Questions and Answers of Elementary Particle Physics

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
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
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)?
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)
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?
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+ +
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)
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
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
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
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
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
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
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:
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
(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.
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
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
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
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
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
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
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
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
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.
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
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
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
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
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 =
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
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
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
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
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
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
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
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
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
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
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
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
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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