qcd and hadronic physics - uni- · pdf filematter/antimatter asymmetry (cp violation), etc....

QCD and hadronic physics

Christoph Klein

University Siegen

Ph.D. seminar07.12.2010

Christoph Klein (University Siegen) QCD and hadronic physics Ph.D. seminar 07.12.2010 1 / 40

Outlook

1 Introduction: Particle physics and the standard model

2 Quantum Chromodynamics (QCD) and the theory of strong interactionHistory of strong interaction physicsQCD - The theory

3 Hadrons and nuclear forceHadrons vs. quarksThe nuclear force

4 Further aspects of QCDNonperturbative MethodsQuark matter and all that . . .

Introduction: Particle physics and the standard model

Outlook

Standard model of Particle physics

Particle physics: What are thefundamental building blocks of natureand how do they interact?

Giant colliders and detectors are builtto answer this question.

Today we know already very much...All matter is made of a few types ofparticles, which are fundamental to theprecision of 10−18 m.

They and their interactions aredescribed by the standard model ofparticle physics.

The fundamental building blocks ofmatter are fermions (quarks andleptons), which appear in threegenerations of different mass.

Three of the fundamental forces ofnature are described by exchangeparticles:

electromagnetic force (γ)weak force (W ,Z )strong (nuclear) force (g)→ subject of this talk.

Our everyday world is just made ofe,u,d ,γ.

Many questions unanswered:mass generation (Higgs,...),mass hierarchy, neutrino masses,matter/antimatter asymmetry(CP violation), etc.

But still the standard model is aremarkable milestone.

Particle physics is a very important interplay between:

experiment ...

... and theory

Particle physics is a very important interplay between:

experiment ... ... and theory

Theoretical particle physics

Goal of theoretical particle physics is to describe particle interactions in a fundamentaltheory and make predictions for experiments.

Particle interactions are discribed by quantum field theories (or more specific: gaugetheories).

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

Simplest example: Quantum electrodynamics (QED)

QED is the fundamental quantum field theory of the electromagnetic interaction.

U(1) gauge theory: symmetric under phase changes of the fermion field.

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

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)

Theoretical particle physics

Goal of theoretical particle physics is to describe particle interactions in a fundamentaltheory and make predictions for experiments.

Particle interactions are discribed by quantum field theories (or more specific: gaugetheories).

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

Simplest example: Quantum electrodynamics (QED)

QED is the fundamental quantum field theory of the electromagnetic interaction.

U(1) gauge theory: symmetric under phase changes of the fermion field.

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

Fµν(x)Fµν(x)

Reminding QED

Lagrangian density of QED

Fµν(x)Fµν(x)

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

� fermion-photon-vertex i e γµ

� photon-propagator −gµν

Reminding QED

Lagrangian density of QED

Fµν(x)Fµν(x)

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

� fermion-photon-vertex i e γµ

� photon-propagator −gµν

Use perturbation theory ...Calculate tree-level processes:

�e−e+

�e−

or higher order corrections like:

... and compare with experiment:

Use perturbation theory ...Calculate tree-level processes:

�e−e+

�e−

or higher order corrections like:

... and compare with experiment:

QCD and the standard model

The standard model: a gauge theory with symmetriesU(1)charge ⊗ SU(2)weak ⊗ SU(3)colour

There is a force between the quarks, described by the SU(3) theory QuantumChromodynamics (QCD).

Problem: the strong force always binds quarks to bound states, called hadrons.

Mesons: π+ = ud , π0 = 1√2

(uu − dd) , K + = us , D0 = cu , · · ·Baryons: p = uud , n = udd ,p = uud , ∆++ = uuu , Λ = uds , · · ·

→ Quarks are unmeasurable and only hadrons can be detected in experiment.

→ It is necessesary to “translate” between the dynamics of quarks and hadrons.This is very difficult and complicated.

→ Main task of a QCD phenomenology theorist.

Mesons: π+ = ud , π0 = 1√2

an electroweak process with myons ... ... and an event with hadrons (jets)

Hadronic interactions mayeven look like this...

On a certain time scaleafter the production ofquarks, they form anumber of hadrons(jets,...).→ Hadronization

a heavy ion collision at ALICE

Hadronic interactions mayeven look like this...

On a certain time scaleafter the production ofquarks, they form anumber of hadrons(jets,...).→ Hadronization

a heavy ion collision at ALICE

Quantum Chromodynamics (QCD) and the theory of strong interaction

Outlook

Quantum Chromodynamics (QCD) and the theory of strong interaction 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

from more and more hadrons are discovered1950 →seemed not to be all fundamental

“Had I foreseen that, I would have gone into botany” - W.Pauli

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) and the theory of strong interaction QCD - The theory

Outlook

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

nf∑q

fits experimental data with Nc = 3

�e−e+

←→�e−e+

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

nf∑q

fits experimental data with Nc = 3

�e−e+

←→�e−e+

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) =

ψR(x)ψG(x)ψB(x)

with 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

ψR(x)ψG(x)ψB(x)

Quarks and colors

ψR(x)ψG(x)ψB(x)

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 color charge themselves like red-antigreen, blue-antired, . . .→ non-linear field theory

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-strength-tensor: Gaµν = ∂µAa

ν − ∂νAaµ + gs f abcAb

µAcν

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

� gluon-propagator −gµν

p2 δab

� fermion-gluon-vertex i gs γµ (λa

� 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

p2 δab

(λa)ik

2γµψk (x)Aa

µ(x)−14

Gaµν(x)Gaµν(x)

µAcν

p2 δab

(λa)ik

2γµψk (x)Aa

µ(x)−14

Gaµν(x)Gaµν(x)

µAcν

p2 δab

(λa)ik

2γµψk (x)Aa

µ(x)−14

Gaµν(x)Gaµν(x)

µAcν

p2 δab

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

→ αem(m2Z ) '

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

→ αem(m2Z ) '

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

Q2@GeV2D

Confinement

asympt. freedom

αs(Q2) =4π

( 113 Nc − 2

3 nf ) ln( Q2

Λ2QCD

with the parameterΛQCD ' 200− 300 MeV experimentally.

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

Q2@GeV2D

Confinement

asympt. freedom

αs(Q2) =4π

( 113 Nc − 2

3 nf ) ln( Q2

Λ2QCD

with the parameterΛQCD ' 200− 300 MeV experimentally.

Hadrons and nuclear force

Outlook

Hadrons and nuclear force Hadrons vs. quarks

Outlook

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. the proton 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+

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 secure knowledgethere.

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

This leads to a confining, linear potential:

V (r) ' −43αs

r+ k · r experimentally: k ' 0.9

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

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

V (r) ' −43αs

Hadrons and nuclear force The nuclear force

Outlook

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

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

Why is there no nucleus consisting of pp or nn, but just e.g. d = pn ?

→ pp or nn only can exchange neutral pions.

→ pn can exchange also charged pions.

→ leads to a stronger binding, but it just suffices to bind the d system.

Nuclear force is short-ranged and only occurs between nearest neighbors.

Interplay between strong and coulomb interaction.

→ This mainly determines the properties of nuclear phyics.

Bethe-Weizsaecker mass formula, drop model, shell model,...

Further aspects of QCD

Outlook

Further aspects of QCD Nonperturbative Methods

Outlook

Nonperturbative Methods

Non-perturbative regime governs the „long-distance“-physics (radius of hadrons) at scales� Λ−1

QCD ∼ 1 fm.

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 the theory.But there exist approximative methods:

QCD sum rules (my working field)Lattice QCDHeavy quark effective theory (HQET)...

Lattice-QCD: The quark and gluon fields are approximated on a discrete space-time lattice. Usingthe Lagrangian of QCD there can be made numerical calculation of hadronic parameters andobservables.

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

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

Intensive work today and in the future.

QCD ∼ 1 fm.

QCD sum rules, invented in 1979, provide ananalytical tool, combinig perturbativecalculations and universal nonperturbativequantities of QCD.

In Siegen: use in flavour physics.

Needed to translate fundamental quarktransitions to hadrons.

Some applications:

B → D meson decay �b

D(∗)

W (q)ℓ

cross section for pp → Λc Λc �u

Some applications:

D(∗)

W (q)ℓ

Some applications:

D(∗)

W (q)ℓ

Further aspects of QCD Quark matter and all that . . .

Outlook

Quark matter

Aside of the normal nuclear matter, there exist a few mostly hypothetical forms of matter tostudy QCD.

For example new bound states, different from the known hadrons:

Bound states of gluons (“glueballs”)5-quark bound states (“tetraquarks”)qq − qq-molecules (“exotic resonances”)...

Other example: Study nuclear matterunder very high density and temperature

→ different phases of matter→ QCD phasediagram

Such matter could exist e.g. in neutronstars (“quark stars”)

Quark matter

Quark-gluon plasma

Important is the quark-gluonplasma, where temperatureand density are so high, thatquarks are no more bound bygluons.

Possible to create and study inheavy-ion collisions. Around10.000 particles produced!

The universe was in this stateshortly after the big bang. Itwas the last phase transition tooccur.→ connection to astrophysics.

In such phases there can occurphenomena from statistical andsolid state physics.

Color superconductivity(quarks build cooper pairs).

Superfluidity

Quark-gluon plasma

Important is the quark-gluonplasma, where temperatureand density are so high, thatquarks are no more bound bygluons.

Possible to create and study inheavy-ion collisions. Around10.000 particles produced!

The universe was in this stateshortly after the big bang. Itwas the last phase transition tooccur.→ connection to astrophysics.

In such phases there can occurphenomena from statistical andsolid state physics.

Color superconductivity(quarks build cooper pairs).

Superfluidity

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.

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

Calculations in non-perturbative range, lattice QCD, . . .

Confinement in QCD still couldn’t be strictly proved.

→ Millenium-problem: Win 1 Mio. $ !

→ QCD still holds many things to do.

Summary

Pion decayConsider the leptonic decay of the pion:

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

M =GF√

⟨µ νµ

∣∣dγµγ5u µγµ(1− γ5)νµ∣∣π ⟩

=GF√

⟨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

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 =∑h

∣∣ h ⟩⟨ h∣∣

�h h

∼∑

⟨0∣∣b(x)γ5q(x)

∣∣ h ⟩⟨ h∣∣ q(0)γ5b(0)

∣∣ 0 ⟩ def .= f 2

B + . . .

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

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 = ei∑aθa λa

a = 1,..,8

with eight arbitrary parameters θa.

Generators are not commutative (non-abelian group):

[λa,λb] =∑a,b

f abcλc

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

qcd and hadronic physics - uni- · pdf filematter/antimatter asymmetry (cp violation), etc....

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