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

1. Physics of Semi-Inclusive DIS2. Consistent Definitions of Transverse

Momentum Dependent (TMD) Parton Distribution and

Fragmentation 3. One-Loop Factorization in SIDIS4. Factorization to all orders in Perturbation

theory.5. Outlook

Content

kPh

P

k'

q

X

• 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┴h┴

1. Physics of Semi-Inclusive DIS

A Brief History• European Muon Collaboration (CERN)

– Measure the flavor dependence of the fragmentation functions (Du

π+ (z), Du

π- (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……..

0Fx

Three cases for measured Ph┴

A. Ph ┴ ~ Q : :

Ph┴ generated from QCD hard scattering, generated from QCD hard scattering, factorization theorem existsfactorization theorem exists. (. (Standard collinear Standard collinear factorizationfactorization))

BB. Q >> . Q >> Ph┴ >> >>ΛΛQCD QCD : : Still perturbative, but resummation is needed.Still perturbative, but resummation is needed.It is important for many processesIt is important for many processes..

CC. . Ph┴ ~ ~ΛΛQCDQCD

Nonperturbative! Nonperturbative! Ph┴ is generated from partons is generated from partons inside inside

of hadrons. of hadrons. Transverse momenta of partons: A transparent Transverse momenta of partons: A transparent

explanation for SSA explanation for SSA It gives a possible way to learnIt gives a possible way to learn 3-dimensional structure of hadrons3-dimensional structure of hadrons!!!!!!!!!!

A factorization theorem is needed for the case Ph┴ ~Λ ~ΛQCDQCD !

Single spin asymmetries observed in many experimentsSingle spin asymmetries observed in many experiments stimulated many theoretical works………stimulated many theoretical works………

1976: Nachtmann discussed SSA in parton fragmentation1976: Nachtmann discussed SSA in parton fragmentation

1992: J. Collins suggested a factorization theorem,1992: J. Collins suggested a factorization theorem, but without a proof and with some mistakes correctedbut without a proof and with some mistakes corrected in 2002. in 2002.

Many people use the theorem……….Many people use the theorem……….

It was also realized:It was also realized:A consistent definition in QCD of TMD parton distributionA consistent definition in QCD of TMD parton distribution was was not therenot there………. , and the factorization theorem?………. , and the factorization theorem?

2. Consistent Definitions of TMD 2. Consistent Definitions of TMD Parton Distribution and Parton Distribution and

FragmentationFragmentationLight cone coordinate system:

Two light cone Two light cone vectors: vectors:

A hadron moves in the z-direction with

Usual parton distribution:The parton distribution is the probability to find The parton distribution is the probability to find a quark with the momentum fraction x, defined a quark with the momentum fraction x, defined asas

A naïve generalization to include TMD would be:A naïve generalization to include TMD would be:

This is This is not consistentnot consistent, because it has the light-cone , because it has the light-cone singularity 1/(1-x) !!!!, and other drawbacks………..singularity 1/(1-x) !!!!, and other drawbacks………..

The singularity is not an I.R. - or collinear singularity. If The singularity is not an I.R. - or collinear singularity. If oneone integrates the transverse momentum, it is cancelled. integrates the transverse momentum, it is cancelled.

QCD DefinitionQCD Definition

t

z b

nnv

v

v is not n to avoid l.c. singularity

Scale EvolutionScale 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

Double logs

Soft contribution

: 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

μ independent!

One can write down an evolution equation in ζ: (Collins and Soper, 1981 )

K and G obey an RG equation:K and G obey an RG equation:

Solve the RG equation:

Double logs have been factorized!

•Solving Collins-Soper equation:

Soft gluon contributions:

• The soft gluon contribution can be factorized

All soft gluon contributions are in the soft factor S:All soft gluon contributions are in the soft factor S:

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

We finally can give a consistent definition of TMD We finally can give a consistent definition of TMD distributiondistribution::

It should be noted:It should be noted:Integration over the transverse-momentum does not usually Integration over the transverse-momentum does not usually yield Feynman distributionyield Feynman distribution

∫∫dd22kk┴┴ q(x, k q(x, k┴┴) = q(x,µ) !!) = q(x,µ) !!

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 It provides all information about the quark inside of the hadron with an arbitrary spin s, it is characterized with hadron with an arbitrary spin s, it is characterized with some scalar distributions. some scalar distributions.

: : certain gauge links……certain gauge links……

H = proton: (uncompleted list)

Quark

Nucleon

Unpol.

Long.

Trans.

Unpol. Long. Trans.

q(x, kq(x, k┴┴)) qqTT(x, k(x, k┴┴))

ΔΔqqLL(x, k(x, k┴┴)) ΔΔqqTT(x, k(x, k┴┴))

δδq(x, kq(x, k┴┴)) δδqqLL(x, k(x, k┴┴)) δδqqTT(x, k(x, k┴┴) )

δδqqTT'(x, k(x, k┴┴))

Boer, Mulders, Tangerman et al.

Cross section

Hadronic Tensor:

3. 3. One-Loop Factorization in SIDIS

At tree-level: At tree-level:

One-loop Factorization (virtual gluon)

• Vertex corrections (single quark target)

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)

p

p′q

k

One-Loop Factorization (real gluon)

• Gluon Radiation (single quark target)

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)

p

p′q

k

• Factorization for the structure function:

Impact parameter space

Factorization TheoremFactorization Theorem: :

with the corrections suppressed bywith the corrections suppressed by ( (P┴, ΛΛQCDQCD / Q)2

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.

4. Factorization to all orders in Perturbation theory

• 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.

Reduced (Cut) DiagramsReduced (Cut) Diagrams

For our case, the reduced diagram looksFor our case, the reduced diagram looks::

Physical picturePhysical pictureColeman & Coleman & NortonNorton

• The most important reduced diagrams are determined from power counting.(Leading region)

The leading region is determined by:

1. No soft fermion lines2. No soft gluon lines attached to the hard part3. Soft gluon line attached to the jets must be

longitudinally polarized4. 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 RegionLeading Region

Factorizing the Collinear Factorizing the Collinear GluonsGluons

• 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 PartFactorizing 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.

FactorizationFactorization• 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┴ ~Λ ~ΛQCD QCD :

when P┴ becomes large so that becomes large so that P┴ >>Λ >>ΛQCD QCD , the , the

famous Collins-Soper-Sterman resummation famous Collins-Soper-Sterman resummation formula can be reproduced from our factorization formula can be reproduced from our factorization theorem. theorem.

The topics discussed here can be found in The topics discussed here can be found in X.D. Ji, J.P. Ma and F. Yuan: X.D. Ji, J.P. Ma and F. Yuan:

Phys.Rev.D71:034005,2005Phys.Rev.D71:034005,2005

▪ Novel distributions that vanish without final state interactions: (Siver’s function, SSA)

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:

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

What we have done: What we have done: We establish a factorization theorem of semi-DIS for We establish a factorization theorem of semi-DIS for the first classes of distributions,the first classes of distributions, JMY: JMY: hep-ph/0404183, hep-ph/0404183, Phys.Rev.D71:034005, 2005Phys.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 and also extend the theorem with TMD gluon distributions, distributions, JMY: JMY: hep-ph/0503015hep-ph/0503015 , , JHEP 0507:020,2005JHEP 0507:020,2005 Outlook: To establish factorization theorem for other Outlook: To establish factorization theorem for other two class distributions, and applications………two class distributions, and applications………

Thank you !