Transverse Momentum Dependent QCD

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Transverse Momentum Dependent QCD
Factorization for Semi-Inclusive DIS
J.P. Ma,
Institute of Theoretical Physics, Beijing
???, ?????, ??
The 6th Particle Physics, Phenomenology Workshop
06.06.2005. I- Lan
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Content

• Physics of Semi-Inclusive DIS
• 2. Consistent Definitions of Transverse Momentum
• Dependent (TMD) Parton Distribution and
  Fragmentation
• 3. One-Loop Factorization in SIDIS
• 4. Factorization to all orders in Perturbation
  theory.
• 5. Outlook

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1. Physics of Semi-Inclusive DIS
k

• Photon momentum q is in the Bjorken limit.
• Final state hadron h can be characterized by
• fraction of parton momentum z and transverse
• momentum Ph-

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A Brief History

• European Muon Collaboration (CERN)
• Measure the flavor dependence of the
  fragmentation functions (Dup (z), Dup- (z))
• H1 and Zeus Collaboration (DESY)
• Topology of the final state hadrons Jet
  structure and energy flow.
• Spin Muon Collaboration (CERN) and HERMES
• Extracting polarized quark dis ?q(x)
• Single Spin Asymmetries
• Long history..

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• Three cases for measured Ph-

A. Ph - Q Ph- generated from QCD
hard scattering, factorization theorem exists.
(Standard collinear factorization) B. Q gtgt Ph-
gtgt?QCD Still perturbative, but resummation is
needed. It is important for many processes. C.
Ph- ?QCD Nonperturbative! Ph- is generated
from partons inside of hadrons. Transverse
momenta of partons A transparent explanation
for SSA It gives a possible way to learn
3-dimensional structure of hadrons!!!!!
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• A factorization theorem is needed
• for the case Ph- ?QCD !

Single spin asymmetries observed in many
experiments stimulated many theoretical works
1992 J. Collins suggested a factorization
theorem, but without a proof and with some
mistakes corrected in 2002.
Many people use the theorem.
It was also realized A consistent definition in
QCD of TMD parton distribution was not there.
, and the factorization theorem?
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2. Consistent Definitions of TMD Parton
Distribution and Fragmentation

• Light cone coordinate system

Two light cone vectors
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• A hadron moves in the z-direction with

Usual parton distribution The parton
distribution is the probability to find a quark
with the momentum fraction x, defined as
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A naïve generalization to include TMD would be
This is not consistent, because it has the
light-cone singularity 1/(1-x) !!!!, and other
drawbacks.. The singularity is not an I.R. -
or collinear singularity. If one integrates the
transverse momentum, it is cancelled.
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QCD Definition
v
v
n
n
t
b?
v is not n to avoid l.c. singularity
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Scale Evolution

• Since the two quark fields are separated in both
  long. and trans. directions, the only UV
  divergences comes from the WF renormalization and
  the gauge links.
• In vA0 gauge, the gauge link vanishes. Thus the
  TMD parton distribution evolve according to the
  anomalous dimension of the quark field in the
  axial gauge

• Integrate over k- generates DGLAP evolution.

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One-Loop Virtual Contribution
Soft contribution
Double logs
Energy of the hadron
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One-Loop Real Contribution
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• The defined TMD distribution has
• 1. No light cone singularity. (good!!!)
• 2. double-logs ln2Q2/?QCD2 for every coupling
  constant.
• (can be resummed with Collins-Soper
  equation)
• 3. Beside collinear divergence, there are also
  infrared
• singularities, i.e., soft gluon
  contributions.
• (can be subtracted ..)

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• For the double logs
• 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 ?
(Collins and Soper, 1981 )
K and G obey an RG equation
µ independent!
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• Solve the RG equation

• Solving Collins-Soper equation

Double logs have been factorized!
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• Soft gluon contributions
• The soft gluon contribution can be factorized

All soft gluon contributions are in the soft
factor S
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We finally can give a consistent definition of
TMD distribution
It should be noted Integration over the
transverse-momentum does not usually yield
Feynman distribution ?d2k- q(x,
k-) q(x,µ) !!

• Similarly, one can perform the same procedure
  to define TMD fragmentation functions.

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• How many TMDs at leading twist?
• In general, in Semi-DIS or other processes, if
  factorization can be proven, one can access the
  quark density matrix in experiment

It provides all information about the quark
inside of the hadron with an arbitrary spin s, it
is characterized with some scalar
distributions.
certain gauge links
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• H proton (uncompleted list)

Nucleon
Unpol.
Long.
Trans.
Quark
Unpol.
q(x, k-)
qT(x, k-)
Long.
?qL(x, k-)
?qT(x, k-)
dqT(x, k-) dqT'(x, k-)
Trans.
dq(x, k-)
dqL(x, k-)
Boer, Mulders, Tangerman et al.
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3. One-Loop Factorization in SIDIS

• Cross section

Hadronic Tensor
At tree-level
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One-loop Factorization (virtual gluon)

• Vertex corrections (single quark target)

q
p'
k
p
Four possible regions of 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)

q
p'
k
p
The dominating topology is the quark carrying
most of the energy and momentum 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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Factorization Theorem

• Factorization for the structure function

with the corrections suppressed by (P-, ?QCD /
Q)2
Impact parameter space
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4. Factorization to all orders in Perturbation
theory

• Main steps for all-order factorization
• Consider an arbitrary Feynman diagram
• Find contributions singular contribution from the
  different regions of the momentum integrations
• (Landau equation, 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

• A Feynamn diagram, if it contains collinear- and
  infrared singularities, will give the leading
  contribution
• These singularities can be analyzed with Landau
  equation, represented by reduced diagram.

For our case, the reduced diagram looks
Physical picture Coleman Norton
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• The most important reduced diagrams are
  determined from power counting.(Leading region)
• The leading region is determined by
• 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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Factorizing the Collinear Gluons

• The collinear gluons are longitudinally polarized
  
• One can 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.
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Factorizing the Soft Part

• The soft part can be factorized from the jet
  using Grammer-Yennie approximation
• Neglect soft momentum in the numerators.
• Neglect k2 in the propagator denominators
• Potential complication in the Glauber region
• Use the ward identity.
• 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 becomes

which corresponds to the factorization formula
stated earlier.
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• An interesting feature of our factorization
  theorem for P- ?QCD
• when P- becomes large so that P- gtgt?QCD , the
  famous Collins-Soper-Sterman resummation formula
  can be reproduced from our factorization theorem.
  
• The topics discussed here can be found in
• X.D. Ji, J.P. Ma and F. Yuan
• Phys.Rev.D71034005,2005

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5 . Summary and outlook
In general there are 3 classes of distributions
to characterize the quark density matrix in a
nucleon
? the ordinary parton distributions
? New effects with the transverse momentum

• ? Novel distributions that vanish without final
  state interactions (Sivers function, SSA)

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• They delivery information about 3-dimentional
  structure, like orbital angular momenta, etc

What we have done We establish a
factorization theorem of semi-DIS for the first
classes of distributions, JMY
Phys.Rev.D71034005, 2005 extend the
theorem of Drell-Yan process, JMY
Phys.Lett.B597299, 2004 and also extend
the theorem with TMD gluon distributions, JMY
hep-ph/0503015 Outlook
To establish factorization theorem for other two
class distributions, and applications
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Thank the organizers of PPP6

• Thank all of you!!!