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Basics of Quantum Chromodynamics (Two Lectures)

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Basics of Quantum Chromodynamics (Two Lectures) Basics of Quantum Chromodynamics (Two lectures) Faisal Akram 5th School on LHC Physics, 15-26 August 2016 NCP Islamabad 2015 Outlines • A brief introduction of QCD Classical QCD Lagrangian Quantization Green functions of QCD and SDE’s • Perturbative QCD Perturbative calculation of QCD Green functions Feynman Rules of QCD Renormalization Running of QCD coupling (Asymptotic freedom) • Non-Perturbative QCD Confinement QCD phase transition Dynamical breaking of chiral symmetry Elementary Particle Physics Today Elementary particles: Quarks Leptons Gauge Bosons Higgs Boson (can interact through (cannot interact through (mediate interactions) (impart mass to the elementary strong interaction) strong interaction) particles) 푢 푐 푡 Meson: 푠 = 0,1,2 … Quarks: Hadrons: 1 3 푑 푠 푏 Baryons: 푠 = , , … 2 2 푣푒 푣휇 푣휏 Meson: Baryon Leptons: 푒 휇 휏 Gauge Bosons: 훾, 푊±, 푍0, and 8 gluons 399 Mesons, 574 Baryons have been discovered Higgs Bosons: 퐻 (God/Mother particle) • Strong interaction (Quantum Chromodynamics) • Electro-weak interaction (Quantum electro-flavor dynamics) The Standard Model Explaining the properties of the Hadrons in terms of QCD’s fundamental degrees of freedom is the Problem laying at the forefront of Hadronic physics. How these elementary particles and the SM is discovered? 1. Scattering Experiments Cross sections, Decay Constants, 2. Decay Processes Masses, Spin , Couplings, Form factors, etc. 3. Study of bound states Models Particle Accelerators Models and detectors Experimental Particle Physics Theoretical Particle Particle Physicist Phenomenologist Physicist Measured values of Calculated values of physical observables ≈ physical observables It doesn’t matter how beautifull your theory is, It is more important to have beauty It doesn’t matter how smart you are. If it doesn’t in one's equations than to have agree with experiment, it’s wrong… R.P. Feynmann them fit experiment..... P.A.M Dirac Decades of observation and calculations show that the Standard Model of particle physics can describe almost every thing which we have observed in the labs of high energy physics. A Quick review of QFT’s • The standard model is a quantum field theory. • In field theories, fields (defined by field functions) act as fundamental dof of the system. • To every different kind of a particle we associate a field. (e.g., electron field, proton field, photon field etc) • Particles appear as quanta of field. Quantization Quanta of field has particle properties. 휙(푥) • Equation of motion of the fields (푠 = 0) (푠 = 1) (푠 = 1) Free Scalar Field: Free abelian Vector Field: Fee Non-abelian Vector Field: 휇 휈 휈 휇 휇 2 휕 휕 퐴 + 휕 (휕 퐴 ) = 0 1 휇휈 (휕휇휕 + 푚 )휙(푥) = 0 휇 휇 ℒ = − 퐹 퐹 1 0 휇휈푎 푎 ℒ = 휕 휙†휕휇휙 − 푚2휙†휙 휇휈 4 0 휇 ℒ0 = − 퐹휇휈퐹 휇휈 휇 휈 휈 휇 휇 휈 4 퐹푎 = 휕 퐴푎 − 휕 퐴푎 + 푔0푓푎푏푐퐴푏퐴푐 (푠 = 1/2) 퐹휇휈 = 휕휇퐴휈 − 휕휈퐴휇 Free Dirac Field: 휇 푖훾 휕휇 − 푚 휓(푥) = 0 휇 ℒ0 = 푖휓 훾 휕휇휓 − 푚휓 휓 A Quick review of QFT’s • Interaction is introduced by the coupling of fields . 휇 For example: ℒ퐼 = 푒휓 훾 휓퐴휇 for QED 휓 훾 휓 • Only Lorentz invariant couplings are allowed. 휇 휇 5 5 휇 ℒ퐼 = 휓 훾 휓퐴휇; 휓 훾 훾 휓퐴휇; 휓 휓휙; 휓 훾 휓휙; 휓 훾 휓휓 훾휇휓; . • In gauge theories coupling are further constrained by the gauge symmetries. i) 푆푈 2 퐿 ⊗ 푈(1)푌 for electroweak interaction. ii) 푆푈(3)푐 for strong interaction. Quantum Chromo Dynamics (QCD) (Foundations) • QCD is the theory of strong interaction of quarks, which is based on SU(3)c color symmetry. • It assumes each flavor of quark comes in three different colors. • The color states of quarks are SU(3) triplets. 푓 = 1,2,3 … 푁 푞 푞 푓 푓 푁 = 6 푞 Quark Fields: 휓퐶 퐶 = 푅, 퐺, 퐵 푓 푅 퐺 퐵 Free Lagrangian of the Quarks: 푁푓 푓 푓 푓 휇 ℒ0 = 휓퐶 푖훾 휕휇 − 푚0 휓퐶 푓=1 퐶=푅,퐺,퐵 The color states of quarks are SU(3)c triplets. 푁 휓푓 푓 푅 푓 푓 휇 푓 푓 푓 ℒ0 = 휓 푖훾 휕휇 − 푚0 휓 휓 = 휓퐺 푓 f iata f 푓=1 휓퐵 e This Lagrangian is invariant under Where 푡푎 are Gell-Mann matrices. global SU(3)c gauge transformation. [ta ,tb ] ifabctc Free and Classical QCD Lagrangian Quantum Chromo Dynamics (QCD) (Foundations) • Extending the global gauge symmetry to local symmetry. Local gauge transformation: f f f ia (x)ta f f U e U Global gauge transformation f f f U i( a )taU Local gauge transformation Free Lagrangian does not possess local symmetry: D ig t A (x) 푓 0 a a 푓 휇 푓 푓 푓 ℒ0 = 휓 푖훾 휕휇휓 + 푚0 휓 휓 f f Such that D U D 1 i 1 ta Aa Uta AaU U U g0 푓 푓 휇 푓 푓 푓 ℒ = 휓 푖훾 퐷휇휓 + 푚0 휓 휓 푓 푓 휇 푓 푓 푓 푓 휇 푓 ℒ = 휓 푖훾 휕휇휓 + 푚0 휓 휓 + 푔0휓 훾 푡푎휓 퐴푎휇 [ta ,tb ] ifabctc Including kinetic energy term of gauge fields 1 ℒ = 휓 푓푖훾휇휕 휓푓 + 푚푓휓 푓휓푓 + 푔 휓 푓훾휇푡 휓푓퐴 − 퐹 퐹휇휈 휇 0 0 푎 푎휇 4 푎휇휈 푎 휇휈 휇 휈 휈 휇 휇 휈 where, 퐹푎 = 휕 퐴푎 − 휕 퐴푎 + 푔0푓푎푏푐퐴푏퐴푐 , (푎 = 1, 2, . , 8) 26 fields = (6 flavors x 3 Color)+(8 gauge bosons); 7 parameter = 6 masses + 1 coupling This complete Lagrangian is invariant under local SU(3)c gauge transformation. Quantum Chromo Dynamics (QCD) (Foundations) Interacting Classical QCD Lagrangian: 1 ℒ = 휓 푓푖훾휇휕 휓푓 + 푚푓휓 푓휓푓 + 푔 휓 푓훾휇푡 휓푓퐴 − 퐹 퐹휇휈 휇 0 0 푎 푎휇 4 휇휈푎 푎 휇휈 휇 휈 휈 휇 휇 휈 where, 퐹푎 = 휕 퐴푎 − 휕 퐴푎 + 푔0푓푎푏푐퐴푏퐴푐 , (푎 = 1, 2, . , 8) 1 1 1 F F ( A A )( A A ) g f ( A A )A A 4 a a 4 a a a a 2 0 abc a a b c g 2 0 f f A A A A . 4 abc ab'c' b c b' c' • Quark-gluon interaction • 3-point gluon interaction • 4-point gluon interaction Quantum Chromo Dynamics (QCD) (Quantization) Green Functions of QFT 퐺 푥1, 푥2, … , 푥푛 ≡ Ω 푇 휙 푥1 휙 푥2 … 휙 푥푛 Ω 1 −푖훿 −푖훿 −푖훿 퐺 푥1, 푥2, … , 푥푛 = . 푍(퐽) 퐽=0 Cross sections 푍(0) 훿퐽(푥1) 훿퐽(푥2) 훿퐽 푥푛 Decay constants 푇 Bound State masses 푖 푑푡 ℒ+퐽(푥)휙(푥) Couplings Generating function: 푍(퐽) = lim 푑휙 푒 −푇 푡→∞(1−푖휖) Form factors Generating function of QCD: i dx4 (L f f f f J A ) Z( ,, J ) d[A ]e QCD a a The path integration over gauge fields diverges due to integrating over gauge equivalent configurations. a a 1 ab b A (x) A (x) D (x) g0 Fixing the gauge means choosing one configuration out of gauge equivalent configurations. Quantum Chromo Dynamics (QCD) (Quantization) Gauge fixing implies i dx4 (L L L f f f A J ) Z( ,, J , , ) d[A ]e QCD GF FPG a a a a a a 1 2 a a abc a b c Not gauge invariant due LGF ( Aa ) ; LFPG ( ) g0 f ( ) A to gauge fixing. 20 The generating function Z must be gauge invariant because gauge transformation amounts to redefine the variables integration. This trivial requirement of Gauge invariance of the generating function translates into non-trivial identities which all Green functions carrying gauge field(s) must satisfy. QED: QCD: Wards-Green-Takahashi identities (WGTI) Slavnov Taylor identities (STI) 2 2 1 q 1 q q D q q D q 0 0 2 1 1 1 iq (k, p)(1 b(k )) (1 B(k, p))S (k) iq (k, p) S (k) S ( p) S 1( p)(1 B(k, p)) Green Functions of QCD 2-point Green functions: Ω 푇 휓(푥1), 휓 (푥2) Ω ≡ 푎 푏 Ω 푇 (퐴휇(푥1), 퐴휐(푥2) Ω ≡ 푎 푏 Ω 푇 (푐 (푥1), 푐 (푥2) Ω ≡ 3-Point Green functions: 4-Point Green functions: Schwinger-Dyson Equations (SDEs) of QCD • In QFT the Green functions satisfy a set of exact mathematical equations called SDEs. • Corresponding to each green function we have a SDE. • These are exact equations and can be used to find Green functions perturbatively as well as non-perturbatively. Schwinger-Dyson Equations of QCD SDE of Quark Propagator: Non Perturbative Truncation: One or more Green functions are modeled subjecting to some general field theoretical constraints. Or SDE of Gluon Propagator: Using the results obtained from lQCD. + 5 terms SDE of quark-gluon vertex: + 7 terms Perturbative Truncation: 푁푓 Feynman Rules of QCD QCD full Lagrangian: f f f f f f f 1 1 a 2 LB i m g0 ta Aa (x) Fa Fa ( A ) 4 20 a a abc a b c 0 0 g0 f 0 0 A0 Where, Vertices: 1 1 1 F F ( A A )( A A ) g f ( A A )A A 4 a a 4 a a a a 2 0 abc a a b c g 2 0 f f A A A A . 4 abc ab'c' b c b' c' ig t a g f abc[( p q) g (q k) g cab ig 2{ f abe f cde (g g g g ) 0 ij 0 g0 f p 0 abe cde (k p) g ] f f (g g g g ) f abe f cde (g g g g )} Propgators: 1 Free Quark propagator : S (k) ij ij k m i 1 q q q q Free gluon propagator : Dab (q) g 0 ab 2 2 2 2 q i q q i q 1 Free ghost propagator : G (k) ab ab k 2 i Color Algebra: 2 a a Nc 1 4 tiktkj CFij , CF 2Nc 3 1 Tr(t at b ) T ab , T F F 2 acd bcd ab f f CA , CA Nc 3 Leading order QCD green functions Quark propagator: dk 4 1 ∞ 푘3푑푘 S 1( p) ( p m ) iC g 2 D (k) ∝ (Linearly divergent) 0 F 4 0 0 0 푘3 (2 ) p k m0 i0 (푘 → ∞) Gluon propagator: 1 1 (quark) (gluon) (ghost) D (q) D0 (q) dk D 4D 1 1 quark iT g 2Tr( ), F D 0 f f f (2 ) k m0 q k m0 4 C dk (2k q) g (k q) g (2q k) g gluon i A g 2 D (k)D (q k) 2 (2 )4 0 0 0 (2k q) g (k q) g (2q k) g dk 4 k (k q) ghost iC g 2 A (2 )4 0 k 2 (k q)2 ∞ ∝ 0 푘 푑푘 (Quadratically divergent) Quark gluon vertex (1PI): ∞ ∝ 0 푑푘 /푘 (Logarithm divergent) If theory is renormalizable then at any order of perturbation, all the divergences can be absorbed by redefinitions of the coupling and fields.
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