A study of B → K∗�+�− decay in soft-collinear effective theory

A. Alia

aTheory Group, Deutsches Elektronen-Synchrotron DESY,Notkestrasse 85, 22603 Hamburg, Germany

A brief account of the study of rare B decay B → K∗�+�− using soft-collinear effective theory (SCET) from [1]is presented here. The main result of this study is a factorization formula, derived to leading power in 1/mb butvalid to all orders in αs. For phenomenological application, the decay amplitude including order αs corrections iscalculated and the logarithms resummed by evolving the matching coefficients from the hard scale O(mb) downto the scale

√mbΛh. The partially integrated branching ratio in the range 1 GeV2 ≤ q2 ≤ 7 GeV2 is calculated

to be (2.92+0.67

−0.61)× 10−7. For the zero-point of the forward-backward asymmetry in the standard model, one getsq20 = (4.07+0.16

−0.13) GeV2. The scale dependence of q20 is improved compared to the earlier estimates of the same.

1. Introduction

The electroweak penguin decay B → K∗�+�−

is loop-suppressed in the Standard Model(SM).It may therefore provide a rigorous test of theSM and also put strong constraints on the flavorphysics beyond the SM. Experimentally, BaBar[2] and Belle [3] Collaborations have observed thisrare decay with a branching ratio of O(10−6).In addition, Belle has published the first mea-surements [3,4] of the so-called forward-backwardasymmetry (FBA) [5]. The best-fit results byBelle for the Wilson coefficient ratios for neg-ative value of A7,

A9

A7

= −15.3+3.4−4.8 ± 1.1 and

A10

A7

= 10.3+5.2−3.5 ± 1.8, are consistent with the SM

values A9/A7 � −13.7 and A10/A7 � +14.9, eval-uated in the NLO approximation. With moredata at the current B factories, and the antici-pated measurements at the LHC, these measure-ments are expected to become very precise.

Theoretically, the exclusive decay B →K∗�+�− has been studied in a number of pa-pers [6]. The emergence of an effective theory,called soft-collinear effective theory (SCET) [7–11], provides a systematic way to deal with theperturbative strong interaction effects in B de-cays in the heavy-quark expansion. SCET hasbeen used extensively in the so-called heavy-to-light transitions in B decays [11–16]; the ex-tension of SCET to two effective theories SCETI

and SCETII , with the two-step matching QCD →SCETI → SCETII was first studied in [17]. Inparticular, this framework was used to prove thefactorization of radiative B → V γ decays at lead-ing power in 1/mb and to all orders in αs [18,19].In [1], the related decay B → K∗�+�− has beenstudied using the SCET approach to the samelevel of theoretical accuracy.

Restricting ourselves to the kinematic regionwhere the light K∗ meson moves fast and can beviewed approximately as a collinear particle, afactorization formula for the decay amplitude ofB → K∗�+�−, to leading power in 1/mb, has beenderived in SCET [1]. This coincides formally withthe formula obtained earlier by Beneke et al. [20],using the QCD factorization approach [21], but isvalid to all orders of αs:

〈K∗a�+�−|Heff |B〉 = T I

a (q2)ζa(q2) +∑±

∫ ∞

0

dω

ω

×φB±(ω)

∫ 1

0

du φaK∗(u)T II

a,±(ω, u, q2) , (1)

where a =‖,⊥ denotes the polarization of the K∗

meson. The functions T I and T II are perturba-tively calculable. ζa(q2) are the soft form factorsdefined in SCET while φB

±(ω) and φaK∗(u) are the

light-cone distribution amplitudes (LCDAs) forthe B and K∗ mesons, respectively. In particular,the location of the zero of the forward-backward

Nuclear Physics B (Proc. Suppl.) 163 (2007) 109–116

0920-5632/$ – see front matter © 2006 Elsevier B.V. All rights reserved.

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doi:10.1016/j.nuclphysbps.2006.09.001

asymmetry in B → K∗�+�−, defined as q20 , which

is one of the most interesting observables in thedecay B → K∗�+�− [22,6], can be predicted moreprecisely in SCET due to the improved theoreticalprecision on the scale dependence of q2

0 . In thistalk, essential aspects of the theoretical deriva-tions and phenomenological analysis from [1] aresummarized.

2. SCET analysis of B → K∗�+�−

For the b → s transitions, the weak effectiveHamiltonian can be written as

Heff = −GF√2

V ∗tsVtb

10∑i=1

Ci(μ)Qi(μ) , (2)

where we have neglected the contribution propor-tional to V ∗

usVub, and have used the unitarity ofthe CKM matrix to factorize the overall CKM-matrix element dependence. The operator basiscan be seen in [23,20].

For convenience, let us assume that the K∗ me-son is moving in the direction of the light-like ref-erence vector n, then its momentum can be de-composed as pμ = n·pnμ/2+pμ

⊥+n·pnμ/2, wherenμ is another light-like reference vector satisfyingn · n = 2. In this light-cone frame, the collinearmomentum of K∗ is expressed as

p = (n · p, n · p, p⊥) ∼ (λ2, 1, λ)mb , (3)

with λ ∼ Λ/mb � 1. In addition to this collinearmode, the soft and hard-collinear modes, withmomenta scaling as (λ, λ, λ)mb and (λ, 1,

√λ)mb,

respectively, are also necessary to correctly repro-duce the infrared behavior of full QCD.

SCET introduces fields for every momentummode and we will encounter the following quarkand gluon fields

ξc ∼ λ, Aμc ∼ (λ2, 1, λ), ξhc ∼ λ1/2, Aμ

hc ∼ (λ, 1, λ1/2),

qs ∼ λ3/2, Aμs ∼ (λ, λ, λ), h ∼ λ3/2 . (4)

In the above, the symbol Aμ stands for the gluonfield, h represents a heavy-quark field, the sym-bols ξ and q stand for the light quark fields, andthe subscripts c, s, hc stand for collinear, soft andhard-collinear modes, respectively. Note that the

momentum q of the lepton pair is taken as a hardcollinear momentum, and hence an extra hard-collinear field ξhc ∼ λ1/2 in the n direction isrequired later. To construct the gauge invariantoperators in SCET, it is more convenient to intro-duce the building blocks, given below, which areobtained by multiplying the fields by the Wilsonlines which run along the light-ray to infinity:

Xc , Aμc , Xhc , Xhc , A

μhc ,

Qs , Aμs , Hs , Qs , Hs . (5)

For example, the field Xhc is defined as

Xhc(x) = W †hc(x)ξhc(x) , with

Whc(x) = P exp

(ig

∫ 0

−∞

dsn · Ahc(x + sn)

),(6)

where Whc(x) is the hard collinear Wilson line.The notations Qs and Hs are used when the as-sociated soft Wilson lines are in the n-direction.For the definitions of the other fields and moretechnical details about SCET, we refer the readerto Ref. [18] and references therein.

Since SCET contains two kinds of collinearfields, i.e. hard-collinear and collinear fields,normally an intermediate effective theory, calledSCETI , is introduced which contains only softand hard-collinear fields. While the final effec-tive theory, called SCETII , contains only soft andcollinear fields. We will then do a two-step match-ing from QCD → SCETI → SCETII .

2.1. QCD to SCETI matching

In SCETI , the K∗ meson is taken as a hard-collinear particle and the relevant building blocksare Xhc, Xhc, A

μhc and h. The velocity of the B

meson is defined as v = PB/mB. The matchingfrom QCD to SCETI at leading power may beexpressed as

Heff → −GF√2

V ∗tsVtb

(4∑

i=1

∫ds CA

i (s)JAi (s)

+

4∑j=1

∫ds

∫dr CB

j (s, r)JBj (s, r)

+

∫ds

∫dr

∫dt CC(s, r, t)JC(s, r, t)

),(7)

where C(A,B)i and CC are Wilson coefficients in

the position space. The relevant SCETI op-

A. Ali / Nuclear Physics B (Proc. Suppl.) 163 (2007) 109–116110

erators for the B → K∗�+�− decay are con-structed by using the building blocks mentionedabove [18]:

JA1 = Xhc(sn)(1 + γ5)γ

μ⊥h(0) �γμ� ,

JA2 = Xhc(sn)(1 + γ5)

nμ

n · vh(0) �γμ� ,

JB1 = Xhc(sn)(1 + γ5)γ

μ⊥ /Ahc⊥(rn)h(0) �γμ� ,

JB2 = Xhc(sn)(1 + γ5) /Ahc⊥(rn)

nμ

n · vh(0) �γμ� ,

JC = Xhc(sn)(1 + γ5)/n

2Xhc(rn)

× Xhc(an)(1 + γ5)/n

2h(0) ,

(8)

where the operators JAi and JB

i represent thecases that the lepton pair is emitted from theb → s transition currents, while JC representsthe diagrams in which the lepton pair is emit-ted from the spectator quark of the B meson.Note that there are altogether four JA

i and JBi

operators; the ones not shown JAi (i = 3, 4) and

JBi (i = 3, 4) are obtained from the ones given

above by the replacement �γμ� → �γμγ5�. Ex-

cept for the lepton pair, the operators JA,Bi have

the same Dirac structures as those of the heavy-to-light transition currents in SCET, which werefirst derived in Ref. [8] for JA

i and in Refs. [16,17]for JB

j (see, also Refs. [15,24]).Since in practice the matching calculations

are done in the momentum space, it is moreconvenient to define the Wilson coefficients inthe momentum space by the respective Fourier-transformations. The corresponding coefficientfunctions are being called CA

i (E), CBj (E, u), and

CC(E, u), with E ≡ n ·vn ·P/2. To get the orderαs corrections to the decay amplitude, we needto calculate the Wilson coefficients CA

i to one-loop level and CB

j and CC to tree level. In the

following we will use ΔjC(A,B,C)i to denote the

matching results from the weak effective opera-tors Qj to the SCET currents JA,B,C

i . With this,the matching coefficients from QCD → SCETI

can be written as

C(A,B,C)i =

10∑j=1

ΔjC(A,B,C)i (μQCD, μ) , (9)

where μQCD is the matching scale and μ is therenormalization scale in SCETI .

Each operator of the weak effective Hamilto-nian, namely Q1−10, will contribute to CA

i at or-der αs level. But due to the small Wilson coef-ficients C3−6, it is numerically reasonable to ne-glect the contributions from Q3−6. The contri-butions from the operators Q1,2, Q7, Q8, Q9 andQ10 are given in [1].

To get the decay amplitude of B → K∗�+�− inorder αs, the tree-level matching of the effectiveweak Hamiltonian (4) onto B-type SCET currents(11) is already enough, and the results are givenin [1]. The diagrams where the virtual (off-shell)photon is emitted from the spectator quark con-tribute to the Wilson coefficients of the C-typeSCET current. The annihilation diagram con-tributing to the matching coefficient CC at orderα0

s is trivial. This, together with the order αs con-tribution of the matching onto the C-type SCETcurrent, are also given in [1].

2.2. SCETI → SCETII matching

As shown in Refs. [17,13], which analyzed theform factors in the framework of SCET, one maysimply define the matrix elements of the A-typeSCETI currents as non-perturbative input sincethe non-factorizable parts of the form factors areall contained in such matrix elements. Thereforethe explicit matching of JA

i to SCETII operatorsis not necessary here.

For B-type SCETI operators, they are matchedonto the following SCETII operators (for the vec-tor lepton current)

OB1 = Xc(sn)(1 + γ5)γ

μ⊥

/n

2Xc(0)

× Qs(tn)(1 − γ5)/n

2Hs(0) �γμ� ,

OB2 = Xc(sn)(1 + γ5)

nμ

n · v/n

2Xc(0)

× Qs(tn)(1 + γ5)/n

2Hs(0) �γμ� .

(10)

A. Ali / Nuclear Physics B (Proc. Suppl.) 163 (2007) 109–116 111

There are two more currents OB3 and OB

4 involv-ing the axial-vector lepton currents obtained fromthe currents defined above by replacing �γμ� →�γμγ5�. Again, it is in practice more convenientto do the matching calculations in the momen-tum space, and the Wilson coefficients DB

i (ω, u)can be defined by Fourier transforming the corre-sponding ones DB

i (s, t) introduced in the positionspace, just like the case in SCETI ,

DBi (ω, u) =

∫ds

∫dt e−iωn·vteiusn·P DB

i (s, t).

(11)

Following the notations of [18], the Wilson coef-ficients DB

i can be expressed as

DBi (ω, u, s, μ) =

1

ω

∫ 1

0

dv Ji

(u, v, ln

mbω(1 − s)

μ2, μ

)× CB

i (v, μ) ,

(12)

where the jet functions Ji arise from theSCETI → SCETII matching and it is clear thatJ1 = J3 ≡ J⊥ and J2 = J4 ≡ J‖. At tree level,using the Fierz transformation in the operator ba-sis, one obtains

J⊥(u, v) = J‖(u, v)

= −4πCF αs

Nc

1

mb(1 − u)(1 − s)δ(u − v) . (13)

Finally, the C-type SCETI current is matchedonto the SCETII operator

OC = Xc(sn)(1 + γ5)/n

2Xc(0)

×Qs(tn)(1 + γ5)/n

2Hs(0)

nμ

n · v �γμ� . (14)

We may similarly define

DC(ω, u) =

∫ds

∫dt e−iωn·vteiusn·P DC(s, t).

(15)

with

DC(ω, u, s, μ) =−eeqs

(ω − q2/mb − iε)

×J C

(ln

mbω(1 − s)

μ2, μ

)CC(E, u, μ) ,(16)

where eq is the electric charge of the spectatorquark in the B meson. Note that the prefactorin the above equation, −eeqs/(ω − q2/mb − iε),has been factored out from the definition of J C .The denominator of this prefactor comes from thepropagator of the hard-collinear spectator quark.With this, at tree level the corresponding jet func-tion is trivial, J C = 1. For later use, we defineDC ≡ DC/(ω − q2/mb − iε).

2.3. Matrix elements of SCET operators

The last step before we can finally get the decayamplitude for the B → K∗�+�− decay is to takethe matrix elements of the relevant SCET oper-ators. Since the soft and collinear parts do notfactorize in the matrix elements of the operatorJ A in SCETII due to an end-point singularity,we will define the matrix elements of the J A cur-rent in SCETI . Following Ref. [18], they may bedefined as follows:

〈M(p)|XhcΓh|B(v)〉 =

−2EζM (E)tr[MM (n)ΓMB(v)] , (17)

where the projection operators are

MB(v) = −1 + /v

2γ5 ,MK∗

⊥(n) = /ε∗⊥

/n/n

4,

MK∗‖(n) = − /n/n

4, (18)

with εμ⊥ being the polarization vector of the K∗

⊥

meson. It is then straightforward to get the ma-trix elements of the SCETI currents JA

i as

〈K∗�+�−|JA1 |B〉 = −2Eζ⊥(gμν

⊥ − iεμν⊥ )ε∗⊥ν �γμ� ,

〈K∗�+�−|JA2 |B〉 = −2Eζ‖

nμ

n · v �γμ� ,

(19)

where gμν⊥ ≡ gμν − (nμnν + nμnν)/2 and

εμν⊥ ≡ εμνρσvρnσ/(n · v), and we use the con-

vention ε0123 = +1. The matrix elements〈K∗�+�−|JA

3 |B〉 and 〈K∗�+�−|JA4 |B〉 are ob-

tained from the above matrix elements by thereplacement �γμ� → �γμγ5�, respectively.

For the B-type SCETII operators (25), therelevant meson LCDAs are defined in [21,25],where two different K∗-distribution amplitudes

(φ‖K∗(u, μ) for Γ = 1 and φ⊥

K∗(u, μ) for Γ = γ⊥)

A. Ali / Nuclear Physics B (Proc. Suppl.) 163 (2007) 109–116112

with their corresponding decay constants f‖K∗

and f⊥K∗(μ), respectively, are involved. With the

above LCDAs, the matrix elements of the opera-tors OB

i can be written as

〈K∗�+�−|CB1 OB

1 |B〉 = −F (μ)m3/2B

4(1 − s)(gμν

⊥ − iεμν⊥ )

× ε∗⊥ν �γμ�

∫ ∞

0

dω

ωφB

+(ω, μ) ×∫ 1

0

du fK∗⊥(μ)

φK∗⊥(u, μ) ×

∫ 1

0

dvJ⊥(u, v, lnmbω(1 − s)

μ2, μ)CB

1 (v, μ)

≡ −F (μ)m3/2B

4(1 − s)(gμν

⊥ − iεμν⊥ )

× ε∗⊥ν �γμ� φB+ ⊗ fK∗

⊥φK∗

⊥⊗ J⊥ ⊗ CB

1 ,

〈K∗�+�−|CB2 OB

2 |B〉 = −F (μ)m3/2B

4(1 − s)

nμ

n · v �γμ�

× φB+ ⊗ fK∗

‖φK∗

‖⊗ J‖ ⊗ CB

2 ,

(20)

F (μ) is related to the B meson decay constant fB

up to higher orders in 1/mb by [26]

fB√

mB = F (μ)

(1 +

CF αs(μ)

4π

(3 ln

mb

μ− 2

)).

(21)

The matrix element of CB3 OB

3 (CB4 OB

4 ) can be ob-tained by simply replacing the lepton current �γμ�on the right hand side of the above equations by�γμγ5� and also replacing CB

1 → CB3 (CB

2 → CB4 ).

The matrix element of OC is obtained likewise,with the result

〈K∗�+�−|DCOC |B〉 = −F (μ)m3/2B

4(1 − s)

nμ

n · v×�γμ�

ωφB−

ω − q2/mb − iε⊗ fK∗

‖φK∗

‖⊗ DC .(22)

Since φB−(ω) does not vanish as ω approaches zero,

the integral∫

dω φB−(ω)/(ω − q2/mb) would be

divergent if q2 → 0. This endpoint singularitywill violate the SCETII factorization, that is whywe should restrict the kinematic region so that theinvariant mass of the lepton pair is not too small,say q2 ≥ 1 GeV2.

2.4. Resummation of logarithms in SCET

In the above analysis a two-step matching pro-cedure QCD → SCETI → SCETII has beenimplemented. This introduces two matchingscales, μh ∼ mb at which QCD is matched ontoSCETI and μl ∼ √

mbΛ at which SCETI ismatched onto SCETII . Thus, with the SCETI

matching coefficients at scale μh, one may usethe renormalization-group equations (RGE) ofSCETI to evolve them down to scale μl and thenmatch onto SCETII . The large logarithms dueto different scales are resummed during this pro-cedure.

Furthermore, one should note that for the A-type SCET currents, only the scale μh is involvedsince it is not necessary to do the second stepmatching of SCETI → SCETII . Similarly, wemay choose the nonperturbative form factors ζ⊥,‖

at the scale μh and avoid the RGE running ofthe A-type SCETI matching coefficients. For theB-type currents, the RGE of SCETI can be ob-tained by calculating the anomalous dimensionsof the relevant SCET operators, which has beendone in [24], where the matching coefficients atany scale μ can be obtained by an evolution fromthe matching scale μh as follows

CBj (E, u, μh, μ) =

(2E

μh

)a(μh,μ)

eS(μh,μ)

×∫ 1

0

dv UΓ(u, v, μh, μ)CBj (E, v, μh)

≡(

2E

μh

)a(μh,μ)

eS(μh,μ) U jΓ(E, u, μh, μ) ,

(23)

with the subscript Γ =⊥, ‖ and the functionsa(μh, μ) and S(μh, μ) are given in Eq. (66) ofRef. [24]. Note that in the above equation oneshould use the subscript Γ =⊥ for j = 1, 3, whileΓ =‖ for j = 2, 4. This equation is solved numer-ically in [1].

Finally, for the C-type SCET current JC , itsanomalous dimension just equals the sum of theanomalous dimensions of the K∗ meson LCDAφK∗ and the B meson LCDA φB

−. However, asthe evolution equation of φB

− is still unknown, wewill not resum the perturbative logarithms for the

A. Ali / Nuclear Physics B (Proc. Suppl.) 163 (2007) 109–116 113

JC current. Numerically the contribution fromthe JC current to the decay amplitude is small.Furthermore, the JC current is completely irrele-vant for the forward-backward asymmetry of thecharged leptons.

3. Numerical analysis of B → K∗�+�−

We are now in the position to write the decayamplitude of B → K∗�+�−, using the similar no-tations adopted in [20],

d2Γ

dq2d cos θ=

G2F |V ∗

tsVtb|2128π3

(αem

4π

)2

m3BλK∗(1 − q2

m2B

)2

×{

2ζ2⊥(1 + cos2 θ)

q2

m2B

(|C⊥9 |2 + (C⊥

10)2) − 8ζ2

⊥ cos θ

× q2

m2B

Re(C⊥9 )C⊥

10 + ζ2‖ (1 − cos2 θ)(|C‖

9 |2 + (C‖10)

2)

},

(24)

with mBλK∗/2 being the 3-momentum of the K∗

meson in the rest frame of the B meson. Theangle θ denotes the angle between the momenta ofthe positively charged lepton and the B meson inthe rest frame of the lepton pair. The ”effective”

Wilson coefficients C⊥,‖9 and C⊥,‖

10 are given by

C⊥9 =

2π

αem

(CA

1 +mB

4

fBφB+ ⊗ f⊥

K∗φ⊥K∗ ⊗ J⊥ ⊗ CB

1

ζ⊥

),

C‖9 =

2π

αem

(CA

2 +mB

4

fBφB+ ⊗ f

‖K∗φ

‖K∗ ⊗ J‖ ⊗ CB

2

ζ‖

− q2

4mB

fBωφB−/(ω − q2/mb − iε) ⊗ f

‖K∗φ

‖K∗ ⊗ DC

ζ‖

),

C⊥10 =

2π

αemCA

3 ,

C‖10 =

2π

αem

(CA

4 +mB

4

fBφB+ ⊗ f

‖K∗φ

‖K∗ ⊗ J‖ ⊗ CB

4

ζ‖

),

(25)

where CA,Bi and DC are defined in Eqs. (13)

and (32), respectively. The above expressions arevalid at leading power in 1/mb and to all ordersin αs. Calculating explicitly the ”effective Wilsoncoefficients” at one-loop order, the SCET-based

results are quite similar to those of [20]. Themain phenomenological improvement here is thatfor the hard scattering part, the matching coeffi-cients CB

i are evolved from the scale μh ∼ O(mb)down to μl ∼

√mbΛh, during which the pertur-

bative logarithms are summed (Λh represents atypical hadronic scale).

3.1. Input parameters

To get the differential distributions numeri-cally, some input parameters have to be spec-ified. The hadronic parameters for the decayB → K∗�+�− include decay constants, light-conedistribution amplitudes (LCDAs) and the softform factors. They are given in Ref. [1] togetherwith the values for the Wilson coefficients.

There are only two independent B → K∗ formfactors in SCET, namely ζ⊥(q2) and ζ‖(q

2). Theyare related to the full QCD form factors as dis-cussed in [24]. Their current knowledge is frag-mentary. We use the radiative B → K∗γ de-cay, which has been measured quite precisely,B(B0 → K∗0γ) = (4.01 ± 0.20) × 10−5, to nor-malize the soft form factor at q2 = 0, obtain-ing ζ⊥(0) = 0.32 ± 0.02, and assume that theq2-dependence of ζ⊥(q2) can be reliably obtainedfrom the LCSRs [27]. For the longitudinal softform factor ζ‖, unfortunately there is no quantita-tive determination from the existing experiments,though this may change in the future with goodquality data available on the decay B → ρ�ν�.Using helicity analysis, one can extract ζρ

‖ (q2);

combined with estimates of the SU(3)-breakingone may determine ζK∗

‖ (q2). Not having this ex-perimental information at hand, one may extractζ‖(q

2) from the full QCD form factor AB→K∗

0 (q2).

LCSRs estimate [27] AB→K∗

0 (0) = 0.374 ± 0.043with the q2-dependence

AB→K∗

0 (q2) =1.364

1 − q2/m2B

− 0.990

1 − q2/36.78GeV2 .

(26)

From which we get ζ‖(0) = 0.40 ± 0.05.

3.2. The dilepton invariant mass spectrum

and the forward-backward asymmetry

Experimentally, the dilepton invariant massspectrum and the forward-backward (FB) asym-

A. Ali / Nuclear Physics B (Proc. Suppl.) 163 (2007) 109–116114

q2[GeV2]

dAFB/dq2

Figure 1. The differential forward-backwardasymmetry dAFB(B → K∗�+�−)/dq2 in therange 1 GeV2 ≤ q2 ≤ 8 GeV2. Here the solid linedenotes the theoretical prediction with the inputparameters taken at their central values, whilethe gray band between two dashed lines reflectsthe uncertainties from input parameters and scaledependence. The dotted line represents the LOpredictions, obtained by dropping the O(αs) cor-rections (from [1]) .

metry are the observables of principal interest.Their theoretical expressions in SCET can be eas-ily derived from Eq. (24):

dBr

dq2= τB

G2F |V ∗

tsVtb|2128π3

(αem

4π

)2

m3B|λK∗ |(1 − q2

m2B

)2

×{

16

3ζ2⊥

q2

m2B

(|C⊥9 |2 + (C⊥

10)2) +

4

3ζ2‖ (|C‖

9 |2 + (C‖10)

2)

},

(27)

dAFB

dq2=

−6(q2/m2B)ζ2

⊥Re(C⊥9 )C⊥

10

4(q2/m2B)ζ2

⊥(|C⊥9 |2 + (C⊥

10)2) + ζ2

‖ (|C‖9 |2 + (C‖

10)2)

.

(28)

Restricting to the integrated branching ratio ofB → K∗�+�− in the range 1 GeV2 ≤ q2 ≤7 GeV2, where the SCET method should work,

we obtain

7 GeV2∫1 GeV2

dq2 dBr(B+ → K∗+�+�−)

dq2=

(2.92+0.57−0.50|ζ‖ +0.30

−0.28|CKM+0.18−0.20) × 10−7 . (29)

Here we have isolated the uncertainties from thesoft form factor ζ‖ and the CKM factor |V ∗

tsVtb|.The last error reflects the uncertainty due tothe variation of the other input parameters andthe residual scale dependence. For B0 decay,the branching ratio is about 7% lower due tothe lifetime difference (ignoring the small isospin-violating corrections from the matrix elements).Experimentally one of the Belle observations [3]of our interest is

8 GeV2∫4 GeV2

dq2 dBr(B → K∗�+�−)

dq2=

(4.8+1.4−1.2|stat. ± 0.3|syst. ± 0.3|model) × 10−7 ,(30)

for which we predict (1.94+0.44−0.40) × 10−7. This

is smaller than the published Belle data by afactor of about 2.5. However, the BaBar col-laboration measures the total branching ratio ofB → K∗�+�− to be [2] (7.8+1.9

−1.7 ± 1.2) × 10−7,which is about twice smaller than the Belle ob-servation [3] (16.5+2.3

−2.2 ± 0.9 ± 0.4) × 10−7. Thisimplies that, if finally the total branching ratioof B → K∗�+�− is found to be closer to theBaBar result, the partially integrated branchingratio over the range 4 GeV2 ≤ q2 ≤ 8 GeV2 couldbe lowered to around 2.3× 10−7, which is consis-tent with our estimate (1.94+0.44

−0.40) × 10−7 withinthe stated errors.

From Eq. (28), it is easy to see that the lo-cation of the vanishing FB asymmetry is deter-mined by Re(C⊥

9 ) = 0. At the leading order, this

leads to the equation C9 +Ceff7 +Re(Y (q2

0)) = 0.Including the order αs corrections, our analysisestimates the zero-point of the FB asymmetry tobe

q20 = (4.07+0.16

−0.13) GeV2 , (31)

of which the scale-related uncertainty isΔ(q2

0)scale =+0.08−0.05 GeV2 for the range mb/2 ≤

A. Ali / Nuclear Physics B (Proc. Suppl.) 163 (2007) 109–116 115

μh ≤ 2mb together with the jet function scaleμl =

√μh × 0.5 GeV. This is to be compared

with the result given in Eq. (74) of [20], alsoobtained in the absence of 1/mb corrections:q20 = (4.39+0.38

−0.35) GeV2. Of this the largest single

uncertainty (about ±0.25 GeV2) is attributed tothe scale dependence. While our central valuefor q2

0 is similar to theirs, the scale dependence inour analysis is significantly smaller than that of[20]. The difference in the estimates of the scaledependence of q2

0 here and in Ref. [20] is both dueto the incorporation of the SCET logarithmic re-summation and the different (scheme-dependent)definitions of the effective form factors for theSCET currents. The SCET resummation hasbeen derived in the existing literature [18,24,28].As power corrections in 1/mb have not been con-sidered here, although they are probably compa-rable to the O(αs) corrections as argued in [20], italso remains to be seen how a model-independentcalculation of the same effect the numerical valueof q2

0 .

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