perturbative quantum chromodynamics...

1

Perturbative Quantum Chromodynamics {QCD}

Von: Adrian Seeger

Alep

h_h

om

e.fnal.go

vIn

side2

_spo

ntan

eou

ssymm

etry.wo

.rdp

ress.co

m

detlas_w

alosch

ek.dec

LEP_R

F_wiki

2


Contents

• Brief Historical Overview

• Experimental results and Perturbative QCD

• Standard Model

• Group Theorie

• Operators in QCD

• Feynman Graphs

• Gauge Theory

• Asymptotic Freedom

3

Brief Historical Overview

• 1950s Invention of bubble chambers and spark chambers lead to observation of a large number of different hadrons

• 1961 Gell-Mann and Yuval Ne‘eman invented the „eightfold way“

• 1963 Gell-Mann proposed, hadrons are build from Quarks [Nobel Prize 1969]

• 1965 Moo-Young Han and other solved Violation of the Pauli Principle (d++Barion) by giving the Quarks an additional degree of freedom

• 1969 Parton model was veryfied for deep inelastic scattering at SLAC

• 1973 David Gross, David Politzer and Frank Wilczek discovered „asymptotic freedom“ [Nobel Prize 2004]

• 1979 Evidence of Gluons was discovered in three jet events at PETRA

8foldway_wikipedia

detlas_w

alosch

ek.dec

4


Experimental results and perturbative QCD

Coupling parameter of the strong force

)/ln()233(

12)(

2

2

2

QNQ

f

S

Peter Sch

mü

ser Feynm

an G

raph

en

5


Experimental results and perturbative QCD

• R ratio of Hadron and Myon in e+e-annihilation

))/(8.12)/(4.1)/(1(3)(

)( 322

SSS

j

j

QED

Qee

HadronseeR

Peter Sch

mü

ser Feynm

an G

raph

en

Standard Model

Is a unified Quantum field theorie consisting of:

• Electroweak theory (Glashow, Salam, Weinberg)

– Electromagnetic theory

– Weak theory

• Quantum chromodynamics (Strong Force)

• A field theory can be described by a Lagrangian density

• Solving the functional S

• Leads to the Euler-Lagrange equations


6

),( iiLL

),( ii

n LxdS

0)(

ii

LL

ni ,...1,0

7


Standard Model

Two important equations:

• Klein-Gordon equation

– Describes bosons (Spin=0, gauge bosons)

– Scalar field φ

• Dirac equation

– Describes fermions (Spin=1/2, leptons, quarks )

– Spinor field ψ

0),(1

2

222

2

2

2

xt

cm

tc

0)(23

0

xmcx

ci

8


Standard Model

Symmetries:

• Electroweak theory

• Quantum Chromodynamics

Standard model is a gauge theory with symmetry:

CYI SUUSU )3()1()2(

CSU )3(

YI USU )1()2(

9


Group Theory

U means unitary:

A unitaryl operator can be written as is hermitean

• A unitary transformation leaves the normalization of the state vectors unchanged

• U(1): Group of all unitary linear operators in one dimension

• Example:

• U(1) is abelian:

SU means specific unitary:

• SU(2): Group of all specific unitary linear operators in two dimensions

• Generators of the SU(2) group: ½*Pauli matrices

• SU(2) is non abelian:

RCz ,;

zezU i

iiii eeee

1 UUIUUUU

0)(1)( TraceUDet

10

01,

0

0,

01

10321

i

i

01

10

01

10

10

01

10

01

01

10

01

10

)exp( i

iiiU i i

10


Group Theory

SU means specific unitary:

• SU(3): Group of all specific unitary linear operators in three dimensions

• Generators of the SU(3) group: ½*Gell-Mann matrices

• SU(3) is non abelian

0)(1)( TraceUDet

000

010

001

,

000

00

00

,

000

001

010

321 i

i

200

010

001

3

1,

00

00

000

010

100

000

,

00

000

00

,

001

000

100

87654

i

i

i

i

11


Operators in QCD

• To change the color state of a quark we can construct operators from the Gell-Man matrices

• Two Gell-Mann matrices remain

• They don‘t change color!

)(2

121 iI

)(2

176 iU

)(2

154 iV

UV

I

)(2

121 iI

)(2

176 iU

)(2

154 iV

B

GR

000

010

001

3

200

010

001

3

18

12


Operators in QCD

Example:

• Transition from green to red

• The quark colour changes from green to red

– a specific gluon is absorbed

• So, you can relate a gluon to every operator!

Rq

Gq

RGI

0

1

0

Gq

B

G

R

RG qi

i

iqI

0

0

1

0

1

0

000

00

00

000

001

010

2

1

13


Gauge Theory

Gauge theory in Electrodynamics:

• Maxwell equations stay invariant under a transformation

Example

• From the second Maxwell equation follows

• Applyed to the third equation yields

t

trX

),('

ABB 0

)),(()(

0)(

trXAtt

XE

t

AE

t

AE

t

AE

t

EjB

t

BE

B

E

EquationsMaxwell

0

_

),(' trXAAA

14


Gauge Theory

Gauge theory in quantum mechanics:

• Schrödinger equation stays invariant under a transformation

• Schrödinger equation for a particel with charge q in electromagnetic field φ

• We can combine φ and A to the four vector

t

trX

),(' ),(' trXAAA )exp(' iqX

tixtqAqi

m

),()(2

1 2

),( AA

15


Gauge Theory

Gauge theory in QCD:

• The Lagrangian density is invariant

• under a global transformation

• If the transformation is local,

• the Lagrangian density is not invariant

)( miL

)2

exp(' jjsg

i

')(')( mimiL

))(2

exp(' xg

i jjs

))))((2

((')(')( mxg

iimimiL jjs

16


Gauge Theory

To make it invariant:

• is replaced by ,where are vector fields

• Gauge transformation of the vector fields

• With the field-strength tensor

• We obtain the Lagrangian density of QCD

• The Dirac equation of a free particle obtained from the Lagrangian density is invariant under a local transformation

jj FFmDiL ,

4

1)(

jkjklsjjj GfgGG '

jjs G

giD

2

jG

lkjklsjjj GGfgGGF

0')'(0)( mDimDi

17


Feynman Graphs

• With the ϒ-matrices

• The four gradient

• We can write the Dirac equation of a particle with charge e in the field

• To solve this inhomogeneous differential equation we define a Green‘s function K

• Solution of the Dirac equation

0010

0001

1000

0100

,

000

000

000

000

,

0001

0010

0100

1000

,

1000

0100

0010

0001

3210

i

i

i

i

A

),(

t

)()()()( xxexmi

')'()'()'()( 4xdxxxxKex

)'()',()( 4 xxxxKmi

onAbbreviati

18


Feynman Graphs

• We can add any solution of the homogeneous Dirac equation to the solution

• Perturbative calculation means, to handle the second term as small perturbation

• Now the only thing we need is the Green‘s function K

• We define the Fourier transform of K

• and find the electron propagator

')'()'()'()()( 4xdxxxxKexx

')'()'()'''()''()''(''')'()'()'(')()(

')'()'()()(

)()(

4424)2(

4)0()1(

)0(

xxxxKxxxKxdxdexxxxKxdexx

xdxxxKexx

xx

iii

))'(()(~

)2()'( 44 xxpipexpKpdxxK

imp

mppK

22

)(~

)(~

pK

19


Feynman Graphs

• With the electron propagator and integration we find the Green‘s function for a particle moving foreward in time to be

• And for a particle moving backwards in time

E

mpEttiExxippexpdixxK

2)]'()'([)2()'(

033

E

mpEttiExxippexpdixxK

2)]'()'([)2()'(

033

20


Scattering in QED

• Defining the scattering matrix as

• We can calculate the matrix elements in first order

• To be

electron

Scat

ectordet

)(xi otonPr

')'()'()( 4)0( xdxxxKexS iScat

)'()'()'()(' 222

34)1( xxxxKxxdxdeS iffi

)'()'()'('4)1( xxxxdieS iffi

f

f

21


Scattering in QED

• Matrix element for this example is

)()()()()2()()( 24213

4

13

4

24

442)1( pupuiq

gpupuqppqppqdieS fi

iq

ig

2

)( 24

4 qppie )( 13

4 qppe

)( 2pu)( 1pu

)( 4pu)( 3pu

mE

pmE

ippNpu

mE

ippmE

pNpu

EquationDiracofSolution

z

yx

yx

z

0

1

)(

0

1

)(

___

2

1

22


Feynman Graphs in QED

• Incoming (outgoing) fermion:

• Incoming (outgoing) photon:

• Virtual photon:

• Virtual electron:

• Lepton-Photon-Vertex

Fermion

Photon

Vertex

)(

)(),(

)(),(

4

22

2

*

qppie

imp

mp

iq

g

kk

pupu

if

23


Feynman Graphs in QED

• Elementary processes

Emission Absorbtion oductionPr onAnnihilati

Time

24


Feynman Graphs in QCD

• Incoming (outgoing) fermion:

• Incoming (outgoing) photon:

• Virtual gluon:

• Virtual electron:

• Quark-Gluon-Vertex

Fermion

Gluon

Vertex

)(

)(),(

)(),(

4

22

2

*

qppig

imp

mp

iq

g

kk

pupu

if

k

ab

g~

2~ g

25


Asymptotic Freedom

• As seen, we can describe processes in QED in low order approximations

• The approximation converges fast, because the coupling constant is small

• In QCD it is not sure, that the perturbative calculation converges.

• Therefore ,a very important discovery was

0

)/ln()233(

12)(

2

2

2

Q

QNQ

f

S

1137

1

4

1 2

0

c

e

26


Asymptotic Freedom

• There is also a Q dependence in the QED

– Not as strong

– inverse behaviour

Example:

• Positive charge in dielectric oil

• Following QED there are vacuum fluctuation creating virtual electron positron pairs, that are causing this effect of shielding

)()( 22 QQ

2

04 r

qE

2

04

'

r

qqF

q

'q


Asymptotic Freedom

• Experiments at LEP support this theory

Feynman diagram

27

9,128

1)( 2 ZM

137

1)( 2 smallQ

28


Asymptotic Freedom

• The Q dependence in QCD is stronger

– α is called running coupling constant

– Two effects contribute

• Quark-Antiquark production

• Emission of a Gluon

• Quark-Antiquark production would dominate only if we had more than 16 quark flavours!

R

R R

R

R

R

R

R

R

R

R

R

R

R

29


Asymptotic Freedom

• Feynman diagram quark-antiquark production

• Feynman diagram emission of a gluon

GluonLoop

30


perturbative quantum chromodynamics...

Documents