Introduction to Elementary Particle Physics 2nd edition Alessandro Bettini - Solutions

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Give the values of the isospin, its third component and the hypercharge for μ+L , τ+R ,t̅L, b̅R.
Give the values of the isospin, its third component and the hypercharge for eL, νμR, uL, dR, and for their antiparticles.
Draw the Feynman diagrams at tree-level for the elastic scattering v̅ee-. What is different in vee–?
The μs have the same interactions, electromagnetic and weak, as the electrons. Why does a μ with energy of a few GeV pass through an iron slab, while an electron of the same energy does not?
What are the differences between a neutrino and an antineutrino? What are the conserved quantities in neutrino scattering? Complete the missing particle in vμ + e–→ μ– + ?. If neutrinos are massless, what is the direction of their spin? And for anti neutrinos? The Universe is full of
Find the value of the Fermi constant GF in S.I. units, knowing that GF/(ℏc)3 = 1:17 x 10-5 GeV-2:
Draw the (lowest-order) Feynman diagrams for (a) ϕ → K+K-, (b) ϕ → K0K̅0,(c) ϕ → π+ π- π0.
A beam of α particles of kinetic energy E = 10 MeV and intensity I = 1 μ A hits a lead target [A = 207, Z = 82, ρ = 1.14 x 104 kg m-3] of thickness t = 0.2 mm. We locate a counter of area S = 1 cm2 at a distance of l = 0.5 m beyond the target at the angle θ = 40°. Neglecting, when necessary,
We observe the elastic scattering of E = 5 GeV electrons from protons [mp = 938 MeV] at the angle θ = 8 and we measure their energy. (1) What is its expected value (neglecting the electron mass)? (2) What is the scattered electron energy in the CM frame?
Consider the processes (1) e+ + e- → μ+ + μ- and (2) e+ + e- → hadrons at the two centre of mass energies √s = 2 GeV and √s = 20 GeV. Calculate the ratio of the cross-section of process (1) at the two energies. Calculate (approximately) the ratios of the cross-sections of the two
As in the previous problem, but for the Λ hyperon.
As in the previous problem, but considering the quarks c and c̅.
How many gluons exist? Give the electric charges of each of them. Give the values of their strangeness, charm and beauty. What is the gluon spin? How many different quarks exist for every flavour? What are their charges? Does QCD define the number of families?
Draw the (lowest-order) Feynman diagrams for (a) π0→γγ, (b) Σ0→→Λγ, (c) e- + e- →e- + e‑, (d) e+ + e+→e+ + e+, (e) e+ + e-→e+ + e-, (f) e+ + e-→μ+ + μ-, (g) e+ + μ+ → e+ + μ+.
Calculate the reciprocal of the fine structure constant at Q2 = 1 TeV2, knowing that α-1 M2Z = 129 and that MZ = 91 GeV. Assume that no particles beyond the known ones exist.
Two photons flying in opposite directions collide. Let E1 and E2 be their energies. (1) Find the minimum value of E1 to allow the process γ1 + γ2 → e+ + e- to occur if E2 = 10 eV. (2) Answer the same question if E1 = 2E2. (3) Find the centre of mass speed in the latter
Draw the tree level diagrams for the Compton scattering γ + e- → γ + e-
Write the scattering amplitudes of the following processes in terms of the total isospin amplitudes: (1) K- p → π - Σ+, (2) K-p → π0Σ0, (3) K-p → π+Σ-, (4) K0p → π0Σ+, (5) K0p → π+Σ0
A baryon decays strongly into Σ+π– and Σ–π+, but not into Σ0π0 nor into Σ+π+, even if all are energetically possible. (1) What can you tell about its isospin? (2) You should check your conclusion looking at the ratio between the widths in the two observed channels. Neglecting
What are the possible electric charges in the quark model of (a) a meson and (b) a baryon?
Suppose you do not know the electric charges of the quarks. Find them using the other columns of Table 4.5. Table 4.5. Quantum numbers and masses of the quarks. I I. S C в т в Y Mass d -1/3 1/2 -1/2 1/3 1/3 4.8+0.7 MeV -0.3 +2/3 1/2 +1/2 1/3 1/3 2,3 +0.7 MeV -0.5 -1/3 -1 1/3 -2/3 95+5 MeV +2/3
Explain why each of the following particles cannot exist according to the quark model: a positive strangeness and negative charm meson; a spin 0 baryon; an antibaryon with charge +2; a positive meson with strangeness –1.
Consider the following three states: π0, π+π+π– and ρ+. Define which of them is a G-parity eigen state and, in this case, give the eigenvalue.
The state Δ(1232) has isospin I = 3/2.(1) What is the ratio between the decay rates Δ0 → pπ- and Δ0 → nπ0?(2) What would it have been if I = 1/2?
An η meson decays into 2γ while moving in the x direction with energy Eη = 5 GeV.(1) If the two γs are emitted in the +x and -x directions, what are their energies?(2) If the two γs are emitted at equal and opposite angles ±θ with x, what is the angle between the two.
For each interaction type, strong (S), electromagnetic (EM) and weak (W), insert a Yor N in the cell of every quantum number, depending on whether it is conserved or not (I = isospin, Iz its third component, S strangeness, B baryon number, L lepton number, T time reversal, C particle?antiparticle
Chamberlain and co-workers employed scintillators to measure the pion lifetime. Why did they not use Geiger counters?
Consider the following particles and their lifetimes. ρ0: 5 x 10-24 s, K+: 1.2 x 10-8 s, η0: 5 x 10-19 s, μ-: 2 x 10-6 s, π0: 8 x 10-17 s. Guess which interaction leads to the following decays: ρ0 → π++π-; K+ 2192 π0+π+; η0 →π++π-+π0; μ- → e- + ν̅e + νμ; π0 → γ + γ
A π+ beam of 12 GeV momentum is sent into a bubble chamber. An event is observed with two positive tracks originating from the primary vertex and two V0s pointing to that vertex. The measurements of the momenta of the two tracks of the first V0 and of the angle between them give p1+ = 0.4 GeV,
Consider the π- capture at rest, giving the reaction π-+ p → n + γ. Calculate the energy of the photon and the kinetic energy of the neutron.
Consider the Ξ0 hyperon produced in the reaction π+ + p →K+ + K+ + Ξ0. Knowing the isospin and its third component for all the other particles, establish I and Iz of the Ξ0.
Which is the X particle at the second members of the following reactions: (a) π-+ p → Σ0 + X,(b) e++ n → p + X, (c) Ξ0 → Λ + X?
Which of the following reactions are forbidden/allowed? Justify your statements.(a) μ-→ e-+ γ,(b) π+ → μ+ + νμ + νμ,(c) Σ0 → Λ + γ,(d) γ + p → π0 + p,(e) p → π0 + e+,(f) π -→ μ-+ γ.
Consider the search for proton decay in a water Cherenkov detector. The refraction index is n=1.33. Find(1) the minimum velocity of a charged particle for emitting radiation,(2) the minimum kinetic energy for an electron and a K meson to do so, and(3) determine whether the charged particle in the
The cosmic radiation also contains neutrinos and antineutrinos in an extremely wide energy range. Neutrinos propagate on cosmological distances without attenuation due to their very small cross-sections with any target. However, at extremely high energies the resonant process ‘νx + ν̅x → Z
76Ge, 130Te and 136Xe are very stable nuclei. If a neutrino is a Majorana particle it might decay via the double beta mechanism without neutrinos. Assume a half life T1/2 = 1027 for each. What is the average number of decays expected in one year in 1 t of each isotope?
The highest-energy cosmic ray protons produce charged pions colliding with the microwave background with the reaction p + γ → n + π+ (Greisen, Zatsepin,Kuzmin effect). The neutrinos from the decays of the pions have extremely high energies, say of the order of 10 EeV (Berezinsky, Zatsepin
The power irradiated by the Earth through its surface is about 40 TW. A large fraction of this energy is due the decay of radioactive isotopes. The most important are 238U, 232Th and 40K. In the ‘standard’ model built by the geologists, the Bulk Silicate Earth Model, radioactivity is assumed to
Consider a possible experiment looking for neutrino oscillations using as a source a nuclear reactor complex of 3 GW, producing 6 x 1020 s-1 electron antineutrinos. You are planning a detector containing free protons (such as a liquid scintillator) to observe the reaction ν̅e + p → e+ + n. In
Electron antineutrinos from natural or artificial sources have typical energies Eν of a few MeV. They can be detected by observing the reaction ν̅e + p → e+ + n in a detector medium containing free protons. The process is immediately followed by the positron annihilation e+ + e- → 2γ and by
In the OPERA experiment a νμ beam is produced at CERN and aimed to the detector located at Gran Sasso at a distance L = 730 km (CNGS = CERN Neutrinos to Gran Sasso). The experiment is designed to detect the appearance of tau neutrino by detecting the CC reaction ντ + n → τ- + p. Neutrino
Consider a neutrino beam produced in a proton accelerator facility. The accelerated proton beam is driven on a target. The emerging pions are focussed in the forward direction and then drift in a vacuum pipe in which they decay. At the end of the decay volume all the charged and neutral particles
One of the source of neutrinos in the Sun is the decay 8B→2α + e+ + νe. The boron neutrinos dominate the energy spectrum around 10 MeV. Consider the energy interval 9 MeV < Eν < 11 MeV. In absence of oscillations their flux at the Earth should be Φ =1010 m–2s–1. The Super-Kamiokande
Consider the muon neutrinos generated by the decays of the mesons produced by the collisions of cosmic rays in the atmosphere (‘atmospheric neutrinos’). Their energy spectrum at the surface of the Earth extends over several orders of magnitudes,decreasing with energy roughly as
In the T2K experiment a νμ is produced at the J-PARC proton accelerator and detected in the water Cherenkov detector, Super-Kamiokande. The direction of the incoming neutrino is known, but not its energy Eν. Consider the ‘quasi-elastic’interaction νμ + n → μ° + p. The energy Eμ of the
In the T2K experiment the proton beam accelerated by the J-PARC high-intensity proton synchrotron is driven on a target. The emerging pions are focussed in the forward direction and then drift in a vacuum pipe in which they decay. At the end of the decay volume all the charged and neutral particles
An important reaction producing electron neutrinos in the Sun is e- + 7Be → νe +7Li In the vast majority of the cases (90%) the Li nucleus is produced in its ground state. Consequently, the ‘Be neutrinos’ energy spectrum is monoenergetic with Eν = 0.862 MeV. The corresponding total neutrino
Atomic electrons interact with the nucleus with not only the electromagnetic interaction (photon exchange) but also with neutral current weak interactions (Z exchange). The latter contribution is extremely small but the interference of the two gives detectable parity violation effects. The main
Find the relative values of the amplitude of couplings of the Vand A currents to Z for νe, e-, u, d, assuming sin2θW = 0.23.
Compare the NC scattering of νe and of νμ of the same energy on the valence quarks of a nucleus. Both cross-sections are proportional to a combination of the Z-charge factors squared of the quarks. Consider the ratio of the contributions of the left and right quarks in the two cases. How large
Consider the NC scattering of ν̅μ on the valence quarks of a nucleus. Write down the Z-charge factors squared of the quarks and their values, assuming sin2θW = 0.23. The total ν̅μ cross-section is proportional to a combination of the Z-charge factors squared of the quarks. Write it and
Consider the NC scattering of νμ on the valence quarks of a nucleus. Write down the Z-charge factors squared of the quarks and their values, assuming sin2θW = 0.23. The total νμ cross-section is proportional to a combination of the Z-charge factors squared of the quarks. Write it and evaluate
The largest fraction of the higher-energy electron neutrinos, the so-called ‘boron neutrinos’ are produced in the Sun by the decay 8B → 2α + e+ + νe. Their maximum energy is Eν, max = 16 MeV. Consider the elastic collision of a boron neutrino with a nucleus of a 76Ge detector. Is the
In 1960, S. Glashow pointed out the possibility of establishing charged weak bosons via the resonant reaction νe + e+ → W+ → μ+ + νμ. Why did the idea not work in practice?
(1) Establish for each of the following decays whether it is allowed or forbidden, giving the reasons. The left upper label is the colour, the right lower one the chirality. (a) W-→BsL+Bu̅R,(b) W-→BdR+Bu̅L,(c) W-→RdL+Bu̅R,(d)
(1) Consider a pair of quarks with colours R and B and a third quark G. Establish whether the force of the pair on G is attractive or repulsive for each of the combinations RB + BR and RB - BR. Considering quarks of the same colour, establish which of the following processes are allowed or
The largest fraction of matter in our Galaxy is invisible. It might consist of particles similar to neutrinos but much more massive, the ‘neutralinos’. Let us indicate them by χ and let mχ be their mass. According to one theory these particles coincide with their antiparticles. The
Calculate the ratio between the CC cross-sections of neutrinos and antineutrinos on nuclei with the same numbers of neutrons and protons, considering only the valence quark contributions. Repeat the calculation for NC interactions.
If there were more than three families, the Z would decay into more neutrino– antineutrino channels. Given the existing limits on the masses, however, the charged leptons and quark channels of the new families would be closed. Consequently, the Z width would increase by Γνν for every extra
Working at the Z with an electron–positron collider and assuming statistical uncertainty only, how many events are needed to exclude at five standard deviations the existence of a fourth neutrino?
What is the percentage variation of the Z total width for an additional neutrino type? What is the variation of the peak hadronic cross-section?
LEP 2 was designed to study the process e+e- → W+W-. If the cross-section at √s = 200 GeV is σ = 17 pb and the luminosity is L = 1032 cm-2s-1, find the number of events produced per day.
Consider the p̅p Tevatron collider working at √s ≈ 2 TeV. For a Z produced at rest, what are the approximate momentum fractions of the annihilating quark and antiquark? Evaluate in which fraction they are sea quarks. If the Z is produced with a longitudinal momentum of 100 GeV what are the
At 1 GeV energy the weak charge g is larger than the electric charge √α by √4π = sin θW ≈ 7:4. Why is the electrostatic force between two electrons at 1 fm distance so large compared to the weak force?
Consider the prediction for the W and Z before their discovery, in round numbers mZ = 90 GeV and MW = 80 GeV. If it is sin2θW ≈ 0.23 with an uncertainty of 20%, what is the uncertainty on MW? If we measure MZ/MW with 1% uncertainty, what is the uncertainty on sin2θW?
Consider the Z production at a proton–antiproton collider and its decay channel Z→ e+e-. The energies of the two electrons as measured by the electromagnetic calorimeters bare E1 = 60 GeV and E2 = 40 GeV. The energy resolution is given by σ(E) / E = 0:15=√E. The measured angle between the
A Z is produced in a p̅p collider working at √s = 540 GeV. The Z moves in the direction of the beams with a momentum pZ = 140 GeV. It decays as Z→e+e- with electrons at 90° to the beams in the Z rest frame. Calculate the two electron energies in the laboratory frame.
Calculate the cross-section σ(e+e- → μ+μ-) at the Z peak and σ(u̅d → e+νe) at the W peak.
Assume that the number of neutrinos with mass << MZ is 3, 4, or 5 in turn, without changing anything else. Evaluate for each case the Z branching ratio into μ+μ- and the ratio Γμμ/ΓZ. Evaluate the ratio of the cross-sections at the peak for e+e- into hadrons for N = 3, 4 and 5.
Assume the cross section value σ(u̅d → e+νe) = 10 nb at the W resonance. Evaluate the total σ(u̅d → q̅q) cross-section at resonance.
Evaluate the ratio gZuu2/gWud′2 and the decay rates ratio Γ(Z → uu̅) = Γ (W → d′ u̅).
Evaluate the ratio gZee2/gWeν2 and the decay rates ratio Γ(Z→e+e-)/Γ(W→ e+νe).
Evaluate the branching ratio for W →e+νe.
Calculate the partial and total widths of the Z for sin2θW = 1/4. Find Γμμ/Γh. [ MZ = 91 GeV.]
A ‘grand unification’ theory assumed the existence at very high energies of a symmetry larger than SU(2) ⊗ U(1), namely SU(5). A prediction of the theory, which was experimentally proved to be false, was the value of the weak angle, sin2θW = 3/8. Find the partial widths of the Z in this
Establish which of the following processes, which might be virtual, are allowed and which are forbidden, and give the reasons: d→W- +uL, Z→eL++eL-, W+→Z+W+,W+ → e- L̅+ v̅L.
Establish which of the following processes, which might be virtual, are allowed and which are forbidden, and give the reasons: W- → dL + u̅L, W- → uL + u̅R, Z → W-+W+, W+ → e+R+ ν̅L.
Consider the CC cross-sections for neutrinos and antineutrinos on nuclei containing the same number of neutrons and protons. Their masses can be neglected in comparison with their energies. Show that the neutrino total cross-section is three times larger than that of the antineutrino.
To produce a narrow band νμ beam, one starts from an almost monochromatic π+ beam (neglecting K+ contamination) and lets the pions decay as π+ → μ++νμ. Assume the pion energy to be Eπ = 200 GeV. Find the neutrino energy in the π+ rest frame. In the laboratory frame, the
Consider an electron of energy Ee = 20 GeV detected by the CHARM2 experiment, produced by an elastic νμ e scattering. How large can the scattering angle be, at most? Evaluate the accuracy that is necessary in the measurement of the electron direction to verify this to be the case. Can we
The CHARM2 experiment at CERN studied the reaction νμe- → νμe- using a ‘narrow band beam’ with mean energy 〈 Eνμ 〉= 24 GeV. The Super Proton Synchrotron provided two pulses Δt = 6 ms apart, with a cycle of T = 14.4 s. The useful mass of the target detector was M = 547 t, the
Consider the decays K01 → π+π-π0 and K02 → π+π-π0. Call l the angular momentum of the π+ π- system, Iππ its isospin, and L the angular momentum of the π0 relative to it. Suppose that the mass of the K were much larger that its actual value and that higher values of l and L were
Prove expression
Prove expression neglecting terms of orders above the first in ε and ε′. )/A(K° →n°n° 0_0 2 1 + 4Reɛ'
Prove expression neglecting terms of orders above the first in ε and ε′. «(* -*)/4«*. x*x)/A(K° (Kº → x*x¯) 1 – 2Rez'. AK → n*n¯
Consider an integrated luminosity of 100 ev/fb of the BaBar experiment. In the laboratory frame the centre of mass moves with the average Lorentz factor βγ = 0.56. How many seconds at a luminosity L = 1034 cm-2s-1 would be needed to collect such sample? Assuming the value ΔR = 3 at the ϒ(4S)
An asymmetric beauty factory operates at the ϒ(41S3), namely at √s = 10580 MeV, to study the process e+e- → B0B̅0. ‘Asymmetric’ means that the centre of mass moves in the reference frame of the collider. Consider an event in which both mesons are produced with the Lorentz factor βγ =
Consider a neutral K-meson beam with momentum pK = 400 MeV, impinging on a liquid hydrogen target and determine the reaction channels open for each component K0 and K̅0. Estimate which has the larger cross-section.
Consider the reactions π- p →K0 + X and π- p → K̅0 + Y and establish the minimum mass of the states X and Y that are compatible with the conservation laws and the two corresponding energy thresholds.
An experiment needs an almost monochromatic K+ beam with momentum p = 2 GeV. We obtain it by building a magnetic spectrometer and a system of slits. However, the total length of the beam is limited by the lifetime of the K+. At what distance is the K+ intensity reduced to 10% of the initial value?
A π– is sent onto a target producing neutral K-mesons and Λ hyperons. Consider the component of the resulting K beam with momentum p = 10 GeV. What is the ratio between KS and KL at the production point? What is it at l = 10 m from the production point? Determine the fraction of decays into 2π
From which of the pp initial states 1S0, 3S1, 1P1, 3P0, 3P1, 3P2 can each of the following reactions, p̅p→ K+ K–, p̅p → K01 K01 and p̅p → K01 K02 proceed?
The DAFNE Φ-factory at Frascati is an e+e– collider at the centre of mass energy equal to the ϕ mass. Calculate the ratio between the annihilation rates into K+K– and K0K̅0 neglecting the mass difference between charged and neutral kaons. Is this a good approximation? Considering the case
Consider the two beta decays of the Σ– hyperon Σ-→n + e- + ve of branching ratio 1.017 ± 0.034 x 10-3 and Σ-→Λ + e- + ve of branching ratio 5.730.27 x 10-5. State qualitatively the reason of the difference.
Give a cascade of ‘Cabibbo favoured’ decays through flavoured hyperons of the following beauty hyperons: Σ+b (uub), Ξ-b (dsb) and Λ0b(udb).
Give a cascade of ‘Cabibbo favoured’ decays through flavoured hyperons of the following charmed hyperons: Σc++(uuc), Ξ+c (usc) and Ω0c (ssc).
In 1959, B. Pontecorvo proposed an experimental idea to establish whether v̅e and v̅μ are different particles or not. To produce v̅μ, a low-energy π+ beam is brought to rest in a target. The μ+s from their decays come to rest too and then decay.(1) What is the lowest energy-threshold
The iron core stars end their life in a supernova explosion, if their mass is large enough. The atomic electrons are absorbed by nuclei by the process e + Z→(Z - 1) + ve. The star core implodes and its density grows enormously. Assume an iron core with density ρ = 100 000 t mm-3. Consider the
Consider a large water Cherenkov detector for solar neutrinos. The electron neutrinos are detected by the reaction ve + e- → ve + e-. Assume the cross-section (at about 10 MeV) σ = 10-47 m2 and the incident flux in the energy range above threshold Φ =1010m-2s-1. What is the water mass in which
A pion with momentum pπ = 500 MeV decays in the channel π+→μ+ + v. Find the minimum and maximum values of the μ momentum. What are the flavour and the chirality of the neutrino?

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Introduction to Elementary Particle Physics 2nd edition Alessandro Bettini - Solutions

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