Transcript Document

Transverse Momentum Dependent QCD
Factorization for Semi-Inclusive DIS
J.P. Ma,
Institute of Theoretical Physics,
Academia Sinica, Beijing
The Sino-German Workshop
21.09.2006 DESY, Hamburg
Content
1. 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
1. Physics of Semi-Inclusive DIS
k'
k
q
Ph
X
P
• 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┴
A Brief History
• European Muon Collaboration (CERN)
– Measure the flavor dependence of the fragmentation
functions (Duπ+ (z), Duπ- (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: Δq(x)
• Single Spin Asymmetries
Long history……..



xF  0
Three cases for measured Ph┴
A. Ph ┴ ~ Q :
Ph┴ generated from QCD hard scattering, factorization
theorem exists. (Standard collinear factorization)
B. Q >> Ph┴ >>ΛQCD :
Still perturbative, but resummation is needed.
It is important for many processes.
C. Ph┴ ~ΛQCD
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!!!!!
A factorization theorem is needed
for the case Ph┴ ~ΛQCD !
Single spin asymmetries observed in many experiments
stimulated many theoretical works………
1976: Nachtmann discussed SSA in parton fragmentation
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?
2. Consistent Definitions of TMD Parton
Distribution and Fragmentation
Light cone coordinate system:
Two light cone vectors:
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
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.
QCD Definition
n
v
t
z
n v
b
v is not n to avoid l.c. singularity
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 v·A=0 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.
One-Loop Virtual Contribution
Soft contribution
Double logs
: Energy of the hadron
One-Loop Real Contribution
The defined TMD distribution has
1. No light cone singularity. (good!!!)
2. double-logs ln2Q2/ΛQCD2 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 ……..)
For the double log’s:
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 ζ:
(Collins and Soper, 1981 )
K and G obey an RG equation:
μ independent!
Solve the RG equation:
•Solving Collins-Soper equation:
Double logs have been factorized!
Soft gluon contributions:
• The soft gluon contribution can be factorized
All soft gluon contributions are in the soft factor S:
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.
How many TMD’s 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……
H = proton: (uncompleted list)
Nucleon
Unpol.
Long.
Trans.
Quark
Unpol.
q(x, k┴)
Long.
Trans.
δq(x, k┴)
qT(x, k┴)
ΔqL(x, k┴)
ΔqT(x, k┴)
δqL(x, k┴)
δqT(x, k┴)
δqT'(x, k┴)
Boer, Mulders, Tangerman et al.
3. One-Loop Factorization in SIDIS
Cross section
Hadronic Tensor:
At tree-level:
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)
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)
Factorization Theorem:
• Factorization for the structure function:
with the corrections suppressed by (P┴, ΛQCD / Q)2
Impact parameter space
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.
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
•
The most important reduced diagrams are
determined from power counting.(Leading region)
The leading region is determined by:
1. No soft fermion lines
2. No soft gluon lines attached to the hard part
3. Soft gluon line attached to the jets must be
longitudinally polarized
4. In each jet, one quark plus arbitrary number of
collinear long.-pol. gluon lines attached to the
hard part.
5. The number of 3-piont vertices must be larger or
equal to the number of soft and long.-pol. gluon
lines.
Leading Region
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.
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.
Factorization
• After soft and collinear factorizations, the reduced
diagram becomes:
which corresponds to the factorization formula stated
earlier.
An interesting feature of our factorization theorem for
P┴ ~ΛQCD :
when P┴ becomes large so that P┴ >>ΛQCD , 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.D71:034005,2005
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: (Siver’s function, SSA)
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:
hep-ph/0404183, Phys.Rev.D71:034005, 2005
extend the theorem of Drell-Yan process,
JMY:
hep-ph/0405085 , Phys.Lett.B597:299, 2004
and also extend the theorem with TMD gluon distributions,
JMY:
hep-ph/0503015 , JHEP 0507:020,2005
Outlook: To establish factorization theorem for other two
class distributions, and applications………
Thank you !