A short overview on strong interaction and quantum chromodynamics

A short overview on strong interaction and quantum chromodynamics Christoph Klein Universität Siegen Doktorandenseminar 08.10.2008 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 QED QCD - The theory Fundamental properties of QCD Hadrons and nuclear force 3 Nonperturbative Ascpects Some fundamental aspects Short summary of non-perturbative methods Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 2 / 28

History of strong interaction physics Outlook 1 History of strong interaction physics 2 Quantum Chromodynamics (QCD) Review of QED QCD - The theory Fundamental properties of QCD Hadrons and nuclear force 3 Nonperturbative Ascpects Some fundamental aspects Short summary of non-perturbative methods Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 3 / 28

History of strong interaction physics History of strong interaction physics - some milestones 1911 Rutherford discovers atomic nucleus Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 4 / 28

History of strong interaction physics History of strong interaction physics - some milestones 1911 Rutherford discovers atomic nucleus 1919 Rutherford discovers the proton as elementary constituent of the nucleus Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 4 / 28

History of strong interaction physics 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 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 4 / 28

History of strong interaction physics 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 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 4 / 28

History of strong interaction physics 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 strong nuclear interaction 1947 Lattes discovers the charged pion in cosmic rays Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 4 / 28

History of strong interaction physics History of strong interaction physics - some milestones 1954 Yang and Mills introduce non-abelian gauge-theories Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 5 / 28

History of strong interaction physics History of strong interaction physics - some milestones 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 of asymptotically 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 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 5 / 28

History of strong interaction physics History of strong interaction physics - some milestones 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 of asymptotically 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 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 5 / 28

History of strong interaction physics History of strong interaction physics - some milestones 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 of asymptotically 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) Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 5 / 28

Review of QED Reminding QED Quantum electrodynamics (QED) Remember: QED is the fundamental quantum field theory of the electromagnetic interaction. Describes interaction of charged fermions ψ(x) (electrons, myons, quarks,...) mediated by the photon A µ (x). The fundamental structure of a quantum field theory is enconded in the Lagrangian density. Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 7 / 28

Review of QED Reminding QED Lagrangian density of QED L QED (x) = ψ(x)(i µγ µ m)ψ(x) + e ψ(x)γ µ ψ(x)a µ(x) 1 4 Fµν(x)F µν (x) el.-mag. field-strenght-tensor: F µν(x) = µa ν(x) νa µ(x) Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 8 / 28

Review of QED Reminding QED Lagrangian density of QED L QED (x) = ψ(x)(i µγ µ m)ψ(x) + e ψ(x)γ µ ψ(x)a µ(x) 1 4 Fµν(x)F µν (x) el.-mag. field-strenght-tensor: F µν(x) = µa ν(x) νa µ(x) fermion-propagator (electrons, quarks, etc.) fermion-photon-vertex i e γ µ photon-propagator g µν p 2 /p+m p 2 m 2 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 8 / 28

Review of QED Reminding QED Calculate tree-level processes: e + µ e e γ γ e µ + e + e + or higher order corrections like: Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 9 / 28

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! Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 10 / 28

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: fits experimental data with N c = 3 σ(e + e Hadrons) σ(e + e µ + µ ) n X f N c q Q 2 q e + µ e + q γ γ e µ + e q Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 10 / 28

QCD - The theory Quarks and colors In QED: electron is described by one fermion field ψ(x) Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 11 / 28

QCD - The theory Quarks and colors In QED: electron is described by one fermion field ψ(x) Quark is described by a 3-vector of fermion fields: ψ i (x) = @ with three colors : red, green, blue (antiquarks have anti-red, anti-green, anti-blue) 0 ψ R (x) ψ G (x) ψ B (x) 1 A Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 11 / 28

QCD - The theory Quarks and colors In QED: electron is described by one fermion field ψ(x) Quark is described by a 3-vector of fermion fields: ψ i (x) = @ with three colors : red, green, blue (antiquarks have anti-red, anti-green, anti-blue) 0 ψ R (x) ψ G (x) ψ B (x) No color observed in nature: Symmetry under SU(3) transformations U (3 3-matrices with U U = 1) 1 A Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 11 / 28

QCD - The theory The group SU(3) SU(3): 3 3-matrices with U U = 1 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 12 / 28

QCD - The theory The group SU(3) SU(3): 3 3-matrices with U U = 1 SU(3) has 8 generators λ a, so that every U can be written as U = e i P θ a λ a a a = 1,..,8 with eight arbitrary parameters θ a. Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 12 / 28

QCD - The theory The group SU(3) SU(3): 3 3-matrices with U U = 1 SU(3) has 8 generators λ a, so that every U can be written as U = e i P θ a λ a a a = 1,..,8 with eight arbitrary parameters θ a. Generators are not commutative (non-abelian group): [λ a,λ b ] = X f abc λ c a,b with the characteristic structure constant f abc (something like ɛ ijk ). Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 12 / 28

QCD - The theory Gauge theories QCD is the non-abelian SU(3)-gauge theory: Lagrangian has to be invariant under SU(3)-transformation of the quarks Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 13 / 28

QCD - The theory 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 A a µ(x) (one for each generator) Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 13 / 28

QCD - The theory 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 A a µ(x) (one for each generator) Gluons carry colors like red-antigreen, blue-antired,... Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 13 / 28

QCD - The theory 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 A a µ(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 A µ(x) Photon doesn t carry electric charge Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 13 / 28

QCD - The theory Lagrangian density of QCD L QCD (x) = ψ k (x)(i µγ µ m)ψ k (x) + g ψi s (x) (λa ) ik γ µ ψ k (x)a a 2 µ(x) 1 4 Ga µν(x)g a µν (x) gluon field-strenght-tensor: G a µν = µa a ν νa a µ + g sf abc A b µa c ν fermion-propagator (quarks) gluon-propagator fermion-gluon-vertex /p+m p 2 m 2 δij g µν δ ab p 2 i g s γ µ ( λa 2 )ij Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 14 / 28

QCD - The theory Lagrangian density of QCD L QCD (x) = ψ k (x)(i µγ µ m)ψ k (x) + g ψi s (x) (λa ) ik γ µ ψ k (x)a a 2 µ(x) 1 4 Ga µν(x)g a µν (x) gluon field-strenght-tensor: G a µν = µa a ν νa a µ + g sf abc A b µa c ν /p+m p 2 m 2 δij fermion-propagator (quarks) gluon-propagator g µν δ ab p 2 fermion-gluon-vertex i g s γ µ ( λa 2 )ij gluon-gluon-vertex g s Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 14 / 28

QCD - The theory Lagrangian density of QCD L QCD (x) = ψ k (x)(i µγ µ m)ψ k (x) + g ψi s (x) (λa ) ik γ µ ψ k (x)a a 2 µ(x) 1 4 Ga µν(x)g a µν (x) gluon field-strenght-tensor: G a µν = µa a ν νa a µ + g sf abc A b µa c ν /p+m p 2 m 2 δij fermion-propagator (quarks) gluon-propagator g µν δ ab p 2 fermion-gluon-vertex i g s γ µ ( λa 2 )ij gluon-gluon-vertex g s 4-gluon-vertex gs 2 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 14 / 28

QCD - The theory Lagrangian density of QCD L QCD (x) = ψ k (x)(i µγ µ m)ψ k (x) + g ψi s (x) (λa ) ik γ µ ψ k (x)a a 2 µ(x) 1 4 Ga µν(x)g a µν (x) gluon field-strenght-tensor: G a µν = µa a ν νa a µ + g sf abc A b µa c ν /p+m p 2 m 2 δij fermion-propagator (quarks) gluon-propagator g µν δ ab p 2 fermion-gluon-vertex i g s γ µ ( λa 2 )ij gluon-gluon-vertex g s 4-gluon-vertex gs 2 (ghost-propagator and -vertex) Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 14 / 28

Fundamental properties of QCD Renormalization in QED Main principle of renormalization: Calculate higher order perturbative contributions like in QED: Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 15 / 28

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) Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 15 / 28

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 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 15 / 28

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 Q 2 in a physical process. Running coupling QED: α em(q 2 = 0) = e2 4π 1 137 α em(m 2 Z ) 1 128 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 15 / 28

Fundamental properties of QCD Running coupling constant QED higher order corrections: But now in QCD: Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 16 / 28

Fundamental properties of QCD Running coupling constant QED higher order corrections: But now in QCD: Now completely different behaviour of running coupling in QCD: Αs Confinement 0.8 0.6 0.4 asympt. freedom 0.2 QCD 0.0 0 2 4 6 8 α s(q 2 ) = 4π ( 11 3 Nc 2 n 3 f ) ln( Q2 ) Λ 2 QCD with the parameter Λ QCD 200 300 MeV experimentally. Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 16 / 28

Hadrons and nuclear force Hadrons and Use of perturbation theory At Q 2 Λ 2 QCD we have small α s and can do perturbative calculations. Here quarks behave like free particles asymptotic freedom. At small Q 2 Λ 2 QCD perturbation theory cannot be used. non-perturbative regime Λ QCD 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 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 17 / 28

Hadrons and nuclear force Hadrons and Use of perturbation theory At Q 2 Λ 2 QCD we have small α s and can do perturbative calculations. Here quarks behave like free particles asymptotic freedom. At small Q 2 Λ 2 QCD perturbation theory cannot be used. non-perturbative regime Λ QCD 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 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 17 / 28

Hadrons and nuclear force Perturbative QCD Perturbative QCD calculations are done in the high-energy regime: Application in hadron-production at high-energy collider experiments e + q e γ Used in description of the production of quarks which later become hadronic jets. Perturbative QCD & hadronization important for calculation of jet properties q Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 18 / 28

Hadrons and nuclear force Long-range interaction The long-range force between quarks is non-perturbative and we have little secure knowledge 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) 4 α s + k r experimentally: k 0.9 GeV 3 r fm Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 19 / 28

Hadrons and nuclear force Long-range interaction The long-range force between quarks is non-perturbative and we have little secure knowledge 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) 4 α s + k r experimentally: k 0.9 GeV 3 r fm Tubes break up when energy is high enough to build new hadrons On this (approximative) basis, hadronization into jets can be described quantitatively Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 19 / 28

Hadrons and nuclear force Yukawa-theory of nuclear interaction What is now the origin of the binding force between protons and neutrons in a nucleus? Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 20 / 28

Hadrons and nuclear force 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) Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 20 / 28

Hadrons and nuclear force 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 by color-neutral (white), virtual pions Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 20 / 28

Hadrons and nuclear force Yukawa-theory of nuclear interaction Pions are bosons and have zero spin (scalar particles). Nuclear interaction by pions can be described as effective theory, by using quantum field theory with a scalar intermediating particle: One can derive the potential between two nucleons, the Yukawa-potential: V Yuk (r) = g 2 e mπr 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.) 1 The potential is localized with a typical range of few fm, what explains the nuclear m π binding force Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 21 / 28

Nonperturbative Ascpects Outlook 1 History of strong interaction physics 2 Quantum Chromodynamics (QCD) Review of QED QCD - The theory Fundamental properties of QCD Hadrons and nuclear force 3 Nonperturbative Ascpects Some fundamental aspects Short summary of non-perturbative methods Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 22 / 28

Nonperturbative Ascpects Some fundamental aspects Nonperturbative Ascpects Now let s consider some nonperturbative aspects of QCD Non-perturbative regime governs the long-distance -physics (radius of hadrons). Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 23 / 28

Nonperturbative Ascpects Some fundamental aspects Nonperturbative Ascpects 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 m p, m n 940 MeV 99% of the nucleon masses comes from non-perturbative quark-gluon interactions Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 23 / 28

Nonperturbative Ascpects Some fundamental aspects Nonperturbative Ascpects 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 m p, m n 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 are some approximate methods. following slides Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 23 / 28

Nonperturbative Ascpects Some fundamental aspects Pion decay Consider the leptonic decay of the pion: ū π d W ν µ µ Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 24 / 28

Nonperturbative Ascpects Some fundamental aspects Pion decay Consider the leptonic decay of the pion: ū π d W ν µ µ The matrix element can be factorized into hadronic and leptonic part: M = G F 2 µ νµ dγµγ 5 u µγ µ (1 γ 5 )ν µ π = G F 2 0 dγµγ 5 u π µ νµ µγ µ (1 γ 5 )ν µ 0 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 24 / 28

Nonperturbative Ascpects Some fundamental aspects Pion decay Consider the leptonic decay of the pion: ū π d W ν µ µ The matrix element can be factorized into hadronic and leptonic part: M = G F 2 µ νµ dγµγ 5 u µγ µ (1 γ 5 )ν µ π = G F 2 0 dγµγ 5 u π µ νµ µγ µ (1 γ 5 )ν µ 0 The hadronic part is not perturbatively calculable and parametrized by the pion decay constant f π, the simplest example of a non-perturbative quantity: 0 dγµγ 5 u π(q) = i f π q µ Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 24 / 28

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-time lattice. Using the Lagrangian of QCD there can be made numerical calculation of hadronic 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. Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 25 / 28

Nonperturbative Ascpects Short summary of non-perturbative methods 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 f B (analog to f π) m b 0 q i γ 5 b B(q) = m 2 B f B Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 26 / 28

Nonperturbative Ascpects Short summary of non-perturbative methods Scetch of the Calculation of f B Consider the 2-point correlator: 0 b(x)γ 5 q(x) q(0)γ 5 b(0) 0 Can be perturbatively calculated with higher orders in α s and estimates of non-perturbative contributions One can put in a full set of intermediate (hadronic) states: 1 = P h h h X h h h 0 b(x)γ5 q(x) h h def. q(0)γ5 b(0) 0 = fb 2 +... Compare both expressions and do some mathematics and approximations... Sum Rule for f B Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 27 / 28

Nonperturbative Ascpects Short summary of non-perturbative methods 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 Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 28 / 28

Nonperturbative Ascpects Short summary of non-perturbative methods 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: 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 New quark matter: pentaquarks, glue-balls, quark-gluon-plasma,... Calculations in non-perturbative range Confinement in QCD still couldn t be strictly proved Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 28 / 28

Nonperturbative Ascpects Short summary of non-perturbative methods 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: 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 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. $! Christoph Klein (Universität Siegen) QCD and strong interaction Doktorandenseminar 08.10.2008 28 / 28