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Quantum Field Theory Theory of Elementary Particles Ahmed Ali DESY, Hamburg DESY Summer Student Programme 2009, DESY, Hamburg August 10 – 14, 2009 Theory of Elementary Particles (page 1) Ahmed Ali DESY, Hamburg August 10 – 14, 2009 Theory of Elementary Particles (page 2) Ahmed Ali DESY, Hamburg Structure of the first generation of leptons and quarks in SM The hypercharge quantum numbers Y are fixed by the relation Q = I + Y • 3 Q are the electric charges (in units of the proton charge): • u +2/3 ν 0 Q = ; Q e = d 1/3 e 1 − − I is the third-component of the weak isospin: • 3 u +1/2 ν +1/2 I = ; I e = 3 d 1/2 3 e 1/2 L − L − I = I = 0 for u , d , e • 3 R R R August 10 – 14, 2009 Theory of Elementary Particles (page 3) Ahmed Ali DESY, Hamburg Plan of these lectures Introduction to Quantum Field Theory (QFT) • Gauge Invariance and Quantum Electrodynamics (QED) • + + An elementary QED process e e− µ µ− • → Gauge Invariance and Quantum Chromodynamics (QCD) • The QCD running coupling constant α (Q2) • s Applications of perturbative QCD to hard processes • The Electroweak Theory • The Higgs Mechanism • CKM Matrix and CP Violation • Precision Tests of the EW theory • Higgs Boson Physics • Gauge coupling unification • Conclusions • August 10 – 14, 2009 Theory of Elementary Particles (page 4) Ahmed Ali DESY, Hamburg Recommended References Books Relativistic Quantum Mechanics & Relativistic Quantum Fields • James D. Bjorken, Sidney D. Drell, McGraw-Hill Book Company (1965) An Introduction to Quantum Field Theory • Michael E. Peskin, Daniel V. Schroeder, ABP, Westview Press (1995) Modern Quantum Field Theory - A Concise Introduction • Thomas Banks, Cambridge University Press (2008) Foundations of Quantum Chromodynamics • Taizo Muta, World Scientific -revised edition (2005) Dynamics of the Standard Model • John F. Donoghue, Eugene Golowich, Barry R. Holstein, Cambridge University Press (Revised version 1994) August 10 – 14, 2009 Theory of Elementary Particles (page 5) Ahmed Ali DESY, Hamburg Review Articles A QCD Primer • G. Altarelli, arXiv Preprint hep-ph/0204179 The Standard Model of Electroweak Interactions • A. Pich, arXiv Preprint 0705.4264 [hep-ph] Concepts of Electroweak Symmetry Breaking and Higgs Physics • M. Gomez-Bock, M. Mondragon, M. M¨uhlleitner, M. Spira, P.M. Zerwas arXiv Preprint 0712.2419 [hep-ph] Essentials of the Muon g 2 − • F. Jegerlehner, DESY Report 07-033, arXiv:hep-ph/0703125 (2007) Review papers on the Standard Model Physics • Particle Data Group (URL:http://pdg.lbl.gov/) I. Hinchliffe (Quantum Chromodynamics) • J. Erler & P. Langacker (Electroweak Model and Constraints on New Physics) • A. Ceccucci, Z. Ligeti, Y. Sakai (The CKM Quark-Mixing Matrix) • G. Bernardi, M. Carena, T. Junk (Higgs Bosons: Theory and Searches) • B. Foster, A.D. Martin, M.G. Vincter (Structure Functions) • August 10 – 14, 2009 Theory of Elementary Particles (page 6) Ahmed Ali DESY, Hamburg Quantum Field Theory - Introduction Physical systems can be characterized by essentially two features, roughly speaking, • size (large or small) and speed (slow or fast) Velocity v c: For slowly moving objects (compared to the speed of light), we ≪ • have the Classical (Newtonian) mechanics to describe this motion Velocity v c: For fast moving system (approaching the speed of light), we have → • Relativistic (Einstein’s) mechanics To describe objects at small distances (say, atoms), energy and other attributes (such • as angular momenta) are quantized in units of the Planck’s constant ~ = 0. For 6 quantum systems ~ = 0 moving with non-relativistic velocities v c, we move 6 ≪ from Classical mechanics to Non-relativistic Quantum mechanics For ~ = 0 and v c, we have Relativistic Quantum mechanics (RQM) • 6 → RQM is insufficient as it does not account for the particle production and • annihilation. However, E = mc2 allows pair creation Quantum Mechanics + Relativity + Particle creation and annihilation = Quantum ⇒ • Field Theory Quantum Field Theory Application of Quantum mechanics to dynamical systems of fields Φ(t, ~x) • August 10 – 14, 2009 Theory of Elementary Particles (page 7) Ahmed Ali DESY, Hamburg Preliminaries Space-time coordinates & Lorentz scalars Natural units (c = ~ = 1). In these units, the Compton wavelength of the electron • 11 is 1/me 3.86 10− cm and the electron mass is me 0.511 MeV. In the same units,≃ m ×938 MeV and m 172 GeV ≃ p ≃ t ≃ Space-time coordinates (t,x,y,z) are denoted by the contravariant four vector • xµ = (x0, x1, x2, x3) (t,x,y,z)=(t, ~r) ≡ The covariant four-vector xµ is obtained by changing the sign of the space • components x = (x , x , x , x ) = g xν (t, x, y, z)=(t, ~r) µ 0 1 2 3 µν ≡ − − − − µν with the metric gµν = g 10 0 0 0 1 0 0 gµν = − 0 0 1 0 − 0 0 0 1 − The inner product, yielding a Lorentz-scalar, is x2 = t2 ~r2 = t2 x2 y2 z2 • − − − − Momentum vectors are defined as: • µ p = (E,px,py,pz) and the inner product (Lorentz scalar) is: p .p = pµp = E E ~p .~p & x.p = tE ~r.~p 1 2 1 2µ 1 2 − 1 2 − August 10 – 14, 2009 Theory of Elementary Particles (page 8) Ahmed Ali DESY, Hamburg Preliminaries Momentum Operator in Coordinate representation ∂ ∂ 1 pµ = i i , ~ i∂µ ∂x ≡ ∂t i ∇ ≡ µ This transforms as a contravariant four-vector • ∂ ∂ ∂ pµp = = ~ 2 ( )2 = µ µ −∂xµ ∂x ∇ − ∂t − The four-vector potential of the electromagnetic field is defined as • µ µν A = (Φ, A~) = g Aν where Φ = Φ(t, ~x) is Scalar potential; A~ = A~(t, ~x) is Vector potential µν ν Maxwell Equations: ∂µF = eJ • 2 where e is the electric charge ; Sommerfeld Constant α = e 1/137 and J ν is 4π ∼ the electromagnetic current The field strength tensor is defined as F µν = ∂ Aµ ∂ Aν , and • ∂xν − ∂xµ E~ = (F 01, F 02, F 03) B~ = (F 23, F 31, F 12) August 10 – 14, 2009 Theory of Elementary Particles (page 9) Ahmed Ali DESY, Hamburg Elements of Classical Field Theory Action S and Lagrangian L Action S = t2 Ldt: Time integral of the Langrangian L • t1 Lagrangian L = T V (Kinetic Energy - Potential Energy); L = L(q(t), q˙(t)) • R − In classical mechanics, the Lagrange function is constructed from the generalized • coordinates q(t) and velcocities q˙(t) Hamilton’s Principle of Least Action: Dynamics of the particle traversing a path q(t) • is determined from the condition δS = δ t2 L(q(t), q˙(t))dt = 0 t1 Also in Quantum mechanics, Classical pathR minimises Action [Feynman; see • Feynman Lectures, Vol. II] Lagrangian in field theory is constructed from Φ(t, ~x), Φ(˙ t, ~x), ~ Φ(t, ~x) • ∇ L = d3x Φ(t, ~x), Φ(˙ t, ~x), ~ Φ(t, ~x) L ∇ Z is the Lagrangian density; S = Ldt = d4x Φ(t, ~x), Φ(˙ t, ~x), ~ Φ(t, ~x) • L | {z L } ∇ Minimising the Action, i.e., δS =R 0 leads toRthe Euler-Lagrange Equation of motion • ∂ ∂ ∂ ∂ L L + ~ L = 0 ∂Φ − ∂t ∂Φ˙ ∇∂(~ φ) ∇ Hamiltonian: H = d3x π(x)φ˙(x) = d3x • − L H R h i R August 10 – 14, 2009 Theory of Elementary Particles (page 10) Ahmed Ali DESY, Hamburg Constructing : The free field case Spin 0 field: Φ(t,L ~x) For a free scalar field with mass m, the Lagrangian density is • 1 ∂ 1 Φ = ( Φ)2 (~ Φ)2 m2Φ2 Lfree 2 ∂t − ∇ − 2 The resulting field equation is called the Klein-Gordon equation • ∂ ( + m2)Φ(t, ~x)=0 with = ( )2 ~ 2 ∂t − ∇ 1 Spin 2 field: Ψ(t, ~x) Ψ(t, ~x)=Ψ (t, ~x) is a 4-component spinor (ψ ,ψ ,ψ ,ψ ) • α 1 2 3 4 Lagrangian density of a free Dirac (=spin 1 ) field with mass m • 2 ψ = ψ¯(iγµ∂ m)ψ = ψ¯(i∂/ m)ψ Lfree µ − − γµ are 4 4 matrices. The resulting field equation is called the Dirac equation • × (i∂/ m)ψ(t, ~x) = 0 − Spin 1 field with m = 0: (Example: Electromagnetic field) Aµ(t, ~x) Lagrangian density of a free spin-1 field: A = 1 F F µν • Lfree − 4 µν Yields Maxwell equations in the absence of charge and current densities • µν ∂µF (t, ~x) = 0 August 10 – 14, 2009 Theory of Elementary Particles (page 11) Ahmed Ali DESY, Hamburg Noether’s Theorem Relationship between Symmetries & Conservation Laws in CFT Noether’s theorem concerns continuous transformations on the field φ(x), which in • the infinitesimal form can be written as φ(x) φ′(x) = φ(x)+ α∆φ(x) → This is a symmetry transformation if it leaves the equation of motion (EOM) • invariant, which is ensured if the action S is invariant More generally, one can allow S to change by a surface term, since such a term • leaves the Euler-Lagrange EOM invariant Thus, for some µ(x), the Lagrangian must be invariant up to a 4-divergence • J (x) (x)+ α∂ µ(x) L → L µJ Let us compute explicitly α∆ , and the result is • ∂L ∂ ∂ α∆ = α∂µ L ∆φ + α L ∂µ L ∆φ L ∂(∂µφ) ∂φ − ∂(∂µφ) µ The second term vanishes due to the EOM. Settingh the first termiequal to α∂µ (x) • µ µ ∂ µ J ∂µJ (x)=0 for J (x) = L ∆φ (x) ∂(∂µφ) −J This result states that the current J µ(x) is conserved • Noether’s theorem can also be stated in the form that the charge • Q J 0(x)d3x ≡ all space is constant in time R August 10 – 14, 2009 Theory of Elementary Particles (page 12) Ahmed Ali DESY, Hamburg Emmy Noether (1882 - 1935) August 10 – 14, 2009 Theory of Elementary Particles (page 13) Ahmed Ali DESY, Hamburg Examples of conserved currents and charges Example 1: Lagrangian of a complex scalar field with mass m • (φ) = ∂ φ 2 m2 φ 2 L | µ | − | | (φ) is invariant under φ(x) eiαφ(x).
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