Semileptonic B-decays into positive-parity charmed mesons

Physics Letters B 269 ( 1991 ) 201-207 PHYSIC S L E T T E R S B North-Holland

Semileptonic B-decays into positive-parity charmed mesons

P. Colangelo a, G. Nardul l i a,b, A.A. Ovch inn ikov c and N. Paver d " INFN, Sezione di Bari, 1-70126 Bari, Italy b Dipartimento di Fisica dell'Universitd di Bari, 1-70126 Bari, Italy c InstituteforNuclearResearch oftheAcademyofSciencesofthe USSR, SU-117312Moscow, USSR d Dipartimento di Fisica Teorica dell'Universitd di Trieste, Trieste, Italy

and INFN, Sezione di Trieste, Trieste, Italy

Received 22 July 1991

By using QCD sum rules we compute the branching ratios BR(B--,D**~9) and BR(B-,D~*£9) where D** and D~* are the positive parity 1 + and 0 + orbital excitations of D* and D respectively. We find BR ( B ~ (D~* + D** ) £9 ) -~ 3 X 10- 3.

I. Introduction

Recent experimental data from both CLEO [ 1,2 ] and ARGUS [3 ] Collaborations have shown evi- dence that the decay modes B~D~9 and B--,D*£9 (~=e, Ix) do not saturate the semileptonic b ~ c inclusive decay. For example, CLEO finds

B R ( B ~ D ~ - 9 ) + BR (B--,D*~-9)

=(6.1_+0.6_+1.1))<10 -2 (1)

and

BR(1)--. (D°+ D+ )X£-9)

= (9.5_+ 1.2_+ 1.2))< 10 -2 , (2)

which means that about one third of the events arises from decay modes different from D£9 and D*~9. Similar results are obtained by ARGUS.

From a theoretical point of view, these data imply rather large corrections to the prediction obtained in the limit mQ--,oo(mQ=mb, me), (mb--mc)/(mb + m c ) ~ 0 ; indeed, in this theoretical limit, one can show [4 ] that the semileptonic width of the B meson can be calculated via a parton model formula and is completely saturated by the exclusive channels with

the pseudoscalar (D) and the vector (D*) s-wave mesons in the final state. This is clearly an appealing result, because in this approximation it would allow to express all semileptonic heavy meson form factors in terms of one universal function [ 5 ] ~. A different approach to exclusive semileptonic decays is pro- vided by potential models [ 8-11 ], which leave some room for additional decay modes. For example in ref. [9] radial and orbital D, D* excitations account for 13% of the inclusive semileptonic branching ratio.

The aim of this letter is to estimate the fraction of the total semileptonic width which is due to the positive parity charmed meson resonances: the D** (2420) 1 ÷÷ meson and the scalar 0 ÷+ that we denote by D~*. These transitions represent a well-defined set of 1/mQ corrections to the mQ-,Oo approximation. Although many non-resonant channels are open (Dn, Dnn, ...), one can expect that resonance behaviour enhances the D** and D~* modes, so that it is sensible to account for their contributions to the inclusive leptonic branching ratio (2). Our calculation will be performed using the QCD sum rules approach [ 12,13 ], along the lines of previous calcula- tions of the B--. D, D* semileptonic modes [ 14,15 ].

Supported in part by the Italian Ministry o f University and of Scientific and Technological Research.

at The relevance of the 1/mQ corrections to the heavy quark limit has been recently discussed, for purely leptonic decays in lat- tice QCD [6] and in the QCD sum rule approach [7].

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved. 201

Volume 269, number 1,2 PHYSICS LETTERS B 24 October 1991

Thus, for completeness, we also compute the latter transitions by the same approach.

2. T h r e e - p o i n t func t ions

We begin by writing down the relevant form factors

1 : (D;*(p)I(y~,~sbIB(p')) 1

=g+(p '+p)~+g_qu , (3)

1 _ (D**(p, e)Ig~,~sbIB(P') ) 1

=if~u,~pe*~(p' + p)~q ~ , (4)

1 _ (D**(p, e ) [gTublB(p ' ) ) 1

- - ~ t . t - f o e , + f + ( e * . p ' ) ( p + p ) , , + f _ ( e .p )qu, (5)

where q = p ' - p , e u is the D** polarization vector and the form factors g+, f, fo, f+ are functions of q2. To- gether with these equations we also consider the form factors for the transitions B-~D and B~D*:

( D ( p ) [gyub[ B(p ' ) )

= a + ( p ' + p ) . + a _ q ~ , , (6)

1 = (D*(p, e) [d7.b[ B(p ' ) ) 1

= iFeu~fle*" (p, + p ) " q # , (7)

1 _ (D*(p, e ) ] (7 .75b[B(p ' ) ) 1

=Foe~, + ~ + (e* 'p ' ) (p'+p)~ + F_ (~ "p )q~, (8)

where G+, F, Fo+ are functions o fq 2. In order to write down QCD sum rules for the form

factors in eqs. ( 3 ) - ( 8 ) we proceed as follows. For B-~ D~* we consider the three-point current correlator

II~,(p, p', q )= i 2 f dx d y e x p ( i p x - i p ' y )

× (0l T{q(x)c(x) , JA(o), 6(Y)75q(Y)} [0) , (9)

while for the B~D** transition we consider the correlators

HV~ A (p, p ', q ) = i 2 j dx dy exp ( ipx - ip'y )

X (OIT{(t(x)y~75c(x), j r 'A(0) , 6(y)y5q(Y)}lO),

ClO)

V A where J~, =(?ub, Ju-(VuYsb and q(x) denotes the light quark operator: q = u, d.

The correlators for the transitions B ~ D and B ~ D* are obtained from eqs. (9), (10) by the changes V,--~A, qc ~qysc and qT~ysc-,qT,c.

The hadronic tensors appearing in (9) and (10) can be decomposed into Lorentz structures as follows:

Hu=/)+ (p+p')u + /)_ q z ,

V ~ t t ! Hu~ =Hogu, +/)1P~P ~ + ]fflzp uP ~

+ H3 P, p, + Ha p l, P, ,

HA, = i/)e~u, p~p 'fl . ( 11 )

We are only interested in the amp l i t udes / )+ , / ) , no and/)p = (/)l +/)2 ) /2, since in the semileptonic decay matrix element the other ones give vanishing contributions.

As is well known, the QCD sum rules method bas- ically consists in matching, by the duality argument, two different expressions of the current correlators. 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 modelled by perturbative QCD. The second expression is obtained by using the short-distance operator product expansion, which consists of the leading perturbative QCD term, plus non perturbative power corrections representing the breaking of asymptotic freedom. The latter ones are proportional to quark and gluon operator vacuum matrix elements, or- dered by increasing dimensionality, which are determined phenomenologically. Using these alternative representations, predictions for hadron masses and couplings can be obtained in terms of quark masses, as 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/7,- any one of the amp l i t udes / )+ , / ) , / ) o and/)p, H, obeys the double dispersion relation

202


1 f pi (s ,s ' ,Q 2) Hi(p2, p,2, Q2)= _ 4n 2 ( s - p 2 ) ( s ' - p '2) ds ds'

+ subtraction terms, ( 12 )

where Q2= _ q 2 > 0. Saturating eq. (12) 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

(P+P')ufi+ (s, s', Q2)

= - 4n2fi+ ( s - m~a. )~+ ( s ' - m ~)

X (0[qcl D~* ) (D~* [jA [B) (BI ~ysq[ 0)

+ [1 -O(so -s)O(s'o - s ' ) ]

× (p+p'),,fi+ (s, s', Q2)lay, (13)

where ~+ (s, s', Q2)IAF is the asymptotic freedom contribution to fi+ (s, s', Q2) which is supposed to represent higher mass states, and so, s; are the continuum effective thresholds in the variables s and s ' respectively. An analogous expression is obtained for H~,, in eq. (10).

Different methods can be used to improve the con- vergence of the dispersion relation (12). We use the double Borel transform to enhance the hadronic ground state contribution, and, at the same time, to minimize the role of the continuum whose threshold cannot be determined a priori, and the number of needed vacuum condensates in the operator product expansion. This method introduces two mass parameters M l and M2; from eq. (12) we obtain

if H,(M 2 ,M 2 , Q 2 ) = _ 4n ~ p,(s,s', Q2)

x e x p ( - s / M 2) e x p ( - s ' / M Z ) d s d s ' , (14)

where H,(M~, M~, Q2) is obtained applying the Borel transform to the operator product expansion of/ / , (p 2, p,2, Q2) .

Before discussing eq. (14) let us consider the vacuum-to meson matrix elements appearing in (13).

3. Leptonic decay constants

In order to compute the hadronic contribution to H~, H ~ we need to estimate separately the following

vacuum-to-meson transition amplitudes ( mq = 0 )"

<0lqclD~* > = f_. L.~,0.m~,0. (15) mc

and

(0l@uyscl D**(P, e) ) = m 2**

G(P) • (16) gD**

The decay constants in (15) and (16) can be obtained by two-point-function QCD sum rules [ 13 ] in the Borel version analogous to that for three-point function sum rules followed in this letter. The method is the same as the one employed by several authors for the determination of the negative-parity mesons D and D* decay constants:

fDm~ ( O l # i y s c l D ) - (17)

mc

and

m~. (01@uclD*(p, e ) ) = - - eu(p) . (18)

gD*

In order to computefD (and its analoguefB) we consider the correlator of two pseudoscalar currents, whereas to compute gD* (and its analogue gB.) we consider the correlator of two vector currents. The corresponding results for fD~* and go** are obtained by considering correlators of two scalar currents and two axial currents respectively, and by changing mc-~ - m c in the explicit expressions of the operator product expansion.

The expressions for the perturbative term (up to O (as ) ) and for the power corrections relevant to our case can be found in the literature [ 7,13,16,17 ], and we shall not report them explicitly. We just mention that we use the following set of parameters [ 16,18 ] : mc=1.35_+0.05 GeV, mb=4.60_+0.10 GeV, (qq) = (--0.250GeV) 3, ( (as /g)G 2 ) = 0.012GeV 4, ( q a , , • ½2aqGl"")=m~(qq) with mg=0 . 8 GeV 2, and AQCD---- 100--150 MeV. With these inputs we find for the leptonic constants the results displayed in table 1. These values are obtained with the continuum thresholds So = 6-8 GeV 2 and s~ = 36-40 GeV 2, and adopting the general criterion that a hierarchy should exist in the contributions from the different operators in the operator product expansion. Specifically, we assume that: (i) the QCD perturbative contribu-

203


Table 1 Leptonic decay constants for different heavy mesons.

Leptonic constant Theoretical result

fD 195 --+20 MeV fa 180 -+ 30 MeV gB- 25 -+ 4 gD" 7.8 -+ 0.8 fD3* 170 -+ 20 MeV go-- 9.8-+ 1.5

tion must be larger than the D = 3 quark condensate contribution, since the former should be the leading term; (ii) the D = 5 operator contribution must be less than 30% of the quark condensate D = 3 contribution (the contributions of D = 4 and D = 6 operators turn out to be negligible). These criteria fix the lower bound for the Borel mass parameter M 2. Moreover we assume that: (iii) the contribution of the low-lying meson to the sum rule must be larger at least by a factor of 2-3 as compared to the continuum, in order to minimize the error induced by the continuum par- ameterization of higher mass resonances. This as- sumption fixes the upper bound for M 2. With these criteria we always find duality regions in the Borel parameter M 2, whose size, together with the spread in the continuum threshold So, determines the theoretical uncertainties of table 1. We also notice that, by taking 1 /M 2 derivatives of the sum rules, we obtain meson masses in agreement with experiment for B, D, D* and D** and the prediction mD3* = 2.5 + 0.1 GeV for the scalar state. In particular, we stress that our results for fa, gB* include also an uncertainty of + 100 MeV in the b-quark mass and of + 50 MeV in the charm quark mass. Our findings are in general agreement with the determinations of the leptonic decay constants obtained by other versions of the QCD sum rule method [18 ], whenever a compari- son is possible.

4. Determination of the form factors

Turning to the determination of the form factors in eqs. ( 3 ) - ( 8 ) we consider the correlators (9) and (10) and saturate them by the low-lying meson states B, D~* and B, D** respectively, similarly to eq. ( 13 ). As already said in section 3, we model higher mass

continuum by the perturbative contribution to the double spectral densities pi(s, s', Q2) above the thresholds So, s0. On the other hand, we compute the correlators by making use of perturbation theory and including non perturbative terms arising from higher dimension operators. After applying the double Borel transform and matching the two expressions, we obtain the following sum rules for the form factors;

mcm2b 2 exp(rnZD~ + m 2 ] g+ (Q2)=--fD~*fBmD~*mB M~ ~22,l

2 2 X/)+ (M~, M 2 , Q 2 ) , ( 1 9 )

rnbgo.. I'm2** m2 ~ f(Q2)= 2fam2**m~ e x p ~ _ 2 + M22, /

X / ) ( M 2, M 2, Q 2 ) , (20)

mbgD.. { m 2.. rn ~'~ f ° ,+(Q2)= fam~..rn 2 e x p ~ - ~ f 2 + ~ /

×/)o,p(M~, M 2, Q2) , (21)

where M1, and Mz are the Borel mass parameters, whereas / ) + ( M 2, M22, Q2), / ) ( M 2, M22, Q2), /)o,p (M 2 , M22, Q2) are the Borel-transform of the Lo- rentz invariant ampl i tudes/ )+, / )and/)o ,p which ap- pear in eq. ( 1 1 ). The corresponding expressions of the form factor for the transitions B ~ D and B ~ D * are obtained by the changes g+ ~G+,f~F, fo,+ --,Fo,+, fDr*o--~fD, g o * * - - ~ g D * * - * g D . , m D $ . - ~ m D and m D . . - * too,. Moreover, one has to replace the amplitudes / )+, /) , / )o,p with the corresponding invariant amplitudes H+, H and Ho,~. Each of these Lorentz invariant amplitudes is expanded as follows:

H,=H~ °) + H [ 3) + H ! s) "{'-H} 6) + . . . . (22)

where H} m are double Borel transformed terms arising from the perturbative contribution ( D = 0 ) and from the condensates of dimension D = 3, 5, 6 respectively. We shall not write down the explicit expressions since they are rather cumbersome. However, they can be easily obtained from the corresponding expressions found in ref. [14] for the transitions B--.D and B~D*, by changing there mc--. - m ~ ~2

HE We confirm the formulae of ref. [ 14] except for the contribution of the operators of dimension D=6. Explicit formulae will be given elsewhere. In any case, the contribution of these operators turns out to be negligible.

204


Table 2 Values at Q2= 0 of the form factors for the different transitions. Theoretical errors only arise from the spread of the Borel mass parameters.

g+(0) or G+(0) f(0) or F(0) (units GeV-' )

f÷(0) or F+ (0) (units GeV -t )

fo(o) or Fo(O) (units OeV)

B--,D(0-) 0.67+_0.19 B-~D*(1-) 0.12+_0.02 -0.05+_0.02 4.35+_0.15 B--, D;* (0 + ) 0.30 _+ 0.03 B~D**(1 + ) 0.10+-0.03 -0.02+_0.02 0.9 +_0.3

Our results for the Q2 = 0 values of the various form factors are reported in table 2. As for the Q2-dependence of the form factors, we assume a simple pole behaviour Fi(Q2)=Ei(O)/(I+Q2/m2), as sug-

gested by dispersion relations, where the pole masses are those of be mesons with appropriate quan tum numbers. The results for the widths are rather insen- sitive to the precise value of these masses: for defi- niteness we take m = 6 . 3 4 GeV for the 1- pole (for

the form factors G+, F, f0 and f+ ) and m = 6 . 7 3 GeV for the 1 + pole (for the form factors g+, f Fo and F+ ). An explicit evaluation of the Q2 dependence of

the form factors by QCD sum rules gives results compatible with this behaviour, although with large errors.

The results presented in table 2 are obtained by adopting the same criteria already employed for the calculation of the leptonic decay constants of table 1; the values of the thresholds are So- 12-15 GeV 2 and s ; -~ 40 GeV 2 (but the results are not sensitive to these

values) whereas the values of the Borel mass parameters M 2 and M 2 are generally within the ranges 2-5 GeV 2 and 5-10 GeV 2. We always find stability regions where our criteria are verified, except for the form factors f+, F+, where the D = 3 operator domi- nates. However, also in this last case the hierarchy among the contr ibut ions of higher d imens ion operators ( D = 3 , 5, 6) is satisfied. The meson masses obtained from three-point functions sum rules are compatible with the masses obtained using the two-point functions employed in section 3.

In order to show the stability of the computed couplings with respect to the Borel parameters, we report in fig. 1 a plot of g+ (0) as a function o f M 2 and M 2.

The depicted surface encompasses the duality region where our criteria are fulfilled. Similar plots are obtained for the other form factors.

We observe that the semileptonic widths depend weakly on the form factors f+ or F+ (they are rather

g+(O 0 .45

0 4

0 .35

0.3

0 .25

0 2

0 .~5

0 1

O 05

z

Fig. 1. Stability surface of the B~ D~* form factor g+ (0) versus the Borel mass parameters M~ and M~. Units of M~ and M~ are GeV 2 The dashed lines show the working region.

important , on the contrary, for the asymmetry parameter) ; therefore the relatively large errors in their values do not affect the final results for the branching

ratios. Our results for the decays into the negative parity

charmed mesons in table 2 differ from those in ref. [ 14 ] because of the different values of the leptonic decay constants employed in the present letter #3.

From the entries of table 2 we can easily compute

the branching ratios for the exclusive decays. The re- suits are reported in table 3, which shows that a good agreement with the experimental data for the transi- t ion B ~ D and B- , D* is obtained with Vcb ~ 0.04.

~3 The authors of ref. [ 14 ] rely for their values offB on ref. [ 19 ] where a larger value of the b-quark mass is considered.

205


Table 3 Branching ratios for the B semileptonic decays into negative- and positive-parity charmed mesons. Experimental data are an average of the ARGUS and CLEO results [20]; rB= 1.24 ps.

Decay modes BR (theory) BR(experiment)

B--,D!~9 1.5 ( Feb/0.04)2 X 10 -2 ( 1.75 +0.42 --+0.35) X 10 -2 B~D*~v 4.6 (Vcb/O.04)2×lO -2 (4.8 -+0.4 _+0.7 ) X I 0 -2 B--*DS*~9 0.15 ( Vcb/0.04)2 X 10 -z B--. D**~9 0.15( Vcn/O.04)2 X 10 -2

As for the positive-parity charmed mesons, our predictions indicate that the sum of the D~* (0 + ) and D**(1 ÷) contribution is about 5% of the sum BR (B--*D~9) + BR(B~D*I~9).

Another experimentally interesting quantity is the asymmetry parameter c~, defined as follows:

cr=2 FL(B-,D*19) 1 (23) FT(B--* D*~9)

where FL and FT are the widths for the production of longitudinal are transverse D* respectively. Imposing the cut p~>~ 1.4 GeV/c as in the CLEO experiment [21 ], we find

c~=0.74, (24)

to be compared to the CLEO [ 20 ] experimental value (similar results are obtained by the ARGUS Collaboration )

ot=0.65_+ 0.66 +0 .25 . (25)

Let us finally mention that the widths for semileptonic B decays into D and D* have been previously computed by potential models [6,7,8,11 ], with results in general good agreement with the data; as for the positive-parity charmed meson final states, the outcome of ref. [ 7 ], indicating a small p-wave (qQ) component in the inclusive spectrum, is compatible with our findings reported in table 3.

In conclusion, the present calculation supports the idea that 1/mQ corrections to the mQ--,oe limit should be individually small with respect to the leading pro- cesses B ~ D and B--,D*; moreover, our results indicate that their size is too small to quantitatively ex- plain the difference between the data ( 1 ) and (2) by just the effect of the transitions into 0 + and 1 + orbit- ally excited charmed mesons. This result should not change by including the so-called bifocal operators [22], since their effect is negligibly small due to the

heavy masses (mb, mc) involved. Therefore, the difference between ( 1 ) and (2) should arise, in addi- tion to the states considered here, from the whole multiplicity of higher radial excitations and of non- resonant background states.

Acknowledgement

One of us (A.A.O.) would like to thank the Inter- national Centre for Theoretical Physics, Trieste, and the Istituto Nazionale di Fisica Nucleare, Italy, for support during the course of this research.

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Semileptonic B-decays into positive-parity charmed mesons