Examination for FY3464 Quantum Field Theory I, NTNU Trondheim, Department of Physics. Contact: Michael Kachelrieß, tel. 99890701 Allowed tools: mathematical tables 1. Miscellaneous and quiz a.) Write down A* for A = $\bar{u}(p_2)\gamma^\mu u(p_1)$. b.) Calculate tr[$\gamma^\mu \gamma^\nu \gamma_\mu \gamma_\nu$]. c.) The covariant derivative of a Yang-Mills theory transforms under a local gauge transformation U(x) as: D → D' = U(x)D U†(x) + i/g (∂_μU(x))U†(x). d.) The field strength of a Yang-Mills theory transforms under a local gauge transformation U(x) as: F(x) → F'(x) = U(x)F(x)U†(x) + i/g (∂_μU(x))U†(x). 2. Scalar field Consider a real, scalar field φ with mass m and self-interaction gφ³. a.) Write down the Lagrange density L, explain your choice of signs and pre-factors. b.) Write down the generating functional for connected Green functions. c.) Determine the mass dimension in d = 4 space-time dimensions of all quantities in the Lagrange density L. d.) Draw the divergent one-loop diagrams and determine their superficial degree of divergence D in d = 4 space-time dimensions. e.) Determine the number d of space-time dimension for which the theory is renormalisable. 3. Fermion with Yukawa interaction Consider a fermion ψ with mass m interacting with real scalar field φ with mass M through a Yukawa interaction, L = -i g $\bar{\psi}\gamma^5 \psi \phi$. a.) Determine the global symmetries of this Lagrangian for m = M = 0 and the resulting Noether charges. b.) Calculate the self-energy Σ(p) of a fermion with momentum p using dimensional regularisation. Express Σ(p) as Σ(p) = A/ε + B ln(D/μ²). d.) Interpret the functional form of A. e.) Interpret the dependence of the self-energy on the parameter μ. 4. Spin-1 fields a.) Use the tensor method to determine the propagator D_{μν}(k) of a massive spin-1 field described by the Proca equation. b.) Give one argument why this method does not work for m = 0. Feynman rules and useful formulas are provided.Examination for FY3464 Quantum Field Theory I, NTNU Trondheim, Department of Physics. Contact: Michael Kachelrieß, tel. 99890701 Allowed tools: mathematical tables 1. Miscellaneous and quiz a.) Write down A* for A = $\bar{u}(p_2)\gamma^\mu u(p_1)$. b.) Calculate tr[$\gamma^\mu \gamma^\nu \gamma_\mu \gamma_\nu$]. c.) The covariant derivative of a Yang-Mills theory transforms under a local gauge transformation U(x) as: D → D' = U(x)D U†(x) + i/g (∂_μU(x))U†(x). d.) The field strength of a Yang-Mills theory transforms under a local gauge transformation U(x) as: F(x) → F'(x) = U(x)F(x)U†(x) + i/g (∂_μU(x))U†(x). 2. Scalar field Consider a real, scalar field φ with mass m and self-interaction gφ³. a.) Write down the Lagrange density L, explain your choice of signs and pre-factors. b.) Write down the generating functional for connected Green functions. c.) Determine the mass dimension in d = 4 space-time dimensions of all quantities in the Lagrange density L. d.) Draw the divergent one-loop diagrams and determine their superficial degree of divergence D in d = 4 space-time dimensions. e.) Determine the number d of space-time dimension for which the theory is renormalisable. 3. Fermion with Yukawa interaction Consider a fermion ψ with mass m interacting with real scalar field φ with mass M through a Yukawa interaction, L = -i g $\bar{\psi}\gamma^5 \psi \phi$. a.) Determine the global symmetries of this Lagrangian for m = M = 0 and the resulting Noether charges. b.) Calculate the self-energy Σ(p) of a fermion with momentum p using dimensional regularisation. Express Σ(p) as Σ(p) = A/ε + B ln(D/μ²). d.) Interpret the functional form of A. e.) Interpret the dependence of the self-energy on the parameter μ. 4. Spin-1 fields a.) Use the tensor method to determine the propagator D_{μν}(k) of a massive spin-1 field described by the Proca equation. b.) Give one argument why this method does not work for m = 0. Feynman rules and useful formulas are provided.