Heavy-to-light transitions on the light cone

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Heavy-to-light transitions on the light cone
Zheng-Tao Wei Nankai University

• Introduction
• Heavy-to-light form factors
• Light cone QCD
• Soft form factors in LC approach
• Summary

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I. Introduction

• B physics had been entered into an exciting era ?
  precision flavor physics.
• CKM angles
• sin2ß0.687/-0.032
• a(9913-8)o
• ?(6315-12)o
• Direct CP violation
• ACP(K p-) -0.108/-0.017
• B-gtVV polarization puzzle
• B-gtFK

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The importance of heavy-to-light form factors in
B physics phenomenology

• CKM parameter Vub,
• QCD, perturbative, non-perturbative
• basic parameters for exclusive decays in QCDF
  or SCET
• new physics,

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• The study of the heavy-to-light form factors
  has been a long history.

mb

• The QCD dynamics is complicated
• (1) many scales mb, ,
• ?QCD
• (2) confinement non-perturbative

?QCD
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Light cone dominance
Light cone vectors
At large recoil region q2ltltmb, the light meson
moves close to the light cone.
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II. Heavy-to-light form factors
Definition
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Hard scattering mechanism
Hard gluon exchange soft spectator quark ?
collinear quark Perturbative QCD is
applicable.
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Endpoint singularity
IR divergence ?01dx/x
endpoint singularity

• Factorization of pertubative contributions from
  the
• non-perturbative part is invalid.
• The soft contribution coming from the endpoint
  
• region is necessary.

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PQCD approach

• The transverse momentum are retained, so no
  endpoint singularity.
• Sudakov double logarithm corrections are
  included.

with
Sudakov factor
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Sudakov suppression effect is about 10-20.
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Soft mechanism

• One parton momentum in the light meson is soft.
• The form factor is dominated by soft
  interactions.
• Methods light cone sum rules,
• light cone quark model

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Spin symmetry for soft form factor
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In the large energy limit,

• The total 10 form factors are reduced to 3
  independent factors.
• There is no flavor symmetry for light mesons.
  3?1 impossible!

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Definition
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QCDF and SCET
In the heavy quark limit, to all orders of as and
leading order in 1/mb,
Sudakov corrections
Soft form factors, with singularity and Spin
symmetry
Perturbative, no singularity

• The factorization proof is more rigorous than
  others.
• The hard contribution (?/mb)3/2,
• soft form factor (?/m b)2/3 (?)
• About the soft form factors, study continues,
• such as zero-bin method

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Zero-bin method by Stewart and Manohar
(hep-ph/0605001)

• The soft component of light meson is contained
  in SCET
• time-ordered products from collinear fields.
  ( complete? )
• A collinear quark have non-zero energy. The
  zero-bin
• contributions should be subtracted out.
• After subtracting the zero-bin contributions,
  the remained is
• finite and can be factorizable.

For example,
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III. LC perturbation theory
Why light cone framework?

• The Lagrangian theory is not suitable to
  describe the bound states.
• For a relativistic Hamiltonian system, the
  definition of time is
• not unique. There are three forms.
• The LC framework is the most possible way to
  understand the
• non-relativistic quark model.
• Now, most people prefer to use the covariant
  form. In history,
• QCD (by Gell-Mann and Fritzsch) and
  perturbative QCD for exclusive
• processes (by Brodsky and Lepage) were
  proposed in the LC form.

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Diracs three forms of Hamiltonian dynamics
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LC Fock space expansion
LC wave functions
LC wave function is the central element in
LCQCD. It depends only on the intrinsic variables
(xi, k-i ).
In principle, wave functions can be solved if we
know the Hamiltonian (TV).
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Advantage of LC framework

• LC Fock space expansion provides a convenient
  description
• of a hadron in terms of the fundamental
  quark and gluon
• degrees of freedom.
• The LC wave functions is Lorentz invariant.
• ?(xi, k-i ) is independent of the bound
  state momentum.
• The vacuum state is simple, and trivial if no
  zero-modes.
• Only dynamical degrees of freedom are
  remained.
• for quark two-component ?,
• for gluon only transverse components
  A-.

Disadvantage

• In perturbation theory, LCQCD provides the
  equivalent results
• as the covariant form but in a complicated
  way.
• Its difficult to solve the LC wave function
  from the first principle.

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Kinetic
Vertex
LC Hamiltonian
Instantaneous interaction

• LCQCD is the full theory compared to SCET.
• Physical gauge is used A0.

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LC time-ordered perturbation theory

• Diagram are LC time x-ordered. (old-fashioned)
• Particles are on-shell.
• The three-momentum rather than four- is
  conserved in each vertex.
• For each internal particle, there are dynamic
  and instantaneous lines.

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Instantaneous, no singularity break spin symmetry
have singularity, conserve spin symmetry
Perturbative contributions

• Only instantaneous interaction in the quark
  propagator.
• The exchanged gluons are transverse polarized.

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III. Soft form factors in LC quark model
Soft overlap mechanism
The form factor is represented by the convolution
of initial and final hadron wave functions.
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Basic assumptions of LC quark model

• Valence quark contribution dominates.
• The quark mass is the constitute mass.

Constituent quark mass
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Melosh rotation
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Decay constants
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Form factors
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Choose Gaussian-type
Power law ?(q2)exp(-?QCD/mb)

• The scaling of the soft form factor depends on
  the light meson
• wave function at the endpoint. Only the
  precise knowledge of
• the wave function at long distance can solve
  it.

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Numerical results
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Comparisons with other approaches
The predictions of V and A0 in LCSR are larger
than other results.
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Summary

• The heavy-to-light form factors reveal rich QCD
  dynamics.
• LC quark model is an appropriate
  non-perturbative method
• to study the heavy-to-light form factors.