P H Y S I C A L R E V I E W D V O L U M E 1 8 , N U M B E R 9 1 N O V E M B E R 1 9 7 8

Contradiction in nonleptonic decays of D mesons

Mituaki Katuya Laboratory of Physics, Shizuoka Women's University, Yada 409, Shizuoka, Japan 422

(Received 30 May 1978)

We point out a contradiction among the uniformly populated Dalitz plot for D ++.rr+.rr'K -, a relation derived by applying the soft-pion theorem to D ++.rr+n+K -, and the estimates of ratios r ( D O + . r r o ~ O)/r(D O +.rr+K -) and r ( D O+all)/r(D ++all) accepted currently.

In this short paper we point out an inevitable pion theorem2 permits us to write for the D+ contradiction among the uniformly populated Da- - a+ n+ K- amplitude, litz plot for D+- a + a + ~ - , ' a relation derived by applying the soft-pion theorem2 to I)+- n+n+K-,

l im a[D+ (q) - a+ (p1)9+ ( p 2 ) ~ - ( p 3 ) 1 P1O'P2-' 0

and the estimate^^-^ of rat ios r (DO- n O ~ O ) / r ( ~ O -n+K-)-& and r (DO-al l ) / r (D+- all)? 1. The =A(+ +-)[I +a(++-) (m,2 -so)] former rat io i s obtained by the quantitative pre- diction7 of quantum chromodynamics (QCD) on the i enhancement of the 20 of SU(4) with respect to the =- [a@+ - a+f?O) -a(DO- T+K-)] 84. The latter rat io i s f rom the naive quark-par- f ff - ton model.

Now we define the D+- a+a+K- amplitude a s

(9' ( p 1 ) ~ + ( p z ) ~ - ( p 3 ) I H , I D+ (4)) Here use i s made of the partially conserved axial- vector current (PCAC) relation defined a s

- - a [ D + ( q ) - ~ ' ( p , ) ~ ' (P,)K-(PS)I , (I) [ (2n)122q02~,2~22~3]1/2

sPn;(x) = &m:y(x), (7) a[D+(q)- fl+ (p1)a' (p2)K-(p3)1

=A (++ -)[l +u(+ + -)(s, -so)] , (2) and the last equality in Eq. (6) comes from the hI=I property8 of the AC = A S = 1 nonleptonic in-

where s,= (q -p,)' and the constant s, i s defined terac tion.

a s From Eq. (6), we can obtain

The decay ra te i s given by

I. (D' - a+ a+K-)

However, since [a(+ + -)(s, - so)( << 1 in the physi- cally allowed region a s reported by Wiss et al.,' we may neglect the second te rm in Eq. (4).

We define the D- nK amplitudes a s

where i, j , and k a r e charge indices. The soft-

X r(DO- all) r (D+ - all) ' (8)

where B means the branching rat io and

1 mD2j/[1 + I u ( + + - ) ( s n - ~ ~ ) l ~ ] d ~ l d ~ z 1 YE- -

4n2 {[(mD + n ~ , ) ~ - m,2][(mD - mff)2 - m,2]}1/2 ft

for f, 130 MeV and u(+ +-) = O . From the study of the color degree of freedom

in the weak interactions, Altarelli et a1 . g and also Ellis et a1 ., have pointed out

( Z U ~ H , , Ixc) =X+C , (10)

and

(67s IH,, luc) =x-c , (11)

where

3510 O 1978 The American Physical Society

18 - C O N T R A D I C T I O N I N N O N L E P T O N I C D E C A Y S O F D M E S O N S 3511

simple quark recombination scheme proposed by Cabibbo and ~ a i a n i , ~ of the hadronic current- current scheme proposed by Koide and the pre-

+~cI~( '~) ( ' c ) +(zd)(Cc)l} 3 ( I2) sent author,6 even in the D*-. F * - D o ~ ~ domin- x ='( * - 3 2 c ~ 4 * ~ 2 0 ) 9 (13) ance model.

and C is a parameter to be determined by experi- From Eqs. (8) and (14), we find

ment. The relation

a[DO- soKO] A?- a[D0- n + ~ - ] - ' (1 4)

x l l + u ( + + - ) ( m , 2 - ~ ~ ) 1 ~ . (15) i s obtained by various ideas: it1 the framework of the factorization approximation method revived The experimental values,10 B(D+- n+ntK-) = (3.9 by Ellis et a1.3 and by Fakirov and Stech,' of the * l . O ) O / o , B(D'- n+K-) = (2.2* 0.6)% lead to

(9.1* 3.4, for u(++-) = 0

(0.59* 0.22, for o(++ -) = l . O GeV-'

where s0=0.730 GeV2 i s used. (This i s obtained analytically in the mass less limit of pions.) The case u(+ + -) = 0 gives the completely uniform Dalitz plot, and the case u(+ + -) = 0.5 G e V 2 gives the Dalitz plot with depletion in the region where the kaon sha re s a large amount of the available kinetic energy. The case u(+ + .-) = 1.0 GeV-2 can be ruled out on about the 3-standard-deviation level.ll

Note that I x - / x + ~ ~ c 1 holds sxnce c2,> 0 and c,,> 0 in QCD, moreover, c,,/c,, = 2.5-4.0. Thus the estimate lx-/~+1~- 10-1-10'-2 i s accepted currently. From this estimate together with a

naive quark-parton model prediction DO - a l l ) / r ( ~ + - all)= 1,

has to be recognized in the Equation (17) i s in contradiction with Eq. (16) and i t is our purpose in this short paper to point out this fact.

The author i s much indebted to Professor Y. Koide, Professor M. Konuma, and Professor Y. Hara f o r helpful discussions and useful comments. This work was supported by the Scientific Research Fund of the Ministry of Education (1978).

'J. W. Wiss et al . , Phys. Rev. Lett. 37, 1531 (1976). 's. L. Adler and R. F. Dashen, C u ~ r e n t Algebra (Ben-

jamin, New York, 1968). F o r application of current algebra to charm decays, see R. L. Kingsley, S. B. Treiman, F. A. Wilczek, and A. Zee, Phys. Rev. D 12, 106 (1975).

3 ~ r ~ l l i s , M. K. Gaillard, and D. V . Nanopoulos, Nucl. Phys. m, 313 (1975).

4 ~ . Cabibbo and L. Maiani, Phys. ]Lett. z, 418 (1978). 5 ~ . Fakirov and B. Stech, Nucl. Phys. M, 315 (1978). 'IV. Katuya and Y. Koide, Shizuoka Women's University

Report No. SH-77-05, revised version (unpublished), 'M. K. Gaillard and B. W. Lee, Phys. Rev. Lett. 33, 108

(1974); G. Altarelli and L. Maiani, Phys. Lett. B, 351 (1974).

8 ~ . L. Kingsley, S. B. Treiman, F. A. Wilczek, and A. Zee, Phys. Rev. D 11, 1919 (1975).

'G. Altarelli, N. Cabibbo, and L. Maiani, Phys. Rev. Lett. 35, 635 (1975).

"1. Peruzzi et a l . , Phys. Rev. Lett. 39, 1301 (1977). " ~ o u g h estimate f rom the figure in Ref. 1.