PHYSICAL REVIEW D VOLUME 42, NUMBER 5 1 SEPTEMBER 1990

Predictions for the quark-mixing doubly suppressed decays of charmed mesons

Ling-Lie Chau Department of Physics, University of California, Davis, California 95616 and Los Alamos National Laboratory, Los Alamos, Neu! Mexico 87545

Hai-Yang Cheng Physics Department, Brookhauen National Laboratory, Upton, New York 11973

and Institute of Physics, Academia Sinica, Taipei, Taiwan 11529 (Received 30 November 1989)

Based on the results from analyzing the currently available data of quark-mixing nonsuppressed or singly suppressed decays of charmed mesons, the branching ratios of the quark-mixing doubly suppressed charm decays are predicted using the quark-diagram scheme.

Recently, considerable progress has been made in the experimental observation of quark-mixing nonsuppressed or singly suppressed charmed meson decays. Their impli- cations, both experimental and theoretical, had been studied using the quark-diagram Here we continue such studies and obtain further new results. I t is now hopeful that the mixing-matrix doubly suppressed charm decays will be within reach by experiments in the near f ~ t u r e . ~ The quark-mixing doubly suppressed de- cays are integrated parts with the quark-mixing non- suppressed and singly suppressed decays in the analyses using the quark-diagram ~ c h e m e . ~ Here we report the in- teresting results that most of the doubly suppressed de- cays can already be predicted based upon the current nonsuppressed and singly suppressed data. These results ought to be very useful for experimental planning, obser- vation, and for furthering our understanding of nonlep- tonic weak decays.

All nonleptonic decays of charmed mesons can be de- scribed by amplitudes from the six distinct types of quark

A , the external- W-emission amplitude; 3, the internal- W-emission amplitude; @, the W-exchange amplitude; a, the W-annihilation amplitude; 6, the hor- izontal W-loop amplitude; and 3, the vertical W-loop am- plitude. These are distinct diagrams with all QCD gluon effects included. The quark-diagram amplitudes for quark-mixing doubly suppressed D', D O , and D: decays are given in Tables 1-111 for vector-pseudoscalar ( V P ) , pseudoscalar-pseudoscalar ( P P ) , and vector-vector ( V V ) decays, respectively. Note that we have included the effects of final-state interactions as well as some SU(3)- breaking effects. Our predicted branching ratios are also given in Tables 1-111. [For decays involving SU(3) sing- lets in the final states, there exist in principle the hairpin diagrams, denoted in these tables by a subscript h. How- ever, we ignore them here since there is no clear experi- mental evidence for their e ~ i s t e n c e . ~ ]

D-VP decays. Our analysis3 of recent data of a charmed meson decaying into a vector meson and a pseu- doscalar meson D- VP have established the following: From the measured decay rates of ~;-q5r+, D f --+4rf, K*OrTT+, and DO-@?O we obtained the values for various quark-diagram amplitudes:

where the primed amplitudes denote the case that the vector meson arises from the decay of the charm quark, and the underlined amplitude refers to the amplitudes of graphs involving the creation of Fs. From the measure- ments of D: -K*OK + and Do-K *-.rrf,K*OaO we get

For unprimed amplitudes, three sets of solutions3 can be derived from the observed rates of D t - p f K O , DO-~KO, p°KO, and p+K- . However, the recent ARGUS result7 B ( D ~ + K ~ ~ * ~ ) < 3 X implies @ - @'. This favors the solution

The other two solutions give too large values for the am- plitude @. The small rate for D,+-p+ r0 indicates3 B=B1. The recent E69 1 result8 for negligible ~ , f -r'w implies a1 >> = a'1. Motivated by @-Set, though not very compelling, we will assume that a)--a)'--+a) for the purpose of giving specific D --+ VP rates.

With the quark-diagram amplitudes and the isospin phase-shift differences given by Eqs. (1)-(3) , we are able to compute the branching fractions for all doubly suppressed D - VP decays except D, -K * K . The re- sults are rhown in Table I. Note that our predictions do

9 1837 @ 1990 The American Physical Society

1838 BRIEF REPORTS

take into account effects of final-state interactions ob- tained from data analyses in the quark-diagram scheme, as indicated by the phase shifts in Eqs. (2) and (3). Without final-state interactions the predicted branching ratio for D +K *'a0 will be smaller by a factor of 3. Such final-state-interaction effects are still beyond the capabili- ty of most theoretical model calculations.

It is also worth emphasizing that our analysis gives non-negligible the so-called nonspectator diagrams, the W-annihilation diagram D and the W-exchange diagram, C and C', as evidenced by Eqs. (2) and (31, in contrast with most model ~ a l c u l a t i o n s . ' ~ ~ ' ~ For example, in our scheme D + +$K + arises solely from the W-annihilation diagram a.

D-PP decays. With the new data of D O - R ~ ~ ' , D: +rf V, and D: -rf v', a unique set of solutions for the D -PP quark-diagram amplitudes A , 53, @, 23, e, and a can now be ~ b t a i n e d . ~ More precisely, the six am- plitudes13 can be determined from the measured decay rates of D++KOT+, ROK +, DO+RO~, ROT', K - T + , K'rO, and D: +KOK +,rf rl (in units of GeV):

Evidently, the nonspectator contributions, especially the W-exchange diagrams, play an essential role in D-PP decays.

Using the quark-diagram amplitudes given by Eq. (41, it is straightforward to compute the branching fractions for quark-mixing doubly suppressed D + P P decays (see Table 11). Again, it should be stressed that nonspectator diagrams and final-state interactions are included in our calculations. In addition to the prediction based upon detailed information of Eq. (41, we have also the following general relations between the doubly suppressed decays and nonsuppressed D o decays by comparing Table I1 with those of nonsuppressed decays given in the tables of Ref. 3:

TABLE I. Doubly suppressed charm-meson decays into a vector boson and a pseudoscalar meson.

Amplitudes with SU(3) Predicted symmetry and no Amplitudes with SU(3) breaking

Channels branching ratios final-state interactions and final-state interactions

0.5 x lo-' -(s112 x {e+ D ~ ) + {e + ~ ~ } e ~ ~ ' ~

2.5 x lo-' - ( l / & ) ( s l ) * X {A' + D' + 2Dh) + { A 1 + D' + 2 ~ ~ ) e ~ ~ ~ ~

- ( l /&) ( s l ) ' x { A + D - 20 ' ) + { A + D - 20' + 20; - 2 ~ ~ ) e ~ ~ ~ " ' - - ( l / & ) ( s ~ ) ~ x { A + D + D' + 30;) + { A + D + D' + 20; + ~ i ) e ~ ~ ~ ' ~ ~ -

1.7 x lo-' t) = t)8 cos B + t)o sin 0; t)' = -08 sin 0 + rlo cos 0; B 1= 20'

2.0 x lo-' -(sl)' x ( 8 + Dl) + { ( B + D l ) + (A' - 8 - 2D1) (1 /3 ) (1 - e - " @ ~ ) ) e ~ ~ [ E

1.0 x lo-' ( 1 / & ) ( ~ 1 ) ' X { A t - D') + {(A' - D') - (A' - 8 - 2D1) (2 /3 ) (1 - e-iApK))ei6:;

1.1 x lo-' -(sI) ' x ( 8 ' + D) + ( ( 8 ' + D ) + i ( A - B' - 2D) (1 - e - i ~ ~ . = ) ) e " J ; '

2.2 x lo-' ( l / f i ) ( s l ) ' x { A - D) + { ( A - D) - ! ( A - 8' - 2D) (1 - e- '~~' .))e 's;; '

D O - 4 K O

+ w K O

+ K*+a-

+ K*OaO

+ K ' O , , ~ - K*Ot)O - K't )

4 p-K+ - p O ~ O

0.2 x lo-' -(sl)' x (2 + Ch) + {C + ~ ~ ) e ~ ~ ' ~

0.9 x lo-' - ( l / & ) ( ~ l ) ' X ( 8 + C' + 2Ch) + { B + C' $ 2ch)eisUK

1.9 x lo-' - ( ~ 1 ) ~ X { A + C ) 4 { ( A + C ) - f ( ~ + ~ ) ( 1 - e-iA~'-))ei6P/;'

1.2 x lo-' - ( ~ / f i ) ( s l ) ~ x { B - C ) + ( ( 8 - C ) - + ( A + ~ ) ( 1 - e - i ~ ~ . * ) ) e ~ ~ J ; ' ,6K-va

- ( l /&)(s l ) ' x (0' + C - 2L') - {B ' + C - 2c' + 2Ck - 2CL)e - - ( ~ / & ) ( s l ) ~ x {B' + C +L1 + 3C;) 4 {B' + C + L' + 2Ci + ~ ; ) e ~ ~ ~ " ' ~ -

0.2 x lo-' t) = cos 0 + t)o sin B ; t)' = -q8 sin 8 + qo cos e; B 1= 200

0.9 x lo-' -(81)' x {A' + C') 4 (A' + C' ) - 2 /3 (A1 + 8 ) ( 1 - eiApK))ei6:E

0.4 x lo-' - ( l / & ) ( s ~ ) ~ x ( 8 - C') - ( ( 8 - C') - 1 /3 (A1 + 8 ) ( 1 - eiApK))ei6:E

K. K

D: - K * + K O 0.7 x lo-' -(si) ' x { A + 8 ) + { ( A + 8 ) + ;(A' + 8 ' - A - 8 ) ( 1 - e - i A ~ ' ~ ) ) e ' 6 '

+ K*'K+ 0.6 x lo-' -(51)' x { A ' + B') + {(A ' + 8') - $(A ' + 8' - A - 8 ) ( 1 - e - i A ~ ' ~ ) ) e i s f ' K

42 BRIEF REPORTS 1839

TABLE II. Doubly suppressed charm-meson decays to two pseudoscalars.

Channels Predicted

branching ratios

Amplitudes with SU(3) symmetry and no

final-state interactions Amplitudes with SU(3) breaking

and final-state interactions

P+ -+K°x+

^K+r,

2.1 x 1 0 - 4

3.7 x 1 0 - 4

1.5 x 10™*

0.1 x 10~*

- ( s i ) 2 x {B + P}

(l/v/2)(5 l)2 x {A - P}

- ( l /^) (s, ) 2 x {A - P} - ( l /v /3)(S l )

J x {A + 2D + 3PJ

r\ = f/g cos 9 + fjo sin 8

TJ' = -t78sinfl-|-fjocos0; 0 w 20°

)}eil& {(8 + V) + (A - B - 2P)(1/3)(1 - e-«A" „ .

{(A-P)-(A-B- 2P)(2/3)(1 - e-iAK')}e</

{A-V+2Dk~2Vk}ei5K"t

{A+2D+2Vk + V£eisK,,°

D°

K°vs

K°r,'

1.2 x 1 0 - 4

0.6 x 10~4

0.4 x 1 0 - 4

0.6 x 1 0 - 4

-(si)2x{A + C}

-(1/V2)(S1)2 x{B-C}

- ( 1 / N / 6 ) ( S I ) 2 X{B-C}

- ( l /v /3) ( 5 l )2 x {B + 2C + 3Ch}

tj = rig cos 6 4- i/o sin 6

rf = -r\% sin9 + r/o cos 0; 0 «s 20°

same as for K~ir+

same as for K°r°

same as for K°r)s

same as for K°rio

P+ ^K+K° 0.3 x 10~4 -(*i)2 x {A + 8} {(» + B)K*

r(D°->/:+7r-)= r ( D u ^ ^ ~ i r + ) , r(Du-^/:uiru) nz>u—tfV) ,

0 „ w r(Du^Kur]) r(Du-+KuT)), r(D»-*Ku'o')= HD0—*V> ,

where

S\~\vi*d\-\vus\> c , « | K c * | w | F „ d | , r] = r]icose + r](fim6, 17'=-778sin0 + T;ocos6>, 0^20° .

(5)

(6)

Channels

TABLE III. Doubly suppressed charm-meson decays into two vector bosons.

Predicted branching ratios

Amplitudes with SU(3) symmetry and no

final-state interactions Amplitudes with SU(3) breaking

and final-state interactions

(j>K*+

uKt+

p+Kt0

p°K>+

- ( * i ) 2 x {£+£>„}

-(l/V2)(s1)ix{A + t> + 2Dh}

- ( 3 i ) 2 x { B + P}

(l/v^)(3i)2 x {A - P}

{B.+Pk)eis*K'

{A + P+ 2Dh}eii"K'

{{B + P) + (A-B- 2P)(1/3)(1 - e - < ^ W ) } e < ^

{(A-P)-(A-B- 2P)(2/3)(1 - «-'*,«:• )}«"&'

<t>K'°

p~Kt+

P°K>°

- ( s i ) 2 x { £ + C , J

6 x l 0 - 5 - ( l / v /2 ) ( s i ) 2 x{B + C + 2CA}

- ( 5 1 ) 2 x {A + C}

- ( l / V 2 ) ( 5 l )2 x { B - C }

6*K' {C + Crfe* {B + C + 2Ch}ei6°*'

{(A + C) - 2/3(A + B)(l - eiA^-))ti6'%

{(B - C) - 1/3(X + B)(l - e ,A'«-)}e"'^

*+K-*0 K*^K -(ai)*x{A + B} {(A + B)}e^'"'

1840 BRIEF REPORTS 3

D - VV Decays. Based on the quark-diagram scheme -+K *Ow) -6 x (see Table 1111, the following predictions can be made: To summarize, branching fractions of the doubly

quark-mixing-matrix suppressed decays of charmed

T ( D O + ~ K * O ) = - r (DO+dK*O) , 1:: j 4 mesons are predicted using the quark-diagram scheme to- gether with the existing data. Measurements of these doubly suppressed charm decays are important in our

T ( D ~ - + ~ K * ~ ) = effort to understand the nonleptonic-decay mechanism.

This work was supported in part by the U.S. Depart- * - ) ' (7) ment of Energy and the National Science Council of

Taiwan. L.L.C. would like to thank the University of California Coordinate Committee for Nonlinear Studies, for an INCOR grant which made her visit at Los Alamos National Laboratory possible, and would also like to thank Dr. David K. Campbell and the members of the Center for Nonlinear Studies for their warm hospitality. H.Y.C. wishes to thank the Theory Group of the Physics

Using the recent ARGUS measurement7 Department of the Brookhaven National Laboratory for B(~~+K*~w)=(2.0f0.5f0.3)%, we predict B ( D O theirhospitality.

'L. L. Chau, in Proceedings of the 1980 Guangzhou Conference on Theoretical Particle Physics, Guangzhou, China, 1980 (Van Nostrand Reinhold, New York, 1981; Science Press, Beijing, 1980); Phys. Rep. 95, 1 (1983).

2 ~ . L. Chau and H. Y. Cheng, Phys. Rev. Lett. 56, 1655 (1986); Phys. Rev. D 36, 137 (1987); 39, 2788 (1989).

3L. L. Chau and H. Y. Cheng, Phys. Lett. B 222,285 (1989). he Tau-Charm Factory Workshop, Stanford, California, 1989

(unpublished). 5 ~ h e s e aspects were emphasized in a talk by L. L. Chau at the

Tau-Charm Factory Workshop (Ref. 4). %ee L. L. Chau, H. Y. Cheng, and T. Huang [Report No.

UCD-89-18 (unpublished)] for an updated analysis. In our previous paper (Ref. 3) an assumption on the nonspectator amplitudes j and a) has to be made in order to determine six quark-diagram amplitudes from five measured decay rates as the data for DO-KO~' were not available at that time.

'P. E. Karchin, in Proceedings of the XIV International Symposi- um on Lepton and Photon Interactions, Stanford, California, 1989, edited by M. Riordan (World Scientific, Singapore, 1990).

8J. C. Anjos et al., Phys. Lett. B 223, 267 (1989). 9The present data for Do-K*-K+ and K * ' K are not accu-

rate enough to determine the phase-shift difference of (S0-6,),,,. Our predictions for D,-K*K rates are ob-

tained in the absence of final-state interactions. 'OI. I. Bigi, in Proceedings of the 16th SLAC Summer Institute,

Stanford, California, 1988 (unpublished); see also T. E. Browder, Report No. SLAC-PUB-5083, 1989 (unpublished).

I'M. Bauer, B. Stech, and M. Wirbel, Z. Phys. C 34, 103 (1987). 1 2 ~ . Yu. Blok and M. A. Shifman, Yad. Fiz. 45, 211 (1987) [Sov.

J. Nucl. Phys. 45, 135 (1987)l; 45, 478 (1987) [45, 301 (1987)l; 45, 841 (1987) [45, 522 (1987)l; see also L. L. Chau and H. Y. Cheng, Mod. Phys. Lett. A 4, 877 (1989), for comments.

I 3 ~ h e other set of solutions is (in units of GeV)

However, this solution leads to the prediction B (D: - T + ~ ' ) -0.1% and hence is ruled out by current ex- periments.