June 5, 2006 Jianwei Qiu, ISU1
QCD and Rescatteringin Nuclear Targets
Lecture 1
Jianwei QiuIowa State University
The 21st Annual Hampton University Graduate Studies Program (HUGS 2006)
June 5-23, 2006Jefferson Lab, Newport News, Virginia
June 5, 2006 Jianwei Qiu, ISU2
The Goal:
From the Parton Model to QCD and pQCDOne lecture
Rescattering in Nuclear TargetsThree lectures
The Plan:
To understand hadron structure, nuclear matter, and strong interaction dynamics in terms of Quantum Chromodynamics (QCD)
Fundamentals of Perturbative QCDOne lecture
June 5, 2006 Jianwei Qiu, ISU3
The Parton Model to QCD and pQCD
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 5, 2006 Jianwei Qiu, ISU4
Nucleons to QuarksProtons, 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 5, 2006 Jianwei Qiu, ISU5
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 the dynamics and a new Q.N. – color
Magnetic moment: ( )3 2 good to %p n p ng gμ μ = = −
June 5, 2006 Jianwei Qiu, ISU6
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)
June 5, 2006 Jianwei Qiu, ISU7
Lepton-hadron DISProcess:
Kinematics:
( , ) ( , ) ( ', ')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: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 5, 2006 Jianwei Qiu, ISU8
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 5, 2006 Jianwei Qiu, ISU9
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 5, 2006 Jianwei Qiu, ISU10
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 5, 2006 Jianwei Qiu, ISU11
Basic Parton Model Relation
( ) ( ) ( )DIS el1
0
ˆ, , eh eparto
fns
ff
dp pq x qx 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
Inelastic hadroniccross section
Partonic elasticcross section fp px=
Probability for= ⊗Nontrivial assumption:
Quantum mechanical incoherence between the large q scattering and the partonic distribution
June 5, 2006 Jianwei Qiu, ISU12
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 5, 2006 Jianwei Qiu, ISU13
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 5, 2006 Jianwei Qiu, ISU14
Drell-Yan Dilepton Production in PM
( ) ( ) ( )' 'p ph Xqh + −+ → +Drell-Yan Process:
PM picture: p
xp
'p
''x p1 fmcollision hadronQ
t t∼∼PM formula:( ) ( ) ( ) ( )
el
2 2
1 1'
0 0'
,
D'
Y
'
'' '
ˆ ,,
, ',
' ff
f ff f
hh x xp pddd d
dq
dx
Qx
Qx
p qx
p σσφ φ= ∑ ∫ ∫
2 2 with q Q=
( ) 22 e
2
2'
el
2m
'
ˆ , 4 1 19
, '' '
'fff ff
x xxx x
dx
q Qd
es
pQ s
pQ
σ παδ δ⎛ ⎞⎛ ⎞
= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
qxp
''x p
June 5, 2006 Jianwei Qiu, ISU15
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 5, 2006 Jianwei Qiu, ISU16
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 gauge symmetryAsymptotic freedom at high energySuccess of QCD perturbation theoryNonperturbative results from Lattice calculations…
June 5, 2006 Jianwei Qiu, ISU17
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 5, 2006 Jianwei Qiu, ISU18
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 5, 2006 Jianwei Qiu, ISU19
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 5, 2006 Jianwei Qiu, ISU20
Feynman rulesPropagators:
Quark:
Ghost:
Gluon:
for covariant gauge
June 5, 2006 Jianwei Qiu, ISU21
Interactions:
∗ ∗
June 5, 2006 Jianwei Qiu, ISU22
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 5, 2006 Jianwei Qiu, ISU23
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 5, 2006 Jianwei Qiu, ISU24
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
11 4 0 3 3 2
for 6fc f
nN nβ = − + ≤<
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 5, 2006 Jianwei Qiu, ISU25
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 5, 2006 Jianwei Qiu, ISU26
Effective quark massRunning quark mass:
[ ]2
1
2 1 2( ) ( ) exp - 1 ( ( )) 0 as mdm m g
μ
μ
λμ μ γ λ μλ
⎡ ⎤= + ⇒ → ∞⎢ ⎥
⎢ ⎥⎣ ⎦∫
Perturbation theory becomes a massless theory when μ →∞
for light quarks, u and d, even s, and QCD( )u dm μ Λ
QCD perturbation theory (Q>>ΛQCD)is effectively a massless theory
June 5, 2006 Jianwei Qiu, ISU27
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
June 5, 2006 Jianwei Qiu, ISU28
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