P H Y S I C A L R E V I E W D V O L U M E 2 1 , N U M B E R I1 1 J U N E 1 9 8 0

Lifetimes of charmed mesons

K. Jagannathan and V. S. Mathur Department of Physics and Astronomy, University of Rochester, Rochester, New York, 14627

(Received 24 October 1979)

We discuss a model of charmed-meson decay in which free-quark pairs rather than free quarks play a central role. This modification, while preserving the simplicity of the earlier model, has the advantage that it can incorporate some recent experimental developments which are difficult to understand in the earlier model.

Recent experiments indicate some unexpected features in the decays of charmed mesons. Among these a r e the preliminary results f rom the emul- sion experiment'

r(Di) ( ~ 8 . 3 x 10-13 sec) >> r(DO) (=0.66 x 10-l3 sec) , (1)

and those from DELCO (Ref. 2)

Since the inclusive semileptonic ra tes for D+ and Do a r e expected to be equal from rather general arguments, the results (1) and (2), in fact, cor- roborate each other. Another recent preliminary result is the measurement3

B(D0-K0?r0)=(2.0 +0.9)%, (3)

which makes it comparable to the other known branching rat ios of D -KT, ~ i z . , ~

If future experiments support the results (1)-(3), our understanding of charmed-meson decays would require considerable modification. This i s be- cause on the basis of the usual picture of free- charmed-quark decay, one expects r(Di) = r(DO) and495

In th is note, we propose a simple modification of the usual picture, based on the idea of color confinement. We show that our model i s consis- tent with (1)-(3).

In the usual model: the effective nonleptonic weak Hamiltonian i s given by

+ Cabibbo-suppressed t e r m s , (6)

where the coefficientsf, contain the renormaliza- tion effects of hard-gluon exchange, which have been calculated from quantum chromodynamics (QCD). The final state of the quarks qFqq must evidently be a color singlet. However, th is can a r i s e ei ther if each qq pair i s itself a singlet o r if each i s an octet, so that together the two pa i rs form an overall singlet. The inclusive nonleptonic (NL) decay is usually obtained by adding these two contributions incoherently, which gives rNL(DO,D+,p+) = (2f+Z+f-2)r0, where

represents the rate6 for the f r ee decay c -qqq. Since al l the observed hadrons a r e color sing-

lets , it i s c lear that a color-octet g q pair cannot manifest itself in t e r m s of ordinary hadrons with- out exchanging color with the other q q color octet. Such an interaction i s not taken into account in the usual analysis. Moreover, if th is effect were con- sidered, the color-octet-octet s tate develops an overlap with the singlet-singlet s tate, leading to interference effects which have been omitted. We conclude that the simple p i ~ t u r e ~ ' ~ leading to equal lifetimes for charmed mesons i s inadequate.

To remedy this , we have to effectively consider only those final s tates where each q q i s a confined color singlet. Since the phenomenon of confine- ment is not well understood from f i r s t principles, for further progress, we proceed phenomenolog- ically.

Very recently, Koide7 has discussed a model with the same basic idea a s outlined below. How- ever, our work differs from his in some impor- tant respects. He concerns himself with inclus- ive and semi-inclustive processes only and does not discuss the two-body decays (3)-(5). Further- more he uses a four-parameter fit compared to our two-parameter fit. Our parametrization and analysis a r e simpler and involve the exclusive

3165 @ 1980 The American Physical Society

K . J A G A N N A T H A N A N D V . S . M A T H U R 2 1 -

FIG. 1. The amplitudes in the nonleptonic decay of the charmed mesons. The quark-antiquark pairs a r e each in a color-singlet state.

modes directly. We write the effective nonleptonic weak Hamil-

tonian in a form similar to Eq. (6), where the quark currents8 now represent confined color-sing- let q? pairs. Furthermore, the known coefficients (f, +f-)/2 should be replaced by unknown para- meters , which we callf, andf,. The nonleptonic amplitudes can be written a s

These amplitudes a r e represented diagramatically, in Fig. 1. Note that the SU(4) s tructure of the non- leptonic Hamiltonian in the present case i s the same a s before. Assuming, a s usual, 20-plet dominance over the 84-plet, then suggests f2/f, <O. Except for this constraint, we shall t rea t f , and f, a s f ree parameters in our analysis.

The inclusive total decay ra tes for charmed mesons can now be calculated easily. We obtain

In Eqs. (10)-(12), the scale for the inclusive non- leptonic contribution has been taken to be the free- c-quark decay rate (7). This can always be done by a suitable redefinition of fl and f,. The inclu- sive semileptonic decays a r e taken to be described in the usual Here the outgoing qq pair can only be in a color-singlet state. The crucial dif- ference among the three ra tes a r i s e s a s follows. Consider, for example, the Cabibbo-dominant

modes of Do and D+. The final s tates [sE][uz] and [uG][sd] in Do a r e distinct, so the two contributions add incoherently. For D+ decay, by contrast, the final s tates a r e indistinguishable; hence, a coher- ent addition of the amplitudes is called for. Elim- inating f, and f, from Eqs. (10)- (12), we obtain a sum ruleg

An experimental test of this relation has t o await a more accurate determination of the lifetimes.

We now turn our attention to the exclustive had- ronic decays. For the Cabibbo-dominant two- body decays of D mesons, the amplitudes can be written aslo

so that

Equation (17) is, in fact, independent of our mo- del and follows directly from isospin consider- ations, if we recall that the AC = AS = 1 piece of the nonleptonic Hamiltonian transforms a s I = 1.

Without any specific assumption on the relative phases of the amplitudes, the sum rule (17) leads to a triangular relation for the decay rates. In t e r m s of the branching ratios, one readily obtains

where

For the experimental results (3)-(5), the bounds on r (D+) / r (DO) cover too wide a range to be of much practical use.

In our model, where the amplitudes (14)-(16) a r e relatively rea l and f2/f1 <0 , Eq. (17) leads t o the equality

It should be noted that relation (20) also obtains in the usual model, where because of the equality of lifetimes Z = 1. Because of the experimental uncertainties we have chosen to represent Eq. (20) in the manner shown in Fig. 2. The box encloses one standard deviation from the reported central values. For B(D+-KOr+) we have used the central value from Eq. (5) and curves of constant Z have been displayed for this number. Any future change in this number can easily be accommodated by an

L I F E T I M E S O F C H A R M E D M E S O N S 3167

B( D ~ - + K - T + )

FIG. 3 . Curves of constant I R / (= Ifi/fi/).

B ( D ' ~ K - T + I

FIG. 2. Curves of constant [= ~ ( D ? / ~ ( D ~ ) I .

appropriate multiplicative change in the labeling of the Z values of the curves. We see that whereas Z = 1 implies a negligible value of B (Do -Ron0), the central values of (3) and (4) pick out a value of

close t o 0.1. Thus our model shows that the large branching ratio (3) i s quite consistent with the small ratio of the lifetimes of Do and D' given by (1).

Another relation can be obtained from Eqs. (14) and (15):

where

7(D0) = 0.70 x 10-13 s e c , r(Dt) =4.9 x 10-l3 s ec , r(F+) =0.72 x 10-13 s e c .

Also the semileptonic branching ra t ios a r e given by

Our resu l t s (23) a r e different from K ~ i d e ' s . ~ In part icular , he finds 7(D0) = 27(Ft). This differ- ence a r i s e s due t o his parametrization which makes the contribution of the annihilation diagrams very large. Mathkr and ~ i z z o " have considered the contribution of annihilation diagrams in a mas- sive-quark model. Thei r calculations do not sub- stantiate Koide's fit for the rat io of the contribu- tions of annihilation diagrams to decay diagrams.

Recent emulsion datal%eem t o indicate r (F+) Figure 3 represents a plot of R against the branch- - 3r(D0). This is based on two F + events and ing ra t ios (4) and (5).

hence i s statistically poor. But in addition, the We have seen that using (3)-(5) a s input, (act- present model14 as well a s Koide's, shows that

ually only tu~o ratios among these three branching the Cabibbo-~~suppressed~ decay mode D+ -ROK +

fract ions a r e used a s input) we can obtain the r e - may not be suppressed and i t s branching rat io su l t s for Z = I'(D')/~?(D') and R =f,/fl. With these may in fact be comparable t o the dominant modes. values of Z and R and using the rat io of Eqs. (10) Hence without a precise momentum measurement, and ( l l ) , we can then solve for the individual the detection of kaons in the final s tate alone i s parameters f, and f,. F o r purposes of illustration, not sufficient t o decide between a D' and an F' de- if we take the values 2 = 0.14 and R = -1.33, we

cay. get I f l / = 3.35, 1 f , 1=4.47. Substituting these val- ues in Eqs. (10)-(12) and using Eq. (7) with'l M, This work was supported by the U. S. Department = 1.66 GeV, we obtain of Energy under Grant No. EY-76-C-02-3065.

3168 K . J A G A N N A T H A N A N D V . S . M A T H U R 21 -

Prentice, talk given a t the 1979 Symposium on lepton and photon interactions at high energies, Fermilab, 1979 (unpublished).

'5. Kirby, talk given a t the 1979 Symposium on lepton and photon interactions at high energies, Fermilab, 1979 (unpublished).

3 ~ . H. Schindler (private communication) ; J. Kirby (see Ref. 2 above).

4 ~ . K. Gaillard, B. W. Lee, and J. L. Rosner, Rev. Mod. Phys. 47, 277 (1975).

5 ~ . Ell is , M. K. Gaillard, and D. V. Nanopoulos, Nucl. Phys. g, 313 (1975); N. Cabibbo and L. Rlaiani, Phys. Lett. F, 418 (1978); D. Fakirov and B. Stech, Nucl. Phys. E, 315 (1978); K. Jagannathan and V. S. Mathur, Phys. Lett. m, 335 (1979).

6 ~ e assume here that the s , u , and d quarks a r e mass- less.

7 ~ . Koide, Phys. Rev. D 20, 1739 (1979). Our attention was drawn to this paper after the present work was completed.

' ~ h e s e quark currents need not have a V-A structure. In fact for the present discussion their space.-time structure is not important.

$Equation (13) is consistent with equal lifetimes for the three mesons, but only if either fi o r fi vanishes. Also since tan2oc< 1 , Eq. (13) implies T(F+) > t a n 2 0 c r ( ~ ' ) .

'O~he constant of proportionality, which is the same in each case, contains a piece that describes the proba- bility of one q7j pair to form a K and another a a . (See last entry of Ref. 5 above).

liA. De Rdjula, H. Georgi, and S. L. Glashow, Phys. Rev. D 12, 147 (1975).

I2v. S. Mathur and T . Rizzo, Phys. Rev. D l(i, 3343 (1977).

I 3 ~ e p o r t e d by J. L. Rosner, at Conference on Color, Flavor, and Unification, University of California a t Irvine, 1979 (unpublished).

1 4 ~ . Jagannathan and V. S. Mathur, Nucl. Phys. B (to be published).