A fresh look at the determination of | V cb | from B → D ⁎ lν

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

A

fresh

look

at

the

determination

of

|

V

_cb

|

from

B

→

D

∗

l

ν

Dante Bigi,

Paolo Gambino,

Stefan Schacht

∗

Dipartimento di Fisica, Università di Torino & INFN, Sezione di Torino, I-10125 Torino, Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Article history:

Received28March2017 Accepted10April2017 Availableonline13April2017 Editor:B.Grinstein

We use recent Belle results on B¯0_→D∗+_l−

_ν

_¯

l decays to extract the CKM element |Vcb| with two

different but well-founded parameterizations of the form factors. We show that the CLN and BGL parameterizations lead to quite different results for _|Vcb| and provide a simple explanation of this

unexpectedbehaviour. Along lastingdiscrepancy betweenthe inclusiveand exclusive determinations of|Vcb|mayhavetobethoroughlyreconsidered.

©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

SemileptonicB decays offerthemostdirectwaytodetermine theelement

|

Vcb

|

oftheCabibbo–KobayashiMaskawa(CKM)quark

mixingmatrix.Thisparticularelementplays a centralrole inthe analyses of the CKM matrix unitarity and in the SM prediction ofFlavour Changing Neutral Current transitions. For a long time thetwoavailablemethodstoextract

|

Vcb

|

fromexperimentaldata,

basedonexclusive(singlehadronicchannel)andinclusive(sumof allhadronicchannels)reconstructionofthesemileptonicB decays,

havebeeninconflict.Thetwo methodsare basedonvery differ-enttheoreticalfoundationsandwhileanewphysicsinterpretation seems currentlydisfavoured ongeneral grounds[1],it is not ex-cluded[2]andisparticularlyinterestinginviewoftheanomalies inB

→

D(∗)

_{τ ν}

_[3].

Atpresent,thetwomostprecisedeterminationsare

|

Vcb| = (38

.

71

±

0

.

75

)

10−3

,

(1)

based on the HFAG global combination of B

→

D∗



ν

results [3]

together withthe FNAL-MILC Collaboration calculation [4] ofthe relevantformfactoratzero-recoil, i.e., whenthe D∗isproducedat restinthe B restframe,and

|

Vcb| = (42

.

00

±

0

.

65

)

10−3

,

(2)

obtainedintheHeavyQuarkExpansionfromafittothemoments ofvariouskinematicdistributionsininclusivesemileptonicdecays

[5]. The difference between (1) and (2) is 3.3

σ

, which becomes 3.1

σ

oncetheQEDcorrectionsaretreatedinthesamewayinboth

*

Correspondingauthor.

E-mail addresses:dante.bigi@to.infn.it(D. Bigi),gambino@to.infn.it(P. Gambino), schacht@to.infn.it(S. Schacht).

cases.Therearealternativecalculationsofthe B

→

D∗ zero-recoil formfactoronthelattice[6]orbasedonHeavyQuarkSumRules

[7,8]buttheyhavelargeruncertainties.

In a recent paper [9]we havereviewed and slightlyupdated the 20 years-old formalism to parameterize the form factors in

B

→

D



ν

in a waythat satisfies important unitarity constraints. Using up-to-date lattice calculations of the form factors and the availableexperimentalresults,wehaveshownthatthe parameter-izationdependenceissmallandobtained

|

Vcb

|

=

40

.

49

(

97

)

10−3,

compatiblewithboth(1)and(2)andonlyslightlylessprecise. The purposeof thisLetteris to performa similar analysisfor the B

→

D∗



ν

decay.WetakeadvantageofthenewBelle prelim-inary results[10] which,for thefirst time, include deconvoluted kinematicand angulardistributions withcomplete statisticaland systematicerrorsandcorrelations,withoutrelying ona particular parameterizationof theformfactors. Wefirst review the formal-ismandthedataandthendescribeourfitsanddiscusstheresults.

2. Formfactorparameterizations

Inthelimitofmasslessleptonsthefullydifferentialdecayrate isgivenby d

( ¯

B

→

D∗l

ν

¯

l

)

dw d cos

θ

vd cos

θ

ld

χ

=

η

2 EW3mBm2D∗ 4

(

4

π

)

4



w2

₋

₁

_×

(

1

−

2wr

+

r2

)

G2_F

|

Vcb|2

×



(

1

−

cl

)

2s2vH+2

+ (

1

+

cl

)

2s2vH−2

+

4s_l2c2_vH2₀

−

2s2_ls2_vcos 2

χ

H₊H₋

−

4sl

(

1

−

cl

)

svcvcos

χ

H+H0 (3)

+

4sl

(

1

+

cl

)

svcvcos

χ

H−H0

} ,

http://dx.doi.org/10.1016/j.physletb.2017.04.022

0370-2693/©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

wherer

=

mD∗

/

mB,cv

≡

cos

θ

v,cl

≡

cos

θ

l,andcorrespondinglyfor

sin

θ

v andsin

θ

l.

θ

v,l and

χ

are thethreeanglesthat characterise

thesemileptonicdecay.Wealsousethekinematicparameter

w

=

m

2

B

+

m2D∗

−

q2

2mBmD∗

,

(4)

whereq2_{istheinvariantmassoftheleptonpair.}

The helicity amplitudes H_±,0 in Eq.(3) are givenin terms of threeformfactors,see e.g. Eqs. (3-5)ofRef.[10].Inthe Caprini– Lellouch–Neubert (CLN) parameterization [11] one employs the formfactor hA1

(

w

)

andthe ratios R1,2

(

w

)

.Traditionally, the ex-perimentalcollaborationsuse

hA1

(

w

)

=

hA1

(

1

)



1

−

8

ρ

2z

+ (

53

ρ

2

−

15

)

z2

− (

231

ρ

2

₋

₉₁

₎

_z3



_,

R1

(

w

)

=

R1

(

1

)

−

0

.

12

(

w

−

1

)

+

0

.

05

(

w

−

1

)

2

,

(5) R2

(

w

)

=

R2

(

1

)

+

0

.

11

(

w

−

1

)

−

0

.

06

(

w

−

1

)

2

,

wherez

= (

√

w

+

1

−

√

2

)/(

√

w

+

1

+

√

2

)

andtherearefour in-dependentparameters intotal.Wewilldiscusstheingredientsof thisparameterization later on. Afterintegration over the angular variables,the w distributionisproportionalto[11]

F

2

₍

_w

₎

₌

_h2 A1

(

w

)



1

+

4 w w

+

1 1

−

2wr

+

r2

(

1

−

r

)

2



−1

×



21

−

2wr

+

r 2

(

1

−

r

)

2



1

+

R2₁

(

w

)

w

−

1 w

+

1



+



1

+ (

1

−

R2

(

w

))

w

−

1 1

−

r



2

.

(6)

An alternative parameterization isdue to Boyd, Grinstein and Lebed(BGL)[16].Intheir notationthehelicityamplitudes Hi are

givenby H0

(

w

)

=

F

1

(

w

)/

q2

_,

H_±

(

w

)

=

f

(

w

)

∓

mBmD∗



w2

₋

_{1 g}

₍

_w

_).

The relations between the relevant form factors in the CLN and BGLnotationare f

=

mBm∗D

(

1

+

w

)

hA1

,

g

=

hV

/

mBm∗D

,

F

1

= (

1

+

w

)(

mB

−

mD∗

)

√

mBmD∗A5

,

and R1

(

w

)

= (

w

+

1

)

mBmD∗ g

(

w

)

f

(

w

)

,

R2

(

w

)

=

w

−

r w

−

1

−

F

1

(

w

)

mB

(

w

−

1

)

f

(

w

)

.

ThethreeBGLformfactorscanbewrittenasseriesinz,

f

(

z

)

=

1 P1+

(

z

)φ

f

(

z

)

∞



n=0 anfzn

,

F

1

(

z

)

=

1 P1+

(

z

)φ

F1

(

z

)

∞



n=0 aF1 n zn

,

(7) g

(

z

)

=

1 P1−

(

z

)φ

g

(

z

)

∞



n=0 angzn

.

Table 1

Relevant

B

(c∗)masses.The1−resonances

areasinRef.[9].

Type Mass (GeV) References 1− 6.329 [12] 1− 6.920 [12] 1− 7.020 [13] 1− 7.280 [14] 1+ 6.739 [12] 1+ 6.750 [13,15] 1+ 7.145 [13,15] 1+ 7.150 [13,15] Table 2

Furthernumericalinputs (uncertain-tiesaresmallandcanbeneglected). Thecalculationofχ˜T 1−(0)andχ1T+(0) followsRefs.[9,17]. Input Value mB0 5.280 GeV mD∗+ 2.010 GeV ηEW 1.0066 ˜ χT 1−(0) 5.131·10−4GeV−2 χT 1+(0) 3.894·10− 4_GeV−2

IntheseequationstheBlaschkefactors P1± aregivenby

P₁±

(

z

)

=

n



P=1 z

−

zP 1

−

zzP

,

(8) wherezP isdefinedas(t±

= (

mB

±

mD∗

)

2) zP

=

t₊

−

m2_P

−

√

t₊

−

t₋

t₊

−

m2_P

+

√

t₊

−

t₋

,

and the product is extended to all the Bc resonances below the

B-D∗ threshold (7.29 GeV) with the appropriate quantum num-bers(1+ for f and

F

1,and1− for g).We usethe Bc resonances

reported inTable 1, butdonot includethe fourth1− resonance, whichistoouncertainandclosetothreshold.

The Bc resonancesalsoenterthe1− unitaritybounds(see

be-low) as single particle contributions. The outer functions

φ

i for

i

=

g

,

f

,

F

1,canbereadfromEq. (4.23)inRef.[16]:

φ

g

(

z

)

=

nI 3

π

χ

˜

T 1−

(

0

)

24r2

(

1

+

z

)

2

(

1

−

z

)

−12

[(

1

+

r

)(

1

−

z

)

+

2

√

r

(

1

+

z

)

]

4

,

φ

f

(

z

)

=

4r m2_B

nI 3

π χ

T 1+

(

0

)

(

1

+

z

)(

1

−

z

)

32

[(

1

+

r

)(

1

−

z

)

+

2

√

r

(

1

+

z

)

]

4

,

φ

_F1

(

z

)

=

4r m3_B

nI 6

π χ

T 1+

(

0

)

(

1

+

z

)(

1

−

z

)

52

[(

1

+

r

)(

1

−

z

)

+

2

√

r

(

1

+

z

)

]

5

,

where

χ

T 1+

(

0

)

and

χ

˜

T

1−

(

0

)

areconstantsgiveninTable 2,andnI

=

2

.

6 representsthe numberofspectator quarks(three), decreased by a large andconservative SU(3) breaking factor. Notice that at zerorecoil(w

=

1 orz

=

0)thereisarelationbetweentwoofthe formfactors

F

1

(

0

)

= (

mB

−

mD∗

)

f

(

0

).

(9)

The coefficients of the expansions (7) are subject to unitarity boundsbased on analyticityandthe OperatorProduct Expansion appliedtocorrelatorsoftwohadronic

¯

cb currents.Theyread[16]

∞



i=0

(

ang

)

2

<

1

,

∞



i=0



(

anf

)

2

+ (

aFn1

)

2



<

1

,

(10)

andensurearapidconvergenceofthez-expansionoverthewhole physical region, 0

<

z

<

0

.

056. Ofcourse, the series (7) need to betruncatedatsomepower N. Ingeneralwefindthat a trunca-tionatN

=

2 issufficientforthe

|

Vcb

|

determination,butwehave

systematicallycheckedtheeffectofhigherordersbyrepeatingthe analysiswithN

=

3

,

4,findingverystableresults.

Theunitarityconstraints(10)canbe madestrongerbyadding otherhadronicchannelswiththesamequantumnumbers.For in-stance,theformfactor f₊enteringthedecayB

→

D



ν

contributes to the left hand side of the first equation in (10). Since lattice calculations and experimental data determine f₊

(

z

)

rather pre-cisely[9],one canreadilyverifythatitscontributionisnegligible. Moregenerally,it iswell-known thatHeavy Quark Symmetry re-latesthe various B(∗)

_→

_D(∗) _{formfactors in a stringentway:in}

theheavyquarklimittheyarealleitherproportionaltotheIsgur– Wisefunctionorvanish.Theserelationscan beusedtomakethe unitarityboundsstronger[11,16],andto decreasethenumberof relevantparameters.TheCLNparameterizationisbuiltoutofthese relations,improved withperturbative and O

(

1

/

m

)

leading Heavy Quark Effective Theory(HQET) power correctionsfromQCD sum rules,andoftheensuingstrong unitaritybounds.Withrespectto the original paper[11], the experimental analyses have an addi-tionalelementofflexibility,astheyfitthezerorecoilvalueofR1,2 directlyfrom data,rather than fixing them at their HQET values

R1

(

1

)

=

1

.

27,R2

(

1

)

=

0

.

80.ItisquiteobviousthattheHQET rela-tionsemployedinRef.[11] haveanon-negligibleuncertainty.We willnotdiscussherehowthiswasestimatedandincludedin[11], butit should be recalled that the accuracy of the parameteriza-tion forhA1

(

w

)

in Eq. (5)was estimatedthere to be betterthan 2%.Suchanuncertainty,completelynegligibleatthetime,isnow quiterelevantascanbeseeninEqs.(1),(2).Howeverithasnever been included in the experimental analyses. Similarly, the slope andcurvatureofR1,2inEq.(5)originatefromacalculationwhich issubjectto O

(

2

_/

_m2

c

)

andO

(

α

s

/

mc

)

correctionsandto

uncer-taintiesintheQCDsumrulesonwhichitisbased.1

TheCLNandBGLparameterizationsarebothconstructedto sat-isfytheunitaritybounds.TheydiffermostlyintheCLNrelianceon next-to-leadingorderHQETrelationsbetweentheformfactors.In thefollowingwearegoingtoverifyhowimportantthisunderlying assumptionisfortheextractionofVcb,remainingmainlyagnostic

onthe validity of the HQET relations,a matter whichultimately willbe decided by lattice QCD calculations.2 _{Ourstrategy in the}

followingwillbe toperformminimum

χ

2 _{fitsto the} experimen-taldatausingtheCLNorBGLparameterizations;inthelattercase wewilllookfor

χ

2_{minimawhichrespecttheconstraints(10)and} evaluate1

σ

uncertaintieslookingfor



χ

2

₌

_{1 deviations.} 3. Fitsandresults

Inour

χ

2_{fitsweusetheunfoldeddifferentialdecayrates} mea-suredinRef.[10].The BelleCollaborationprovides the w,cos

θ

v,

cos

θ

l, and

χ

distributions, measured in10 bins each, fora total

of40observables, andtherelativecovariancematrix.Inaddition, likeRef.[10],inthefollowingwealwaysusethevalueoftheform factorhA1 calculatedatzero-recoilonthelattice[4],

hA1

(

1

)

=

0

.

906

±

0

.

013

.

(11)

1 _{Thesepointsarealsoemphasizedin[18],whichappearedaswewereaboutto}

publishthispaperontheArXiv.

2 _{Asnotedin[9],recentlatticecalculationsdifferfromtheHQETratiosofform}

factorsatthelevelof10%.

Fig. 1. Comparison of fit results with different parametrizations.

This is the only form factor relevant at zero-recoil, and to the best ofour knowledge thisFermilab/MILCcalculation is the only publishedunquenchedcalculation.Amongolderquenched calcula-tions, Ref.[19] extendsup to w

=

1

.

1,butwewill notemployit herebecauseoftheuncontrolledquenchinguncertainty.

Asfar asthedetermination of

|

Vcb

|

isconcerned, thepurpose

ofafittoB

→

D∗



ν

observablesisthereforesimplytoextrapolate

the measurements to the zero-recoil point, where (11) provides thenormalization. Asthedifferentialwidthvanisheslike

√

w

−

1 as w

→

1,seeEq.(3) andFig. 1(a),theextrapolation isnot triv-ial. Like in the caseof B

→

D



ν

,the situation isset to improve significantly assoon aslattice calculationsof theform factors at non-zero recoil will become available, but for the moment it is importantto keepin mindthat theextrapolationshould be con-trolled by the low recoil behaviour of the form factors. In this contexttheangularobservablesprovideverylittleinformation,as they are integrated over the full w range andreceive negligible contributionfromthesuppressedlow-recoilregion.Ofcourse,the angularobservablesare very importantto constrainnewphysics, seeforinstanceRef.[20],buttheircontributioninthe determina-tionof

|

Vcb

|

ismarginal.

The results of our BGL and CLN fits to the full data set and to (11) are given in the first columns of Tables 3(a) and 3(b). The resultsof theCLN fitare in perfectagreement withtheone inAppendix BofRef. [10]. Incidentally,Belle’s paperalsoreports

Table 3

FitresultsusingtheBGL(a)andCLN(b)parameterizations.IntheBGLfits

a

F1

0 isfixedbythevalueof a

f

0,seeEq.(9).

(a)

BGL Fit: Data+lattice Data+lattice+LCSR

χ2_{/dof 27.9/32 31.4/35} |Vcb| 0.0417 ₊₂₀ −21  0.0404+₋1617  a0f 0.01223(18) 0.01224(18) a1f −0.054  +58 −43  −0.052  +27 −15  a2f 0.2 ₊₇ −12  1.0+₋05  aF1 1 −0.0100 ₊₆₁ −56  −0.0070+₋5452  aF1 2 0.12(10) 0.089  +96 −100  a0g 0.012  +11 −8  0.0289  +57 −37  a1g 0.7 ₊₃ −4  0.08+₋822  a2g 0.8  +2 −17  −1.0  +20 −0  (b)

CLN Fit: Data+lattice Data+lattice+LCSR

χ2_{/dof 34.3/36 34.8/39} |Vcb| 0.0382(15) 0.0382(14) ρ2 D∗ 1.17 ₊₁₅ −16  1.16(14) R1(1) 1.391  +92 −88  1.372(36) R2(1) 0.913 ₊₇₃ −80  0.916+₋6570  hA1(1) 0.906(13) 0.906(13) Table 4

Additionalfits.Thelatticeinput(11)isalwaysincludedandLCSRconstraintsarenever in-cluded.

Additional fits χ2

/dof |Vcb|

CLN without angular bins 7.1/6 0.0409+16

−17



BGL (N=2) without angular bins 5.1/2 0.0428+21

−22



CLN only angular bins 23.0/26 0.074+4

−37



BGL (N=2) only angular bins 22.3/32 0.058+25

−31



CLN with R1,2slopes let free 28.1/34 0.0415(19)

BGL (N=2) fit with R1,2(w=1.4)=HQET±20% (CLN Eq. (36)) 31.7/34 0.0407

₊₁₇

−18



theresults ofafit performedwithoutunfoldingthe distributions whichgives

|

Vcb

|

=

0

.

0374

(

13

)

.TheBGLfitinTable 3(a),left

col-umn, hasa 9% highercentral value anda 40% larger uncertainty thantheCLNfit.The fitsareboth good,andsuch alargeshiftin

|

Vcb

|

comes quiteunexpected.Webelieve itisrelatedtothefact

thatthe CLNparameterizationhaslimitedflexibility andthatthe angularobservablesdilutethe sensitivitytothelow recoilregion, whichiscrucialforacorrectextrapolation(theyalsodecreasethe overall normalisationof the rateby 0.8%). This isclearly seen in

Fig. 1b,where the bands corresponding to the BGL andCLN fits arecomparedwiththedata,andonecannoticethattheCLNband underestimatesallthethreelowrecoilpoints.Table 4shows

|

Vcb

|

obtained fromfits to the w distribution and(11) only: the CLN fitis7%higherandthetwoparameterizations giveconsistent re-sults.Anotherfitwhichsupports thesimpleexplanation aboveis onewherewegivemoreflexibilitytotheCLNparameterization,by floatingtheslopesofthe R1,2ratios.Theresult,showninTable 4, isagainveryclosetotheBGLone.

Concerningthequalityofthefitsweshow,oneshouldtakeinto account that all BGLfits are constrainedfits where (10)are em-ployedaftertruncationatorderN.Theeffectivenumberofdegrees offreedomisthereforelarger thanthenaivecountingshowninthe Tables(thenumberofdegreesoffreedomisnotwell-definedina constrainedfit,astheparametersarenotallowedtotakeany pos-siblevalue). This is well illustrated by thesecond fit in Table 4, whose

χ

2

_/

_dof

₌

₅

_.

₁

_/

_{2 maylook suspect. However, the unitarity} constraintsplayanimportantrolehere:withoutthemthebestfit wouldhave

χ

2

_/

_dof

₌

₁

_.

₂

_/

_2.

ItcanbereasonablyarguedthattheHQETinputusedin devis-ingtheCLNparameterizationisimportanttheoretical information thatoneshouldnotneglect.A simplewaytodothatistoinclude HQETconstraintson R1,2 atspecificvaluesof w witha

conserva-Table 5

Fitsincludinganhypotheticalfuturelattice calcu-lationgiving∂F ∂w|w=1= −1.44±0.07. Future lattice fits χ2 /dof |Vcb| CLN 56.4/37 0.0407(12) CLN+LCSR 59.3/40 0.0406(12) BGL 28.2/33 0.0409(15) BGL+LCSR 31.4/36 0.0404(13)

tiveuncertainty.Asanexample,wehaveusedtheHQETvaluesof

R1,2atw

=

1

.

4 witha20%uncertaintyintheBGLfitandobserved a downwardshiftin the Vcb central value, seeTable 4.Lowering

theuncertaintyweobserveverylittleeffect:fora10%uncertainty,

|

Vcb

|

=

0

.

0407

(

+₋1720

)

. It turnsout that thevalue of

|

Vcb

|

depends

mostsensitivelyonthatofR1 atlarge w.

Alternatively,onecanavoidHQETinputsaltogetherandemploy instead information on the form factors at maximal recoil from LightConeSumRules[21]:

hA1

(

wmax

)

=

0

.

65

(

18

),

(12)

R1

(

wmax

)

=

1

.

32

(

4

),

R2

(

wmax

)

=

0

.

91

(

17

).

The results of the BGL and CLN fits with the complete Belle’s datasetandEqs.(11),(12)aregiveninTables 3(a)and3(b),right column. The CLN fit is unaffected by the LCSRconstraints, while the BGL fit givesa smaller

|

Vcb

|

.Now the two fits are

compati-ble, but the difference between their

|

Vcb

|

central values is still

larger than5%. Itis interestingto compare R1,2

(

w

)

derived from the BGLfit (bandsin Fig. 1(c)) withtheHQET predictions ofthe samequantities[11](straightlines).Theyareperfectlycompatible ifoneassumesa

∼

10% uncertaintyforthelatter.

Finally, we show in Table 5 what would happen if a 5% de-terminationofthe slopeof theformfactor

F(

w

)

, seeEq.(6), at

w

=

1 were available fromthe lattice. For the central value we take the central value of the BGL fit with LCSR constraints. The resultsdemonstratetheimportanceofapreciselattice determina-tionof the slopeto control the zero-recoil extrapolation. Indeed, theparameterizationdependencebecomes minimalandtheLCSR constraintsbecome much less important.The quality of the CLN fitsdeteriorates,whiletheBGLuncertaintyisstillsomewhatlarger.

4. Finalremarks

Wehaveperformedfits totherecent B

→

D∗



ν

databyBelle

[10]withtheCLNandBGLparameterizations.TheBGLresultsfor

|

Vcb

|

areconsistentlyhigherthanthoseobtainedwiththeCLN

pa-rameterization.One cannot avoidnoticing that thecentral values ofallourBGLfitsareperfectlycompatiblewith(2).However,one shouldbe very carefulininterpreting our results:we simply ob-servedthattheBelledatawehaveemployedleadtodifferent

|

Vcb

|

whenthey are analysed withtwo parameterizations whichdiffer mainlyintheirrelianceonHQETrelations.Thedatadonot show anypreferenceforaparticularparameterization(bothgive accept-ablefits), but inthe absenceof newinformation fromlattice on the slope and zero-recoil value of the form factors the BGL pa-rameterizationoffersamoreconservativeandreliablechoice.Itis possible,even likely, that thebehaviour we have observed is ac-cidentallyrelatedto thenewBelledata only,andthat Babar and previousBelledatawouldleadtoasmallerdifferencebetweenthe CLNandBGLfits.Still,webelievethataparameterizationthatdoes not incorporate HQET relations but satisfies important unitarity bounds,suchasBGLinthewaywe useditabove,wouldprovide amorereliableestimateofthecurrentuncertaintyon

|

Vcb

|

.

Whileourfindingsdonotprovideaclearresolutionofthe

|

Vcb

|

puzzle,they strongly question the reliability of the current B

→

D∗



ν

averages[3]andcallforareanalysisofoldexperimentaldata beforeBelle-IIcomesintoaction.

Acknowledgements

WearegratefultoFlorianBernlochnerandChristophSchwanda forusefulcommunications concerning theBelle Collaboration re-sults.

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