1. 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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2. I. Introduction
B physics had been entered into an exciting era → precision flavor physics.
CKM angles
sin2β=0.687+/-0.032;
α=( )o;
γ=( )o
Direct CP violation
ACP(K+ π-)= /-0.017
B->VV polarization puzzle
B->ΦK*
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3. 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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4. The study of the heavy-to-light form factors has been a long history.mb
The QCD dynamics is complicated:
(1) many scales: mb, ,
ΛQCD;
(2) confinement: non-perturbative
ΛQCD
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5. Light cone dominance Light cone vectors close to the light cone.
At large recoil region q2<<mb, the light meson moves
close to the light cone.
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6. II. Heavy-to-light form factors
Definition
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7. Hard scattering mechanism
Hard gluon exchange: soft spectator quark → collinear quark
Perturbative QCD is applicable.
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8. 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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9. PQCD approach Sudakov double logarithm corrections are included. with
The transverse momentum are retained, so no endpoint singularity.
Sudakov double logarithm corrections are included.
with
Sudakov factor
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10. Sudakov suppression effect is about 10-20%.
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11. 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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12. Spin symmetry for soft form factor
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13. 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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14. Definition
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15. QCDF and SCET
In the heavy quark limit, to all orders of αs 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 ~ (Λ/mb)3/2,
soft form factor ~ (Λ/m b)2/3 (?)
About the soft form factors, study continues,
such as zero-bin method…
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16. 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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17. 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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18. Dirac’s three forms of Hamiltonian dynamics
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19. 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 (T+V).
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20. 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.
Ψ(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 ξ,
for gluon: only transverse components A┴.
Disadvantage
In perturbation theory, LCQCD provides the equivalent results
as the covariant form but in a complicated way.
It’s difficult to solve the LC wave function from the first principle.
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21. Instantaneous interaction
Kinetic
Vertex
LC Hamiltonian
Instantaneous interaction
LCQCD is the full theory compared to SCET.
Physical gauge is used A+=0.
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22. 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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23. Perturbative contributions:
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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24. 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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25. Basic assumptions of LC quark model
Valence quark contribution dominates.
The quark mass is the constitute mass.
Constituent quark mass:
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26. Melosh rotation
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27. Decay constants
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28. Form factors
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29. Power law: ξ(q2)~exp(-ΛQCD/mb)
Choose Gaussian-type
Power law: ξ(q2)~exp(-ΛQCD/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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30. Numerical results
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31. Comparisons with other approaches
The predictions of V and A0 in LCSR are larger than other results.
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32. 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.
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