Transcript The Flavor of a little Higgs with T

SCET Workshop 2006
Model-independent properties of
B-meson LCDA
Seung J. Lee
Cornell University
3/3/2006
S.J.L. , Matthias Neubert
Phys. Rev. D 72, 094028
(2005)
Outline
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Introduction
Moment Analysis
Elimination of the pole mass
Renormalization-group evolution
Phenomenological model
Estimates of inverse moments
Conclusions
Introduction
 Exclusive decays of B-meson such as:
and
SCET for calculations of heavy-to-light form factors.
Factorization theorem:
Soft-overlap, spin-symmetry
preserving contribution
Hard-scattering, spin-symmetry violating contribution
Understanding B-meson
LCDA important!
(Also applicable for semi-inclusive
decays- recall C. Kim’s talk)
 Most previous studies of B-meson LCDA and its properties have
been limited to model-dependent analysis, based on QCD sum rules.
Introduction
 The leading-twist, two particle LCDA ,
Bilocal HQET operator
defined as:
of the B-meson is
A.G. Grozin, M.
Neubert
Phys. Rev. D 55, 272
(1997)
and momentum–space LCDA is given by:
 For most cases, two particle LCDAs are enough, since current
with extra soft gluons are power suppressed.
 We use OPE in our moments analysis of model-independent BLCDA. (basic idea is the same as that used in shape function
analysis of inclusive decay )
Moment Analysis
 Define regularized moments of B-meson LCDA as
 Taking the derivative of the zeroth moment M0 with respect to
cutoff, we can obtain a model-independent description of the
asymptotic behavior of B-meson LCDA:
 Using OPE: for a sufficiently large value of
the moments
MN can be expanded in a series of B-meson matrix elements of
local operators.
Moment Analysis
 Finding all possible local operators of dimension D < 5:
-from the structure of the bilocal HQET operator and the Feynman rules of
HQET it follows that the resulting local operators have Dirac structure
massless light quark
even numbers, so that there’s no open Dirac indices
 by using equations of motion:
D=3: no derivative
D=4: one derivative
 But, the Wilson coefficients of Q1c and Q1d are zero, because residual
momentum k of the external heavy quark field only appears as v.k in the
leading order HQET diagrams.
Moment Analysis
 The resulting expansion of the moments is:
 Matching at one-loop level (calculate Wilson coefficients)
k
p
-employ partonic expression
-use on-shell conditions: p2 = v.k = 0
-expand Feynman amplitude to linear order in p
Moment Analysis
 Matching at one-loop level (calculate Wilson coefficients):
The results for the one-loop local operators becomes simple:
The result for the one-loop matrix element of the bilocal HQET
operator is still nontrivial:
Moment Analysis
 The star distributions are generalized plus distributions are defined as:
where F(w) is a smooth test function
 Wilson coefficients obtained using partonic calculations are reliable, since
they do not depend on long-distance physics.
 Matrix-elements depend on long-distance physics, but they are constrained
by heavy-quark symmetry:
Moment Analysis
 Wilson coefficients:
We have also checked our results
with a different IR regularization
scheme, obtaining the same
Wilson coefficients.
 First two moments of the renormalized B-meson LCDA
Moment Analysis
 Asymptotic behavior
For sufficiently large values of the cutoff, the moments of the B-meson LCDA
can be calculated using the OPE. Taking the derivative of the zeroth moment
M0 with respect to the cutoff, we can obtain a model-independent description
of the asymptotic behavior, i.e.
 This relation holds for
up to power correction of order
 Radiation tail: becomes negative at
for a sufficiently large value of
Elimination of the pole mass
 So far, calculations have been performed in on-shell (pole)
scheme, where
is defined in terms of the
b-quark pole mass.
 But, the pole mass suffers from renormalon ambiguities.
eliminate the pole scheme parameter in favor of
new, short distance parameter
defined in some
renormalization scheme.
(for example: “kinetic mass”, “potential-subtracted mass”,
“1S mass”, and “shape-function mass” schemes)
 We can define a new physical subtraction scheme, guided
by tree level moments relations
define a running parameter to all order in
perturbation theory:
By taking the ratio of M1 and M0, the double logarithmic radiative corrections are eliminated
Elimination of the pole mass
 From our next-leading order Moments calculations, it
follows that:
 Perturbative relations can be used to transform it into
other mass-definition schemes. For example, in “shapefunction mass” scheme
 In numerical analysis, we can use well known values of
certain scheme. i.e. we use a rather precise value for
that has been
extracted from moment analysis of various spectra in the inclusive decays:
Renormalization-group evolution
 The renormalization group can be used to obtain modelindependent description of how B-meson LCDA changes under
various scales.
 The integro-differential evolution equation of B-meson LCDA
was previously derived, where an analytic solution was
presented in the form of a double integral:
B.O. Lange and M. Neubert
Phys. Rev. Lett. 91, 102001
(2003)
Renormalization-group evolution
 Integrand
has poles:
at imaginary axis, t = -i(n-g) ; at positive axis, t = in with n positive integer
 We take a step further by performing analytical integration,
using theorem of residues, obtaining the most compact
Hypergeometric function
expression:
 We believe that this is the exact
solution to Renormalization-group
evolution equation for the LCDA, valid
up to all orders in perturbation theory. c.f.
shape function
 But, it’s proven only for the leading
order evolution.
Phenomenological model
 For phenomenological purposes, we need a model for Bmeson LCDA at scale
“hard-collinear” scale
for hard spectator scattering.
 Realistic models should satisfy the model-independent
properties we have found.
 Our model consists of two-component ansatz:
Normalization
Based on
exponential model
Radiation tail added for correct asymptotic behavior
The tail is “glued” at a position wt , such that resulting function
is continuous.
Phenomenological model
The normalization constant N and parameter w0 can be fixed by
matching the expression for first two moments:
 Our model ansatz is to a good approximation preserved under RGE.
model ansatz at 1 GeV
evolving from 1Gev to 2.5 GeV
model ansatz
at 2.5 GeV
Phenomenological model
Comparison with other model:
QCD sum-rule based model at next-leading order in
:
V.M. Braun, D. Y. Ivvanov,
and G. P. Korchemsky
at
Phys. Rev. D 69, 034014
(2004)
Exhibit a similar
asymptotic behavior!
Phenomenological model
Moments comparison:
Comparison of model results (black) and OPE predictions (gray) for the first
two moments of the LCDA, evaluated at
and for different values of
the cutoff. The solid black curves are obtained in our model, the dashed ones in the
QCD sum-rule base model.
For small value of cutoff, we see large deviation. (OPE valid for
)
Estimates for inverse moments
The inverse moments
play an important role in the
analysis of many exclusive B-meson decays.
They control the strength of the leading-power spectator
interactions.
The quantity
and
enters these analyses when one
goes beyond tree approximation.
With our realistic model, we can provide estimates for these
parameters.
Estimates for inverse moments
 Our findings are in good agreement with QCD sum-rule estimate
at next-to-leading order in
, indicated by the data points.
Do not point to a large non-perturbative enhancement of hard-scattering contribution.
 The error band reflects the variation of results with the input
parameter
lighter bands are an estimate of total theoretical error (neglected higher order terms in OPE,
and non-perturbative uncertainties in the precise shape of LCDA for small value of w.)
Estimates for inverse moments
 hard-scattering contributions:
(non-perturbative enhancement)
R.J. Hill, T. Becher, S.J.L, M. Neubert
J. High Energy Phys. 07 (2004) 081
 for pseudoscalar and longitudinally-polarized vector meson:
leading order correction
 for perpendicularly-polarized vector meson:
where
enters for single and
double logarithms
Estimates for inverse moments
 hard-scattering contributions:
Pseudoscalar and
longitudinallypolarized vector
meson
Perpendicularlypolarized vector
meson
No large non-perturbative enhancement of hard-scattering contribution.
small for
(soft-overlap contributions larger than that
of hard-scattering- recall Richard Hill’s talk)
Conclusions
 Model-independent properties of renormalized B-meson LCDA
have been obtained using rigorous methods based on OPE.
 Base on model-independent analysis, we have proposed a
realistic model of the B-meson LCDA , which is consistent with
the moment relations, and invariant under RGE in a good
approximation.
 With the help of this function, we have obtained estimate for the
inverse moment parameters
. At
we
find that
and
with conservative errors.
 We hope that our analysis open a new strategy for further , more
detailed studies of
using a systematic short-distance
approach. Ultimately, this may help reduce theoretical
uncertainties in predictions for exclusive B-decays.