Factorization in Multibody Radiative B Decays
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  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. We use this method to includethe non-resonant contribution in the calculationof B K and B Ke+e, 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
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. FACTORIZATION AND HM-PT
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
120 + 1
6 + K+
120 + 1
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
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:
L = f2
] 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
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 wit