Factorization in Multibody Radiative B Decays

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  • Factorization in Multibody Radiative B Decays

    Benjamn Grinsteina

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

    We study the radiative decays B K and B Ke+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 K and for the forward-backward asymmetry inB Ke+e.


    There is considerable interest in exclusive ra-diative B-meson decays. In particular the ratesfor the decays B K and B Ke+eare determined by the CKM element Vts andthe corresponding decays with substituted forK are determined by Vtd. Moreover, CP viola-tion (CPV) in B K and Forward-BackwardAsymmetry (FBA) in B Ke+e have beenextensively studied since they are sensitive probesof physics beyond the standard model. In theseprocesses the K is observed through its decayinto K. Theoretical studies of these decayslargely neglect the non-resonant K 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 dierences with large cancellations among thedominant resonant contributions, one must becareful not to neglect other, possibly signicantcontributions. Indeed, it was shown in [1] thatCPV is absent in resonant B (K)K in thelimit of ms 0, while there is a several per-centCP asymmetry in non-resonant B K.

    In this talk we report on progress towards anunderstanding of the non-resonant decays B K and B Ke+e. More generally, we givea method for computing amplitudes that involvean energetic K and a soft in the nal 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 K. 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 -hard K,hard -hard K, and hard -soft K, respectively.

    of a B decay[2]. We use this method to includethe non-resonant contribution in the calculationof B K[3] and B Ke+e[4], for whichwe rst use Soft Collinear Eective 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 benetsfrom this approach.

    Nuclear Physics B (Proc. Suppl.) 163 (2007) 121126

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



  • 0 20 40 60 80 100mB





    E ,m


    Figure 2. Minimum energy from B Kfollowed by K K.

    1.1. KinematicsFigure 1 shows the Dalitz plot for the decay

    B K in the pion energy (E) vs. invariantK mass (MK) plane. A vertical line is drawnat E = 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 t). We have divided the remaining region,accessible experimentally, into three:

    I. Soft and hard K, E , EK Q.II. Hard and K, E EK Q.III. Hard and soft K, E Q, EK .

    We have arbitrarily placed the boundary of regionI at E = 1 GeV. For power counting the scalingis E , EK Q, where 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 dened by EK < 1 GeV. We also indicatein the gure a shaded area for E < 500 MeV,dening 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 K. Thiswould not be the case in a world with a muchheavier b-quark: we show in Fig. 2 the minimum

    E from B K K 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).


    2.1. 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 eectively 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 rst 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 = /Q.

    The non-factorizable matrix elements, whiledicult to compute, satisfy spin symmetry re-lations. This has important consequences. ForB K these symmetry relations give the re-sult, mentioned above, that CP asymmetries van-ish at zero s-quark mass. And for B Ke+ethey 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 coecient 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 OCbe small, MX . Region I in Fig. 1 hasMX

    Q, 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 comes from OS and K from OC .

    B. Grinstein / Nuclear Physics B (Proc. Suppl.) 163 (2007) 121126122

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

    OC q(x)W (x, 0) q(0) . (2)

    Here the quark elds are separated along thelight-cone, x = x0 +x3, and W is a Wilson line,preserving gauge invariance. stands for any ofthe basis -matrices. In our problem the relevantamplitude is the matrix element of this betweenthe vacuum and a K or K state. They are given(by denition), 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) 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 gives a contribution in region I to the fac-torizable amplitude for B K. Which begs thequestion, what is the OS matrix element betweenB and meson states?

    2.2. HM-PTThe 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 a-vor symmetry, while realizing linearly the unbro-ken, vector avor symmetry. The Lagrangian isgiven in terms of = exp(iM/f) (and = 2),where

    M =

    120 + 1

    6 + K+

    120 + 1

    6 K0

    K K0 26

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

    LR , LU = UR (4)under the avor group SU(3)L SU(3)R. Uis a non-linear matrix function of M implicitlydened by the 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 avor symmetries, but also the spin symme-try of Heavy Quark Eective Theory (after all,we are expanding in 1/Q 1/mb). This can beaccomplished by combining the B and B mesonswith denite velocity v into matrices of elds,

    Ha =1 + v/2


    Ba5], (5)

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

    H H U . (6)The eective Lagrangian describing interactionsof the M -elds with themselves and with H eldsis xed by symmetry considerations alone. Tolowest order in the derivative and heavy massexpansions[57]:

    L = f2


    )+ 0Tr

    [mq+ mq

    ] iTrHvH + i2TrHHv

    [ +




    [ ]+ (7)

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

    PLQ in QCD can bewritten in the low energy chiral theory as

    La =i


    ba] + , (8)

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

    B. Grinstein / Nuclear Physics B (Proc. Suppl.) 163 (2007) 121126 123

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

    (R)S and

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


    and O(L)S transform as (1L,3R) and (3L,1R), re-spectively. We see that O(L)S is very similar tothe local current (8), but have now additional xdependence. As for La, avor symmetry impliesthat the eective theory from of O(L)Sa must in-volve the product Hb ba. But are these eldslocalized at 0 or at x? To answer this questionpromote the vector avor symmetry to a localsymmetry. This requires introducing gauge eldsfor SU(3)V . We do not make these elds 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 =


    4Tr[L(x) PR Hb(0)

    ba(x) ], (9)

    and similarly for O(R)S . The Fourier transformof the matrix element of O(L)S between the vac-uum and B state xes the Fourier transform ofL(x), and is determined in terms of the light-cone wave-functions of the B, (k+); see Refs.[2,8] for details:

    L(k+) = R(k+) = fBmB [n/+(k+)+n/(k+)](10)

    where n = (1, 0, 0, 1) and n = (1, 0, 0,1).This is a remarkable result: the matrix elementof the non-local operators O(L,R)S between a Bmeson 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 veriedin 1+1 QCD at large Nc[8].

    3. MODEL

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

    location of the FBA zero in B Ke+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 nalK 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 . Moreover, the region is fairlysmall.

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

    H(B K) = H(B K)W(m2K) (11)where W is the Breit-Wigner function

    W(m2K) =gKK((K

    ) p)m2K m2K + imKK

    . (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 , are then given by rela-tions analogous to (11), e.g.,

    BK (mK, E) = BK npKW(m2K). (13)

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

    4. CPV IN B KThe time dependent dierential decay rate for

    B K isd2(B0(t) KS0i)




    12et {1 + Ci cosmt Si sinmt} ,


    B. Grinstein / Nuclear Physics B (Proc. Suppl.) 163 (2007) 121126124

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

    with i = L,R the photon polarization and

    Ci(E,MK) =|Ai|2 |Ai|2|Ai|2 + |Ai|2 , (15)

    Si(E,MK) = 2Im (e2iAiAi )|Ai|2 + |Ai|2 , (16)

    given in terms of time-independent amplitudes:

    AL = H(B0 KS0L) , (17)AR = H(B0 KS0R) , (18)AL = H(B0 KS0L) = + p

    p AR , (19)

    AR = H(B0 KS0R) = p+ p AL . (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 . No complete analysis of sub-leading corrections exists at present, so we incor-









    0 0.5 1 1.5 2 2.5 3 3.5

    q2 0


    BK* (0) = 0.3


    = 300 MeVE

    cut= 500 MeV


    = 700 MeV = 4.8 GeV

    Figure 4. Location of the FBA zero as a functionof the K mass. The leading, non-factorizableamplitude has BK

    (0) = 0.3. The curve labeled

    = 4.8 GeV shows the result if the factorizableand spectator contributions are neglected.

    porate them phenomenologically through[1,3]






    + hseis (21)

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

    hs 13C2C7


    0.09 (22)

    The factor 1/3 is a color suppression, while theenhancement |C2/C7| 3.2 arises from the coef-cients of the eective Hamiltonian for weak de-cays (with O7 = mbe/162 sLFbR).

    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 mK. Our result is shown in Fig. 3. The thicksolid line corresponds to hs cos(s) = 0, and thegray region is obtained by allowing hs cos(s) 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 ter...