Decays of charmed mesons via color-octet decays of the c-quark

Volume 109B, number 3 PHYSICS LETTERS 18 February 1982

DECAYS OF CHARMED MESONS VIA COLOR-OCTET DECAYS OF THE c-QUARK "~

Jan FINJORD lnstitute for Theoretical Physics, University of Bern, CH.3012 Berne, Switzerland

Received 9 June 1981

We unify the color-octet weak transitions first studied in c~ annihilation models with the coherence structure of c-quark decay models. We find then, with a dominant c-quark decay contribution, 2.3 ~< r(D+)/r (D °) ~< 3.0 and r (D +) ~ 7.5 X 10 -13 s, not inconsistent with the trends in the experimental results.

The purpose of this note is to point out that the D+/D 0 life-time ratio, as well as the absolute life-times, may find their basic quantitative explanation in terms of c-quark decay, however with color-octet weak transitions as well as the coherence properties introduced in a consistent way.

The early estimates of charm-meson decays [ 1 - 3 ] , based on color-singlet weak decay of the c-quark and one-loop strong interaction renormalization of the weak vertex [4,5], were contradicted by the early experimental result [6]

Fto t (D0)/Pt ot (D +) ~ 4 - 1 0 , (1)

which has since then been drastically revised [7]"

Ptot (D0)/Ptot(D+) ~ 3.0 + 3.3 - 1 . 8 , (2)

"r(D +) ~ (8.0 _+4:0) X 1 0 - 1 3 s . (3)

In (2) and (3) an average of the reported results has been a t tempted, giving emulsion results less weight than those from bubble chambers or counters. Efforts to understand large values for the life-time ratio have de- veloped along two competing lines. One approach 18] still concentrated on c-quark decay, but by t~.,i, ~g tile spectator antiquark into consideration the coherence properties for D + and D O decay were argued to be different, and the ratio I'nl(DO)/Pnl(D+ ) between the non- leptonic rates could be larger than unity. The other line of approach was based on the idea that the cq sys-

¢~ Work in part supported by Schweizerischer Nationalfonds.

tern could be in a spin-1 state after gluon radiation [9, 10] or alternatively if there were a component of con- stituent gluons in the meson wave-function [9,11,12]. A Cabibbo-unsuppressed weak interaction could then take place between the c-quark and the antiquark in the D O and F + cases (generically denoted by "annihi- lat ion"), notably without the helicity suppression of a spin-0 state.

In a previous paper [13] we have shown that even without hellcity suppression, the annihilation contribution to the D O rates comes out one order of magnitude too small to explain the result (1), and even the smaller value (2) can only with difficulty be understood in terms of a dominant annihilation contribution. Such a conclu- sion was also reached earlier for color-singlet transitions [14]. In the treatment of the annihilation mechanism of ref. [ 13], also color-octet c~t annihilation was considered, and first at tempts were made to incorporate color-octet effects explicitly also in c-quark decays. An independent analysis [15,16] included color-octet weak transitions, however with coefficients adjusted so as to explain K- meson decays without Penguin diagrams, in contradic- tion to the philosophy of ref. [13]. More recently, a study of color-octet transitions and other soft gluon effects in explicit decay channels has appeared [17].

Here we will present a fuller study of color-octet effects in the total rates for pseudoscalar charm-meson decays via c-quark decay. The consistent application of coherence arguments and the concept of color-octet transitions, will permit a unified approach to the results of Ellis et al. [1] and those of Guberina et al. [8], and

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our differences from those treatments can be discerned. For an estimate of the total D O width we will then as- sign the annihilation channel a relative importance of at most 30% [ 13,14]. Because of the present non-conver- gence of the experimental situation evident from eqs. (2) and (3), we will not attempt to fit the annihilation contribution by considering exclusive channels, which would be of at most dubious value [ 18]. However, we will see that our assumptions give a reasonable repro- duction of the results in eqs. (2)and (3), which would imply that the life-time asymmetry is basically a coherence effect in the c-quark decay channel.

One should notice that the value 30% may be a rather liberal one. It is based on the estimate fD ~ 350 MeV; however, some recent theoretical evidence [19] points to 250 MeV or smaller, in which case it would be increasingly difficult to argue for important annihilation effects. An experimental determination of f o is thus overdue. (For clarity we mention that annihilation can anyway be expected to be important in baryon decays [17], where no helicity suppression takes place.)

1 In terms of the left-handed quark fields qL - 7( 1 -- 75)q the Cabibbo-unsuppressed part of the renor- realized weak hamiltonian is

He'af~ =-1 = 2X/2 G v cos20c

X (fl SL'YUCL t~LYudL +f2fitTtaCg-gLYtzdt ) ' (4)

with

1 1 f l =2(c84 + c 2 0 ) ' f2 = 2 ( c 8 4 - C20)' (5)

where the one-loop calculation gave [ 1 ]

c20 = (c84) -2 ~ 2 .15, (6)

compared to c20 = 1 without strong-interaction renormalization. Without renormalization one finds the basic rate for c-quark decay, in the approximation where mu, rod, m s and lepton masses are neglected,

3 cos20 c N~ ,

['0 = 3 coS40c G2FmSc/192rr3

= 4.6 X 10-13(mc/1.5 GeV) 5 GeV, (7)

where Nq (= 1) and N e (= 2) are the numbers of SU(2) quark and lepton doublets which can couple to a W in charm-quark decay.

We then estimate the c-quark decay contributions to D 0, D + and F ÷ decays by means of eq. (7), modi- fied by the additional assumption that the spectator antiquark combines with one of the final-state quarks. The four possible diagrams on the basis of eq. (4) are shown in fig. 1. The coupling strengths are indicated. By Fierz transformations, ( lc) and (1 d) can be brought into the same topological forms as ( lb) and (la), re- spectively. Because I/M w "~ typical hadronic distances, it is a good approximation to add ( ld) to ( la) coher- ently, as well as ( lc) to (lb). In treating those subsums, for the D + case we will also add them together coher- ently, since for D + all four diagrams may lead to the same final-state configurations. This is analogous to the color clustering arguments of ref. [8] : The coefficient c20 of the 20 representation of SU(4) will cancel.

In the treatment of the subsums in the cases o f D O and F +, it will now be most consistent to take the physical content [1] of the interference rules into account also here: We will add the (ad) and (bc) subsums inco- herently, since they in general lead to different final- state configurations. At the basis of the above consid- erations lies of course the assumption that the decay- ing system has a physical description in terms of two quark-ant iquark pairs at some intermediate stage fol- lowing the weak decay. This physical content of the notions of coherence versus incoherence will be sup- posed to hold, so that our results should be essentially relevant even with the oversimplified assumptions. Further discussion can be found in ref. [8].

One immediate consequence of the above considera- tions is, up to kinematical differences and SU(3) break- ings,

r (D 0) ~ r ( F + ) , (8)

c f l ~ d c f2 ~ d

.,.0, \ \ , ° o , \ \ o u , d , s ' \ u , d , s ' \

) (e)

, . U,d,S

Fig. 1. Quarkqine diagrams for D °, D + and F + decays (spectatc ~t, d, g, iespectively).

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also with the possibility of color-octet transitions included, provided c-quark decay is dominant in the decays of these mesons. The result (8) is consistent with experiment [7]. To compare D O and D + decays, study first the effective hamihonians corresponding to the (ad) and (bc) subsums. Written so as to give a factor- izable matrix element, the two cases give

Ha eff = 2x/~GF c°S20c [(fl + ~-f2) SL')'UCL fiL"/udL

+ 2 f 2 - 1 a - ' V d L] (9) SLTU~ ;k CL UL~u~

and

Hl~c ff = 2X/~ G F cos20c [(~-fl + f2) flLT/~CLgL~//adL

- I a - 1 ~adL] (10) + 2 f 1 UL7~*~2 CLSL~'#~

The rate (7) is based on the unrenormalized current bilinear gL~,UCL fiL')'udL. One finds then after some algebra that the evident generalizations of (7) to obtain total c-decay widths are, in the D O and F + cases

Nq-~Nq~[f 2 +f22 +~flf2 +~( f2+#)O/S] ( l l )

and in the D + case

Nq " N q ~ [(fl + / 2 ) 2 + ~-(fl + f2 ) 2 O/S] . (12)

The parameters S and O measure the absolute squares of the color singlet and color octet matrix elements, as defined in eqs. (9) and (10), but with the factor 2V'2~ GF cos20 c and the coupling combinations fl

1 1 + ~f2, ~fl + f2, 2f2 and 2f l removed. S and O can be considered analogous to the corresponding quantities of refs. [11 ] and [13], where a random distribution of values of the quark color indices led to O/S = 2/9. In the present rate calculation one sums independently over all color indices of the squared matrix element. It is then not difficult to show that there is no singlet- octet interference, and that also for this c-decay case one finds O/S = 2/9 - not unexpectedly because the independent sum over colors corresponds to the sta- tistical average in the annihilation case, where soft gluons were included. Then, the presence of gluons in the c-decay case is not likely to change the O/S value. One may however introduce a scaled O/S'ratio:

O/S = ~x.2 (13)

The point of view of ref. [1] corresponds to x = 0, and with (11) inserted into (7) one reproduces exactly the

result obtained there, which was however also incon- sistently assumed to hold for D + decays. With x = 1 in (1 l) and (12), inserted into (7), non-leptonic rates are obtained which are exactly twice those for D O and D + of ref. [8]. Here lies the principal difference between ref. [8] and the present work; with a consistent treatment of the coherence properties one necessarily obtains this factor, which appeared in ref. [ 1 ]. For D O it means that the non-leptonic rate from c-quark decay is doubled even in the case of no short-distance renor- realization of c20 and c84 [13]. While the ratio of non- leptonic rates is not changed compared to ref. [8], the individual total rates and their ratio are - and deci- sively so, since the c-decay rate can then account for a larger value of the life-time ratio.

From eqs. (7), (11), (12) and the relations (5) and (6) between the coefficients, one finds for the D O and D + non-leptonic and total widths from c-decay

rd(D°)/rnl(D +1 = + ~ C3o , (14)

Ftdot(D0)/Ftot(D +) = 1+ (15) _ 4 + 2x +~c20 2 0 ] ,

with cos20c = 1 in eq. (15). It is immediately seen that, irrespective of the value of x, a 20-plet strong interaction enhancement of the weak interaction is a pre- requisite for having Fnl(D 0) > Pnl(D+), since O/S is non-negative *1. Withx = 1 and c20 = 2.15, our ratio of total rates is larger than that of ref. [8] since we find 0.36 for the coefficient of c230, instead of 0.29. With x = 0 and c20 = 2.15 one would get the values 1.74 and 1.41 for the non-leptonic and total rate ratios. Using the "canonical" value x = 1, one finds instead

Vndl(D0)/Fnl(D +) ~ 3 , (16)

so octet inclusion is seen to be necessary to obtain large life-time ratios in the c-quark decay model. Intro- ducing now the %30% effect in the D O total rate due to cc~ annihilation, we have

2.3 % Ptot(D°)/Ytot(D+ ) % 3.0 . (17)

The lower limit based on ref. [8] would instead be 1.9. Thus, should the importance of annihilation be less than estimated, the present experimental value would

#1 These ratios differ from unity even if C2o = 1.

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be less off target with our treatment of c-quark decay. As an additional basis for checks of the model, we

derive the D + life-time from eqs. (7) and (12),

1 3 T(D +) ~ CO 1 9C20/(1 + gX + gC20 )

- 1 3 ( 1"5 GeV) 5 C20 1 3 s ( 1 8 ) = 8 . 1 X 10 ~ / 1 + i x + g c 2 0

and the semi-leptonic branching ratio

BR(D + - e + v e X ) ~ ] 1 + ----3c20 !" (19)

These two quantities are seen to depend less upon the precise amount of octet transition involved, but still crucially upon the amount of 84-plet renormalization and 20-plet cancellation. The branching ratio obtained from eq. (19) is 18%, compared to 22% with no octet inclusion. The life-time given by (18) is 7.5 X 10 -13 X (1.5 GeV/mc)5 s, within experimental limits [7] for the D + life-time if m c = 1.5 GeV is used. (The result of ref. [8] corresponds to 1.1 X 10 -12 s, giving less good agreement though still within experimental limits.) The predicted life-time does not differ very much from what would be obtained directly from eq. (7) without refine- ments due to coherence and renormalization; the non- leptonic rate is 24% larger than the corresponding part of (7). The cancellation of the large coefficient c20 because of the assumption of full coherence did not lead to drastic changes of the absolute rate. We conclude that the remaining small discrepancy with experiment may be less due to the specific set of assumptions about coherence, than to the arbitrary simulation of bound- state effects and final-state effective masses implicit in our choice o f the effective c-quark mass in eq. (18). One notes that the predictions of eqs. (8), (14), (15) and (16) are independent of the choice of m c.

In summary: The present experimental value for z (D+) / r (D 0) is compatible with predictions from a c- quark decay model, eventually with small corrections due to cq annihilation. It is shown how a consistent t reatment of the assumptions about the coherence properties, as well as the inclusion of color-octet decays, lead to a higher value for the ratio of pure c-decay rates than found previously, and also to a somewhat better agreement with the measured absolute rate for D + decay. For D + the life-time prediction was shown not to

depend strongly on the assumption of full coherence. When the experimental ratio (2) converges towards some value, it may serve to determine the amount of annihilation involved in D O and F + decays; on the basis of present data, however, it seems to be on the 30% level. The formalism with explicit color singlet and color octet transitions provides a particularly suitable basis for analyses of the quark decay as well as the annihilation contributions.

It is a pleasure to thank G. Altarelli, I.I.Y. Bigi, H. Fritzsch, M. Gronau and P. Minkowski for discussions.

References

[ 1 ] J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B100 (1975) 313.

[2] D. Fakirov and B. Stech, Nucl. Phys. B133 (1978) 315. [3] N. Cabbibo and L. Maiani, Phys. Lett. 73B (1978) 418. [4] M.K. Galliard and B.W. Lee, Phys. Rev. Lett. 33 (1974)

108. [5] G. AltareUi and L. Maiani, Phys. Lett. 52B (1974) 351. [6] J. Kixkby, Intern. Syrup. on Lepton and photon inter-

actions at high energies (FNAL, Batavia, IL, 1979); V. Liith, Intern. Symp. on Lepton and photon interactions at high energies (FNAL, Batavia, IL, 1979).

[7] L. Fo~, in: Proc. 1981 Intern. Syrup. on Lepton and photon interactions at high energies (Bonn, 1981), to be published.

[8] B. Guberina, S. Nussinov, R.D. Peccei and R. Rtickl, Phy! Lett. 89B (1979) 111 ; see ref. [13] for further references

[9] H. Fritzsch and P. Minkowski, Phys. Lett. 90B (1980) 455. [10] M. Bander, D. Silvermann and A. Soni, Phys. Rev. Lett.

44 (1979) 7,962 (E). [11 ] H. Fritzsch and P. Minkowski, Nucl. Phys. B171 (1980)

413. [12] W. Bernreuther, O. Nachtmann and B. Stech, Z. Phys. C4

(1980) 257. [ 13] J. Finjord, Nucl. Phys. BI 81 (1981) 74; erratum/addendu:

to be published. [14] H. Krasemann, Phys. Lett. 96B (1980) 397. [15] Y. Igaxashi, L. Kitakado and M. Kuroda, Phys. Lett. 93B

(1980) 125. [ 16] Y. Igarashi, M. Kuroda and k. Kitakado, Bielefeld preprfi

BI-TP 80/06 (1980). [17] M. Gronau and D,G. Sutherland, Nucl. Phys. B183 (198

367. [18] Q. Ho-Kim and H.C. Lee, Phys. Lett. 100B (1981) 420. [19] M.A. Shifman, in: Proc. 1981 Intern. Syrup. on Lepton

and photon interactions at high energies (Bonn, 1981), be published.

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Decays of charmed mesons via color-octet decays of the c-quark