PHYSICAL REVIEW D VOLUME 3 1, NUMBER 5 1 MARCH 1985

Phenomenological Lagrangian for nonleptonic charmed-meson decays

A. N. Kamal, Lo Chong-Huah, and M. Rahman Theoretical Physics Institute and Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1

(Received 29 May 1984)

A gauged chiral-Lagrangian model for charmed-meson decays is proposed. The O+ uncharmed mesons are included in the model to account for S-wave hadronic final-state interactions. All the hadron fields are incorporated into the phenomenological Lagrangian through the general frame- work proposed by Coleman, Wess, and Zumino. The model correctly predicts the branching ratios of various two-body D-meson decay modes in both Cabibbo-angle-favored and Cabibbo-angle- suppressed channels, with the only exception of B ( ~ O - K +K- ) / B (4'-&a-). We have also shown that the prediction of S-wave threshold parameters in an- scattering can be in good agree- ment with data, if the SU(3)-symmetry-breaking effect in 0+ uncharmed-meson decay vertices is suitably adjusted.

I. INTRODUCTION

It is clear by now' that final-state interactions must be considered in studying the nonleptonic charm decays. Such hadronic final-state interactions are incorporated into the theory through Muskhelishvili-Omnes (MO) equations2 or through Watson's t h e ~ r e m . ~ The inhomo- geneous term in the MO equation is generally taken to be the quark-model matrix element corrected for short-range gluonic effects. The solution of the MO equation then generates the physical amplitudes. This amplitude is in general complex, and its phase plays an important role in describing the decay process.

In the two-body charm decays the weak transition am- plitude is expressed as a linear combination of final-state isospin amplitudes. Each of these isospin amplitudes is found to be a product (matrix product for coupled- channel final states) of a quark-model amplitude and the strong-interaction "enhancement factorw-the inverse D(v) function. Here v is the center-of-mass momentum squared of the two-body system. The function D(v) is the denominator of the two-body S-wave scattering amplitude TO(v ) written in the N(v)/D(v) form. The denominator function is normalized to unity at infinity. It can be writ- ten as D(v) = 1 + f (v) with f (v) -+O asymptotically. In nonexotic channels the denominator function has the Breit-Wigner zeros corresponding to resonances. On the other hand, the denominator function is represented by an effective-range formula, with relatively mild variations with energy in the exotic isospin channels. Final-state in- teractions in many-body final states are dominated by the interactions in the various two-body subsystems of the de- cay products. The MO equation can be used to deal with many-body final states, but in general it proves to be rath- er unwieldy.3

In this paper, we propose a phenomenological- Lagrangian model for charmed-meson decays. The two- body decay amplitude gets a contribution from two pro- cesses: direct decay into two pseudoscalar mesons and a decay via a O+ uncharmed intermediate state to the final two-body state. The amplitude for the latter process is

complex. This model takes into account resonant interac- tions in the final state in a manner that mimics the N/D technique. This physical picture of charmed-meson de- cays requires our weak-interaction Lagrangian to contain the two-body "charmed meson-0+ uncharmed meson" direct-conversion vertices in addition to the three-body de- cay vertices. Furthermore, the observation of vector mesons in multiparticle decay modes of charmed particles suggests that vector and axial-vector mesons should also be included in the phenomenological Lagrangian. These observations determine the particle spectrum of our phenomenological Lagrangian.

We use the chiral Lagrangian to describe strong- interaction dynamics of pseudoscalar mesons, where the chiral symmetry SUL (3) X SUR (3) is realized nonlinearly by the pseudoscalar-meson fields. The "vector-meson universality" and low-energy phenomenology of strong in- teractions [such as the Kawarabayashi-suzuki- Riazuddin-Fayazuddin (KSRF) relation4], suggest that spin-1 vector fields should be the gauge fields of chiral symmetry. Charmed mesons and the scalar mesons form a triplet and an octet, respectively, of the diagonal sub- group SUV(3) of the chiral group. They are incorporated into the chiral Lagrangian by the general framework pro- posed by Coleman, Wess, and ~ u m i n o . ~ We emphasize that we classify particle multiplets by SU(3) and not by SU(41, since only SU(3) has been proven to be a good (ap- proximate) symmetry of strong interactions.

In our model the chiral Lagrangian for the nonleptonic charmed-meson decay is assumed to have the same chiral-transformation property as the / AC / = 1 current X current interaction in the Weinberg-Salam model. Our model has four parameters. Two of them, A and B (to be defined later), are associated with direct- decay vertices and the other two, C and D (to be defined later), are associated with weak-interaction vertices con- taining the scalar-meson fields. We have studied various two-body decay modes in Cabibbo-angle-favored and Cabibbo-angle-suppressed channels.

We find that when the weak vertices involving the sca- lar mesons are ignored (the predicted branching ratios de-

1055 @ 1985 The American Physical Society

1056 A. N. KAMAL, LO CHONG-HUAH, AND M. RAHMAN

pend on one parameter only, B/A), all the predicted branching ratios agree with data except the prediction

which is different from the measured value 3.4k1.9 (Ref. 6; a value of 3.42::; is quoted by Abbott, Sikivie, and wise7). However, from two-body data alone, we have failed to find a set of values for the four parameters that correctly describe the two-body data. This happens due to particular combinations of the parameters appearing in the description of the decay modes. We expect that a better control over the parameters will result from the three-body data. The analysis of the three-body decay modes is in progress. The results will be reported in a fu- ture publication.

It should be noted that our model Lagrangian has the ( V - A ) X ( V - A ) structure. However, at the quark level different chiral structures arise from penguin diagrams8 and charged Higgs mesons in the extended version of the Weinberg-Salam model.9 The implications of such in- teractions are not considered in our work.

The paper is organized as follows. In Sec. 11, we dis- cuss the construction of the phenomenological Lagrangian in detail. We calculate the low-energy parameters for TT

and TK scattering and estimate the symmetry-breaking ef- fects. In Sec. 111, we study the various two-body decay modes of charmed particles. Final conclusions and a summary of results is given in Sec. IV.

11. CONSTRUCTION OF PHENOMENOLOGICAL LAGRANGIAN

The fundamental fields in our chiral model are the fol- lowing.

(1) Pseudoscalar Goldstone bosons. These are the eight pseudoscalar Goldstone bosons Ma ( x ) ( a = 1,. . . ,8 ) associ- ated with the spontaneous breakdown of chiral SUL (3) x SUR (3) symmetry. They transform linearly like an octet under broken SUv(3) [the diagonal subgroup of SUL (3) x SUR (3)] and nonlinearly under SUL (3) X SUR (3) in such a way that Z(x) which is the "local orientation of vacuum" as defined below transforms like (3,3*):

2iM(x) X( x ) = exp -------- , f o

where

ha are the eight Gell-Mann 3 x 3 matrices, and f o is the "unrenormalized" pion decay constant.

(2) Spin-1 gauge bosons. There are sixteen gauge bo- sons; eight of them, 1; ( a = 1, . . . , 8) associated with lo- cal SUL(3) transformation, the other eight gauge bosons r; ( a = 1, . . . , 8) associated with local SUR (3) transfor- mations.

(3) Pseudoscalar charmed mesons. We assume that the charmed mesons, Do, D-, I;- transform linearly like a triplet under SUy(3) and nonlinearly under SUL (3) x SUR (3). The following functions of Goldstone bosons and charmed mesons defined as

q=gzC, and q l = f + $ ,

where

transform linearly like (3,l) and (1,3) under SUL(3) x SUR (3) according to the Coleman-Wess-Zumino t h e ~ r e m . ~

(4) Scalar mesons. The scalar mesons are J P = O +

quark-anti uark bound states. The isotriplet S(980), iso- 'I doublets K-( 1350) and KO,RO( 1350), and two isoscalars S"(975) and ~(1300) form an SU(3) octet and an SU(3) singlet. The octet-singlet mixing angle of S* and E , which appear as resonant states in rrrr and KK scattering pro- cesses, is a strong function of The ~(1300) is "mostly octet" below the KK threshold and becomes an SU(3) singlet when the energy is above 1200 MeV. The scalar-meson matrix has the form

where

The "chiral-covariant derivatives" P,, D,o, and D, 1Ct are generally used to construct chirally invariant or co- variant quantities which contain field derivatives. Pp, D,o, and D,I) are defined as

and

where the "gauge connection" is defined as

The gauge coupling constant fp can be identified as a vector-meson coupling.

P, and D,o transform like octets and D,$ transforms like a triplet under SUV(3). All of them transform non- linearly under SUL (3) X SUR (3). However, we can con- struct various functions of P,, D,u, and D,$J which transform linearly under the chiral group. Some of these, which will be needed in the weak-transition Lagrangian of our model, are listed below along with their transforma- tion properties:

3 1 - PHENOMENOLOGICAL LAGRANGIAN FOR NONLEPTONIC . . .

The Lagrangian density in our model can be written as The SUv(3)-invariant mass term a sum of several terms,

L = L O + L M + L ~ + L W . (2.10) mo2

LM=--Tr(lpip+r,,rp) L o is invariant under local SUL (3) X SUR (3) symmetry. It 2 contains kinetic-energy terms and strong-interaction ver- tices of various fields:

breaks local chiral symmetry to the global chiral symme- try. It prevents the pseudoscalar Goldstone bosons from being "gauged away" through the Higgs mechanism.

The global chiral symmetry, as well as the SUV(3) sym- + ( ~ , a ) ~ ( k ) ~ a ) - m ~ ~ $ ~ $ - m ~ ~ r ( a a ) metry is broken by the "medium-strong" interaction LAM

which has the same transformation property as the quark-mass term in the QCD Lagrangian and belongs to

- ~[Tr(F~, .F1p") + T ~ ( F ~ , . F ~ ~ " ) ] . (2.1 the (3,3*)+(3*,3) representation of S U ~ ( ~ ) X S U ~ ( ~ ) : " " ~

+ (intermultiplet mass-splitting interactions for charmed mesons ,

scalar mesons, vector mesons, and axial-vector mesons) . (2.13)

f T 2 2 2 f.R2 2 a=-(m, +2mK ) and b=-(mV2-mK),

24 12

Y and [m ] are the constant matrices mA2 1 f o 2 f p 2

which gives the Okubo mass relation for pseudoscalar mesons and the PCAC (partial conservation of axial- vector current) condition. m and m, in the "quark-mass matrix" need not have the "current-quark" mass values. They are taken to be arbitrary parameters whose ratio determines the SUv(3)-breaking effect of the second term in (2.13). From the decay widths of X+ and 6+ [r(G+)/r(X+)=$], we find that the reasonable value for m,/m should be in the interval (1,2.8). The upper bound of the interval m,/m=2.8 is obtained if the total width of the K + meson is saturated by the partial widths of various two-body decay modes.

The gauge bosons associated with the generators of SUv(3) are the physical J'= 1- vector mesons and mo is very close to the mass of the p vector meson. The longitu- dinal components of axial-vector gauge bosons mixed with the Goldstone bosons by the bilinear coupling

Y =

This interaction produces a mass difference between the p vector meson and the A 1 axial-vector meson,

and

so that the physical pseudoscalar mesons and axial-vector Parameters a,b can be determined by the masses of mesons are related to Goldstone bosons and gauge bosons

pseudoscalar mesons. Their values are as

The physical pion decay constant is

f , = * f o . (2.19)

With the choice Z = t, we can obtain both Weinberg's mass ratioJ3 between p and Al and the KSRF relation4 from (2.16).

The construction of the chiral Lagrangian for the non- leptonic charm-changing weak interaction is achieved by converting the / AC / = 1 four-quark interaction in the standard model into one which involves charmed mesons and other phenomenological hadron fields. The chiral Lagrangian for the charm-changing nonleptonic weak in- teraction should have the same transformation properties under SUL (3) x SUR (3) and CP transformation as the four-quark operators. In the standard model, the Cabibbo-angle-favored nonleptonic charm-changing weak transition is characterized by the selection rule AS/AC= 1, and the Cabibbo-angle-suppressed decay is characterized by the selection rule AS /AC = 0. The four-fermion effective Lagrangians for these two channels are

- - l + ~ , (2.16) mo2 Z

. (2.14) 2mo

and a "renormalization" of pseudoscalar Goldstone fields,

1 0 0 0 1 0 0 0 -2

a n d [ m ] =

'm 0 0 O m 0 0 0 m,

1058 A. N. KAMAL, LO CHONG-HUAH, AND M. RAHMAN 31

and

GFsine cose O ( A S / A C = O ) =

Ir2 ( f + Q l s + f - Q , * + H . c . ) ,

The indices i and j are color indices; repeated indices are summed. Coefficients f +, f - are QCD renormalization fac- tors. The one-loop-order perturbation calculation gives'4 f + e - 0 . 7 and f - e 2 . O I 5 and Q I 5 transform like Ti3 and T12- ~4~ states of (15 , l ) ; 06* and Q6* transform like A22 and A23 states of (6*, 1 ).

The chiral Lagrangians of O ( AS/AC = 1 ) and O ( AS/AC = 0 ) are given as

and

where

The constant column matrices and square matrices in Eqs. (2.28M2.35) are

3 1 - PHENOMENOLOGICAL LAGRANGIAN FOR NONLEPTONIC . . .

We define the charge-conjugation phase of the charmed-meson triplet to be - 1 (i.e., C$= - qt), so that L (AS/AC = 1 ) and L(AS/AC=O) are CPeven.

In the remaining part of this section, we shall calculate the threshold parameters of meson-scattering processes, and try to get information about the parameter m,/m from the analysis of low-energy scattering data. It is known that the tree-graph analysis of the chiral (nongauged) ~ a ~ r a n ~ i a n ' ~ reproduces the current-algebra results'62" for the S-wave scattering lengths of various meson-scattering processes. These results are15

They are not in good agreement with the experimental data, which arel8,I9

ao(rr)=(0.26i0.05)m,- ' , a2(rr)=(-0.02810.012)m,-' ,

a1/2(~n-)=0.336m,-', and a3 l2 (Kr )= -0.14m,-' . The S-wave scattering length a1 is simply related to the scattering amplitude Fl(s,t,u) when all the Mandelstam vari-

ables have the threshold values. There are four types of interaction vertices in our model that contribute to the meson- scattering amplitudes.

(a) The four-meson contact interaction

(b) The vector-current interaction

(c) The scalar-current interaction

(d) The So tadpole interaction

They contribute to the meson-scattering processes as shown by Feynman diagrams in Figs. I (a)- 1 (d), respectively. The S-wave scattering lengths are calculated to be

A. N. KAMAL, LO CHONG-HUAH, AND M. RAHMAN

The current-algebra results can be obtained by setting Z = 1 and h=O in Eqs. (2.40a)-(2.40d). We take cos2+=f, since the octet-singlet mixing of the (S*,E) sys- tem is very close to the ideal mixing at threshold.1° The "strength constant" of the scalar-meson-mediated interac- tion h2m '/f ,4 can be determined from the decay width of the 6+--q071-+ mode; one finds h2m 2/f , 4 = ~ . 4 ~ 8 2 . NU- merical values of these scattering lengths, corresponding to Z = and various values for m,/m are given in Table I. In I3ig. 2 we have plotted the four scattering lengths as a function of m,/m.

The contribution of the scalar-current interaction in- creases the predicted values of ao(r71-) and a l/2(K71-), and decreases the magnitude of a2(71-T). This is in the right direction to reduce the discrepancy between the theoretical predictions and experimental data. In fact, when m,/m lies between 2.5 and 3, the predicted ao(71-71-1 and a2(71-71-) are in good agreement with data. Our prediction of

a3/2(K.ir) is even worse than that of current algebra; both the SUy(3)-symmetry-breaking effect introduced by the vector-current interaction and the contribution of the scalar-current interaction tend to make a3/2 (KT) more positive. It is possible that the "unitary correction" to the scattering lengths in exotic channels is non-negligible, but the calculation of this correction is too complicated to be carried out here. We also believe that the unitary effect may not drastically change the predictions in nonexotic channels. Thus our analysis of low-energy scattering data gives us enough evidence that the symmetry-breaking pa- rameter m, /m should be less than 3.

111. TWO-BODY CHARMED-MESON DECAYS

The weak-interaction vertices in the phenomenological Lagrangian which contribute to two-body charmed-meson decay modes are of the following three types:

31 - PHENOMENOLOGICAL LAGRANGIAN FOR NONLEPTONIC . . .

Figure 3(a) shows graphs arising from L1. The two- body conversion vertices in L2 are the AC=l weak- interaction vertices that give rise to the scalar-meson poles shown schematically in Fig. 3(b). The four-body weak- interaction vertices in L3 result in the processes shown in Fig. 3(c), where the SO meson (J'=o+) decays "strongly" into the vacuum through the tadpole interaction

The amplitudes for the three Cabibbo-angle-favored D- meson decay modes are

M(DO-+K+T-)=~A,/,+ f ~ , / ~ [ 1 + f(mD2)] , (3.5a)

where

and

A( C - D )( 3m + m, )cos28 g= (3.6d)

f T 2 The amplitudes satisfy the isospin sum rule

M(D-+KOT-)=M(DO-+K+T-) + ~ ~ M ( ~ ~ - K O T ~ ) . (3.7)

A. N. KAMAL, LO CHONG-HUAH, AND M. RAHMAN 31 -

' , ' . 0 0 . 0

0

:)l**~: + crossed 0

0 4

. 0

. 8 .

\

FIG. 2. Graphs of various S-wave scattering lengths as func- tions of m,/m. Vertical axes are scattering lengths in m,-'. The horizontal axis represents m, /m.

. 0- ' . ' + crossed graph

FIG. 1. Various Feynman diagrams that contribute to meson-meson scattering processes. The pseudoscalar, vector, and scalar mesons are represented by dashed, wavy, and solid lines, respectively.

If we neglect the scalar-meson-mediated interactions in (3.5a) and (3.5b), there are only two parameters A, B in the decay amplitudes. The branching ratio of each ex- clusive mode is completely determined by the ratio x = B / A which measures the relative strength of L6, and LI5 . Its value should be simultaneously determined by the branching ratios of the 5' decay modes,6

and the lifetime ratio6 of charged and neutral D mesons,

FIG. 3. Feynman diagrams that contribute to D-+2 pseudoscalar-meson decay processes. The pseudoscalar and sca- lar mesons are represented by a dashed and a solid line, respec- tively. The black square box in Fig. 3(b) represents the weak charmed meson -+O+ uncharmed meson vertex. The open cir- cle in Fig. 3(c) represents the S O tadpole vertex.

31 - PHENOMENOLOGICAL LAGRANGIAN FOR NONLEPTONIC . . . 1063

Using the interaction Lagrangian L I , the branching ra- ZAmD2 tio of (3.8) is calculated as M(D --+r-rO) = -sine cos&------ .

f i f a2 (3.14)

r( X) = 0.732( 1 -x )' (3.10) The predicted branching ratio

( 0 . 7 9 + 1 . 0 7 x ) ~ ' and the lifetime ratio as

B ( D - + T - T O ) / B ( D - - ~ - K O ) -- d D - ) - ( 0 . 7 5 k 0 . 2 4 ) R ( x ) , (3.11) dB0) =0 .06(B/A= -7 ,C=D=O) , (3.15)

where the Particle Data Group value6

well below the experimental lower bound of 0 . 2 7 . ~ (3 '12) The experimental data on Cabibbo-angle-suppressed Do

and D - two-body decays are is used, and

r ( x ) and R ( x ) are plotted in Figs. 4 and 5, respectively. B(fi0-t~+~-)/~(D0+~+r-)=0.033~0.017, The acceptable values of the parameter x are in the

neighborhood of x = -7 . When x = - 7 , the prediction (3.16b) for the branching ratio is B(DO+ K ~ ~ ~ ) / B ( ~ ~ B(D---+K~K-)/B(D--+K~~-)=O.~~+O.~~ . -+K+r- )=1 .04 and the prediction for the lifetime ratio is ~ ( D - ) / ~ ( f i O ) ~ 3 . 8 1 f r 1.2 which agrees with experimental (3.164 data in (3.9).

The decay amplitude of the Cabibbo-angle-suppressed The decay amplitudes of these Cabibbo-angle-suppressed mode D -+.sr-rO is given as modes are given as

TABLE I. Numerical values of S-wave scattering lengths ar, corresponding to various values of symmetry-breaking parameters m,/m. The scattering lengths are in units of m,-I.

1 1.5 2 2.5 3

ao(.rr.rr) 0.3 0.27 0.25 0.22 0.20 az(v.rr) 0.02 0.01 0.0005 -0.01 - 0.02 al/z(K.rr) 0.151 0.157 0.163 0.17 0.178 as/z(K.rr) - 0.02 -0.017 -0.013 -0.009 - 0.004

M ( ~ - K - K O ) = sine cose z ( A + B ) ~ D ~ 2 h ( m + m , ) ( C + D ) 4 h ( C + D ) ( m - m , )

- - f 2f r2 f T 2 ( m ~ 2 + ~ s 2 + i M s r s ) f a 2 ~ S * 2

A. N. KAMAL, LO CHONG-HUAH, AND M. FL4HMAN 3 1 -

FIG. 4. ~(X)=B(DO-LR~~O)/B(D~- K+T-) as a function of x; x=B/A.

The octet-singlet mixing angle 4 of the (S*,E) system is a function of energy. With S * as the isoscalar member of the SU(3) octet, the mixing angle is 90" in the D-meson mass region.'' The masses and halfwidths of S* and 6' are almost equal; MS, =975 MeV, M 8 =980 MeV, r,, = 15 MeV, and r8,=27 MeV. Because of this, the propagators of S* and 6' do not introduce significant SU(3bbreaking effects into the decay amplitudes. If the scalar-meson-mediated interactions are turned off be., C = D =O), the amplitudes in (3.17)-(3.19) are of the same size, that is,

= -M(D--K-K')

Z ( A +B)mD2 = sin0 cos . (3.20)

2fa2

The predicted branching ratios are then

B(D0-K+K-)/B(D0-~+~-)=0.85 , (3.21a)

FIG. 5 . R (x) = T ( D O + K + ~ - ) / T ( D - + K ~ ~ - ) as a func- tion of x; x =B / A .

The branching ratios in (3.21b) and (3 .21~) are in good agreement with data. Thus the weak-interaction Lagrang- ian L1 itself can predict all the two-body decay except the branching ratio in (3.16a).

When the contribution of scalar mesons is taken into account, the Do-decay amplitude becomes complex. The prediction of the branching ratio and lifetime ratio are determined by the complex number

a and b are solutions of the following equation:

Since the two-body decay data have large errors, we are not able to fine tune the parameters C and D for small values. However, we found that the sets of parameters with C and D sufficiently different from zero do not yield correct predictions of Cabibbo-angle-suppressed two-body decay modes.

If the charmed-meson decay processes are calculated in the six-quark model, the mixing matrix elements Vcd VZd and V,yls which are the coefficients of the operators FL yWL d L y P u ~ and Q ypsL FL y P u ~ , respectively, are gen- erally different. New operators which transform like (3,l) would appear. Their contributions may be significant to Cabibbo-angle-suppressed charmed decay modes. Other intera~tions,"~ such as the charged-Higgs-boson-mediated interactions in the extended Glashow-Weinberg-Salam model or penguin diagrams, may also be responsible for the observed enhancement of the D'-K+K- decay rate. Our chiral Lagrangian based on the simple four-quark model may be too simple to account for the Cabibbo- angle-suppressed two-body decays. Perhaps the three- body-decay data will lead us to a better determination of the parameters in our model.

IV. CONCLUSION

In this paper, we have constructed a gauged chiral La- grangian to study charmed-meson decays. Vector and axial-vector mesons appear both in strong and weak in- teractions. Their couplings to other hadron fields are determined by the gauge invariance of the chiral- symmetry group. The interaction mediated by O + un- charmed mesons not only accounts for the S-wave final- state interaction of decay products, but also improves the prediction of threshold parameters in TT scattering. We have found that, when the two-body charmed-meson de- cays are assumed to be dominated by the direct-decay ver- tices, our model can correctly predict all the branching ra- tios of the two-body D-meson decays in both Cabibbo- angle-favored and Cabibbo-angle-suppressed channels with the only exception of ~(6'- K +K - )/B(DO -T+T-). It should be pointed out that the neglect of scalar-meson pole diagrams does not mean that the ha- dronic final-state interaction is unimportant. In fact, the strong final-state interaction in two-body Cabibbo-angle- favored D-meson decays introduces a phase difference,

3 1 - PHENOMENOLOGICAL LAGRANGIAN FOR NONLEPTONIC . . . 1065

which is roughly 180", between I = and I = % weak- interaction amplitudes.' It does not, however, produce large enhancement to isospin amplitudes. The desired rel- ative sign between I = and I = 1 decay amplitudes cal- culated in our model can be obtained by choosing the proper sign of the parameter B / A where A and B are the effective coupling constants in the direct-decay vertices. The scalar-meson pole diagrams result in only a small enhancement in the I = decay amplitude. However, the contribution of scalar mesons can be significant in three- body decay processes. The absence of the resonance in the Dalitz plot of the DO-K~T+T- mode may be due to

the destructive interference of the broad resonant state p and the narrow resonant state S*. A detailed investiga- tion of the problem is in progress.

We conclude that our chiral-Lagrangian model gives a rather promising description of two-body D-meson de- cays, and it is worthwhile to further study the phenomenology of this model.

ACKNOWLEDGMENTS

This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

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