Z. Phys. C - Particles and Fields 32, 243-248 (1986) Particle, s f~r Physik C

and �9 Springer-Verlag 1986

Nonresonant Three-Body Decays of Charmed Mesons in Chiral Perturbation Theory Hai-Yang Cheng* Physics Department, Indiana University, Bloomington, IN 47405, USA

Received 23 March 1986

Abstract. Direct (nonresonant) 3-body decays of charmed mesons are calculated in chiral perturbation theory. The magnitudes of the 20 and 84 represent- ations of the effective chiral S U(4) x S U(4) Lagran- gian are determined from the measured 2-body D Krc rate. For decay modes which do not contain non- spectator contributions, the agreement of theoretical predictions with the data is satisfactory. Nevertheless, the large discrepancy between theory and experiment for decays which can proceed through the W-exchange or W-annihilation might imply the importance of nonperturbative corrections to the nonspectator dia- gram and the existence of final-state interactions.

I. Introduction

It is well known that in kaon decays the nonleptonic decay amplitude of K ~ 37r can be related to that of K ~ 2~z through the soft-pion theorem. The prediction of K ~ 37r decay rate using current algebra is good to an accuracy of about 15%. To use the soft-pion tech- nique, the energy dependence of the on-shell amplitude of K --* 3 zt has to be assumed. The energy dependence is usually assumed to be linear and is only linear in odd-pion energy due to the symmetry arguments (see, for instance, Ref. [1].) There is another different approach dealing with the weak decays of pseudo- scalar mesons, that is the effective chiral perturbation theory. Given a weak Hamiltonian in terms of quark fields, it is generally difficult to evaluate the matrix element (3 P I Hw[ M ) directly. In the framework of current algebra the three-body matrix element can be reduced to two-body matrix elements which are easier to calculate. On the other hand, in the nonlinear chiral approach the idea is to replace the weak Hamiltonian by an effective Hamiltonian in terms of the meson fields. More precisely,

q,7.( 1 -- 75)qj ~ if(U O. U +)U2 (1)

* Chester Davis Fellow

where the 3 x 3 matrix U = exp(i22"4)"/f) describes the eight Goldstone bosons, g,K, and t/, 2's are the Gel l -Mann matrices, f is the common decay constant ~130MeV. For kaon decays, the A S = I weak transition is dominated by the octet interaction due to the A I = 1/2 rule. Therefore the leading-order effective chiral Lagrangian for the nonleptonic K decays is given by [2]

= c Tr (2 6 0 # g ~/~ U +). (2)

This chiral Lagrangian transforms as (8, 1) under S UL(3 ) x S UR(3 ) symmetry. The only unknown para- meter c is determined from K --* 27z decay rates; K --* 37r is then computed using (2). Apart from the usual tree diagrams there are additional pole contributions which involve four-point strong-interaction vertices [2]. The amplitudes and the slope parameters of K ~ 3 lr described by (2) agree well with experiment as shown by Cronin [2] almost two decades ago!

Current-algebra results are therefore reproduced to lowest order in the low energy expansion of chiral perturbation theory. However, the effective chiral Lagrangian approach is not merely an alternative to current algebra, it does more. For instance, the mom- entum dependence of the off-shell amplitude can be easily figured out in the effective Lagrangian theory but is in general fairly tedious for current algebra. Fur- thermore, the chiral Lagrangian provides a systematic way of studying the loop corrections and higher-order tree-diagram contributions arising from higher- derivative interactions [3].

For three-body decays of charmed mesons, the on- shell amplitude is in general linear in all three final- particle energies, which makes the use of current algebra quite complicated. Hence, in this paper, we would like to study the three pseudoscalar meson decays of D meson using chiral perturbation theory. The analysis of D ~ 3P is, however, complicated by another new feature, namely, the three-body decays of the D meson are dominated by the vector-meson pole contributions, i.e. the quasi-two-body decays contain- ing a vector particle (p or K*) and a pseudoscalar (K or

244

Table 1. Branching ratios of nonresonant D--* 3P decays. Theoret- ical predictions and the Mark II1 data are exhibited in columns 2 and 3 respectively. Fractions ofD ~ 3 P from nonresonant decays are also tabulated in the fourth and fifth columns.

Mode Br (~o)theory Br (~o)exr , Fraction(%) theory experiment

D~ ~/~~ - 0,15 1.5 +0.7+0.3 2.0 19.3_+9.3 ~K-n+n ~ 0.22 1.0 _+0.8_+0.6 1.1 5.5_+5.3 ~f f ,~ 0.03 - - 1.9 - - -~n-n+n ~ 0.05 - - 4.1 - -

D+ --*/(~ ~ 0.94 0.9 _+0.8_+0.6 4.7 6.5_+6.8 ~ K n+n + 2.03 - - 18.3 - - ~t+n+n - 0.17 - - 36.2 - - ~ + K + K 0 . 0 2 0.66_+0.30_+0.12 - - - -

n). The D ~ ~ mode shows a very strong K - p + contr ibut ion whereas D ~ "~ + n - is dominated by K * - n + . Direct three-body decays are generally only small fractions (see Table 1) of total D ~ 3 P rates. But it is the nonresonant decay channel which is to be computed in current algebra or chiral per turbat ion theory. Experimentally, only a few channels of direct D ~ 3P decays have been measured [4-6] .

The Q C D corrected weak Hamil tonian responsible for nonleptonic charmed meson decay is of the form

Hef t = GF/ ~ Vcs Vud(C _ 0 _ "q- C+ 0 +)

O+ = gy.(1 - ys)ct~yu(1 - ys)d + gTu -(1 - 75)aayu(1 - y5)c (3)

where 0 _ ( 0 + ) transforms as _6 (15) of SU(3), which is contained in the 20 (84) representation of SU(4). For c_ > c+ the _5 contr ibut ion dominates over the 15 contr ibut ion (i.e. the so-called sextet dominance [7]). It has been realized that the _6 representation for the charm-changing reaction definitely does not dominate to the same extent as _8 does for the strangeness- changing interaction. (For a review, see [8].) In other words, the 15 representation is also impor tant and cannot be neglected at all.

II. Chiral Perturbat ion Theory

Nonleptonic decays of charmed mesons have been examined in chiral per turbat ion theory by several authors [9, 10]. The framework which we are going to use is essentially the one proposed by Singer [9]. However, these authors only considered the 20 repres- entat ion of S U(4) for the effective chiral Lagrangian. Consequently, the D + ~ / ( ~ mode is virtually for- bidden since in the S U(3) limit it receives contr ibut ion only from the 0 + operator. Taking into account the 84 representation we write the effect Lagrangian as

, . ~ = C1 ~(~(20) _1_ C2 ,.~O(84) (4)

where the coefficients Ca and c2 are determined from the measured D ~ K n rates as described later.

H.-Y. Cheng: Nonresonant Three-Body Decays

In order to construct the effective chiral S U ( 4 ) x S U(4) Lagrangian, we first rewrite the nonleptonic weak Hamil tonian in terms of the representation of SU(4). Let 0,p be a 4 • 4 matrix with nonvanishing elements 0 1 2 = 0 4 3 = C O S 0 C and 0 1 3 = - - 0 4 2 = sin Oc (with 0 c the Cabibbo angle), so that the weak charged current becomes [1 1]

J = O,pj~

j~ = (r q,~) - �88 (5)

where ( f fP f f . )=~ .7 . (1 -Ts )qp (For convenience the Lorentz index has been omitted in j~), ~k" = (u, d, s, c), ff~ = ~ + is the ant iquark field. It has been shown by Einhorn and Quigg [11] that the S U(4) representation structure of the nonleptonic weak Hamil tonian is given by

H w = ( ~ ( D 1 + pt, x + Pl, + ~ 0 , p 0 ~ o ~ , ~ + ~ 0 , p 0 ~ p ~ , ~ (6)

where the explicit expressions of (1), (20), (84) are given in [11]. After some manipulat ion the 20 and 84 representations of the Hamil tonian for A c = 1 tran- sitions reduce to

G ~ �9 2 "4 - - "2 '4 Hw(20 , 84) = 2~22 { c~ ']-J3J1)

% /

-- sin2 Oc(j3 j 4 Yr J 2J1)'3 . 4 ,

+ sin Occos O c ( j 3 j 4 -T- j a j ~

�9 2.4 ~ .2.4 --J1J2 J2J1)} (7)

where - ( + ) is for 20 (84) representation. Perturbative Q C D corrections modify the relative magnitudes of Hw(20) and H~(84) to

Hw = c_ Hw(20) + c+ Hw 8(8(8(8(8(8(8(8(8(8~ (8)

where c_ = 2.09 and c+ = 0.69 for AqcD = 500MeV. Using (1) to replace the bilinear quark fields by the corresponding meson fields we obtain the following leading-order effective SU(4) • SU(4) chiral Lagran- gian for A c = 1 transitions*

Gv s - 2, , f2 {c~ + c2)JZlJ'~ - (cx - c2)j~j]]

+ sin 0 c cos Oc [(Cl + c2)( j3 j 4 - jZ l j4 )

- - (C 1 "3 "4 '2 "J, -- c2)(J3J1 --J2J1)] -3.4 -- sin 20c[(Cl + c2)jlJ2 + (cl -- c2)j3j~] } (9)

where the coefficients c_ and c+ have been replaced by the parameters ca and c2 respectively. Similarly the

* Both [9] and [10] used c1=1, and C 2 = 0 . In addition, a huge enhancement factor s = 2.14/sin 0 c cos 0 c was introduced by Singer [9] in order to describe the observed K ~2n decays. However, as indicated by (18), experiment does not show such huge enhancement in charmed-meson decays

H-Y. Cheng: Nonresonant Three-Body Decays

current j~ becomes

(j~)=~-(Uc3~,U )~--�88 (UOuU+){. (10)

In the chiral SU(4) case, pseudoscalar mesons are contained in the following matrix

~ o tl rl ~ - + - + - - rc +

,266 ,/S _ xo rt tl~

+ + - - K ~ ,266

D O D + F +

m

K + ~o

D -

F -

ro ~43y]c m

(11)

The unitary matrix U = exp(2 iqS / f ) transforms under chiral SU(4) • SU(4) as

U-~ VL U V ~ (12)

for VLeSUL(4), VReSUR(4) . The unknown parameters ct and c2 are determined

from the Cabibbo allowed D ~ K n decays. Using the effective Lagrangian (9) we find that on mass-shell the amplitudes are

A(DO__+ K - rt+) = 2[c1(D z _ 71.2) + c2(D 2 -- 2 K 2 + z~2)]

A ( D ~ 1 7 6 ~ = _ 2 [ - c 1 ( D 2 + K 2 - 2x 2)

+ c2(O 2 + K 2 - 2nz) ] /x /2 (13)

A(D + __./~o n +) = 2[ - el (K 2 - rc 2)

+ c2(2D z -- K 2 __ rc2)]

where 2=- i f cos2OcGv/ (2xf2 ) , and masses are de- noted by the particle symbols. The following sum rule is satisfied by (13)

, ~ 2 A ( D ~ ~ K~ ~ = A (D + -~ ~~ +)

- A ( D ~ (14)

We note that the decay D ~ no is forbidden when c I =c2 , since quark currents are replaced by the effective meson currents (1) in chiral perturbation theory. Consequently, a meson cannot be formed from two quarks associated with different color-singlet currents; the color suppression is infinite rather than �89 as in the usual quark model. It is thus very interesting to note that the recent 1IN approach [12] for charmed- meson decays and the model of Bauer and Stech [ 13] all require an infinite suppression on color mixing in order to explain the experimental data of two pseudo- scalar meson decays of charmed mesons.

The branching ratios of D ~ K n measured by the Mark III [5] are given by

245

Br(D ~ ~ K - zt +) = (4.9 _+ 0.4 _+ 0.4)~o

Br(D o ~ / ~ o ~o) _ (2.2 _+ 0.4 _+ 0.2)~o

Br(D + ~ / ( ~ = (3.5 + 0.5 + 0.4)~o. (15)

It is easy to check that if all weak decay amplitudes of D ---, Krc are real the sum rule (14) cannot be satisfied and the data (15) cannot be fitted by any real Cl and c2 [13 15]. This is due to ignoring final-state inter- actions;* the final-state interaction is particularly important for charmed-meson two-body decays since strangeness one resonances are known to exist at 1.3-1.5 GeV. The effect of final-state hadronic inter- actions can be accounted for by the complex phase- shift e x p ( - r h - igt),where r//and 6 t denote the s-wave inelasticity and phase shift respectively for the isospin I amplitude A t (For details, see [14]). It should be emphasized, from the outset, that the phase shifts 6t appearing in the weak amplitudes are generally not the same as the ones occurring in the purely strong- interaction scattering amplitudes; this point is often ignored in the literature [14]. Writing

A(D ~ ~ K - rc +) -- �89

+ A3/2 e-~s/2+i'~3/2) (16)

A(D 0 ~ Korco) = �89 _ Aa/ze- , , /2 + i~,/2

+ xf2A3/Ee-"3/2+in3/2)

A (D + ~ K~ +) = A3/2e-,3/~+i~3/2

we find the best fit to Mark III data occurs when**

6 1 / 2 - - 6 3 / z = 7 9 ~ A 1 / z e - " ' / 2 / A 3 / z e "3/2=3.8. (17)

For the inelasticity one can argue that q3/2 = 0 due to the absence of isospin 3/2 resonances. For the isospin I /2Krc amplitude e x p ( - ql/2 ) ~ 0.8 has been used [13]. Identifying At/2 and A 3/2 from (13), we obtain from (17) that

c I = 1.96, c z = 0.51 (18)

which are close to the QCD corrected coefficients c_ = 2.09 and c+ = 0.69.

Having determined the magnitudes of the 20 and 84 representations, we next proceed to discuss the 3-body decays. Expanding U up to fourth-order in ~b, we have

U = 1 + 2 i f f ) / f - 2~b=//2 - ia3ff)3/f 3

+ 2(a3 -- 1)~b4/f 4 (19)

* This reminds us of the same feature in the kaon system. Without taking into account the final-state interaction the isospin 3/2 amplitude determined from K ~ is inconsistent with the same amplitude measured from K + --,2n ** The other solution is 61/2 - 63/2 = rc - 79 ~ and A~I2 e ,l~/2/A3/2e ,~3/2 = _ 3.8. This solution corresponds to c~ and c 2 which are in opposite sign and hence is not favored since the corresponding QCD corrected coefficients c_ and c+ in the quark model are of the same sign. Realistic-model calculations also prefer the solution (17) 1-14]

246

where a3 = 4/3. Different nonlinear realization of chiral symmetry corresponds to different a3.* It has been observed by Chisholm [16] that S-matrix elements (i.e. on-mass-shell amplitudes) are independent of a3; this serves as a useful check for our calculations. From (19) and (10) the relevant terms for D ~ 3 P decays are

j.j = _ {6q qg(�89 q53 -- 2qSc3uqS2 + 2q~263uq~ )

+ (a30u~93/2 -- 2qSCOuq52 + 2 ~ 2 c3uqS)cOuq~

+ 4(4 8.~b)(qSc~uqS) - 2(8.~bz)(~b O.~b) - 2(~ au~b)(auq52) + (6~#(]~2)(~/~q~2) } (20)

where the flavor indices have been omitted for convenience. Apart from the direct contribution com- ing from four-point weak vertices there are also pseudoscalar pole contributions for D--*3P decay, which arise from the combination of a two-point weak vertex and a four-point strong vertex. The strong interaction Lagrangian including chiral symmetry breaking by the meson masses is given by

~ s = [Tr(~?vUc3"U+)+TrM(U+ g+)] (21)

where M u = 0 (i # j ) and 1-93

n 2 = M l l =M22, K2 = (Mll + M33)/2 D 2 = (Mii + M44)/2, F 2 = (M33 + M44)/2

Retaining only fourth-order terms in q5 we have

~CPs = -- ; ; 2 Tr(c~uqSc?u q~3 + 0U~3~, ul#)

1 + 2f~Tr auq~2Ouq~2 + (a2f21) TrMq54. (22)

For Cabibbo favored decays there are only two possible weak vertices in the pole diagrams, namely, D ~ ~ and F + - n +. For Cabibbo suppressed channels, the allowed weak vertices are D + - n +, D ~ o , and F + - - K +.

Before proceeding, we would like to address the question as to whether chiral perturbation theory is a reasonable approach to describe charm weak decays. One concern is that the subleading expansion of the chiral Lagrangian might be larger than the leading term due to the large momentum in charm decays. To begin with, we note that the chiral-symmetry breaking scale A = 2 x / ~ f [17] is about 1.4GeV for the SU(4) case (It is 1.1 GeV for the S U (3) case.). Since we are only interested in the partial decay rate, we may focus on the average momentum of particles (about 0.8 GeV) in a typical Dalitz plot of three-body decays. It is clear that contributions from the quartic-derivative Lagrangian

* Not all current-algebra results are obviously respected by any nonlinear chiral model. One example is the bilinear quark operator g(1-Ts)d; the soft-pion relation ( n n [ g ( 1 - 7 5 ) d l K ) = - i x / 2 / f (n]g(1 - 75)dl K ) (Strong interaction is not considered here.) is only realized in the chiral model with a a = 2

H.-Y. Cheng: Nonresonant Three-Body Decays

are suppressed by factors of p 2 / A 2 (0.8/1.4)2 from naive dimensional argument. Hence, as far as the decay rate of charm mesons is concerned, chiral perturbation theory is as applicable as in the kaon system. Further- more, this is supported by the 1/N expansion (N being the number of colors) for charm decays, as advocated recently by Buras, G6rard and Rtickl [12]. The lowest-order chiral Lagrangian's prediction for D ~ Kn, (13), is exactly equivalent to the leading terms of the 1IN expansion for the same decays; all other nonleading contributions are suppressed by at least factors of 1IN. Indeed, the new Mark lII data of D ~ P P (P: pseudoscalar) can be reproduced to an accuracy of 1 2 standard deviations by the leading terms of 1IN [12]. Thereby, the result (13) obtained in chiral perturbation theory is in fact the dominant contribution. Again, this implies that it is justified to use the chiral Lagrangian as a perturbation theory for weak charm decays, and the higher-derivative Lagran- gian and loop corrections cannot be that important.

The calculation of D --* 3 P amplitudes is somewhat tedious but otherwise straightforward. In the following we only list the results for those 3-body decays of D mesons which have been measured by the Mark III (five Cabibbo-allowed and three Cabibbo-suppressed) [52:

2 2F2 A(D + + K - n + n + ) = - a (C 1 -1- C2)(S 1 - 2 n )F~-~C2

+ (C 1 - - c2)(F 2 + 3 x 2 _ 3Sl) }

A(D+ ~K~176 a (Cx +C2)(Sx-2x2)F2_n2

(C 1 - - C2) I - - S I -'[- S 3 +

- - (S 2 - - $3)D2 (23)

A(D + --, n + n + n-) = b {2(cl + c2)(s 3 - - ~2)

+ (cl -c2) (D 2 + 3n 2 - 3s3) }

A(D + --*x + K + K - ) = b { ( c l + c2)I~2 - D2 +

+ (D 2 + 2K 2 _ $3)

. D 2 + K 2 K2D~+F2~ D 2 _ ~z 2 D 2 _ x 2 J

21- (C 1 - - C2)($3 -- $2) t

A(D~ ~ g~ n-)

f F 2 al(c l+ c2)(2x 2 -- Sl)F2 _ x 2

H.-Y. Cheng: Nonresonant Three-Body Decays

+(q-c2) s2N---K2 s3 _K rd

A(D~ ~ K - ~ + z~ ~

- a (cl + c2)(2~z 2 - s l ) ~ _ zc 2

+(C 1 - c 2 ) S1--S 2 + ( S 2 - S 3 ) D 2 Z X 2

A(DO--. ~O K + K -)

= a{(c 1 + cz)(st -- s2) + (c, - cz)(s 2 -- K2)}

A(D~ --. rc- Tz + rc ~

= ~ 2 b { ( c l +c2)[D2 + 37z2- 3s2

~2 -1 + gr2 _ - s3)J

- - (C 1 - - C 2 ) 7~ 2 "3 t- S 1 - - 2s2 + D y ~ _ ~2 (sl - s3)

a = GF_cos20c, b = G~sinOccosOc 2 x / 2 2x/2

where s~ = (pD-p32, and p~ is the 4 -momentum o f the i-th meson. There are several checks which can be made on the results (23): (a) All physical amplitudes are independent of a 3, as they should be. (b) For c 1 = 1, c2 = 0, our results for D + --* K - lr + lr + and D o ~ zc- ~r + K ~ agree with [9]. (c) Since the d c = 0 part of the 20 representation has 8 only, the Dalitz plots for D+~rt+r~+Tr - and L)~ ~ should have the same energy dependence as that o f K + ~ lr + 7r + lr- and K ~ --* 7r- 7r + 7r ~ when c2 = 0. In fact, for c2 = 0 we obtain

A(D + --* r~ + rc + 7z-) oc (D 2 + rc 2 - s3) A(D ~ --* 7z- 7r § zc ~ oc (re 2 - s3) (24)

as expected. Finally, in order to compute the decay rate we have

to integrate the amplitude squared over all phase space. To do this we note first that the phase-space integral

= [d3p1 d3P2 d3P36 4 p I O2E 1 ~ 2 2 ~ 3 3 ( o - P 1 - P 2 - P a ) - ~ d R

(25)

can be reduced to a one-dimensional integral

7~2 (m~ dsl 1 / 2 2 2

= - - +~ ~ (Sl , ran, ml) I 4m2(,,~ m~)~ Si

1 / 2 2 2 "2 (sl,m2,m3) (26)

2(x,y,z) = x 2 + yZ + z 2 _ 2 x y - 2yz - 2xz.

Since the phase-space integral I is invariant under the permutat ion of the indices 1, 2, 3, the integrations of s~

247

and s~ over the phase space are performed by choosing the same integration variables. The integral of, say, sl s3 over the phase space is done by virtue of the following formula*

7~ 2 {rno mr) 2

Ssl s3dR - 4m20 ,,.2j,,3,2 21/2(st,mZ, m2)

1/2 2 2 �9 2 (sl,m2, m3)

.[m~ + m ~ - 2 ~ l ( s l - m Z + m2)

�9 ( s t + - ( 2 7 )

Phase-space integrals of s2 s3 and st s2 are obtained in a similar way.

The results of calculations for the branching ratios of direct decays o fD ~ 3 P are exhibited in Table 1. Use of z ( D ~ and z ( D + ) = 8 . 9 x 10-13s has been made. Experimentally, only a few channels of nonresonant 3-body decays have been measured. Frac- tions of D ~ 3 P from nonresonant decays are also computed using the experimental results for the 3-body decay rates [5]. The Mark I I I data are shown in the third and fifth columns of Table 1. While the nonreso- nant decay of D o is predicted to be only a few percents of total 3-body decays, fractions of D + ~ 3 P from direct decays can be as large as 36~o as in the channel D + ~Tr + 7r + 7r-. I t is evident from Table i that the theoretical predictions for the final s t a t es / (on+ 7to and K - rc § rc ~ are consistent with experiment in spite of the fact that the SU(4) chiral symmetry is expected to be valid only to 3 0 - 5 0 ~ accuracy. On the other hand the predicted K ~ zt + 7r- and rc + K § K - nonresonant rates are too small when compared with the data. Since these decay modes contain nonspecta tor contr ibut ions (i.e. W-exchange or W-annihilation), it is conceivable that the nonspecta tor ampli tude is greatly enhanced by some nonper turbat ive effects, presumably the soft- gluon corrections [19]. That is to say that the " ~ q + g l u o n " state mainly transforms into hadron states with more than two particles. Because D~ K ~ + K - and ~ - ~z + ~z ~ can also proceed through W- exchange, it is of interest to see experimentally whether or not these modes are enhanced by nonper turbat ive corrections.

There are still some features of the 3-body data which need to be elaborated upon:

(a) The experimental fact that the nonresonant rate of D + ~ 7r + K + K - is larger than that of 7r + zc + 7r- (The branching ratio (including resonances) of the latter is (0.47 _+ 0.19 _+ 0.02)~ [5]) presents a difficulty. Owing to the available phase space (The phase space of the former is about three times smaller than that of the

*This may be derived from (V.2.16) of [18] **The predicted rate for the nonresonant D O ~ K- rc +/r ~ is in strong disagreement with the TPS measurements [6] which contain large statistical errors. Errors are substantially improved by Mark III [5]

248

latter.) and the identical-particle effect, the ratio ~z § K § K-/Tz § zc § n - is naively expected to be 1/6. Since W-annihilation contributes to both modes, it seems the aforementioned nonperturbative effects cannot explain the observed ratio. This is analogous to the difficulty with the ratio K § K - I n + n - in D O decays. Apart from S U(3) breaking, the large inelasticity in the final-state interactions of nn so that n+n - is converted into K § K - can be a sizable effect 1-20] for the enhancement of K/s relative to n z~. For three-body decays, one might conjecture that similar final-state interactions are responsible for the relative enhancement of n § K § K - to n+ n+ n -.

(b) From the branching ratio Br(D § ~ g ' * ~ (3.0+ 1.9 + 1.7)% determined from the Dalitz plot of D § 1 7 6 ~ I-5] together with Br(D+--*K - n § n § = (11.1 + 1.4 + 1.2)%, one can induce that the final K - n + n § states consists of only 9% K'*~ § (Branching ratio of K '*~163176176 is �89 However, the nonresonant contribution is predicted to be of 18%. Indeed, the majority of the Dalitz plot of Mark III data shows the structure away from the /~,o region is inconsistent with a flat nonresonant contribution 1-21]. It is conceivable that the remaining 70% might be attributed to S-wave xn or D-wave/~* (1430)n state. It immediately raises the question as to why these S- and D-wave states are not manifested themselves in other channels. This has to be answered both theoretically and experimentally.

III. Conclusions

We have discussed nonresonant three-body decays of charmed mesons within the framework of chiral per- turbation theory. The use of the chiral Lagrangian for charm decays is reasonable and justified since the contribution from the quartic-derivative Lagrangian is suppressed by factors of p 2 / A 2 (0.8/1.4)2, based on the dimensional argument. Moreover, the fact that the lowest-order effect Lagrangian is equivalent to the leading terms of the 1IN expansion indicates that this approach should give reasonable description of weak D decays.

The two unknown coefficients of the effective chiral Lagrangian are determined from the observed D ~ K'n rates. We then apply this approach to three-body charm decays. Apart from the direct weak decay, the nonresonant 3-body decay amplitude also receives pole contributions involving four-point strong vert- ices. Our calculations show that nonresonant contri- butions are generally small fractions of total 3-body decay rates. Theoretical predictions for the final states

H.-Y. Cheng: Nonresonant Three-Body Decays

/~~ ~ and K - n + n ~ are consistent with the data. However, the large discrepancy between theory and experiment for /~on § and the ratio z~ + K + K -/Tz + 7r + re- may imply the importance of(soft- gluon) nonperturbative corrections to the W-exchange or W-annihilation amplitude, and the importance of final-state interactions. Finally, we point out the difficulty with the decay D + ~ K - n + z+: where the dominant quasi-two-body contribution is unknown.

Acknowledgement. Part of this work was done while the author was at Brandeis University. I am indebted to Dr. Ling-Lie Chau for valuable discussions and to Dr. D. Coward for explaining to me the Mark III data. This work was supported in part by DOE contract DE-AC03-ER-03230 (Brandeis University) and contract DE-AC02- 84-ER40125 (Indiana University).

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