Factorization in Multibody Radiative B Decays

Benjamın Grinsteina∗

aUniversity of California, San Diego9500 Gilman Drive 0319, La Jolla, CA 92093-0319, USA

We study the radiative decays B → Kπγ and B → Kπe+e−, including both K∗ resonant contributions andnon-resonant ones. We describe new soft pion theorems with which we compute certain non-resonant multibodyamplitudes. We present results for CP asymmetry in B → Kπγ and for the forward-backward asymmetry inB → Kπe+e−.

1. INTRODUCTION

There is considerable interest in exclusive ra-diative B-meson decays. In particular the ratesfor the decays B → K∗γ and B → K∗e+e−

are determined by the CKM element Vts andthe corresponding decays with ρ substituted forK∗ are determined by Vtd. Moreover, CP viola-tion (CPV) in B → K∗γ and Forward-BackwardAsymmetry (FBA) in B → K∗e+e− have beenextensively studied since they are sensitive probesof physics beyond the standard model. In theseprocesses the K∗ is observed through its decayinto Kπ. Theoretical studies of these decayslargely neglect the non-resonant Kπ contribu-tions to the rate. This is appropriate for totalrates, since the resonant contribution is domi-nant. But in the case of asymmetries, involv-ing differences with large cancellations among thedominant resonant contributions, one must becareful not to neglect other, possibly significantcontributions. Indeed, it was shown in [1] thatCPV is absent in resonant B → (Kπ)K∗γ in thelimit of ms → 0, while there is a several per-centCP asymmetry in non-resonant B → Kπγ.

In this talk we report on progress towards anunderstanding of the non-resonant decays B →Kπγ and B → Kπe+e−. More generally, we givea method for computing amplitudes that involvean energetic K and a soft π in the final state

∗Work in collaboration with Daan Pirjol. This work wassupported in part by the DOE under grant DE-FG03-97ER40546.

Figure 1. Dalitz plot showing regions of phasespace available in the decay B → Kπγ. Quanti-ties are in units of GeV. To the right of the verti-cal line the photon energy exceeds 2.3 GeV. Theregions I, II and III, correspond to soft π-hard K,hard π-hard K, and hard π-soft K, respectively.

of a B decay[2]. We use this method to includethe non-resonant contribution in the calculationof B → Kπγ[3] and B → Kπe+e−[4], for whichwe first use Soft Collinear Effective Theory to es-tablish factorization theorems.

Technical details can be found aplenty in thereferences (where further references to originalwork, which were omitted here due to space lim-itations, can also be found). We wish to use thisspace to present those results in a more accessible,dare we say, pedagogical manner. We apologizeto the expert and hope the non-expert benefitsfrom this approach.

Nuclear Physics B (Proc. Suppl.) 163 (2007) 121–126

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

www.elsevierphysics.com

doi:10.1016/j.nuclphysbps.2006.09.010

0 20 40 60 80 100mB

0.4

0.8

1.2

1.6

EΠ

,min

Figure 2. Minimum π energy from B → K∗γfollowed by K∗ → Kπ.

1.1. KinematicsFigure 1 shows the Dalitz plot for the decay

B → Kπγ in the pion energy (Eπ) vs. invariantKπ mass (MKπ) plane. A vertical line is drawnat Eγ = 2.3 GeV. Photons in the region to theright of the line are too soft to be detected exper-imentally (experiments change this cut as theysee fit). We have divided the remaining region,accessible experimentally, into three:

I. Soft π and hard K, Eπ ∼ Λ, EK ∼ Q.

II. Hard π and K, Eπ ∼ EK ∼ Q.

III. Hard π and soft K, Eπ ∼ Q, EK ∼ Λ.

We have arbitrarily placed the boundary of regionI at Eπ = 1 GeV. For power counting the scalingis Eπ ∼ Λ, EK ∼ Q, where Λ denotes a typicalhadronic scale and Q a typical “hard scale”, inthis case, Q ∼ mb. In region I the K recoils nearlyin the opposite direction of the photon, so we alsorefer to the K as “collinear.” Similarly, regionIII is defined by EK < 1 GeV. We also indicatein the figure a shaded area for Eπ < 500 MeV,defining roughly the region of validity for soft piontheorems.

As we will see below, we have found a methodto reliably compute the non-resonant process inthe shaded area of region I. However, there is alarge contribution to the decay in region I fromthe resonant process, B → K∗γ → Kπγ. Thiswould not be the case in a world with a muchheavier b-quark: we show in Fig. 2 the minimum

Eπ from B → K∗γ → Kπγ as a function of the Bmass. For mB ≥ 60 GeV there is no contributionto region I from the resonant decay. On the otherhand, for mB = 5 GeV the resonant decay canproduce arbitrarily soft pions (that is, at rest inthe B rest-frame).

2. FACTORIZATION AND HM-χPT

2.1. SCET: FactorizationBoth in the Standard Model of electroweak in-

teraction and in extensions that include new par-ticles at the TeV scale, the amplitude for radiativeB decays proceeds through an effectively local in-teraction represented in Field Theory by an op-erator O of dimension six. One can use the tech-niques of SCET to show that these amplitudesfactorize. Schematically,

O → T ⊗ OS ⊗ OC + Onf + · · · (1)

where the first term is a convolution of a shortdistance factor T , and soft and collinear matrixelements of operators OS and OC , respectively,the second term is a “non-factorizable” matrixelement and the ellipsis denote correction termsof relative order of the small parameter λ = Λ/Q.

The non-factorizable matrix elements, whiledifficult to compute, satisfy “spin” symmetry re-lations. This has important consequences. ForB → K∗γ these symmetry relations give the re-sult, mentioned above, that CP asymmetries van-ish at zero s-quark mass. And for B → K∗e+e−

they give that the FBA vanishes at a particu-lar value of q2 (the invariant e+e− mass), andthe predicted location of this zero is largely in-sensitive to non-perturbative physics, dependingonly on the coefficient of the operator O in theweak Hamiltonian. One can therefore view thefactorizable contributions as corrections to thesepredictions.

The validity of the SCET requires that themass of the hadronic state X produced by OC

be small, MX ∼ Λ. Region I in Fig. 1 hasMX ∼ √

ΛQ, and it would appear it cannot be de-scribed by SCET. However, if the pion is a purelysoft object and the K is collinear one is clearlywithin the domain of SCET. This is particularlyobvious when π comes from OS and K from OC .

B. Grinstein / Nuclear Physics B (Proc. Suppl.) 163 (2007) 121–126122

The OC amplitudes typically involve matrix el-ements of operators of the following form:

OC ∼ q(x−)W (x−, 0)Γ q(0) . (2)

Here the quark fields are separated along thelight-cone, x− = x0 +x3, and W is a Wilson line,preserving gauge invariance. Γ stands for any ofthe basis γ-matrices. In our problem the relevantamplitude is the matrix element of this betweenthe vacuum and a K or K∗ state. They are given(by definition), after Fourier transform in x−, bythe light-come wave-functions of these mesons.

The OS amplitudes are similar. They corre-spond to matrix elements between soft states ofoperators of the form

OS ∼ q(x−)W (x−, 0)Γ bv(0) . (3)

Computing this between a B meson and the vac-uum gives, as in the K case, the B-light-conewave-function. This can be combined with thecollinear matrix element for (vac.) → K∗ to givethe factorizable contribution for B → K∗ transi-tions. Also, combining the collinear matrix ele-ment for (vac.) → K with the soft amplitude forB → π gives a contribution in region I to the fac-torizable amplitude for B → Kπ. Which begs thequestion, what is the OS matrix element betweenB and π meson states?

2.2. HM-χPTThe answer is supplied by a judicious applica-

tion of soft pion theorems, which generally allowus to compute a one pion matrix element from azero pion amplitude. However, the devil is in thedetails, that we now proceed to describe.

Chiral Lagrangians provide a simple and prac-tical way of deriving soft pion theorems. As iswell known they are constructed to provide a non-linear realization of the spontaneously broken fla-vor symmetry, while realizing linearly the unbro-ken, vector flavor symmetry. The Lagrangian isgiven in terms of ξ = exp(iM/f) (and Σ = ξ2),where

M =

⎛⎜⎝

1√2π0 + 1√

6η π+ K+

π− − 1√2π0 + 1√

6η K0

K− K0 − 2√6η

⎞⎟⎠

and f � 135 MeV is the pion/Kaon decay con-stant at lowest order in the chiral expansion.These fields transform as

Σ → LΣR† , ξ → LξU† = UξR† (4)

under the flavor group SU(3)L × SU(3)R. Uis a non-linear matrix function of M implicitlydefined by the ξ transformation law. For ourpurposes we need to describe soft interactions ofthese particles with B mesons (the interactionsare soft in the rest frame of the B mesons). Wewould like these interactions to respect not onlythe flavor symmetries, but also the spin symme-try of Heavy Quark Effective Theory (after all,we are expanding in 1/Q ∼ 1/mb). This can beaccomplished by combining the B and B∗ mesonswith definite velocity v into matrices of fields,

Ha =1 + v/

2[B∗

aμγμ − Baγ5

], (5)

labeled by a = 1, 2, 3 = u, d, s. Under the flavorsymmetry the field Ha transforms as

H → H U †. (6)

The effective Lagrangian describing interactionsof the M -fields with themselves and with H fieldsis fixed by symmetry considerations alone. Tolowest order in the derivative and heavy massexpansions[5–7]:

L =f2

8Tr

(∂μΣ∂μΣ†) + λ0Tr

[mqΣ + mqΣ†]

− iTrHvμ∂μH +i

2TrHHvμ

[ξ†∂μξ + ξ∂μξ†

]+

ig

2TrHHγνγ5

[ξ†∂νξ − ξ∂νξ†

]+ · · · (7)

Symmetries constrain also the form of oper-ators such as currents. For example, the lefthanded current Lν

a = qaγνPLQ in QCD can bewritten in the low energy chiral theory as

Lνa =

iα

2Tr[γνPLHbξ

†ba] + · · · , (8)

The parameter α is fixed from the vacuum to Bmatrix element of the current, which gives α =fBmB .

B. Grinstein / Nuclear Physics B (Proc. Suppl.) 163 (2007) 121–126 123

We are finally ready to apply this machinery tothe computation of the matrix elements of non-local operators OS . It is convenient to considerseparately operators with PLΓ and PRΓ in placeof Γ in OS in (3), and we denote these as O

(R)S and

O(L)S , respectively. Under the flavor group O

(R)S

and O(L)S transform as (1L,3R) and (3L,1R), re-

spectively. We see that O(L)S is very similar to

the local current (8), but have now additional x−dependence. As for Lν

a, flavor symmetry impliesthat the effective theory from of O

(L)Sa must in-

volve the product Hb × ξ†ba. But are these fieldslocalized at 0 or at x−? To answer this questionpromote the vector flavor symmetry to a localsymmetry. This requires introducing gauge fieldsfor SU(3)V . We do not make these fields dynam-ical and we set them to zero at the end of thecomputation. Similarly we also promote the b-number to a local U(1) symmetry. Performinglocalized symmetry transformations we see that

O(L)Sa =

i

4Tr[αL(x−) PR Γ Hb(0) ξ†ba(x−) ], (9)

and similarly for O(R)S . The Fourier transform

of the matrix element of O(L)S between the vac-

uum and B state fixes the Fourier transform ofαL(x−), and is determined in terms of the light-cone wave-functions of the B, φ±(k+); see Refs.[2,8] for details:

αL(k+) = αR(k+) = fBmB [n/φ+(k+)+n/φ−(k+)](10)

where nμ = (1, 0, 0, 1) and nμ = (1, 0, 0,−1).This is a remarkable result: the matrix elementof the non-local operators O

(L,R)S between a B

meson and any number of pions is completely de-termined (in the chiral limit and to lowest orderin the chiral and 1/mb expansions) by the light-cone wave-functions! The result has been verifiedin 1+1 QCD at large Nc[8].

3. MODEL

It is time to use the results of the previous sec-tion to estimate the effects of factorizable correc-tions on CP asymmetries in B → Kπγ and on the

location of the FBA zero in B → Kπe+e−. Tothis end we will need to compute the matrix el-ements of both factorizable and non-factorizableoperators between the initial B state and the finalKπ state. However, it is not known how to com-pute these exactly. We adopt a well motivated,simple phenomenological model.

First we ignore the contributions from regionIII. Formally, for a collinear pion and a soft K theamplitude is sub-leading in the SCET expansion,i.e., it is order λ. Moreover, the region is fairlysmall.

In region II we use a resonant approximation.That is, if Hσ, σ = +, 0,−, denotes a generichelicity amplitude, we take

Hσ(B → Kπ) = Hσ(B → K∗)Wσ(m2Kπ) (11)

where Wσ is the Breit-Wigner function

Wσ(m2Kπ) =

gK∗Kπ(ε∗σ(K∗) · pπ)m2

Kπ − m2K∗ + imK∗ΓK∗

. (12)

Finally, in region I we have can compute non-resonant contributions to the factorizable am-plitudes using the methods of the previous sec-tion. We also include a resonant contributionto the non-factorizable amplitudes using a res-onant approximation, as in Eq. (11). For thenon-factorizable amplitudes we neglect the non-resonant contribution and again use a Breit-Wigner form for the resonant ones. The non-factorizable amplitudes, often called soft func-tions and denoted by “ζ,” are then given by rela-tions analogous to (11), e.g.,

ζBKπ⊥ (mKπ, Eπ) = ζBK∗

⊥ n·pK∗Wσ(m2Kπ). (13)

The precise definition of the soft functions can befound in [3,4].

4. CPV IN B → Kπγ

The time dependent differential decay rate forB → Kπγ is

d2Γ(B0(t) → KSπ0γi)dEπdM2

Kπ

=1

2(4π)3m2B

(|Ai|2+|Ai|2)

× 12e−Γt {1 + Ci cos Δmt − Si sinΔmt} ,

(14)

B. Grinstein / Nuclear Physics B (Proc. Suppl.) 163 (2007) 121–126124

Figure 3. CP asymmetry in B → Kπγ in theStandard Model, as a function of the Kπ invari-ant mass. The gray area corresponds to varyingthe unknown sub-leading correction hs cos(φs)between -0.05 and +0.05, and the dark line cor-responds to hs cos(φs) = 0.

with i = L,R the photon polarization and

Ci(Eπ,MKπ) =|Ai|2 − |Ai|2|Ai|2 + |Ai|2 , (15)

Si(Eπ,MKπ) = 2Im (e−2iβAiA

∗i )

|Ai|2 + |Ai|2 , (16)

given in terms of time-independent amplitudes:

AL = H(B0 → KSπ0γL) , (17)

AR = H(B0 → KSπ0γR) , (18)

AL = H(B0 → KSπ0γL) =ε+ · pπ

ε− · pπAR , (19)

AR = H(B0 → KSπ0γR) =ε− · pπ

ε+ · pπAL . (20)

In the last two lines we have used CP conservationof the b → s transition to relate the B to B decayamplitudes.

We now use the model of the above sectionto compute these amplitudes. Neglecting mo-mentarily factorizable contributions, spin sym-metry of the non-factorizable amplitudes givesHBK∗

+ = 0 in the limit that ms = 0. More pre-cisely, we have HBK∗

+ /HBK∗− = ms/mb. This is

a small number and therefore we cannot neglectterms of order λ. No complete analysis of sub-leading corrections exists at present, so we incor-

2

2.5

3

3.5

4

4.5

5

5.5

0 0.5 1 1.5 2 2.5 3 3.5

q2 0

M2Kπ

ζBK*⊥ (0) = 0.3

Eπcut= 300 MeV

Eπcut= 500 MeV

Eπcut= 700 MeV

μ = 4.8 GeV

Figure 4. Location of the FBA zero as a functionof the Kπ mass. The leading, non-factorizableamplitude has ζBK∗

⊥ (0) = 0.3. The curve labeledμ = 4.8 GeV shows the result if the factorizableand spectator contributions are neglected.

porate them phenomenologically through[1,3]

HBK∗+

HBK∗−

=ms

mb+ hse

iφs (21)

The leading contribution to the correctionarises from the four quark operator O2 =(sc)V −A(cb)V −A through an internal charm loop.We therefore estimate

hs ∼ 13

C2

C7

Λmb

∼ 0.09 (22)

The factor 1/3 is a color suppression, while theenhancement |C2/C7| ∼ 3.2 arises from the coef-ficients of the effective Hamiltonian for weak de-cays (with O7 = mbe/16π2 sLσμνFμνbR).

To this we add the contribution in region I fromfactorizable operators using the method describedabove. This introduces into the CP asymmetry(the parameter S in Eq. (14)) a mild dependenceon mKπ. Our result is shown in Fig. 3. The thicksolid line corresponds to hs cos(φs) = 0, and thegray region is obtained by allowing hs cos(φs) tovary between -0.05 and 0.05, which in light of (22)is a conservative estimate.

In the absence of a reliable computation of thesub-leading terms we conclude that only a verysubstantial CP asymmetry could be construed as

B. Grinstein / Nuclear Physics B (Proc. Suppl.) 163 (2007) 121–126 125

2

2.5

3

3.5

4

4.5

5

5.5

0 0.5 1 1.5 2 2.5 3 3.5

q2 0

M2Kπ

ζBK*⊥ (0) = 0.1

Eπcut= 300 MeV

Eπcut= 500 MeV

Eπcut= 700 MeV

μ = 4.8 GeV

Figure 5. Same as Fig. 4 but with ζBK∗⊥ (0) = 0.1

arising from new physics. Significant deviation inthe dependence on mKπ from what is shown inFig. 3 could also be a sign of new physics (a topicthat perhaps deserves some study).

5. FBA-ZERO in B → Kπe+e−

The FBA is defined in terms of the differentialdecay rate as

AFB(q2) =

[ ∫ 1

0

d cos θ+dΓ(q2, θ+)dq2d cos θ+

−∫ 0

−1

d cos θ+dΓ(q2, θ+)dq2d cos θ+

]/dΓ(q2)

dq2,

(23)

where q2 stands for the invariant mass of the e+e−

pair. In terms of helicity amplitudes HV,Aσ for

vector (V) and axial (A) currents, it is

AFB ∝ Re (HV−HA∗

− − HV+ HA∗

+ ) (24)

Let us neglect, for now, the factorizable contribu-tion to these amplitudes. Spin symmetries of thenon-factorizable amplitudes give H+ = 0. A zeroin the FBA must come from a point where an H−vanishes. Now, H is the product of a matching(short distance) coefficient and a hadronic ma-trix element. The latter is non-vanishing, so welook for zeroes of the coefficients. The coefficientfor the axial current amplitude is constant. De-noting the coefficient of the non-factorizable vec-

tor amplitude by c(V )1 the condition for a zero

is Re c(V )1 (mKπ, q2) = 0. When we include the

effects of factorizable amplitudes we will needthe hadronic matrix element (the soft function)ζBK∗⊥ (q2) in the non-factorizable amplitude, for

which we adopt a modified pole shape, and con-sider its normalization to two values ζBK∗

⊥ (0) =0.3 (conservative) and 0.1 (aggressive).

Including the factorizable corrections the con-dition for a FBA zero is

Re( c(V )1 − af − asp) = 0, (25)

where af and asp denote the factorizable and spec-tator amplitudes, respectively, normalized to thenon-factorizable soft function. These can be com-puted in terms of light-cone wave functions usingthe model of Sec. 3. The results are shown inFigs. 4 and 5, which show the value of q2 for whichthere is a zero in the FBA as a function of mKπ,for ζBK∗

⊥ (0) = 0.3 and 0.1, respectively. For eachcurve the factorizable contribution in region I isintegrated up to a maximum pion energy Ecut

π

(recall we use HM-χPT, which applies for softpions only). The results display mild, but non-negligibe dependence on mKπ. This dependencehad not been considered previously since mKπ isfixed to mK∗ when only the resonant contributionis included (see, e.g., Ref. [9]).

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B. Grinstein / Nuclear Physics B (Proc. Suppl.) 163 (2007) 121–126126