Vacuum insertion and nonperturbative effects for exclusive nonleptonic decays of charmed mesons and kaons

Z. Phys. C - Particles and Fields 32, 237-242 (1986) Partkztes ~r Physik C

and F = Js �9 Springer-Verlag 1986

Vacuum Insertion and Nonperturbative Effects for Exclusive Nonleptonic Decays of Charmed Mesons and Kaons Hai-Yang Cheng* Physics Department, Indiana University, Bloomington, IN 47405, USA

Received 10 March 1986

Abstract. The inconsistency of vacuum insertion with the symmetry properties of W-exchange amplitudes and the fact that not all vacuum-saturation terms are genuine contributions from the vacuum intermediate state all reflect the necessity of modifying the traditional vacuum-insertion method. As suggested by the 1/N expansion (N: number of colors), final-state interactions and the nonperturbative effects due to soft- gluon exchange are indispensible for a satisfactory explanation of nonleptonic decays of mesons. From the data we determine the parameter e, which measures the contribution from color octet currents relative to that from the corresponding color singlet currents, to be - 0.47, - 1.0, - 1.1 respectively for D ~ PP, VP (P: pseudoscalar meson, V: vector meson) and K ~ 2 r t decays. Presumably this can be tested by the lattice Monte Carlo calculations.

A standard approach of evaluating weak matrix elements is based on the valence quark assumption and vacuum-insertion (or factorization) approximation in which the matrix elements of two quark bilinear operators are saturated by the vacuum intermediate states in all possible ways [1]. As we shall see later, this approximation leads to an internal inconsistency, and some vacuum-saturation terms actually arise from the contributions of one-particle intermediate states. Con- sequently, the traditional vacuum-insertion method is not completely logical. As an example for demonstrat- ing the use of vacuum-saturation, consider the matrix element (z t+K-l ( f tc ) (~d) lD~ where (~lqz) = qlY.( 1 - - ~5)q2; one has

( n + K-[(ftc)(gd)[D ~

= (re + K-I(gd)10) (01(ac) lO ~

+ �89 + I(ad)10)(g-I(gc)lO ~ ) (1)

where use has been made of the Fierz identity

(ft c) (gd) = ~(O. d)(gc) + �89 2" d)(g2" c). (2)

* Chester Davis Fellow

The color octet currents in (2) do not contribute to the weak decay so long as the intermediate states are color singlets.

Although the utilization of vacuum insertion is considerably simple, it is also well known that this method of matrix element estimation fails to explain the A I =�89 rule for kaons and the A I = 3/2 hyperon decay amplitudes [2]. Furthermore, it leads to a serious discrepancy between theory and experiment for nonleptonic charm decays. The most noticeable example is the decay D ~ 1 7 6 ~ which is naively expected to be both color and QCD-correction suppressed; experimentally it proceeds at a rate comparable to the decay mode K-7r +.

Here we would like to point out another less known fact, namely that the vacuum-insertion approximation is not internally consistent with the SU(3) or S U(6) symmetry properties of the charm decay amplitudes when it applies to the following W-exchange processes: (1) In the limit of U-spin symmetry, the W-exchange in D ~ K - p + , K *zt + is forbidden [6]. However, the W-exchange amplitude for D ~ V P decays (V: vector meson, P: pseudoscalar meson) in vacuum saturation is of the form ( V P ] A u l O ) ( O I A U l D ) , which does not vanish in the S U(6) limit since the axial-vector current is not conserved. By contrast, owing to the conserved vector current, the W-exchange amplitude in D ~ K" rc dimin- ishes in the limit of S U(3) symmetry, whereas U-spin symmetry implies an enhancement of W-exchange for K-re + relative to K - p + and K-*zt +. (2) In the SU(3) limit, quark-diagram amplitudes for D o ~ K "~ ~/a, K~ r/o, where r/s, qo are S U(3) octet and singlet states respectively, are given by [-3-5].

A(D ~ ~ K~ = l_cosZ Oc(b - c) ,/6

1 A (D O ~ /~o r/o) = ~ cos 20c(b + 2 c) (3)

where b is the internal W-emission diagram and c is the W-exchange diagram. On the other hand, vacuum

238 H.-Y. Cheng: Vacuum Insertion and Nonperturbative Effects

insertion gives [7]

A (D o ~ / ( o t/8) = ~ cos z Oc(b + 3 c) `/6

A (D O ~ / ( o r/o) = 1 _ cos20c(b) (4) ,/3

which is not in accord with (3) derived from the S U(3) symmetry. These inconsistencies and the aforemen- tioned failure of the vacuum-saturation method when compared with experiment all reflect the necessity of modifying the traditional vacuum-insertion method.

The justification and improvement of vacuum saturation might be guided by a recent systematic analysis of the 1/N expansion (N being the number of colors) for meson weak decays by Buras et al. [8]. In the large N limit, it is pointed out in [8] that the leading contribution in the 1/N expansion corresponds to a factorizable diagram (Fig. la) and is of order N U2. There are two different kinds of subleading diagrams (both of order N- 1/2): the one-particle intermediate state nonfactorizable diagram (Fig. lb) and the diagram for final-state interactions (Fig. lc). At first glance, a calculation of Fig. lb due to a complete set of single intermediate states seems to be a formidable task. Thanks of the Fierz identity, the nonfactorizable diagram (Fig. lb) can be decomposed into two pieces: one of them becomes factorizable to the order of N- x/2; the other piece, which contains color octet currents, is not factorizable. More precisely, for the decay D ~ + K - we have

Fig. la = ( n + K-[(Od)(gc)lD~ ('/~)

= ( n + 107d)10)(g-I(gc)lO ~ (5)

Fig. lb = < n + K - I(~d)(gc)lD~ ~1/'/~1

1 = ~ ( n + K-l(gd)10 ) <Ol(ac)lO ~ )

+ �89 n + K " I(g2"d)(aA"c)lD ~ >. (6)

Now soft-gluon exchange between the two bilinear quark operators (gX~ and (t72"c) certainly can contribute to D--+Kn decays. Hence to be subleading order of the 1IN expansion, the effect of all single- particle intermediate states is equivalent to the sum of a calculable next-to-leading factorizable term and a nonperturbative soft-gluon contribution, which is beyond our present ability to estimate (except perhaps in the lattice gauge theory [9]). From Fig. la (or (5)) it is obvious that the leading contribution to D -- , Kn does come from the vacuum intermediate states. It is also clear from (6) that the subleading vacuum- insertion terms are actually part of the single intermediate states' contributions. Thereby the traditional use of the vacuum-saturation method is not quite consistent. For instance, an inclusion of the one- particle contribution, such as n, t/ . . . . etc., to the B- parameter

B = < K~ I(ds)21/s ~ ) / (4 f2mr/3) (7)

_ 7 r ~ "rr+ 7/-+

( o ) + (c)

Fig. 1.a-c. Example of leading a and subleading contributions b, e to D O -~K-n +. For details see [8]

would cause a double counting problem for the contributions from physical single particle intermediate states. Hence a consistent way of employing vacuum saturation is to insert vacuum intermediate states (and other physical states) directly between the two bilinear operators rather than to insert them in all possible ways. That is to say that only those vacuum- insertion terms which arise from color-matched operators are genuine contributions from the physical vacuum intermediate states. Empirically, it has been observed by many authors [10-12] that the discrepancy between theory and experiment for K ~ 2n and D --* PP decays is greatly improved (but not completely consistent) if contributions from colour-mismatched currents are dropped.

As already emphasized by Buras, G6rard, and Riickl, to the subleading order in l /N, it is mandatory to include all other next-to-leading contributions besides the non-leading vacuum-saturation terms. In fact, it has been shown that both soft-gluon nonperturbative effects and final-state interactions are absolutely indispensible for a satisfactory explanation of D ~/(" n data [3-5,13]. This means that in order to have a quantita- tive agreement with data, one has to go beyond the leading order in the 1/N expansion. This is also necessary for solving the apparent internal inconsistency as discussed in passing.

Despite the fact that final-state interactions and nonperturbative QCD corrections are in practice very difficult to estimate in terms of present attainable technique, nevertheless they can be extracted from the available data. In this paper we wish to estimate the soft-gluon effect to weak decays from D ~ PP, VP and K ~ 2 n data. Before proceeding, we would like to emphasize that all exclusive nonleptonic decays of a meson can be expressed in terms of six distinct quark diagrams [14, 15]: external W-emission a, internal W- emission b, W-exchange c, W-annihilation d, the penguin diagram e, and the side-ways penguin diagram f, as depicted in [14] Fig. 2.4.

1. The Decays D --, P P

The quark diagram amplitudes for D - , K n decays with final-state interactions are given in Table 1, where 61(t/i ) is the phase shift (inelasticity) for the isospin I amplitude. (For details, see [4].) A fit to the Mark III data for D ~ / ( n [16] (with r/3/2 = 0 due to the absence

H.-Y. Cheng: Vacuum Insertion and Nonperturbative Effects 239

Table 1. D ~ / ( ~ t decays, where the isospin amplitudes are related to quark-diagram amplitudes by All 2 = (2a - b + 3 c ) / ~ and h3/2 = (a + b). Use of z(D ~ = 4.3 x 10- ~3s, and z(D +) = 9.0 x 10- ~3s has been made for calculating the branching ratios. Experimental data are taken from [16]

Decay mode amplitudes with final-state interaction Br(~o)th~o~y Br(~o)exp

D o ~ K - n + lcos20c(x/2A1/zei6,/~- ~,/2 + A3/2 eia~/~ ,,/=) 11.9 4.9 + 0.4 + 0.4

--, R7 ~ ,-r ~ 31 cos 20c( - A 1/2 e~a'/=- "'/" + ~//22A 3/2 e~a'"~ - ""~) 0.4 2.2 + 0.4 + 0.2

D + -* K'~ + COS 20cAsl2e ~a~'~ ~'~ 14.3 3.5 + 0.5 + 0.4

ofisospin ~ resonances, and e -el/2 = 0.8 [1 1]) gives the solution*

(b - c)/(a + b) = - 1.35 _+ 0.17,

C51/2 - -63 /2 7 9 _ +1~176 = 1 4 o . (8) The effective QCD-corrected weak Hamiltonian

responsible for Cabibbo-allowed charm decay is given by

GF Hat - 2,,//2 V,a Vc~[(c+ + c_ )fftd)(gc)

+ (c+ - c )(~ic)(gd)]. (9)

The standard calculation based on the vacuum- insertion approximation yields

_ GF (c a - x/~ 1 + lc2)f~[(m2 -- mZ)f~ + mZfD-K(m2)]

(10a)

Gv b = 7~(C2 + ~ c O f r [ ( m 2 - m2)f~ + m2fD_~(m2)]

v -

(10b)

GF 1 2 2 ~ K 2 c - , ~ ( c 2 + ~COfD[(mK-- m , ) f + (mo)

- - m 2 f = - K ( m 2 ) ] (10c)

where use of

( P(q)IAuIO ) = - i f pq, (pI(p2) I WuIpIC(PO)

= ifi jk[f+(Pl + P2)U + f - ( P l -- PZ)u] (11)

has been made, and cl = (c+ + c_)/2, cz = (c+ - c_)/2. For form factors at q2 = 0, we use the results obtained from the Isgur model [17]**

fo+== 1.33, fO+K= 1.15, T~_K = 1.03

f ~ f~ f " f = 0 . 1 3 . (12)

* In [3-5], rh/2 = 0, c+ = 0.69 and c_ = 2.09 are chosen. Here we take the more realistic value for q~/2 from [11]. The other solution ( b - c)/(a + b)= 2.68 _+ 0.17 is ruled out on the grounds that the theoretical calculation always gives a negative value for ( b - c ) / (a + b) ** For the form factor f,K we take 0.13 as determined from the effective chiral Lagrangian [18] rather than -0 .30 as calculated in [17]. The negative sign for f fK (not fK_=!) is inconsistent with experiment [19]

Except for gK, the quark-model calculations of the form factor f _ are consistent with the U(2,2) quark model's prediction [20]

f _ (q2) __ mt -- m2 ~ 2 f + (q2). (13)

ml

Assuming the dipole form for the q2 dependence of the form factors, we obtain (with c_ = 1.99, c+ = 0.71)

a = 4 . 4 • 10-6GeV, b = - l . 1 • 10-6GeV,

c = - 6.3 x 10 -8 GeV. (14)

The W-exchange amplitude is both color and helicity suppressed. From (14) and the decay amplitudes given in Table 1 (without final-state interactions), it is straightforward to compute the branching ratios for D-~RTt decays. It is evident from Table 1 that the standard approach based on the vacuum-insertion method fails. From (14) it follows ( b - c)/(a + b)= - 0 . 3 0 , which is not compatible with (8).

As noted in passing, to the subleading order in 1IN not only the next-to-leading vacuum-insertion contribution but also the nonperturbative QCD correction must be included. Denoting e as the contribution from the matrix elements of color octet currents relative to that from the corresponding color singlet currents, we have

<~z+K - I(a2ad)(g2ac)lD~

e. - <~z+ K_ I(~d)(ffc)lD~

(~+ ~.~ ) = (~+R~ + )vao (15)

( ~o/~o i (g2a d)(t~ 2a c) lD o ) e b - (jtOKOl(gd)(~c)lDO>~

( rt + K~ c)lD + )

( ~ + K~ d)(ac)l D+ >vao

for the amplitudes a and b respectively. A similar quantity is not defined for W-exchange since e~ becomes divergent in the S U(3) limit. (10) are then modified to

a=(10a) with (c 1 -~-1C2)-"+1-C 1 - ' k C 2 ( l + e a / 2 ) ] (16a)

b = (10b) with (c 2 + 1 Cl )__+ [c2 + ca (�89 + ~b/2)] (16b)

c~- caGf(~-K+l(g2d) (g t2ac) lD~ ). (16C) 2,,/2

240

Although (10c) vanishes in the S U(3) limit, the nonper: turbative term (16c) does not. Therefore, the inconsistencies of vacuum-saturation for W-exchange with the S U(3) or S U(6) symmetry properties for decay amplitudes as discussed before are presumably taken care of by the soft-gluon contribution. Historically there are two extreme improved models: (i) Chernyak- Zhitnitsky (CZ) model [21] in which W-exchange is enhanced by the radiative gluon which is emitted from the initial light quark and creates the final quark- antiquark pair, but ~, = eb = 0, (ii) Deshpande-Gronau- Sutherland (DGS) model [22] in which W-exchange is negligible but e, and e b are sizable. The CZ model is apparently not favored since there is no reason to neglect the soft-gluon effect for the amplitudes a and b. Indeed, if e~ = eb = 0, the predicted decay rate for D + ~ K ' ~ + is still the same as that in vacuum- saturation (see Table 1) which is three times too large. Furthermore, this model is also disfavored by the Mark III data for D + ~ / ( ~ and particularly for D ~ got /decays [3]. In the DGS model W-exchange is still suppressed based on the assertion that since the final two quarks in the W-exchange process are energetic, the effect of soft-gluon exchange might be neglected due to the large relative momentum between the two quarks. Unfortunately, the role of W-exchange still cannot be decisively determined by the present Mark III data. Nevertheless, we know that c cannot be as important as advocated in CZ model. As elabo- rated in [3] the prediction of the relative decay rate for D + ~ / ( ~ and D~176 in the DGS model is consistent with the data.

Since in the limit of S U(3) symmetry a, = eb, we will let e = ea = eb in (16) for later convenience. (Or, e may be regarded as the average value of ea and eb. ) Now in the DGS model e can be determined from (8) and (16) and it turns out that

e (D ~ P P) = - 0.47 (17)

and

a = 5 . 0 x 10-6GeV, b = - 2 . 9 x 10-6GeV. (18)

Comparing with (14), it is evident that due to the nonperturbative QCD effect, the internal W-emission amplitude is enhanced by a factor of about 3, and hence no longer color suppressed. As shown in (16), the subleading contributions are associated with the parameter (1/N + e/2), where the 1/N comes from the next- to-leading vacuum-insertion term and the e/2 measures the nonperturbative correction. Since the subleading vacuum-insertion term is largely cancelled by the nonfactorizable soft-gluon contribution so that (1/3 + e/2) ,~0.1, this explains why the leading term in the 1/N expansion already drammatically improves the traditional vacuum-saturation method for D--*PP decays.*

* Keeping only the leading terms in the 1/N expansion corresponds to e = - 2/3

H.-Y. Cheng: Vacuum Insertion and Nonperturbative Effects

Three remarks are in order. First of all, although the relative magnitudes of D ~ / ( = amplitudes are repro- duced with e = -0 .47 , the absolute magnitudes are overestimated by 28% as one can check from (18) and Table 1. Presumably the nonleading contribution of order N-s/2 in 1/N will take care of the discrepancy. Second, the DGS model mimicks the sextet dominance model [23] in the sense that the ratio (e-/c+)eff ~ 7.4 is substantially larger than the conventional value ~ 3. Third, to the leading order in l /N, the ratio of D o /(o=o to K - = + is predicted to be about �88 which is to be compared with the experimental value 0.45 + 0.08 _+ 0.05 [16]. Hence, although the discrepancy between theory and experiment is greatly improved, the inclusion of the next-to-leading corrections is obviously inevitable.

2. The Decays D, F-* VP

For D, F--* VP decays there are two different amplitudes for each quark-diagram depending on whether the vector meson comes from the charmed-quark decay or not. We denote the primed amplitudes for the case that the vector meson arises from the decay of the c quark. The amplitude for the Pc--* VP decay is of the form

M(Pc--* V P) = eUpuA (19)

where eu is the polarization vector and Pu is the 4- momentum of the charmed meson Pc. The dimension- less amplitudes A for D ~ / ( * ~ and / (p are similar to that of D--,K'1t given in Table 1 except that unprimed amplitudes are replaced by primed amplitudes for D~/ (*1t . As pointed out in [4,5], the primed amplitudes a' and b' can be uniquely determined from F + ~b~z +, D + --* ~b~z + and D ~ / ( * l z decays. With the phase convention that a' is positive, we obtain the model-independent results [5]

a' = (2.50 _+ 0.42) x 10 -6,

b' = - (3.67 + 0.51) x 10 -6. (20)

The present Mark III data for D ~ / ( * ~ and/( 'p decays are not accurate enough to the determine the phase- shift difference precisely [5]. However, from (20) it is obvious that there is a severe destructive interference in the decay mode A(D § _, / (o , t /+) = cos20r , + b').

Now we turn to the vacuum-insertion calculations of the quark-diagram amplitudes a' and b'

a'= Gv(c 1 + �89 + I(~d)lO><K~ + >/~/2

b'= GF(c2 + �89 F, ~ > <~+ I(t~c)lO + >/v/2

(21)

Parametrizing the matrix elements as

<V(q)[Vu[O> - 2 - fvmvS. < Vi(Pz)lA~lP~(pl)>

= --fiJKeV(F~ 9uv + FzPl~,PI~ + F3p2~,pl~) (22)


and using the form factors derived from the consti tuent quark model [20]*

2 F l ( O ) = m l + m 2 , F 2 = 0 , F 3 - (m l+m2)2F1 (23)

together with fK* = - f } f o = 0.287 determined from �9 0 N / - P

the wtdth of the leptomc decay p -* e + e - [4], we obtain

a ' = 2 . 1 3 • -6 , b ' = - 0 . 8 7 x 1 0 -6 (24)

where (11) and (13) have also been used. Compar ing (24) with the experimental value (20), it is obvious that b' is underest imated by a factor of 4. It should be noticed that wi thout taking into account the structure cons tan t f i j k in (11) and (22), as did in some literature, b' would be parallel to a'.

As in the case of D ~/( ' re decays, this difficulty with the vacuum-insert ion method can be circumvented by including the nonperturbat ive Q C D correction. As before, a fit to (b'/a')~xp (i.e. (20)) yields**

e(D --*/~* 7t) ~ - 1.0. (25)

A compar ison of (25) with (17) shows that the soft- gluon effect is two times as large as that in the P P decays, an expected result owing to the small relative m o m e n t u m between the V P final states, For D ~ V P decays (1/3 + e/2) ~ - 0.17 ( ,~ 0.1 for the Krc case); this implies that the subleading terms give destructive contributions.

3. The Decay K ~ 2 n

The A I = �89 rule for kaons is one of the oldest and most mysterious problems in weak interactions. The evaluat ion of the matrix elements of the operators 0~...04 [24] in vacuum-satura t ion is rather standard; the calculation for the penguin operators 05,6 is, however, somewhat controversial�9 It was originally realized by Shifman etal. [24] that the amplitude (2r~I05,61K) would vanish in the chiral limit if it is evaluated in vacuum-insert ion without momentum- dependence (i.e. only vertices independent of momentum are kept). Therefore, in the chiral limit it is impor tant to retain the momentum-dependent terms in the matrix elements in order to obtain nonvanishing penguin amplitudes�9 However, the inclusion of these terms is beyond the current-algebra technique�9 In the literature, momentum-dependent contr ibutions are usually put in by hand. Correct momen tum dependence of the matrix elements can be obtained from the next-order expansion in chiral per turbat ion theory�9

* There are many different parametrizations for the form factors Fa and F 3. Fortunately, the amplitudes a' and b' are insensitive to the choice of F~ and F 3 due to the smatl q2 ** From the available Mark III data for D --*/(p, co/(, one can have three different solutions for the unprimed amplitudes a and b [5]. One of the solutions a=(4.06+0.38)x 10 -6, b=-(1.88+0.28) x 10 6 yields e(D --,/(p) ~ - 1.1. Hence we believe this is the correct solution since it is in accordance with (25)

241

The chiral representation for the quark bilinear field is [25]

qiRq~ = A U,i + Bf2(UduU63u U+)ij/8 + " " (26)

where

U = exp(2i~b"2"/f), Tr()~,2b) = bah, f ~ O . 9 5 m ~

A = - - f2v /4 , B = - 4m2/(msm 2) ~- - 4v /A 2

m2+ mZK +

m u d- rrl d m u + ms �9

The parameter A .~ 1 GeV is the scale ofchiral symmetry breaking [26]. F rom (26) it follows that

(0lgLdRI K~ ) ---- if v~2, (~+ I~ iRdLI0> ---- - / f v / 2

( Tr + (q + )rt- (q_ )l dRdLlO ) = 2 ( l ' ' q++ '+~T- ) "q- "~

( ~ r - ( q ) l g L u R l K ~ (27,

to be compared with

(re + re-I~Rdgl0) = 211 + (q+ + q-)2/m2] (28)

in [24], where the momentum-dependen t piece is put in by hand�9 Standard vacuum-insert ion calculation then gives*

A ( K + ~ n + n ~

= 3/4GvsinOccosOcf~c4(mZ-mZ~)(1 +�89 (29)

A(K o __. re+ re-)a1 = 1/2

Gv - 2x /~ , s inOccosOcf~{ (m~-m~)

"(cl - c2 - c3)(1 - ~) + v z [4c5(1 - 1/9) + 2c6(1)]

. ( I , ,_ 1 I,,m -2m. m,, \ f ~ ~ f~ A~ t - ~

Using c 1 = - 2�9 c 2 = 0�9 c 3 = 0�9 c4 = 0�9

c 5 = - 0.097, c 6 = - 0�9 [27], and f r / f~ = 1.22 [28], we obtain

A(K + -*n+ 7z ~ = 3�9 x 10-8 GeV,

A ( K o ~ n+ n-)a~= 1/2 = 1.20 x 10- 7 GeV (30)

to be compared with the experimental values [19]

IA(K + --*n+n~ = 1.83 x 10-8 GeV,

IA(K~ =2.71 x 10 -7GeV. (31)

To take into account the soft-gluon corrections, we note that because the penguin opera tor is of the

* For on-shell amplitudes, it makes no difference whether or not the operators O~ are normal ordered

242

(S - P) x (S + P) structure (before summing the quark flavors) the nonperturbative effects for the penguin diagrams might be quite different from that for the ( V - A) x ( V - A) interaction characterized by

( n + n~ + ) e(K ~ 2n) =

( n+ n~ K + )~a~

( n ~ n~ "d)(ti2au)lK~ ) = (nOnOl(gd)(fiu)lKO)va c (32)

We thus cannot determine the parameter ~ solely from the ratio ao/a2. Nevertheless, it is instructive to determine e by requiring that the discrepancy between theory and experiment for A (K § ~ n § n ~ be saturated by the nonperturbative soft-gluon corrections [22]. This amounts to assuming that other nonleading contributions in the 1/N expansion (i.e. of order N-3/2 or smaller) are negligible. Now replacing �89 by (-~ + 5/2) for ( V - A) x ( V - A) interactions in (29), it is clear that if the soft-gluon effect suppresses K + ~ 2 n , it would enhance the amplitude K ~ ~ 2n. From (29) and (31) we find

e(K ~ 2 n ) ~ -- 1.1 (33) where the other solution e ~ - 4.3 is not acceptable in the spirit of the 1IN expansion. Equation (33) was first obtained in [22]. A realistic value of e should be in the neighborhood of - 1.1. From (29) and (33), it turns out the usual ( V - A ) ( V - A) interactions give

A(KO ~ , + ~-~at=l/2 - ~ ~ . . . . ] ( V _ A ) ( V _ A ) - - ~ , ~ x 10-8GeV,

(ao/a2)(V_A)(V_A) = 6.8 (34)

whereas experimentally

a o 3 A ( K ~ aI=l/2 - - =

a 2 2 A(K+ ~ n +no ) = 22.2. (35)

This indicates the possibility that 70% of the A I -- �89 amplitude might be attributed to penguin and W- annihilation contributions.

It is also intriguing to consider the B parameter (7) which, to the subleading order in 1/N, is given by [12]

('2) B= 3/4 1 + ~ + . (36)

Nonperturbative soft-gluon effects reduce the B- parameter from the vacuum-insertion value 1 to 0.6.* This is compatible with the PCAC and S U(3) estimate, 0.33 _+ 0.33 [29] and a recent estimate 0.2 < B < 0.6 [30].

In summary, we have determined the parameter e, which measures the soft-gluon corrections to nonleptonic weak decays, to be about -0 .47 , - 1.0, and - 1.1 respectively for D ~ PP, VP and K ~ 2n decays. This might be tested by the lattice Monte Carlo calculations [93. Since the subleading contributions in the 1/N expansion are characterized by the parameter (1/N + e/2), it is clear that the leading terms in 1/N

* To the leading contribution in 1/N,B = 3/4 1-12]


work reasonably well for two-body decays of charmed mesons and kaons. The inclusion of nonleading contributions are, however, indispensible for a satisfactory explanation of nonleptonic decays of mesons. Soft- gluon effects are presumably weaker for B and T mesons owing to the large relative momentum between the final state of heavy-meson decays. In this work, we have also pointed out the correct treatment for the evaluation of matrix elements of penguin operators in vacuum-insertion for kaon decays.

Acknowledgement. I wish to thank Dr. Ling-Lie Chau for the collaboration in the study of charm decays on which this work is based, and Professor A.W. Hendry for reading the manuscript. This work was supported in part by the Department of Energy.

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Vacuum insertion and nonperturbative effects for exclusive nonleptonic decays of charmed mesons and kaons