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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
June 25, 2007 Jianwei Qiu, ISU/ANL2
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
June 25, 2007 Jianwei Qiu, ISU/ANL3
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
June 25, 2007 Jianwei Qiu, ISU/ANL4
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)
June 25, 2007 Jianwei Qiu, ISU/ANL5
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
June 25, 2007 Jianwei Qiu, ISU/ANL6
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
June 25, 2007 Jianwei Qiu, ISU/ANL7
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:
June 25, 2007 Jianwei Qiu, ISU/ANL8
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
June 25, 2007 Jianwei Qiu, ISU/ANL9
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
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
June 25, 2007 Jianwei Qiu, ISU/ANL10
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∼∼
June 25, 2007 Jianwei Qiu, ISU/ANL11
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
June 25, 2007 Jianwei Qiu, ISU/ANL12
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
June 25, 2007 Jianwei Qiu, ISU/ANL13
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:
June 25, 2007 Jianwei Qiu, ISU/ANL14
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
=
⎛ ⎞= =⎜ ⎟⎝ ⎠
∑
June 25, 2007 Jianwei Qiu, ISU/ANL15
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?
...
June 25, 2007 Jianwei Qiu, ISU/ANL16
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
June 25, 2007 Jianwei Qiu, ISU/ANL17
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=
June 25, 2007 Jianwei Qiu, ISU/ANL18
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
June 25, 2007 Jianwei Qiu, ISU/ANL19
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
June 25, 2007 Jianwei Qiu, ISU/ANL20
Feynman rulesPropagators:
Quark:
Ghost:
Gluon:
for covariant gauge
June 25, 2007 Jianwei Qiu, ISU/ANL21
Interactions:
∗ ∗
June 25, 2007 Jianwei Qiu, ISU/ANL22
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!
June 25, 2007 Jianwei Qiu, ISU/ANL23
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
June 25, 2007 Jianwei Qiu, ISU/ANL24
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
June 25, 2007 Jianwei Qiu, ISU/ANL25
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
June 25, 2007 Jianwei Qiu, ISU/ANL26
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 ∼
June 25, 2007 Jianwei Qiu, ISU/ANL27
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
June 25, 2007 Jianwei Qiu, ISU/ANL28
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
June 25, 2007 Jianwei Qiu, ISU/ANL29
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
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