Two-body decays of charmed mesons

Volume 73B, number 4, 5 PHYSICS LETTERS 13 March 1978

TWO-BODY DECAYS OF CHARMED MESONS

N. CABIBBO 1 Laboratoire de Physique Thdorique et Hautes Energies, Universitd de Paris VL Paris, France

and

L. MAIANI 1 Laboratoire de Physique Thdorique de l'Ecole Normale Supdrieure 2, Paris, France

Received 27 October 1977

We compute the rates of two-body weak decays of charmed mesons, on the basis of a simple quark recombination scheme. The ratio of rates for different modes is determined directly in terms of the relative strength of the "20" and "84" SU(4) components of the effective weak lagrangian. Present data, in particular on D +~ ~o n+ and D O ~ K-Tr ÷, support the quan- titative prediction of QCD on the enhancement of the "20" component with respect to the "84".

In this paper we study the simplest exclusive decay modes of charmed mesons, namely the PP and PV modes (P and V denoting a pseudoscalar or vector meson). Our treatment is based on previous analyses on inclusive decays [ 1 - 4 ] , which indicate that the dominant decay mechanism for D +, D O and F is the disintegration of the c-quark (c ~ sud), the uncharmed antiquark acting as a spectator. Extending this picture to exclusive modes, we will assume that the PP and PV final states arise from a rearrangement of the four fermions (quarks and antiquarks) present after c-disinte-

gration * A closely related discussion of the two body decays

1 On leave of absence from: Istituto di Fisica dell' Universita, Roma, Italy.

2 Laboratoire propre du C.N.R.S. associ6 h l'Ecole Normale Sup6rieure et/t l'Universit6 de Paris-Sud. Postal address: 24 rue Lhomond, 75231 Paris Cedex 05, France.

, l In the parton picture, an alternative decay mechanism is given by the interaction of c with the spectator to yield an uncharmed qq pak. The multiplicity distribution arising from this process is expected to be very similar to that arising from c disintegration. The contribution of the former-mechanism, which is negligible in total rates, should therefore be also very small in exclusive channels. The situ- ation is different from strange particle decays, where the two mechanisms can give comparable amplitudes [5].

of charmed mesons has been given by Fakirov and Stech [6]. These authors aim at an estimate of the absolute rates of the different channels. To do so, they introduce further assumptions whose validity is, at present, difficult to assess. The point we want to stress in this paper is that in the SU(3) limit, the ratios of rates for PP modes can be directly predicted in the rearrangement scheme from the structure of the c ~ sud amplitude. Thus, these decays provide a clear test for the current theoretical picture. Ratios between different PV rates can be predicted in terms of a single phenomenological parameter.

The present theoretical picture of c-quark decay ,2

can be summarized as follows [7]: (i) The c ~ sud amplitude is derived from the effec-

tive lagrangian [4] (a and b are color indices and the sum over repeated indices is understood):

27w= (G/x /2) (1)

× [ l ( f+ + f _ ) (~a3, ( 1 _ 3"5)aa) (s-b3'u( 1 -- 3"5) Cb)

+ ½ (f+ - f - ) (s-a3". ( 1 - 3"5 )da) (~b3""(1 - 75 )Cb)]"

(ii) The coefficients f+_ embody the renormalization

,2 For simplicity we restrict to AS = AC processes and we shall set 0c= 0.

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Table 1 Amplitudes and relative rates for PP decay modes. The amplitudes are given in units of A (see eq. (8)). Rates are normalized to F(D ° ~ K-n+), and are computed using the values of X_+ given in eq. (4). A phase space correction proportional to the decay momentum has been introduced. We have assumed and r/' to be pure octet and singlet respectively.

Decay mode Amplitude Rate relative to D ° ~ K - #

D + -~ ~ n+ 8f. _ 3 +=-2(X++X ) 0.60

D O ~ K-Tr + 2X+ 1.00 D O --* K°n° x/2X_ 0.025 D O ~ K°n x / ~ X _ 0.0073 D O --, K°r/' (2/,,/3)X_ 0.010

F + ~ n + - 2 2 x ~ X + 0.73 F + ~ n 'n + (2/,,/3) X+ 0.30 F + ~ K+g, ° 2 X_ 0.050

effects due to ha rd-g luon exchange ( f + = f _ = 1 in the

l imit o f free quarks ) and have been c o m p u t e d to be

[81: M2 "112/(33-2F)

f = [ 1 3 3 - 2 F 2 ln~'~J + ~ a ( m c ) me (2)

f+ = ( f _ ) - 1 / 2 .

a ( m 2) is the r u n n i n g q u a r k - g l u o n fine s t ruc tu re con-

s tan t eva lua ted at the mass o f the c h a r m e d quark , and

F the n u m b e r o f qua rk flavors. Assuming a ( m 2) = 0.7

as suggested by the s tudies o f scaling v io la t ions in

deep inelast ic processes [9] , and F = 6, leads to :

f _ ~ 2 . 1 5 , f+ ~ 0 . 6 8 . (3)

In wha t fol lows, we shall use the two c o m b i n a t i o n s

X_+, given by:

X+ = ½ ( 2 f + + f _ ) - ~ 1 . 1 7 ,

(4) X_ = ½ (2f+ - f _ ) -~ - 0 . 2 6 ,

where the numer i ca l values fo l low f rom eq. (3) .

(iii) Since the u n c h a r m e d a n t i q u a r k acts as a spec-

ta tor , one is led to p red ic t equal l i fe t imes ,3 and equal semi- lep tonic b r a n c h i n g ra t ios for the th ree mesons (C = D 0, D +, F+):

F ( C -~ la+p u h a d r o n s ) 1

BSL = P(C ~ all) 2 + 2 f 2 + f 2 (5)

Table 2 Amplitudes and relative rates for PV decay modes. The amplitudes are given in units orB (see eq. (10)). The rates have been computed with z = 1 and are normalized to F ( D ° ~ p+K-). A phase space correction proportional to p3 (p being the decay momentum) has been included.

Decay mode Amplitude Rate relative to D o ~ 0+K - (z= 1)

D + ~ o+I( ° 2 (X+ + zX_) 0.61 D + ~ ~ ,o n+ 2 (zX+ + X_) 0.71

D o ~ K*-n + 2zX+ 1.2 D O ~ o+K - 2)(+ 1.0 D O ~ K*°n° ,,/2X_ 0.029 D O __, ~*o~ 2- , /~X_ 0.0052 D O ~ o°K ° x/'2zX_ 0.025 D O -+ coK ° x/2zX_ 0.024

F + ~ ~o*r + 2 zX+ 1.4 F + ~ 0 % / - 2 2x/~X+ 0.97 F +--" K*+I( ° 2zX_ 0.061 F + ~ K,*° K+ 2X_ 0.061

Using eq. (3) , one f inds BSL ~ 13%, to be c o m p a r e d

wi th the e x p e r i m e n t a l value [10] : BSL = 10--12%.

Our p red ic t ions for PP and PV decay m o d e s are

r epor t ed in tables 1 and 2. The c leanest case is t h a t

o f PP modes , where , up to a c o m m o n n o r m a l i z a t i o n ,

all ra tes ( and the re fo re all b r a n c h i n g rat ios , 3 )

depend on ly u p o n f+ and f _ . In par t icu la r , we ob- ta in , 4 .

B ( O ° -~ K - n + ) / B ( D +-+ K0n+) = ¼(1 + I"_ /22+) 2 . (6)

A d e t e r m i n a t i o n o f th is ra t io , c o m b i n e d w i th eq. (5) ,

allows for a comple te d e t e r m i n a t i o n o f f + and f _ ,

,3

, 4

An important practical consequence of the prediction of equal lifetimes is that the ratio of the rates of different decay modes is expected to coincide with the ratio of the corresponding branching fractions, even for different charmed mesons. This allows a comparison of the predic- tions with the experimental data, even prior to a determi- nation of the absolute rates. We recall that f+ and f_ are the coefficients of terms in "QW transforming respectively, as SU (4) "20" and "84". The effect of a "20" enhancement (f_/f+ ~ 3.2) is not reflected in a large suppression of D + ~ g,°n+ in respect to D O K-n +. The reason is that, although the first channel is pure "84", the second has a relatively small projection on the "20" and a larger one on the "84".

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(P) (q) P

D D

g

p (~) p (D)

S N

Fig. 1. Diagrams contributing to D ÷ ~ K.°n+. Diagrams (a) and (b) arise from the first term in eq. (1) of the text, and are proportional to (f÷ + f_); diagrams (c) and (d) arise from the second term and are proportional to (f÷ - f_).

and therefore gives a very stringent test of eqs. (1) and

(2). Observe that the r.h.s, of eq. (6) equals 0.56 for

f+ = f _ = 1 (free theory), while it equals 1.6 for the theoretically preferred values off_+, eq. (3). The latter value compares very well with the present data [11 ] which indicate a ratio of about 1.5. Table 1 predicts furthermore a very distinctive pattern, with a substan- tial suppression of all the PP decay modes which have not yet been identified.

Decays into two pseudoscalar mesons. As an illus- tration we compute explicitly the amplitude for D + K°Tr +. In fig. 1 we have reported the relevant diagrams. Diagrams (a) and (b) are associated to the first current X current term in eq. (1), proportional to (f+ + f_ ) , diagrams (c) and (d) correspond to the term proportional to (f+ - f_ ) .

Let us define as (f+ +f_ )A the contribution of diagram (a) to the amplitude of the process under consideration. If we neglect SU(3) breaking, the contribution of diagram (c) is obviously: ( f + - f _ ) A . To compare diagrams (b) and (c), we Fierz rearrange the four fermion vertex in diagram (b), according to (for simplicity we omit color indices):

[ u T u ( 1 - 7 5 ) d ] [svU(1 - T5)c]

: 1 [~u (1 _ 75 )d ] [fiTu(1 _ T5)c ]

(7)

where X a are the color SU(3), Gell-Mann matrices. When inserted in diagram (b), the second term in

eq. (7) would create a sd pair in a color octet state, so that it does not contribute to our process. The first term gives a contribution which is obviously related to diagram (c). One gets therefore the amplitude" 1 -~(f+ + f _ ) A from diagram (b). Similarly, the

1 amplitude from diagram (d) is: g ( f + - f _ ) A and the total amplitude results:

A(D + ~ ~0rr+) = 8 -~ f+A . (8)

To compute the amplitude for D O -> K - u +, one has to replace, in fig. l, the spectator d with ft. One sees immediately that only diagrams (a) and (d) contribute, with the same amplitudes as before. Thus:

A(D0 ~ K - u + ) = 4 z (-~f+ + x f _ ) A ==-2X+A. (9)

A comparison of eqs. (8) and (9) leads to the result given in eq. (6). With similar arguments, one computes the amplitudes for all decay modes into two pseudo- scalars. The results are given in table 1. A computation of the amplitude A would require a knowledge of the recombination factors. However, ratios of rates are uniquely determined by the structure of the lagrangian eq. (1).

Decays into a pseudoscalar and a vector meson. As before, we consider first a specific channel, namely: D + ~ ~0p+. The amplitude is given again by the diagrams of fig. 1, where now the ud pair goes into a p+. The contribution of:diagram (b) is related to the contribution of diagram (c) by the same relation as before, and similarly for diagrams (d) and (a). Comparing diagrams (a) and (c), we observe that the two amplitudes are not obviously related as before. In fact diagrams (a) and (c) differ not only for an u ~ s exchange (as in the previous case) but also in that while in diagram (c) the p + is formed using the spectator d, in diagram (a) the 0 + is formed from a pair of fermions, both originating from c decay. We will therefore indicate * s

For PV decays there is clearly only one amplitude, corresponding to the emission of a longitudinally polarized vector meson.

420


by (f+ + f _ ) B the contr ibution of diagram (a), and by ( f+- f_)Bz the contr ibution of diagram (c), where z is a new phenomenological parameter, related to the effect described above. The total amplitude is thus:

A(D+ + EOp+)= 4 } f ( 1 - - z ) ] B [ -~ f+ ( l+z )+ _

-2B(X+ +zX_) . (10)

With similar arguments one may compute all the other amplitudes ,6 , with the results reported in table 2. We observe that, as before, a number of amplitudes are proportional to X_ , leading to a strong suppression of the corresponding decay modes. The magnitude of z can be directly measured from the relative rates of the dominant PV decay modes such as D O -~ K * - ~ + versus p + K - or F + -> ~ + versus p+r/. The sign of z can be obtained from D + decay modes.

Although we prefer to consider z as a free parameter, to be determined from experiments, the following argument leads to expect z ~ 1. Since the D + meson is an S-wave state, we may consider, in an approximate sense, the spectator d to be localized at the position of the c-quark. In this case we may consider the four outgoing fermions to be created at the same point, as if created from the vacuum by a local four-fermion operator. The form of the operator is fixed by the requirement that the field of the spectator d combines with the other three fields to give a spinless operator. The result is that the final state is created by the operator obtained by replacing i n , w , eq. (1), the c with a d field. At this point there is no difference between the spectator d and the newly created d, which results in z = 1. We will use this value in our numerical esti- mates given in table 2 in spite of the fact that it is difficult to assess its accuracy.

The relative magnitude of PP and PV rates. It is possible to give a reasonable estimate of the ratio of the amplitudes A and B, introduced above. Both these amplitudes refer to diagram (a) of-fig. 1, with the sd pair combined into a ~0, while the ud pair gives a n + (A) or a O + (B). I f we consider the latter particles as being created from the vacuum by the weak V - A current involved, we get * 7 :

• 6 We have not included in the table the Zweig suppressed modes, like e.g. F+-* con+; see, e.g. ref. [ 12].

,7 It is interesting to recall that the combination of ms, fO andfrr in eq. (11) is the same which appears in the KSFR relation [13].

B _ v~m2eU(F{IJul D) - v ~m2 2eU(PD) u

A fofnp~(~lJulD) fofn m2 D

_ v ~ m 2 2 Ip o I 1.2, (11)

IoY mD where m2/fo is the coupling of the p ° to the e.m. current (f2/47r ~ 2), eU and Po are the 0 + polarization and momentum. Eq. (11) can be used to normalize the PV rates to the D O -+ K - a + rate, as we did for PP rates. We get, e.g.:

F(D0-+ P(DU -+ K-rr+)

~- 1.1 p(DO-+ K-Tr+). (12)

PV rates thus normalized and with z = 1 are reported in table 2. Assuming eq. (12), the relative rates reported in table 2 can be transformed directly into the corresponding branching ratios, using the experimental value [11] for

B(D 0 + K - a +) = (2.2 -+ 0.6)%. (13)

In this way, we may obtain the predicted contribution of PV decay modes to the decay modes of D + and D 0. Defining Bpv to be the contribution of PV channels to the branching ratio, we find (z = I) :

B(D 0 ~ K 0 7 r + ~ - ) p v = 1.9% (4.0 + 1.3) ,

B(D 0 ~ K-Tr+Tr0)p v -~ 3.4% (9 + 5 ) ,

B(D + -+ K-rr+Tr+)p V ~ 1.1% (3.9 + 1.0) . (14)

For each KTnr channel we have reported in parentheses the present experimental value for the overall branching ratio of that channel [11]. While we expect PV modes to account for a sizeable fraction o f Krnr modes, eq. (14) may be compatible with the approximately uniformly populated Dalitz plot for D + -+ K-rr+Tr + reported in ref. [14].

Concluding remarks. The first data made available on KTr and KTrzr decays indicate that the theoretical ideas on the structure of the c ~ sud amplitude, em- bodied in eqs. (1) and (2) may pass successfully the test o f correctly describing the simplest PP and PV exclusive decays. A more precise conclusion can be drawn only from more accurate values for PP branching fractions, and from the values for individual PV modes.

421


On the theoret ical side much work remains to be done.

One can obtain a s t ra ightforward extension o f our pre-

dictions to VV modes. Another obvious application

of the present ideas would be to two-body decays o f

charmed baryons.

At a more diff icult level, one might wish to a t t empt

an absolute de te rmina t ion o f the recombina t ion pro-

babil i ty, and to improve the predict ion on the value

of z. This might result e i ther in sounder arguments

for z = 1 or in a cor rec ted est imate . Another interest ing

problem is that o f the relation o f the partial rates

(such as the ones discussed here), to the total rate.

The lat ter is expec ted to be the same for D +, D 0, F +,

while i f we sum the PP and PV entries in tables 1 and

2, we obtain di f ferent funct ions o f f_. for each o f the mesons. The si tuation is no t inconsis tent since, if

we take the central value f rom eq. (13), our numerical

results predict that PP and PV channels account for

only a relatively small fract ion o f the total non lepton-

ic rate (i.e. about 4% for D +, 8% for D 0, 8% for F+).

We would like to thank Prof. G. Altarelli for many

discussions on the subject.

References

[1] B.W. Lee, M.K. Gaillard and J. Rosner, Rev. Mod. Phys. 47 (1975) 277.

[2] G. Altarelli, N. Cabibbo and L. Maiani, Nucl. Phys. B88 (1975) 285.

[3] S.R. Kingsley, S. Treiman, F. Wilczek and A. Zee, Phys. Rev. D l l (1975) 1919.

[4] J. Ellis, M.K. Gaillard and D. Nanopoulos, Nucl. Phys. B100 (1975) 313; G. Altarelli, N. Cabibbo and L. Maiani, Phys. Rev. Lett. 35 (1975) 635.

[5] C. Schmid, Phys. Lett. 66B (1977) 353. [6] D. Fakirov and B. Stech, Heidelberg preprint, H.D.

THEP 77-8. [7] For a more extensive discussion see the talks by

L. Maiani and by T. Walsh, Proc. 1977 Internat. Symp. on Lepton and photon interactions at high energies (Hamburg).

[8] B.W. Lee and M.K. Gaillard, Phys. Rev. Lett. 33 (1974) 108; G. Altarelli and L. Maiani, Phys. Lett. 52B (1974) 351.

[9] A. De Rujula, H. Georgi and D. Politzer, Ann. Phys. 103 (1977) 315; J. Kogut and J. Shigemitsu, Cornell preprint (1977); A. Buras, E. Floratos, D.A. Ross and C.T. Sachrajda, CERN preprint TH 2340 (1977).

[10] J. Kirby, DELCO results; S. Yamada, Results from DASP, reported at the 1977 Internat. Syrup. on Lepton and photon interactions at high energies (Hamburg).

[11] I. Peruzzi et al., Phys. Rev. Lett. 39 (1977) 1301; A. Barbaro Galtieri, SPEAR results SP 26, reported at the 1977 Internat. Symp. on Lepton and photon interactions at high energies (Hamburg).

[12] S. Nussinov, Phys. Rev. D15 (1977) 2025. [13] K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16

(1966) 255; Riazuddin, Fayazuddin, Phys. Rev. 147 (1966) 1071.

[14] J.E. Wiss et al., Phys. Rev. Lett. 37 (1976) 1531 ; M. Piccolo et al., Phys. Lett. 70B (1977) 260.

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Two-body decays of charmed mesons