Particle Physics I Lecture Exam Question Sheet
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1 Particle Physics I Lecture Exam Question Sheet Five out of these 16 questions will be given to you at the beginning of the exam. (1) (a) Which are the different fundamental interactions that exist in Nature? (b) Name important processes or quantities whose dynamics or properties are mostly influenced by each of these four fundamental interactions. Name one for each fundamental interaction. (c) Which of the fundamental interactions are described by the Standard Model? (d) Name all elementary particles contained in the Standard Model and distinguish them into the groups hadronic, leptonic, gauge bosons and scalars. Indicate which particles and antiparticles are different/identical. (e) Give the electric charges of all elementary particles. (You do not have to explicitly quote the charges of the corresponding antiparticles.) (f) What is the content of the spin-statistics theorem? (2) (a) Write down the defining property of Lorentz transformations and show that the product (contraction) of two four-vectors is Lorentz-invariant. (b) What is an inertial frame? (c) Consider two inertial frames S and S where the frame S moves with velocity v in the y-direction with respect to the frame S. Write down the Lorentz transformation of the space-time coordinates of the two frames. Use primed coordinates for frame S. (d) A stick of length L is lying at rest along the x-axis of frame S. Calculate how long the stick is in frame S. (e) A stick of length L is lying at rest along the y-axis of frame S. Calculate how long the stick is in frame S. (f) An astronaut sitting in a space ship that is travelling with velocity v < 1, is throwing a ball with velocity v into the direction of flight and perpendicular to the direction of flight. What is the velocity of the ball in both cases for an observer that sees the space ship having velocity v? (3) (a) Under which condition is a collection of four quantities called a 4-vector? (b) Write down three examples for physical 4-vectors and explicitly quote the entries. (c) What are covariant and contravariant 4-vectors and how are they related? (d) Explain time-like, space-like, light-like. (e) How is the scalar product of two 4-vectors defined? (f) Write down the entries of the Lorentz transformation matrix Λ µ ν for a contravariant 4-vector describing the relation of the space-time coordinates of frames S and S, where frame S moves with velocity v in the y-direction with respect to frame S. Use primed coordinates for frame S. (g) Write down the entries of the Lorentz transformation matrix Λµ ν for covariant 4-vectors for the case explained in (f). Use primed coordinates for frame S. (h) How is the 4-momentum p µ of a particle defined in terms of the space-time coordinate x µ (t) = (t, x(t))? Which important inertial frame-independent equation does p µ satisfy? (4) (a) Show that the partial derivative µ = / x µ is a 4-vector (of course based on the assumptioin that x µ is a 4-vector satisfying the correct Lorentz transformation property). So you need to 1
2 prove that µ Λ µ ν ν for a Lorentz transformation given that x µ Λ µ νx ν holds. (b) Consider a fixed-target experiment, where particle A has been accelerated and hits particle B at rest. In the reaction the particles C 1, C 2,... C n are being produced. Determine the minimal energy E min that particle A needs to have such that the reaction can take place. The particle masses are m A, m B, m C1, m C2, etc. (c) Consider the 3-body decay A B + C + D. A is at rest and the masses of the decay particles are given. Determine the minimal and maximal possible energy of particle B. Draw pictures of the kinematic configurations of the particles for these two situations. (5) (a) Consider the particle reaction A + B C + D + E + F. Which conditions satisfy the 4-momenta p µ i, with i = A,..., F? What is the physical meaning of this relation? (b) Consider the pion decay π µ ν µ in the muon rest frame and determine the velocities of the decaying pion and the antineutrino. Assume that (anti)neutrinos are massless. (One way to proceed is to first consider the reaction in the pion rest frame and then think of the situation in the muon rest frame.) (c) Consider the scattering reaction p + p p + p + p + p in the fixed-target frame where one initial-state proton is at rest. Which minimal energy does the moving initial-state proton need to have such that the reaction can take place? (d) Assume that you do the experiment with exactly the minimal required energy such that the fixed-target scattering reaction of (c) takes place. Write down the 4-momenta p µ i (i = 1, 2, 3, 4) of the four final-state particles assuming that the moving proton has 4-momentum p µ p = (E p, p). (6) (a) Specify the meaning of the fundamental space-time transformations P, C and T. (b) How do P and T act on the quantities x µ = (t, x), L = x p, A µ = (V, A) and on helicity s p/ p? (The term s stands for the spin vector and p for the 3-momentum of a particle.) (c) What is the content of the CPT-Theorem? (d) Make a table which shows for which of the fundamental interactions (electromagnetic, strong, weak, gravitational), the transformations P, C and T are symmetries. (e) What is a pseudoscalar particle? Name 5 observable particles, which do not need to be fundamental, and quote their parity eigenvalues. What is the parity eigenvalue of antiquarks if you assume that the quarks are eigenstates of parity with eigenvalue 1? (f) Consider the two-state neutral kaon system containing a K 0 and K 0 at rest. Write down the P and C transformations for the K 0 and K 0 states and determine the eigenstates in the K 0 - K 0 space with respect to the combined CP -transformation. (g) Determine the C, P and CP eigenvalues of the π 0 π 0, π 0 π 0 π 0, π + π and π + π π 0 multiparticle states, where you can ignore a possible relative angular momentum. Explain which of the CP eigenstates from (f) can decay into a final state with exactly 2 or 3 pions. (These decay channels of the kaons are completely dominated by the strong interaction.) (7) (a) Write down the Klein-Gordon equation. Provide the definition of all quantities that appear in the Klein-Gordon equation. Which types of particles does the Klein-Gordon equation describe? Write down the particle and antiparticle solutions. (b) Write down the the Dirac equation. Provide the definition of all quantities that appear in the Dirac equation. Which types of particles does the Dirac equation describe? Write down the particle and antiparticle solutions. (c) Show how the Klein-Gordon and the Dirac equations transform under a generic Lorentz trans- 2
3 formation Λ. Show the intermediate steps in the case of the Dirac equation. (d) Write down how the Dirac field and matrices transform under a Lorentz transformation. Show that the Lagrangian for a Dirac field is manifestly Lorentz-invariant and that the electric current j µ for a Dirac field is a four-vector. (8) (a) Write down the QED Lagrangian (general fermion with mass m and electric charge Q) and explicitly provide the definitions for the field strength tensor F µν and the covariant derivative D µ. (b) Work out the mass dimensions of the Dirac fermion field and the photon field. (c) Write the QED Lagrangian in terms of right-handed and left-handed fields using the 4-component notation. Identify the terms that induce a change in helicity. (d) Calculate the Euler-Lagrange equations for the QED Lagrangian from part (a). (e) Write down the U(1) gauge transformations of Dirac and photon fields. How does the QED Lagrangian transform under these (just quote the result)? Argue about the absence of a photon mass term. (f) Write down the QCD Lagrangian and give the definitions of the field strength tensor F µν,a and the covariant derivative D µ. Write explicitly flavor, color and Dirac indices and summations (including summation limits) over them. Why are there eight different gluons? (9) (a) Give the definition of the decay rate Γ and discuss the experimental decay law through which Γ can be determined. What is the mass dimension of Γ? What is the relation between Γ and the life-time τ? (b) Define the (total) cross section σ: explain each term in your definition. Derive from your definition the mass dimension of σ. Define the differential cross section dσ/dω. Is the differential cross section dσ/dω Lorentz-invariant? (c) Consider a decay channel A B 1 +B 2 + +B N. Write down the formula for the partial decay width Γ(A B 1 + B B N ) and define each term in this expression. Why is it necessary to introduce a symmetry factor? (d) Consider a scattering channel A 1 + A 2 B 1 + B B N. Give the formula for the cross section σ(a 1 + A 2 B 1 + B B N ) and define each term in this expression. (10) (a) Assume an unstable particle with the decay width Γ. Assume that you have N 0 of these particles at time t = 0. Formulate the differential equation for N(t) for t > 0 and determine the N(t). (b) Determine the probability distribution p(t) that expresses the decay probability of each one of the particles as a function of time for t > 0. (c) Calculate the average lifetime τ from the result of (b) as a function of Γ. (d) Write down the general formula for the decay rate of the particle decay A B + C assuming that the matrix element (amplitude) for the reaction is M(A B + C). Note that B and C are distinguishable. (e) Assume that the particles A, B and C from (d) are scalar particles. Explain why M does not depend on the 3-momenta of the particles, but only on the particle masses. (f) Assume that particles B and C are in addition massless. Explain why the modulus square of the matrix element for A B +C must have the form M 2 = a m 2 A, where a is a positive real number. (11) (a) Compute the two-body phase space dπ 2 and work out the decay rate Γ(A B 1 + B 2 ) 3
4 in the case that there is no spin-dependence in the corresponding matrix element M. (b) Consider a generic 2 2 scattering process without spin dependence. Work out the expression of the differential cross section dσ/d cos θ in the center-of-mass frame. (c) Consider the three-body phase space in terms of the variables m 2 12 (p 1 +p 2 ) 2 and m 2 23 (p 2 + p 3 ) 2, where p 1, p 2 and p 3 are the momenta of the decay products. Show that all possible final-state kinematic configurations must be contained in the region defined by (m 1 +m 2 ) 2 m 2 12 (M m 3) 2 and (m 2 + m 3 ) 2 m 2 23 (M m 1) 2. (12) (a) Write down the four γ matrices γ µ, µ = 0, 1, 2, 3 in the Dirac representation. (b) Write the following expressions in terms of the four γ matrices γ µ or the unit matrix: (γ 0 ), (γ k ), (γ µ ). (c) Show the following equalities using the anticommutation relation {γ µ, γ ν } = 2g µν : γ α γ α = 4, γ α γ µ γ α = 2γ µ, γ α γ µ γ ν γ α = 4g µν, γ α γ µ γ ν γ λ γ α = 2γ λ γ ν γ µ (d) Show the following equalities: Tr[γ µ ] = 0, Tr[γ µ γ ν ] = 4g µν, Tr[γ µ γ ν γ ρ γ σ ] = 4(g µν g ρσ g µρ g νσ + g µσ g νρ ), Tr[γ µ γ ν γ σ γ 5 ] = 0. (13) (a) Write down the four Maxwell equations in terms of electric and magnetic fields, E and B, in the presence of a charge density ρ( x) and an electric current j( x). (b) Write the Maxwell equations in covariant form using the 4-vector-potential A µ = (V, A), where V is the electric potential and A is the vector potential, and the 4-current j µ = (ρ, j). (c) Write down relations between V and A and the electric and magnetic fields and determine from them the entries of the 4-tensor F µν = µ A ν ν A µ. (d) Show that the Maxwell equations in the covariant form is invariant under the gauge transformation A µ A µ + µ λ, where µ = / x µ and λ = λ(t, x) is an arbitrary smooth function of space and time. (e) Show that E and B fields are invariant under a gauge transformation. s in e,s in µ (14) (a) Draw the (leading order) Feynman diagrams in QED for the following processes: e + e µ + µ, e µ e µ and e + e e + e resulting from the current-current interaction rule. Specify the momentum carried by each line. (b) Write down, using the current-current interaction rule, the non-spin-summed matrix element for electron-muon scattering, M(e µ e µ ), (i) in terms of the QED currents sandwiched between appropriate initial and final states; (ii) in terms of the corresponding spinors u i ( p i, s i ), v j ( p j, s j ) etc., i.e. using 4-component notation. (c) Calculate the complex conjugate M = M using the relations (γ µ ) = γ 0 γ µ γ 0 und ū = u γ 0. (d) Write the modulus square of the amplitude, summing over the final state particle spins and averaging over the initial state spins, 1 4 s out e,s out M 2 using the relation µ s u( p, s)ū( p, s) = /p + m, where m is the mass of the particle with spinor u( p, s). You can write the result as traces in Dirac space. (e) Carry out the traces using the trace relations Tr[γ µ γ ν ] = 4g µν, Tr[γ µ γ ν γ ρ γ σ ] = 4(g µν g ρσ g µρ g νσ + g µσ g νρ ). (15) (a) The renormalization group equation (RGE) for the energy scale dependent QED coupling 4
5 α = α(µ) that accounts for the effects of the electron, is given by the form µ 2 dα dµ = dα d ln µ 2 = α2 4π β α, with β α 4 3. For which energy scales µ is this formula valid? (b) In what energy range is the QED coupling independent of µ (i.e. β α = 0) and which value does it have for these energy scales? (c) Explain (text plus picture) the basic physical mechanism behind the energy dependence of the QED coupling. (d) Solve the RGE. Quote the result writing α(µ 1 ) as a function of α(µ 0 ). You may neglect the effect of any flavor threshold due to particle heavier than the electron. (e) The solution from (d) for α(µ 1 ) appears to become infinite at some scale µ 1. Give the analytic expression for the scale where this happens. Why is the result not physical? (f) What do we mean by asymptotic freedom in QCD and why is this property so important to perform calculations? (16) Consider The Lagrangian L = 1 4 F µν(x)f µν (x) + (D µ φ(x)) (D µ φ(x)) V (φ (x), φ(x)) where φ(x) is a complex (charged) Klein-Gordon field, F µν (x) = µ A ν (x) ν A µ (x) (where A µ (x) is a U(1)-gauge field), D µ = µ + iqea µ (x) and the potential V (φ (x), φ(x)) = µ 2 φ (x)φ(x) + λ 2 (φ (x)φ(x)) 2. (a) Write down the U(1) gauge transformation for each field and show explicitly (be economic!) that this Lagrangian is invariant under these U(1) transformations. (b) What physical condition is satisfied by the system described by this Lagrangian if one chooses λ > 0? (c) Given λ > 0, what physical pictures correspond to the two cases µ 2 < 0 and µ 2 > 0? (d) For µ 2 > 0, (i) what is the vacuum expectation value of φ; (ii) what do we mean by saying that in this case U(1) gauge symmetry is spontaneously broken; (iii) which interactions are contained in the potential V after spontaneous symmetry breaking; (iv) what is the mass of the physical massive scalar (Higgs) boson of this theory? 5
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