Transverse Momentum Dependent Factorization

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Transverse Momentum Dependent Factorization

• Feng Yuan
• Lawrence Berkeley National Laboratory
• RBRC, Brookhaven National Laboratory

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Outline

• Encode the final state interaction phase in the
  light-cone wave function amplitudes
• Transverse momentum dependent factorization
• Outlook

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Encode the phase in light-cone wave function

• Motivations
• Explicitly show the light-cone wave function
  contains the final state interaction phase needed
  to generate nonzero Sivers function
• Model building for the T-odd TMDs
• Brodsky,Pasquini,Xiao,Yuan, to be sumitted

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Light-cone wave function amplitudes

• A0 doesnt fix the gauge completely, we have
  to specify the boundary condition at the spatial
  infinity
• At(8)At(-8)0, wave function real
• Advanced boundary condition, At(8)0
• Retarded boundary condition, At(-8)0

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To generate a phase

• Time-order perturbation theory

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Lowest order
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Polarization sum for the gluon

• Antisymmetric boundary, principal value
  prescription for v.q
• Advanced boundary condition

Provides a phase
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Final result is very simple
Light-cone energy factors
Imaginary part of the wave function
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Some examples

• Three quark state for nucleon

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Quark-antiquark for Pion

• Similar result holds for quark-diquark model, and
  can reproduce Brodsky-Hwang-Schmidt model
  calculation for the Sivers function

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Large pt for pion wave function

• Depends on the distribution amplitudes, like the
  real part, the same power behavior
• Phenomenological applications can be carried out
  following these wave functions

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TMD factorization
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Why Worry about Factorization?

• Safely extract nonperturbative information
• Theoretically under control
• No breakdown because of un-
• cancelled divergence
• NLO correction calculable
• Estimate the high order corrections

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What to Worry for Factorization?

• Correct definition of TMD parton distributions
• Gauge Invariance?
• Soft divergence gets cancelled
• Hard Part can be calculated perturbatively
• The cross section can be separated into Parton
  Distribution, Fragmentation Function, Hard and
  Soft factors

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Previous Works on Factorization(basis of the
present work)

• Factorization for back-back jet production in
  ee- annihilation (in axial gauge)
• -- Collins-Soper, NPB, 1981
• Factorization for inclusive processes
• -- Collins, Soper, Sterman, NPB, 1985
• -- Bodwin, PRD, 1985
• -- Collins, Soper, Sterman,
• in Perturbative QCD, Mueller ed., 1989

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TMD Naïve Factorization

• SIDIS Cross section
• Naïve factorization (unpolarized structure
  function)

Hadron tensor
TMD distr.
TMD frag.
Mulders, Tangelman, Boer (96 98)
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TMD Factorization

• Collins-Soper, 81
• Ji-Ma-Yuan, 04
• Collins-Metz 04
• Scherednikov-Stefanis, 07

• Leading order in pt/Q
• Additional soft factor

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One-loop Factorization
Purpose

• Verify the factorization
• Deduce the correct definition of TMD parton dis.
• Estimate of one-loop correction to H

Procedure

• Take an on-shell quark as target
• Calculate dis. and frag. to one-loop order
• Define and calculate the soft factor
• Full QCD calculation at one-loop order
• Extract the relevant hard part

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TMD the definition
In Feynman Gauge, the gauge link
v is not n to avoid l.c. singularity !!
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• TMDs are process dependent (Fragmentation is
  different)
• Gauge link direction changes from DIS to
  Drell-Yan process
• More complicated structure for dijet-correlation
  in pp collisions, standard factorization breaks
• Light-cone singularity beyond Born diagram
• Transverse momentum resummation

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One-Loop Real Contribution
energy dep.
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Energy Dependence

• The TMD distributions depend on the energy of the
  hadron! (or Q in DIS)
• Introduce the impact parameter representation
• One can write down an evolution equation in ?
• K and G obey an RG equation,

Collins and Soper (1981)
µ independent!
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TMD Fragmentation functions

• Can be defined in a similar way as the parton
  distribution
• Have similar properties as TMD dis.

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One-loop Factorization (virtual gluon)

• Vertex corrections (single quark target)

q
p'
k
p
Four possible regions for the gluon momentum k
1) k is collinear to p (parton distribution) 2)
k is collinear to p' (fragmentation) 3) k is
soft (Wilson line) 4) k is hard (pQCD
correction)
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One-Loop Factorization (real gluon)

• Gluon Radiation (single quark target)

p'
q
k
p
Three possible regions for the gluon momentum k
1) k is collinear to p (parton distribution)
2) k is collinear to p' (fragmentation) 3) k is
soft (Wilson line)
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At one-loop order, we verified the factorization
The hard part at one-loop order,
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All Orders in Perturbation Theory

• Consider an arbitrary Feynman diagram
• Find the singular contributions from different
  regions of the momentum integrations
• (reduced diagrams)
• Power counting to determine the leading regions
• Factorize the soft and collinear gluons
  contributions
• Factorization theorem.

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Reduced (Cut) Diagrams

• Leading contribution to a cross section from a
  diagram.
• Can be pictured as real spacetime process
  (Coleman and Norton)

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Leading Regions

• The most important reduced diagrams are
  determined from power counting.
• No soft fermion lines
• No soft gluon lines attached to the hard part
• Soft gluon line attached to the jets must be
  longitudinally polarized
• In each jet, one quark plus arbitrary number of
  collinear long.-pol. gluon lines attached to the
  hard part.
• The number of 3-piont vertices must be larger or
  equal to the number of soft and long.-pol. gluon
  lines.

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Leading Region
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Collinear And Soft Gluons

• The collinear gluons are longitudinally polarized
  
• Use the Ward identity to factorize it off the
  hard part.
• The result is that all collinear gluons from the
  initial nucleon only see the direction and charge
  of the current jet. The effect can be reproduced
  by a Wilson line along the jet (or v) direction.
  
• The soft part can be factorized from the jet
  using Grammer-Yennie approximation
• The result of the soft factorization is a soft
  factor in the cross section, in which the target
  current jets appear as the eikonal lines.

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Factorization

• After soft and collinear factorizations, the
  reduced diagram become
• which corresponds to the factorization
  formula stated earlier.

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Compared to the collinear factorization

• Simplification
• Of the cross section in the region of ptltltQ, only
  keep the leading term
• Extension
• To the small pt region, where the collinear
  factorization suffer large logarithms

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Transverse momentum dependence

• QCD Dynamics

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Transition from Perturbative region to
Nonperturbative region

• Compare different region of PT

Nonperturbative TMD
Perturbative region
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Summary

• The TMD factorization has been shown for the
  semi-inclusive DIS process, and the hard factor
  been calculated for some observables
• Experiments should be able to test this
  factorization
• Sign change between DIS and Drell-Yan for Sivers
  effects
• Universality of the Fragmentation effects

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Perturbative tail is calculable

• Transverse momentum dependence

Power counting, Brodsky-Farrar, 1973
Integrated Parton Distributions Twist-three
functions
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A unified picture (leading pt/Q)
Transverse momentum dependent
Collinear/ longitudinal
PT
?QCD
Q
PT
ltlt
ltlt
Ji-Qiu-Vogelsang-Yuan,2006 Yuan-Zhou, 2009
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Subtract the soft factor in the Dis.

• TMD distribution contains the soft contribution
• Subtract the soft contribution

Zero bin subtraction Monahar, Stewart, 06
Lee, Sterman, 06
Idilbi, Mehen, 07