1

Topical Seminar on Frontier of Particle Physics 2004: QCD and

Light Hadrons Lecture 1

Wei ZhuEast China Normal University

2

Outline of my three lectures

1. What is the structure function: definition and tools

2. Factorization, parton distributions and evolution equations

Definition Time Ordered Perturbation TheoryCollinear Factorization Scheme Parton(Scattering) and Dipole pictures

DGLAP Equations BFKL Equations

3

3. Small x physics

Introduction Modified DGLAP Equations JIMWLK Equation Phenomenology of Saturation A Geometric Nuclear Effect

4

Outline of Lecture One

Time Ordered Perturbation Theory

Definition

Collinear Factorization Scheme

Parton(Scattering) and Dipole pictures

5

1.Definition

EE

m

Q

qp

Qx

EEqQ

22

02

sin4

22

222

WLE

E

Q

eM

E

E

dEd

d

42

42

2

2

1616

1

Leptonic tensor:

)(2 2

2

gkkkkkk

eL

Hadronic tensor:

,)()(,

2

1

4

1 4 pzJzJpzedW ziq

6

Structure Functions:

Wμν has total 16 components

Parity Invariance

Time-reversal InvarianceCurrent conservation

Wμν= Wν μ for spin-averaged symmetric

Wμν= Wν μ real

Ə μJ μem =0

22

212

,2

1

2

1, QxW

xqp

xqpQxW

q

qqgW

Dimensionless Structure Functions:

2

22

2

21

21

,,

,,

QxqWpQxF

QxWQxF

7

Polarized Structure Functions:

longitudinal structure function

transverse structure function

projection operators

8

9

The kinematic domains probed by the various experiments, shown together with the partons that they constrain

10

fCF 2

Coefficient function

Universal parton distribution

PQCD

γT*

γT*

PQCD

11

Many Interesting Subjects Relating to SFsFactorizatio

nEvolution DynamicsShadowing, Anti-shadowingSaturation, Color Glass CondensationHigher Twist EffectsNuclear EffectsSpin Problem, Polarized SFs

Asymmetry of Quark Distributions

Diffractive SFs

Large Rapidity Gap

Generalized (skewed) Parton Distributions

……

12

Research Tools

Operator Product Expansion

Renormalization Group Theory

Covariant Perturbation Theory

Time Ordered Perturbation Theory (TOPT)

Parton (Scattering) Model

Dipole Model

Pomeron Theory

……

13

2.TOPTHistory

Old-fashioned TOPT

Feynman covariant perturbation theory~1949

14

15

CVPT:

0

),(2

0

24

l

lll

il

lld

CVPT

TOPT

After contour integral

l0=ω (F)

or =- ω (B)

16

17

)(2

ˆ

21

3

EE

kkd

x

F

t

12

t

x

B1

2

)(2

ˆ

21

3

EE

kkd

220 kk

),(ˆ kk

0ˆ2 k

18

General Rule For TOPT

19

Propagating momentum

CVPT

TOPT

k

k̂

Off-mass-shellOn-energy-shell

On-mass-shellOff-energy-shell

TOPTCVPT

20

Application: Weizsäcker-Williams(equivalent particle) Approximation

13

1

2 2

3

13

2 2

13

21

Collinear TOPT (massless)W.Zhu, H.W.Xiong and J.H.Ruan P.R.D60(1999)094006

F F

F

F

F

B

suppressed finite

22

F

F

B

B

k

k

k

k

23

Elementary Vertices of QCD

Elementary Vertices of QED

24

25

Propagating Momentum is but not k !k̂

FF

B

B

y

y

26

Application:Eikonal approximation

Emission of absorption of soft particle cause hardly any recoil to a fast moving source.

The eikonal approximation origins in the application of Maxwell electromagnetism theory to geometric optics by Bruns (1895).

In the quantum electrodynamics field theory, the eikonal approximation implies that the denominator of the relativistic propagator, which connecting with the soft photon can be linearized. In this case, the contributions from the soft photos to the hard source can be summed as an exponential. Therefore, the eikonal approximation is an idea tool in the treatment of the corrections of the soft gluons to the high energy processes.

27

A massless quark moving along light-cone y+- direction with a large momentum.nPP

Assuming a soft gluon collinear attaches to this hard quark with the momentum k <<p.

F F

BP

k

P+k

F

F

B

P P+k

k

A+=0

=0

Therefore, we can only keep the forward- and backward-components for a fast quark and soft gluon, respectively.

28

A similar conclusion holds for a fast gluonF F

B

P P+k

k

α

νμ

β

y

y

F

F

F

F

B

B

B

F

F

F

F

B

B

B

A fast parton moving along the y--direction can not collinear couple with any gluons in the light-cone gauge since the vertex with two collinear backward partons are inhibited.

Wilson Line

29

3. Collinear Factorization Schemeγ

*γ*

γ* γ*

30

B B

B

BB

B B B

F F F

F

F F

31

F FF F

γ* γ* γ*

F

F

F

F

F

FB

knxpknxp

nxp

nxp

nxp

knxp

Collins, Soper, Sterman

32

4. Parton(Scattering) and Dipole pictures

33

The transverse coefficient function with one quark-loop correction are described by the absorptive part of the amplitudes

34

Sudakov variables

Transverse coefficient functions

35

LLA

TOPT

36

p+ >> q-, Figure (a)

37

q->>p+, figure (b)

38