Zhongbo Kang Los Alamos National Laboratory

QCD structure of the nucleon and spin physics Lecture 2 & 3: QCD collinear factorization and evolution

HUGS 2015, Jefferson Lab June 2, 2015

QED: the fundamental theory of electo-magnetic interaction

§  QED Lagrangian:

§  Feynman rule: photon has no charge, thus does not self-interact

2

L = ψ̄(iγµ∂µ −m)ψ + eψ̄γµψAµ − 1

4FµνF

µν

Fµν ≡ ∂µAν − ∂νAµ

QCD: the fundamental theory of the strong interaction

§  As the fundamental theory, QCD describes the interaction between quarks and gluons (not hadrons directly)

§  Feynman rules: gluon carries color, thus can self-interact

3

L = ψ̄(iγµ∂µ −m)ψ + gψ̄γµtaψGaµ − 1

4Ga

µνGµνa

Gaµν ≡ ∂µG

aν − ∂νG

aµ − gfabcG

bµG

cν

Experimental verification of the color

§  The color does exist: color of quarks Nc = 3 (low energy R=2/3 v.s. 2)

4

Re+e− =σ(e+e− → qq̄)

e+e− → µ+µ− = Nc

�

q

e2q

Understanding QCD: running coupling (asymptotic freedom)

§  Rough qualitative picture: due to gluon carrying color charges §  Value of the strong coupling αs depends on the distance (i.e., energy)

5

Screening: Anti-screening: αem(r) ↑ as r ↓ αs(r) ↓ as r ↓

Why does the coupling constant run?

§  Leading order calculation is simple: tree diagrams – always finite

§  Study a higher order Feynman diagram: one-loop, the diagram is divergent as q → ∞

§  Make sense of the result: redefine the coupling constant to be physical

6

q

Renormalization (Redefine the coupling constant)

§  Renormalization §  UV divergence due to “high momentum” states

§  Experiments cannot resolve the details of these states

§  Combine the “high momentum” states with leading order

7

Low momentum state High momentum state

LO:

NLO:

Renormalized coupling

No UV divergence!

Simple study of Deep Inelastic Scattering (DIS): parton model

§  DIS has been used a lot in extracting hadron structure

§  Leptonic and hadronic tensor

8

N

(E, p)

hπ

π

q

e

+

(E, p )’ ’

π

u

du

*γ

Structure functions

§  Hadronic tensor: Lorentz decomposition + parity invariance (for photon case) + time-reversal invariance + gauge invariance

§  All the information about hadron structure is contained in the structure functions

9

Lµν = 2��µ�

�ν + ��µ�ν − � · ��gµν

�

dσ

dxBdQ2=

4πα2em

xBQ4

�(1− y)F2(xB , Q

2) + y2xBF1(xB , Q2)�

The paradigm of perturbative QCD

§  The common wisdom: to trace back what’s inside the proton from the outcome of the collisions, we rely on QCD factorization

§  Hadron structure: encoded in PDFs

§  QCD dynamics at short-distance: partonic cross section, perturbatively calculable

10

xP

ee

X

q

P

parton

σ̂parton

fparton(x)Parton Distribution Functions (PDFs): Probability density for finding a parton in a proton with momentum fraction x

σproton(Q) = fparton(x)⊗ σ̂parton(Q)Universal (measured) calculable

Universality of PDFs: extraction from DIS

11

σproton(Q) = fparton(x)⊗ σ̂parton(Q)Universal (measured) calculable ZEUS

0

1

2

3

4

5

1 10 102

103

104

105

F2

em-l

og

10(x

)

Q2(GeV

2)

ZEUS NLO QCD fit

tot. error

ZEUS 96/97

BCDMS

E665

NMC

x=6.32E-5x=0.000102

x=0.000161x=0.000253

x=0.0004x=0.0005

x=0.000632x=0.0008

x=0.0013

x=0.0021

x=0.0032

x=0.005

x=0.008

x=0.013

x=0.021

x=0.032

x=0.05

x=0.08

x=0.13

x=0.18

x=0.25

x=0.4

x=0.653

What Do We Know About Glue in Matter?

• Scaling violation: dF2/dlnQ2 and

linear DGLAP Evolution !

G(x,Q2)!

Deep Inelastic Scattering : d

2" ep#eX

dxdQ2

=4$%e.m.

2

xQ4

1& y +y

2

2

'

( )

*

+ , F2(x,Q2) &

y2

2FL (x,Q2)

-

. /

0

1 2

Gluons dominate low-x wave function

)20

1( !xG

)20

1( !xS

vxu

vxd

!

What about higher order?

§  pQCD calculations: understand and make sense of all kinds of divergences §  Ultraviolet (UV) divergence : renormalization (redefine coupling

constant)

§  Collinear divergence : redefine the PDFs and FFs

§  Soft divergence : usually cancel between real and virtual diagrams for collinear PDFs/FFs; do not cancel for TMDs, leads to new evolution equations

§  Going beyond the leading order of the DIS, we face another divergence

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k → ∞

k//Pk → 0

= +Wµν

q

k1

!

P

qk

P

q q

k

!

QCD dynamics beyond tree level

§  Going beyond leading order calculation

13

QCD factorization: beyond parton model

§  Systematic remove all the long-distance physics into PDFs

14

Scale-dependence of PDFs

§  Logarithmic contributions into parton distributions

§  Going to even higher orders: QCD resummation of single logs

15

P

kk1

k

P

q q!

φ(x, µ2)C(Q2/µ2)

+ + ...

...

P

kk1

k

P

kP

= + + + ...φ(x, µ2)

αs lnµ2

Λ2

�αs ln

µ2

Λ2

�2

DGLAP evolution = resummation of single logs

§  Evolution = Resum all the gluon radiation

§  By solving the evolution equation, one resums all the single logarithms of type

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P

kk1

k

P

kP

= + + + ...φ(x, µ2)

DGLAP Equation

P

k-P

kk1

k

P+ + ...=φ(x, µ2)

∂

∂ lnµ2φi(x, µ2) =

�

j

Pij(x

x� )⊗ φj(x�, µ2)

Evolution kernel splitting function

�αs ln

µ2

Λ2

�n

PDFs also depends on the scale of the probe

§  Increase the energy scale, one sees parton picture differently

17

Q20 Q2 > Q2

0

ZEUS

0

1

2

3

4

5

1 10 102

103

104

105

F2

em-l

og

10(x

)

Q2(GeV

2)

ZEUS NLO QCD fit

tot. error

ZEUS 96/97

BCDMS

E665

NMC

x=6.32E-5x=0.000102

x=0.000161x=0.000253

x=0.0004x=0.0005

x=0.000632x=0.0008

x=0.0013

x=0.0021

x=0.0032

x=0.005

x=0.008

x=0.013

x=0.021

x=0.032

x=0.05

x=0.08

x=0.13

x=0.18

x=0.25

x=0.4

x=0.65

Q20 Q2 > Q2

0

Evolutions of PDFs

§  Perturbative change:

§  Feynman diagrams for unpolarized PDFs

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q(x, µF )∂lnµ2

F

=αs

2π

� 1

x

dξ

ξ

�Pqq(z) q(ξ, µF )

�

Pqq(z) = CF

�1 + z2

(1− z)++

32δ(1− z)

�

Example

§  We will now study a detailed example §  SIDIS at both LO and NLO

§  We can understand QCD collinear factorization and evolution of PDFs and FFs

§  SIDIS and hadron frame

19

�+ p↑ → �� + h(Ph) +X

Computer program

§  A little bit on the Dirac gamma matrix computation §  FeynCalc (mathematica): http://www.feyncalc.org

§  Tracer (mathematica): http://library.wolfram.com/infocenter/MathSource/2987/

§  Reduce: http://reduce-algebra.sourceforge.net/

§  FORM: http://www.nikhef.nl/~form/

§  The detailed calculations are provided on scanned notes as well as white board presentation

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Summary

§  Asymptotic freedom: allow one to calculate partonic cross sections

§  Parton distribution functions

§  Renormalization scale and factorization scale

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