Semileptonic decays of charmed mesons produced in electron-positron annihilation

P H Y S I C A L R E V I E W D V O L U M E 1 6 , N U M B E R 3 1 A U G U S T 1 9 7 7

Sernileptonic decays of charmed mesons produced in electron-positron annihilation*

F. Bletzacker and H. T. Nieh Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 1 1 794

A. Soni Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11 794

and Department of Physics, University of California, Santa Barbara, California 93106 (Received 30 November 1976)

In this paper we study a few models for the inclusive semileptonic decays as well as a few exclusive decays of the charmed meson D(1865), recently discovered at SPEAR. We calculate the electron distributions for the sequence e + + e 3 b * + D -+ b + a + D followed by D --I e + v + . . . , at the center-of-mass energy 4.1 GeV. The results are compared with the recent findings of a DESY experiment carried out at center-of-mass energies between 4.0 and 4.2 GeV by Braunschweig et a l . We conclude that D --t e + v + K is most likely the dominant semileptonic decay mode of the D meson. Furthermore, the decay structure for D -+ e + v + K* favors a phenomenological ( V + A ) ( V - A ) form, as is dictated by an underlying charm scheme. This may be viewed as an indirect, and another partial confirmation of the weak-interaction scheme of charm. We also present results for dilepton production, for the afore-mentioned sequence of production and decay. The lepton distributions for the sequence of e + + e -+ D + + D - followed by D --t e + v + K * (or K), at the center-of-mass energy 3.9 GeV, is also calculated.

I. INTRODUCTION

The newly discovered weakly decaying hadronsl*' fit v e r y well with predicted spectroscopy of the charm ~ c h e m e . ~ However, urgently needed i s the experimental verification of the weak-interaction proper t i es and, in par t i cu la r , the semileptonic decay proper t i es of the charm scheme. Also needed i s an understanding of the inclusive semileptonic decays and the identification of dominant semileptonic decay modes of the charmed hadrons.

Since charmed hadrons a r e mos t likely to be the source of the multileptonic events observed in high-energy neutrino and muon scat ter ing pro- cesses; information can be derived from these events concerning the decay proper t i es of the charmed hadrons. Recently, evidence was re- ported by Braunschweig e f ~ 1 . ~ of semileptonic decays of the charmed meson D(1965) produced in e + e - annihilation. Since the production of D in e'e- annihilation, slightly beyond threshold, i s a s im- ple mechanism theoretically, the lepton events ob- se rved in e'e- annihilation should in principle pro- vide a bet ter chance for a clean analysis of the semileptonic decay modes of the charmed D meson.

In this paper we study the semileptonic decays of D(1865). We consider two models for the inclusive semileptonic decay

a s well a s the exclusive semileptonic decays

and

T o compare with the recent DESY resu l t of Braun- schweig et a1. ,5 we have calculated the electron energy spec t rum, fo r these decays, in the p r o c e s s

e + + e - - B * + D

e ( p ) + v + . \ ' D+n (or y) \

for the e'e- center-of-mass energy a t 4.1 GeV. Our resu l t suggests that D - e + v +K * i s the dominant s e m i l e ~ t o n i c decay mode of D, and that the s tructure of the decay waatrix element is in accord with the underlying weak -current structure%f the charm scheme.

In Sec. I1 we presen t the models fo r the inclusive semileptonic decays of D. In Sec. I11 we consider D - e + v + K , and in Sec. IV, D - e + v +K*. We discuss the production of D and D* in e'e- annihilation in Sec. V. Numerical resu l t s for single- lepton and dilepton distributions for the p r o c e s s (4) a r e presented in Sec. VI. In Sec. VII, we present the resu l t s for

a t center-of-mass energy 3.9 GeV. Some discus- s ions a r e given in Sec. VIII.

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11. MODELS FOR INCLUSNE SEMILEPTONIC DECAYS OF D C,,, = ( G / ~ ) [ G , ya(l - y,) - cos6chC ya(l + y,)Ce 1 Models A and A'

The simplest model for the inclulsive semileptonic decays of D corresponds to the decay of the underlying charmed quark according to the charm scheme,'

- F- X+ e-(g-) + P,

with, however, the quark masses for F and X taken to be the physical masses M, and AT,, respectively. According to the charm scheme, the effective Lagrangian for the two decays

g-- v,+em+Ve

and

It i s clear from (6') that the effective coupling for - the decay F- X+e-+F is of a (V+A)(V-A) structure, corresponding to a Michel parameter of a, instead of a s for muon decay, for the electron momentum spectrum.

In this model, the lepton spectrum for the inclusive process

i s thus given by (model A)

X-c+e-+Pe where

i s given by 8 % ~ ; - Z ( P ~ . P , ) .

get,= (G/f i ) [P , y ~ ( 1 - y5)p + ~ o s O ~ F y ~ ( 1 - y5)h] For completeness, the width l? is given below:

X F Y ~ ( ~ - y 5 ) ~ , 1 (6) r = G~ C O S ~ O ~ , 16a4ED

which can also be written, in terms of the charge conjugates cC and XC, where

To contrast with the "effective" (v+A)/(v-A) structure implied by the charm scheme,= we also give the formula for the (V-A)(V-A) structure in the following (model A'):

where again Q2 = MDZ- 2(PD -pe). The corresponding width I' i s identical to (7).

We remark that the spectrum (A), a s compared with the spectrum (Af), contains more low-energy electrons. We also remark that according to the numerical results to be presented in Sec. VI both (A) and (A') contain excessive high-energy electrons a s compared with the DESY results, and are , therefore, not good models for the inclusive process D - e + v +X.

Models B and B'

In models A and At, we have essentially assumed that the invariant mass M, of the final hadrons in D - e + v + X i s fixed at M,. This i s definitely a simple- minded assumption. Excessive high- en- ergy electrons contained in these models, a s compared with experiment (to be discussed Sec. VI), means that this simple-minded assumption is phys- ically not a good one. To imp-'ove on this, we allow for a spectrum of the final-hadron invariant mass. Let p(m2) be the spectral weight. Then the electron distribution is given, in the effective (V+A)(V- A) case, by (model B)

and, in the case of effective (V - A)(V- A), by (model Bf)

734 F . B L E T Z A C K E R , H . T . N I E H , A N D A . S O N 1

where

The spec t ra l function p(m2) according to our de- finition has the dimension of (mass)". In the sp i r i t of scal ing, we assume

With this assumption, the m2 integration can be c a r r i e d out in (B) and (B'). However, the resu l t s will not be presented.

With the incorporation of the final-hadron m a s s spec t rum, the agreement with DESY resul ts , ' a s will be seen in Sec. VI, becomes improved. Again, the effective (V+A)(V- A) s t ruc ture offers bet ter agreement than ( V - A)(V- A ) , although the agreement i s s t i l l not too good.

The decays D - e + v + K a r e s imi la r to K - e + v + n. Simple SU(4) consideration7 leads to, according to the charm scheme, the following ra t ios of the decay coupling constants:

The mat r ix elements for both D decays a r e identical. With the momentum variables a s indicated in

the mat r ix element i s of the fo rm

( D - K ) = C O S O , ( G : ' J Z - ) ~ ( ~ ~ ) ( P + ~ ) ~ Z , , (12)

where I , represen ts the lepton cur ren t , and f(q2) represen ts the fo rm fac tor , which we choose to be

f (q2)= (1 - q ' / ~ ~ * ~ ) - ' , MF** 2.2 GeV . (13)

We r e m a r k that we have neglected a (P- k) t e r m in the mat r ix element, which gives r i s e to a con- tribution proportional to the lepton mass .

In the r e s t f r a m e of D, the double energy spec- t r u m i s given by

x MD[4Ee(MD- E, - EK) - q2] . (14) where

The l imi t s for the energy variables a r e the fol-

lowing:

The energy spec t ra will be presented in Sec. VI, with the production distribution of D taken into account. We repor t here only that numerical computation yields the following decay width:

- 9.5 X 10-I' MeV. (17)

F o r comparison, we have a l so calculated D- e + v + n. We only repor t the total width here. Using f @) = ( 1 - q 2 / ~ D * 2 ) - 1 with MD* = 2 GeV, we obtain the following numerical resu l t , according to the charm scheme:

- 9 . 4 ~ 10-l2 MeV. (18)

The phase space favors the r a t e of Do- e ' v i over Do-e 'vK by a factor of (0.39)/(0.19)= 2.05.

Of a l l the semileptonic decays of D, this i s prob- ably the mos t important one. The arguments a r e the following: (i) In the e'e- annihilation experiment , ' in which the charmed meson Disd iscovered , it i s found that the production of DD* dominates over the production of DD. This means that the BD* coupling i s much stronger t h a n D ~ . We therefore expect that the weak ver t i ces would behave likewise. That i s , D- evK* should be much m o r e important than D-evK. (ii) T h e r e i s the theoretical argument, ' based on current-algebra and soft-pion hypotheses, suggesting that (D- e vK + n's) i s not important. (iii) Although there i s no theo- re t i ca l indication concerning the s t rength of the weak D- K** (1420) transition, one may expect a suppression of the r a t e because of the s m a l l phase space. (iv) Phenomenologically, as we shal l s e e in Sec. VI, the observed electron energy spectrum of Braunschweig e t a1.' a g r e e s well with the calculated spectrum for D- e + v + K*, if the t ransi- tion mat r ix element i s chosen in accordance with the underlying charm scheme.

A s we have pointed out in Sec. 11, according to the c h a r m scheme6 the quark decay T - X+ e-+ V has an effective (V+A)(V-A) s tructure. At the phenomenological level, this should be reflected in the

16 - S E M I L E P T O N I C D E C A Y S O F C H A R M E D M E S O N S P R O D U C E D ... 735

decays of the charmed hadrons. That i s , the c h a r m scheme pred ic t s a (V+A), o r a right-handed cur ren t s t ruc ture for the weak transition D- K*. We shal l in the following r e f e r t o this a s "normal," and the (V-A) o r the left-handed combination a s "anomalous."

F o r the two decays

the mat r ix elements a r e identical. With the momentum variables a s indicated in (and c , the po- lar izat ion vector K*)

the mat r ix element we wr i te in the form

X(D- K*) = COSB,(G/\%)

where q = P- k, 1, i s the leptonic cur ren t , and

F o r a discussion of the mat r ix element, s e e Ap- pendix A. We shal l neglect the symmetry-breaking and renormalization effects, and note the following correspondence:

f, , right- handed (normal) (23)

- f,, left- handed (anomalous) . This correspondence we shal l explicitly i l lustrate in Appendix B.

We shal l take, with M , t = 2.2 GeV,

fv (q2) = / jA(q2) / = (1 - q2 , '~F*2) -1 . (24)

The parameter a in (21) can be est imated by relat- ing, through S U ( ~ ) , to r ( w -ny). We thus obtain

The differential decay spec t rum, in the r e s t f r a m e of D, i s given by

where

The total decay width i s given in the r e s t f r a m e of D, for bothf, = f, and f A = -f,, by

r ( D - evK*) -- 3.2 x 1 0 - ~ 0 s ~ e , ( a h . 1 ~ ) ~ [ G ~ 1 ~ 4 ~ ~ i ( 1 9 2 ? 1 ~ ) ] -- ( a ~ , ) ~ x 1.6 x 10-l2 MeV.

If we use the est imated value a - 6 GeV" given in (25), the calculated width i s -2 X 10"' MeV. Recall that, according to (17), r (D- - e'57K0) " 10-l' MeV. T h e r e i s some room for adjusting the parameter a (see, however, Appendix A).

V. PRODUCTION OF D AND D* IN e'e ANNIHILATION

The m a s s of the charmed pseudoscalar meson D is 1.86 GeV, and the m a s s of the charmed vector meson D* i s 2.0 GeV. The DESY experiment5 i s c a r r i e d out a t the center-of-mass energies between 4.0 t o 4.2 GeV. The conceivable charm- producing channels a r e BD,B*D,B*D*, and the emission of accompanying pions. F r o m the r e s u l t s of a SLAC experiment, run a t s imi la r energies , i t s e e m s that DD* i s the mos t prominent production channel. We will therefore a s s u m e the domin-

anceg of the D*D production channel i n the DESY e'e' annihilation experiment, the energy range for which i s f rom 4.0 to 4.2 GeV, and calculate the following sequence of reactionslO:

B,+F (or y )

F o r simplicity, we will only give the numerical r e s u l t fo r a fixed center-of-mass energy 4.1 GeV, to be shown in Sec. VI, and a s s u m e that it is r e p r e - sentat ive of the (4.0-4.2)-GeV range.

With the yS*D vertex described by

F . B L E T Z A C K E R , H . T . N I E H . AND A . S O N 1

the production c r o s s sect ion can b e easily calculated. I t is given by

where, with s represent ing (center-of-mass en- ergyI2,

0' = [s - (M,* +M,)'] [s- (nilD* - IM,)'] /s2 . (31)

In the following, we a l so give the expression for e'e- - D+ D':

where the corresponding R i s (E' - i ~ D 2 ) 1 ' 2 / ~ . In o rder to study the decay proper t i es of D, it i s

des i rab le t o have an exclusive production channel fo r which the production mechanism i s f i rmly understood and t h e r e i s no contamination from other production channels. The production of DID- a t center-of-mass energies l e s s than 2M,+M = 3.87 GeV i s such a case . We will p resen t the numer ica l resu l t , in Sec. VII, for

The resu l t s of the l a s t few sections can b e easily combined" to yield the single-lepton and dilepton distribution for

e ' + e - - B * + D \ ' e (p )+v+*a*

q + n (or Y )

A s h a s been discussed in Sec. V, we assume the dominance of the production channel B*D for the energy range 4.0-4.2 GeV of the DESY experiment.' We have chosen 4.1 GeV a s a representa- tive energy, and presen t numerical resu l t s in the e + e' center-of-mass f r a m e for the various models and decays discussed i n Secs. 11-IV. In the numer- i ca l computation, we have a l so imposed an angular cut, 1 cose (5 0.6, corresponding t o the angular ac - ceptance of the DESY experiment.'

In Fig. 1 i s shown the calculated electron spectrum" f o r the two inclusive decay models A and B given in Sec. 11. Both of these models have the normal (V+A)(V-A) s t ruc ture . The data points a r e taken f rom Braunschweig e t a1 .5 It i s seen that the agreement is not good. The experimental spec t rum peaks toward the low-energy region.

0.0 0.5 1. 0 Ee (GeV)

FIG. 1. Electron energy spec t rum in the semilept- tonic decay of D lor 5) . The data points a r e taken from Ref. 5 . The two curves A and B correspond to the inclusive decay models A and B, respectively, considered in Sec. 11, for the D o r 5 mesons produced in e - + e - - D + D * - D ~ u + n with the beam energy at 2.05 Gel.'. Normalization of data points is a rb i t ra ry .

We have a l so computed the spectrum for the models A' and B', both of which have a n anomalous (V -A)(V-A) s t ruc ture . The calculated s p e c t r a a r e m o r e energet ic than the corresponding normal (V+A)(V -A) spectra . Their agreement with the measured spectrum i s worse. We have not p re - sented the numerical resu l t s fo r these anomalous models A' and B' .

In Fig. 2 is shown the calculated electron spectrum'' resul t ing f rom the exclusive decay D - e + v + K, which we presented in Sec. 111. As ex- pected, i t s agreement with experiment is not good.

We presen t the results1' fo r the exclusive decay D - e + v +K* i n Fig. 3. The two curves correspond to the two choices, the normal (v+A)(v-A) s t ruc- t u r e and the anomalous (V -A)(V -A) s t ructure. The spectrum corresponding to the normal s t ruc- t u r e peaks more toward the low-energy end and is seen to agree very well with the measured expec- t r u m toward the lower energy. In contrast , the spec t rum corresponding to the anomalous s t ruc- t u r e peaks a t higher energy and d i sagrees with the experimental spectrum. The s t ruc ture around 0.6 GeV i s conceivably due t o D - e + v +K.

In Fig. 4 we superimpose D - e + V+K* and D -e + v + K with a relat ive branching rat io of 3:l. The data seem to be well represented by this combination.

We have calculated the angular distribution of the electron1' for D - e + v + ~ * in (34). The d i s -

S E M I L E P T O N I C D E C A Y S O F C H A R M E D M E S O N S P R O D U C E D ... 73 7

0.2 0.6 1.0 1.4

E, (GeV)

FIG. 2 . Calculated electron energy spectrum for D - e + v + K, where the charmed mesons a r e produced in e' +e--D+D*-E+D + n , with the beam energy at 2.05 GeV. The data points a r e taken from Ref. 5.

tribution is essentially flat in cos8, consistent with the experimental finding. It is shown in Fig. 5.

It thus s e e m s that D - e + v +K* i s likely to be the dominant semileptonic decay mode of D. There i s

FIG. 3 . Calculated electron energy spectrum for D - e + v + K*, where the charmed mesons a r e produced in e+ + e - -E+D*-E+D +?r, with the beam energy at 2.05 GeV. Curve (a) corresponds to the normal, i.e., effective (V + A ) ( V - A ) , structure. Curve (b) corresponds to the anomalous, i.e., effective (V - A)(V - A ) , structure. The data points a r e taken from Ref. 5, normalization being arbitrary.

E, (GeV)

FIG. 4. Calculated electron spectrum for D -evK* and D -evK with a relative branching ratio 3 : l . Only the normal couplings a r e considered. Data points a r e taken from Ref. 5, normalization being arbitrary.

other available experimental information that a l so supports this asser t ion, namely, the hadronic mul- tiplic ity distribution given by Braunsc hweig et a1 .5

F o r one of the D mesons in (34) to decay into e +v+K*, while the other D i s decaying, mos t likely, into KT, Knn, and Knn channels, the final hadron multiplicity in (34) should be 5 o r 6 (with K * count- ing for 2). The experimental distribution i s shown in Fig. 6. The measured hadron multiplicity indeed peaks a t 5, in good agreement .

We have calculated the dilepton correlat ion f o r

FIG. 5 . Angular distribution of the electron, calculated for D -e + v+ K*, where the charmed mesons a r e produced in e + + e - - D + D * -D+D + n with the beam energy at 2.05 GeV. The coupling for the decay i s the normal, i.e., effective (V+A)(V -A) . The histogram i s taken from Ref. 5.

738 F . B L E T Z A C K E R , H. T. N I E H , A N D A. S O N 1 - 16

N E U T R A L T R A C K S

FIG. 6. Multiplicity distribution taken from Ref. 5 .

e + + e - - b * + ~ 1 \ @ ( e ) + v + ~ *

E,+n (or 7)

The distribution in the collinear angle, defined by

is given in Fig. 7 . The angular acceptance lcosa I 0.6 has been imposed.

As has been discussed in Sec. V, the exc lus i ve production of D'D- a t center-of-mass energies l e s s than 2rl lD+M,=3.87 GeV offers a c l e a r setting for studying the decay proper t i es of D . We have there - f o r e calculated the lepton distributions, a t center- of-mass energy 3.9 GeV, for

followed by

In Fig. 8 we presen t the numerical resul t , which a l so incorporates the angular acceptance /cos6 / 5 0.6 , fo r the lepton spec t rum. The calculated dilepton angular distribution i s shown in Fig. 9 (for the K* mode). The collinear angle a,,,, is defined by ( 3 6 ) .

VIII. DISCUSSIONS

F r o m our study we a r e reasonably convinced that D - e + v + K * i s likely to b e the dominant semileptonic decay mode. Several arguments a r e given in Sec. IV in support of this asser t ion. The most important support i s , of course , the agreement between the calculated and observed distributions, presented in Sec . VI.

Also significant f rom our study i s that the decay s t r u c t u r e fo r Dm- e - + i J + K f favors , quite convinc-

FIG. 7. Angular correlations between e and P , which FIG. 8. (a) Calculated electron spectrum for a r e due to the production and subsequent decay of the D -e + v+ K*, where the charmed mesons a r e produced charmed mesons: e* + e - -D+D*, followed by in e+ + e - -D+D, with the beam energy at 1.95 GeV. (b) D - e + v + K * a n d B - p + v + K * . The beamenergy i s Calculated electron energy spectrum for D - e + v + K, taken to be 2.05 GeV. under the same conditions.

S E V I L E P T O N I C D E C A Y S O F C H A R M E D M E S O N S P R O D U C E D . . . 739

c o d e p

FIG. 9. Angular correlations between e andp, which are due to the production and subsequent decay of charmed mesons: et +e- -D + D , followed by D-e+u+K*andD-p+v+K*. Thebearnenergyis taken to be 1.95 GeV.

ingly, the phenon~enological (V+A)(V - A ) , a s i s dictated by the underlying charm scheme. This may be viewed as a n indirect confirmation of this scheme.

In Sec. VII we have presented the calculation of the exclusive production of D'D- and their subsequent semileptonic decays. It s e e m s to us that this offers a clean setting for studying the weak-decay proper t i es of the D meson.

We have a l so presented resu l t s fo r dilepton production, which will be studied i n a forthcoming ex- per iment now being prepared a t SPEAR.

Based on the DESY resul t5

u (e'e-- e + hadrons) > 1 nb , and that the upper bound of the D production c r o s s section i s estimated to be given by

which is, at 2E = 4.1 GeV, roughly 8 nb, the branching ra t io fo r semileptonic decays of D i s

We conclude by making a r e m a r k about the s t range and charmed meson F in the c h a r m scheme. The F meson is believed to have a m a s s of about 2 GeV, and therefore should have been produced, in pa i r s , in the DESY experiment. Based on our study on D, we expect that the dominant semileptonic decay mode of E' i s likely to be

The decay charac te r i s t i cs , however, would be quite s imi la r to those due to the D decay in the DESY experiment , offering no g r e a t hope f o r un- raveling the F meson f r o m a study of the lepton distributions.

During the preparat ion of the p resen t paper , we learned of s i m i l a r works done by Ali and Yang,13 and by B a r g e r , Gottschalk, and Phillips.14 We thank J. Smith f o r bringing the l a s t work to our at ten- tion.

APPENDIX A

We consider the decay D-- em + F + K * in Sec. N . The form of the mat r ix element (21) we adopted i s not the most genera l one. In th i s appendix we presen t a m o r e genera l discussion of the mat r ix element.

F o r the t ransi t ion

the mat r ix element of the vector cur ren t i s unique, and is of the fo rm -

KuYqV , (A2 ) - where K u Y i s defined in (22) and q = P - k . This fo rm of the mat r ix element, incidentally, c o r r e s - ponds to a conserved vector current .

The mos t genera l fo rm of the t ransi t ion mat r ix element of the axial-vector c u r r e n t i s a l inear combination of E U, E ~ P , ~ ~ , and E ' P , ~ ~ . However, for obvious reasons , we r e s t r i c t ourse lves to a l e s s general f o r m of the mat r ix element, which we requi re to correspond to a part ia l ly conserved axial-vector cur ren t . By partially conserved axial-vector c u r r e n t we mean that the axial-vector cur ren t is conserved in the l imi t of z e r o m a s s e s . We a r e thus limited to the fo rm

where K ~ ~ = ~ ~ E ~ - ~ ~ E ~ . We note that the q t e r m in (A3) corresponds to a conserved axial-vector c u r - rent, while the b t e r m corresponds to a partially conserved axial-vector c u r r e n t (IPI,* - 0).

The mat r ix element for D- K* transi t ion i s a l inear combination of (A2) and (A3). To make con- tact with the underlying s t ruc ture of the weak c u r - rent , we sha l l fur ther r e s t r i c t the f o r m of the ma- t r ix element s o that i t has definite helicity p ro- jections in the m a s s l e s s l imit . A s we shal l explicitly i l lustrate in Appendix B, zpY i K u Y a r e the right-handed (+) and left-handed (-) helicity p ro- jections f o r K*, provided K* i s mass less . Noting that

we will adopt the following mat r ix element fo r D ' ( P ) - e ' ( p ) + i ~ ( p ' ) + ~ * ( k , ~ " ) :

where the leptonic mat r ix element is given by

F . R L E T Z A C K E R , H . T. N I E H , A N D A . S O N 1

FIG. 10. Electron energy spec t rum f rom D--e-GK* for D at r e s t , fo r two values of the parameter b = 0 and -0.25. Curves (a) correspond to normal coupling, and curves (b) to abnormal coupling.

and

fa=* f v .

As to the parameter b, we do not know exactly what value it should take. However, we can est i - mate i t s value by making the assumption, which we believe i s reasonable, that the vector and axial- vector c u r r e n t s have roughly equal contributions to the total decay ra te . By actual computation, we find th i s l imi t s b to be smal l and negative. Equal contributions f rom vector and axial-vector cur ren ts correspond to

Over this range of b values, we a l s o find that the peaking charac te r i s t i cs of the electron spectrum, fo r f, = z f v , a r e essentially the s a m e . We show in Fig. 10 the electron spec t ra , for'^ decaying a t r e s t , f o r b = 0 and-0.25. Since thepeaking charac te r i s t i cs a r e our p r i m e in te res t in this paper , we will for simplicity assume b=O. F o r b= -0.25, the r a t e becomes

r@ -evK*) = (a 1~.1,)~0.63 x 10-l2 MeV,

which is to be compared with (28) for b = 0 .

APPENDIX B

In this appendix we explicitly demonstrate the handedness of K ~ ~ * K ~ ~ f o r the transition D- K*,

*Work partially supported by the National Science Foun- dation under Grant No. P H Y . 76-15328.

T ~ e s e a r c h supported by the National Science Foundation. ' G . Goldhaber et al., Phys. Rev. Lett. 37, 252 (1976);

which we have considered in Sec. IV. Denoting by k" and E " the 4-momentum and the polarization vector of K*, respectively, we have defined in Sec. JY

We shal l explicitly demonstrate that K@" +I?"'' contains only right-handed K*, while KuV -@@" contains only left-handed K*, if K* is massless:

K'LV * ~ U Y : right-handed (+)

left-handed (-) . The spin-1 mat r ices S j a r e given by

( S j ) i b= ic i i k r

which satisfy

[S i t S j ] =ie i j kSR ,

F o r a m a s s l e s s vector par t i c le , o r in the high- energy approximation of neglecting the mass , the 4-momentum k u can be chosen to b e

k u = ( k 0 , k 1 , k 2 , k 3 ) = ~ ( 1 , 0 , 0 , 1 ) .

The helicity operator i s

It can b e easily checked that the helicity + 1 (right- handed) and helicity -1 (left-handed) polarization vectors a r e

Corresponding to these polarization vectors EY*,, KU' +@ i s given by

That i s , ~ ~ " + k ' " ~ is z e r o for E';-, and, therefore, a right-handed corlbination.

Simiarly, K"" -Ku" can b e easily demonstrated to be a left-handed combination.

I. Peruzz i et al., ibid. 37, 569 (1976). *B. Knapp et al., Phys. Rev. Lett. 37, 882 (1976). 3 ~ . De R6jula et al ., Phys. Rev. D 12, 147 (1975). 4 ~ x p e r i m e n t a l and theoretical re fe rences can be found

16 - S E M I L E P T O N I C D E C A Y S O F

in F. Bletzacker et a,, Phys. Rev. Lett. 37, 1316 (1976).

5 ~ . Braunschweig et al., Phys. Lett. -, 471 11976). 6 ~ . Hara, Phys. Rev. 134, B701 (1964); J. D. Bjorken

and S. L. Glashow, Phys. Lett. 2, 255 (1964); S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970).

'invoking the Ademollo-Gatto theorem on symmetry- breaking effect, we do not expect large SU(4)-symmetry breaking for the matrix elements of the weak vector currents.

'&I. K. Gaillard et uL. , Rev. Mod. Phys. 47, 277 (1975). he D*D* channel could conceivably be important at

energies higher than the DESY experiment (Ref. 5).

C H A R M E D M E S O N S P R O D U C E D ... 74 1

We have carried out the calculation for e + + e - - 8 * - D * followed by D * - D + n and D - e + V + . The results differ very little, actually, from the D * D case.

'O~he re i s experimental indication that D*-Dy is com- parable to D * -Dr. Since both a r e two-body decays, and the pion mass is negligibly small compared to 'MU, essentially the same kinematics applies to both cases in (29).

orre relation effect due to the spin of D * i s neglected. "we average over the two final electrons in (34). ''A. Ali and T. C. Yang, Phys. Lett. -, 295 (1976). '*v. Barger, T. Gottschalk, and R. J. N. Phillips,

University of Wisconsin report (unpublished).

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Semileptonic decays of charmed mesons produced in electron-positron annihilation