Ž .

Physics Letters B 455 1999 283–290

*Semileptonic and rare B meson decays*

### into a light pseudoscalar meson

### M. Ladisa

a,b### , G. Nardulli

a,b### , P. Santorelli

c,d a*Dipartimento di Fisica dell’UniÕersita di Bari, Italy*`

b

*Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy*
c

*Dipartimento di Scienze Fisiche, UniÕersita ‘‘Federico II’’ di Napoli, Italy*`

d

*Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Italy*

Received 9 March 1999 Editor: R. Gatto

**Abstract**

In the framework of a QCD relativistic potential model we evaluate the form factors describing the exclusive decays
*B ™ p*_{ll}*n and B ™ K*_{ll}q

lly. The present calculation extends a previous analysis of B meson decays into light vector mesons. We find results in agreement with the data, when available, and with the theoretical constraints imposed by the Callan-Treiman relation and the infinite heavy quark mass limit. q 1999 Published by Elsevier Science B.V. All rights reserved.

The study of the decays

*B ™ p*

### ll

n ,_{ll}

### Ž .

1*B ™ K*

### ll

q### ll

y### Ž .

2represents a significant part of the experimental
pro-grammes at the next proton-proton accelerators and
*at the future B-factories at SLAC and KEK. The*
importance of these processes arises from the

follow-Ž .

ing reasons. The decay 1 allows to measure the

Ž .

product of the Kobayashi-Maskawa KM matrix
*element Vu b* and the form factor describing the decay

1 _{Ž .}

process ; similarly, the decay 2 will give access,
in appropriate regions of phase space, to the KM
*matrix element V ; therefore these processes wouldt s*

allow to measure fundamental parameters of the

Ž .

*Standard Model SM of the fundamental *

interac-1

Ž .

There is one form factor contributing to 1 for a massless lepton.

tions, to say nothing of the possibility to explore, in both cases, new effects beyond the SM.

It is fair to say, however, that, in spite of the

Ž . Ž .

fundamental relevance of the processes 1 and 2 , the basic theory of the hadronic interactions,

Quan-Ž .

tum-Chromo-Dynamics QCD , is still unable to
pro-duce clear predictions for the hadronic matrix
*ele-ments B ™ p , B ™ K involved in these decays.*
This is due to the lack of a theoretical tool, as
powerful as perturbation theory, able to produce
predictions for the nonperturbative quantities
in-volved in these processes. The most frequently used
theoretical methods to deal with these problems are
based on approximation schemes such as lattice QCD
or QCD sum rules. These approaches have however
their own limitations. In the former method the finite
lattice size introduces a cut-off in the small
mo-menta, which precludes the possibility to make
reli-able predictions in the small momentum transfer

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.

Ž .

Ž 2 2. Ž

*region Q F 15 GeV* for recent reviews of lattice
*QCD predictions for B into light meson transitions*

w x.

see e.g. 1 . In the case of QCD sum rules or their variant, light cone sum rules, the theoretical uncer-tainties are dominated by the peculiar theoretical

Ž

tools employed by this method criteria for stability, hierarchic role of the different nonperturbative

con-.

tributions parametrized by the various condensates and cannot be reduced by adding new terms in the

Ž

Operator Product Expansion for a discussion see

w x.2 .

w x

On the basis of these considerations, in 3 we
have presented an analysis of semileptonic and rare
*transitions between the meson B and a light vector*

w x

meson in a QCD relativistic potential model. In 3 we argued that, because of its simplicity, this model might be used as a viable alternative to the more fundamental, but still limited theoretical approaches we have discussed above. It is the aim of this paper

Ž . Ž .

to extend this analysis to the decays 1 and 2 .
To begin with, we review the main features of the
*QCD relativistic potential model. It is a potential*

*model because the mesons are described as bound*

states of constituent quarks and antiquarks tied by an

Ž .

*instantaneous potential V r . It is a QCD model*
because the potential is modelled according to the
theory of the hadronic interactions, i.e. it has a
confining linear behaviour at large interquark

dis-Ž .

*tances r and a Coulombic behaviour , ya r rrs*

Ž .

*at small distances, with a rs* the running strong
coupling constant: in practice the interpolating

Ž . w x

*Richardson’s potential V r* is used 4 , cut-off at

Ž

very small distances of the order of the inverse

.

heavy meson mass to take care of unphysical

singu-w x

larities introduced by the relativistic kinematics 5 .
*Finally it is a relatiÕistic model because the wave*
equation used to obtain the meson wave function C
is the Salpeter equation embodying the relativistic
kinematics:

2 2 2 2

### (

y*= q m q y = q m q V r*1

### (

2### Ž .

**C r**### Ž .

s_{MC r ,}

_{Ž .}

_{Ž .}

_{3}

where, for heavy mesons made up by a heavy quark

*Q and a light antiquark, 1 refers the heavy quark and*

2 to the light antiquark. The relativistic kinematics plays an important role when at least one of the two

quarks constituting the meson is light, as in our case, and represents an improvement in comparison with the approach based on the non-relativistic quark

Ž .

model. In 3 *M is the heavy meson mass that is*

obtained by fitting the various parameters of the
model, in particular the b-quark mass, that is fitted to
*the value m s 4890 MeV, and the light quark massesb*

*m , m s 38 MeV, m s 115 MeVu* *d* *s* 2*. The *

B-me-son wave function in its rest frame is obtained by

Ž .

solving 3 ; a useful representation in the momentum

w x

space was obtained in 3 and is as follows

3 y*a k*

### (

*c k s 4p m a e*

### Ž .

*B*,,

### Ž .

4y_{1} _{<} _{<}

with a s 2.4 GeV **and k s k the quark ***momen-tum in the B rest frame.*

The constituent quark picture used in the model is
well suited for the mesons comprising at least a
heavy quark; for light mesons other dynamical
fea-tures, not accounted for by this simple picture, should
be incorporated, e.g. the nature of pseudo
*Nambu-Goldstone bosons of p ’s and K ’s and the presence*

Ž .

*of important spin–spin terms in V r , not included*

Ž

in the Richardson’s potential their neglect for heavy mesons is justified by the spin symmetry of the

Ž . w x

Heavy Quark Effective Theory HQET 6 which is

.

*valid in the limit m ™ ` . The solution adopted in _{Q}*

w x_{3 was to avoid, for light mesons, the constituent}

quark picture and to describe their couplings to the quark degrees of freedom by effective vertices. This assumption produces a set of rules that are used to compute the quark loop of Fig. 1, i.e. the diagram by which the hadronic amplitudes describing the decays

Ž .1 and 2 are evaluated. They are as follows.Ž .

. Ž " .

*1 For a light pseudoscalar meson M s p , K*
*of momentum p*X

we write the coupling

*N N*X

*q* *q* X

y _{p}

### u

_{g ,}

_{Ž .}

_{5}

5

*fM*

*where f s f s 130 MeV or f s f s 160 MeV.M* p *M* *K*

*The normalization factors N , N*X _{for the quark }

*cou-q* *q*

pled to the meson are discussed below.

2

Data on the heavy meson spectra are not of great help in
fitting light quark masses, which, therefore, are not accurately
determined in the model; its predictions, however, are not
*sensi-tive to m , m , m values in most of the available kinematicalu* *d* *s*

*M. Ladisa et al.r Physics Letters B 455 1999 283–290* 285

Fig. 1. Quark loop diagram describing the matrix element

X m X m

²*M p*Ž .<*J* <*B p ; M is a light pseudoscalar meson, J s q G b*Ž .: m

is the current inducing the decay and Gm is a combination of Dirac matrices.

.

*2 For the heavy meson B in the initial state one*
introduces the matrix:

1 *m mq* *b*
*B s* *c k*

### Ž .

## (

### '

3*m m q q P qq*

*b*1 2 =

*q*

### u

q*y*

_{m}

_{q}### u

q*1*

_{m}*b*2

*q*y

_{i g}_{,}

_{6}

### Ž

5### .

### Ž .

*2 m*

_{b}*2 m*

_{q}*where m _{b}*

*and m*are the heavy and light quark

_{q}*masses, q*m

*, q*m

their 4-momenta. The normalization

1 2

² < :

factor corresponds to the normalization *B B s*

*2 m and _{B}*

*d*3

*2 <*

_{k}

_{c k}_{Ž .}

< s

_{2 m}### H

3*B*2p

### Ž

### .

Ž .already embodied in 6 . One assumes that the
*4-momentum is conserved at the vertex Bqb, i.e.*

*q*_{1}mq_{q}_{2}ms* _{p}*ms

_{B meson 4-momentum. Therefore}m

Ž . m Ž .

* q s E , k , q s E ,y k and*1

*b*2

*q*

*E q E s m .b* *q* *B*

### Ž .

7.

3 To take into account the off-shell effects due to the quarks interacting in the meson, one introduces

Ž .

*running quark masses m k , to enforce the condition*

2
2 _{<} _{<}

### (

*E s m*

### Ž .

**k q k**### Ž .

8for the constituent quarks. For the kinematics of the

Ž . Ž .

decays 1 and 2 it is sufficient to introduce the running mass only for the heavy quark 3

*m s mb* *b*

### Ž .

*k ,*

### Ž .

93

² Ž .:

By this choice, the average *m kb* does not differ

signifi-w x

*cantly from the value mb* fitted from the spectrum, see 3 for
details.

defined by the condition

2 2

2 < <

### (

2 < <* m q k* q

*s*

_{m q k}

_{m .}_{Ž}

_{10}

_{.}

### (

*q*

*b*

*B*

. 2

*4 The condition m G 0 implies the constraintb*

*m*2_{y}* _{m}*2

*B* *q*

*0 F k F k sM* ,

### Ž

11### .

*2 m _{B}*

*on the integration over the loop momentum k*

*d*3*k*
.

### Ž

12### .

### H

3 2p### Ž

### .

.*5 For each quark line with momentum q and not*
representing a constituent quark one introduces the
factor
*i*
2
*= G q*

### Ž

### .

,### Ž

13### .

X*q*

### u

y

_{m}*q*Ž 2.

*where G q* is a shape function that modifies the
*free propagation of the quark of mass m* X _{in the}

*q*

hadronic matter. The shape function

*m*2y* _{m}*2X

*G*

*q*2

*G q*

_{Ž}

_{.}

s _{Ž}

_{14}

_{.}

2 2
*m y qG*w x

was adopted in 3 ; the value of the mass parameter

w x

*mG* was determined in 3 by the experimental data

) _{w} _{x} 2

*on the B ™ K g decay. A range 1.2,7.6 GeV* of
*possible values of m*2*G* was obtained.

.

6 For the hadronic current in Fig. 1 one puts the factor

*N N*X_{G}m _{,} _{15}

### Ž

### .

*q* *q*

where Gm is a 4 = 4 matrix. We shall consider

m m m mn

Ž mn

G sg _{and} G ss _{q}_{n} _{with} s s_{ir}

w m n_{x.}

2 g ,g *. The normalization factor N is as fol-q*

lows:
*m*

### °

_{q}*if q s constituent quark*

### Ž

### .

## (

### ~

_{E}*q*

*N s*

_{q}### ¢

1### Ž

otherwise .### .

16### Ž

### .

.7 For each quark loop one puts a colour factor of 3 and performs a trace over Dirac matrices.

This set of rules can now be applied to the

eva-X X m ² Ž .<

luation of the matrix element *M p* *qG b =*
<_{B p}_{Ž} _{.:}_{with the result:}

X m
²*M p*

### Ž

### .

<*qG b B p*<

### Ž

### .

: 3*d k*

### '

### w

### x

s_{3}

_{H}

_{u k y k c k}_{Ž .}

*M*3 2p

### Ž

### .

=*m mq*

*b*

## (

*m m q q P q*

_{q}

_{b}_{1}

_{2}=

*q*

### u

q*y*

_{m}

_{q}### u

q

_{m}*1*

_{m m}*b*2

*q*

*b*

*q*Tr

_{Ž}

y

_{i g}_{.}

5 ## (

*2 m*

_{b}*2 m*

_{q}*E E*

_{b}*= 2*

_{q}*i G*

### Ž

*q y q*

### .

y_{1}

_{1}m

*p*

### u

y

_{q}### u

g G .### Ž

### .

5 X*f*

*q*

### u

y

_{q}### u

y

_{m q i e}*M*1

*q*17

### Ž

### .

From this expression one can obtain the relevant
*formulae for the various form factors. With q s p y*

*p*X
, we write
X X m
²*M p*

### Ž

### .

<*qg b B p*<

### Ž

### .

: m X 2 2 m s

_{f}_{q}

_{Ž}

_{q}_{.}

_{Ž}

_{p q p}_{.}

q

_{f}_{y}

_{Ž}

_{q}_{.}

*m X 2 s*

_{q}

_{F q}_{1}

_{Ž}

_{.}

_{Ž}

_{p q p}_{.}

*m*2

_{y}

*2*

_{m}*B*

*M*m 2 2 q

_{q}_{Ž}

_{F}_{Ž}

_{q}_{.}

y

_{F q}_{Ž}

_{.}

_{.}

_{,}

_{Ž}

_{18}

_{.}

0 1
2
*q*X X mn ²

*M p*

### Ž

### .

<*i qs*

*q b B p*n <

### Ž

### .

: 2*f*

_{T}_{Ž}

*q*

_{.}

X m _{2}

_{2}

_{2}m s

_{Ž}

_{p q p}_{.}

_{q y m y m}_{Ž}

_{.}

_{q}_{,}

*B*

*M*

*m q m*

_{B}*19*

_{M}### Ž

### .

where*F q*

_{Ž}

2_{.}

s

_{f}_{Ž}

*2*

_{q}_{.}

_{,}1 q

*q*2 2 2 2

*F*

_{0}

_{Ž}

*q*

_{.}

s

_{f}_{q}

_{Ž}

_{q}_{.}

q

_{f}_{y}

_{Ž}

_{q}_{.}

_{.}

_{Ž}

_{20}

_{.}

2 2
*m y mB*

*M*Ž . Ž .

*In 18 and 19 we shall consider M s p or M s K*
since both cases are of physical interest if we wish to
consider not only semileptonic and radiative
transi-tions, but also nonleptonic decays.

The calculation of the trace and the integral in

Ž17. is straightforward and is similar to the one

w x )

*obtained in 3 for B ™ r, B ™ K* transitions. For

Ž 2. Ž 2 X_{.}

*all the form factors we write F q* s* _{F q , m}* y

*q*

Ž 2 .

*F q , m _{G}* , where, for the various form factors, we

have
*F* *q*2* _{, x}*
Ž .
0

### '

6 s 2 2 2*4p fM*Ž

*m y mB*

*M*. 2 Ž .

*dk k c k*

*kM*=

_{H}

2
2
0 _{(}

_{E E}_{q}

_{b}### w

_{m y m y m}_{B}_{Ž}

_{b}

_{q}_{.}

### x

1 1 =_{H}

*2 0 2 2*

_{dz}_{<}

_{<}

*m y2 E*

*m y q*q

*y1*

**m y x q2 q k z***M*

*q*Ž

*B*.

*q*0 2

_{<}

_{<}=

_{½}

w*2 E*

_{q}_{Ž}

*m y q*

_{B}_{.}y

*x 0*

**m y2 q k z**_{M}_{Ž}

_{.}< < =w

_{Ž}

_{m E q m E}_{q}

_{b}

_{b}

_{q}_{.}

_{q q m y m}_{b}

_{q}*x*

_{q k z}*m q m*X

*q*

*q*2 0 2 2 Ž . q

### w

_{Ž}

*q y m q*

_{B}_{.}

### w

*m y m y m*

_{B}

_{b}

_{q}### x

2 2< < q

_{2 m}_{B}

_{q k z}### x

_{5}

_{,}

_{Ž}

_{21}

_{.}

*F q*2

*Ž . 1 2*

_{, x}### '

6*kM*

*dk k c k*Ž . s

_{H}

2 _{2}

_{2}

*8p fM*0

### (

*E Eq*

*b*

### w

*m y m y mB*Ž

*b*

*q*.

### x

1 1 =_{H}

*2 0 2 2*

_{dz}_{<}

_{<}

*m y2 E*

*m y q*q

*y1*

**m y x q2 q k z***M*

*q*Ž

*B*.

*q*0 2 < < =

_{½}

w*2 E*

_{q}_{Ž}

*m y q*

_{B}_{.}y

*x < < 0 X Ž .*

**m y2 q k z**_{M}*Ž*

**q m E q m E***q*

*b*

*b*

*q*.q

*q k z m y mb*

*q*

*m q mq*

*q*= q < <

*mB*

*2 2 0 2*

**q**_{<}

_{<}2 Ž .

*2 k z m q y q*

_{Ž}

_{B}_{.}y

_{q}### w

_{m y m y m}_{B}

_{b}

_{q}### x

=_{<}

_{<}

### 5

,### Ž

22### .

**q**### '

6*m q m*

_{B}*2*

_{M}*fT*Ž

*q , x s y*.

*2*

_{4p f}_{2}

*M*2

_{Ž}

_{.}

*dk k c k*

*kM*=

_{H}

2
2
0 _{(}

_{E E}_{q}*w*

_{b}

_{m y m y m}_{B}_{Ž}

_{b}

_{q}_{.}x 1 1 =

_{H}

*dz*

_{2}

_{0}

_{2}

_{2}< <

*m y 2 E*

*m y q*q

*y1*

**m y x q 2 q k z**

_{M}

_{q}_{Ž}

_{B}_{.}

_{q}*k z*0 2

_{<}

_{<}X Ž . = w

*2 E*

_{q}_{Ž}

*m y q*

_{B}_{.}y

*x q*

**m y 2 q k z**_{M}*m q m*

_{q}

_{q}### ½

<*< < < 0*

**q**_{Ž}

_{.}yŽ

*m E q m Eb*

*q*

*q*

*b*.

*Ž*

**q q m y q***B*.

*k z m y mb*

*q*=

### 5

. < <*m*23

_{B}**q**### Ž

### .

*M. Ladisa et al.r Physics Letters B 455 1999 283–290* 287

*In these equations q*0 is the time component of
*four-momentum q*m,

*z s cos u ,*

### Ž

### .

### Ž

24### .

**with u the angle between k and the direction of****transferred momentum q. We note that, for M s p ,**

*m s m* X_{s}_{m and f s f , while, for M s K, m}

*q* *q* *u* *M* p *q*

s* _{m , m}* X

_{s}

_{m and f s f .}*u* *q* *s* *M* *K*

Before discussing our numerical results in detail

Ž 2. 2 2 2

*let us compute F q*0 *for q s m y m ; in theB* *M*

Ž 2 2 . Ž 2.

*chiral limit F m y m* ,_{F m}_{must obey the}

0 *B* *M* 0 *B*
w x
*Callan–Treiman relation 7*
*f _{B}*
2

*F*

_{Ž}

*m*

_{.}

s _{.}

_{Ž}

_{25}

_{.}

0 *B*

*f*p

This is therefore a consistency test to be satisfied by

*Bp*_{Ž} 2_{.}

*the model. We have numerically evaluated F*0 *mB*

*for different values of the parameter mG* and we

*Bp*_{Ž} 2_{.}

*have obtained the result F* *m* ,_{1.48, almost}

0 *B*

*independent of m . This result should be comparedG*

*to f rf , 1.58, which is obtained using f s 0.2 _{B}* p

*B*

w x

GeV, i.e. the value computed in 3 using the present model. The small discrepancy in the Callan-Treiman

Ž .

relation , 6% may be attributed to the deviations
*induced in the B meson wave function by the chiral*
limit that are not accounted for by this calculation.
We expect however that these differences vanish if,
in addition to the chiral limit, one also takes the
infinite heavy quark mass limit; as a matter of fact
one can verify rather easily, using the previous

for-w x

*mula for F and the expression in 3 for f , that the*0 *B*

*Callan-Treiman exactly holds in the combined m ™b*

*` and m ™ 0 limit. _{M}*

Ž 2. Ž

*Let us now consider the form factor F q*1

re-Ž 2..

*spectively fT* *q* . Our numerical results for the
*central value of m , i.e. m s 1.77 GeV, show thatG* *G*

*the q*2_{-behaviour of this form factor is increasing}

Žresp. decreasing for both small and moderate val-.

*ues of q*2_{, independently of the value of the mass}

Ž .

*parameter m _{G}* introduced in Eq. 13 . This behaviour

2 _{Ž} 2 2_{.}

*should hold also at large q* *q G 15 GeV* due to
*the effect, in this region, of a pole in the q*2 _{}

func-tional dependence, predicted by the dispersion

rela-Ž 2.

*tion. Differently from our analysis of F q , we*0

cannot pretend, however, to extend the validity of

Ž 2. Ž 2.

*our predictions for F q*1 *and fT* *q* at the extreme
*values of q*2. The difference between the two cases

Ž

*is as follows. In the case of F , the pole with*0

*P* q

.

*J s 0* contribution to this form factor vanishes in

the chiral limit and has therefore a minor impact on
*the q*2 behaviour. On the contrary the form factors

*F and f , have a non vanishing polar contribution*1 *T*

*which becomes larger and larger with increasing q*2.
While we expect that this behaviour become visible
*well before the pole, at extreme values of q*2 the
diverging behaviour induced by such a contribution
cannot be reproduced by the model. As a matter of

2 _{<} _{<}

* fact, for larger values of q , q becomes smaller and*
smaller, and, therefore, the model becomes sensitive
to the actual values of the parameters, in particular
the light quark masses that are not accurately fitted

Ž .

by the available experimental data see above . Therefore we can consider that our predictions are

Ž . 2 2

*reliable in the range 0, 15 GeV ; at q s 0 we get*

*FBp* _{0 s F}Bp_{0 s 0.37 " 0.12 ,}

### Ž .

### Ž .

0 1*FB K*

_{0 s F}B K_{0 s 0.26 " 0.08 ,}

### Ž .

### Ž .

0 1*fBp*

_{0 s y0.14 " 0.02 ,}

### Ž .

*T*

*fB K*

_{0 s y0.09}q

_{0 .05}. 26

### Ž .

### Ž

### .

*T*y

_{0 .02}

*The central values are obtained for m s 1.77 GeV,G*

w x

*which is the best fit of the parameter mG* found in 3

Ž )

.

by the experimental branching ratio B*B B ™ K g ,*

whereas the theoretical uncertainty is obtained by

w x

*varying mG* in the range 1.1, 2.8 GeV. The results
*for B ™ p refer to charged pions.*

*Let us now consider the q*2_{-behaviour of the form}

factors. We introduce the two-parameter function for the three form factors

*F 0*

### Ž .

2*F q*

_{Ž}

_{.}

s _{;}

_{Ž}

_{27}

_{.}

2
2 2
*q*

*q*

*1 y a*q

_{F}*2 2*

_{b}_{F}### ž /

*mB*

### ž /

*mB*

*here a , b are parameters to be fitted by means ofF* *F*

Ž . Ž .

*the numerical analysis and F 0 is given in Eq. 26 ;*
to allow a comparison with other approaches we

Table 1

Ž .

*Parameters appearing in Eq. 27 for different B form factors*

Ž . Ž .
*F 0* *aF* *bF* *F 0* *aF* *bF*
*Bp* *B K*
*F*1 0.37 0.60 0.065 0.26 0.50 0.39 *F*1
*Bp* *B K*
*F*0 0.37 1.1 0.44 0.26 1.2 0.56 *F*0
*Bp* *B K*
*fT* y0.14 0.92 0.21 y0.09 0.76 0.76 *fT*

Ž 2. Ž 2.

*Fig. 2. F q , F q*0 1 *for B ™ p and B ™ K transitions.*

*perform the analysis up to q*2s_{15 GeV}2_{, both for p}

*and K mesons. We collect the fitted values in Table*
*1 and report the q*2_{-dependence in Figs. 2 and 3.}

From Table 1 and from Figs. 2 and 3, one can see

Ž 2. *Bp*Ž 2. *Bp*Ž 2. 2

*that F q , F*_{0} _{1} *q* *and f _{T}*

*q*

*have a q*

*be-B K*_{Ž} 2_{.}

*haviour similar to a single pole. For F*_{1} *q* and

*B K*_{Ž} 2_{.}

*f _{T}*

*q*there are significant deviations from this

Ž 2.

*Fig. 3. fT* *q* *for B ™p and B ™ K transitions.*

behaviour. We do not have yet experimental data to
test these predictions and we shall limit to compare
our results with other theoretical approaches; before
doing that, let us discuss the infinite heavy quark
mass limit of the model. In the framework of the
Heavy Quark Effective Theory, which corresponds
*to the m ™ ` limit, there is a constraint to be _{b}*
satisfied by the three form factors, i.e., the relation

w x
originally found in 8 :
*m q mB* *M*
2 2
*f _{T}*

_{Ž}

*q*

_{.}

s y

_{F q}_{1}

_{Ž}

_{.}

*2 mB*2 2

*F*

_{0}

_{Ž}

*q*

_{.}

y

_{F q}_{1}

_{Ž}

_{.}

2 2
y_{Ž}

_{m y m}_{B}

_{M}_{.}

_{.}

_{Ž}

_{28}

_{.}

2
*q*

*This relation holds in the limit m ™ ` and for high _{b}*

2 _{Ž} 2 2 _{.}

*q* *q , q*_{max} . We have checked that this relation

formally holds in our model in the infinite heavy
*quark mass limit and for q*2_{,}* _{q}*2

_{. For the actual}

ma x

Ž .

*value of m _{B}* s

_{5.28 GeV and for the transition}

*B ™ p the situation is as follows. We choose q*2_{s}

*M*

15 GeV2_{, i.e. the maximum value at which we}

can trust our predictions and we find numerically

Ž 2. Ž 2. Ž 2.

*F q*_{1} * _{M}* ,

_{0.54,}

_{F q}_{0}

*,*

_{M}_{0.71,}

_{f}_{T}*, y*

_{q}_{M}_{0.27.}

Ž .

Therefore the relation 28 has a significant violation of 50%, that may be attributed to the fact that we are

*M. Ladisa et al.r Physics Letters B 455 1999 283–290* 289

Table 2

Comparison of the results coming from different approach to evaluate form factors

w x w x w x w x

This work LCSR 12 SR 13 Latt. 1 Latt. q LCSR 14

*Bp*
Ž .
*F*1 0 0.37 " 0.12 0.30 " 0.04 0.24 0.27 " 0.11 0.27 " 0.11
*Bp*
Ž .
*fT* 0 y0.14 " 0.02 y0.19 " 0.02 – – –
*B K*
Ž .
*F*1 0 0.26 " 0.08 0.35 " 0.05 0.25 – –
*B K* q0.05
Ž .
*fT* 0 y0.09y0.02 y0.15 " 0.02 y0.14 – –
2 _{Ž} _{.}

*still far from q*max and O*O 1rmb* corrections are
*large. Similar results are obtained for the B ™ K*
transition.

Let us finally comment on the scaling laws of the
*form factors at large q*2 _{that can be helpful in using}

the heavy flavour symmetry to relate the form

fac-w x _{Ž} _{.}

*tors of B and D mesons 10 . From Eqs. 22 and*

Ž23 the following behaviours can be formally de-.

rived:
2
*F q*

_{Ž}

_{.}

f_{(}

_{m ,}_{Ž}

_{29}

_{.}

1 max *b*2

*f*

_{Ž}

*q*

_{.}

f_{(}

_{m ;}_{Ž}

_{30}

_{.}

*T*max

*b*Ž .

they are in agreement with the pole vector meson

w x

dominance of the form factors observed in 9,10 ;

Ž .

moreover in the chiral limit, from Eq. 21 one gets: 1

2

*F*_{0}

_{Ž}

*q*

_{max}

_{.}

f _{.}

_{Ž}

_{31}

_{.}

*m*

### (

*b*

Let us now compare our work with other
theoreti-cal approaches. In Table 2 we compare our outcome
*for the values at q*2_{s}_{0 with the results of QCD sum}

Ž

rules and lattice QCD calculations for other work on

w _{x.}

this subject see, e.g. 11 . We observe that our results are in agreement, within the theoretical uncer-tainties, with the determinations obtained by light

Ž . w x w x

cone sum rules LCSR 12 , lattice 1 and lattice q

w x

LCSR 14 .

*As for the q*2 dependence, we have not reported
the predictions of other theoretical approaches, since
they qualitatively agree with our calculations. In
absence of detailed experimental data on the form
factors, the best we can do to test the model is to use

Ž .

*data on the partial width G B ™ p*

### ll

n . To performthis comparison we must, however, extrapolate the

2 _{Ž} _{.}

*q -behaviour obtained by Eq. 27 and Tables 1 and*

Ž . 2

2, and valid in the region 0,15 GeV , to the whole

*q*2 _{range. This procedure implies an uncertainty}

which is difficult to assess, but should not be

ex-tremely large due to the phase space limitation at
*high q*2_{. We obtain}
2
<* _{V}* <

*u b*y 0 q y4 B

*BR B ™ p*

### Ž

### ll

n s 1.03### .

### ž

y3### /

10 , 3.2 10 32### Ž

### .

Žto be compared to the experimental value B*BR B ™*

. Ž . y4 w x

p

_{ll}

n s _{1.8 " 0.6 10}

_{15 . Therefore our }

re-exp.

sult is compatible with the present range of the KM

w x y_{3}

*matrix element V s 1.8, 4.5 = 10 _{u b}* ; the preferred
range of values selected by the model and by the

Ž

*present experimental limits on Vu b* *is V s 4.0 "u b*

. y3

0.5 = 10 .

We conclude our analysis by summarizing our results. We have used a QCD relativistic potential

w x

model, introduced in 3 , to study the weak and
*radiative transitions B ™ p , K. We have computed*
the relevant form factors and tested the
Callan-Trei-man and the Isgur-Wise relation. The former
rela-tion, valid in the chiral limit, is satisfied at the 6%
*level, while the latter, valid in the m ™ ` limit, hasb*

Ž .

significant violations, due to O*O 1rmb* corrections.

Ž .

Our result for the branching ratio B*BR B ™ p*

### ll

nagrees with the experimental data.

**Acknowledgements**

We thank P. Colangelo and F. De Fazio for the collaboration in the early stage of this work.

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