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Nuclear Physics B (Proc. Suppl.) 13 (1990) 255-260 255 North-Holland SEMILEPTONIC DECAYS OF HEAVY MESONS Manfred Wiebel Institut fiir Physik, UniversitAt Dortmund, Postfach 500500, 4600 Dortmund 50, West-Germany Abstract. We study semiJeptonic decays of heavy mesons into hadrons and ev e or 9e. p~irs respectively, with the emphasis on exclusive decays. We compare the predictions of various~els and discuss the theoretical uncertainties. 1. Introduction Semileptonic decays of hadrons have played and still play an important role for our understanding of the interplay between weak and strong interactions: They ar~ essential for testing the standard model and determining its fundamental parameters. They also provide valuable information on the bound state structure of hadrons not yet calculable from QCD. Semileptonic decays of heavy mesons have therefore been extensively studied experimentally 1-12 as well as theoretically 13-33 2. Inclusive Decays It is theoretically simplest to ~t~rt the analysis of semileptonic decays at the quark level where the heavy quark decays while the light spectator quark goes along unaffected. As long as one is not interested in seperating into exclusive final states one may try a free quark calculation where the spectator quark is irrelevant. The total rates as well as the shapes of the lepton spectra depend on unknown quark masses which occur in the amplitude and - most important - determine the allowed phase space. In addition the simple quark model has to be modified by taking into account radiative corrections due to the emission of virtual and real gluons. These QCD corrections have been calculated 34 and one obtains for the semileptonic width Fs/: 2 5 GFmQ fqQ |v 12 rsl(Q~qlvl ) - - - - ,.qQ <1) 192 m 3 where fqQ is the product of phase space and QCD correction factors and VqQ is the Kobayashi- Maskawa matrix element. The predictions for the semileptonic width are also modified considerably by considering bound state effects for the initial meson. The classical model of this type 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 energy-momentum conservation with its invariant mass given 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 adjustable width PF has been assumed in this model. The lepton spectrum and the total width then result by folding this distribution with the decay spectrum of the heavy b quark with effective mass W. The allowed phase space thus depends on the mass of the spectator 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 respects has been recently proposed by Bareiss and Paschos to describe the decays of B mesons 17. They visualize the decay as taking place in an infinite 0920-5632/90/$03.50 © Elsevier Science Publishers B.V. (North-tlolland)

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Page 1: Semileptonic decays of heavy mesons

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

SEMILEPTONIC DECAYS OF HEAVY MESONS

Manfred Wiebel

Ins t i tu 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 spec t ive ly , with t h e emphasis on exclus ive decays. We compare the predictions of v a r i o u s ~ e l s and discuss the theore t ica l uncer ta int ies .

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

Semileptonic decays of hadrons have p layed

and st i l l p lay an important role for our

unders tanding of the in te rp lay be tween weak and

strong interact ions: They ar~ es sen t i a l for t es t ing

the s tandard model and determining i ts

fundamental parameters . They also provide

va luab le information on the bound s t a t e s tructure of

hadrons not ye t calculable from QCD. Semileptonic

decays of h e a v y mesons have therefore been

ex t ens ive ly s tudied exper imenta l ly 1 -12 as well as theore t ica l ly 13-33

2. I n c l u s i v e D e c a y s

It is theore t ica l ly s implest to ~t~rt the analys is

of semileptonic decays a t the quark leve l where

the h e a v y quark decays while the l ight spec ta tor

quark goes along unaffected. As long as one is

not i n t e r e s t ed in sepera t ing into exclusive f inal

s t a t e s one may try a f ree quark calculat ion where

the spec ta to r quark is i r r e levan t . The tota l r a tes

as well as the shapes of the lepton spect ra

depend on unknown quark masses which occur in

the ampli tude and - most important - determine

the al lowed phase space. In addi t ion the simple

quark model has to be modified by taking into

account r ad i a t i ve corrections due to the emission

of v i r tua l and rea l gluons. These QCD corrections

have been calcula ted 34 and one obtains for the

semileptonic width Fs/:

2 5 GFmQ fqQ | v 12

r s l (Q~q lv l ) - - - - , . q Q < 1 )

192 m 3

where fqQ is the product of phase space and QCD

correction fac tors and VqQ is the Kobayashi -

Maskawa matr ix element. The predict ions for the

semileptonic width are also modified considerably

by considering bound s ta te e f fec t s for the ini t ial

meson. The classical model of this type is the

nonrela t iv is t ic quark model of Al tare t l i and

co-workers 13. The light specta tor quark has a

def ini te mass rasp in this model and the decaying

heavy quark is o f f - she l l because of

energy-momentum conservation with i ts invar iant

mass g iven by

W"? = m~l + m~p - 2mM -V/~'z + m~p

m M being the mass of the decaying meson and

the momentum of the specta tor quark. A Oaussian

distribution for I ~ I with an ad jus tab le width PF has been assumed in this model. The lepton

spectrum and the total width then resul t by

folding this distr ibution with the decay spectrum

of the h e a v y b quark with e f fec t ive mass W. The

allowed phase space thus depends on the mass of

the spec ta tor quark and the 'Fermi' momentum PF'

as well as the mass of the final quark.

A model which is complementary to the

Altarel l i model in many respects has been

recent ly proposed by Bareiss and Paschos to

describe the decays of B mesons 17. They visual ize

the decay as taking place in an infinite

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

Page 2: Semileptonic decays of heavy mesons

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 decays of free b quarks

carrying a fraction z of the B meson momentum

Pb~ = z PBp" The decay of the B meson is then

obtained by calculating the decay of a quas i - f r ee

b quark folded with the probabil i ty of finding a b

q u ~ k carrying a fraction z of the longitudinal

momentum of the B meson. The fragmentat ion

function for a b quark to fragment into a B meson

is taken as the distribution function in the

infinite momentum frame.

Many other models have been suggested for the

description of inclusive semleptonic decays of

heavy mesons all of which give a good fit to the

experimentally measured spectra and rates. The

uncertaint ies of the theoret ical calculations may

be summarized by comparing the results obtained

for fqQ defined in eq. (1). Typical values for fcb

and rub are given 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 consti tuent

quark masses (mb=5.2 GeV, mc=l.8 GeV, mu=0.34

GeV), respectively.

Bareiss and Paschos give va lues for fc which range from

f c b = 0.24 f o r m c = 1.3 OeV

to

f c b = 0.15 f o r m c = 1.8OeV

with m b = m B 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 Altarell i 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 values for fcb which are considerably

smaller than those obtained in the free quark

model. The corresponding values for [Vcb I will

therefore b~ larger with, however, large

uncertaint ies due to the choice of d i f ferent

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 va lues include n e x t - t o - l e a d i n g - l o g corrections.

mode current const i tuent masses masses

fcb = 0.56 0.42 b - + c l v l

f c b = 0.48 0.36

rub = 1.00 0.97 b - ~ u l v I

rub = 0.87 0.84

models and the var ia t ion of parameters within the

models.

3. Exclus ive Semileptonic B Decays

The inclusive approach described above is

probably appropr ia te if the final hadronic state

consists of a +continuum' of hadrons. The endpoint

region of the lepton spectrum must, however~ be

dominated by low-mass hadrons from kinematical

reasons, since the maximum value of the lepton energy is g iven by"

2 2 E~ax_ mM-mx

2 m M

with the hadronic mass m X determined by:

m~= (PM - q)2.

PM is the momentum of the decaying meson and q

is the momentum transfer. Near the endpoint m 2 ~X is given by the mass of discrete s ta tes , like m~,

2 2 m 2 for semileptonic B decays, roD.,.., or mnj p,...

whereas i t is continuous in the free quark decay

models. In addit ion it has been found

experimental ly - thereby confirming the

theoret ical picture which has developed through

the last yea r s - that the b ~ c l v I and c ~ s l p l

t ransi t ions are dominated by few exclusive

channels. The inclusive t reatment of semileptonic

Page 3: Semileptonic decays of heavy mesons

M. Wirbel / Semileptonic decays of heavy mesons 257

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 initial and

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 + q2 q~ F0lq2)

w i t h -

tpg = (PB + PP)g q2 qg (21

and

m B + m V

e.q +itvg + i ~ qg2 m VA0(q 2)

with ##

. e . q tvp = leg - - ~ - qp) (m B + m V) A 11q2)

q2

- (m 2 - m~ (PM + PV)~t - q~t)

e . q * 4-

mB+ mv (PB PV)g A2 (q2) 13)

eg is the polarisat ion vector of the final meson V.

The formfactor decomposition has been wri t ten in

such a way tha t

qP tpg = qg tvg = 0 (4)

Various approaches have been suggested to est imate the invar iant formfactors 17-31 and i t is

impossible to discuss all of them in this context.

Instead I will concentrate on the resul ts o~/.~h~ed

by two models which use quite d i f ferent

assumptions to calculate the ra tes and spectra: i) The BSW model 20 as~ames neares t pole

dominance for the qZ-dependence of the

formfactors:

h I F l (q 21 = etc. (5)

2 1 - q21mpole

The unknown constants h i - i.e. the formfactors a t q2 _ 0 - are es t imated by describing the mesons

as re la t iv is t ic bound s ta tes of a quark-ant iquark

pair in the infinite -momentum limit. The

constants h i a re then given by over lap integrals of the wave functions of the ini t ial and fina!

meson. A qui te successful description of D and B meson decay da ta has been possible 20,30s36,37

This method has the advantage of using a fully

re la t ivis t ic formalism, but there are ~lso several

diff icult ies connected with this model: It isj for

example, diff icult to define e igens ta tes of JP

using inf ini te-momentum-frame wavefunctions.

ii) The GISW model 21,31 uses the non- re la t iv i s t ic

quark potent ia l method to make a correspondence

between the Lorentz- invar iant fornffactors

defined in eqs. (10) to (14) and those appearing

in a quark-model calculation ('mock-meson

method'). These formfactors are ident i f ied 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 l inear potent ia l have been chosen as the wave functions of the init ial

and final mesons. In extrapolat ing away from

zero-recoi l the q2 dependence of the formfactors

is not calculable accurately and terms of order 2 _ q~)Z (qmz have been dropped. This procedure

results in an exponential q2 dependence of the

formfactors:

Fl(q2 ) = Fllq2max) exp(_~lq2ax _ q2)) etc. (6)

with, for example, ~.---0.12 GeV -2 for B -~ lv and X~0.03 GeV -2 for B-~DI~. If Y.q2~'l for the whole physical q2 range we can write

2 2 Fl lq2) _ F l l q 2 ax ) 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

Page 4: Semileptonic decays of heavy mesons

258 M. W~rbd/Semileptonic decays of heavy mesons

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

Decay BSW GISW mode

D°-~K - 8.3 8.4

D+->~ 9.5 9.1

D ° - ~ - 0.7 0.~

V°~p - 0.7 0.5

Experiment

9 . 0 -+ 1.1 -+ 1 .2 11

4.1 +- 0.7 ± 0.5 11

÷ 0.7 + 0.2 10 0 . 9 0 . 3 -

D ° -~ X/+ 17.8 -+ 3.9 9

D + - ~ X / + 1 5 . 6 -+ 1 . 9 9

(inclusive}

Table 3. $enfileptonic decay rates of B mesons. All rates are given in units of I0 I0 sec -1. I Vcb |-- 0.05 has been used.

Decay BSW GISW Experiment mode

B-~D 2.0 2.8 = 2 -+ 1 8

B-*D* 5.5 6.2 5.8 -+ 1.0 -+ 1.6 4

Vub Vub

~o_,p+ 6.5 IVubl 2 2.1 IVubl ~ 7 .%o ,

-~ | - - X 9 . 6 + 0 . 8 7 (inclusive)

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

The two models described above differ

considerably in their assumptions and therefore

give an impression of the theoret ical

uncertainties connected with the prediction of

semileptonic rates and spectra. The theoretical

results for total semileptonic decay rates of D and

B mesons are summarized in t-~bles 2 and 3 and

compared with experimental da ta ~s far as they

are available. It is evident l'rom these tables tha t

the theoretical predictions roughly agree with

each other except for the B -~ nl-~ and B -~ pl-~

= I l . , I ' i . , I ' '

5ISW

°" D

o.s / / \

0.1 ! I I ! m

0.2 1 2

Et I GeY

Fig. 1 Energy spect ra 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.

L o

L ¢ , ,

. - I t -

• • • ! • | a | I I I I | I

I. BSW p -.v t • ....~ ~ / BSW , ....... ;

...... , , , ,

0.1

2 10 2O q 2 1 G e V 2

Fig. 2 Energy_ spectra of the charged 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

determination of the so far unknown Kobayashi-

Maskawa matrix element ]Vub [ . Another point of

concern is the disagreement between theoretical

predictions to t the D ~ K*eu decay and

experimental results• Unfortunately the

experimental resul ts still disagree on the size of

Page 5: Semileptonic decays of heavy mesons

M. Wirbel /Semileptonic decays of heavy mesons 259

¢' i~,

% V~

P

3i ~ I !

a

0 ~

""I

2,5 S ?,S 10 qZl6eV

w 2

Itl

o

-- I

! • I !

/ f' ~ . . . . I I

2 5 5 7.5 10

q2 / 5eV 2

b

Fig. 3 qZ dependence of the semileptonic decays

B-~ Dl pandB-* D ! v~in =hea) B~Wandb)GISW model, respectively. Dt~raasv and 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

lepton in semileptonic ~-~ D,D s 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 Vub/Vcb ~ £ 0.20

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* -~ D~. The angular distribution in the

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

dUal(B, D =* D~) ~- ( l + ~B c°sZe') (8)

dcosO =

where =B measures the ratio of longitudinal to

transverse polarization:

=

rs¢(~ * Dl°ng) (9) ~B=2 - 1

Us/{B * D~ransv) A first measurement of ~B has been performed by

the ARGUS collaboration . Their result is

=B = 0.7 -* 0.9 (10)

from which one easily deduces

Plong/Ctransv = 0.85 ± 0.45 (11)

This finding agrees nicely with the theoretical expectation from the GISW 31 K6rner avd Schuler 27

and BSW 20 modeis~ respectively:

Flonglrtransv = 0.94, 1,06 and 1.07.

Page 6: Semileptonic decays of heavy mesons

260 ~L Wirbel/Semilepton:.c decays of heavy mesons

The polar izat ion of the f ina l vectormeson has also been measured in semileptonic D decays

where large longitudinal polar iza t ion has been

found 11:

+ 1.7 ± 0.2 rlong/Ftrunsv "- 2 . 4 0.9

whereas the predict ions of the BSW and GISW

models are: FlonglPtransv = l.Ojl.1. An ad hoc modification of the BSW model, where some

formfactors have been scaled, g a v e again good

agreement be tween theory and exper iment for the semilep~onic D-~K* decays 30

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