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P H Y S I C A L R E V I E W D V O L U M E 2 2 , N U M B E R 1 1 1 D E C E M B E R 1 9 8 0

N ~ n l e p t ~ n i c decays of charm mesons

A. I. Sanda Rockefeller Univerrity, New York, New York 10021

(Received 25 February 1980; revised manuscript received 5 May 1980)

We investigate the possibility that both nonleptonic strange-particle and charm-particle decays can be understood in terms of one effective nonleptonic Elamiltonian N,,=H,+H,,, . H , is a usuzl four-fermion current-current interaction term and the H,,, is the color-radius interaction term. The color-radius interaction is known to explain the A1 = 1/2 rule in strange-particle decays. The effects of H,,, as well as those of final-state interaction on two-body charm-meson decays are examined. The theory is in qualitative agreement with all measured branching ratios. Predictions for charm-meson decays into two pseudoscalar mesons are presented.

I. INTRODUCTION

It h a s been over 20 y e a r s s ince the hI = + rule t o descr ibe the nonleptonic decays of hyperons and K mesons was introduced.' Hundreds of papers have been written attempting to descr ibe the or i - gin of the amazingly successful rule . Recently, a very s imple and natural explanation of the A 1 = + ru le was put for th by Shifman, Vainshtein, and Zakharov.' They have shown that the short-distance behavior of the strangeness-changing effective four-fermion interaction computed f r o m the Lagrangian

~ = C Q F D +&QCD (1)

possesses a t e r m which is purely A1 =$ [ ~ n what follows, we shal l r e f e r t o thls interaction a s color-radius (COR) interactlonj and that the contribution f r o m th i s t e r m t o the decay a m - plitude i s l a r g e f o r s t range-part ic le decays. Here CQFD is the Lagrangian f o r the Weinberg-Salam model [quantum flavor dynamics (QFD)] with quark multiplets

U C

- s ~ 0 . ~ 0 , + d sin8,

and CQ,, is the Lagrangian f o r quantum chromo- dynamics (QCD). Indeed the effective Hamiltonian which follows f r o m the Lagrangian given in (1) is capable of explaining a l l two-body decays of K mesons and hyperons.

It is interesting t o attempt to extend the theore- t ical considerations t o descr ibe decays of D mesons. Also, it is important to understand the role of these new interactions before a detailed tes t of gQ,, i s made f o r AC# O decays. Unfortunately, urtlike in the c a s e of K decays, the s t rength of COR interaction f o r D decays cannot be computed reliably. While the COR interaction f o r K-meson decay i s short range, ( I I M , ) , compared t o the Compton wavelength of the K meson, the COR in-

teract ion f o r D-meson decay i s long range, ( L I M , ), compared t o the Compton wavelength of the D meson. This invalidates the use of the operator- product expansion and renormalization-group equation in obtaining the strength of COR interaction f o r D decays.

In th i s paper , we consider nonleptonic charm- meson decays. In o r d e r to remedy the sbove-mentioned difficulty, we extract the general fea tures of COR interact ions by examining the s t ruc ture of Feynman d iagrams which contributes to the interaction. We then paramet r ize the strength of the interaction and determine the parameter f r o m the experimental rat io r ( D a - K ' K - ) / r ( D O - st s'). Predict ions f o r various two-body decays of pseudoscalar c h a r m mesons a r e given. The importance of the COK interaction a s well a s the validity of the parametr izat ion used r e s t on the varifica- tion of these predictions.

A number of authors have considered two-body decays of c h a r m part ic les f rom various points of view, f o r example, f ree-quark model with QCD corrections, ' SU(3) together with heavy-quark m i ~ i n g . ~ - ~ The following summar izes the difference between this investigation and e a r l i e r ones. We have included contributions f rom the color- radius i n t e r a ~ t i o n ~ ' ~ (see Sec. 111). We have used a n approximation ( see Sec. IV) to compute al l r e l - evant reduced matr ix elements . We have est ima- ted the effect of heavy -quark mixing and concluded that it i s negligible within our framework ( see Appendix B). And finally me have included effects of final-state interactions.

Our resu l t s a r e tabulated in Tables 1-111. We a r e hopeful that the effective Hamiltonian based on the Lagrangian given in (1) offers a unified understanding of nonleptonic K decays, hyperon decays, and charm-meson decays. While it i s not yet con- clusive, our picture i s i n sat isfactory agreement with experiments. Fur ther experimental s tudies of D decays a r e crucial in understanding the the- o ry of nonleptonic decays. Decays of par t icular

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

interest a r e Experimentally this decay is not forbiddenlo:

Careful measurements of branching ratios for these decays will be crucial in understanding non- ieptonlc decays.

In Sec. I1 we discuss some puzzles which we encounter if decays a r e analyzed with a free-quark model. In Sec. I11 we discuss the nonleptonic effective Hamiltonian. In Sec. IV, we discuss a procedure used to evaluate matrix elements of current-current operators. In Sec. V we discuss an evaluation of matrix elements for H,,. In Sec. VI we discuss numerical results . In Sec. VII we discuss a relationship between the COR interactions for K and D decays. In Sec. VIII we state our conclusion. In Appendix A we discuss effects of final-state interaction on the decay amplitudes. In Appendix B we discuss effects of heavy-quark mixing. In Appendix C we present a theoretical estimate of the COR interaction strength. In Ap- pendix D we present details of the calculation which leads to H,,, presented in Sec. 111.

11. PUZZLES

Consider the standard Weinberg-Salam model with doublets of quarks given in (2). If the strong interaction i s ignored, the following effective four-fermion interactions can be derived from the Hamiltonian:

H & =* sine, c o s ~ ( ~ u ) ( ~ d ) + ~ . c . , 4-2

Unless otherwise stated (qlq,) (i&q,)=&y,y-q2ij3ypy. y* = i(l + y,). If physics can be deduced naively from this Hamiltonian, we immediately encounter many puzzles.

Puzzle 1. H~~ given in (4) does not have a mechanism for suppression of K+- r+irO, a purely AI = % transition. Experimentally, this decay is highly forbiddenx0:

(D' -ROKK') -4.9 (experiment), (7) r (D+ -Ron' ) sin2Bc

B(DO-K"n+) *, ---- B(D' - ~ o n ' ) 1-33 (experiment)

[sin28, in (7) i s introduced to adjust fo r the fact that D' -ROK+ i s expected to be suppressed by sin20, compared to Cabibbo--allowed decays]. Why do D decays violate the AI = rule?

Puzzle 3. It i s easily seen that the operators which multiply cos28,, sine, cosQ,, and sin2@, in Hnc of (4) form a U-spin triplet. Since ( T ' , K ' ) form a U-spin doublet the relation r (DO -v+vm) = r (DO - K+K'-) immediately follows. Expesiment- ally this relation i s violatedlO:

Why do D decays violate the U-sfiin symmetry? PuZZbe 4. The Hamiltonian given in (4) and a

naive quark-model calculation of the matrix element leads us to conclude that"

the amplitude for Do-KOrO i s suppressed by a factor of from the Clebsch-Gordan coefficient and by a factor of 3 from the color factor. The experimental resultlo for this rat io is =I. Why do D decays contradict the naive quark-model prediction?

111. AN EFITECTIVE NQNLEPTONIC HAMILTQNIAN

In this section, we discuss t h e effective Ham- iltonian density for charm-changing nonleptonic decay with specific application to two-body decays of charm meson in mind. Consider the once-Ca- bibbo-suppressed AC + 0 Ilamlltonian in the ab- sence of the strong interaction:

( a ) ( b ) (6)

FIG. 1. The leading-order QCD corrections to coefficient functions of the short-distance expansion. (a)

Why do K decays obey the A1 = i rule? Diagrams contributing to the coefficient functions in Puzzle 2. If the AI =+ rule i s applicable to the (12a) and (12b). (b) The COR interaction gjving r ise to

charm decays, the decay D+ -ROT+ i s forbidden. the coefficient function ( 1 2 ~ ) .

2816 A . I . S A N D A 22 -

It is well known that the strong-interaction cor rec t ions , fo r example those shown in Fig. l ( a ) , lead to correct ions of ~ ( a , l n ~ , ~ / p ~ ) and that when these cor rec t ions a r e summed, the Hamiltonian density becomesll

where

b = 11 - 2 / 3 N f , Nf is the number of quarks with iWq < M,. Similarly,

It should be kept in mind that only the cor rec t ions which a r e dominant a t shor t distance, leading t e r m s in powers of l n ~ , ' / p ' , have been kept. The ignored t e r m s may be of the s a m e order of magnitude a s the nonperturbative effects which cannot be computed reliably. The uncertainties introduced by ignoring the long-distance interaction a r e expected to be l e s s than 25%.

Other opera tors which might contribute to the effective Hamiltonian can be determined by study- ing the perturbation expansion of the charm-quark decay amplitude in powers of a,. In analogy with the K-decay amplitude discussed in Ref. 2, the

I

COR interaction, shown in Fig. 1 (b), contributes to c h a r m decays. Unlike in the s-quark decay amplitude, the amplitude f o r c-quark decay cannot be computed reliably. Consider the interaction in coordinate space a s shown in Fig. 2. The dis tances between points x,. . . ,x4 a r e a s follows:

Since ix , - x , / is l a r g e compared to the Compton wavelength of the decaying part ic le , the COR interact ion h a s a long-distance range.

In addition to the diagram shown in Fig. l ( b ) , a l l d iagrams of the fo rm of Fig. 3 contribute to the interaction. Such a diagram can be written in gen- e r a l a s

sinec c o s 8, - M - ~ ( ~ ' ) y + y ~ O " l a " ' j"nyay-

Mw2

where ~ l ' l * . . ~ , Pn and 0 ' ~ 1 ~ - " * !+ a r e operators

consisting of propagators and vertex factors , and ?'a1. . . , Q,, and ~ ' 0 , ~ . . . a n a r e operators in color space. In Appendix C, we show that M c a n be written a s

where F:(q2), F: (q2) a r e f o r m factors . In the s a m e appendix, we a l so show that the amplitude M c a n be generated by an effective Hamiltonian:

N O N L E P T O N I C D E C A Y S O F C H A R M M E S O N S

FIG. 2. The color-radius interaction in the coordinate FIG. 3. The general structure of color-radius space. interactions.

where Vf and V; a r e Fourier t ransforms of F,8 (q2) and F: (q2), respectively! representing potentials for nonlocal interactions. In contrast to the complicated nonlocal effective Hamiltonian, a local one i s extremely simple. For example, ~ $ 0 " ~ i s obtained by the following substitutions:

V;(X -Y) -0 for i+1,

V:(x-y)-0 for al l i ,

The cos9, sine, transition i s thus given by an effective Hamiltonian

where H,"; and H$ER a r e given in Eqs. (12) and

(17), respectively. At f i r s t sight H$&(x) s eems hopelessly com-

plicated. A close examination of (171, however, shows that for pseudoscalar-charm-meson decays to two pseudoscalar mesons, the effective Ham- iltonian takes on a very simple form. We shall s ee this in Sec. V and in Appendix C.

1V. C O S ~ O , AND S I N ~ O , AMPLITUDES

Cabibbo-allowed amplitudes can be determined by taking matrix elements of H:: given in (13). Here we will consider the decay D+ -ROT+ a s an example, and describe a simple approximation fo r evaluating the matrix element. The final-state interaction between two mesons a t center-of-mass energy M , modifies the matrix element considerably. This will be considered below. For the time being, we ignore the final-state interaction and obtain

where we have approximated the matrix element by saturating it by the lowest intermediate state. This approximation will be refer red to a s the vacuum saturation approximation. Using

(n'lG(0)~ "Y, 4 0 ) / 0) =ip;fr,

(n+ ~ G ( O ) Y ~ ~ ( O ) ~ D + ) = i l ( ~ , +P,)"f+

+(PD -~,)"f-l, and SU(3) symmetry we obtain

where we have ignored t e rms of O ( M K 2 / ~ D 2 ) . It is shown in Appendix A that the final-state interaction modifies the amplitudes for

(PP)27-plet, 'e' (PP)octet, and '0- (PP)8ing1et amplitudes by multiplicative factors

respectively. The le t te rs P and PC stand for a pseudoscalar meson and a charm meson, r e - spectively. Also, a = - 1 GeV, r, = - 1.76 GeV-', y =0.575 GeV, M , =1.42 GeV, and

We then obtain

(24)

This amplitude is related to the decay r a t e by

This can be compared to the semileptonic decayL2

281% A . I . S A N D A 22 -

r(D'-Bon) ] '" ~ 0 . 5 6 (experiment 0.36 10.29) L ( D + - Koe+v)

This i s in a sat.isfactory agreement with experiment, In p a ~ s i n p we nv+c hbai the? frrc quark - model prediction (c+62,, = 1 ) for the rat io i s 1.8. The short -distance correction and the final-state - interaction correction a r e c, =0.50 and / Q,, / ~ 0 . 5 4 , respectively. Other amplitudes can be computed in the same manner. It is convenient to present our results in t e rms of reduced rnatrix elements defined 1n13 Ref. 6 ,

s = ( ~ , / f ( ~ ) = - ~ c - n , ,

E=(P,/~-?% 18) =$c+n, ,

F = ( P ~ / Z * 18) =+(c+ +c-)n8, (27)

G = ( P , I : * I ~ ) = ~ ( c + + c - ) s ~ , , T = { P ~ I ~ I ~ ~ ) = & C + ~ ~ ~ - ~

In Table I we give the amplitudes in t e rms of the reduced matrur elements,

V. codc sinec AMPLITUDES

The once -Cabibbo-suppressed amplitude can be obtained by taking a matrix element of

AC = H A C +NAC *cs ocs COR

given in Eqs. (12), (17'). and (18). Matrix elements of HRC", can be evaluated in the manner described in Sec. IV. Matrix elements of H;:, require a further discussion. We shall evaluate ( K ' K - ~ H ~ ~ (Do) a s an example. F i r s t note that ~z~ with the ordering of the quark field a s shown in (17) has a vanishing matrix element in the vacilum saturation approximation. This i s because each t e rm in HZ, i s a prodact of two operators which a r e color octets. The contribution to the matrix element comes from the Fierz-transformed ordering of H%,. Denote / x, ,x , ,X;P) a s a state

TABLE I. Amplitudes for charm-meson decays i n t o kvo pseudoscalar particles. These amplitudes which a r e proportional to and sin2bc are not affected by the COR ihteraction. See Ref. 13 for conventions. --

&I

D 0 - - ~ - r ' [ Z T + (E- $1 cos2Qc

D O - - O 0 K n 1 - [3T- ( E - 31 cas2.1c

gO-KOriO - 1 fi [3T- (E - 91 C O S ~ B ~

D f - KOn+ -5T cos2ec

F t & - O t K K [ZT + ( E + S ) I cos2eC

F +- ria+ (+)i '2[3~- (E- 91 ~ 0 s ~ ~ ~

D o - ~ + n - [2T+ (I?-S)] sin28,

D O - K O T " 1 - [3T- (E - S)] sin20c JZ

D ~ - K " V O I - a [3T- ( E - ~ ) l s i n ~ Q ~

D +--K~IT+ - [ 2 ~ + (E +s)] sin2Qc

D + - K + V O 1 -- iiT [W- ( E + s ) ] s i 1 1 ~ 8 ~

of pseudoscalar meson a t position X and valence quarks q, and & at x, and x,, ~.espectively, and / ~ ; p ) a s an on-shell P state with momentum P . We have shown in Appendix C that

x { K - ; y , x ,x- IE(x)q(y) / 0 ) 1-(8- (28)

where Gy(x - y ) and G;(X - y) a r e functions which can be written in t e rms of VS(x -y), and (8 - 1) denotes

N O N L E P T O N I C D E C A Y S O F C H A R M M E S O N S 2819

replacing 8 by 1 representing the singlet interaction in (17). Now compare this result with the short-distance limit which can be obtained by setting

G ~ ( x - y ) = $ ~ : ( x - y ) = 6 ~ ( x - y ) . , G ~ ( x - ~ ) = & G ~ ( x - ~ ) = - ~ ~ ~ ( x - ~ ) ,

Since H;g,(x) and [HAC(x)],, have similar Lorentz structures and exactly the same flavor-SU(3) structure, it i s reasonable to assume

where f i s independent of SU(3) indices a and b . In principle, the dependence on external masses may a r i s e from the detail s tructure of the wave functions. We expect such an effect to be much smaller than the e r r o r introduced in using the vacuum saturation approximation.

The matrix element of H,, given in (28) can be reproduced by a simple effective Hamiltonian,

where K~ and K~ a r e parameters to be obtained from experiments. After performing the Fierz transformation we obtain

where K = K ~ +:K'. All other t e rms do not contribute to the matrix element since they involve either color-octet currents or flavor-singlet cur - rents. It i s easily seen that the particular linear combination of K, and K, denoted by K appears for a l l matrix elements of H,,, between a charm meson and two-octet pseudoscalar -meson states. Thus a l l D and F decays into two psuedoscalar mesons can be described in t e rms of one parameter K. The scalar and pseudoscalar matrix element in (30) can be obtained a s follows:

given in (30) contains a t e rm of the form Gy-ciZy+u which gives r i s e to a matrix element (an /iZu 1 0)(0 I ay5c / 0). For the K-decay analysis,' this type of annihilation diagram played a crucial role. We have no reliable way to estimate this contribution for D decay. We argue however that the an form factor falls rapidly. For example, a t q2 =mD2, Fy(mD2) -mp2/(mD2 -mp2) - $. Also, the scalar form factor may behave a s m?/(mD2 -m:) -- :. Thus the contribution of this matrix element will be roughly 10°/0 of the COR interaction contribution computed above and we shall ignore them.

We then obtain

where we have used x ( - $ ) ( ~ T + E - S tc,) , (31) i

u(x) Y,S (x) = - - a ,E(X)Y "7,s (x) , ma + m u where

2820 A . I . S A N D A

Other amplitudes can be computed in a similar manner.

Note that consistency in writing (28) and (29) requires that when the above computation i s r e - peated, for example, for D O - s+a- decay, the resulting amplitude i s related to that of (31) by SU(3) Clebsch-Gordan coefficients. Using

this indeed can be verified. Also note that H,,, gives r i s e to a new reduced matrix element which has not been included in previous sU(3) a n a l ~ s i s . ~ If our approximation in evaluating the matrix elements turns out t o be too naive to explain all the experimental observations, a general SU(3) analy- s i s including the H,,, operator should be necessary.

As i s seen from Table 11, we have

Relative signs of these amplitudes can easily be understood by decomposing n+n- and K'K- states

TABLE 11. Amplitudes for charm-meson decays into two pseudoscalar particles. These amplitudes which a r e proportional to cosec and sinBc may be affected by the COR interaction. See Ref. 13 for conventions.

in t e rms of U-spin eigenstates:

Noting that ~t~ is a component of the U-spin tr ip- let, H $:, i s a U-spin singlet, and DO i s a U-spin singlet, we have

We determine 5 by demanding that

This gives

< = 0.28 k0.12 o r 3.6 i 1 . 6

and

For the smaller solution for 5, we have

leading to a value

sin8, = 0.26 i0.04

in good agreement with 0.22, the value determined f rom K decays. For this reason we choose the smal ler solution for t .

In Appendix C we have given a naive theoretical estimate for the strength of COR interaction:

where M, 's a r e constituent quark masses, E, i s a mean value of the running coupling constant for 0 i Q' <Mc '. The experimental value for K leads to dis=4.6. Considering the fact that the long-distance interaction contributes to the COR interaction, this value of 15, i s not an unreasonable one. Note that sN coupling constant g 2 / 4 s - 14. While validity of the estimate (39) can be questioned for such a large value of 5, , we find it very satisfy- ing that neither an ad hoc enhancement nor a suppression of COR interaction i s needed to explain the experimental observation (35).

VI. NUMERICAL RESULTS

Numerical results for two-body pseudoscalar meson decays of charm mesons a r e given in Table 111. We have also given branching ratios

N O N L E P T O N I C D E C A Y S O F C H A R M M E S O N S 2821

TABLE III. Our prediction for relative decay rates. The uncertainties in the predictions for D - Kev and F - qev a r e roughly a factor of 4. The relative decay ra tes for two pseudoscalar modes a r e considerably more reliable. The values given under "without COR" a r e mod- ified values obtained by setting K = 0.

Theory

Do -K-T+ 1 K-ev 0.52 - K0r0 0.99 KOqo 0.33 K+ n- 0.002 6 KO*' 0.0026 K0q 0.0009 KCK- 0.084 n+s- 0.025 nono 0.061 qOIO 0.11 n0q0 0.02 K'ZO o

Experiment

normalized

0.71.0.5

0.11 1.0.05 0.032 1. 0.02

normalized 7*5

0.24k0.17

normalized

Without COR

without the COR interaction contribution. In o r d e r to obtain these numerical resu l t s , we used

where the exponents correspond to N = 6,

The K T scat ter ing length and the effective range f o r the I = channel is given by l4 a = - 1 k0.05 GeV, yo = - 1.76 h0.3 GeV-l. The ~ ( 1 4 0 0 ) reso- nance p a r a m e t e r s a r e y = 0.575 GeV, M , = 1.42 GeV. These yield

A s explained i n the preceding section we u s e

The predictions given c a n be compared with four existing experimental values and a r e i n sat isfactory agreement . In addition to the predictions given in Table 111, the theory gives

The experimental value f o r the branching ra t io isLn

r(DO-K-n+)/ r ( D t - R n ~ + ) = 1.33 *0.53 . (43) I'(DO - a l l ) P(D' -al l )

This l eads to

I'(DO - al l ) r ( D + -all)

=4.5 k1.8 (exp> 5.8 k1.5) , (44)

a value consistent with the lower limit. The experimental lower l imit was deduced using the measurementlo

r(DO-e v + anything) I'(DO - a l l )

<4% (95yoc.l.) . (45)

The theory indicates that th i s branching ra t io should not be very smal l compared t o the upper limit. Including the effects of the f inal-s tate interact ion i s crucial in understanding charm-meson decays. F o r example, r ( ~ ~ - ~ ~ n ~ ) / r ( ~ ~ - K - r + ) = 0.1 if the final-state interaction is ignored. Such a l a r g e difference is due to the fact that the re la - tive sign of octet and 27-plet amplitudes if flipped (i.e., 08 /027=-2 .0) .

VII. THE COR INTERACTION IN STRANGE- PARTICLE DECAYS

We have seen that the COR interact ion contr i - butes to once-Cabibbo-suppressed amplitudes. Our analysis shows that while it makes a considerable difference i n the overall picture of charm-meson decays, i t is not the dominant contribution to these decays. In contrast , the COR interaction is the dominant contributor to K mesons and hyperon decays, thus explaining the I = 4 rule.2 How can the COR interaction give la rge contributions to K mesons and hyperon decays and much s m a l l e r contributions to charm-meson decays? In this s e c - tion we sha l l examine this question.'

The COR interaction f o r the AC = 0, A S # 0 Ham- iltonian i s given by

In Appendix B, we have given naive es t imates f o r the s t rength of COR interactions:

where M , a r e constituent quark m a s s e s . Using this Hamiltonian, the rat io of K - r n amplitudes f o r AI = 5 and 4 is given by

Using the value f o r K given in (36) and the theoret ical est imate given in (47) we obtain

K : = -0.75 , (50)

r ( K S - T'T-) l I 2

r ( K + - n+ r O ) = 3.7 + 16.8 (experiment 21) :

where the la t t e r number corresponds t o the COR contribution. The relat ive importance of the COR interaction contribution t o K decays compared to that to D decays comes f r o m the enhancement fac- t o r of quark-mass ra t ios shown in (49). We s e e f r o m (31) and (49) that this enhancement factor i s

COR interaction needed to explain the rat io of decay r a t e s f o r Do-K'K- and Do- a t r - is in reasonable agreemwnt with naive theoretical expecta- tions.

The color-radius interaction a r i s e s naturally f r o m QCD perturbation theory. Inclusion of this t e r m solves many puzzles mentioned in Sec. 11: (a ) It explains why N decays obey the A I = ~ rule , (b) It explains why D decay violates the A I = i rule , and (c) it explains why D decays violate the U-spin symmetry. Fur ther experimental studies of c h a r m decays, however, a r e necessary in o rder to establ ish the COR interaction a s the integral par t of the nonleptonic Hamiltonian. Examining Table 111, i t is seen that the theory i s in qualitative agreement with experiments . Aside f rom the rat io J?(DO-K+K-)/ r ( D O - n'r-) the re is no c lea r - cut evidence f o r o r against the presence of COR interaction effects. Fur ther experimental s tudies of D+-KiKO, n'nO, r'qO a r e crucial in understanding the role of COR interaction in c h a r m decays.

The final-state interactions play a very important role in understanding exclusive decays of c h a r m part ic les .

Note added. After completion of this paper , the author received papers by J.F. Donoghue and B. R. Holstein, [phys. Rev. D21 , 1334 (1980)] and by H. Lipkin [ ~ h y s . Rev. ~ x t . 44. 710 (1980)l. These authors a l s o point o u t t h e importance of final-state interact ions, but assume that the effects of COR interactions a r e negligible.

The qualitative feature that th i s rat io is l a r g e follows merely f r o m the m a s s ra t ios and is relat ive- ly insensitive t o uncertainties in theoret ical es t i - mates of K' and K.

To summar ize , the effective Hamiltonian H:& and H @ ~ have a s i m i l a r fo rm. They a r e neither enhanced nor suppressed. The mat r ix elements of (TT-A) ( V + A ) color-octet cur ren ts depend on cur ren t quark m a s s e s . The m a s s dependence is such that the matr ix elements f o r s t range-part ic le decays a r e much l a r g e r than those f o r charined- par t i c le decays.

ACKNOWLEDGMENTS

VIII. CONCLUSION

Starting f r o m a nonleptonic Hamiltonian which includes the color-radius interaction in addition t o the usual current-current interaction, we have studied D- and F-meson decays, The mat r ix elements were computed by saturat ing the intermed- ia te s t a t e s by only a lowest available s tate . Our resu l t s were compared with four presently available branching-ratio measurements . They a r e in reasonable agreement . Also, the s t rength of the

This investigation is a n outgrowth of a n investigation by M. Fukugita, T . Hagiwara, and the auth- o r (Ref. 8). The author acknowledges stimulating and enjoyable conversations with M. Fukugita, T. Hagiwara, M. A. B. B6g, A. P a i s , R. Phillips, H.-S. Tsao, L . L . Wang, -L. Wolfenstein, and V. Zakharow. P a r t of this investigation was p e r - formed at Rutherford Laboratory, Aspen Center f o r Physics , and Los Alamos Scientific Labora- tor ies . The author thanks R. Phillips and G. West f o r the i r hospitalities a t Rutherford and Los Al- amos , respectively. This work was supported i n par t by the U. S. Department of ~ n e r ~ ~ - u n d e r Contract No. EY-76-C-02-2232B.*000.

APPENDIX A: EFFECTS OF FINAL-STATE INTERACTION

Within a framework of the saturat ion approximation, we can est imate the final-state interaction using the Omnes equation.14 The S-wave p a r - tial-wave amplitude for the KT sys tem c a n be approximated by

22 - N O N L E P T B N I C D E C A Y S O F C H A R M M E S O N S 2823

a, / , (s)

where the isospin is specified by the subscript and l5 k Z = [s - ( M ~ + M , ) ~ ] [ s - (MK - M , ) ~ ] / ~ s , a = -1 GeV, y o = -1.76 GeV"', y = 0.575 GeV, and M,= 1.42 GeV. We take a3/,(s) and a,,,(s) to give the S-wave partial-wave amplitudes for the 27- plet and the octet s ta tes , respectively of pseudo- sca lar meson-meson scattering. This is reasonable since a,, ,(s) reproduces the I = 2 nr partial- wave amplitude to within 25% and al,,(s) i s dom- inated by ~ ( 1 4 2 0 ) which is a member of O'octet along with ~ ( 1 4 0 0 ) and S'(960). With the partial- wave amplitudes given in (Al) , the Omnes functions a r e

where these functions a r e normalized to 1 at the threshold s = (MK+ M,),. The final-state interaction of an octet (a 27-plet) pseudoscalar meson-meson state i s taken into account by multiplying the ap- propriate amplitude hy 52,(52,,).

APPENDIX B: EFFECTS OF t- AND h-QUARK MIXINGI6

We have analyzed the effects of IJ-spin violation due to the COR interaction. Tt has been pointed out that the U-spin violation can also be introduced by a heavy-quark m i ~ i n g . ~ Fo r example, consider the six-quark generalization of Mobayashi and Maskawa.17 The weak current i s

(A21 If heavyquark mixing is ignored, V,, = V,, = cosG,, and V12= -V,, = sine,:

where C =~(V12V,,+ VllV2,) and A = ~ ( V , ~ V , ~ - v,,V,,). Note that the t e rm proportional to C (A) i s antisymmetric (symmetric) under the interchange d- s . These modifications to H,",", lead to an amplitude

where

It i s reasonable to assume16

1 A/Z I s 0.02 034)

and thus expect

A < 0.022.

I

An estimate of X generated by the heavy-quark- mixing effect is an order-of-magnitude smal ler than the value needed to explain the experimental measurement.'' Since our approximation used in evaluating the reduced matrix element is not expected to be much better than 1@0, we shall ignore al l heavy-quark-mixing effects.

APPENDIX C: A N A N E ESTIMATE

In this appendix, we will present a naive est i- mate of K, using a lowest-order perturbation theory.' While we realize fully that higher-order diagrams and nonperturbative effects may contribute to K , we think that the comparison of the lowest-order perturbative contribution with experiment i s not entirely meaningless. A gross disagreement between the lowest-order perturbative prediction and experiment will imply that we a r e on a wrong track.

Consider the lowest-order COR contribution in coordinate space. The short-distance operator- product expansion implies that X,=X,=X, in the limit Mw - m. It does not, however, res t r ic t X,. This X, dependence of the coefficient function be-

comes a Q2 dependence in the momentum space. In principle, the D-decay rate i s computed by integrating over Q 2 after folding in the wave func-

tions. Thus the Q2 dependence of the coefficient function must be studied. The lowest-order contribution gives

(C1)

I

where estimate that

~ ( y ) = + ( y + Z ) ( y - 1 ) 2 1 n ( ~ 2 - 1 ) - ~ ( ~ 2 -3)ln(y+ 1 )+y2 . The Q2 dependence of [ I ( ~ , ~ / Q ~ ) - I(md2 'Q2)] i s shown in Fig. 4 for 1M,= 1500 MeV, M,= 150 MeV, and m, = 10 MeV. The average value i s

We also note that K(Q') i s proportional to u(Q2). The running coupling constant has a large Q2 variation in the region Q2<MC2; it increases rather rapidly a s Q2 decreases. This Q2 variation of as(Q2) tends to favor smaller Q2, thus increasing our (K). With these discussions, we roughly est i- mate K(Q') as

where 6, is some effective coupling constant. In Ref. 19, it has been argued that the COR con-

tribution is negligible. This i s based on a rough

and

We see from Fig. 3 that the estimate (C6) i s off by a factor of 6 and furthermore, (I) i s considerably larger than 0.01. While the lowest-order computation of K i s not convincing. our estimate suggests that the COR interaction may play an important role in charm-meson decays. In any case , it i s dangerous to conclude that the COR interaction is negligible based on the lowest-order computation.

APPENDIX D: EFFECTS OF NONLOCAL INTERACTIONS

Starting from a general expression for the diagram shown in Fig. 3

0-1 I I I -I--- .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

CJ2/rn:

FIG. 4. The Q~ dependence of the diagram shown in Fig. lb).

22 - N O N L E P T O N I C D E C A Y S O F C H A R M M E S O N S 2825

we wish to derive an effective Hamiltonian given Using these relations, we can simplify M to in Eq. (17). Owing to the presence of y, and Y - factors, we can restrict ourselves to the terms in 1 Oull.e.*un which consist of an odd number of y ma- Me-

Mw , [ ~ ( p ~ ) ~ ~ ~ - ~ ~ c ( p ) q ( k ' ) r , A l : q(k)

trices. First note two identities,

(yOO!J1'... * o n Y 'Y- )~ ,= -Tr(O~l '""~n + ~ ( p f ) r ~ - c ( p ) ~ ( k ' ) ~ A q ( k ) ~ , 034) YAy+)(Y?-)i,

where

and (D2) r,= - Tr(O"l*...*" " ~ X Y - ) ~ ' ~ ~ , . ~ . , P, 9

( ~ a ~ ~ ~ . , a , , ) -pi,...,ana bl - (ha),, +Gall*..*%G kl 7 =~a~.....a~.a~a~...~.a~, ha

where Fal'".*a and Gal*..'<'n are constructed out of f*,, and d, , , defined by A = G ~ I . . . . . ~ ~ Talt...tan .

(D3) Writing a general form for the quark current r,q, we have Eq. (16):

+ ~(P')Y%-c(P) x ~(k')k:(q2)rx+~~(q2)uX~qT+~~(q2)q,k(k) . (D5 q=u,d .s

Above, we have ignored SU(3) breaking in Fit8(q2) which we expect to be of order 10%. Using a relation

we see that the amplitude can be reproduced by an effective Hamilitonian:

Before we discuss the matrix element of H,,,, it is convenitent to prove some properties of matrix elements of nonlocal operators which will be useful below. Let Ix,, xz,X;P) be a state of pseudoscalar meson P at position X and valence quarks q, and q2 at x, and x,, respectively. (In defining such an object, we have explicitly as- sumed a quark model and the operations that follow should be considered within the context of such a model.) We will prove that

For O = 1, y,, a,,, cr,, y,. This is obvious if x , =x,, but it requires some discussion in the case x,f x,. Consider the problem in the rest frame of P. The q,, q, pair is in a state which is symmetric under the interchange 3,- k, = -El.

I

We can then write

( e x , x1, x, lq l (~, )~q,(x , ) 10) =${p;x,x, ,x2 J ~ ~ , ( x , ) ~ q , ( x , ) + ~ , ( x ~ ) ~ q ~ ( ~ ~ ) l ) ~ ) .

It is clear that the right-hand side has definite transformation properties. For example, if 6 = 1 and y,, the matrix element i s a pseudoscalar and an axial vector, respectively. The available three-vectors a re 2 and gF/aS where F is a scalar function which depends on x:, xz2, XZ, x, . x,, X . x,, Xex,. It is however, clear that B F / ~ $ is again proportional to $. With only one three-vector, it i s impossible to construct the rest-frame equivalent of the pseudoscalar, axial vector, or antisymmetric second-rank tensor.

Let I P ; ~ ) be an on-shell state with momentum p. Consider an evaluation of

A . 1. S A N D A

The f i rs t te rm in H,,,(x), Eq. (D7), gives

where we have performed a Fierz transformation on the f i rs t term, and used the vacuum saturation approximation and the property of the matrix element discussed above. The matrix element of the form

(P,;Y, x,,x, k b ) y X ~ , c ( x ) px,x,,xc;pc> Cpi;y,x,X, I U(X)Y,Y,Y~) 19)

will not contribute to the integral. This can be shown in the res t frame of P, using the parity operation and the fact that P, P' a r e in the S wave.

Under the Fierz transformation of the second term, which has a lorm ;,a,, y , ~ " ~ , ~ ~ u ~ ~ ~ ~ ~ ~ , color-sing- - - let terms generated a r e <,o,A,y,q,q,owwy,q, and q,y,q,y,Y,q,, Using the above discussion on the matrix element of a,,, we conclude that the second term will give a contribution of the form

The third term which contains a totally antisymmetric tensor E""~ will not contribute to a two-body decay matrix element.

The fourth term is of the structure already encountered above and we have

where Gy(x - y) and Gi(x -y) a r e functions which can be written in terms of Vj(x- y).

'M. Gell-Mann and A. Pais , Phys. Rev. 7 , 1385 (1975). 'A. 1. Vainshtein, V. I. Zakharov, and M. A. Shifman,

Zh. Eksp. Teor. E'iz. 2, 1275 (1977) [Sov. Phys.- JETP 45, 670 (1977)l; M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys, E, 316 (1977).

'J. Ell is , M. K. Gaillard, and D. V. Nanopoulos, Nucl. Phys. z 0 , 313 (1975); N. Cabibbo and L. Maiani, Phys. Lett. E B , 418 (1978).

4k. L. Kingsley, S. B. Treiman, F. Wilczek, and A. Zee, Phys. Rev. D s, 1919 (1975); ivI. B. Einhorn and C. Quigg, ibid. 12, 2015 (1975).

'L. L. Wang and F. Wilczek, Phys. Rev. Lett. 43, 816 (1979); M. Suzuki, ibid. 2, 818 (1979); L. Wolfenstein, Carnegie Mellon report (unpublished).

6 ~ . Quigg, 2. Phys. C 4 , 55 (1980). 'other investigations include V. Barger and S. Pakvasa,

Phys. Rev. Lett. 2, 812 (1979); S. B. Treiman and F. Wilczek, ibid. 43, 1059 (1979); G. L. Kane,SLAC Report No. SLAC-PUB-2326 (unpublished); K. Ishikawa,

UCLA Report No. U C L A / ~ S / T E P / ~ ~ (unpublished). his paper is an extension of an investigation described

in M. Fukugita, T. Hagiwara, and A. I. Sanda, Ruther- ford Report No. RL-79-052/~.048 (unpublished).

' ~ n addition to the authors of Ref. 8, the possible importance of color-radius interaction in charm-meson decays has also been pointed out independently by M. Suzuki (Ref. 5) and K. Ishikawa (Ref. 7).

'O~he experimental numbers for charm decays used m this paper a r e f rom talks presented by J. Kiskby and V. Luth, in Proceedings of the 1979 Sympos ium on Lepton and Photon Interactions a t High Energ ies , F e r m i l a b , edited by T. B. W. Kirk and H. D. I. Abar- banel (Fermilab, Ratavia, Illinois, 1979) and those on K-meson decays a r e f rom Particle Data Group, Phys. Lett. E, 1 (1978).

"M. K. Gaillard and B. W. Lee, Phys. Rev. Lett. 3, 108 (1974); G. Altarelli and L. Maiani, Phys. Lett. 52B, 351 (1974). -

22 - N O N L E P T O N I C D E C A Y S O F C H A R M M E S O N S 2827

12see, for example, a lecture by J. D. Jackson, in Weak Interactions a t High Energy and the Production of New P a r t i c l e s , proceedings of the SLAC Summer Institute on Particle Physics, 1976, edited by &I. Zipf (SLAC, Stanford, 1977).

13we have used the phase conventions of J. J. de Swart, Rev. Mod. Phys. 35, 916 (1963). Our resul t differs from that of Ref. 6 by a sign for each n+ and K present in the decay product. For the decay with two identical particles in the final state, our amplitudes differ f rom those of Ref. 6 by a factor of 2. This i s compensated by the factor of 4 difference in our definition of decay

ra te and that of Ref. 6. I 4 ~ o r a background in this subject, see, for example,

G. Barton, Introduction to Dirpersion Techniques in Field Theory (Benjamin, New York, 1965.

I5p. Estabrooks et a l . Nucl. Phys. H, 490 (1978). 1 6 ~ e r e we follow the notation of Ref. 6. '?M. Kobayashi and K. Maslrawa, Prog. Theor. Phys.

49, 652 (1973). I8Fhis conclusion i s consistent with a much more gen-

e r a l analysis of Suzuki (Ref. 5). '$L. F. Abbott, P. Sikivie, and M. B. Wise, Phys. Rev.

D 21, 768 (1980).