Semileptonic decays of heavy mesons

Nuclear Physics B (Proc. Suppl.) 13 (1990) 255-260 North-Holland

255

SEMILEPTONIC DECAYS OF HEAVY MESONS Manfred Wiebel I n s t i t u t fiir Physik, UniversitAt Dortmund, Postfach 500500, 4600 Dortmund 50, West-Germany A b s t r a c t . We study semiJeptonic decays of h e a v y mesons into hadrons and ev e or 9e. p~irs r e s p e c t i v e l y , with t h e emphasis on e x c l u s i v e decays. We compare the predictions of v a r i o u s ~ e l s a n d discuss the t h e o r e t i c a l uncertainties.

2 1. I n t r o d u c t i o n

Semileptonic decays of hadrons h a v e p l a y e d and still p l a y an important role for our understanding of the i n t e r p l a y b e t w e e n weak and strong interactions: They ar~ e s s e n t i a l for t e s t i n g the standard model and determining its fundamental parameters. They also provide v a l u a b l e information on the bound s t a t e structure of hadrons not y e t calculable from QCD. Semileptonic decays of h e a v y mesons h a v e t h e r e f o r e b e e n e x t e n s i v e l y studied e x p e r i m e n t a l l y 1 - 1 2 as well as t h e o r e t i c a l l y 13-33 2. I n c l u s i v e

Decays

It is t h e o r e t i c a l l y simplest to ~t~rt the a n a l y s i s of semileptonic decays a t the quark l e v e l where the h e a v y quark decays while the light s p e c t a t o r quark goes along unaffected. As long as one is not i n t e r e s t e d in s e p e r a t i n g into exclusive final s t a t e s one may try a free quark calculation where the s p e c t a t o r quark is i r r e l e v a n t . The total r a t e s as well a s the shapes of the lepton spectra depend on unknown quark masses which occur in the amplitude and - most important - determine the allowed p h a s e space. In addition the simple quark model has to be modified by taking into account r a d i a t i v e corrections due to the emission of v i r t u a l and r e a l gluons. These QCD corrections h a v e b e e n calculated 34 and one obtains for the semileptonic width Fs/:

0920-5632/90/$03.50 © Elsevier Science Publishers B.V. (North-tlolland)

rsl(Q~qlvl )

-

5

GFmQ fqQ |, .vq Q - - 192 m3

12

<1)

where fqQ is the product of p h a s e space and QCD correction factors and VqQ is the K o b a y a s h i Maskawa matrix element. The predictions for the semileptonic width are also modified considerably by considering bound s t a t e e f f e c t s for the initial meson. The classical model of this t y p e is the nonrelativistic quark model of Altaretli and co-workers 13. The light spectator quark has a definite mass rasp in this model and the decaying heavy quark is off-shell because of e n e r g y - m o m e n t u m conservation with its invariant mass g i v e n by W"? = m~l + m~p - 2mM -V/~'z + m~p mM being the mass of the decaying meson and the momentum of the spectator quark. A Oaussian distribution for I ~ I with an a d j u s t a b l e width PF has been assumed in this model. The lepton spectrum and the total width then result by folding this distribution with the d e c a y spectrum of the h e a v y b quark with e f f e c t i v e mass W. The allowed phase space thus depends on the mass of the s p e c t a t o r quark and the 'Fermi' momentum PF' as well as the mass of the final quark. A model which is complementary to the Altarelli model in many r e s p e c t s has been recently proposed by Bareiss and Paschos to describe the d e c a y s of B mesons 17. They visualize the decay as taking place in an infinite

256

M. Wirbel / Semileptonic decays of heavy mesons

momentum frame where the B meson moves with large momentum. The decay of the meson is the incoherent sum of the d e c a y s of free b quarks carrying a fraction z of the B meson momentum Pb~ = z PBp" The decay of the B meson is t h e n obtained by calculating the d e c a y of a q u a s i - f r e e b quark folded with the probability of finding a b q u ~ k carrying a fraction z of the longitudinal

Table 1. Correction factors f_~ and rub defined in eq. (1) (from 35, table 7).~UThe upper v~lue~ are for ~s = O, the lower v a l u e s include n e x t - t o - l e a d i n g - l o g corrections. mode

b-+cl

current masses

function in

the

fcb = 0.56

0.42

f c b = 0.48

0.36

rub = 1.00

0.97

rub = 0.87

0.84

vl

momentum of the B meson. The fragmentation function for a b quark to fragment into a B meson is t a k e n as the distribution infinite momentum frame.

constituent masses

b-~ul

vI

Many other models have been suggested for the description of inclusive semleptonic decays of h e a v y mesons all of which g i v e a good fit to the experimentally measured spectra and rates. The uncertainties of the theoretical calculations may be summarized by comparing the results obtained

models and the v a r i a t i o n of p a r a m e t e r s within the models.

for fqQ defined in eq. (1). Typical v a l u e s for fcb and rub are g i v e n in table 1 for the free quark model for current quark masses (mb=4.8 GeV, mc=1.35 GeV, mu=O.O06 GeV) and constituent quark masses (mb=5.2 GeV, mc=l.8 GeV, mu=0.34 GeV), respectively. Bareiss and Paschos give v a l u e s for fc which range from f c b = 0.24

f o r m c = 1.3 OeV

f c b = 0.15

f o r m c = 1.8OeV

to with mb = mB in eq. (1). QCD corrections are included. (From table 4 of ref. 17 with the central value of the parameter e which determines the fragmentation function.) Typical results for feb using the model of Altarelli and co-workers are the following: fcb = 0.23 for msp= 0.15 GeV, PF = 0.30 GeV, and feb = 0.29 for rasp= 0.15 OeV, PF = 0.15 OeVj including QCD corrections. We note that those models which include bound state corrections predict v a l u e s for fcb which are considerably smaller than those obtained in the free quark

3. E x c l u s i v e Semileptonic B Decays The inclusive approach described above is probably a p p r o p r i a t e if the final hadronic state consists of a +continuum' of hadrons. The endpoint region of the lepton spectrum must, however~ be dominated by l o w - m a s s hadrons from kinematical reasons, since the maximum v a l u e of the lepton energy is g i v e n by" 2 2 E ~ a x _ mM-mx 2 mM with the hadronic mass mX determined by: m~=

(PM -

q)2.

PM is the momentum of the decaying meson and q is the momentum transfer. Near the endpoint m~X 2 is g i v e n by the mass of discrete s t a t e s , like m~, 2 2 m2p,... for semileptonic B decays, roD.,.., or mnj whereas it is continuous in the free quark decay models. In addition it has been found experimentally thereby confirming the theoretical picture which has d e v e l o p e d through

model. The corresponding v a l u e s for [Vcb I will therefore b~ larger with, however, large

the last y e a r s - that the b ~ c l v I and c ~ s l p l transitions are dominated by few exclusive

uncertainties

channels. The inclusive t r e a t m e n t of semileptonic

due

to

the

choice

of d i f f e r e n t

M. Wirbel / Semileptonic decays of heavy mesons b~c and c-~s transitions is for these reasons questionable 31 and the theoretical study of exclusive decay modes is therefore very important. In an exclusive treatment the decay distributions are given in terms of matrix elements of the weak currents between initialand final meson states. We will consider the transitions involving a pseudoscaiar (X=P) or vectormeson (X=V) in the following. From Lurentz invariance one finds' the decomposition cf the hadronic matrix elements in terms of unknown formfactors:

~ P [ J p l 0 l [ M ~ = tp~ Fl(q 21 + with

q2

q~ F0lq2)

-

tpg = (PB + PP)g

q2

qg

(21

and mB + mV +itvg

e.q + i ~

q g 2 m VA0(q 2)

with ##

e.q t v p = leg - - ~ - qp) (mB + mV) A 11q2) .

q2 - ( m 2 - m~ (PM + PV)~t - q~t) e.q *

mB+ mv (PB

4-

PV)g A2 (q2) 13)

eg is the polarisation vector of the final meson V. The formfactor decomposition has been written in such a way t h a t qP tpg = qg t v g = 0

(4)

Various approaches h a v e been suggested to estimate the invariant formfactors 17-31 and it is impossible to discuss all of them in this context. Instead I will concentrate on the results o~/.~h~ed by two models which use quite different assumptions to calculate the r a t e s and spectra: i) The BSW model 20 as~ames nearest pole dominance for the qZ-dependence of the

257

formfactors: Fl(q 21 =

hI 2

etc.

(5)

1 - q21mpole The unknown constants h i - i.e. the formfactors a t q2 _ 0 - are estimated by describing the mesons as relativistic bound states of a q u a r k - a n t i q u a r k pair in the infinite -momentum limit. The constants h i are then given by o v e r l a p integrals of the wave functions of the initial and fina! meson. A quite successful description of D and B meson decay d a t a has been possible 20,30s36,37 This method has the advantage of using a fully relativistic formalism, but there are ~lso several difficulties connected with this model: It isj for example, difficult to define e i g e n s t a t e s of JP using infinite-momentum-frame wavefunctions. ii) The GISW model 21,31 uses the n o n - r e l a t i v i s t i c quark potential method to make a correspondence between the Lorentz-invariant fornffactors defined in eqs. (10) to (14) and those appearing in a quark-model calculation ('mock-meson method'). These formfactors are identified near 2 = ( m M _ mx)2. zero recoil, i.e. a t maximal qmax Variational solutions of the $chri~dinger equation with the usual Coulomb plus linear potential have been chosen as the wave functions of the initial and final mesons. In extrapolating away from zero-recoil the q2 dependence of the formfactors is not calculable accurately and terms of order 2 _ q~)Z h a v e been dropped. This procedure (qmz results in an exponential q2 dependence of the formfactors: Fl(q2 ) = Fllq2max) e x p ( _ ~ l q 2 a x _ q2))

etc. (6)

with, for example, ~.---0.12 GeV- 2 for B - ~ l v and X~0.03 GeV - 2 for B-~DI~. If Y.q2~'l for the whole physical q2 range we can write 2 2 Fllq2) _ F l l q 2 a x ) 1 -qmax/mpole~

(7)

1 - q2/m~ole with m 2 l e ffi 1/~. We therefore expect rough agreement between the two models for semileptonic B-~D(D*) and D-~K(K*) decays and

258

M. W~rbd/Semileptonic

decaysof heavy mesons

Table 2. Semileptonic decay r@tes of .D mesons. All rates are given in units of I0 ~u sec -~. GISW

D°-~K -

8.3

8.4

9.0

D+->~

9.5

9.1

4.1 +- 0.7 ± 0.5 11 ÷ 0.7 + 0.2 10

0.~

V°~p -

0.7

0.5

l

.

,

I

'

i

.

-+ 1.1

0.9

,

I

'

'

5ISW

BSW

0.7

I

Experiment

Decay mode

D°-~ -

=

-+ 1.2 11

0.3

°" D

-

//

o.s

D° -~ X/+

17.8 -+ 3.9 9

D+-~ X/+

15.6

-+ 1 . 9 9

\

0.1

(inclusive}

!

0.2

I

I

!

m

1

2

Et I GeY

Table 3. $enfileptonic decay rates of Bmesons. All rates are given in units of I0 I0 sec -1. IVcb 0.05 has been used.

|--

Decay mode

BSW

GISW

B-~D

2.0

2.8

B-*D*

5.5

6.2

Fig. 1 Energy s p e c t r a of the cha~l~ed lepton in semileptonic B ~ D l - v and B -~ D 1 u decays as predicted by the BSW and GISW models (in lhe B rest ~ m e ) . The s~ectra for B -~ D / - ~ a~e nearly identical.

Experiment •

Vub

•

•

!

•

|

a

|

I

I

I

I

|

I

= 2 -+1 8 5.8 -+1.0 -+1.6 4

Vub

I.•....~ BSW

p~ -.v /

.......

Lo

BSW t,

;

L¢,,

~o_,p+

6.5 IVubl 2

2.1

IVubl~

.%o

,

7

.-It-

...... ,,,,

0.1 -~ | - - X

9.6 + 0.8 7

(inclusive)

2

10

2O q21GeV2

large discrepencies for B - ~ and B-*O. The

two

models

described

above

differ

considerably in their assumptions and therefore give an impression of the theoretical uncertainties connected with the prediction of semileptonic rates and spectra. The theoretical

Fig. 2 Energy_ s p e c t r a of the c h a r g e d lepton in semileptonic 1:3 -* ~ l - ~ and B -~ pl v decays as predicted by the BSW and GISW models (in the B rest frame).

decays

whic:~

are

most

important

for

the

results for total semileptonic decay rates of D and

determination of the so far unknown Kobayashi-

B mesons are summarized in t-~bles 2 and 3 and compared with experimental d a t a ~s far as t h e y

Maskawa matrix element ]Vub [ . Another point of concern is the disagreement b e t w e e n theoretical

are available. It is evident l'rom these tables t h a t the theoretical predictions roughly agree with each other except for the B -~ nl-~ and B -~ pl-~

predictions t o t the D experimental results•

~ K*eu decay Unfortunately

and the

experimental results still disagree on the size of

M. Wirbel/Semileptonicdecaysof heavy mesons

3i

¢'i~,

~

lepton in semileptonic ~-~ D,Ds and ~ 4 ~ decays are shown in figures I and 2, respectively (in the B rest frame}. The BSW and GISW models not only predict different rates but also the form of the spectrum is quite different for the b -* ul-~ transitions. The upper limit on ]Vub/Vcb ~ obtained by this method therefore depends significantly on the theoretical model and it is very difficult to guess the theoretical error. A conservative estimate of the upper limit is

!

I

a 0 ~

% V~ P

259

""I

IVub/Vcb ~ £ 0.20

2,5

S

?,S

10

qZl6eV !

•

I

!

b

w

2

Itl

The q2 dependence of the b -~ cl-~ dec~y is shown in figure 3 where we have included the qZ dependence of the production of transversely and longitudinally polarized D* mesons. The polarization of the D can be determined by measuring the angular distribution of the strong decay D= -~ D~. The angular distribution in the D* rest frame is proportional to cosZB* for longitudinally polarized D* and sinZ8~ for transversely polarized D*, respectively. 0* is the angle between the n meson and the momentum direction of the D*. The total decay distribution can therefore be parametrized by

o

--

dUal(B, D=* D~)

I

~- ( l + ~B c°sZe')

(8)

dcosO =

where =B measures the ratio of longitudinal to transverse polarization:

/

f'

~. . . .

25

I

I

5

7.5

10

q2 / 5eV 2

=

~B=2

rs¢(~

*

Us/{B *

Fig. 3 qZ dependence of the semileptonic decays B-~ Dl p a n d B - * D ! v~in =hea) B ~ W a n d b ) G I S W model, respectively. Dt~raasvand DI~_ denote the contributio=~ of transversely and l~ngitndin~ily polarised D mesons.

the nonresonant D-~ (Kn)l-v decays, i.e. {K~) pairs not coming from D -* K*[-v -~ (Kn)I-v 9,10 Limits on [Vub/Vcb ~ have been obtained in the past by ~ study of the eudpoint region of the lepton ,,omentum spectrum 7j38 The theoretical predictions for the energy spectra of the char¢ed

Dl°ng)

D~ransv)

- 1

(9)

A first measurement of ~B has been performed by the ARGUS collaboration . Their result is

=B = 0.7 -* 0.9

(10)

from which o n e easily d e d u c e s

Plong/Ctransv = 0.85 ± 0.45

(11)

This finding agrees nicely with the theoretical expectation from the GISW 3 1 K6rner avd Schuler 27 and BSW 20 modeis~ respectively:

Flonglrtransv

= 0.94, 1,06 and 1.07.

260

~L Wirbel/Semilepton:.cdecays of heavy mesons

The polarization of the f i n a l vectormeson h a s also b e e n measured in semileptonic D d e c a y s where large longitudinal polarization h a s b e e n found 11:

rlong/Ftrunsv "- 2 . 4

+ 1.7 ± 0.2 0.9

w h e r e a s the predictions of t h e

20. M. Wirbel, B. Stech and M. Bauer, Z. Phys. C29 (1985) 637. 21. B. Grins~ein, M.B. Wise and N. Isgur, Phys. Rev. Lett. 5_6_6(1986) 298. 22. F. Sch~iberl a n d H. Pietschmann, Europhysics L e t t e r s 2 (1986) 583.

BSW and GISW

23. M. Shiflnan and M. Voloshin, ITEP-64 (1987).

models are: FlonglPtransv = l.Ojl.1. An ad hoc modification of the BSW model, where some formfactors h a v e been scaled, g a v e a g a i n good a g r e e m e n t b e t w e e n theory a n d e x p e r i m e n t for t h e semilep~onic D-~K* decays 30

24. 5. Nussinov a n d W. Wetzel, Phys. Rev. D36 {1987) 130.

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2.

S. Behrends et al., Phys. Rev. Lett. 59(1987).

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3.

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