Introduction to

Quantum Chromodynamics (QCD)

Jianwei Qiu Theory Center, Jefferson Lab

May 29 – June 15, 2018

Lecture Three

Observables with ONE identified hadron

q DIS cross section is infrared divergent, and nonperturbative!

�DIS`p!`0X(everything)

Identified initial-state hadron-proton!

�DIS`p!`0X(everything) / … + + +

+

q QCD factorization (approximation!) Color entanglement Approximation

Quantum Probabilities Structure

Controllable Probe

Physical Observable

bT

kTxp

bT

kTxp

⊗1 OQR⎛ ⎞

+ ⎜ ⎟⎝ ⎠xP, kT

�DIS`p!`0X(everything) =

q  Scattering amplitude:

( ) ( ) ( )', '; ' , q u k ie u kλ µ λλ λ σ γ⎡ ⎤⎣ ⎦=Μ −

( )'2

i gq

µµ⎛ ⎞−⎜ ⎟

⎝ ⎠

( )' 0 ,emX eJ pµ σ

*

*

q  Cross section:

( )( ) ( )

2 3 32DIS

3 3, ', 1

1 1 ', '; ,2 2 2 2 2 2 '

Xi

X i i

d l d kd qs E Eλ λ σ

σ λ λ σπ π=

⎡ ⎤⎛ ⎞= Μ ⎢ ⎥⎜ ⎟⎝ ⎠ ⎢ ⎥⎣ ⎦

∑ ∑ ∏ ( )4 4

1

2 'X

iil k p kπ δ

=

⎛ ⎞+ − −⎜ ⎟

⎝ ⎠∑

( ) ( )2DIS

3 2

1 1 , '''

,2

dE W qd k s

k pQ

L kµνµν

σ ⎛ ⎞= ⎜ ⎟

⎝ ⎠

q  Leptonic tensor:

( )2

' ' '2( , ')

2eL k k k k k k k k gµν µ ν ν µ µν

π= + − ⋅– known from QED

μ

μ’ μ’

μ

Inclusive lepton-hadron DIS – one hadron

q  Hadronic tensor:

q  Symmetries:

( ) ( )

( ) ( ) ( )( )

( )

2 21 22 2 2

222 2

1

1

. .+

.

, ,

, ,.

B B

p B B

q q p q p qW g p q p qq p

S SS

q q q

p q q piM q

p

F x Q F x Q

g xq

g x Qp q

Q

µ νµν µν µ µ ν ν

σ σµνρσ σρε

⎛ ⎞ ⎛ ⎞⎛ ⎞⋅ ⋅= − − + − −⎜ ⎟ ⎜ ⎟⎜ ⎟⋅ ⎝ ⎠

⎡ ⎤⎢ ⎥⎢ ⎥⎣

⎝⎠

−+

⎦

⎠⎝

q  Structure functions – infrared sensitive:

( ) ( ) ( ) ( )2 2 2 21 2 1 2, , , , , , ,B B B BF x Q F x Q g x Q g x Q

4 †1( , , ) e , ( ) (0) ,4

iq zS S SW q p d z p J z J pµν µ νπ⋅= ∫

²  Parity invariance (EM current) ²  Time-reversal invariance

²  Current conservation

*

sysmetric for spin avg.

real

0

W W

W W

q W q W

µν νµ

µν µν

µ νµν µν

=

=

= =

No QCD parton dynamics used in above derivation!

DIS structure functions

xB =Q

2

2p · q

Q2 = �q2

Long-lived parton states

q  Feynman diagram representation of the hadronic tensor:

W µν ∝ …

q  Perturbative pinched poles:

42 2

0

1 1 perturbati1T( , vely( , ) ) Hr

Qd k k kk i k iε ε⎛ ⎞⎛ ⎞ ⇒⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∞+ −∫

q  Perturbative factorization: 2 2

2T

Tk kk xx

p n kp n

µ µ µ µ+= + +

⋅

22 2

0

2 2H( , 0) 1 1 ( 1T , )T dkdx d k ki ik

Qr

kkx ε ε

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=+ −∫ ∫

Short-distance Nonperturbative matrix element

+O✓ hk2i

Q2

◆

Light-cone coordinate:

v± =1p2(v0 ± v3)vµ = (v+, v�, v?),

q  Collinear approximation, if Q ∼ xp ⋅n ≫ kT , k 2

Parton’s transverse momentum is integrated into parton distributions, and provides a scale of power corrections

Scheme dependence

≈2

2TkOQ⎛

+⎞

⎜ ⎟⎝ ⎠

⊗ +UVCT

– Lowest order:

Same as an elastic x-section

q  Corrections:

q  DIS limit: 2 whil, , e fixedBQ xν →∞

Feynman’s parton model and Bjorken scaling

Spin-½ parton!

Collinear factorization – further approximation

Parton distribution functions (PDFs)

q  PDFs as matrix elements of two parton fields: – combine the amplitude & its complex-conjugate

But, it is NOT gauge invariant!

can be a hadron, or a nucleus, or a parton state!

ZO(µ2)

– need a gauge link:

ZO(µ2)

– corresponding diagram in momentum space: Z

d

4k

(2⇡)4�(x� k

+/p

+)

μ-dependence

Universality – process independence – predictive power

+UVCT(µ2)

Gauge link – 1st order in coupling “g”

q  Longitudinal gluon:

q  Left diagram:

= g

Zdy

�1

2⇡n ·Aa(y�1 )t

a

Zdx1

1

�x1 + i✏

eix1p+(y�

1 �y

�)

�M = �ig

Z 1

y�dy�1 n ·A(y�1 )M

Zdx1

Zp

+dy

�1

2⇡eix1p

+(y�1 �y

�)n ·Aa(y�1 )

�M(�igt

a)� · pp

+

i((x� x1 � x

B

)� · p+ (Q2/2x

B

p

+)� · n)(x� x1 � x

B

)Q2/x

B

+ i✏

q  Right diagram: Zdx1

Zp

+dy

�1

2⇡eix1p

+y

�1n ·Aa(y�1 )

��i((x+ x1 � x

B

)� · p+ (Q2/2x

B

p

+)� · n)(x+ x1 � x

B

)Q2/x

B

� i✏

(+igt

a)� · pp

+M

= g

Zdy

�1

2⇡n ·Aa(y�1 )t

a

Zdx1

1

x1 � i✏

eix1p+y

�1

�M = ig

Z 1

0dy�1 n ·A(y�1 )M

q  Total contribution: �ig

Z 1

0

�Z 1

y�

�dy�1 n ·A(y�1 )MLO

O(g)-term of the gauge link!

q  NLO partonic diagram to structure functions:

2 212

10

Q dkk

−

∝ ∫ Dominated by

k12 0

tAB →∞

Diagram has both long- and short-distance physics

q  Factorization, separation of short- from long-distance:

QCD high order corrections

q  QCD corrections: pinch singularities in 4id k∫

+ + + …

q  Logarithmic contributions into parton distributions:

( )22

22QCD2

2 2( , ) , , ,fB

FFf

B fsx QF x Q C x Ox Q

α ϕ µµ

⎛ ⎞Λ⎛ ⎞= ⊗ + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑

⊗ + +…+UVCT

q  Factorization scale: 2Fµ

To separate the collinear from non-collinear contribution

Recall: renormalization scale to separate local from non-local contribution

QCD leading power factorization

q  Time evolution:

Now “Future” “Past” Time:

Long-lived parton state

q  Unitarity – summing over all hard jets:

DIStot Imσ ∝ t ∼ 1

Q

t ∼ R Not IR safe

Interaction between the “past” and “now” are suppressed!

Picture of factorization for DIS

q  Use DIS structure function F2 as an example:

( )22QCD2

2 2,

2/2( , ) , , ,B

h B q f sq f

f hx QF x Q C x Ox Q

α ϕ µµ

⎛ ⎞Λ⎛ ⎞= ⊗ + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑

h q→²  Apply the factorized formula to parton states:

( )2

22

,

2/2( , ) , , ,B

q B q f sq

ff

qx QF x Q C xx

ϕ µµ

α⎛ ⎞

= ⊗⎜ ⎟⎝ ⎠

∑Feynman diagrams

Feynman diagrams

²  Express both SFs and PDFs in terms of powers of αs:

0th order: ( ) ( ) ( ) ( )02 2/

0 02 22 ( , ) ( / , / ) ,q qq B q BF x Q C x x Q xµ ϕ µ= ⊗

( ) ( )0 02( ) ( )q qC x F x= ( ) ( ) ( )0

/ 1q q qqx xδ δϕ = −

1th order: ( ) ( ) ( ) ( )( ) ( ) ( )

1 12 02 222

0 2

/

12 2/

( , ) ( / , / ) ,

( / , / ) ,

q B q B

q B

q q

q q

F x Q C x x Q x

C x x Q x

µ ϕ µ

µ ϕ µ

= ⊗

+ ⊗

( ) ( ) ( ) ( ) ( )1 1 02 2 2 12 2/2 2( , / ) ( , ) ( , ) ,q q qq qC x Q F x Q F x Q xµ ϕ µ= − ⊗

How to calculate the perturbative parts?

q  Change the state without changing the operator:

– given by Feynman diagrams

q  Lowest order quark distribution:

²  From the operator definition:

PDFs of a parton

q  Leading order in αs quark distribution:

²  Expand to (gs)2 – logarithmic divergent:

UV and CO divergence

q  Projection operators for SFs:

( ) ( )2 21 22 2 2

1, ,q q p q p qW g F x Q p q p q F x Qq p q q qµ ν

µν µν µ µ ν ν

⎛ ⎞ ⎛ ⎞⎛ ⎞⋅ ⋅= − − + − −⎜ ⎟ ⎜ ⎟⎜ ⎟⋅ ⎝ ⎠⎝ ⎠⎝ ⎠

22 2

1 2

22 2

2 2

1 4( , ) ( , )2

12( , ) ( , )

xF x Q g p p W x QQ

xF x Q x g p p W x QQ

µν µ νµν

µν µ νµν

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

q  0th order: ( )

( ) ( ) ( )

0(0)2 ,

22

2

1( ) 4

1Tr 2 ( )4 2

(1 )

q q

q

q

F x xg W xg

exg p p q p q

e x x

µν µνµν

µνµ ν

π

γ γ γ γ πδπ

δ

= =

⎡ ⎤= ⋅ ⋅ + +⎢ ⎥⎣ ⎦

= −( )0 2( ) (1 )q qC x e x xδ= −

Partonic cross sections

( ) ( ) ( ) ( ) ( )1 1 02 2 2 12 2/2 2( , / ) ( , ) ( , ) ,q q qq qC x Q F x Q F x Q xµ ϕ µ= − ⊗

q  Projection operators in n-dimension: 4 2g g nµνµν ε= ≡ −

( )2

2 2

41 (3 2 ) xF x g p p WQ

µν µ νµνε ε

⎛ ⎞− = − + −⎜ ⎟

⎝ ⎠

q  Feynman diagrams:

q  Calculation: ( ) ( )1 1, ,and q qg W p p Wµν µ ν

µν µν−

( )1,qWµν

+ UV CT

+ + +

+ + c.c. + + c.c.

Real

Virtual

NLO coefficient function – complete example

q  Lowest order in n-dimension:

( )0 2, (1 ) (1 )q qg W e xµν

µν ε δ− = − −

q  NLO virtual contribution:

( )1 2,

2 2

2 2

(1 ) (1 )

4 (1 ) (1 ) 1 3 1 * 4(1 2 ) 2

Vq q

sF

g W e x

QC

µνµν

ε

ε δ

α π ε επ ε ε ε

µ

− = − −

⎡ ⎤ Γ + Γ −⎛ ⎞ ⎡ ⎤− + +⎢ ⎥⎜ ⎟ ⎢ ⎥Γ − ⎣ ⎦⎝ ⎠ ⎣ ⎦

q  NLO real contribution:

( )2

1 2, 2

4 (1 )(1 )2 (1 2 )

1 2 1 1 2 * 11 1 2 2(1 2 )(1 ) 1 2

R sq q Fg W e

Q

xxx

C

x

ε

µνµν

α π εε

π ε

ε ε εε ε ε ε

µ⎡ ⎤ Γ +⎛ ⎞− = − − ⎢ ⎥⎜ ⎟ Γ −⎝ ⎠ ⎣ ⎦

⎧ ⎫− ⎡ ⎤ −⎛ ⎞⎛ ⎞− − + + +⎨ ⎬⎜ ⎟⎜ ⎟⎢ ⎥− − − − −⎝ ⎠⎝ ⎠⎣ ⎦⎩ ⎭

Contribution from the trace of Wμν

q  The “+” distribution:

( )1

21 1 1 (1 )(1 )1 (1 ) 1

n xx Ox x x

ε

δ ε εε

+

++

−⎛ ⎞ ⎛ ⎞= − − + + +⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠

l

1 1( ) ( ) (1) (1 ) (1)(1 ) 1z z

f x f x fdx dx n z fx x+

−≡ + −

− −∫ ∫ l

q  One loop contribution to the trace of Wμν:

( )2

1 2, 2

1(1 ) ( ) ( )2 (4 )Es

qq qqq q P Qg W e x x ne

Pµνµν γ

αε

π ε µ π −

⎛ ⎞⎛ ⎞⎧− = − − +⎨ ⎜ ⎟⎜ ⎟⎩⎝ ⎠ ⎝ ⎠

l

( )2

2 (1 ) 3 1 11 ( )1 2 1 1Fn x xx n

x xC x

x ++

⎡ − +⎛ ⎞ ⎛ ⎞+ + − −⎜ ⎟ ⎜ ⎟⎢ − − −⎝ ⎠ ⎝ ⎠⎣

l l

293 (1 )2 3

x xπδ

⎫⎤⎛ ⎞ ⎪+ − − + − ⎬⎥⎜ ⎟

⎝ ⎠ ⎪⎦⎭q  Splitting function:

21 3( ) (1 )(1 ) 2qq F

xx xx

P C δ+

⎡ ⎤+= + −⎢ ⎥−⎣ ⎦

q  One loop contribution to pμpν Wμν:

( )1, 0Vqp p Wµ ν

µν = ( )2

1 2, 2 4R sq q F

Qp p W ex

Cµ νµν

απ

=

q  One loop contribution to F2 of a quark:

( ) ( )2

1 2 22 2

CO

1( , ) ( ) 1 (4 ) ( )2

Eqq q

sq qq

QF x Q e x x n eP x nPγεε

απ

π µ− ⎛ ⎞⎧⎛ ⎞= − + +⎨ ⎜ ⎟⎜ ⎟

⎝ ⎠⎩ ⎝ ⎠l l

2 22 (1 ) 3 1 1 9(1 ) ( ) 3 2 (1 )

1 2 1 1 2 3Fn x xx n x x x

x x xC π

δ+ +

⎫⎡ ⎤⎛ ⎞− + ⎪⎛ ⎞ ⎛ ⎞+ + − − + + − + − ⎬⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪⎣ ⎦⎭

l l

as 0ε⇒ ∞ →

q  One loop contribution to quark PDF of a quark:

( )1/

V CO

2

U

1 1( , ) ( ) UV-CT2s

q q qqPx xαµ

εϕ

π ε⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎩ ⎭

Different UV-CT = different factorization scheme!

– in the dimensional regularization

q  Common UV-CT terms:

²  MS scheme: MSUV

1UV-CT ( )2 qs

qP xαπ ε

⎛ ⎞= − ⎜ ⎟⎝ ⎠

²  DIS scheme: choose a UV-CT, such that ( )1 2 2

DIS( , / ) 0qC x Q µ =

²  MS scheme: ( )MSUV

1UV-CT ( ) 1 (4 )2

Esqq x nP e γα

ε πεπ

−⎛ ⎞= − +⎜ ⎟⎝ ⎠

l

q  One loop coefficient function:

( ) ( ) ( ) ( ) ( )1 1 02 2 2 12 2/2 2( , / ) ( , ) ( , ) ,q q qq qC x Q F x Q F x Q xµ ϕ µ= − ⊗

2 22 (1 ) 3 1 1 9(1 ) ( ) 3 2 (1 )

1 2 1 1 2 3Fn x xx n x x x

x x xC π

δ+ +

⎫⎡ ⎤⎛ ⎞− + ⎪⎛ ⎞ ⎛ ⎞+ + − − + + − + − ⎬⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪⎣ ⎦⎭

l l

( )2

1 2 2 22MS

( , / ) ( )2 qq q qs P QC x Q e x x nα

µπ µ

⎧ ⎛ ⎞⎪= ⎜ ⎟⎨ ⎜ ⎟⎪ ⎝ ⎠⎩

l

q  Physical cross sections should not depend on the factorization scale

22

22 ( , ) 0BFF

d F x Qd

µµ

=

µF2 ddµF

2C f

xBx,Q

2

µF2,αs

!

"##

$

%&&

'

())

*

+,,f

∑ ⊗ϕ f x,µF2( )+ C f

xBx,Q

2

µF2,αs

!

"##

$

%&&

f∑ ⊗µF

2 ddµF

2ϕ f x,µF

2( ) = 0Evolution (differential-integral) equation for PDFs

DGLAP evolution equation: 2

/2 2

2 ( , ) ( ', ),'i i j jF F s F

F j

xPx

x xµ ϕ µ α ϕ µµ

⎛ ⎞⎜= ⎟⎝ ⎠

∂⊗

∂ ∑

q  PDFs and coefficient functions share the same logarithms

PDFs:

Coefficient functions:

( ) ( )2 2 2 20 QCDlog or logF Fµ µ µ Λ

( ) ( )2 2 2 2log or logFQ Qµ µ

Evolution

F2(xB , Q2) =

X

f

Cf (xB/x,Q2/µ

2F ,↵s)�f (x, µ

2F )

Calculation of evolution kernels

q  Evolution kernels are process independent

²  Parton distribution functions are universal

²  Could be derived in many different ways

q  Extract from calculating parton PDFs’ scale dependence

P

kk

P p p

k k

PP

p-kp

kpPP

pk

p-k

PP

+1

2+

1

2

“Gain” “Loss” Change

Collins, Qiu, 1989

²  Same is true for gluon evolution, and mixing flavor terms

q One can also extract the kernels from the CO divergence of partonic cross sections

σ totDIS ∼ ⊗

1 OQR⎛ ⎞

+ ⎜ ⎟⎝ ⎠

From one hadron to two hadrons

q  One hadron: e p

Hard-part Probe

Parton-distribution Structure

Power corrections Approximation

q  Two hadrons:

σ totDY ∼ ⊗

1 OQR⎛ ⎞

+ ⎜ ⎟⎝ ⎠

s

Predictive power: Universal Parton Distributions

q

Drell-Yan process – two hadrons

q  Drell-Yan mechanism: S.D. Drell and T.-M. Yan Phys. Rev. Lett. 25, 316 (1970)

q2 ⌘ Q2 � ⇤2QCD ⇠ 1/fm2

with

q  Original Drell-Yan formula:

2 2

2

2

⌦ ⌦

Right shape – But – not normalization

No color yet! Rapidity:

Lepton pair – from decay of a virtual photon, or in general, a massive boson, e.g., W, Z, H0, … (called Drell-Yan like processes)

Drell-Yan process in QCD – factorization

q  Beyond the lowest order: ²  Soft-gluon interaction takes

place all the time ²  Long-range gluon interaction

before the hard collision

Break the Universality of PDFs Loss the predictive power

q  Factorization – power suppression of soft gluon interaction:

Collins, Soper and Sterman, Review in QCD, edited by AH Mueller 1989

Nayak, Qiu, Sterman, 2006

dσ AB→C+X pA, pB , p( )dydpT

2= φA→a x,µF

2( )a,b,c∑ ⊗φB→b x ',µF

2( )

⊗dσ̂ ab→c+X x,x ', z, y, pT

2µF2( )

dydpT2

⊗ Dc→C z,µF2( )

²  Fragmentation function: ( )2,c C FzD µ→

²  Choice of the scales: 2 2Fac r n

2e Tpµ µ≈ ≈

To minimize the size of logs in the coefficient functions

q  Factorization for high pT single hadron:

Factorization for more than two hadrons

�,W/Z, `(s), jet(s)B,D,⌥, J/ ,⇡, ...

pT � m & ⇤QCD

+ O (1/PT2)

q  Factorization for observables with identified hadrons:

@f(x, µ2)

@ lnµ2= ⌃f 0

Pff 0(x/x0)⌦ f

0(x0, µ

2)

F2(xB , Q2) = ⌃fCf (xB/x, µ

2/Q

2)⌦ f(x, µ2)

d�

dydp

2T

= ⌃ff 0f(x)⌦ d�̂ff 0

dydp

2T

⌦ f

0(x0)

²  DGLAP Evolution:

²  One-hadron (DIS):

²  Two-hadrons (DY, Jets, W/Z, …) :

q  Input for QCD Global analysis/fitting:

²  World data with “Q” > 2 GeV

²  PDFs at an input scale: �f/h(x, µ20, {↵j})

Fitting paramters

Input scale ~ GeV

Testing QCD: Universal PDFs for all cross sections?

Global QCD analysis – Testing QCD

PDFs from DIS

q Q2-dependence is a prediction of pQCD calculation:

0.2

0.4

0.6

0.8

1

-410 -310 -210 -110 1

0.2

0.4

0.6

0.8

1

HERAPDF1.5 (prel.)

exp. uncert.

model uncert. parametrization uncert.

x

xf 2 = 2 GeV2Q

vxu

vxd

0.05)×xS (

0.05)×xg (

HER

A St

ruct

ure F

unct

ions

Wor

king

Gro

upJu

ly 2

010

H1 and ZEUS HERA I+II Combined PDF Fit

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

-410 -310 -210 -110 1

0.2

0.4

0.6

0.8

1

HERAPDF1.5 (prel.) exp. uncert. model uncert. parametrization uncert.

HERAPDF1.0

x

xf 2 = 10000 GeV2Q

vxu

vxd

0.05)×xS (

0.05)×xg (

HER

A St

ruct

ure F

unct

ions

Wor

king

Gro

upJu

ly 2

010

H1 and ZEUS HERA I+II Combined PDF Fit

0.2

0.4

0.6

0.8

1

q Physics interpretation of PDFs: Probability density to find a parton of flavor “f” carrying momentum fraction “x”, probed at a scale of “Q2”

f(x,Q2) :

² Number of partons: Z 1

0dxuv(x,Q

2) = 2,

Z 1

0dx dv(x,Q

2) = 1

² Momentum fraction: hx(Q2)if =

Z 1

0dx x f(x,Q2)

X

f

hx(Q2)i = 1

Scaling and scaling violation

Q2-dependence is a prediction of pQCD calculation

Jet production from the LHC

Hard probes from high energy collisions

q  Lepton-lepton collisions:

QCD phase Diagram!

Hadrons

² No hadron in the initial-state ² Hadrons are emerged from energy ² Not ideal for studying hadron structure

q Hadron-hadron collisions:

Hadrons

² Hadron structure – motion of quarks, … ²  Emergence of hadrons, … ²  Initial hadrons broken – collision effect, …

q  Lepton-hadron collisions:

Hard collision without breaking the initial-state hadron – spatial imaging, …

Why a lepton-hadron facility is special?

Q2 àMeasure of resolution

y à Measure of inelasticity

x à Measure of momentum fraction

of the struck quark in a proton Q2 = S x y

q  Many complementary probes at one facility:

Semi-Inclusive events: e+p/A à e’+h(π,K,p,jet)+X Detect the scattered lepton in coincidence with identified hadrons/jets

(Initial hadron is broken – confined motion! – cleaner than h-h collisions)

Exclusive events: e+p/A à e’+ p’/A’+ h(π,K,p,jet) Detect every things including scattered proton/nucleus (or its fragments)

(Initial hadron is NOT broken – tomography! – almost impossible for h-h collisions)

Inclusive events: e+p/A à e’+X Detect only the scattered lepton in the detector

(Modern Rutherford experiment!)

The Electron-Ion Collider (EIC) – the Future!

q  A sharpest “CT” – “imagine” quark/gluon structure without breaking the hadron

– “cat-scan” the nucleon and nuclei with a better than 1/10 fm resolution

– “see” proton “radius” of quark/gluon density comparing with the radius of EM charge density

To discover color confining radius, hints on confining mechanism!

q  A giant “Microscope” – “see” quarks and gluons by breaking the hadron

e p γ*, Z0, ..

1/Q < 1/10 fm Q

To discover/study color entanglement of the non-linear dynamics of the glue!

Backup slides

Drell-Yan process in QCD – factorization

q  Factorization – approximation:

²  Suppression of quantum interference between short-distance (1/Q) and long-distance (fm ~ 1/ΛQCD) physics

Need “long-lived” active parton states linking the two

Perturbatively pinched at p2a = 0

Active parton is effectively on-shell for the hard collision

²  Maintain the universality of PDFs:

Long-range soft gluon interaction has to be power suppressed

²  Infrared safe of partonic parts:

Cancelation of IR behavior Absorb all CO divergences into PDFs

Collins, Soper, Sterman, 1988

Drell-Yan process in QCD – factorization

q  Leading singular integration regions (pinch surface):

Hard: all lines off-shell by Q

Collinear: ²  lines collinear to A and B ²  One “physical parton”

per hadron

Soft: all components are soft

q  Collinear gluons:

²  Collinear gluons have the

polarization vector:

²  The sum of the effect can be

represented by the eikonal lines,

which are needed to make the PDFs gauge invariant!

Drell-Yan process in QCD – factorization

q  Trouble with soft gluons:

²  Soft gluon exchanged between a spectator quark of hadron B and

the active quark of hadron A could rotate the quark’s color and

keep it from annihilating with the antiquark of hadron B

k±

k±²  The soft gluon approximations (with the eikonal lines) need not

too small. But, could be trapped in “too small” region due to the

pinch from spectator interaction: k± ⇠ M2/Q ⌧ k? ⇠ M

Need to show that soft-gluon interactions are power suppressed

Drell-Yan process in QCD – factorization

q  Most difficult part of factorization:

0?

0? y?

y?

²  Sum over all final states to remove all poles in one-half plane

– no more pinch poles

²  Deform the k± integration out of the trapped soft region

²  Eikonal approximation soft gluons to eikonal lines

– gauge links

²  Collinear factorization: Unitarity soft factor = 1

All identified leading integration regions are factorizable!

Factorized Drell-Yan cross section

q  TMD factorization ( ):

The soft factor, , is universal, could be absorbed into the definition of TMD parton distribution

q  Collinear factorization ( ):

q? ⌧ Q

q? ⇠ Q

+O(1/Q)

q  Spin dependence:

The factorization arguments are independent of the spin states of the colliding hadrons

same formula with polarized PDFs for γ*,W/Z, H0…

Global QCD analysis – Testing QCD

Input DPFs at Q0

{ }( ), jf h x aϕ

DGLAP

( )f h xϕ at Q>Q0 Comparison with Data at various x and Q

Minimize Chi2 Vary { }ja

QCD calculation

Procedure: Iterate to find the best set of {aj} for the input DPFs

Uncertainties of PDFs

“non-singlet” sector

“singlet” sector

Partonic luminosities

q - qbar g - g