Semileptonic decays of charmed particles

Volume 83B, number 3,4 PHYSICS LETTERS 21 May 1979

SEMILEPTONIC DECAYS OF CHARMED PARTICLES

M.B. GAVELA Laboratoire de Physique Thdorique et Hautes Energies 1, Orsay, France

Received 2 February 1979

Semileptonic rates of ground state charmed particles are calculated. A quark model is used, including an accurate treatment of SU(4) breaking effects. Semileptonic lifetimes come out relatively large; rs(D°) = 6 × l0 -12 s, leading to an estima- tion o f the total lifetime r (D °) ~ 1.2 × 10-12s.

Semileptonic branching ratio of D decays has been measured [1]. Controversial indications on charmed particle lifetimes have also been given [2,3]. This opens the way to a determination of absolute semileptonic rates and therefore to a test of quark models. At the same time, the controversies about the lifetime em- phasize the need for quantitiative estimates.

Up to now, the main trends have been to use either i) quark parton models for inclusive rates [ 4 - 8 ] or ii) quark SU(4) symmetry arguments plus suitably meson dominated form factors [ 9 - 1 4 ] . Complete quark model calculations have not retained much at- tention although the power of quark models, imple- mented by realistic bound state wave functions, has been very often exemplified in classical hadron decays. They can specially handle SU(4) breaking effects in a realistic manner. A previous calculation of the process vN-+/~C [15] along the line of the Orsay group quark model of neutrino production processes [ 16], show- ed significant effects coming from the m c - rn quark mass difference. Notable discrepancies appeared with respect to other calculations of the same process [17].

Although we do not want to overlook the large un- certainties which are necessarily present in the extra- polation of our semi-relativistic treatment to such large mass differences, we think it is worth calculating the decays of charmed baryons and mesons along the same line. We start from the decay width formula for the

1 Laboratoire associ~ au Centre National de la Recherche Scientifique.

process i -~ fev (lepton mass is neglected) (~ = rn i + mf, A = m i -- mf, s = ( p f - - p i ) 2)

p _ ( o 2 / 2 ) cos20 1 mf (1)

(2n)3 3 m i

A2

× f , ,/UZs,/Z-s(lIs 12 + JIR 12 + 12) 0

where the form factors f (a) are defined and calculated exactly as in ref. [15]. The general formalism being extensively discussed in ref. [16] and the specific SU(4) breaking effects being discussed in ref. [15], we refer the reader to these papers and we limit our- selves to a few comments on specific features of the decay:

i) The main process is now Cabbibo conserving and leads to strange quarks c -~ s~v. Therefore, in the quark form factor, the dominance D* D** mesons should be replaced in that case by F* F** dominance.

ii) We need the wave function of mesons in addi- tion to baryons. It writes

xp ~ exp ( - R 2p2/4), (2)

where R 2 is fixed by the Regge slope of classical mesons t o R 2 = 8 GeV 2 [,18] and for charmed mesons R2c = (m + mc/2mc)l /~R 2 (see ref. [15] ).

iii) Especially for meson decays, SU(3) and SU(6) breakings in the hadron spectrum may have important consequences for the estimate of the rate. We account for SU(3) breaking by taking m s 4= m u = m d. On the other hand, it is very difficult to account consistently

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for the SU(6) breaking. Therefore, we keep m K = inK* = m S + m u in the quark model hadronic matrix ele- ment, except in the expression of the transfer q; but we treat exactly the leptonic tensor (this is the usual prescription of the quark model) and the phase space.

iv) In the decay process q2 = s ~ (pf - pi) 2 > 0. Therefore the overlap 1 > I > I (q = 0) has a less marked effect than in production where I < I(q 2 = 0). So for example, in D -~ K£v, the form factor writes

(KI V u ID) = (pD + pK)uf+(q2), (3)

_v q 2 1 f+(q2) _ I ~ - [fS(q2)l -

1 -q2/m2*

X (1 - q ° 4 1 - /32X) i ( q 2 ) , (4)

the Lorentz contraction factor being

1 - - /32 = 1 , ( 5 ) 1 + q2/(2m + m s + m c ) 2

q2 = ~ 2 ( A 2 _ q2)/4mDm K , q0 = X//~q2 + q 2 , (6)

m /1 il+ ) X=mK + m ~ ~--~- ~-~ R2 + R 2 (7)

and the overlap integral writes here

( 2 R R c ~ 3/2

i(q2) = (~K IqSD) = (1 --/32)1/2 \ R 2 ~ R 2 ] (8 )

Xexp - ~ - ( 1 -/32 ) m + m s + m c (R 2 + R 2)

(See fig. 1). The results are given in tables 1,2. Taking a branch-

ing ratio F ( D - , Xev)/r(D) ~ 10% [1], we are led to a typical lifetime:

r D ' - 1 . 2 X 10 - 1 / 2 s ,

w h i c h is h ighe r t h a n t h e usua l ly q u o t e d 10 - 1 3 s [2]

b u t c lo se r to t he m o s t p r o b a b l e l o w e r b o u n d o f ref .

[ 3 ] : 10 - 1 2 s.

The transition to excited states (L 4: 0) are found small, as exemplified by C~ ~ A(1405)£v which is 10% of the main decay mode.

One should also notice the qualitative fact that

r ( c ~ ~ Xg_v) > r ( D -~ X~v).

~ . 0 ~ 50 425

"".... ........... ...~.. 075

05~ ~ ....

02, 92o

~7~GEV2) 2 3 -~ -1 0 1'

Fig. 1. Effects of spatial overlap and quark form factors in D --, K~ue. - - : Overlap integral I(q 2) . . . . : Product of I(q 2) with the quark form factor I(q 2) X 1/(1 - q2/m~)*).

Table 1 Ground state charmed meson decays. When calculating widths I', Q+ stands for e + or ~+. In the semileptonic lifetimes we sum up e + and ~+ decay channels. Constituent quark masses have been taken: m u = m d = 0.385, m s = 0.507, m c = 1.6.

F (GeV) F (GeV) Partial semileptonic lifetime (s)

K-

Do ~ K * - /l-- p -

r :o

L: o

,7 q~

F + ~ r/' K o K*o

~+ v~ 4.43 X 10 -14 ~* vQ 8.31 X 10 - i s ~+v~ 1.44 × 10 -15 ~6.01 × 10 -12

~+ v~ 6.37 × 10 -16

~÷ v~ 4.47 X 10 -14 ~÷ vQ 8.51 X 10 - l s ~+ v~ 7.15 X 10 -16 ~+ v12 4.45 X 10 -16 ~5 .96 X 10 -12 ~÷v£ 1.36 X 10 -16 ~+ v~ 3.26 X 10 -16 ~+v~ 3.11 X 10 -16

~+ v~ 5.38 X 10 -14 ~+ v~ 1.08 X 10 -14 ~+ v~ 5.55 X 10 -15 ~4 .50 X 10 -12 ~+v~ 2.14 X 10 -15 ~+v~ 8.13 X 10 -16

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Table 2 Ground state charmed baryon decays. When calculating widths I', ~÷ stands for e ÷ or i~ +. In the semileptonic lifetimes we sum up e÷and la ÷ decay channels. Constituent quark masses have been taken: m u = m d = 0.313, m s = 0.5, m c = 1.6.

F(GeV) Partial S.L. lifetime P(GeV) Partial S.L. lifetime (s) (s)

+ { A ~+ v~ 9.84 X 10 -14 [ ~2 ~+ v~ 3.20 X 10 - la Co "-* n 2+v~ 1.21 X 10 -14 ~2.97 X 10 -12 T O --* X*- 2*v~ 4.76 X 10 -15 ~1.00 X 10 -12

t --'- ~÷v~ 3.19 X 10 -15 {1;* ~+v~ 4.69 X 10 -14 Co "-* X*- ~+v 2 4.57 X 10 -14 ~3.55 X 10-12 { A ° ~+vi~ 9.88 X 10 -15

t1;o ~+v2 4.75 X 10 -14 ) SO 2+v2 9.83 X 10 -15 + 1;,0 2+v2 4.61 X 10 -14 ~C ° ~+v 2 1.63 X 10 -15 ~1.42X 10 -11

C 1 ~ / ~ 0 2÷v2 8.70X 10 -15 ~ 3 . 1 7 X 10 -12 X d --* |(cds)*~+vQ 1.32× 10 -15

2÷v~ 1.58 X 10 -15 I,C~° Q+v~ 4.59 X 10 -16 ~n I A 2 + v~ 9.88 X 10 -15

X 2 ÷ v~ 4.79 X 10 -14 S + C~÷ I; +* 2*v2 4.66 X 10 -14 2+v2 9.83 X 10 -15

~]zX + 2÷u~ 4.35 X 10 -15 ~3.22 X 10 -12 C 1 2+v2 1.37 X 10 -15 Q+v~ 3.19X 10 -15 Xu -'* |(cus)*Q÷v2 1.32X 10 -15 ~1.42X 10 -11 kp

I c ~ I~+v~ 5.78 X 10 -16 { ~ - ~+v~ 1.81X 10 -13 [C~ + ~+v 2 2 . 3 0 X 10 -16

A° ~ 2 + vQ 1.32 X 10 -14 ~1.69 X 10 -12 T O ~+ v~ 2.64 X 10 -14

X 2+v~ 1.85 X 10 -13 A+-* 1;o 2+v~ 6.70 X 10 -15 ~1.69 X 10 -12 (css)* 2+v~ 7.08 X 10 -15

X s ~ {S 0 2+v2 2.01 X 10 -15 ~8.78 X 10 -12

~*- Q+vl2 1.17 X 10-13 [ A ° 2+u~ 1.63 X 10 -15 so --* I X " ~+vQ 3.06 X 10 -14 (cds)* 2+v2 3.99 X 10 -16

1;*- ~+v 2 1.09 X 10 -14 ~2.82 X 10 -12 ( X s ~+v 2 9.08 X 10 -13 t Z- £+ v~ 1.58 X 10 -15 J Xd ~+vI~ 1.02 X 10 -13

(ccc) ~ ~4.67 X 10 -15 1.18 X 10 -13 ~ (ccs)* 2 + vQ 2.96 X 10 -14

~,(ccd)* 2 + v~ -N 0 Q+ v~ X ° ~+v 2 3.12 X 10 -14 1.61 X 10 -14

S+--* X *° 2+v~ 5.57 X 10 - i s ~2.11 X 10 -12

1;o Q+v2 7.99 X 10 -16 A ~+v!~ 7.16 X 10 -16

Comparison with o ther estimates. We first consider

inclusive predict ions in quark par ton models , the crudest es t imate P = (o2/1921r 3) (me)5 = 2.4 X 10 -13

GeV, lies well above our predict ions. However , mass

correct ions account ing for m s 4 : 0 have been in t roduced

by Mathur and Rizzo [7] , Pham and Nabavi [8] ,

Cabbibo and Maiani [6] and give a strong reduct ion,

down to ~ 1 0 -13 GeV, close to our model , which at-

t empts to treat masses realistically. On the o ther hand Pham and Nabavi have proposed to take into account

the s tructure funct ions o f the decaying charmed par-

ticle; they are able then to different ia te be tween the C~" and D, and obtain F ( D -* X~v) = 0.79 × 10 -13

GeV, r(C~- ~ X£v) = 0.53 X 10 -13 GeV. This could

part ly parallel our t r ea tment wi th bound state wave

functions. One notices that a cut m x > mK has no t

been included by Pham and Nabavi and it would lead

to P ( D -+ X~v) = 0.47 X 10 -13 GeV very close to our

0.5 × 10 -13 GeV. But there should be also a correc-

t ion to their formula x = q2 /2M v coming f rom the

quark mass difference m c - m s (cf. discussion in ref.

[ 15]). Final ly, gluon correct ions as discussed by Cabbibo and Maiani, and Suzuki [6,19] lead to an ad- dit ional reduct ion; here, a compar ison will have to be

made with our effect ive quark couplings as discussed in ref. [15 ,16] .

Exclusive predict ions have been given for D -~ Key ,

K*ev , based mainly on SU(4) , i.e. wi thou t considera- t ion o f the spatial wave funct ions [ 9 - 1 3 ] . For D

K e y , the results are close to 10 -13 GeV. On the con-

trary the D -~ K* ev predict ions are much more dispers-

ed, ranging f rom 0.268 to 4.18 X 10 -13 GeV. Our

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F(D -+ Key) is smaller by a factor 2 which can be at- tr ibuted to the overlap function, not considered in these approaches. For P(D -+ K*ev), our very small value is compatible only with All and Yang, while the

other models state that F (D -+ K*ev) > F (D ~ Key) (except in ref. [12]) . The smallness of our result is

partly due to the assumption that (gh)q = l/X/2-, which is inspired by the nucleon axial current [16] , while other models are implicitly setting (gA)q = 1. A large value for F (D -~ K*ev) /F(D -~ Key) has often been considered desirable in order to explain the elec- tronic decay spectrum. But Suzuki [19] has recently suggested that the shift of the maximum towards low E e could be due to QCD effects. There remains however to understand how QCD perturbative effects are to be combined with such quark model calculations.

Finally, for baryons, there is the work of Buras [14] which yields very large widths ~ 5 X 10 -13 GeV for

F(C -~ X~v) as read from his fig. 2 (at MC+ = 2.26). This very large number is understandable ~ecause of the neglect of the overlap function, and the dipole-like form factors (in contrast with simple pole used for mesons). On the opposite, Bucella et al. [20] obtain very small widths through strong breaking of SU(4) for consti tuent quarks matrix elements. Once again we are more in agreement with the corrected parton models

[151.

I owe very much to discussions with the other mem- bers of the Orsay quark group: A. Le Yaouanc, L. Oliver, O. P~ne, J.C. Raynal and S. Sood. I am specially indebted to A. Le Yaouanc for his help with some calculations and friendly encouragement.

References

[1 ] J. Peruzzi and M. Piccolo, Frascati Report LNF 78/12(P). [2] G. Bertrand, in: Leptons and multileptons, Proc. XIIth

Rencontre de Moriond, ed. J. Tran Thanh Van (les Arcs, 1977).

[3] D.J. Crennell et al., Phys. Lett. 78B (1978) 171. [4] M.K. Gaillard, B.W. Lee and J. Rosner, Rev. Mod. Phys.

47 (1975) 277. [5] J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys.

B100 (1975) 313. [6] G. Altarelli, N. Cabbibo and L. Maiani, Phys. Rev. Lett.

35 (1975) 635; N. Cabbibo and L. Maiani, Phys. Lett. 73B (1978) 418; 79B (1978) 109.

[7] V.S. Mathur and T. Rizzo, Phys. Rev. D16 (1977) 3343. [8] X.Y. Pham and R.P. Nabavi, Phys. Rev. D18 (1978) 220. [9] A. Ali and T.C. Yang, Phys. Lett. 65B (1976) 275.

[10] F. Bletzacker, H.T. Nieh and A. Soni, Phys. Rev. D16 (1977) 732.

[11] G.L. Kane, Phys. Lett. 70B (1977) 272. [12] D. Fakirov and B. Stech, Nucl. Phys. B133 (1978) 315. [13] X.Y. Pham and J.M. Richard, Nucl. Phys. B138 (1978)

453. [14] A.J. Buras, Nucl. Phys. B109 (1976) 373. [15] A. Amer, M.B. Gavela, A. Le Yaouanc and L. Oliver,

Phys. Lett. 81B (1979)48. [16] P. Andreadis et al., Ann. Phys. 88 (1974) 242. [17] J. Finjord and F. Ravndal, Phys. Lett. 58B (1975) 61 ;

R.E. Shrock and B.W. Lee, Phys. Rev. D13 (1976) 2539; C. Avilez, T. Kobayashi and J.C. K6rner, Phys. Rev. D17 (1978) 7O9.

[18] A. Le Yaouanc, L. Oliver, O. P~ne and J.C. Raynal, Orsay report LPTHE 72/6. This value is used throughout all our calculations.

[19] M. Suzuki, Berkeley report LBL 7948 (1978). [20] F. Buccella, A. Sciarrino and P. Sorba, Phys. Rev. D18

(1978) 814.

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Semileptonic decays of charmed particles