Z. Phys. C 75, 657–664 (1997) ZEITSCHRIFTFUR PHYSIK Cc© Springer-Verlag 1997

B → πlν form factors calculated on the light-front

Chi-Yee Cheung,1 Chien-Wen Hwang,1,2 Wei-Min Zhang1

1 Institute of Physics, Academia Sinica, Taipei 11529, Taiwan, ROC2 Department of Physics, National Taiwan University, Taipei 10764, Taiwan, ROC

Received: 6 November 1996

Abstract. A consistent treatment ofB → πlν decay isgiven on the light-front. TheB to π transition form fac-tors are calculated in the entire physical range of momentumtransfer for the first time. The valence-quark contribution isobtained using relativistic light-front wave functions. Higherquark-antiquark Fock-state of theB-meson bound state isrepresented effectively by the|B∗π〉 configuration, and itseffect is calculated in the chiral perturbation theory. Wavefunction renormalization is taken into account consistently.The |B∗π〉 contribution dominates near the zero-recoil point(q2 ' 25 GeV2), and decreases rapidly as the recoil mo-mentum increases. The calculated form factorf+(q2) followsapproximately a dipoleq2-dependence in the entire range ofmomentum transfer. We estimate that|Vub|=0.003.

I Introduction

The study of exclusive semileptonic decays of heavy mesonshas attracted much interest in recent years. Semileptonicdecays of heavy mesons to heavy mesons, such asB →D(D∗)lν, provide an ideal testing ground for heavy-quarksymmetry and heavy-quark effective theory. By comparison,weak decays of heavy mesons to light mesons are muchmore complicated theoretically, since in general there existsno symmetry principle for guidance. Nevertheless, it is es-sential to understand the reaction mechanisms of these decaymodes, because they are the main sources of information onthe CKM mixing matrix elements between heavy and lightquarks. In particular, the study of theB → πlν decay is im-portant for the determination of matrix elementVub whosevalue is only poorly known[1].

Recently, theB → πlν decay has been investigatedby many groups [2–17] using various different approaches,such as quark models, QCD sum rules, heavy-quark sym-metry, perturbative QCD, and so on. In most studies, transi-tion form factors are calculated only at one kinematic point,q2 = (PB − Pπ)2 = 0, so that extra assumptions are neededto extrapolate the form factors to cover the entire range ofmomentum transfer. In [13–16] chiral perturbation is em-ployed, so that the results are valid for soft pion emission

only. In [17] perturbative QCD is applied, in conjunctionwith hadronic wave functions obtained from QCD sum rule,to calculate the decay amplitudes; therefore the result is validonly when the final pion is energetic.

In this study, we first calculate theB → πlν decay formfactors using relativistic light-front hadronic wave functions(Fig. 1a), corresponding to the so-called valence-quark con-tribution. The parameters in these wave functions are de-termined from other informations, and Melosh transforma-tion is used to construct meson states of definite spins. Al-though light-front wave functions and quark model in theinfinite momentum frame have been used in the past to studyB → πlν decay and other heavy-light transitions [2, 4, 18],the decay form factors were only calculated forq2 = 0. Inthis work, we directly evaluate the valence contribution inthe entire kinematic region. Within the framework of light-front quark model, one only has to calculate this valence con-tribution if q2 ≤ 0, since then one can choose the light-front“+” component of the momentum transferq+ = 0. Howeverthe B → π transition involves time-like momentum trans-fers. That meansq+ > 0, and it is well known that one mustalso consider the so-called nondiagonal light-front diagram(or Z-graph) [18–21], as depicted in Fig. 1b. This contribu-tion is generated by the quark-antiquark (qq) excitation orhigher Fock-states in the light-front bound state wave func-tion. We shall effectively represent theqq- configuration oftheB-meson by the mesonic|B∗π〉 state, giving rise to theB∗-pole contribution shown in Fig. 1c. It was first noted byIsgur and Wise [22] that theB∗-pole effect is important inthe zero-recoil region. Previous investigations either simplyadded this effect to the valence-quark contribution [22, 23],or totally ignored it. In our more unified approach, theB∗-pole contribution arises from the|B∗π〉-component of theB-meson bound state. The mixing of different Fock-stateconfigurations naturally requires a consistent wave functionrenormalization which has not been mentioned in previousworks.

The basic philosophy of this work is as follows. Sinceone does not know how to calculate exactly the hadronicform factors in the entire momentum transfer range, wesuperimpose consistently the two contributions mentionedabove. One of which (B∗-contribution) dominates in the

658

(a)

π

π

(b)

B

B

(c)

πB

B

b u

*

Fig. 1. aB → πlν: Valence-quark contribution;b B → πlν: Non-valencecontribution;c B → πlν: B∗-pole contribution

zero-recoil region,q2 ≈ q2max, and the other (valence-quark

contribution) dominates near the maximum recoil point,q2 = 0. Both contributions are calculated in the entire mo-mentum transfer range. Although not completely satisfac-tory, our approach represents a significant step beyond theusual practice of evaluating the form factor at one singlepoint (q2 = 0).

This paper is organised as follows. In Sect. II, the basictheoretical formalism is given. In Sect. III, numerical resultsare present and discussed, and finally a summary is given inSect. IV.

II General formalism

The weak current that is responsible forb-quark decay isgiven by

Jµ = qγµ(1− γ5)b, (2.1)

where q stands for a light quark. ForB → πlν, only thevector current contributes, and our main task is to evaluatethe hadronic matrix element,Mµ

Bπ = 〈π(Pπ)|Jµ|B(PB)〉,which can be parametrized as

MµBπ = 〈π(Pπ)|Jµ|B(PB)〉

= f+(q2)(PB + Pπ)µ + f−(q2)(PB − Pπ)µ. (2.2)

In the previous calculations of hadronic matrix elementsin the infinite momentum frame or with light-front wavefunctions, one usually setq+ = P +

B − P +π = 0. This leads to

q2 = −q2⊥, implying a space-like momentum transfer. How-

ever, momentum transfers in real decay processes are alwaystime-like. Hence matrix elements calculated withq+ = 0 isrelevant only at the maximum-recoil point withq2 = 0, andone needs an extrapolation ansatz to extend the result toother physical momentum transfers. A direct calculation ofthe form factors for the whole momentum transfer rangehas not been performed. In this work, we work in a frame

where the “⊥”- components ofPB and Pπ vanish, so thatq2 = q+q− ≥ 0, and we can evaluate the form factors inthe entire physical range of momentum transfer. As men-tioned earlier, in a frame withq+ > 0, there are two distinctcontributions to the matrix element. Apart from the usualvalence contribution (Fig. 1a) which is calculated with rel-ativistic light-front wave functions, one must also includethe nondiagonal light-front diagram as depicted in Fig. 1b.Here, such non-valence effects are taken into account effec-tively by theB∗-pole contribution shown in Fig. 1c. Simpleperturbation theory in the light-front approach gives [25]

|B(PB)〉 =√Z2

{|B0(PB)〉 (2.3)

+∫

[d3k][d3q]〈B∗(q)π(k)|HI |B0(PB)〉

P−B − q− − k−

|B∗(q)π(k)〉},

where |B0〉 represents the valence configuration describedby a light-front bound-state wave function, while|B∗ π〉 isthe most important higher-Fock-state configuration, as willbe explained later;HI is the interaction Hamiltonian fortheBB∗π-vertex obtainable from chiral perturbation theory[13, 14, 24],

[d3k] ≡ dk+d2k⊥2(2π)3k+

,

〈P ′|P 〉 = 2(2π)3P +δ(P ′+ − P +)δ2(P ′⊥ − P⊥), (2.4)

and√Z2 is the wave function renormalization constant.

All particles in (2.3) are on the mass-shells so thatP−B =

P 2B⊥+M2

B

P +B

, q− = q2⊥+M2

B∗q+ , andk− = k2

⊥+m2π

k+ .From Eqs. (2.2) and (2.3), we have formally

MµBπ =

√Z2

{ (fv+ (q2) + fB

∗+ (q2)

)(PB + Pπ)µ

+(fv−(q2) + fB

∗− (q2)

)(PB − Pπ)µ

}, (2.5)

wherefv+ , fv− represent the valence-configuration contribu-

tions, andfB∗

+ , fB∗

− theB∗-pole contributions. The resultantB → πlν decay form factors are therefore given by

f+(q2) =√Z2

[fv+ (q2) + fB

∗+ (q2)

],

f−(q2) =√Z2

[fv−(q2) + fB

∗− (q2)

]. (2.6)

Most of the previous investigations only calculatedfv± atq2 = 0, some also included the contribution offB

∗± , but

none has taken into account the effect of√Z2.

In the following, we shall first calculate the valence con-tribution using light-front bound state wave functions. Sub-sequently, theB∗-pole contribution, as well as the effect ofwave function renormalization, will be discussed in moredetails.

A Valence configuration contribution

A meson bound state consisting of a quarkq1 and an anti-quark ¯q2 with total momentumP and spinS can be writtenas

659

|M (P, S, Sz)〉 =∫{d3p1}{d3p2} 2(2π)3δ3(P − p1 − p2)

×∑λ1,λ2

ΨSSz (p1, p2, λ1, λ2) |q1(p1, λ1)q2(p2, λ2)〉, (2.7)

wherep1 andp2 are the on-mass-shell light-front momenta,

p = (p+, p⊥) , p⊥ = (p1, p2) , p− =m2 + p2

⊥p+

, (2.8)

and

{d3p} ≡ dp+d2p⊥2(2π)3

|q(p1, λ1)q(p2, λ2)〉 = b†λ1(p1)d†λ2

(p2)|0〉, (2.9)

{bλ′ (p′), b†λ(p)} = {dλ′ (p′), d†λ(p)} = 2(2π)3 δ3(p′ − p) δλ′λ.

In terms of the light-front relative momentum variables(x, k⊥) defined by

p+1 = x1P

+, p+2 = x2P

+, x1 + x2 = 1,

p1⊥ = x1P⊥ + k⊥, p2⊥ = x2P⊥ − k⊥, (2.10)

the momentum-space wave-functionΨSSz can be expressedas

ΨSSz (p1, p2, λ1, λ2) = RSSzλ1λ2

(x, k⊥) φ(x, k⊥), (2.11)

whereφ(x, k⊥) describes the momentum distribution of theconstituents in the bound state, andRSSz

λ1λ2constructs a state

of definite spin (S, Sz) out of light-front helicity (λ1, λ2)eigenstates. Explicitly,

RSSzλ1λ2

(x, k⊥) =∑s1,s2

〈λ1|R†M (1− x, k⊥,m1)|s1〉 (2.12)

×〈λ2|R†M (x,−k⊥,m2)|s2〉〈1

2s1

12s2|SSz〉,

where|si〉 are the usual Pauli spinor, andRM is the Meloshtransformation operator:

RM (x, k⊥,mi) =mi + xiM0 + iσ · k⊥ × n√

(mi + xiM0)2 + k2⊥

, (2.13)

with n = (0, 0, 1), a unit vector in thez-direction, and

M20 =

m21 + k2

⊥x1

+m2

2 + k2⊥

x2. (2.14)

It is possible to rewrite the transformation matrixRSSzλ1λ2

in acovariant form [18], which is useful in practical calculations:

RSSzλ1λ2

(x, k⊥) =

√p+

1p+2√

2 M0u(p1, s1)Γv(p2, s2), (2.15)

where

M0 ≡√M2

0 − (m1 −m2)2, (2.16)

u(p, s)u(p, s′) =2mp+

δs,s′ ,∑s

u(p, s)u(p, s) =6p +mp+

,

v(p, s)v(p, s′) = −2mp+

δs,s′ ,∑s

v(p, s)v(p, s) =6p−m

p+,

Γ = γ5 (pseudoscalar, S = 0),

Γ = − 6ε(Sz) +ε · (p1 − p2)M0 +m1 +m2

(vector, S = 1),

with

εµ(±1) =

[2P +

ε⊥(±1) · P⊥, 0, ε⊥(±1)

],

ε⊥(±1) =∓(1,±i)/√

2, (2.17)

εµ(0) =1M0

(−M2

0 + P 2⊥

P +, P +, P⊥),

We normalize the meson state as

〈M (P ′, S′, S′z)|M (P, S, Sz)〉= 2(2π)3P +δ3(P ′ − P )δS′SδS′zSz , (2.18)

so that∫dxd2k⊥2(2π)3

|φ(x, k⊥)|2 = 1. (2.19)

In principle, the momentum distribution amplitudeφ(x, k⊥)can be obtained by solving the light-front QCD bound stateequation [25, 26]. However, before such first-principle so-lutions are available, we would have to be contented withphenomenological amplitudes. One example that has beenoften used in the literature for heavy mesons is the so-calledBSW amplitude [2], which for aB(bq)-meson is given by

φB(x, k⊥) = NB

√x(1− x) exp

(−k2⊥

2ω2B

)exp

[−M2

B

2ω2B

(x− x0)2

], (2.20)

whereNB is the renormalization constant,x is the longi-tudinal momentum fraction carried by the light anti-quark,

x0 = (12−

m2b−m2

q

2M2B

), mb=b-quark mass,mq= light-quark mass,

andωB is a parameter of orderΛQCD.For the pion, we shall adopt the Gaussian type wave

function,

φπ(x, k⊥) = Nπ

√dkzdx

exp

(− k2

2ω2π

), (2.21)

wherek = (k⊥, kz),

x =E1 + kzE1 +E2

, 1− x =E2 − kzE1 +E2

, (2.22)

with Ei =√m2i + k2. We then have

M0 = E1 +E2, (2.23)

kz =

(x− 1

2

)M0 − m2

1 −m22

2M0(2.24)

and

dkzdx

=E1E2

x(1− x)M0(2.25)

is the Jacobian of transformation from (x, k⊥) to k. Thiswave function has been also used in many other studies ofhadronic transitions. In particular, with appropriate param-eters, it describes satisfactorily the pion elastic form factorup toQ2 ∼ 10 GeV2 [27].

With the light-front wave functions given above, andtaking a Lorentz frame wherePB⊥ = Pπ⊥ = 0 (i.e.,q⊥ = 0),we readily obtain (forB0 → π+l−ν)

660

〈π(Pπ)|Jµ|B0(PB)〉 =∑λ′s

∫{d3pd}φ∗π(x′, k⊥)φB(x, k⊥)

×R00†λuλd

(x′, k⊥) u(pu, λu)γµu(pb, λb)R00λbλd

(x, k⊥), (2.26)

wherex(x′) is the momentum fraction carried by the spec-tator anti-quark (d) in the initial(final) state, such that

xP +B = x′P +

π ; (2.27)

the meaning of all the other variables should be self obvi-ous. The valence-quark part of the reaction mechanism isdepicted in Fig. 1a. Substituting the covariant form given in(2.15) into (2.26), we get

〈π(Pπ)|Jµ|B0(PB)〉 =√P +B

P +π

∫ dxd2k⊥2(2π)3

φ∗π(x′, k⊥)φB(x, k⊥)

× −1

2M0πM0B√

(1− x′)(1− x)

×Tr[γ5(6pu +mu)γµ(6pb +mb)γ5(6pd −md)

].(2.28)

The trace in the above expression can be easily carried out.For the “good” component,µ = +, we have

−Tr[γ5(6pu +mu)γ+(6pb +mb)γ5(6pd −md)

]= 4

[m2q + k2

⊥x

+mq(mb −mq)

]P +π , (2.29)

where we have setmu = md = mq. From Eq. (2.27),x andx′ are related byx = R±x′, with

R± ≡ P +π

P +B

=1

2M2B

[M2

B +M2π − q2 (2.30)

±√

(M2B +M2

π − q2)2 − 4M2BM

2π

].

R± correspond to the pion recoiling in the positive and neg-ative z-direction respectively relative to the B meson. Thephysical kinematic rangeq2 : 0→ (MB−Mπ)2 correspondsto

R+ : 1→Mπ/MB ,

R− : (Mπ/MB)2 →Mπ/MB . (2.31)

As pointed out in Ref. 26, the choice of the positivez-axis is immaterial, and the matrix elements calculated inboth reference frames (call them the “+” and “−” frame)should produce the same form factorsf±(q2) as defined in(2.2). In the “+” frame, we write

fv+ (q2)(P +B + P +

π ) + fv−(q2)(P +B − P +

π ) = F (R+)P +π , (2.32)

or equivalently,

fv+ (q2)(1 +R+) + fv−(q2)(1−R+) = F (R+)R+; (2.33)

similarly, in the “−” frame,

fv+ (q2)(1 +R−) + fv−(q2)(1−R−) = F (R−)R−, (2.34)

where

F (R±) =

√R±

(2π)3

∫ 1

0dx′

∫d2k⊥φ∗π(x′, k⊥)φB(R±x′, k⊥)

× 1

M0πM0B

√(1− x′)(1−R±x′)

×[m2q + k2

⊥R±x′

+mq(mb −mq)

]. (2.35)

Solving for fv±(q2) from Eqs. (2.33, 2.34), we finally arriveat

fv+ (q2) =F (R+)R+(1−R−)− F (R−)R−(1−R+)

2(R+ −R−)(2.36)

and

fv−(q2) =−F (R+)R+(1 +R−) + F (R−)R−(1 +R+)

2(R+ −R−). (2.37)

These are the valence-configuration contributions to theB → πlν decay form factors, valid in the entire range ofmomentum transferq2 = [0, (MB −Mπ)2].

B Higher-Fock-state contribution

It is often not adequate to describe the internal structure ofa hadron solely by its valence configuration. As we havediscussed, for time-like momentum transfers, one must alsoconsider the effects of higher Fock-states corresponding toconfigurations containing quark-antiquark pairs in additionto the valence particles. Such contributions are shown inFig. 1b. Unfortunately these higher-Fork-state wave func-tions are not available, and we shall estimate their contri-butions in an effective mesonic picture. In the case of theB-meson, it is expected that the effective higher-Fock-stateconfiguration|B∗π〉 is the most important if the relative mo-mentum is small. The reason is as follows. TheB andB∗masses are almost degenerate due to heavy-quark symmetry,and also the pion mass is small. Consequently, when the rel-ative momentum is small, the|B∗π〉 configuration is close tothe energy shell (i.e., the energy denominator is small), andis thus enhanced. This can be readily seen in (2.3), where theinteraction HamiltonianHI describing theBB∗π couplingis given in chiral perturbative theory by [24]

HI = − g

fπ

√MBMB∗

∫dx−d2x⊥B∗†

µ (i∂µπa)τaB, (2.38)

with fπ = 93 MeV being the pion decay constant.The contribution to the decay matrix element from the

|B∗π〉 configuration [see Fig. 1c] is given by

〈π(Pπ)|Jµ|B(PB)〉B∗ =∫[d3q]〈0|Jµ|B∗(q)〉 〈B

∗(q)π(Pπ)|HI |B0(PB)〉P−B − q− − P−

π. (2.39)

From

〈0|Jµ|B∗(q)〉 = MB∗fB∗εµ, (2.40)

〈B∗(q)π−(Pπ)|HI |B0(PB)〉 = (2.41)

2(2π)3δ3(PB − q − Pπ)

√2gfπ

√MBMB∗ ε∗ · Pπ,

and

εµε∗ν = −gµν +PµB∗P ν

B∗

M2B∗

, (2.42)

we readily obtain

661

〈π−(Pπ)|Jµ|B(PB)〉B∗ = (2.43)√

2gfB∗

fπ

PB∗ · PπPµB − (M2

B∗ + PB∗ · Pπ)Pµπ

P +B∗ (P−

B − P−B∗ − P−

π )

√MB

MB∗.

Therefore,

fB∗

+ (q2) = gfB∗√2fπ

−M2B∗

P +B∗ (P−

B − P−B∗ − P−

π )

√MB

MB∗(2.44)

fB∗

− (q2) = gfB∗√2fπ

M2B∗ + 2PB∗ · Pπ

P +B∗ (P−

B − P−B∗ − P−

π )

√MB

MB∗(2.45)

whereP +B∗ = P +

B − P +π > 0 andP−

B∗ = M2B∗+(PB⊥−Pπ⊥)2

P +B−P +

π. It

is easy to see thatfB∗

± are functions ofq2 because

1q2 −M2

B∗=

1P +B∗

1

P−B − P−

B∗ − P−π

(q+ = P +B∗ ). (2.46)

However the above results are not quite complete, be-cause chiral perturbation theory is a low-energy effectivetheory, such that the chiralBB∗π-vertex given in (2.38) isvalid only for soft pions. A suppression factor is generallyexpected when the pion momentum increases. This can alsobe understood in the quark picture by the following rea-soning. The higher Fock-state|B∗π〉 arises from quark-paircreation which is a predominently soft process. First of all,there is not much probability for producing harduu pairs.Moreover, once produced, a hardu has little chance of form-ing a pion with a slow ¯u, likewise a hardu is not likely toform aB∗-meson with a slowb quark. Therefore configura-tions with a high-momentum pion in|B∗π〉 is expected to besuppressed. This physical requirement can be implementedphenomenologically by introducing a damping form factor,F (q2), to the chiralBB∗π-vertex, such that,

g → gF (q2). (2.47)

We shall take

F (q2) = exp

(− vB · (pπ −Mπ)

Λ

), (2.48)

wherevB = PB/MB is the 4-velocity of theB-meson, andΛis the cutoff momentum which is expected to be of the orderof the chiral symmetry breaking scaleΛχ ' 1 GeV. Theform of the suppression factor is similar to that proposed byWolfenstein [15] in the soft-pion region, and is normalizedto unity at the zero-recoil point whereq2 = (MB −Mπ)2.The substitution indicated in (2.47) is to be understood forall of the equations derived in this subsection.

C Wave function renormalization

Finally we calculate the wave function renormalization con-stantZ2 which is required for consistency, as indicated in(2.3). The result is

Z−12 = 1 +

12(2π)3P +

B

(2.49)∫[d3k][d3q]

|〈B∗(q)π(k)|HI |B0(PB)〉|2(P−

B − q− − k−)2

= 1 +3g2

f2π

MBMB∗

P +B

Fig. 2. Valence-quark contribution tof+(q2). Solid line:mq = 0.30 GeV,ωπ = 0.29 GeV;dashed line:mq = 0.25 GeV,ωπ = 0.33 GeV

∫[d3q]

1k+

[ε · k]2

[P−B − q− − k−]2

F 2[(PB − k)2],

whereq andk are on the mass-shells,k+ = P +B − q+, k⊥ =

−q⊥, and we have included the suppression form factor asgiven in (2.47, 2.48).

Combining Eqs. (2.36), (2.37), (2.44), (2.45) and (2.49)together, we obtain a relatively more consistent treatment forB → πlν decay form factors in comparison to the previousstudies.

III Numerical results

In this section, we present the numerical results forB →πlν decay form factors. We first present contributions fromthe valence-quark configuration. The parameters in the light-front wave functions are fixed by fitting to other hadronicproperties. For the pion wave function, the parametersωπand mq(= mu,d) are fitted to the pion decay constantfπand also the pion elastic form factor for momentum transferQ2 = 0∼ 10 GeV2 [27]:

ωπ = 0.29 GeV, mq = 0.30 GeV. (3.1)

For the light-frontB-meson wave function, we take

ωB = 0.57 GeV, mb = 4.8 GeV, (3.2)

which were determined from theB decay constantfB =0.187 GeV, and other decay data [2, 28]. With these param-eters, the decay form factorfv+ due to the valence config-uration is presented in Fig. 2. At the maximum recoil point(q2 = 0),

fv+ (0) = 0.24, (3.3)

which is consistent with results from most other calculations[2–12, 17]. A slightly different set of parameters for the pionwave function

ωπ = 0.33 GeV, mq = 0.25 GeV (3.4)

also fits the pion data quite well. The result is not qualita-tively changed by using this set of numbers, as can be seenalso in Fig. 2. The value of the form factor atq2 = 0 ishowever increased by 20% to

fv+ (0) = 0.29. (3.5)

662

Fig. 3. Solid line: Same as that in Fig. 2;dashed line:Gaussian wave func-tion used forB, with ωB = 0.55 GeV

Fig. 4.Solid line:Same as that in Fig. 2;dashed line:Eq. (3.6), withα = 1.6andMpole = 5.32 GeV

To test the sensitivity to theB-meson wave functionused, we plot in Fig. 3 the valence contribution, using aGaussian wave function of the same form as shown in (2.21),with ωB = 0.55 GeV. We find that the result is not changedsignificantly except in the zero-recoil region,q2 > 20 GeV2.The discrepency can be attributed to the different shapes ofthe two wave functions. Specifically, the BSW wave functionpeaks atx ' 0.15, while the Gaussian wave function has amuch narrower peak atx ' 0.01. As we shall see below, inthe zero-recoil region, theB∗-pole contribution dominates,and the valence contribution is relatively not important.

If we extrapolatef+(q2) away from theq2 = 0 point bythe formula

fpole+ (q2) =f+(0)

(1− q2/M2pole)α

, (3.6)

we find that the valence contribution is rather well describedwith α = 1.6 andMpole = 5.32 GeV, except in the zero-recoil region wheref+(q2) decreases asq2 increases (Fig. 3).

Next we consider theB∗-pole (or higher-Fock-state)contribution given by (2.44, 2.45). We take

fB = 0.187 GeV, fB∗/fB = 1.1,g = 0.48, Λ = 1.0 GeV, (3.7)

where the values taken forfB and the ratiofB/fB∗ areconsistent with QCD sum rule and lattice QCD results[11, 29, 30]. However the magnitude ofg is less certain;

Fig. 5. a Solid lines: B∗-pole contribution withΛ = 1 GeV; dashedline: valence-quark contribution (same as solid line in Fig. 2);b combinedvalence-quark andB∗-pole contribution withΛ = 1 GeV

it is not directly measurable, sinceB∗ → Bπ is kine-matically forbidden. In the infinite-heavy-quark-mass limit,heavy-quark symmetry predicts thatg is independent ofheavy quark flavor, so thatBB∗π is equal toDD∗π whichis experimentally measurable. However,they have different1/mQ corrections. From the upper limit of the total width ofD∗+(< 131 MeV) [31], we know that,g should be less than0.7. From non-relativistic quark-model constrained by theaxial coupling constantgA = 1.25 of the nucleon, one gets[24, 32] g = 0.75 in the symmetry limit, which some whatovershoots the experimental upper limit. QCD-sum-rule cal-culations [33, 34, 35] usually obtain smaller values, namely,g ' 0.2 ∼ 0.4. Results from various other approaches tendto fall between these two limits [28, 36–39]. In Eq. (3.7), wehave simply taken the average of the non-relativistic quarkmodel prediction and the lowest of the QCD sum-rule re-sults. The cutoff momentumΛ in (2.47) is also not wellknown, and we have assumedΛ to be equal to the chiralsymmetry breaking scaleΛχ ' 1 GeV. Sensitivity of theresults to the values ofg andΛ will be shown below.

Finally, we need to compute the renormalization con-stantZ2 given by (2.49). The result is

√Z2 = 0.85, 0.93,

0.98, forg = 0.75, 0.48, 0.20 respectively. Hence, depend-ing on the coupling constantg, the effect of wave functionrenormalization can be quite significant (2-15%).

TheB∗-pole contribution is plotted in Fig. 5a withΛ =1.0 GeV and three values ofg as indicated. The combinedvalence andB∗-pole contribution is plotted in Fig. 5b. We

663

Fig. 6. a Solid line: same as theg = 0.48 line in Fig. 5b;dashed line:Eq.(3.6) withα = 2.0 andMpole = 5.6 GeV; dash-dotted line:Eq. (3.6) withα = 1.0 andMpole = 5.32 GeV; b solid line: g = 0.20 line from Fig. 5b;dashed line:Eq. (3.6) withα = 2.0 andMpole = 5.8 GeV; dash-dottedline: Eq. (3.6) withα = 1.0 andMpole = 5.32 GeV

Fig. 7. Same as Fig. 5b except thatΛ = 1.50 GeV

see that theB∗-pole contribution dominates in the zero-recoil region, q2 ' 25 GeV2, and decreases rapidly asq2 decreases. With theB∗-contribution included, we findf+(0) = 0.26, 0.28, 0.29 for g = 0.20, 0.48, 0.75 respectively;therefore the effect of theB∗ contribution atq2 = 0 is withinthe theoretical error caused by the uncertainty in the pionwave function (see Eqs. (3.3, 3.5)). Forg = 0.48, the theo-retical result can be approximately described by (3.6), withα = 2.0 andMpole = 5.6 GeV (see Fig. 6a). Theg = 0.2curve also shows approximately a dipole behavior (α = 2.0),with a slightly different pole massMpole = 5.8 GeV (see

Fig. 6b). Hence our results indicate thatf+(q2) does not fol-low a simple monopoleq2-dependence (i.e.α = 1.0).

In order to study the sensitivity of our results to the cut-off parameterΛ, we have plotted in Fig. 7 the same curvesas in Fig. 5b, but withΛ=1.5 GeV. From Eqs. (2.44, 2.47,2.48), it is easily seen that theB∗-pole contributionfB

∗+

increases withΛ; this effect is however counter balanced bya smallerZ2. Since we have taken into account the effectof wave function renormalization, our results are relativelyinsensitive to the precise value ofΛ used. It thus seems thata measurement off+(q2) near zero-recoil would give a rathergood estimate of theBB∗π coupling constantg, which isnot obtainable from direct strong decays.

Finally, we can calculate the branching ratio forB →πlν. Comparing with the latest experimental result from theCLEO collaboration [40], we obtain|Vub|=0.003 forg=0.48.Predictions from other models range from|Vub|=0.002-0.004.

IV Summary

The exclusiveB → πlν decay is studied in this paper.We have calculated two different contributions to the decayprocess: The valence-quark contribution is calculated usinglight-front wave functions for mesons, while theB∗-polecontribution is evaluated with the help of chiral perturba-tion theory. In contrast to previous works using relativisticwave functions, the decay form factors have been calcu-lated directly in the entire range of momentum transfer, sothat extrapolation fromq2 = 0 is no longer required. Fur-thermore, in a more unified approach, theB∗-pole contribu-tion is calculated from the higher-Fock-state configuration|B∗π〉 of the B-meson wave function. The mixing of dif-ferent configurations requires a consistent renormalizationof the B-meson wave function, which is a 7% effect forg=0.48. We find that the form factorf+(q2) does not fol-low a monopoleq2-dependence, as is customarily assumedin the literature. Instead, our result is closer to a dipoleq2-dependence. At the maximum recoil point, we find thatf+(0) = 0.24−0.29, which is consistent with most other cal-culations. We observe that, since theB∗-pole contributiondominates in the zero-recoil region, a measurement off+(q2)near zero-recoil would be helpful in determining the valueof the chiralBB∗π coupling constantg. Finally we obtain|Vub|=0.003 forg=0.48.

Acknowledgements.It is a pleasure to thank H.Y. Cheng for helpful dis-cussions. We would also like to thank S.J. Brodsky for his comment onlight-front calculations with time-like momentum transfers. One of the au-thors (CYC) thanks the MIT Department of Mathematics for hospitality,where part of this work was completed. This work was supported in part bythe National Science Council of the Republic of China under Grant Nos.NSC84-2112-M-001-036, NSC85-2112-M-001-023, and NSC85-2816-M-001-001L.

References

1. Particle Data Group, Phys. Rev. D50, 1315 (1994)2. M. Wirbel, S. Stech, and M. Bauer, Z. Phys. C29, 637 (1985); M.

Bauer and M. Wirbel, ibid.42, 671 (1989)

664

3. N. Isgur, D. Scora, B. Grinstein, and M. Wise, Phys. Rev. D39, 799(1989)

4. P.J. O’Donnell, Q.P. Xu, and K.K. Tung, Preprint UTPT-95-2 (Univ.of Toronto)

5. R.N. Faustov, V.O. Galkin, and a. Yu. Mishurov, Preprint (RussianAcademy of Science)

6. P. Ball, V.M. Braun, H.G. Dosch, Phys. Lett. B273, 316 (1991); Phys.Rev. D 44, 3567 (1991); P. Ball, Phys. Rev. D48, 3190 (1993)

7. V.L. Chernyak and I.R. Zhitnitsky, Nucl. Phys. B345, 137 (1990);A.A. Ovchinnikov, Sov. J. Nucl. Phys.50, 519 (1989); C.A.Dominguez and N. Paver, Z. Phys. C41, 217 (1988)

8. V.M. Belyaev, A. Khodjamirian, and R. Ruckl, Z. Phys. C60, 349(1993)

9. P. Colangelo and P. Santorelli, Phys. Lett. B327, 123 (1994)10. A. Ali, V. M. Braun, and H. G. Dosch, Z. Phys. C63, 437 (1994)11. H.Y. Cheng, Chin. J. Phys.32, 425 (1994)12. G. Burdman, Z. Ligeti, M. Neubert, and Y. Nir, Phys. Rev. D49, 2331

(1994)13. M.B. Wise, Phys. Rev. D45, 2188 (1992)14. G. Burdman and J.F. Donoghue, Phys. Lett. B280, 287 (1992)15. L. Wolfenstein, Phys. Lett. B291, 177 (1992)16. R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio,

and G. Nardulli, Phys. Lett. B299, 139 (1993)17. H.N. Li and H.L. Yu, Phys. Rev. Lett.74, 4388 (1995)18. W. Jaus, Phys. Rev. D41, 3394 (1990)19. S.J. Brodsky, Private communication20. M. Sawicki, Phys. Rev. D44, 433 (1991)21. A. Dubin and A. Kaidalov, Yad. Fiz.56, 164 (1993), [Phys. At. Nucl.

56, 237 (1993)]

22. N. Isgur and M. B. Wise, Phys. Rev. D41, 151 (1990)23. G. Burdman and J. F. Donoghue, Phys. Rev. Lett.68, 2887 (1992)24. T.M. Yan, H.Y. Cheng, C.Y. Cheung, G.L. Lin, Y.C. Lin, and H.L.

Yu, Phys. Rev. D46, 1148 (1992)25. W. M. Zhang, Chinese. J. Phys.31, 717 (1994); Preprint IP-ASTP-19-

95 (hep-ph/9510428)26. C.Y. Cheung, W.M. Zhang, and G.L. Lin, Phys. Rev. D52, 2915

(1995)27. P.L. Chung, F. Coester, and W.N. Polyzou, Phys. Lett. B205, 545

(1988)28. P.J. O’Donnell and Q.P. Xu, Phys. Lett. B336, 113 (1994)29. M. Neubert, Phys. Rev. D46, 1076 (1992)30. A. Abada, et al., Nucl. Phys. B376, 172 (1992)31. S. Barlarg et al., Phys. Lett. B278, 480 (1992)32. H.Y. Cheng, C.Y. Cheung, G.L. Lin, Y.C. Lin, T.M. Yan, and H.L.

Yu, Phys. Rev. D49, 5857 (1994)33. T.M. Aliev, D.A. Demir, E. Iltan, and N.K. Pak, Phys. Lett. B351,

339 (1995)34. P. Colangelo, G. Nardulli, A. Deandrea, N. di Bartolomeo, R.Gatto,

and F. Fergulio, Phys. Lett. B339, 151 (1994)35. V.M. Belyaev, V.M. Braun, A. Khodjamirian, and R. Ruckl, Preprint

(MPI-PhT/94-62)36. P. Cho and H. Georgi, Phys. Lett. B296, 408 (1992)37. J.F. Amundson et al., Phys. Lett. B296 415 (1992)38. W.A. Bardeen and C.T. Hill, Phys. Rev. D49, 409 (1994)39. E. Eichten et al., Phys. Rev. D21, 203 (1988)40. CLEO collaboration, CLEO CONF 96-16