1 Practical task for exam in QFT (2) 1. Operator algebra and Feynman diagrams. Consider the ϕ4 -theory,- the theory with a single real scalar field ϕ(x). The interaction Lagrangian reads Lint = λ 4 ϕ (x). 4! - Given the (normal-ordered) operator O(z) =: ϕ2 (x) : consider the following vacuum average V1 = ⟨Ω|T (ϕ(x)O(z)ϕ(y))|Ω⟩ - Applying Wick theorem derive the leading (∼ λ0 ) and next-to-leading (∼ λ1 ) contributions to V1 . Plot diagrams and write down the expressions for them. ˜ = ϕ2 ? - What will change if the operator would be just a product of fields, O(z) - Which diagrams will contribute to the amplitude ⟨ϕ|O(z)|ϕ⟩? * Consider the ϕ4 -theory with complex field ϕ, Lint = λ4 (ϕ∗ ϕ)2 . Derive the leading and nextto-leading diagrams for the vacuum average V2 = ⟨Ω|T (ϕ∗ (x)O2 (z)ϕ(y))|Ω⟩ with O2 (z) =: ϕ∗ (z)ϕ(z) :. Describe the difference between V1 and V2 . 2. Muon lifetime. The main mode of the muon decay is given by the process µ− → νµ + ν¯e + e− , where e− is electron, νµ is muon-neutrino and ν¯e is electron-anti-neutrino. At the leading order it is given by the diagram, where W − is W-boson. In the following we suppose that the electron is massless (unless integrals would be more difficult), neutrinos are also massless. 2 - Write down the expression for this diagram in the limit MW ≫(all momenta). For that you need the fermion-fermion-W vertex which is given by (the only difference from the QED photon-fermion interaction is the matrix 1 − γ 5 , and the coupling constant) iγ µ 1 − γ 5 GF √ . 2 2 The propagator of W -boson in the limit MW ≫(all momenta) is −ig µν 2 +i0 p2 −MW ≃ ig µν 2 . MW - Calculate the matrix element-squared (averaged over polarization of µ and summed over all final polarizations ⟨M2 ⟩. - Calculate the decay rate Γ which is given by the expression Γ= (2π)4 2mµ ∫ dp3e (2π)3 (2Ee ) ∫ dp3νe (2π)3 (2Eνe ) ∫ dp3νµ (2π)3 (2Eνµ ) ( ) δ (4) pµ − pe − pνe − pνµ ⟨M2 ⟩ also evaluate the differential decay rate with respect to electron energy dΓ/dEe . - Calculate the muon lifetime τ = Γ~ , using that GF = 1.166 × 10−5 GeV−2 , mµ = 105.65MeV, ~ = 6.5821 × 10−16 s eV. The experimental value of muon lifetime is τ = 2.197 × 10−6 s. HINT 1 : During the integration over the phase space for Γ you can use the following integral (for massless k1 and k2 ) ∫ d 3 k1 2E1 ∫ ) d3 k2 µ ν (4) π ( 2 µν k1 k2 δ (X − k1 − k2 ) = X g + 2X µ X ν . 2E2 24 HINT 2 : For the calculation of the Γ you can chouse any frame, e.g. the frame where µ is at rest. 3 List of topics 1. Elements of classical field theory: • Least action principle,Euler-Lagrange equation. • Hamiltonian formulation 2. Transformation properties of fields • Lorentz transformation of fields, scalar, vector, spinor fields. • Conserved currents, Noether’s theorem. • Energy-momentum tensor, angular-momentum tensor. 3. Free Klein-Gordon (KG) field • KG equation, Lagrangian for KG equation, Hamiltonian formulation • Quantization, field decomposition. • Creation and annihilation operators, Hamiltonian in operator form, energy eigenstates. • Covariant commutation relation, KG propagators (Schwinger and Feynman) • Complex KG field, charge operator. 4. Interacting fields • ϕ4 theory: Lagrangian, symmetries, equation of motions • Interaction representation, time-evolution operator and its properties. • Time- and normal-ordering, first- and second- Wick’s theorem • S-matrix, initial and final states, cross-section • Feynman diagrams, Feynman rules for ϕ4 -theory 5. Dirac field • Dirac equation, properties of Dirac field and Dirac equation under Lorentz transformations • Gamma-matrices: algebra, traces, complete basis of gamma-matrices, etc. • Solution of Dirac equation, properties of bi-spinors, completeness relation. 4 • Quantization of the Dirac field, anti-commutation relations, propagator of Dirac field • Wick’s theorem for Dirac field, Yukawa interaction 6. Electro-magnetic field • Field-strength tensor and Maxwell equations, gauge invariance. • Massive vector field: plane-wave solution, quantization, propagator. • Quantization of electro-magnetic field in Lorentz gauge, photon propagator. • Vacuum of gauge theory, Gupta-Bleuler condition, initial- and final-state photons. 7. Quantum electrodynamics (QED) • Feynman rules of QED. • Calculation of (un)polarized processes. • e+ e− annihilation, and e+ e− → µ+ µ− . 8. Miscellaneous • Mandelstam variables, crossing symmetry, and kinematic constrains. • Discrete symmetries: P,T, and C transformation properties of fields, CPT-theorem. • Loops: Feynman parameters, ultraviolet and infrared divergences, ultraviolet regularization, radiation of soft photons.
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