A047W 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 05 Thursday, 8 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.
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 = 0.3894 mb GeV = 0.973 fm GeV = 6.58 0 5 s G F =.66 0 5 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. 0.975(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. A047W
Clebsch Gordan coefficients J 3 3 3 3 m m M + 3 + + 3 + + + /3 /3 0 + /3 /3 0 /3 /3 + /3 /3 J 0 m m M + + + 0 0 0 + + + 0 / / 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. A047W 3 [Turn over]
. The lowest order differential cross section for the charged current reaction ν e e ν e e is given by the formula dσ dy = G s F π ( y), () where G F is the Fermi coupling constant, s is the square of the centre-of-mass energy and y = (E E )/E, where E (E) is the energy of the outgoing (incoming) ν e in the laboratory frame of reference in which the electron is initially at rest. Determine the dependence of the differential cross section on the centre-of-mass scattering angle θ, clearly stating your assumptions. Using the above definition of y, show that y = ( cos θ ) and hence explain the y-dependence in equation. [0] Assuming that the proton is composed of quarks, but neglecting anti-quarks and explaining any further assumptions, show that in the parton model the differential cross section for ν µ scattering on protons is given by dσ(ν µ p) dx dy = G F s x π ( y) q(x), where q(x) is the parton density function for quarks and x is the momentum fraction carried by a quark in the proton. Show that the total cross section for deep inelastic scattering of ν µ p scales as E, where E is the energy of the ν µ in the laboratory frame of reference. Explain the physical significance of this scaling behaviour. [7] Explain how intense beams of high energy ν µ can be produced. Describe the key features of a suitable detector for studying deep inelastic scattering using beams of ν µ, and explain how the momentum of the outgoing muon and the energy of the scattered quark can be measured. [8] A047W 4
. Draw Feynman diagrams showing the allowed Standard Model couplings of gauge bosons to high-energy fermions. Indicate on your diagrams the helicity states of the fermions involved, and identify the diagrams and gauge bosons that are relevant for high-energy e + e µ + µ. [6] For e + e µ + µ, at high centre-of-mass energies but considering QED processes only, find the dependence of the interaction cross section on θ, the centre-ofmass scattering angle between same-charge incoming and outgoing fermions. List any assumption you make. [7] Experimental data from e + e collisions at centre-of-mass energies near the Z 0 mass show that the angular distribution of e + e µ + µ is not symmetric. Which processes beyond QED-only have to be included to explain this result? Calculate the angular distribution and comment on the difference with respect to the pure QED result. Explain briefly how this angular distribution could be applied to measure an important parameter of the Standard Model. [] A047W 5 [Turn over]
3. The left figure below shows data collected from e + e collisions with a centre-ofmass energy of s 3. GeV. The resonant state, known as J/ψ, is observed decaying to hadronic, muonic and electronic final states. Branching Fraction J/ψ e + e 6.0 0 J/ψ µ + µ 6.0 0 J/ψ ρ + π 5.6 0 3 J/ψ ρ 0 π 0 5.6 0 3 J/ψ ρ π + 5.6 0 3 J/ψ K 0 S K0 L. 0 4 J/ψ K 0 S K0 S not seen J/ψ p p. 0 3 η c p p.5 0 3 Explain how the isospin and J PC of the J/ψ may be inferred from the data and the information provided in the table and figure to the right. [4] Write down the Breit-Wigner formula for the cross section of the initial e + e state going to an e + e final state via the J/ψ resonance. Integrate and find an expression relating the total width of the resonance to the cross section. [6] Using this result and the data, estimate the natural width of the J/ψ resonance, given the acceptance efficiency of the experiment is 50%. With reference to the OZI rule, explain how the quark content of this state is inferred. [6] After the J/ψ discovery, several resonances with the same quark content were observed. Copy the energy level diagram above and, for each state, explain what its angular momentum quantum numbers are and hence deduce their J PC. [4] Four of these states are observed decaying to two photons. State which ones they are and why. The mass spectrum shows that the lowest mass state is η c. Why was the J/ψ discovered first? Both the η c and the J/ψ decay to a proton-antiproton pair. In each case, is there an angular dependence on the proton s direction? [5] A047W 6
4. In the two-flavour model of neutrino mixing, the mass eigenstates are related to the flavour eigenstates by one mixing angle, θ, ( ) ν = ν 3 ( cos θ sin θ sin θ cos θ ) ( ) νµ ν τ Using this model, and stating any assumptions made, derive the transition probability P (ν µ ν τ ), and hence the survival probability ( P (ν µ ν µ ) = sin (θ) sin.7 m L ) E where L [km] is the distance from creation of the muon neutrino of energy E [GeV], and m [ev ] is the difference between the squared masses of the two eigenstates. { Hint: consider a behaviour of the form ν a (t) = ν a exp [ i(e a t p a x)/ h ]. } [0] The minos detector is located 730 km from Fermilab in a ν µ beam. It consists of alternate layers of scintillation counters and magnetised iron plates oriented perpendicular to the beam. Describe how the ν µ is produced at Fermilab from a proton accelerator, and how the energy of the interacting ν µ is estimated by minos. [7]., The figure above shows the energy distribution of ν µ observed in the detector and the expected distribution with and without oscillations. Estimate m [ev ] given that θ π/4. What limits the precision of this experiment? How does your result compare to the m measured in solar neutrino experiments? What is the evidence from other experiments that in this regime of L/E, the dominant oscillation mode is ν µ ν τ? [8] A047W 7 [Turn over]
5. Compute the threshold energy for a ν τ to produce a charged current interaction when colliding with a stationary target. [4] The DONUT experiment was the first to observe tau-neutrinos directly. The ν τ beam was produced by colliding the 800 GeV Fermilab Tevatron beam into a block of material large enough to stop the charged pions and kaons. Why is it helpful to stop the π ± and K ±? The ν τ was made mainly from the decay of the D S meson. Draw Feynman diagrams for each interaction in the production sequence from the beam protons to the ν τ. Describe briefly how the CKM matrix can be used to explain the relative decay rates of different quarks and give the underlying CKM formalism. Explain why the production rate of ν τ from D S is larger than that from D ± and D 0 or any other particle produced in the interactions, given that the production cross sections are σ (D ± ) = µb/nucleon, σ ( D 0) = 7 µb/nucleon and σ (D S ) = 5. µb/nucleon. [4] The DONUT neutrino beam had a ν τ fraction of 0%, and 45% each of ν e and ν µ. Explain why it is not possible to get a beam close to 00% pure in ν τ and why it is plausible for the ν e and ν µ fractions to be similar. Briefly explain how it is possible to have near 00% pure sources of ν e and ν µ with specially optimised beam lines or accelerators. [7] A047W 8
6. Write down the Dirac equation for free electrons. Derive the adjoint Dirac equation and an expression for the probability current 4-vector. [6] Free electron positive energy solutions of the Dirac equation in the standard representation are ( ) where ψ s (r, t) = N χ s σ p E+m χs s =,, χ = ( 0 ) exp [ i(p r Et)/ h ],, χ = N is a normalization factor, and the other symbols have their usual meaning. Find N and explain why these wave functions are not normalized to one particle per unit volume. [6] The operator for spin projection on the x-axis is Σ = ( ) σ 0. 0 σ Find its eigenvalues and eigenvectors, and show that in the non-relativistic limit one can form linear combinations of ψ (r, t) and ψ (r, t) that are eigenvectors of Σ, but that at relativistic energies this is impossible. [8] Briefly describe how to produce an electron beam and how to make a polarised electron beam? [5] ( 0 ), A047W 9 [Turn over]
7. Draw Feynman diagrams for the decays of τ into leptons and hadrons. Find the branching fraction for τ -decays into hadrons, stating all assumptions you make. How is it possible to have final states involving both leptons and hadrons? Give examples and discuss whether these have a significant branching fraction. [7] Pairs of tau (τ + τ ) are produced at an e + e collider running at a centre-ofmass energy s = m Z c. What other pairs are produced? How can (τ + τ ) events be identified? [6] How can τ decays be utilized to (i) show that the τ carries a conserved tau-lepton number and has an associated neutrino; (ii) test lepton coupling universality; (iii) determine the mass of the τ neutrino? [4] Final states with charged hadrons with an average invariant mass of.5 GeV/c have been observed. Estimate the range of corresponding hadronic jet energies measured in a detector. Show that the sharing of energy between the hadronic products in the detector frame depends on the helicity state of the τ. Which helicity state gives on average the higher hadronic energy? [8] A047W 0
8. Write short accounts on THREE of the following topics. (a) The experimental determination of the helicity of the neutrino. (b) The theoretical background and experimental detail related to the detection of solar neutrino oscillation. (c) The motivation for the existence of a top-quark in the Standard Model and its discovery. (d) How values of the elements in the CKM matrix are measured. (e) The detector sub-systems (purpose and principle of operation) of a general purpose detector for colliding beams. [5] A047W [LAST PAGE]