B-meson semileptonic exclusive decays

Nuclear Physics B (Proc. Suppl.) 27 (1992) 47-57 North-Holland

L.L LûL[~_L'v

`

B-MESON SEMILEPTONIC EXCLUSIVE DECAYS . Giuseppe NARDULLI Dipartimento di Fisica, Universita' di Bari and I.N.F.N ., Sezione di Bari, Italy. Recent theoretical progresses in B-meson semileptonic exclusive decays are reviewed . They include new results obtained within the effective quark theory, QCD sum rules and potential models . Among the many interesting results that have been 1 . INTRODUCTION In the last two years there have been interesting obtained in this limit, let us mention the following ones: progresses in heavy quark physics, both theoretical a) The semileptonic width of the bq meson can be and experimental. Here we shall only discuss exclucomputed by the parton formula (neglecting IVubI2 « sive semileptonic B-meson decays and, in particular, I Vcb l2 ): decays into charmed mesons, which have been exten5 sively studied in the last few years by both the CLEO G2F(m r(b -} ctvl) = (2) and ARGUS collaborations . 15 -70 ) IVcbi 2 On the theoretical side much attention has been and is completely saturated by two exclusive channels: paid to the so-called heavy quark effective theory2,s,4, but some new interesting results have been obtained (3) B -+ _n f !,,in the framework of QCD sum rules5,6 and potential B -+ D* P ve (4) models? as well. In this section we shall review some results of the heavy quark effective theory, whereas the with r(B D*fv) other approaches will be discussed in the next sections . 3 r(B -~ Dfv) The basic ideas of the heavy quark effective theory are rather simple: b) By using the SU(2) x SU(2) spin-flavour sym1) If one considers the limit mQ-+ oo (Q = heavy metry, one can relate all the following form factors : quark), then, during the interactions, the B meson will not change its velocity vP = pP/mQ (mQ ^_- MB) . To < B(p)IV,IB(P) >= (n+?;),F(q2) a good extent this will be true also for the D meson. 2) The spin degree of freedom is decoupled from the dynamics, as the colour hyperfine interaction is pro< D(P )I V I B(p) >= f(P + j), portional to mQ l . As a consequence, a spin SU(2) symmetry arises . MB MB MD q)F(g 3) Since one takes the limit Mb) mc --> oo, then a flavour , iw 2) + q-2 q-2 MDgFo(q2) SU(2) symmetry can be assumed as well. Several analyses can be performed in the heavy 2V (q 2 ) quark limit, better defined as follows 2 : < D* (P,E)IV»IB(p) >= E,,,PQE* "p'p° (mb + mc)AQCD « (mb - MC)2 « (mb + rnc)2 0920-5632/'92/$05 .00 0 1992 - Elsevier Science Publishers B .V

2 MB + MD*

(1) All rights reserved .

(8)

universal the D*(P Egs aBthe these Fi(g2), =though matter and ,E)IA,IB(p) Isgur-Wise 1t~form relations Aà(92) D(*) +q-++~(v (g2) ofMD p®)A factors =MD* ==fact velocities =pbetween 1(0) Ao(g2) satisfy MB function -->= As(g2) D cannot one M MBMD p', MB can MB = =+ +MDVv + +Ai +iE* Fi(0) MD Ao(0) )MD MD,,, F(0) _ MD the has, MD MD and be (q2 -the [iUTB +gz be f(v AO(g2) expressed )F(q2 relations ZMD computed in various ~(v =-,+ v v') )s-the )q1MD MD where NardullilB-meson v') q~A(g2 and heavy form jA,(g2) inwithin AZ2 the terms v) factors and quark form ) the of se»iileptonicthat v' exclusive the decays the be relevance theory, corrections Pbe order non which parity and form momentum ve adecay means original there in made heavy computed the are analysisii The most in dramatic non -* --> resonant The decays Dô* terms can in the factor, compatible As should (D° O(mQ') other (D 1+ allowing theory ismodes perturbative also that m-' important of 2candidates quark to be + no limit apredictivity and =+ the the to of theorem9,lo, matter D*) vgmax reduction for used should hand clear characterized within gives D+)XPve) about background lead, the is 0+ radially -O(mQi On one are effective v' that (1) aheavy Pvery with orbital to ve) computation =it the =way heavy function present, in shortcoming and of be one to the prove (MB are 1ispredictive functions, =principle, )of other fact, baryon introduced the of to fill excited corrections easily (6 taking At _ third theory theory, stating the different excitations quark the deal -(9 in athis prediction by All which whereas thand, MD( generalization model this number Similar 0with decays& of to fseems D* these of them beyond to point no limit of at in the 1be gap fB means from in resonances for asince the much this this the 1contributions experimental semileptonic f of convinced very semileptonic events are into (5), apredictions of the are q2 1D* maximum the O(1lmQ approach hard case they D paramethat the 0a-recent remains predicflavour account of PIsgurand of gmax arises ve that posican task sec(19) the adand D of)l or

G.

48 and

.

< -i(E* q) In factors

(p 1B [ ~1B

.

A2(g2)

I

.(6-9)

: Fo

(10)

A(g2) with

A3(0)

(12)

a are

. MB 2NID.

. MB Ai(g2 MD .

All

(g

" (13)

" v

2

-

2

2MB1ti7D, " ,

(14)

As limit : Fo(g2) qz

Fl(g2)

=

- MB 2` = MB 2 V(q2)

AZ

Even theory,

MB 2

.VI)=

(15)

.

" ))2

corresponding symmetry Ademollo-Gatto the ond . corrections ditional Wise not of On the B . results CLEO

(16)

_

1 _ (MB+MD " MB . = . 2 " v')

(17)

(18)

.

; .

BR(B

.1

.6

.1)10-2

whereas BRCS

. .

.

9

(11)

and A3(g2)

imply ters, tive decays can The is effects . recoil

This from D* . tive D** the vanish in

.5

.2

.2)10-2 . (20)

: ;

. .

G. Nardulli/B-meson semileptonic exclusivedecays

Some hints on the size of these effects can be obtained by QCD sum rules and potential models . This will be discussed in the next sections. 2. QCD SUM RULES We begin by writing down the form factors that are rele rant for the transitions to posive parity charmed me., ons : 1i < DO** (P)I ~y~ysbl B(p) >= g+ (p + P), + g- qP

(21)

1i < D**(P, E)Icy,ysbjB(P) >= ifE~vapE*v(P +P)'qp (22) < D** (P, E)I Z7wbl B(p) >= = fo EP, + f+

(E* -

P)(P' + P)P. + f-

(E* -

P' )q,

(23)

where q = P' - P, El` is the D** polarization vector and the form factors g±, f, fo, ff are functions of q2 . Together with these equations we also consider the form factors for the transitions B -> D and B --+ D* that we rewrite as follows : < D(P)I Zy~bl B(P') >= G+

(P + P), + G- qa (24)

1 - < D*(P, E)I&y,bl B(Pf) >= iF i

Ep.v«pc

*v

(P + P)agp

(25)

>= i < D* (P,E)I &y,y5bl B(P') (E* = Fo E*A + F+ - P)(P' + P)o + F_ (E * - P )qo

(26)

where the form factors G±, F, Fo ,+ are related to those introduced in the previous section by the formulae : G+ = Fl, G- = (Ms - MD)(Fo - Fl )l q2 , F = VI (MB + MD.), Fo = (MB + MD. )A I , F+ _ . -A2(NIB + MD* ), F- = 2MD " (Ao - A3)/q2

49

In order to write down QCD sum rules for the form factors in Egs.(21-26) we proceed as follows . For B -~ D** we consider the three-point current correlator II, (P,P~, q) = i2

I dx dy

e`P z-tP' v

< OIT{i(x)c(x),J,~(0),b(y)ysq(y)}1O > , (27)

while for the B --> D** transition we consider the correlators IIAv (P,P , q)

=

Z2

dx dy e :Px - =Py

< OIT{q(x)yvysc(x),JK'A(0),b(y)ysq(y)}1O > (28)

where JV = c"y,b, J,A = cy,y5b and q(x) denotes the light quark operator : q = u, d. The correlators for the transitions B -* D and B -i D* are obtained from Egs.(27,28) by the changes V 4-4 A, qc --+ gy5c and gyvy5c --+ gYvc . The hadronic tensors appearing in (27) and (28) can be decomposed into Lorentz structures as follows: II11=ÎI+(P+P),+IÎ-q, II

II Av

= IIo9~.v + II1P~.P,. + II2PpPv + II3PuPv+ + II4PgP,. = i II

(29)

Ea0Kv PaP P

We are only interested in the amplitudes II+, fl, IIo and IIP = (ÎI1 + f12)/2, since in the semileptonic decay matrix element the other ones give vanishing contributions. As well known, the QCD sum rules method 12 basically consists in computing the current correlators in two different ways. The first one is obtained by saturating the correlator by the lowest lying hadronic states plus a continuum of states, starting at some threshold, which is usually modeled by perturbative QCD . The second expression is obtained by using the short distance Operator Product Expansion (OPE), which consists of the leading perturbative QCD term, plus

50

G. NardullilB-n:eson semileptonic exclusive decays

non perturbative power corrections arising from operators of higher dimension . The latter ones are proportional to quark and gluon operator vacuum matrix elements that are determined phenomenologically. Using, these alternative representations, predictions for hadron masses and couplings can be obtained in terms of quark masses, a, and non perturbative vacuum condensates. The representation of current correlators in terms of lowest hadronic states plus a continuum is obtained by means of a dispersion relation. Thus, in the case of interest here, denoting by II= any one of the amplitudes II+, II, Ho and Up, III obeys the double dispersion relation ®2

2

1

pà(s, S', Q') P2 )(S' - P'2) subtraction terms

47[2 J

where Q2

(S -

(P+Pf )e P+(s,st ,Q 2 ) -41r 26+ ( s _ 1,D2

=

M2) X

x < OlgclD ** >< Dô*IJA IB >< BIb-y5 gl0 > + + (1 - 9(so - S) 0(s0 - s')) (p+ pi),.

im ii, f2

x

(31

2

1

p=(s, s', Q2) exp(-s/M1) exp(-.9 1 /

)ds ds' (32)

p12'

Q2 ). In order to compute the hadronic contribution to II,,,, II,,,  we need to separately estimate the following vacuum-to-meson transition amplitudes (mq = 0): IIi(P2'

fD°" MDö . < OlgclD ** >= mc

(33)

and 2 MD .. * < 04Y,-Yscl D * (P, E) >= E'(P) 9D "

(34)

The decay constants in (33) and (34) can be obtained by two-point-function QCD sum rules. The method is the same as the one employed by several authors for the determination of the negative parity ruesons D and D* decay constants 13 : < Olg iys

p+(S,S',Q2)IAF

where p+ (s, s', Q2) I AF is the Asymptotic Freedom contribution to p+ (s, s', Q"- ) which is supposed to represent higher mass states, and so , sô are the continuum effective thresholds in the variables s and s' respectively . An analogous expression is obtained for III, in Eq.(28). Different methods can be used to improve the convergence of these dispersion relations . In particolar one can use the double Borel transform which enhances the hadronic ground state contribution, minimizes the role of the continuum, which is rather model dependent,

_,

where II=( 1 , MZ , Q2) is obtained applying the Borel transform to the Operator Product Expansion of

(30)

0. Saturating Eq.(30) by the 'lowest lying and higher mass hadron states in the p2, P'2 channels we get the hadronic contribution . For example, in the case of B -+ D** transition one gets: = -q2 >

and ensures that only a finite number of non perturbative terms are needed in the Operator Product Expansion . This method introduces two mass parameters Ml and M2 ; from eq.(30) we obtain :

cI

M2 D >= fD D

(35.a)

2 >= Mk E. .(P)

(35 .b)

mc

and < Ol g y, cl D*(P, E)

9D "

In order to compute fD (and its analogue fB) we consider the correlator of two pseudoscalar currents, whereas to compute 9D " (and its analogue 9B " ) we consider the correlator of two vector currents . The corresponding results for fD-" and 9D ". are obtained by considering correlators of two scalar currents and two axial currents respectively, and by changing m, --+ -m c in

G. Nardulli/B-meson semileptonic exclusive decays

the explicit expressions of the Operator Product Expansion . In Ref. 5 the following set of parameters has been used: m e = 1.35 ± 0.05 GeV, Mb = 4.60 ± 0.10 GeV, < qq >= (-0 .250 GeV )3 , < ~ G2 >= 0.012 GeV4, Z < gQEty gGll`' >= mô < qq > with mô = 0.8 GeV2, and AQCD = 100 - 150 MeV. With these inputs one obtains the following results 5: fD = 195 ± 20

MeV

fB = 180 ± 30

MeV

=25±4

9B*

(36)

,qD" = 7.8 0.8 fDö" = 170

s 20

MeV

9D-" = 9.8 ± 1 .5

These values are obtained with the continuum thresholds so = 6 - 8 GeV 2 and so = 36 - 40 GeV2 . Turning to the determination of the form factors, one has to saturate the correlators (27) and (28) by the low-lying meson states B, D** and B, D** respectively and to model the continuum by QCD. On the other hand one cornpuies the correlators by malting use of the Operator Product Expansion . After applying the double Borel transform and matching the two expressions, one obtains the following sum rules for the form factors:

9+(Q2 ) _ -

Mc Mb

fDô , fB

2

MD,~,

exp( MDô' + MB x Mi M2) Mâ

x IÎ+(Mi ~M2~Q 2 )

f(Q2)

_-

mb 9D*'

2 fB

MD . .

x n(M2 , M2,

2

MB

Q2)

(37)

exp(

MD2 .. Mi

2

+ MB )x 1V12

(38)

fo ,+(Q 2 )

= -

51

2

mb 9D""

-- + MB ) exp( MD x MÎ fs MD. . MB M2

x IIo,P(Mi , M2, Q2 )

(39)

where Ml , and M2 are the Borel mass parameters, whereas fl+(Ml , M2 , Q2) , II(M1, M2 , Q2) and ft 0,P(M1, M2, Q2 ) are Borel-transform of the Lorentz invariant amplitudes II+ , fl and IIo,p. Similar expressions can be found for the form factors ofthe transitions B -+ D and B --> D* . Each of the Lorentz invariant amplitudes is expanded as follows: II,

) + IIi3 ) + ]1(5) + IIL 6, + . . . = II=o

(40)

where II~D) are double Borel transformed terms arising from the perturbative contribution (D = 0) and from the condensates of dimension D = 3, 5, 6 respectively The results of this analysis for the Q2 = 0 values of the various form factors are as follows (theoretical errors only arising from the spread of the Borel mass parameters) : G+(0) = 0.68 ± 0.19 ; g+(0) = 0.30 ± 0.03 F(0) = 0.12 ± 0.19GeV-1 ;

(41)

= 0.10 ± 0.03GeV-1 (42) F+(0) = -0.05 ± 0.02GeV-1 ; f+ (0) = -0.02 ± 0.02 (43) Fo(0) = 4.35 ± 0.15GeV ; fo(0) = 0.9 ± 0.3GeV (44) f(O)

The values of the thresholds are so ^-- 12 -15 GeV2 and sô ^-" 40 GeV2 , whereas the values of the Borel mass parameters Mi and M2 are generally within the ranges 2-5 GeV2 and 5-10 GeV2 . To obtain the previous results one has imposed that a hierarchy ejdsts among the different terms of the OPE, so that operators of higher dimension give smaller contributions; in this way one can always obtain find stability regions where such hierarchy criterion is satisfied and the final

52

G. Nardullil B-n:eson semileptonic exclusive decays

Another experimentally interesting quantity is the results do not depend too strongly on Borel parameters asymmetry parameter a, defined as follows : and effective thresholds . the As for the Q''-dependence of the form factors, _ rL(B --} D*Pv) 1 (49) a 2 results of Ref. 5 are compatible with a simple pole rT(B --; D*fv)_ behaviour Fi(Q2) = Fi(0)/(1 + ~), as suggested by where PL and rT are the widths for the production of dispersion relations, where the pole masses are those longitudinal and transverse D* respectively. Imposing of be mesons with appropriate quantum numbers and the cut pt > 1.4 GeV/c as in the CLEO experiment" their values agree with the predictions of the potential one obtains : models (e.g.m = 6.34 GeV for the 1- pole (for the form a = 0.74 , factors (50) G+, F, fo and f+ ) and m = 6.73 GeV for the 1+ pole (for the form factors g+ , f, Fo and F+). to be compared to the CLEO experimental value (simiFrom the results (41-44) one can easily compute the lar results are obtained by the ARGUS Collaboration) : branching ratios for the exclusive decays. The results are as follows : a=0 .65±0.66±0 .25 . (51) BR(B - Dî-e) =1 .5 (i )2 x 10' 2 (Theory) 0.04 b

=1 .75, ± 0.42 ± 0.35 x 10-` (Exp.) (45)

BR(B --" D* fv) =4 .6 (

VCb

0 04 )2

x

10-2 (Theory)

=4.8 ± 0.4 ± 0 .7 x 10-2 (Exp.)

(46)

1"`b

(47) BR(B --, D**fv) = 0.15 ( )2 x 10-2 0.04

BR(B ---" D* *fv) = 0.15 ( VVCb 04 )2 x 10-2

(48)

where experimental data are an average of the ARGUS and CLEO resultsl 4 . These results shows that a good agreement with the experimental data for the transition B -> D and B - D* is obtained with Vc b ^_- 0.04 . As for the positive parity charmed mesons, the predictions (47) and (48) indicate that the sum of the D**(0+) and D**(1+) contribution is about 5% of the sum BR(B --+ Dfv) + BR(B --} D*fv).

In conclusion, by using QCD sum rules one is able to compute form factors for semileptonic B decays into negative and positive parity charmed mesons . As we shall see, the results for negative parity decays are in agreement with other calculations, mainly based on potential models . As for the positive parity case, these results seem to indicate a minor role of these resonances in the semileptonic inclusive branching ratio. Therefore, the difference between (19) and (20) should arise, in addition to the states considered here, from the whole multiplicity of higher radial excitations and of non-resonant background states . Let us finally mention some results obtained by QCD slim rules for b --} u semileptonic exclusive decays. In Ref. 6 the method has been applied to the computation of form factors describing the matrix elements < 7r + (

P )Itii,, 1B0 (p)

>

(52)

and < po (p' ,E)IJ~IB

(p)

(53)

The results are BR(B0 --~ 7r+ e

ve ) _ (2-0 ± 0.7) ( Vub )2 x 10 -4 0.005

(54a)

53

G. Nardulli /B-meson semileptonic exclusive decays

BR(B - --, p° e- ve )

_

(2.5 f 1.4) ( Vub

0.005

)2 X

10 -3

(54b)

The second result should be compared with recent data from the ARGUS Collaboration' BR(B - --> p° Q vt) = (1.13 f0.36f0.30) x 10-3 , (55)

which would indicate a value of Vub around 3-4x10`However, due to the large uncertainties both in the theoretical calculations and in experiment, we should look at this determination as only a very preliminary indication of the value of Vu b . 3. POTENTIAL MODELS Charmed semileptonic B meson exclusive decays can be studied by potential models's -'s, namely by assuming potential interaction between the valence quarks comprising the mesons . Harmonic oscillator wave functions in the infinite meson momentum frame are adopted in Refs. 16 and 17, while in Ref. 18 approximations in the treatment of the relativistic quark kinematics produce an exponentially decreasing q2 behaviour of the form factors. Here we wish to describe in some detail a recent calculation which tries to improve such computations by taking into account the relativistic kinematics of the Qq pair and a realistic description of the interquark interaction' . The model uses meson wave functions that are solutions of the equation (k+xF )2 +mi+ -

V(_ k +

(1-~)P)2+mi-

M2 +p 2 1 XV,(k+Xp,-k+(1 - xP ))+

+ f dk'V(gi,k,P)ik(P,gi-P)=0 .

(56)

Eq.(56) is valid in a moving frame where the meson of mass M has momentum 7i; x is the fraction of the

meson momentum carried by the quark of mass mi . V is the instantaneous potential that in the meson rest frame coincides with the Richardson potental20; in the r-space it takes the form V(r)

33 8 2n A (Ar_

r) f(Ar) )

(57)

where A is a parameter, nf is the number of flavours, and ~~ dg sin(g f(t) = 4 Tr u q

t)

1 - 1 ln(1 +T) q2 ]

.

(58)

This potential behaves linearly when r - oo, whereas at small distances it follows perturbative QCD predictions. For r near the origin one assumes a constant potential V(r) = V(rm)

'' < rM

= a 3M'

(59)

in order to avoid unphysical singularities ; A is a parameter whose fitted value is A=0 .6. The values of the other parameters are m.=md=38 MeV, m,=115 MeV, m,=1452 MeV, mb=4890 MeV and A=397 MeV ; using these parameters, the mass spectrumfor the Qq system has been derived21 . The calculation of the form factors requires the meson wave functions with 1 0 0; in other terms we have H to boost the solutions 0(k) = 0(k, 0(~, -k) obtained in the rest frame by Eq.(56). This is an an old problem, whose general solution is unknown22 but for special cases. A possible approach is to assume transformation properties of the potential and the wavefunctions, but this procedure is unsatisfactory due to his arbitrariness . In Ref. 6 the following strategy was adopted . One limits himself to infinitesimal Lorentz boost, so as to neglect terms 0(p') in the following. As a consequence, one writes

G. NardullilB-meson sendleptonic exclusive decays

54

where 0(k) is the solution of (56) in the meson rest frame and g(k, x) is an unknown function. Instead of trying to derive it from transformation properties of the potential V, one chooses to minimize its effect by an appropriate choice of the parameter x. As a matter of the fact one puts where x is the value which minimizes the function g(x) = maxk Jg(k-' X)J ,

(62)

then g(k,~) will be uniformly small in k and one does not make a large error in neglecting it altogether in the final formulae . As a consequence physical quantities are expressed only in terms of the known rest frame wavefunctions J 0(k). Clearly this procedure is justified provided it is compatible with (at least) approximately relativistic invariance; in other words one has to check that scalar quantities computed in two different reference frames by this method are approximately equal. In order to determine t for B and D mesons one proceeds as follows . Since, due to the approximation, the potential V(r) does not contain spin terms, both D and D* (B and B*) have the same x, namely XD (XB) . Let us now consider the current particle matrix elements (35) that we rewrite as follows : < 01A (0)IM(p) >= ip"fp < 0117i;(0)IV(p,E) >=

ba ob '9

di,~,(k+xp,-k+(1-x)P)x

xP, -k + (1 - x)P-)

(66) where Fij is the flavour matrix, a and ,ß are colour indices, r and s are spin indices . We can now compute Eqs. (63) and (64) in two different frames: the meson rest frame (P 0) and a moving frame with infinitesimal 1pl . By using canonical anticommutation relations for annihilation and creation operators appearing in (65),(66) one gets, in the rest frame:

fp =~ 2 x

fv =e

M

ao 0

dkkz,(k)N~(k)x

k2 (Ei +mi)(Ej +m') [1 00

2

dk k û(k) N I (k)x

k2 x 1+ 3(Ei + mi)(Ej + m.i)

(67)

(68)

where û(k) = k0(k)/(%,F2ir), M is the meson mass, ~ = Tr(JF) is a Clebsch-Gordan coefficient and N(k) _ ~ Ei(k)+m, ~ E,(k)+m, ~ . E, (k) ) E, (k)

On the other hand, in the moving frame with infinitesimal I p I we get fp = E(x) + 6E(x) = E(x) + bE'(x)

(69) (70)

where E(x) can be explicitly computed' and bE(x), bE'(x) are correcting factors containing the integral of the unknown function g(k, x) . We now impose that g(k, x) is computed for x = x, or, in other terms, that [bE(x)]2+[bE'(x)]2 has a minimum ; in this way we obtain for the fractional momentum carried by the heavy quark in the (ed) and (bit.) system respectively: XD

(65)

I dkV,(k +

x bi (k+xP,r,a)di (-k+(1 - x)e,s,Q)10 >

MV

E"fv ,

x b±(k+xi,r,a)d, (-k+(1 - x)~i,s,13)10 >

)rs __ F~ bap 03 (-E'"C"

(63)

where the pseudoscalar and vector meson states are given by

= iFij

IV(P,E) >=

=0.82 ;

xB=0.88 .

(71)

G. Nardulli lB-meson semileptonic exclusive decays

We note that these large values of the fractional momenta are reasonable, since we expect that in heavylight quark mesons the meson momentum is carried in average more by the heavy than by the light quark . If we now neglect SE(XD)7 bE'(XD), bE(XB), bE'(XB), given their expected smallness, we obtain the results: fD ^-' E(XD) ;

fD " ^-' MD . E(xD) ;

fB fs ^-' E(Xs) ; " ^-' MB . E(xB ) -

(1 -

X B)P =

(1 -

XD)p' -

masses of the form factors Fo and Fl. This constraint is satisfied for masses that differ by 15% from the quark model predictions . All these results are as follows (in parentheses are the pole masses) : Fo(0) = 0.69 (m = 7.60GeV) Fl(0) = 0.69 (m = 5.54GeV) V(O) = 0.84 (m = 5.54GeV )

(72)

Ao(0) = 0.81 (m = 6.30GeY) Al (0) = 0.65 (m = 6.73GeV)

(73)

Numerically one finds' E(XD) = 0.222 GeV, E(XB) = 0.257 GeV; by comparing these findings with the results obtained in the meson rest frame, i.e. from Egs .(63),(64) (fD = 0.17 GeV, fD" = 0.53 GeV', fB = 0.23 GeV, fB" = 1 .51 GeV')", one gets deviations by 10% in the B case and by 18% in the D case. These are the typical errors introduced by the neglect of the function g(k, x) in Eq.(60) ; we expect similar errors in the computation of the form factors. These form factors are written as overlaps of the B and D (or D*) wave functions computed for different values of their argument. There is a particular frame where these expressions become simpler and both wave functions are expressed as ik (k + xyi, - k -!- (1 - x)p-), where P is the meson momentum and x the active quark fractional momentum. This happens when (74)

where p" and p'~ refer to B and D* respectively. In this reference frame one can use the approximation (60) and the results (71) for the fractional momenta XB and XD carried by the active quarks in the B and D (or D*) mesons. By considering the limit. 1pl, lp' l -> 0 one obtains the form factors computed at q z = q~,.a.x . Their expression for any value of q' can be obtained by assuming a simple pole behaviour as predicted by single pole dominated dispersion relations, with mesons masses computed in the relativistic potential model. One h.ns : o observe that the condition Fo(0) = Fl(0) poses a strong constraint or tle pole

55

;75)

A2(0) = 0.45 (m = 6.73GeV) A3(0) = 0.81 (m = 6.73GeV ) It is worth stressing that, as shown in Ref. 7, when computed in the heavy quark limit, this relativistic potential model reproduces all the constraints obtained by the effective theory, in particular the absence of O(mQl ) corrections for Fl and A1 at the maximum recoil point q' = q,'nax . Another interesting aspect of the potential models that one can easily compute the decays of B mesons is to excited charmed mesons such as 2S or 1P states . For example, in Ref. 18 a fraction of 3/5 of the inclusive b --+ c semileptonic branching ratio is due to the decay B -+ D* Q vi, another fraction 0.27 is due to B -+ D Q vi and the remaining 13% is due to 2S or 1P states . The method of Ref. 7 can be easily applied to the radially excited resonances 2S: D' and D*'. One finds that

In both cases the contribution of non resonant background seems necessary in order to explain the experimental results (19) and (20).

4. SUMMARY AND CONCLUSIONS We conclude by listing predictions of the different models for the B semileptonic decays into D or D* . For B --+ Dev one has (in parentheses we indicate the model)

G. Nardulli lB-meson sendleptonic exclusive decays

56

BR(B -~ DCP) =1 .6

(0 .04)2

x 10-2 (WBSis )

)2 x 10 -2

=1 .6 (6.04 0.04

=1.7(0 .04)Y

(E'Si7 )

x 10 -2 (CNT7 )

=2.2( Q 04)2 x 10-2 (ISGW i8 ) =2.5 ( Vcb )2 x 10-2 (AWi9) 0.04 x 10-2 (CNOP 5 ) =1.5 ( 0.04 )2 14 =1.75 ± 0.42 ± 0.35 x 10-2 (Exp. ) (77) For the semileptonic decay B --> D*eP the results are:

BR(B --+ D* fo) =4.3( 0

=5.1(

04)2

x 10-2 (WBS is )

Vcb x 10-2 (K S17) 0.04 )2

=5.2 Kcb )2 x 10-2 (CNT7 ) (0 04 =4.9 ( Vcb x 10-2 (ISGW i8 ) Q.04)2 =4.7 ( Vcb )2 x 10-2 (AW i9 ) 0.04 =4.6 ( Vcb )2 x 10-2 (CNOP 5 ) 0.04 =4.8±0.4±0.7 x 10-` (Exp. "') (78) Finally we list the results for the asymmetry pa-

rameter a defined in Eq. (49). Imposing the cut pt > 1.4 GeV/c as in the CLEO experiment" one obtains: a = 0.44 (WBS is ) = 0.37 = 0.71 = 0.13

(KS1 7 )

(CNT7 )

(ISGW i8 )

= 0 .21

(AW i9 )

= 0.74

(CNOP5 )

(79)

= 0.65 ± 0.66 ± 0.25 (CLE0 1 5 ) = 0.7 ± 0.9

(ARGUS i5 )

(the ARGUS result is obtained with a cut in the lepton momentum at pe > 1 .0 GeVlc). From these results it appears clear that the most sensitive parameter to discriminate between different models is the asymmetry parameter. Better quality of the data on branching ratios would improve the discrimination only marginally ; on the other hand precise data on a would certainly falsify most of (hopefully not all) the models so far invented to describe B semileptonic decays into semileptonic charmed mesons .

ACKNOWLEDGMENTS It is a pleasure to thank P. Colangelo and N. Paver for their collaboration on the themes -discussed in this talk and for many helpful discussions.

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G. Nardulli /B-meson sendleptonic exclusive decays

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57

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