comment on the heavy→light form factors

13 April 2000

Ž .Physics Letters B 478 2000 417–423

Comment on the heavy™ light form factors

Damir Becirevic a, Alexei B. Kaidalov b,c

a Dip. di Fisica, UniÕersita ‘‘La Sapienza’’ and INFN, Sezione di Roma, P.le Aldo Moro 2, I-00185 Rome, Italy`b Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117526 Moscow, Russiac ( ) 1Laboratoire de Physique Theorique Bat. 210 UniÕersite de Paris XI, 91405 Orsay Cedex, France´ ˆ ´

Received 4 May 1999; received in revised form 7 December 1999; accepted 29 February 2000Editor: R. Gatto

Abstract

We propose a simple parameterization for the form factors in heavy to light decays, which satisfies the heavy quarkscaling laws and avoids introducing the explicit ‘‘dipole’’ form. The illustration is provided with the set of lattice data forB™p semileptonic decay. The resulting shape of the form factor also agrees with the QCD sum rule predictions. q 2000Elsevier Science B.V. All rights reserved.

1. Preliminaries

A lack of the precise information about the shapesof various form factors from the first principles ofQCD was, and still is the main source of uncertain-ties in extraction of the CKM parameters from the

w xexperimentally measured exclusive decay modes 1 .This problem is particularly pronounced in the heaÕy™ light processes because the kinematically acces-sible region is very large. In such a situation it is notonly important to know the absolute value of a

Žparticular form factor at one specific point tradition-2 . 2ally at q s0 , but also its behaviour in q , which

Ž .when suitably integrated over the whole phase spacegives the physically measurable branching ratio. Sofar, there were very many studies devoted to thatissue. In spite of the recent progress, there is still no

1 Unite Mixte de Recherche C7644 du Centre National de la´Recherche Scientifique.

method based on QCD only, which could be used todescribe the complicated nonperturbative dynamicsin the whole physical region 2. Most of the ap-proaches agree that the functional dependence of theheaÕy ™ light form factors is ‘distorted’ in the

2 Žregion of not so high q or equivalently, for moder-2 .ately and very large transfers, q . In other words, it

is different from what one would obtain by invokingthe nearest pole dominance. Although somewhat in-triguing, this discrepancy with the pole dominancehypothesis is however expected: the low q2 region isvery far from the first pole, so that many singulari-

Ž .ties higher excitations and multiparticle states be-come ‘visible’ too, i.e. their contribution becomes

Ž .sensible comparable to that of the first pole . Onewould obviously like to quantify these effects ofhigher states.

2 The recent reviews about the form factors in heavy to lightw xdecays can be found in Refs. 2,3 . They also contain an exhaus-

tive list of references.

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00290-2

( )D. BecireÕic, A.B. KaidaloÕrPhysics Letters B 478 2000 417–423418

For definiteness, we consider the form factorsq,0Ž 2 .f q which parametrize B™p lln decay as

² X < < :p p V 0 B pŽ . Ž . Ž .m

m2 ym2B pX q 2s pqp yq f qŽ .2ž /q

m

m2 ym2B p 0 2q q f q , 1Ž .Ž .m2q

q 2Ž .where V sug b. f q is the dominant form fac-m m

tor, i.e. the only one needed for the decay rate

dGy0 qB ™p ll nŽ .ll2dq

2 < < 2G VF ub 23r2 2 q 2< <s l q f q , 2Ž .Ž . Ž .3 3192p mB

3 Ž .in the case of a massless lepton llse,m . The0Ž 2 .subdominant form factor f q , measures the diver-

gence of the vector current and its contribution isproportional to m2 in the decay rate, thus it isll

w xirrelevant for present experiments 1 . Its knowledgeis however important in lattice analyses, as well as in

Žthe approaches relying on the soft pion theorem in2 .which the form factors are calculated at large q ,

qŽ 2 .because it can help to constrain the dominant f q ,through the kinematic condition

f q 0 s f 0 0 . 3Ž . Ž . Ž .

2. Constraints

Ž .Beside this important constraint 3 , in the heavyŽ .quark limit m ™` and near the zero-recoil pointB

Ž 2 2 2 .q ,0, or equivalently q ,q where the pion ismax

soft, one encounters the well known Isgur-Wise scal-w x 4ing law 4

X² < < :p p V B p A m 1qOO 1rm ,Ž . Ž . Ž .( Ž .m B B

3 Ž . Ž 2 2 .2 2 2l t s tq m y m y4m m , is the usual triangle func-B p B p

tion.4 The overall logarithmic correction proportional to

y2 r b 0 Ža m ra m arising from the perturbative matchingŽ . Ž .Ž .s b s.between the QCD and the heavy quark vector current is irrelevant

for our discussion.

which in terms of form factors reads:

q 2 2f q ,q ,m ; m ,(Ž .max B B

10 2 2f q ,q ,m ; . 4Ž .Ž .max B m( B

w xAs first noticed in Ref. 5 , one may use the lowenergy theorem to get one more constraint, ofteninvoked in K decays. Namely, in the soft pionll 3

Ž X 2 . Ž .limit p ™0 and m ™0 , the r.h.s. of 1 simplym p0Ž 2 .becomes f p p , while the l.h.s. from the LSZm

reduction of the pion amounts to f rf p , so thatŽ .B p m

one deduces:

fB0 2f m s . 5Ž .Ž .B fp

Next scaling law comes from the asymptotic limit ofthe light cone QCD sum rules which are applicable

2 w xin the low q region. It was first shown in Ref. 6 ,that the form factors in heavy to light decays satisfy

f q0 q2 ,0,m ;my3r2 . 6Ž .Ž .B B

This scaling law was recently derived also in thew xframework of the large energy effective theory 7 .

Ž .6 will provide us an important additional constraintin what follows.

Our knowledge of the form factors goes a little bitbeyond. We know that the Lorentz invariant formfactors are analytic functions satisfying the disper-sion relations:

`01 Im f tŽ .

0 2f q s dt , 7Ž .Ž . H 2p tyq y iet0

Res 2 2 f q q2Ž .q sm )Bq 2f q sŽ . 2 2)m yqB

`q1 Im f tŽ .

q dt . 8Ž .H 2p tyq y iet0

The imaginary parts are consisting of all single andmultiparticle states with the same quantum numbers

qŽ . Ž 0Ž .. P y Ž q.as f t f t , i.e. J s1 0 , thus a multitudeof poles and cuts situated above the Bp production

( )D. BecireÕic, A.B. KaidaloÕrPhysics Letters B 478 2000 417–423 419

Ž .2 Žthreshold, t s m qm . However, below t but0 B p 0. )above the semileptonic B™p region there is a B

Ž .)vector meson m s5.325 GeV contributing to theB

qŽ .f t form factor. The residue at that pole is theproduct of the B)Bp coupling and the coupling ofB) to the vector current:

1q 22 2 ) ) )Res f q s m f g , 9Ž .Ž .q sm B B B Bp2)B

where the standard definitions were used:

² < < ) :) )0 V B p ,e s f m e ,Ž .m B B m

² y q < ) 0 :)B p p q B pqq ,e sg qPe .Ž . Ž . Ž . Ž .B Bp

10Ž .

While the lattice is providing us with better andw x

) )better estimate for f 8 , the value of gB B Bp

remains vague. From the above definitions, one im-qŽ 2 .2 2mediately finds that Res f q scales asq sm )B

m3r2.B

The simple pole dominance would mean that onlyŽ .the first term in the r.h.s. of 8 survives, whereas all

the other states which can couple to the vectorcurrent and are above t , would eventually cancel.0

Moreover, one would encounter the discrepancywhen trying to reconcile the pole dominance ansatz

Ž . qŽ .with 6 , because the pole dominance gives f 0 ;

my1r2.B

The easiest way out was to suppose the pole0Ž 2 . Ž .behavior for f q , and the double pole dipole

qŽ 2 . w xone for f q 9,10 :

f 0Ž .0 2f q s ,Ž . 2 2

q1yq rm0

f 0Ž .q 2f q s . 11Ž .Ž . 22 2

y1yq rmŽ .1

This choice basically satisfies all constraints but it isnot informative at all: we know that we have a pole

) Ž .at B and more importantly, we know its positionwhich we trade for a better fit with data, withoutgetting any supplementary physical information. Forthat reason, we wanted to propose a newparametrization which would satisfy all the con-straints mentioned above, and use the fact that weknow where the first pole is. In this way the fit withdata will turn to be far more informative.

3. Proposal

qŽ 2 .One should always start from the fact that f qdoes haÕe a pole at m )

2 . The contribution of otherBŽhigher states for which we do not know positions,.nor residues , can be parametrized as an effective

pole whose position is to be determined relatively tothe first pole. In other words, we propose:

1 aqf x sc y , 12Ž . Ž .B x1yx� 01y

g

where we introduced xsq2rm )

2 , and parametersB

a and g - positive constants which scale with m .B

Obviously, c m )

2 is the residue of the form factorB B

at B). Note that g m )

2 is the squared mass of anB

effective 1y excited B)

X

-state, which is equal toŽ .2 2

) ) )m qD ,m q2 Dm , where D does not de-B B B

pend on the heavy quark in the heavy quark limit.Therefore, g,1q2 Drm ) . We now check theB

scaling laws. At q2 , one has:max

agqf x ,c m 1q 13Ž . Ž .max B B ž /1yg mŽ . B

Since c scales as ;my1r2, then in order to fulfillB BŽ . qŽ . 1r2the heavy quark scaling law 4 f x ;m ,max B

1yg can at most scale as 1rm . That is consistentB

with the above observation on g . It is also clear thatin this counting, ag scales as a constant. On theother hand, at q2 s0 we have:

f q 0 sc 1ya ;my1r2 1ya ;my3r2 ,Ž . Ž . Ž .B B B

14Ž .

Ž .which means that 1ya scales as ;1rm . ThisB

shows why we have chosen the sign ‘‘y’’ betweenŽ .the two terms in 12 . In fact, it is easy to see that

both a and g scale as 1qconst.rm q PPP . ToB0 0Ž . qŽ .build f , we take into account f 0 s f 0 . For

convenience, we can measure the position of theeffective pole to f 0 in terms of xsq2rm )

2 , andB

write

c 1yaŽ .B0f x s . 15Ž . Ž .x1y

b


0Ž 2 . y1r2One can easily see that f q ;m as itmax B0Ž . y3r2should, while f 0 ;m by construction. UsingB

the same argument as for g , it is obvious that b alsoŽ .scales like 1qOO 1rm .B

Ž .These four parameters c , a , b , g contain aB

considerable information on the process: c gives aB

residue of the form factor at the pole B) , a mea-sures the contribution of the higher states which areparameterized by an effective pole at m )

2 sgPm )

2 .B Beff

Contributions to the scalar form factor are parameter-ized by another effective pole whose position is atm )

2 sbPm )

2 . The situation with the form factorB B0 eff

f q resembles the situation with the electromagneticform factor of a nucleon, where the simplest r-domi-nance is not sufficient to describe a dipole behaviorof the form factor at large q2, and one needs tointroduce at least one extra excited vector meson.

Present lattice analyses have limited statisticalaccuracy and accessible kinematical region for theform factors, and for a description of lattice data amore constrained parametrization is required. In the

Žlimit of large energy of the light meson large re-.coils , in the rest frame of the heavy meson

2 2 m ™`Bm q m mB p BEs 1y q , 1yx .Ž .2 2ž /2 2m mB B

16Ž .w xIn Ref. 7 , it was shown that in the heavy quark

Ž .limit and for large recoils m ™` and E™` ,B

there exists one nontrivial relation between the twoform factors, namely:

2 E0 qf s f . 17Ž .

mB

In this limit, and by using the above counting argu-ment with as1ya rm , gs1qg rm , and b0 B 0 B

s1qb rm , we have0 B

c a g m0 0 0 Bqf s 1y 1yž /a 2 E2 E m( 0B

qOO 1rm q . . .Ž .B

c a0 00f s 1qOO 1rm q . . . . 18Ž . Ž .B2 E m( B

Ž .Therefore, to satisfy the relation 17 it is necessaryto have g sa . It is reasonable then to take, aqg0 0

Table 1w xLattice results obtained by the Hiroshima group, Hi-KEK 11

2 2 0 2 q 2w x Ž . Ž .q GeV x f q f qB™p B™p

Ž . Ž .20.7 0.70 0.72 3 2.09 8Ž . Ž .19.2 0.65 0.71 3 1.52 8Ž . Ž .18.3 0.62 0.67 2 1.41 7Ž . Ž .17.4 0.59 0.66 4 1.28 10Ž . Ž .16.4 0.55 0.65 4 1.20 12Ž . Ž .15.4 0.52 0.60 7 1.11 17

Ž .s2, but also as1rg , which satisfy 17 up toŽ 2 .terms OO 1rm . In what follows, we use as1rg .B

To summarize, the recipe is the following one: inŽ

)lattice analyses, one first calculates m from theB. 5corresponding two-point correlation function to

measure x, and then fits the data according to:1 1 x

s 1y ,0 ž /c 1ya bf x Ž .Ž . B

1 1s 1yx 1ya x , 19Ž . Ž . Ž .qf x c 1yaŽ . Ž .B

Ž .which makes altogether three parameters c , a , bB

to perform this fit.

4. Illustration

Now, we would like to show how it works on thespecific example with the set of lattice data. Before

Ž .the new systematically improved results are re-leased 6, we decided to use the most recent available

w x Ž .results, those of Hi-KEK group 11 see Table 1 .Since our purpose is only to illustrate the

Ž .parametrization 19 , we will not further comment onthe technical details used in the calculation of Hi-KEK. However, one should note that the latticegroups reported results by assuming the negligible

Ž .SU 3 breaking effects, which is also supported by7 w xthe QCD sum rule calculation 12 , as well as by

5 In experiment, one fixes it to the physical mass of the vectormeson B).

6 APE and UKQCD groups, are already analyzing their newrespective data.

7 Ž .Note that the LCQCD sum rules predict larger SU 3 breakingw x13 , which is the consequence of the normalization of the light

Ž q Ž . q Ž .meson wave functions implying that f 0 r f 0 ,B™ K B™p

.f r f . This is in conflict with the findings of lattice calculationsK p

w x2 . Further research clarifying that issue is needed.


Table 2Ž . Ž .Results of fits for a fixed g . The last column is obtained using Eqs. 5 and 15 in which we took xs0.983

Ž .) ) ) ) ) ) )m rm a f 0 g f rm b m rm f rfB B B Bp B B B B B peff 0 eff

Ž . Ž . Ž . Ž .1.15 0.76 0.30 1 2.45 5 1.12 6 1.06 3 2.5 " 1.0Ž . Ž . Ž . Ž .1.20 0.69 0.32 1 2.10 4 0.19 7 1.09 3 1.8 " 0.6Ž . Ž . Ž . Ž .1.25 0.64 0.34 1 1.90 4 1.26 8 1.12 4 1.6 " 0.4Ž . Ž . Ž . Ž .1.30 0.59 0.36 1 1.76 4 1.32 9 1.15 4 1.4 " 0.3Ž . Ž . Ž . Ž .1.35 0.55 0.38 1 1.65 4 1.39 10 1.18 4 1.3 " 0.3

w xexperiment with D-mesons 14 . That means that thepion used in the Hi-KEK calculation is consisted ofdegenerate quarks of mass near the strange one.Also, the spectator quark in B meson is the ‘‘s’’quark 8.

First, we fix the mass of m ) s5.46 GeV, asB

obtained in the same calculation of Hi-KEK. Inrealistic situation, one of course takes the experimen-tal m ) s5.32 GeV. Then we fit the data with theB

Ž .expressions 19 , i.e. by minimizing:21 xi0° 1r f y 1yŽ . i ž /Ž .c 1y a bB2 ~x sÝ 0d 1r fŽ .ž /ii ¢

21q ¶w xŽ Ž . . Ž .1r f 1y x y 1y a xi iŽ .c 1y aB •q qw xŽ Ž . .d 1r f 1y xž /i ß

from which we obtain the following result:

f q,0 0 s0.38 8 , as0.54 17 , bs1.4 9 .Ž . Ž . Ž . Ž .20Ž .

The best fit and the lattice data are depicted in Fig.Ž .1. These conservative errors in 20 are estimated

from the correlated fit, corresponding to Dx 2 s3.53Ž . w x68% C.L. 17 , which defines the ellipsoid of errorsŽ . Ž .three parameters . The errors we quote in 20correspond to the extrema of that ellipsoid. As wealready mentioned, by our parametrization, besides

qŽ .the first pole in f x all the other singularities arereplaced by an effective pole, which is situated at

q0.3') ) )m sm g s1.3 m . The effective pole forB B y0.2 Beff0 q0.3Ž . ') ) )f x is at m sm b s1.2 m .B B y0.5 B0 eff

8 To our knowledge, among recent studies, only the APEw xcollaboration 15 made the extrapolation to the chiral limit.

An important advantage of this parametrization isits transparent physical meaning. For example, after

Ž . Ž .identifying f 0 sc 1ya , we may also extractBŽ .within the proposed parameterization the informa-tion on the residue of the form factor f q at B) , i.e.about the coupling g ) :B Bp

c s0.8 3 ´g ) s1.6 6 m )rf ) . 21Ž . Ž . Ž .B B Bp B B

If we take the physical value m ) s5.32 GeV, andB

f ) ,0.2 GeV, we see that the resulting g ) s42B B Bp

"16, is compatible with most of today’s estimatesŽ w x.of this coupling see Table 2 in Ref. 3 . A statistical

Ž .and more importantly, a systematic improvement ofthe lattice data will provide more reliable estimate ofthis coupling.

The ellipsoid of errors is of the cigar shape so thatone can reduce the errors and consequently get someadditional information, by making reasonable as-sumptions. One can for instance make the following

Ž .Fig. 1. Fitting the lattice data using the parametrization 19 . Notethat the fit of f q form factor is constrained by the precise data

0 Žfor f . For easiness, only the central curves without errors in.parameters are displayed.


Fig. 2. The shape of the form factor from our parametrizationŽ . 219 , constrained by the lattice data at large q , is compared to the

2 Ž Ž . w xQCD sum rule predictions at ‘‘low’’ q Eq. 132 in Ref. 3 forŽ . w xthe dashed curve, and Eq. 12 in Ref. 13 for the dotted

10 .curve . A very good agreement is found.

exercise. We saw that the position of the effectiÕeqŽ 2 . q0.3

)pole of the f q function is at 1.3 m . If wey0.2 BŽ .

)now fix it to 1.15,1.20,1.25,1.30,1.35 =m , weB

obtain the results listed in Table 2. We notice thatthe data tend towards larger values of the mass of thescalar effective pole as compared to the expectedmass of the lowest scalar meson 9, b,1.17, thoughthey are consistent with this value. Note that anexpected mass of the first excited vector state ac-cording to theoretical estimates should be close to 6GeV, which again corresponds to a smaller value of

Ž .gf1.3 or af0.7 , but within the errors agreeswith the values found in our analysis.

Now, when we have fixed the parameters by theŽ .lattice data 20 , we would like to compare the

qŽ 2 .resulting shape of the form factor f q to theresult of the light cone QCD sum rule, which isapplicable in the low q2 region. To do that we define

qŽ 2 . qŽ .f q rf 0 , and plot them both in Fig. 2. We seethat our parametrization, constrained by the latticedata at large q2 ’s, indeed reproduces the shape ofthe form factor at low q2 ’s as predicted by QCDsum rule. To be consistent, in the comparison weused the physical m ) s5.32 GeV.B

9 Ž q .The scalar 0 meson is expected in the region of thew x ) )

))measured bump 16 of B states: m s5.73 GeV.B10 Ž . w x qSee also Eq. 23 in Ref. 18 for f form factor.

5. Summary

In this letter we proposed the physically more infor-mative way to fit the data for the form factors inheavy to light decays. This is illustrated on theexample of B™p semileptonic decay, and can bealso used in D™p , as well as in the other heaÕy™ light transitions 11. Initially, the parametrizationfor two form factors contains four parametersŽ . Ž . Ž .12 , 15 , which we further reduced to three 19 . Allparameters scale with the heavy mass as const.qŽ .OO 1rm , except for c . The proposed parameteriza-B B

tion gives us the information on the singularitieswhich decisively influence the shape of the formfactor in the semileptonic region, and allows us toextract the value of the residue at the nearest poleŽ .

)i.e. to determine the value of g -coupling .B Bp

On the specific example, we have shown thatwhen fixing the parameters in our parametrization bythe lattice data in the large q2 region, one repro-

Ž .duces to a good precision the shape of the formfactor obtained by QCD sum rules at low and inter-mediate q2 ’s.

We believe that this parametrization may be use-ful in the forthcoming analyses of systematicallyimproved lattice data, as well as in experimentalstudies of exclusive heaÕy ™ light decays.

Acknowledgements

We are very grateful to J. Charles, G. Martinelliand O. Pene for valuable comments on the`manuscript, and to H. Matufuru for correspondence.Remarks by P. Ball are also acknowledged.

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comment on the heavy→light form factors

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