PHY304 Particle Physics Dr C N Booth

Particle Physics  -  Homeworks

Roughly every two weeks, you will be given an exercise, which should be attempted by the following week. The purpose of the exercises is to help your revision of the lecture material and to give you practice in problem solving. The work may take one of two forms:

Three of the exercises are assessed homeworks, which must be handed in the following week. These will be marked (the total homework mark counting 15% towards this module), and will be returned to you, with comments, before the next homework is set. The other exercises are no less important to your understanding of the course, and should also be attempted seriously. Solutions will be provided two weeks after each non-assessed exercise is set.

As with the handouts, if you wish to print out these problems, you may be better with the PDF version below.


Assessed Homework 1 - Form Factors & Kinematics

1) In the lectures, we showed that the form factor F(q) is the 3-D Fourier transform of the normalised charge distribution ρ(r)
.

For a simplified model of a proton's charge distribution, :

  1. Find the constant of proportionality required to normalise ρ correctly.
  2. Show that .
You must show full working for the problems. All of the integrals required in the course are ones which you should be able to evaluate yourself. You should not use "standard" integrals from the internet, text books or other sources. (One of the main purposes of the homeworks is to prepare you for similar questions in the examination, and you will not have access to such material in an exam.) You may use standard integrals that appear on the exam formula sheet. If you find something appropriate there, that is fine, but you must make it explicit HOW you have used the standard form ‐ i.e. what substitutions you have made.

Note:
  1. The charge distribution falls away gradually as r → ∞. R can be considered as some characteristic "size" of the proton, setting the rate at which the charge dies away, but does not constitute a hard edge to the particle.
  2. This could be used in a scattering experiment to determine R.  If the value of q is determined at which the scattering amplitude drops to  (intensity drops to ), and is denoted qhalf, then clearly qhalf2R2/ħ2 = 1, so R = ħ/qhalf.
  3. This form for the charge distribution is in fact not very realistic, but was chosen to simplify the maths!  We will consider a slightly more likely nuclear charge distribution later, so keep your working for this problem to hand.

2) Electrons of energy 4.00 GeV are incident upon stationary target protons and a search is made for neutral particles Q0 produced in the reaction

e + p → e + p + Q0
What is the maximum mass of Q0 which can be produced? Use any reasonable approximations, but justify your arguments. Make clear which statements you make are generally valid, and which are specific to the case of maximum Q0 mass.

[Mass of electron is 0.511 MeV/c2;  that of proton is 938.3 MeV/c2.]
 

Hint for Problem 2

Consider how the three particles will appear in the centre-of-mass frame in the special case of Q0 being at its maximum mass. Note that E2p2 (whether for an individual particle or a system of particles) is both a Lorentz invariant, so the same in all frames, and also the difference between two conserved quantities, so the same before and after an interaction. Use this to relate the initial state in the laboratory frame to the final state in the c.m.s. frame. (Remember, as shown in the lectures, that while E2p2 = m2 for a single particle, you cannot assume that ETot2pTot2 = mTot2, since m is not a conserved quantity.)

(This is only one of many ways of attempting the problem. Use whichever method you find easiest.)


Exercise A (not assessed)

Fermions & Bosons

1) Classify the following into fermions and bosons.  Look up any particles you have not met previously.
e, p, γ, νe, π0, K+, Λ, Z, Σ+.


Leptons

2) Which of the following reactions are allowed by lepton number conservation, and which are forbidden?  Explain.

  1. τ+ → μ+ νμ ν̅τ
  2. π+ → μ+ γ
  3. π+ → μ+ νμ
  4. π0 → e+ e γ
  5. τ+ → e+ γ
  6. τ → π ντ

Yukawa Potential

2) Earlier in the course, we used the Born approximation to show that in the case of scattering with a momentum transfer q from a spherically symmetric potential V(r), the matrix element is given by
.

For the case of the Yukawa potential,

 
(with Rħ/mc, and m the mass of the exchanged boson mediating the force) show that the matrix element evaluates to
.


When an electron scatters electromagnetically off a nucleus, the exchanged boson is the massless photon, and at low energies the nucleus can be considered to remain effectively at rest.  Starting from the definition of q,   q = pipf, derive an expression for q2 in terms of θ for elastic scattering, and show that the angular dependence of the scattering is then given simply by the Rutherford formula

.


Assessed Homework 2

Note. Marks are obtained for a proper explanation of your working, not simply for obtaining the correct numerical or algebraic results!

Kinematics

1) The neutral rho meson often decays into two charged pions, ρ0 → π+ π.  In a monoenergetic beam of these mesons, some decays are observed where one pion is at rest.  What is the energy of the particles in the beam?

2) In the final stage of experiments at LEP, the energy was raised until pairs of W bosons could be produced. Electrons and positrons, each of energy 80.5 GeV, were collided head-on, and had exactly the right energy to produce W pairs at rest.

e+ e → W+ W
To what energy would a positron have to be accelerated if in collision with a stationary atomic electron it were to produce a W+ W pair?

3) A high energy photon can excite the quarks in a proton, producing a short-lived state which rapidly forms a nucleon and a pion. Calculate the minimum photon energy required for the following reaction to occur when the target is a stationary proton

γ p → n π+
Hint: consider the final state in the centre of mass frame.

[Particle masses: ρ0 769.3 MeV/c2, π± 139.6 MeV/c2, electron 0.511 MeV/c2, proton 938.3 MeV/c2, neutron 939.6 MeV/c2.]

The Propagator term

4) In Exercise A, we saw that the Yukawa potential leads to an expression for the amplitude of
.

In the scattering high energy neutrinos off electrons, it is observed that (after correcting for changing phase space or density of states effects), the differential cross-section falls by 10% as the momentum transfer q increases from small values to 20 GeV/c. Use this information to estimate the mass of the exchanged boson.


Exercise B (not assessed)

Parity

1) We saw earlier in the course that the ρ0 is a spin 1 meson which decays into π+ π. The spin of the pion is 0, and parity is conserved in the decay. By considering conservation of angular momentum, and equating the overall parities of the initial and final states, determine the intrinsic parity of the ρ0.
 (Note that it is not necessary to know the intrinsic parity of the pion, just that it is the same for π+ and π.)

2) The η meson has spin 0.  Conservation of parity prevents it decaying into two pions.  What does this tell us about the intrinsic parity of the η? The η does decay into three pions, η → π+ π π0. Can you deduce the intrinsic parity of the pion?

Pion Decay

3) The negative pion decays according to
 .
 If the pion decays at rest, derive and simplify an expression for the muon's kinetic energy in terms of the rest masses of the pion and muon. (The neutrino should be treated as massless.) State the physical principles you use.


Unassessed Exercise C - Quarks, Isospin, Strangeness and Charm; Gluon colours

(Questions 1 to 3 are straightforward, but you may find question 4 challenging! It has some similarity to the lecture material on wavefunction symmetries so working through that may help your understanding.)

1) The particles for this exercise are all taken from the baryon decuplet, with symmetric isospin functions. Write down the quark content of the following:

  1. Ω
  2. Δ+
  3. Σ0
  4. Ξ
 Given that a change in I3 value within a multiplet is associated with a change between d and u quarks, it should be obvious in each case what the value of I is from the number of distinct quark states in the multiplet. Verify this for the 4 multiplets given above.

2) The Σc baryons contain one c quark, the others being u or d. What charge states of the Σc exist? What is its isospin?

3) Which of the following particles exist? Explain your answer in each case. (For those that do exist, you could try to find the name.)

  1. a positive baryon with c = 1 and S = 1
  2. a positive baryon with c = 1 and S = −1
  3. a negative baryon with c = 1 and S = −1
  4. a neutral baryon with |c| = 2
  5. a doubly-charged baryon with |c| = 2.
  6. a positive meson with c = 1
  7. a neutral meson with c = 1
  8. a negative meson with c = 1
  9. a meson with c = 1 and S = 1
  10. a meson with c = 1 and S = −1
(Here c is the charm quantum number and S is the strangeness.)

4) You may have read that there is an octet of coloured gluons. In this exercise, we will demonstrate why there are 8 such states. Remember that there are 3 colours, which we label r, g and b (and 3 corresponding anticolours r-bar, g-bar and b-bar).

  1. Each gluon carries a colour and an anticolour. Write down the nine possible composite states.
  2. Six of your composites should be explicitly coloured. Identify these, and label them G1 to G6. (To avoid confusion with "green", we will use capital G to represent gluons!) We can define "colour swap operations", e.g. g ↔ r which changes any occurence of g into r and vice versa while simultaneously changing g-bar into r-bar and vice versa. Show that any of the three possible colour swap operations applied to a member of G1 to G6 results in another member of the same set.
The other three composites have no net colour, and can therefore mix. This results in states with different symmetry properties. (This is somewhat similar to the mixing of ↑↓ and ↓↑ spin states to form s = 0 and s = 1 combinations.) The rest of the exercise involves only these three states.
  1. Find the combination of the three states which is unchanged by any of the colour swap operations. Normalise it correctly. This is the colour singlet, denoted G0.
There must be two other independent (orthogonal) combinations of these states.
  1. Write down a normalised combination of two of the three states which is orthogonal to G0. Denote this G7.
  2. Finally find another combination of the three states which is orthogonal to both G0 and G7. Normalise this properly to find G8. (There are different possibilities for G7, and the definition of G8 will depend on which you chose.)
  3. Demonstrate that applying one of the colour swap operations to G7 simply reverses its sign.
  4. Show that applying either of the other two swap operators to G7 produces a properly normalised superposition of G7 and G8.
A proper application of group theory shows that all G1 to G8 have the same coupling in quark strong interactions, and correspond to the octet of physical gluons. G0 is a singlet with no coupling and therefore does not correspond to a gluon.


Assessed Homework 3 - Form Factors & Kinematics of 2-Body Decays

This final homework revisits two topics which have already been covered in earlier exercises and problems done in class.  The physical principles are the same but the calculations are somewhat more involved.  It may therefore be useful for you to look back at your previous work, and the feedback received when it was marked.

Note that full marks will only be obtained if you explain your calculations!

Form Factors

1) Use the fact that the form factor, F(q), is the Fourier transform of the normalised charge distribution ρ(r), which in the spherically symmetric case gives
,
to find an expression for F(q) for a simple model of the proton considered as a uniform spherical charge distribution of radius R.

Show that the requirement that the wavelength associated with q be much greater than the proton size is equivalent to the condition . (Ignore factors like 2π.) Using your calculated expression for F(q), demonstrate that in this limit the form factor reduces to 1. (Note that this condition must be satisfied by any form factor).

Kinematics of 2-Body Decays

In the lectures, we examined the decay of a moving particle into 2 products, one of which was massless. Here we remove that restriction, allowing both daughter particles to have a finite rest mass.

2) The D meson decays (through the weak interaction) according to D → K0 π. If a D is travelling with a total energy of 2.85 GeV, calculate the range of possible energies the produced K0 may have. (i.e. determine the minimum and maximum values of kaon energy.) Justify why the values you calculate are the minimum and maximum possible.

[Mass of D is 1.869 GeV/c2, mass of K0 is 0.498 GeV/c2, mass of πis 0.140 GeV/c2]


PDF versions of the problems are available here:
Homework 1   Exercise A    Homework 2    Exercise B    Exercise C    Homework 3


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