SECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS

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1 754 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS TRINITY TERM 04 Thursday, 9 June,.30 pm 5.45 pm 5 minutes reading time Answer four questions. Start the answer to each question in a fresh book. A list of physical constants and conversion factors accompanies this paper. The numbers in the margin indicate the weight that the Examiners anticipate assigning to each part of the question. Do NOT turn over until told that you may do so. Pages and 3 contain particle physics formulae and data for this paper. The questions start on page 4.

2 C4 Particle Physics formulae and data Unless otherwise indicated, the questions on this paper use natural units with h = c =. The energy unit is GeV. Cross sections Length Time Fermi constant GeV = mb GeV = fm GeV = s G F = GeV Dirac (Dirac-Pauli representation) and Pauli matrices ( ) ( ) γ 0 I 0 0 σ =, γ =, 0 I σ 0 γ 5 = ( ) ( ) 0 0 i σ =, σ 0 =, i 0 σ 3 = ( 0 I I 0 ( 0 0 ) ) Rotation matrices j, m e ijyθ j, m = d j m m (θ) d / ++ = d / = cos(θ/) ; d / + = d / + = sin(θ/). d = d = ( + cos θ)/ ; d = d = ( cos θ)/ ; d 00 = cos θ ; d 0 = d 0 = d 0 = d 0 = sin θ/. Spherical harmonics Y m l (θ, φ) 3 4π 5 Y0 0 = 4π ; Y 0 = 5 Y 0 = 6π (3 cos θ ) ; Y ± = ± 3 cos θ ; Y = 8π sin θ e±iφ. 8π sin θ cos θ e±iφ ; Y ± = 5 3π sin θ e ±iφ. CKM quark mixing matrix The mixing of the charge e/3 quark mass eigenstates (d, s, b) is expressed in a 3 3 unitary matrix V : d V ud V us V ub s = V cd V cs V cb b V td V ts V tb The magnitudes of the elements, derived from the Particle Data Group 004 tables, are given below. The number in brackets gives an estimate of the uncertainty in the last digit. Note that these values may not give an exactly unitary matrix, but this has no significance (0) 0.4(3) 0.004() V = 0.4(3) 0.974() 0.04() 0.009(5) 0.040(3) 0.999(0) d s b. 754

3 Clebsch Gordan coefficients J m m M /3 /3 0 + /3 /3 0 /3 /3 + /3 /3 J 0 m m M / / 0 + / / + /6 / /3 0 0 /3 0 /3 + /6 / /3 0 / / 0 / / Breit-Wigner resonance formula The formula represents the energy dependence of the total cross-section σ(i f) for unpolarised scattering between a two-body initial state i to a final state f, in the vicinity of a resonance of rest-mass energy M, spin J and total width Γ. σ(i f) = π λ Γ i Γ f g [(E M) + Γ /4], where λ = hc pc, g = J +, p is the magnitude of the centre-of-mass momentum of the initial state particles, s a, s b are their spins and Γ i, Γ f the initial and (s a + )(s b + ) final state partial widths [Turn over]

4 . Explain how Young s tableaux are used to represent the combination of two spin- particles as = 3. Add a third tableau to derive a representation for the combination of three spin- particles. [5] Starting with the symmetric state, 3, +3 =, derive the eight orthogonal combinations of three spin- particles grouping them according to the representation derived in the first part of the question. [6] What is isospin? What assumptions underpin its usefulness? What is its relation to angular momentum spin? Hence write down the eight combinations of three isospin- particles with an appropriate notation. [6] The baryon wavefunction consists of four parts, ψ = ψ spatial ψ spin ψ flavour ψ colour. Briefly explain how the observation of the J=S= 3, ++ baryon motivates the inclusion of ψ colour. Deduce the normalised spin-flavour wavefunction for the proton in the static quark model and clearly state any important assumption. [8]. A free particle solution of the Dirac equation is Ψ(x) = C(p) exp( ip x), where p = (E, 0, 0, p 3 ). Find the normalized E > 0 spinors C + (p) and C (p) for positive and negative helicity states in the standard Dirac representation and using covariant normalization. [5] The operator for spin projection on the x-axis is Σ = ( ) σ 0. 0 σ Show that neither C + (p) nor C (p) is an eigenstate of Σ. [3] The expectation value of Σ is Σ = E C (p)σ C(p), where C = C + (p) cos α + C (p) sin α, with α a real constant, and normalization is to unit volume. Calculate Σ and interpret the result in the non-relativistic and the ultra-relativistic limits for α = ±π/4. [7] 754 4

5 3. In the simple quark-parton model the cross section for deep inelastic electron proton scattering is given by d σ dx dy = i xf i (x) πα s [ Q 4 qi + ( y) ] for Q MZc, () where s is the square of the e-p centre-of-mass energy, Q is the negative of the squared 4-momentum transfer between the electron and the proton, and y is a function of the scattering angle θ in the e-q centre-of-mass frame, y = ( cos θ). Justify qualitatively the form of the above expression, explaining the form of the dependence on y, s and α. What do the variable x, the parameter q i and the function f i (x) represent in the quark-parton model? [0] What is meant by the term Bjorken-scaling? How does Quantum Chromodynamics modify the predictions of the quark-parton model? [3] Starting from equation (), but restricting your model to u and d flavours only, show that the proton structure function F ep may be written as F ep = 4 9 x[u(x) + ū(x)] + x[d(x) + d(x)], 9 and define the terms. If scattering is performed using a deuteron target how would this formula be modified? [4] Deep inelastic scattering can also be performed using neutrino probes. The quarkparton model expression for neutrino-proton scattering through the exchange of a W boson is given by d σ dx dy = sg F π i xf i (x) + ( y) ī x f i (x), for Q MWc. () Explain the form of this expression, paying particular attention to the y-dependence and the flavours entering the sums. How would the flavour structure be modified for neutrino-deuteron scattering? By comparing your expression to the general form d νd σ = sg F dx dy 4π ( ) M [ ] W MW + ( + ( y) )F νd Q + ( ( y) )xf3 νd, write down the quark-parton model prediction for the F νd structure function and hence derive the relationship F ed = 5 8 F νd. [8] [Turn over]

6 4. What is meant by the universality of charged current weak interactions as applied to leptons? Explain how the CKM matrix is used to generalize the universality to the quark sector. Illustrate your answer by listing the possible decay modes of a W boson at the lepton or quark level and giving the applicable vertex factors. [7] When the final state of W decays include quarks, there can be effects due to second order strong interactions. Illustrate what such a second order strong interaction is with Feynman diagrams of W decays. The total effect of all these interactions has been computed to increase the partial width of the W to quark states by a factor ( + α/π), where α is measured from other experiments to be 0.9 ± By using the factors obtained above and assuming lepton universality, obtain an expression for each of the decay modes of the W in terms of the branching ratio B (W lν l ), where l is a lepton. Hence obtain an expression for B (W lν l ) in terms of relevant CKM matrix elements and α. [5] The current best measurements of V cs are less accurate than the other elements of the CKM matrix involved in the decay of the W. Use your expression from above, along with the measurement of B (W lν l ) = (0.83±0.07±0.07)%, to obtain a value for V cs. Obtain an improved value under the assumption that the CKM matrix is unitary. [5] Explain how the following are sensitive to one or more elements of the CKM matrix and may be used to make a measurement of CKM elements: (a) The ratio of the decay rates of K + π 0 e + ν e and π + π 0 e + ν e. (b) The rate at which muon pairs are produced in interactions of muon neutrinos with nuclei. (c) The muon energy spectrum of B decays, in particular near the high-energy end of the spectrum. [8] 754 6

7 5. Explain what the observation of both K ± π ± π 0 and K ± π ± π + π decays says about parity invariance in weak decays. Explain further, why two neutral kaons are observed with very different lifetimes despite almost identical mass: τ(k ) = 90 ps; τ(k ) = 5 ns. [6] Neutral kaons are produced in a state of definite flavour by a pion beam incident on a proton target. The time evolution of the kaons strangeness is examined by reconstructing semileptonic K π ± l ν decays. Using Feynman diagrams, illustrate and explain these two statements. [4] The time evolution of the neutral kaons may be written as [( ) ] [( ) ] K 0 (t) = exp im Γ t K + exp im Γ t K, where M () and Γ () are the mass and decay constant of the K () state. Derive expressions for the intensity of the K 0 and K 0 components as a function of time in terms of the mass difference, M = M M. Taking M = ps, sketch both the K 0 and K 0 intensity curves on a graph of fraction of the initial intensity versus proper time during the first 000 picoseconds. [8] The positive sign of M is known from measuring the rate of K regeneration in a K beam after it is passed through a block of material. Explain qualitatively how this regeneration can occur. [3] The average lifetimes and mass differences from other neutral meson systems are given in the table. Comment on the observability of their flavour oscillations. D B d B s τ ps M ps [4] [Turn over]

8 6. For each of the following processes, give an argument (quantitative, where possible) whether the final state given is the most likely final state and in case where it is clearly not, give an example of a more likely one. For processes that are forbidden, rare or strongly suppressed, explain why. Detailed phase space calculations are not required. For the W, t and H consider the decays into quarks and ignore the step where hadrons are formed. (a) + n π + (b) K 0 S π+ e ν e (c) H ZZ (d) D 0 π + π (e) t Wb (f) W e ν (g) K + π + ν e ν e (h) K 0 L µ+ µ (i) J/Ψ e + e (j) π + e + ν e [5] 754 8

9 7. Fermi s Golden rule for a system with total energy E and total momentum P is m Γ fi = (π) 4 n M fi d 3 p j E i= i (π) j= 3 (E j ) δ P n n p j δ E E j, j= j= }{{} Lorentz invariant phase space for m initial particles with energy E i and n final state particles with energy and momentum E j and p j respectively. By switching to polar coordinates and making use of the Dirac delta-function identity, δ(f(x)) = df dx a δ(x a), prove that the partial width of a two-body decay of a particle is Γ fi = p 3π m i M fi dω, where dω is the solid angle infinitesimal, m i is the mass of the decaying state, and p is the magnitude of the momentum of the final state particles in the centre of mass frame. Furthermore show that by energy conservation, p = [m i m (m + m ) ][m i (m m ) ] i [] [6] Give p for the case m = 0. Hence calculate the ratio of phase space factors for the decays π e ν e and π µ ν µ. The measured value for the branching ratio Γ (π e ν e ) /Γ (π µ ν µ ) is Explain what needs to be included in the calculation of the branching ratio to achieve better agreement with the experimental value. What would be the lifetime of the charged pions if the leptons were massless? [8] 8. Write short accounts on three of the following topics. (a) The determination of sin Θ W, where Θ W is the weak mixing angle. (b) The experimental evidence for gluons. (c) The determination of the helicity of the neutrino. (d) Electroweak symmetry breaking and the search for the Higgs boson. (e) The number of different neutrino types is 3. [5] [LAST PAGE]