Theoretical Physics Homework Problems #2 in SI2400 Theoretical Particle Physics, 7.5 credits Spring 2015 Deadline: Teachers: Examiner: GOOD LUCK! 1. May 06, 2015 @ 17:00 Dr. Sushant Raut ([email protected]) Dr. Juan Herrero ([email protected]) Stella Riad ([email protected]) Prof. Tommy Ohlsson ([email protected]) a) For a Dirac field ψ(x), show that the 4-vector current j µ = ψ(x)γ µ ψ(x) is conserved. b) Let (j 5 )µ = ψ(x)γ µ γ 5 ψ(x). Calculate ∂µ (j 5 )µ . c) Given the Dirac equation, prove the identity 1 u(p, s) [(p + q)µ + iσ µν (p − q)ν ] u(q, s) ; 2m for u(p, s)γ µ γ5 u(q, s), derive a similar identity. u(p, s)γ µ u(q, s) = 2. The bilinear form of a Dirac spinor can be constructed for an arbitrary 4 × 4 matrix M , namely ψM ψ. However, only sixteen of them ψΓa ψ (for a = S, V, T, A, and P) are linearly independent: T A P ΓS = I4 , ΓV µ = γµ , Γµν = σµν , Γµ = γµ γ5 , Γ = iγ5 where I4 stands for the 4 × 4 identity matrix and σ µν ≡ i[γ µ , γ ν ]/2. Prove the following properties of Γa matrices: a) (Γa )2 = ±I4 ; b) For any matrix Γa (a 6= S), there exists a matrix Γb , leading to Γa Γb = −Γb Γa ; c) For any matrix Γa (a 6= S), Tr[Γa ] = 0 holds; d) For any pair of matrices Γa and Γb with a 6= b, a matrix Γc with c 6= S can be found to satisfy Γa Γb = Γc up to a factor of ±1 or ±i; X e) The identity λa Γa = 0 holds if and only if all the constants λa are vanishing. a 3. Consider a universe with only the following particles: (a) A lepton f (and its antiparticle f ), (b) A self-conjugate scalar particle h, (c) A self-conjugate vector boson A which does not interact with itself. Based on spin and lepton number conservation laws, draw all possible vertices (up to mass dimension 4) using these particles that you think might exist. Using these vertices, draw all possible Feynman diagrams for the process f f → hh with two and three vertices. 4. Prove that the following properties hold for the gamma matrices: a) Tr(odd number of γ matrices) = 0, b) Tr(γ µ γ ν γ ρ γ σ ) = 4(g µν g ρσ − g µρ g νσ + g µσ g νρ ), c) γ µ γ ν γµ = −2γ ν , i d) γ 5 = − εµνρσ γ µ γ ν γ ρ γ σ , 4! µ ν ρ σ 5 e) Tr(γ γ γ γ γ ) = −i4εµνρσ . 5. In a parity-invariant theory, it is not possible to add a scalar to a pseudoscalar. However, in the weak interactions, which do not conserve parity, one frequently encounters expressions of the type 1 ∓ γ5 1 ψψ ∓ ψγ 5 ψ = ψ ψ ≡ ψPL/R ψ, 2 2 i.e., adding a scalar Dirac field bilinear to a pseudoscalar one. The operators PL/R = 1 ∓ γ5 2 are actually projection operators, projecting out the left/right chiral parts of a fermion. Starting from the Dirac equation and using the so-called chiral representation of the gamma matrices 0 σi −1 0 0 1 0 i 5 γ = , γ = , γ = 1 0 −σ i 0 0 1 show that, for a massless plane-wave solution ψ to the Dirac equation, ψL/R ≡ PL/R ψ are eigenstates of the helicity operator, h= p ·Σ |p| with eigenvalues ∓1. Note that Σj = γ 0 γ j γ 5 . 6. For the process e+ γ → e+ γ in QED, compute the unpolarized differential cross section dσ/d cos θ in the center-of-mass frame is given to lowest order in the coupling constant e. Express your answer in terms of the Mandelstam variables s and t. You may assume that the electron is massless. 7. In so-called scalar QED, the electron is replaced by a charged scalar particle φ (and its corresponding antiparticle φ∗ ), which couple to the photon through the following interaction vertices: φ p′ φ Aµ Aµ = −ie(p − p0 )µ φ p = 2ie2 g µν φ Aν Calculate, to lowest order in e and setting mφ = 0, the differential cross section dσ/d cos θ in the center-of-mass frame for the process φ∗ γ → φ∗ γ. Compare your result to the result of problem 6. 8. The one-loop running of the strong coupling constant in the regime q 2 µ2 is given by αs (µ2 ) 2 αs (q ) = , 1 + αs (µ2 )/(12π)(11nc − 2nf ) log(q 2 /µ2 ) where αs = gs2 /4π, nc is the number of colors, nf is the number of quark flavors and αs (MZ2 ) = 0.118. a) If nc = 3, as in the Standard Model, assuming that the new generations of fermions are perturbative, what is the maximum number of quark flavors that is consistent with the asymptotic freedom of QCD? b) Compute the value of αs (q 2 ) at each quark mass threshold. Explain the result and draw it qualitatively. Where does perturbation theory break? Is this an indication of the confinement of quarks? c) Compute ΛQCD as the pole in the one loop result. d) Compute αs at LHC typical centre of mass energies, 1 TeV2 . 9. Assume a world with two colors, black and white, invariant under the gauge group SU(2) (recall that it has 3 generators, the Pauli matrices), and suppose a new color carrier, the coloron. a) Explain the decomposition of states into irreducible singlet (antisymmetric) and triplet (symmetric) representations under SU(2), using ladder operators. b) Compute the SU(2) coloron exchange for the singlet state. c) Compute the SU(2) coloron exchange for the triplet state. Rules and guidelines for homework problems in SI2400, Theoretical Particle Physics When solving the homework problems, you are allowed to use books or other sources of information, as well as to discuss the problems with one another. However, the solutions that you hand in have to reflect your own knowledge. Therefore, make sure that you motivate the steps in your solutions. If we receive nearly identical solutions, we might ask you to orally describe what you have done. Also, you should follow the following simple guidelines: • Your solutions should be handwritten • Motivate your computations, it should be clear that you understand what you are doing and why! • Start each problem on a separate sheet of paper • Hand in the problems stapled together, in the correct order • Write clearly, so that your solutions are readable • Do not forget to put your name on your solutions • Hand in the solutions in time—if, for some reason, you cannot do this, please talk to one of the teachers rather than just handing in the solutions too late. Failure to follow these rules might result in a deduction of points.