1.3 Decays and Scattering Flashcards
State Fermi’s Golden Rule
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)
Describe ρ(E_f) in Fermi’s golden rule
How many configurations of the final state particles can you generate given a final state energy E_f
What is the general particle structure for a decay?
1 -> n
What do most particles decay into?
Lighter states
What do we know about the timings of a decay?
They are random, but occur with an average decay time, τ
What is the decay rate?
1/τ = Γ = LHS of Fermi’s golden rule
Where is τ defined with regards to the system?
In the rest frame of the decaying particle
What is the branching fraction/ratio?
BR_i = Γ_i / Γ
What is Γ_i?
The decay width for a particle decaying to a specific channel
Describe the general process for measuring Γ in terms of particles in a detector
- Measure the length the parent particle travelled as we know when it was created and decayed
- Measure momenta of daughter products and the decaying particle momentum is the sum
- τ = t_lab / γ = L m / |p|
- Repeat to form an exponential decay distribution of the form of e^-(Γt) and solve
What is the critical problem with trying to measure Γ by using particles in a detector?
Some particles will decay too fast and not leave a trace in a detector
What is another method of measuring Γ for particles that decay too quickly?
Measure energies instead as energy and time are a Fourier pair
Is it easier to measure the position and time of a particle decay, or the energies of the particles in the system?
Energies are much easier to measure
Describe the components of the wavefunction of the particle when we are measuring its energy
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/)
How are the energy and time domain related?
Via a fourier transform
Φ(E) = FT (Ψ)
How can we get the probability density of the state?
|Φ(E)| ^ 2
How can we measure the decay rate, Γ, once we have the probability density of the state?
Plot a graph of |Φ(E)| ^ 2 against E
- Get a bell curve centered about the mass
- FWHM = Γ
Why is the bell curve on the graph of |Φ(E)| ^ 2 against E centered about the particle mass?
The central energy is the rest mass of the decaying particle
What is the Breit Wigner function?
The bell curve for the graph of |Φ(E)| ^ 2 against E
- Peak located at rest mass energy
- FWHM = decay rate Γ
How does the shape of the Breit Wigner function change for very fast and slow decaying particles?
Very quick decaying particles have a very wide distribution as FWHM = decay rate = 1/τ and vv
Describe the steps of measuring Γ from using the Breit Wigner function
- Measure 4 momentum of decay daughters
- Calculate the invariant mass of the system
- Histogram and fit a Breit Wigner Function
What is the name for the bell curve (lineshapes) of the Breit Wigner function?
Resonances = the decaying particle
What is the general particle structure for a scatter?
2 -> n
How is the scattering of a particle characterised?
By the cross section
Describe the cross section of a particle scatter
It is a MEASURE of the probability for 2 particles to interact to a given final state
What is the units for the scattering of a cross section?
Barns = 10^-28 m^2
- It is a Lorentz invariant measure of a unit of area
What is the EXPERIMENTAL definition of the cross section?
σ = Interaction rate (dN/dt) / Incident particle flux
- Measure interaction rate
- Control incident particle flux
What is the THEORETICAL definition of the cross section?
dN/dt = interaction rate = transition rate in Fermi's golden rule σ = Γ (i -> f) / flux
What is the differential cross section?
The cross section for scattering a final state particle into a particular differential solid angle dΩ
What is dΩ?
d(cos theta) d(phi)
= area subtended on the unit sphere by a particle with scattering angle theta
