Introduction to Perturbative QCD · Lecture 1 Jianwei Qiu Iowa State University/Argonne National...

June 25, 2007 Jianwei Qiu, ISU/ANL1

Introduction to Perturbative QCDLecture 1

Jianwei QiuIowa State University/Argonne National Laboratory

PHENIX Spinfest at RIKEN 2007June 11 - July 27, 2007

RIKEN Wako Campus, Wako, Japan


The Goal:To understand hadron structure, and strong interaction dynamics in terms of Quantum Chromodynamics (QCD)

From the Parton Model to QCD and pQCDOne lecture

Cross sections with identified hadrons in pQCDTwo lectures

The Plan:

Purely infrared safe observables in pQCDOne lecture


Outline for Lecture 1

Excellent resource – CTEQ summer school websitehttp://www.phys.psu.edu/~cteq

Nucleons to Quarks

Deep Inelastic Scattering (DIS)

The Parton Model

Extensions of Parton Model beyond DIS

Quantum Chromodynamics (QCD)

Asymptotic freedom and perturbative QCD


Nucleons to Quarks

Protons, Neutrons, and Pions

3

938.3 MeV 1 2

1 2

mp S

I

=== + 3

939.6 MeV 1 2

1 2

mn S

I

=== −

pN

n⎛ ⎞

= ⎜ ⎟⎝ ⎠

Isospindoublet

3

139.6 MeV 0

1

mSI

π ±

=

== ±

0

3

135.0 MeV 0

0

mSI

π

=

==

Isospintriplet

0

π

π π

π

+

−

⎛ ⎞⎜ ⎟

= ⎜ ⎟⎜ ⎟⎝ ⎠

“Historic” – as bound statesNNπ

( ) ( ) ( )0 1, , 2

pn np pp nnπ π π+ −= = = +

Fermi and Yang, 1952; Nambu and Jona-Lasinio, 1960 (dynamics)


Nucleons not point-like spin ½ Dirac particlesProton magnetic moment: Neutron magnetic moment:

2pg ≠0ng ≠

“Modern” – common substructure: quarks, Nπ– Gell Mann, Zweig, 1964Quark Model

Quarks:

3

2 3 1 2

1 2

Q eu S

I

=== + 3

1 3 1 2

1 2

Q ed S

I

= −== − 3

1 3 1 2

0

Q es S

I

= −==

( ) ( ) ( )0 1, , 2

ud du uu ddπ π π+ −= = = +

( ) ( ) ( ) ( )++, , ,..., ,...p uud n udd K u uuus+= = = Δ =

But, need a new quantum number – color and the dynamics!

Magnetic moment: ( )3 2 good to %p n p ng gμ μ = = −

Han, Nambu, 1965


How to “see” substructure of a nucleon?

Rutherford experiment:– to see the substructure of an atom

α

α Atom

NucleusHigh energy α bounce off something very hard!

Discovery of nucleus inside an atom

SLAC experiment (1969):

θe

eNucleon

partonScattering information

on the θ-distribution

Discovery of the point-like spin-1/2 “partons”

Lepton-nucleon deeplyinelastic scattering (DIS)

Callan-Gross relation


Lepton-hadron DISProcess: ( , ) ( , ) ( ', ')e k P p e k Xλ σ λ+ → +

Charged current (CC)

W-

( '), 'kν λ

Neutral current (NC)

γ∗,Z0

2

2B qQxp

=⋅

Bjorken variable:

Kinematics:4-momentum transfer:

Squared CMS energy:

Final-state hadronic mass:

22Q q= −

( )2

2

B

Qsy

p kx

= + =

( ) ( )2

22 1 BB

QW xqx

p= + ≈ −

yk

pp

q⋅=

⋅Inelasticity:


Lepton-hadron DIS – general analysisScattering amplitude:

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

( )'2

i gq

μμ⎛ ⎞−⎜ ⎟

⎝ ⎠

( )' 0 ,emX eJ pμ σ

∗

∗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

ii

l k p kπ δ=

⎛ ⎞+ − −⎜ ⎟⎝ ⎠∑

( ) ( )2DIS

3 2

1 1 , '''

,2

dE W qd k s

k pQ

L kμνμν

σ ⎛ ⎞= ⎜ ⎟

⎝ ⎠

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


Hadronic tensor (No QCD has been used):4 †1 1( , ) e , ( ) (0) ,

4 2iq zW q p d z p J z J pμν μ ν

σ

σ σπ

⋅⎧ ⎫= ⎨ ⎬

⎩ ⎭∑∫

Structure functions:Parity invariance (EM current)Time-reversal invarianceCurrent conservation

*

sysmetric for spin avg.

real

0

W W

W W

q W q W

μν νμ

μν μν

μ νμν μν

=

=

= =

( ) ( )2 22 2

1 22, ,1B B

q q p q p qW g p qF x Q F xp qq p q q

Qq

μ νμν μν μ μ ν ν

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

Reduced to two dimensionless scalar structure functions for spin-avgeraged DIS

Two more structure functions for spin-dependent DIS

Note: No explicit QCD was used in above derivation!

Measure cross sections extraction of structure functions


Before the collision: Feynman, 1969, 1972

“Deeply inelastic scattering”

in e-–parton cm frame:

The Parton Model

( )e k−p

ix p

0 1ix≤ ≤1i

ix =∑

Lorentz contractedTime dilated

Effectively frozen

After the collision:

( )'e k−

ix p q+ ( )2 0ix p q+ ≈2. ., qi e⎡ ⎤⎣ ⎦fragments

elastic collision

1 fmcollision hadronQt t∼∼


Inelastic hadroniccross section

Partonic elasticcross section fp xp=

Probability for= ⊗

Basic Parton Model Relation

( ) ( ) ( )DIS el1

0

ˆ, , eh eparto

fns

ff

dp x xpq q xσ σ φ−

= ∑ ∫where

( )h p

( )f xp

( )DIS ,eh qpσ

( )elˆ ,ef qxpσ

( )f xφ

DIS cross section for hadron:

Elastic cross section for parton:

Probability for to have - PDFf xp

Nontrivial assumption:

Quantum mechanical incoherence between the large q scattering and the partonic distribution


Structure Functions in Parton Model

( ) ( )DIS 2

3 2 ,1 , '2

1''

eh

sW q pdE L

d k Qk k ν

μνμ

σ ⎛ ⎞= ⎜ ⎟

⎝ ⎠

Recall:

PM formula: ( ) ( ) ( )1

0

elˆ, 1 , part

ffons

W q p W qx x xx

pdμν μν φ−

⎡= ⎤⎢ ⎥⎣ ⎦

∑ ∫

( ) ( ) ( ) ( ) ( )el 2 21 1ˆ , Tr 2 ( )4 2f

fW q p e p q px x px qxμν μ νγ γ γ γ π δ

π⎡ ⎤= ⋅ + ⋅ +⎣ ⎦∑

22

22 2

1 12

1 1

f

f

B

B

q q xg eq

p q p q xp q p q ep q q

x

qx

x

μ νμν

μ μ ν ν

δ

δ

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

( ) ( ) ( )2 2 22 1, 2 ,B B Bf f

fB BF x Q e x x x F x Qφ= =∑

2QCallan-Gross Relation spin ½ parton Bjorken scaling - independent universal PDFs


Fragmentation Functions in PMCross from DIS:

p

q

X

“crossing”

p

Xq

Single particle inclusive (1PI)

Cross from Parton Model:p'

qX

p( )fD zp z

Fragmentation function( )f xφ

xp

Xp

q

p'

Parton distribution

( ) ( ) ( )1PI 1P1

0

Iˆ, , hparto

fns

ff

dz zq zq Dp pσ σ−

= ∑ ∫PM formula for 1PI:


Drell-Yan Dilepton Production in PM

( ) ( ) ( )' 'p ph Xqh + −+ → +Drell-Yan Process:

PM picture: p

x p

'p

' 'x p1 fmcollision hadronQ

t t∼∼

2 2 with q Q=

qxp

' 'x p

PM formula:( ) ( ) ( ) ( )

el1 1'

0 0

''

, '2

Y

2

D , ' ', '' '

ˆ ,, ff

f ff f

hh xp x pddd d

dq

dp p

xQ Q

xq

x xσσ

φ φ= ∑ ∫ ∫( ) 2

2 e2

2 2

el'

'm' ' 1

3 ' 'ˆ , 1 1

3, 4ff

f ff

de

s sxp x p

xq

d Q xQxxQ

σ παδ δ⎛ ⎞⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

,

2

1

31 1ColorFactor3 3ij ji

i jδ δ

=

⎛ ⎞= =⎜ ⎟⎝ ⎠

∑


Need to Improve the PM

Drell-Yan cross section:DY

Exp/Thy Ex

D

p'hy

'Y

T2hh hhK σ σ= ≥

Need a better dynamical theory!

Total momentum carried by the partons:

( )1

0

0.5ff

qF x x xd φ≡ ∑∫ ∼

missing momentum particles not directly interactwith photon (or EM charge) the gluon

Scaling violation Q-dependence of structure functions?

...


Quantum Chromodynamics (QCD)Known Fundamental Interactions:

“Strong” – QCD WeakElectromagnetic - QED

…

Gravity

Electro –Weak

StandardModel

QCD – stands as a very solid building block of the SM:Unbroken SU(3) color gauge symmetryAsymptotic freedom at high energySuccess of QCD perturbation theoryNonperturbative results from Lattice calculations… Not many surprises so far

AdS

/CFT

co

rres

pond

ence


QCD as a field theoryFields:

( )fi xψ

( ),aA xμ

Quark fields, Dirac fermions (like electron)Color triplet: i = 1,2,3=NCFlavor: f = u,d,s,c,b,t

Gluon fields, spin-1 vector field (like photon)Color octet: a = 1,2,…,8 =NC

2-1

Lagrangian density:

Color matrix:

( ) ( )( ),,f f

ff

QCD a a ii ijL A i A t mg μ

μ μψ ψ γ ψ⎡ ⎤= ∂ − −⎣ ⎦∑2

, , , ,14 a a c bv ab cgA A C A Aμ ν μ μ ν⎡ ⎤− ∂ − ∂ −⎣ ⎦

+ gauge fixing + ghost terms

[ , ]a b abc ct t iC t=


Gauge invariance:( )'i j ji iU xψ ψ ψ→ =

1 1' ( ) ( ) ( ) ( )iA A U x A U x U x U xgμ μ μ μ

− −⎡ ⎤→ = + ∂⎣ ⎦

, ,a aA A tμ μ=where ( ) unitary [ det = 1, SU(3)]ijU x

Gauge fixing:

Allow us to define a propagator:

with Feynman gauge


Ghost:

so that optical theorem (and hence unitarity) may be respected:

ghost fields

2 Im

…

= Σ 2

Fail without the ghost loop

Sum over all physical polarizations


Feynman rulesPropagators:

Quark:

Ghost:

Gluon:

for covariant gauge


Interactions:

∗ ∗


Renormalization in QCDScattering amplitude:

= +

+ ...+

Ei EiEI

= 1 ... + ... i

II

PSEE

⎛ ⎞+⎜ ⎟

⎝ ⎠⇒

−∞∫

UV divergence “Sum” over states of “high mass”

Uncertainty principle: high mass states = “Local” interaction

No experiment has an infinite resolution!


Renormalization:UV divergence due to “high mass” statesExperiments cannot resolve the details of these states

combine the “high mass” states with LO

= +

“Low mass” state “High mass” states

−

NLO: − + ... No UV divergence!

LO: + =Renomalized

coupling

Renormalization = re-parameterization of the expansion parameter in perturbation theory


Renormalization GroupPhysical quantities can’t depend on therenormalization scale - μ:

2 ( )( )4s

gα μπ

μ =2 ( ) 2 2phy

( )( ) ( , )2s

nn

n

Q Q μασ σπ

μ ⎛ ⎞= ⎜ ⎟⎝ ⎠

∑

22

phy2 2 , ( ), 0d Qd

gμ μ μσμ μ

⎛ ⎞=⎜ ⎟

⎝ ⎠

The β-function:3 51

2

( )( ) ( )16

gg g O gμ ββ μμ π

∂= = + +

∂1

4 0 3 2

11 for 63 c f

fN nn

β −= < ≤+

QCD running coupling constant:

12 2 12

1 21 2

1

( )( ) 0 as for 01 ( )

4

ss

s n

α μα μ μ ββ μα μπ μ

= ⇒ → ∞ <⎛ ⎞

− ⎜ ⎟⎝ ⎠ Asymptotic freedom


QCD running coupling constant1

2 2 21 2 2

1 2Q C

2 11 D

( ) 4( ) 1 ( )

4

ss

s n n

α μ πα μβ μ μα μ βπ μ

=⎛ ⎞ ⎛ ⎞

− −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎝ Λ⎠

≡

⎠

ΛQCD:

μ2 and μ1 not independent


Effective quark massRunning quark mass:

[ ]2

1

2 1 2( ) ( ) exp - 1 ( ( )) 0 as mdm m g

μ

μ

μ μλ

μλ γ λ⎡ ⎤

= + ⇒ → ∞⎢ ⎥⎢ ⎥⎣ ⎦

∫

Perturbation theory becomes a massless theory when μ →∞

QCD perturbation theory (Q>>ΛQCD)is effectively a massless theory

for light quarks, u and d, even s, and QCD( )u dm μ Λ

Choice of renormalization scale: Qμ ∼


Infrared SafetyInfrared safety:

2 2 2 2 2 22 2

phy 2 2 2 2

( ) ( )ˆ, ( ), , ( )s sQ m Q mO

κμ μσ α μ σ α μ

μ μ μ μ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⇒ + ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

0κ >Infrared safe =

Asymptotic freedom is useful only for

quantities that are infrared safe

Asymptotic freedom + Infrared safety = perturbative QCD


Foundation of perturbative QCD

Renormalization – QCD is renormalizable

Asymptotic freedom – weaker interaction at a shorter distance

Infrared safety – pQCD factorization and

calculable short distance dynamics


Summary

QCD is a SU(3) color non-Abelian gauge theory of quark and gluon fields

QCD perturbation theory works at high energybecause of the asymptotic freedom

Perturbative QCD calculations make sense onlyfor infrared safe (IRS) quantities

QCD perturbation theory is effectively a masslesstheory – renormalization group equation for the

parton mass

Look for IR safe quantities

Introduction to Perturbative QCD · Lecture 1 Jianwei Qiu Iowa State University/Argonne National...

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