QCD and Rescattering in Nuclear Targets Lecture 1 · Lecture 1 Jianwei Qiu Iowa State University The 21st Annual Hampton University Graduate Studies Program (HUGS 2006) June 5-23,

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


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


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


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)


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 = = −


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)


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:


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∼∼


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


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

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


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 gauge symmetryAsymptotic freedom at high energySuccess of QCD perturbation theoryNonperturbative results from Lattice calculations…


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

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


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

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


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


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

Documents

QCD and Rescattering in Nuclear Targets Lecture 1 · Lecture 1 Jianwei Qiu Iowa State University The 21st Annual Hampton University Graduate Studies Program (HUGS 2006) June 5-23,