A short overview on strong interaction and quantum ..._Uni... · A short overview on strong interaction and quantum chromodynamics Christoph Klein Universität Siegen Doktorandenseminar

A short overview on strong interaction and quantum chromodynamics

Christoph Klein

Universität Siegen

Doktorandenseminar08.10.2008

Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 1 / 28

Outlook

1 History of strong interaction physics

2 Quantum Chromodynamics (QCD)Review of QEDQCD - The theoryFundamental properties of QCDHadrons and nuclear force

3 Nonperturbative AscpectsSome fundamental aspectsShort summary of non-perturbative methods


History of strong interaction physics

Outlook






History of strong interaction physics - some milestones

1911 Rutherford discovers atomic nucleus

1919 Rutherford discovers the proton as elementary constituent of the nucleus

→ Because of Coulomb force between the protons,there has to be a strong nuclear interaction between them,that holds the nucleus together

1932 Chadwick discovers neutron

1935 Yukawa postulates the π-Meson as force-carrying particle of the strongnuclear interaction

1947 Lattes discovers the charged pion in cosmic rays








































1954 Yang and Mills introduce non-abelian gauge-theories

1964 Gell-Mann and Zweig postulate quarks

1969 Bjorken discovers in collider experiments, that protons consist ofasymptotically free particles (partons: quarks and gluons)

1971 QCD is proposed by Fritzsch, Gell-Mann, t’Hooft, et al.-1973 as fundamental theory of strong interaction

1974 Discovery of J/Ψ, the bound state of two charm quarks,good agreement with the new QCD theory

1979 The existence of the gluon is verified by a three-jet-event at PETRA (DESY)















































Quantum Chromodynamics (QCD)

Outlook





Quantum Chromodynamics (QCD) Review of QED

Reminding QED

Quantum electrodynamics (QED)

Remember: QED is the fundamental quantum field theory of the electromagneticinteraction.

Describes interaction of charged fermions ψ(x) (electrons, myons, quarks,...)mediated by the photon Aµ(x).

The fundamental structure of a quantum field theory is enconded in theLagrangian density.



Reminding QED

Lagrangian density of QED

LQED(x) = ψ(x)(i∂µγµ −m)ψ(x) + eψ(x)γµψ(x)Aµ(x)− 14

Fµν(x)Fµν(x)

el.-mag. field-strenght-tensor: Fµν(x) = ∂µAν(x)− ∂νAµ(x)

� fermion-propagator (electrons, quarks, etc.) /p+mp2−m2

� fermion-photon-vertex i e γµ

� photon-propagator −gµν

p2



Reminding QED

Lagrangian density of QED

LQED(x) = ψ(x)(i∂µγµ −m)ψ(x) + eψ(x)γµψ(x)Aµ(x)− 14

Fµν(x)Fµν(x)

el.-mag. field-strenght-tensor: Fµν(x) = ∂µAν(x)− ∂νAµ(x)

� fermion-propagator (electrons, quarks, etc.) /p+mp2−m2

� fermion-photon-vertex i e γµ

� photon-propagator −gµν

p2



Reminding QED

Calculate tree-level processes:

�e−e+

γ

µ+

µ−

�e−

e−

γ

e+

e+

or higher order corrections like:

�Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 9 / 28

Quantum Chromodynamics (QCD) QCD - The theory

Why colors?

Nucleons build of quarks: p: | u u d > , n: | u d d >

Also e.g. ∆++: | u u u >→ Pauli-principle: no particles with same quantum numbers!

Also experimental evidendence:

σ(e+e− → Hadrons)

σ(e+e− → µ+µ−)∼ Nc

nfXq

Q2q

fits experimental data with Nc = 3

�e−e+

γ

µ+

µ−

←→�e−e+

γ

q

q



Why colors?

Nucleons build of quarks: p: | u u d > , n: | u d d >

Also e.g. ∆++: | u u u >→ Pauli-principle: no particles with same quantum numbers!

Also experimental evidendence:

σ(e+e− → Hadrons)

σ(e+e− → µ+µ−)∼ Nc

nfXq

Q2q

fits experimental data with Nc = 3

�e−e+

γ

µ+

µ−

←→�e−e+

γ

q

q



Quarks and colors

In QED: electron is described by one fermion field ψ(x)

Quark is described by a 3-vector of fermion fields: ψi (x) =

0@ ψR(x)

ψG(x)

ψB(x)

1Awith three „colors“: red, green, blue(antiquarks have anti-red, anti-green, anti-blue)

No color observed in nature:→ Symmetry under SU(3) transformations U (3× 3-matrices with U† U = 1)



Quarks and colors



0@ ψR(x)

ψG(x)

ψB(x)





Quarks and colors



0@ ψR(x)

ψG(x)

ψB(x)





The group SU(3)

SU(3): 3× 3-matrices with U† U = 1

SU(3) has 8 generators λa, so that every U can be written as

U = eiPaθa λa

a = 1,..,8

with eight arbitrary parameters θa.

Generators are not commutative (non-abelian group):

[λa,λb] =Xa,b

f abcλc

with the characteristic structure constant f abc (something like εijk ).



The group SU(3)

SU(3): 3× 3-matrices with U† U = 1SU(3) has 8 generators λa, so that every U can be written as

U = eiPaθa λa

a = 1,..,8



[λa,λb] =Xa,b

f abcλc




The group SU(3)

SU(3): 3× 3-matrices with U† U = 1SU(3) has 8 generators λa, so that every U can be written as

U = eiPaθa λa

a = 1,..,8



[λa,λb] =Xa,b

f abcλc




Gauge theories

QCD is the non-abelian SU(3)-gauge theory:Lagrangian has to be invariant under SU(3)-transformation of the quarks

→ Leads to introduction of eight new gauge fields, the gluons Aaµ(x)

(one for each generator)

Gluons carry colors like red-antigreen, blue-antired, . . .

Compare QED:

U(1)-gauge theory: only 1 generator→ commutes with itself(abelian gauge theory)

→ one gauge field, the photon Aµ(x)

Photon doesn’t carry electric charge



Gauge theories





Compare QED:






Gauge theories





Compare QED:






Gauge theories





Compare QED:






Lagrangian density of QCD

LQCD(x) = ψk (x)(i∂µγµ −m)ψk (x) + gsψi (x)

(λa)ik

2γµψk (x)Aa

µ(x)−14

Gaµν(x)Gaµν(x)

gluon field-strenght-tensor: Gaµν = ∂µAa

ν − ∂νAaµ + gs f abcAb

µAcν

� fermion-propagator (quarks) /p+mp2−m2 δ

ij

� gluon-propagator −gµν

p2 δab

� fermion-gluon-vertex i gs γµ (λa

2 )ij

� gluon-gluon-vertex ∼ gs

� 4-gluon-vertex ∼ g2s

� (ghost-propagator and -vertex)�





(λa)ik

2γµψk (x)Aa

µ(x)−14

Gaµν(x)Gaµν(x)



µAcν


ij


p2 δab


2 )ij








(λa)ik

2γµψk (x)Aa

µ(x)−14

Gaµν(x)Gaµν(x)



µAcν


ij


p2 δab


2 )ij








(λa)ik

2γµψk (x)Aa

µ(x)−14

Gaµν(x)Gaµν(x)



µAcν


ij


p2 δab


2 )ij








(λa)ik

2γµψk (x)Aa

µ(x)−14

Gaµν(x)Gaµν(x)



µAcν


ij


p2 δab


2 )ij





Quantum Chromodynamics (QCD) Fundamental properties of QCD

Renormalization in QED

Main principle of renormalization:

Calculate higher order perturbative contributions like in QED:

�

This can be split up into a finite and an infinite part (Regularization)

→ Finite part gives corrections to physical processes

→ Infinite part is defined into the parameters of the theory,like the coupling constant g = e

This makes the coupling constant dependent of the considered energy scale Q2

in a physical process.−→ Running coupling

QED: αem(Q2 = 0) =e2

4π'

1137

→ αem(m2Z ) '

1128






�This can be split up into a finite and an infinite part (Regularization)






4π'

1137

→ αem(m2Z ) '

1128












4π'

1137

→ αem(m2Z ) '

1128












4π'

1137

→ αem(m2Z ) '

1128












4π'

1137

→ αem(m2Z ) '

1128



Running coupling constant

QED higher order corrections:

�But now in QCD:

� � �

Now completely different behaviour of running coupling in QCD:

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Q2@GeV2D

Αs

LQCD

Confinement

asympt. freedom

αs(Q2) =4π

( 113 Nc − 2

3 nf ) ln( Q2

Λ2QCD

)

with the parameterΛQCD ' 200− 300 MeVexperimentally.



Running coupling constant

QED higher order corrections:

�But now in QCD:

� � �Now completely different behaviour of running coupling in QCD:

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Q2@GeV2D

Αs

LQCD

Confinement

asympt. freedom

αs(Q2) =4π

( 113 Nc − 2

3 nf ) ln( Q2

Λ2QCD

)

with the parameterΛQCD ' 200− 300 MeVexperimentally.


Quantum Chromodynamics (QCD) Hadrons and nuclear force

Hadrons and Use of perturbation theory

At Q2 � Λ2QCD we have small αs and can do perturbative calculations.

Here quarks behave like free particles→ asymptotic freedom.

At small Q2 ∼ Λ2QCD perturbation theory cannot be used.

→ non-perturbative regime

ΛQCD corresponds to length scales of ∼ 1fm = 10−15 m, the scale of e.g. theproton radius.

These energy scale is characteristic for interactions in hadrons,the bound states of quarks. There are to possible types:

Mesons: quark-antiquark bound state (color-anticolor)Baryons: 3-quark bound state (one of each color)

→ seen „white“ from outside

strong force between quarks gets linearly bigger, when they are seperated

quarks cannot be seperated from each other−→ strong interaction forms immedeately new quark-antiquark-pairs, which bind

to hadrons → confinement



























































Perturbative QCD

Perturbative QCD calculations are done in the high-energy regime:

→ Application in hadron-production at high-energy collider experiments

�e−e+

γ

q

q

Used in description of the production of quarks which later become hadronic jets.→ Perturbative QCD & „hadronization“ important for calculation of jet properties



Long-range interaction

The long-range force between quarks is non-perturbative and we have little secureknowledge there.

Acknowledged model: gluons build a flux-tube between the quarks:

This is also supported by numerical lattice calculations.

This leads to a confining, linear potential:

V (r) ' −43αs

r+ k · r experimentally: k ' 0.9

GeVfm

Tubes break up when energy is high enough to build new hadrons

On this (approximative) basis, hadronization into jets can be describedquantitatively






This is also supported by numerical lattice calculations.This leads to a confining, linear potential:

V (r) ' −43αs


GeVfm








This is also supported by numerical lattice calculations.This leads to a confining, linear potential:

V (r) ' −43αs


GeVfm





Yukawa-theory of nuclear interaction

What is now the origin of the binding force between protons and neutrons in a nucleus?

force-carriers are not gluons(nucleons would interchange color→ problems with confinement)

nuclear binding force is intermediated bycolor-neutral (white), virtual pions
















Pions are bosons and have zero spin (scalar particles).

Nuclear interaction by pions can be described as effective theory, by using quantum fieldtheory with a scalar intermediating particle:

One can derive the potential between two nucleons, the Yukawa-potential:

VYuk (r) = −g2 e−mπ r

r

So nuclear interaction is a „remnant“ of the strong QCD-force, that binds quarks to hadrons.(This is somehow analog to the el.-mag. van-der-Waals force between two atoms.)

The potential is localized with a typical range of 1mπ∼ few fm, what explains the nuclear

binding force








r



binding force








r



binding force


Nonperturbative Ascpects

Outlook





Nonperturbative Ascpects Some fundamental aspects


Now let’s consider some nonperturbative aspects of QCD

Non-perturbative regime governs the „long-distance“-physics (radius of hadrons).

Take e.g. proton and neutron:build of three quarks with masses ∼ few MeV , but mp,mn ' 940 MeV→ 99% of the nucleon masses comes from non-perturbative quark-gluon

interactions

→ Properties of hadrons can (still) not be calculated in a fundamental way from thetheory.

But there are some approximate methods.→ following slides







interactions









interactions





Pion decay

Consider the leptonic decay of the pion:

�u

d

W−

νµ

µ−

π−

The matrix element can be factorized into hadronic and leptonic part:

M =GF√

2

˙µ νµ

˛dγµγ5u µγµ(1− γ5)νµ

˛π¸

=GF√

2

˙0˛dγµγ5u

˛π¸·˙µ νµ

˛µγµ(1− γ5)νµ

˛0¸

The hadronic part is not perturbatively calculable and parametrized by the pion decay constant fπ ,the simplest example of a non-perturbative quantity:˙

0˛dγµγ5u

˛π(q)

¸= i fπ qµ



Pion decay


�u

d

W−

νµ

µ−

π−


M =GF√

2

˙µ νµ


˛π¸

=GF√

2

˙0˛dγµγ5u

˛π¸·˙µ νµ


˛0¸


0˛dγµγ5u

˛π(q)

¸= i fπ qµ



Pion decay


�u

d

W−

νµ

µ−

π−


M =GF√

2

˙µ νµ


˛π¸

=GF√

2

˙0˛dγµγ5u

˛π¸·˙µ νµ


˛0¸


0˛dγµγ5u

˛π(q)

¸= i fπ qµ


Nonperturbative Ascpects Short summary of non-perturbative methods

Used methods

Today there are mainly two methods used for non-perturbative calculations:

Lattice-QCD

QCD Sum Rules

Lattice-QCD: The quark and gluon fields are approximated on a discrete space-timelattice. Using the Lagrangian of QCD there can be made numerical calculation ofhadronic parameters and observables.

Needs very much computer power to calculate.(So could only be done since computers became fast enough.)

Good results, but difficult and long work.Reliable error estimates are still a problem.

Intensive work today and in the future.



Used methods


Lattice-QCD

QCD Sum Rules







Used methods


Lattice-QCD

QCD Sum Rules







QCD Sum Rules

My working field: QCD Sum Rules

A longer practiced method (since 1979) than Lattice-QCD.

Analytical calculation of non-perturbative parameters.

Easiest example: Calculation of the B-Meson decay constant fB (analog to fπ)

mb 〈 0 | q i γ5 b |B(q) 〉 = m2B fB



Scetch of the Calculation of fB

Consider the 2-point correlator:˙0˛b(x)γ5q(x) q(0)γ5b(0)

˛0¸

�Can be perturbatively calculated with higher orders in αs and estimates of non-perturbativecontributions

One can put in a full set of intermediate (hadronic) states: 1 =Ph

˛h¸˙

h˛

�h h

∼X

h

˙0˛b(x)γ5q(x)

˛h¸˙

h˛q(0)γ5b(0)

˛0¸ def .

= f 2B + . . .

Compare both expressions and do some mathematics and approximations...→ Sum Rule for fB



Scetch of the Calculation of fB

Consider the 2-point correlator:˙0˛b(x)γ5q(x) q(0)γ5b(0)

˛0¸

�Can be perturbatively calculated with higher orders in αs and estimates of non-perturbativecontributions

One can put in a full set of intermediate (hadronic) states: 1 =Ph

˛h¸˙

h˛

�h h

∼X

h

˙0˛b(x)γ5q(x)

˛h¸˙

h˛q(0)γ5b(0)

˛0¸ def .

= f 2B + . . .

Compare both expressions and do some mathematics and approximations...→ Sum Rule for fB



Summary

Summary:QCD is the fundamental theory of the strong interaction

→ One of the two parts of the Standard Model (beside the Electroweak theory)

Completely other structure like QED, asymptotic freedom and confinement.

Non-perturbative effects make many difficulties.

Nevertheless the theory is extremely successful.

„If the Lord Almighty had consulted me before embarking upon creation,I would have recommended something simpler.“

- King Alphonse X. of Castille and Léon (1221-1284),

on having the Ptolemaic system of epicycles explained to him

Unsolved problems:New quark matter: pentaquarks, glue-balls, quark-gluon-plasma, . . .

Calculations in non-perturbative range

Confinement in QCD still couldn’t be strictly proved

→ Millenium-problem: Win 1 Mio. $ !



Summary















Summary















Summary














A short overview on strong interaction and quantum ..._Uni... · A short overview on strong interaction and quantum chromodynamics Christoph Klein Universität Siegen Doktorandenseminar

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