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B ± → K ± π - π + and B 0 → K 0 π - π + amplitudes from factorization
and analyticity
IPNO Participation: B. Moussallam
Collaboration : B. El Bennich (ANL, Argonne), A. Furman, B. Kaminski, L. Lesniak (INP, Cracow)
L’idée de factorisation naive, suggérée depuis longtemps pour les désintégrations faibles non leptoniques
des mésons lourds ,a été validée en QCD pour les désintégrations en deux corps comme approximation de
premier ordre. Nous supposons ici sa validité pour les désintégrations à trois corps dans des configurations
cinématiques spécifiques, et l’appliquons a B ± → K ± π - π + and B 0 → K 0 π - π + . Dans ce cas , les amplitudes
s’écrivent comme produit d’un facteur de forme Bπ et d’un facteur de forme Kπ. Nous incluons des corrections
de premier ordre en α s aux coefficients mais montrons la nécessité de corrections supplémentaires
que nous paramétrons. Nous construisons les facteurs de forme en utilisant leur propriétés d’analyticité et
des propriétés expérimentales de diffusion. Nous faisons une comparaison détaillée avec plus de 300
points expérimentaux de Belle et de Babar. Nous utilisons aussi l’analyticité pour donner une définition de
la désintégration en quasi deux corps B→K^*(1430)π et déterminer le taux de branchement.
Introduction
The idea of naive factorization has been applied
for a long time to weak non leptonic decays of heavy
mesons. In the case of two body decays it was
shown, in particular by Beneke, to be a leading
order approximation in an expansion in α s and in
inverse powers of m b . We assume here its validity
in the case of three body decays, in specific kinematical
configurations where two mesons are
quasi aligned (i.e. near one of the borders of the
Dalitz plot) and perform detailed comparisons with
experimental results from Belle and Babar involving
energy distributions as well as angular distributions.
More specifically, we will consider the amplitudes
B ± → K ± π - π + and B 0 → K 0 π - π + . In this
case, factorization predicts a particularly simple
form involving a product of a B to π form-factor and
a K to π form factor. We computed the coefficients
of these products including corrections to first order
in αs. We show that additional contributions to
these coefficients are necessary, which we determine
phenomenologically. The form-factors are
determined using analyticity and scattering experimental
data. As an application, we provide an ambiguity
free definition (and determination) of the
quasi two-body decay mode K^*(1430)π.
Factorized decay amplitudes
The hamiltonian which controls weak non-leptonic
decays for processes involving masses much
smaller than M W as a sum of four quark operators.
Using naive factorization the matrix elements relevant
for the decays B ± → K ± π - π + and B 0 → K 0 π - π +
involve a product of B to π vector of scalar formfactor
and the analogous K to π form factor and the
coefficients have simple expressions in terms of
the CKM matrix elements products V cb V* cs and
V ub V* us . These naïve coefficients, however, need
corrections as they fail to obey the correct renormalization
scale dependence.
Bπ and Kπ form-factors and analyticity
The Bπ form-factor is needed near zero energy
and this can be determined from lattice QCD using
extrapolation based on analyticity. The energy dependence
in the Dalitz plot is controlled in our approach
by the Kπ form-factor. In this case as well,
analyticity is a key ingredient, which can be combined
with information on its asymptotic behaviour
from QCD and with unitarity on the real cut. The
scalar and vector form-factors can both be deduced
from coupled-channel Muskhelishvili equations
owing to known properties of Kπ scattering.
Tests of these form-factors can also be performed
in tau semi-hadronic decays.
Results
As mentioned above, additional corrections to the
numerical coefficients of the form-factors are
needed which introduces eight parameters into our
amplitudes. With these we are able to describe
approximately 300 data points: energy as well as
angular distributions of charged and neutral B decays
as well as CP asymmetries. The description
of the data is illustrated below.
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