only six pieces are independen t in un i t a ry space. (Note 2__00~ | 2_00~ | 15 = 7 • 8 4 Q 6 • 2 O O 9 • G . . . . ) W e give discussion in this Le t t e r only for the charmed- 1+ baryons ~r I t ~ , = 0, Y = 0, C = 1, 3*) and ~+.01~+ 7 , o ,.~ , ~ = �89 Y = - - 1, C = 1, 3*) and charmed pseudoscalar mesons D + , 3 ( 0 - , I = � 8 9 Y = 0 , C = I , 3 * ) and F+(0 - , I = 0 , Y = I , C = 1 ,3" ) , since these s tates are probable candidates to be stable against the s t rong in te rac t ion (6).

(*) Y. IWASAKI: Phys. Rev. Left. , 34, 1407 (1975); G. ALT.~RELLI, N. CABIBBO a n d L. MAIANI: Phys. Lell., 57 B, 277 (1975). (t) Y . IWASAKI: Phys. Rev. Lett. , 35, 749 (1975); G. ALTARELLI, N. CABIBBO a n d L. MAIANI: Nucl. Phys. , a8 B, 285 (1975); see a l so re f . (3). (~) T. H&YASHI, V. KOIDE, S. OOAWA, H . OKONOGI a n d M. u Prog. Theor. Phys., Suppl. , E x t r a N u m b e r , 381 (1968). (0) Y. IWASAKI: Phys. Rev. Left. , 34, 1407 (1975).

)IONLEPTONIC DECAYS OF CHARMED PARTICLES 359

Sum rules for A + =+ and ~o decays are:

(4)

5 /(A+-+ Arc ~) = - - c t g O { ~ [ - - / ( E n ) - - ~ ] ( A _ ) ] + --4- v/~/(E o + ) / ,

f l [ ~ 1 ] ? / ( Z + ) } ,

J

3 V~ E+ /

1 /(A +_+ EOK § =/(Eco_+ Z+K-) = ctg 0 ~ / ( E ~ ) ,

[ ' ] /(E+-+ E~ +) = - - / ( E +-+ E+K ~ = - - ctg 0 /(A_) - - ~ / ( Z ~ ) ,

1 %/6 + 1 + / ( E o ~ E~ ~ = c t g O { ~ [ - - /(E=) - - ~ /(A )] + - ~ - / ( Z o ) - - ~ / ( Z + ) } ,

/(,=)r176 = - - e r g O ~ ~/ (A_) + - ~ - / ( Z o ) + ~ / ( Z ; ) ,

1 + / ( E ~ ~ = c t g 0 { ~ - - ~ [ - - / ( E 2 ) - - / ( A _ ) ] + ~ / ( Z o ) } ,

1 L/(~o +~o~o)=-ctgO~/(A ),

where ] means /v, /v and /A- Subs t i tu t ing the exper imenta l values for /v and /a from the nonleptonic decays of

hyperons, we predict decay rates of charmed baryons as a funct ion of mass, which are shown in fig. 1 and 2.

I t is wor th while to note tha t the rates for A + ~ Z+= ~ and E~ are two orders of magn i tude smaller t h a n the others. This is due to the fact t ha t the Lee-Sugawara relat ion for hyperons holds precisely bo th for /v and /~ (:).

Next we discuss nonleptonic decays of charmed mesons, 15(0-)-+ 15(0-) + 15(0-}. The CP- inva r i an t effective Hami l ton ian which t ransforms as 20 and 84 vanishes if i t involves no der ivat ive operators (s). However, if we int roduce the effective Hami l ton ian with derivat ive operators

(5) ~l~rP[c[ab] ~ locipi~t~pd ~#7:)tpa t H~f

(7) If the nonderivative type-Hamiltonian is employed, we can get ]v(A +->y,+~~ and Y)~+)~_ 0, however, the decay rates for them are not so suppressed because of the considerable s-wave amplitude. (s) In the SU, limit a pseudoscalar meson cannot decay to two pseudoscalar mesons for the same reason as the K ~ meson caalnot decay to 2~ in the SU, limit. See M. GELL-MANN: Phys. Rev. Left., 12, 62 (1964).

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NONLEPTONIC DECAYS OF CHARMED PARTICLES 3 6 1

we can obtain nonvanishing amplitudes for 0--+ O- + 0- , for example, ](D~ K - n +) = = - - ] cos 20(m~-- m~). From experiment for Ks-+ n+u-, we get

I/I = 3 . 9 2 9 . 1 0 -4 MeVl(m~- m~).

Using this value we predict the decay rates for charmed mesons as a function of mass, which are shown in fig. 3.

Note that the decay rate for D+-> K~ + is two orders of magnitudes smaller than the others in our model, which is pointed out by several authors (9).

In anticipation of discoveries of charmed particles, we predicted the decay rates for them by assuming the 20 dominance and the derivative-type Hamiltonian. We con- clude by emphasizing tha t the crucial test of our model is the relative suppression of the decay rates for A+-~Z+r: ~ and E~ + as well as D+-+ K~ +.

We would like to express our sincere thanks to Dr. T. KARI•O for useful and sti- mulating discussions.

( ') T. HAYASttI, !~/[. NAKtLGAWA, H. NITTO a n4 S. 0(~AWA: Prog. Theor. P h y s . , 49, 351 (1973); M . K . GAILLARD, B. W. LEE aD.(]. J . ~ . ROSNER: ]:~ev. Mod . P h y s . , 47, 277 (1975).