1.3 Decays and Scattering

Question 1

State Fermi’s Golden Rule

Answer 1

The transition probability per unit time from an initial state i to a final state f is given by:
Γ (i -> f) = (2pi/h bar) * | < f | H | i > |^2 * ρ(E_f)

Question 2

Describe ρ(E_f) in Fermi’s golden rule

Answer 2

How many configurations of the final state particles can you generate given a final state energy E_f

Question 3

What is the general particle structure for a decay?

Answer 3

1 -> n

Question 4

What do most particles decay into?

Answer 4

Lighter states

Question 5

What do we know about the timings of a decay?

Answer 5

They are random, but occur with an average decay time, τ

Question 6

What is the decay rate?

Answer 6

1/τ = Γ = LHS of Fermi’s golden rule

Question 7

Where is τ defined with regards to the system?

Answer 7

In the rest frame of the decaying particle

Question 8

What is the branching fraction/ratio?

Answer 8

BR_i = Γ_i / Γ

Question 9

What is Γ_i?

Answer 9

The decay width for a particle decaying to a specific channel

Question 10

Describe the general process for measuring Γ in terms of particles in a detector

Answer 10

1. Measure the length the parent particle travelled as we know when it was created and decayed
2. Measure momenta of daughter products and the decaying particle momentum is the sum
3. τ = t_lab / γ = L m / |p|
4. Repeat to form an exponential decay distribution of the form of e^-(Γt) and solve

Question 11

What is the critical problem with trying to measure Γ by using particles in a detector?

Answer 11

Some particles will decay too fast and not leave a trace in a detector

Question 12

What is another method of measuring Γ for particles that decay too quickly?

Answer 12

Measure energies instead as energy and time are a Fourier pair

Question 13

Is it easier to measure the position and time of a particle decay, or the energies of the particles in the system?

Answer 13

Energies are much easier to measure

Question 14

Describe the components of the wavefunction of the particle when we are measuring its energy

Answer 14

Ψ(t) = exp(imt) * exp(-Γ t/2)

• exp(imt) is a phase factor from the TDSE that vanishes upon taking the probability of the wavefunction
• exp(-Γ t/2) gives the decay rate upon taking the probability of exp(-Γ t/)

Question 15

How are the energy and time domain related?

Answer 15

Via a fourier transform

Φ(E) = FT (Ψ)

Question 16

How can we get the probability density of the state?

Answer 16

|Φ(E)| ^ 2

Question 17

How can we measure the decay rate, Γ, once we have the probability density of the state?

Answer 17

Plot a graph of |Φ(E)| ^ 2 against E

• Get a bell curve centered about the mass
• FWHM = Γ

Question 18

Why is the bell curve on the graph of |Φ(E)| ^ 2 against E centered about the particle mass?

Answer 18

The central energy is the rest mass of the decaying particle

Question 19

What is the Breit Wigner function?

Answer 19

The bell curve for the graph of |Φ(E)| ^ 2 against E

• Peak located at rest mass energy
• FWHM = decay rate Γ

Question 20

How does the shape of the Breit Wigner function change for very fast and slow decaying particles?

Answer 20

Very quick decaying particles have a very wide distribution as FWHM = decay rate = 1/τ and vv

Question 21

Describe the steps of measuring Γ from using the Breit Wigner function

Answer 21

1. Measure 4 momentum of decay daughters
2. Calculate the invariant mass of the system
3. Histogram and fit a Breit Wigner Function

Question 22

What is the name for the bell curve (lineshapes) of the Breit Wigner function?

Answer 22

Resonances = the decaying particle

Question 23

What is the general particle structure for a scatter?

Answer 23

2 -> n

Question 24

How is the scattering of a particle characterised?

Answer 24

By the cross section

Question 25

Describe the cross section of a particle scatter

Answer 25

It is a MEASURE of the probability for 2 particles to interact to a given final state

Question 26

What is the units for the scattering of a cross section?

Answer 26

Barns = 10^-28 m^2

- It is a Lorentz invariant measure of a unit of area

Question 27

What is the EXPERIMENTAL definition of the cross section?

Answer 27

σ = Interaction rate (dN/dt) / Incident particle flux

• Measure interaction rate
• Control incident particle flux

Question 28

What is the THEORETICAL definition of the cross section?

Answer 28

dN/dt = interaction rate = transition rate in Fermi's golden rule
σ = Γ (i -> f) / flux

Question 29

What is the differential cross section?

Answer 29

The cross section for scattering a final state particle into a particular differential solid angle dΩ

Question 30

What is dΩ?

Answer 30

d(cos theta) d(phi)

= area subtended on the unit sphere by a particle with scattering angle theta