Quantum Field Theory and Particle Physics Badis Ydri Département de Physique, Faculté des Sciences Université d’Annaba Annaba, Algerie 2 YDRI QFT Contents 1 Introduction and References 1 I Path Integrals, Gauge Fields and Renormalization Group 3 2 Path Integral Quantization of Scalar Fields 5 2.1 FeynmanPathIntegral. .. .. .. .. .. .. .. .. .. .. .. 5 2.2 ScalarFieldTheory............................... 9 2.2.1 PathIntegral .............................. 9 2.2.2 The Free 2 Point Function . 12 − 2.2.3 Lattice Regularization . 13 2.3 TheEffectiveAction .............................. 16 2.3.1 Formalism................................ 16 2.3.2 Perturbation Theory . 20 2.3.3 Analogy with Statistical Mechanics . 22 2.4 The O(N) Model................................ 23 2.4.1 The 2 Point and 4 Point Proper Vertices . 24 − − 2.4.2 Momentum Space Feynman Graphs . 25 2.4.3 Cut-off Regularization . 27 2.4.4 Renormalization at 1 Loop...................... 29 − 2.5 Two-Loop Calculations . 31 2.5.1 The Effective Action at 2 Loop ................... 31 − 2.5.2 The Linear Sigma Model at 2 Loop ................. 32 − 2.5.3 The 2 Loop Renormalization of the 2 Point Proper Vertex . 35 − − 2.5.4 The 2 Loop Renormalization of the 4 Point Proper Vertex . 40 − − 2.6 Renormalized Perturbation Theory . 42 2.7 Effective Potential and Dimensional Regularization . .......... 45 2.8 Spontaneous Symmetry Breaking . 49 2.8.1 Example: The O(N) Model ...................... 49 2.8.2 Glodstone’s Theorem . 55 3 Path Integral Quantization of Dirac and Vector Fields 57 3.1 FreeDiracField ................................ 57 3.1.1 Canonical Quantization . 57 3.1.2 Fermionic Path Integral and Grassmann Numbers . 58 4 YDRI QFT 3.1.3 The Electron Propagator . 64 3.2 Free Abelian Vector Field . 65 3.2.1 Maxwell’s Action . 65 3.2.2 Gauge Invariance and Canonical Quantization . ..... 67 3.2.3 Path Integral Quantization and the Faddeev-Popov Method . 69 3.2.4 The Photon Propagator . 71 3.3 GaugeInteractions............................... 72 3.3.1 Spinor and Scalar Electrodynamics: Minimal Coupling ....... 72 3.3.2 The Geometry of U(1) Gauge Invariance . 74 3.3.3 Generalization: SU(N) Yang-Mills Theory . 79 3.4 Quantization and Renormalization at 1 Loop................ 85 − 3.4.1 The Fadeev-Popov Gauge Fixing and Ghost Fields . 85 3.4.2 Perturbative Renormalization and Feynman Rules . ..... 89 3.4.3 The Gluon Field Self-Energy at 1 Loop............... 92 − 3.4.4 The Quark Field Self-Energy at 1 Loop...............103 − 3.4.5 The Vertex at 1 Loop.........................105 − 4 The Renormalization Group 113 4.1 Critical Phenomena and The φ4 Theory ...................113 4.1.1 Critical Line and Continuum Limit . 113 4.1.2 Mean Field Theory . 118 4.1.3 Critical Exponents in Mean Field . 123 4.2 The Callan-Symanzik Renormalization Group Equation . ........128 4.2.1 Power Counting Theorems . 128 4.2.2 Renormalization Constants and Renormalization Conditions . 133 4.2.3 Renormalization Group Functions and Minimal Subtraction . 136 4.2.4 CS Renormalization Group Equation in φ4 Theory . 140 4.2.5 Summary ................................146 4.3 Renormalization Constants and Renormalization Functions at Two-Loop . 149 4.3.1 The Divergent Part of the Effective Action . 149 4.3.2 Renormalization Constants . 154 4.3.3 Renormalization Functions . 157 4.4 CriticalExponents ............................... 158 4.4.1 Critical Theory and Fixed Points . 158 4.4.2 Scaling Domain (T > Tc) .......................164 4.4.3 Scaling Below Tc ............................168 4.4.4 Critical Exponents from 2 Loop and Comparison with Experiment 171 − 4.5 The Wilson Approximate Recursion Formulas . 175 4.5.1 Kadanoff-Wilson Phase Space Analysis . 175 4.5.2 Recursion Formulas . 177 4.5.3 The Wilson-Fisher Fixed Point . 187 4.5.4 The Critical Exponents ν .......................192 4.5.5 The Critical Exponent η ........................196 YDRI QFT 5 A Exercises 199 B References 205 6 YDRI QFT 1 Introduction and References • M.E.Peskin, D.V.Schroeder, An Introduction to Quantum Field Theory. • J.Strathdee,Course on Quantum Electrodynamics, ICTP lecture notes. • C.Itzykson, J-B.Zuber, Quantum Field Theory. • J.Zinn-Justin,Quantum Field Theory and Critical Phenomena. • A.M.Polyakov, Gauge Fields and Strings. • C.Itzykson, J-M.Drouffe, From Brownian Motion to Renormalization and Lattice Gauge Theory. • D.Griffiths, Introduction to Elementary Particles. • W.Greiner, Relativistic Quantum Mechanics.Wave Equations. • W.Greiner, J.Reinhardt, Field Quantization. • W.Greiner, J.Reinhardt, Quantum Electrodynamics. • S.Randjbar-Daemi, Course on Quantum Field Theory, ICTP lecture notes. • V.Radovanovic, Problem Book QFT. 2 YDRI QFT Part I Path Integrals, Gauge Fields and Renormalization Group 2 Path Integral Quantization of Scalar Fields 2.1 Feynman Path Integral We consider a dynamical system consisting of a single free particle moving in one dimen- sion. The coordinate is x and the canonical momentum is p = mx˙. The Hamiltonian is H = p2/(2m). Quantization means that we replace x and p with operators X and P satisfying the canonical commutation relation [X, P ] = i~. The Hamiltonian becomes H = P 2/(2m). These operators act in a Hilbert space . The quantum states which de- H scribe the dynamical system are vectors on this Hilbert space whereas observables which describe physical quantities are hermitian operators acting in this Hilbert space. This is the canonical or operator quantization. We recall that in the Schrödinger picture states depend on time while operators are independent of time. The states satisfy the Schrödinger equation, viz ∂ H ψ (t) >= i~ ψ (t) > . (2.1) | s ∂t| s Equivalently i ψ (t) >= e− ~ H(t−t0) ψ (t ) > . (2.2) | s | s 0 Let x > be the eigenstates of X, i.e X x >= x x >. The completness relation is | | | dx x><x = 1. The components of ψ (t) > in this basis are <x ψ (t) >. Thus | | | s | s R ψ (t) > = dx<x ψ (t) > x > . (2.3) | s | s | Z i <x ψ (t) > = <x e− ~ H(t−t0) ψ (t ) > | s | | s 0 = dx G(x,t; x ,t ) <x ψ (t ) > . (2.4) 0 0 0 0| s 0 Z 6 YDRI QFT In above we have used the completness relation in the form dx x >< x = 1. The 0| 0 0| Green function G(x,t; x ,t ) is defined by 0 0 R i G(x,t; x ,t ) = <x e− ~ H(t−t0) x > . (2.5) 0 0 | | 0 In the Heisenberg picture states are independent of time while operators are dependent of time. The Heisenberg states are related to the Schrödinger states by the relation i ψ >= e ~ H(t−t0) ψ (t) > . (2.6) | H | s We can clearly make the identification ψ >= ψ (t ) >. Let X(t) be the position | H | s 0 operator in the Heisenberg picture. Let x,t > be the eigenstates of X(t) at time t, i.e | X(t) x,t >= x x,t >. We set | | i i x,t >= e ~ Ht x > , x ,t >= e ~ Ht0 x > . (2.7) | | | 0 0 | 0 From the facts X(t) x,t >= x x,t > and X x>= x x> we conclude that the Heisenberg | | | | operators are related to the Schrödinger operators by the relation i i X(t)= e ~ HtXe− ~ Ht. (2.8) We immediately obtain the Heisenberg equation of motion dX(t) i ∂X i i = e ~ Ht e− ~ Ht + [H, X(t)]. (2.9) dt ∂t ~ The Green function (2.5) can be put into the form G(x,t; x ,t ) = < x,t x ,t > . (2.10) 0 0 | 0 0 This is the transition amplitude from the point x0 at time t0 to the point x at time t which is the most basic object in the quantum theory. We discretize the time interval [t ,t] such that t = t + jǫ, ǫ = (t t )/N, j = 0 j 0 − 0 0, 1, ..., N, tN = t0 + Nǫ = t. The corresponding coordinates are x0,x1, ..., xN with xN = x. The corresponding momenta are p0,p1, ..., pN−1. The momentum pj corresponds to the interval [xj,xj+1]. We can show G(x,t; x ,t ) = < x,t x ,t > 0 0 | 0 0 = dx < x,t x ,t >< x ,t x ,t > 1 | 1 1 1 1| 0 0 Z N−1 = dx1dx2...dxN−1 <xj+1,tj+1 xj,tj > . (2.11) j=0 | Z Y YDRI QFT 7 We compute (with <p x>= exp( ipx/~)/√2π~) | − i <x ,t x ,t > = <x (1 Hǫ) x > j+1 j+1| j j j+1| − ~ | j i = dp <x p >< p (1 Hǫ) x > j j+1| j j| − ~ | j Z i = dp (1 H(p ,x )ǫ) <x p >< p x > j − ~ j j j+1| j j| j Z dpj i i i = (1 H(p ,x )ǫ) e ~ pj xj+1 e− ~ pj xj 2π~ − ~ j j Z dpj i = e ~ (pj x˙ j −H(xj ,pj))ǫ. (2.12) 2π~ Z In above x˙ = (x x )/ǫ. Therefore by taking the limit N , ǫ 0 keeping j j+1 − j −→ ∞ −→ t t = fixed we obtain − 0 − dp0 dp1dx1 dpN−1dxN−1 i N 1(p x˙ −H(p ,x ))ǫ G(x,t; x ,t ) = ... e ~ Pj=0 j j j j 0 0 2π~ 2π~ 2π~ Z i t ~ R ds(px˙−H(p,x)) = p x e t0 . (2.13) D D Z Now x˙ = dx/ds. In our case the Hamiltonian is given by H = p2/(2m). Thus by performing the Gaussian integral over p we obtain 1 i t ~ R dsL(˙x,x) G(x,t; x ,t ) = x e t0 0 0 N D Z i = x e ~ S[x]. (2.14) N D Z In the above equation S[x] = dt L(x, x˙) = m dt x˙ 2/2 is the action of the particle.