Sudakov and heavy-to-light form factors in SCET

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Sudakov and heavy-to-light form factors in
SCET

• Zheng-Tao Wei
• Nankai University

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• Introduction to SCET
• Sudakov form factor
• Heavy-to-light transition form factors
• Summary

Wei, PLB586 (2004) 282, Wei, hep-ph/0403069, Lu,
et al., PRD (2007)
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I. Soft-Collinear Effective Theory

• The soft-collinear effective theory is a low
  energy
• effective theory for collinear and soft
  particles.
• (Bauer, Stewart , et al. Beneke,
  Neubert.)
• (1) It simplifies the proof of
  factorization theorem
• at the Lagrangian and operator level.
  
• (2) The summation of large-logs can be
  performed
• in a new way.

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• Diagrammatic analysis and effective Lagrangian

eikonal approximation
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Transforming the diagrammatic analysis into an
effective Lagrangian
LEET
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• Degrees of freedom

Reproduce the full IR physics

• Power counting

Field
Momentum
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The effective Lagrangian

• The effective interaction is non-local in
  position space.

• Two different formulae hybrid
  momentum-position space
• and position space representation.

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Gauge invariance
The collinear and ultrasoft gauge transformation
are constrained in corresponding regions,

• The ultrasoft field acts as backgroud field
  compared to
• collinear field.

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No interaction with usoft gluons
Wilson lines
Gauge invariant operators (basic building blocks)
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• Matching mismatch? New mode, such as
  soft-collinear mode
• proposed by Neubert et al.?
• Endpoint singularity?

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Two step matching

• Integrate out the high momentum
• fluctuations of order Q,
• 2. Integrate out the intermediate scale
  (hard-collinear field)

SCET(I)
SCET(II)
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III. Sudakov form factor

• Form factor

• The matrix elements of current operator
  between initial
• and final states are represented by
  different form factors.
• Form factors are important dynamical quantity
  for
• describing the inner properties of a
  fundamental or
• composite particle.

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2009.9.9, KITPC, Beijing
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The interaction of a fermion with EM current is
represented by
At q20 , the g-factor is given by and the
anomalous magnetic moment is
The form factor (only the first term F1(Q2)) in
the asymptotic limit q2?8 is called Sudakov
form factor. (in 1956)

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The naïve power counting is strongly modified
(at tree level F1).

• The large double-logarithm spoils the
  convergence of pertubative
• expansion.
• The summation to all orders is an exponential
  function.
• The form factor is strongly suppressed when Q
  is large.
• In phenomenology, it relates to most high
  energy process in certain
• momentum regions, DIS, Drell-Yan, pion form
  factor, etc.

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• Methods of momentum regions (by Beneke,
  Smirnov, etc)

The basic idea is to expand the Feynman diagram
integrand in the momentum regions which give
contributions in dimensional regularization.
Each region is involved by one scale.

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Bauer (2003)
Regularization method
Introduce a cutoff scale d in both k and k-.
DOF
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(No Transcript)
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Factorization
Step 1 the separation of hard from collinear
contributions.
Step 2 the separation of soft from collinear
functions.
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Evolution
Two-step running

• The anomalous dimension depends on the
  renormalization scale.
• The exponentiation is due to the RGE.
• The suppression is caused by the positive
  anomalous dimension.

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• Exponentiation and scaling

Exponential of logs can be considered as a
generalized scaling.
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Comparisons with other works

• The leading logarithmic approximation method sums
  the leading
• contributions from ladder graphs to all
  orders. The ladder graphs
• constitutes a cascade chain qq-gtqq-gt-gtqq.
  There are orderings
• for Sudakov parameters.
• Korchemsky et al. used the RGE for a soft
  function whose evolution
• is determined by cusp dimension. The cusp
  dimension contains a
• geometrical meaning.

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3. CSS use a diagrammatic analysis to prove the
factorization. The RGEs are derived from
gauge-dependence of the jet and hard function.
The choice of gauge is analogous to the
renormalization scheme.
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IV. Heavy-to-light transition form factor

• The importance of heavy-to-light form factors
• CKM parameter Vub
• QCD, perturbative, non-perturbative
• basic parameters for exclusive decays in QCDF
  or SCET
• new physics

Light cone dominance
At large recoil region q2ltltmb2, the light meson
moves close to the light cone.
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Hard scattering
Hard gluon exchange soft spectator quark ?
collinear quark Perturbative QCD is
applicable.
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Endpoint singularity
endpoint singularity

• Factorization of pertubative contributions from
  the
• non-perturbative part is invalid.
• There are soft contributions coming from the
  endpoint region.

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Hard mechanism -- PQCD approach

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

Soft mechanism

• Momentum of one parton in the light meson is
  small (x-gt0).
• Soft interactions between spectator quark
  in B and soft
• quark in light meson.
• Methods light cone sum rules, light cone
  quark model
• (lattice QCD is not applicable.)

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Spin symmetry for soft form factor
In the large energy limit (in leading order of
1/mb),
J. Charles, et al., PRD60 (1999) 014001.

• The total 10 form factors are reduced to 3
  independent factors.
• 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 rigorous.
• 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

• 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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Soft overlap mechanism
The soft part form factor is represented by the
convolution of initial and final hadron wave
functions.
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Diracs three forms of Hamiltonian dynamics( S.
Brodsky et al., Phys.Rep.301(1998) 299 )
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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

• Diagrams 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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Basic assumptions of LC quark model

• Valence quark contribution dominates.
• The quark mass is constitute mass which absorbs
• some dynamic effects.
• LC wave functions are Gaussian.

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LC wave functions
In principle, wave functions can be solved if
we know the Hamiltonian (TV).
Choose Gaussian-type
Power law

• The scaling of soft form factor depends on the
  light meson
• wave function at the endpoint.

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Melosh rotation
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Numerical results
The values of the three form factors are very
close, but they are quite different in
formulations.
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Comparisons with other approaches
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Summary

• SCET provides a model-independent analysis of
  processes with
• energetic hadrons proof of factorization
  theorem, Sudakov
• resummation, power corrections.
• SCET analysis of Sudakov form factor
  emphasizes the scale
• point of view.
• LC quark model is an appropriate
  non-perturbative method
• to study the soft part heavy-to-light form
  factors at large recoil.
• How to treat the endpoint singularity is still
  a challenge.

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Thanks