Measurable Quantities, Conservation Laws and Hadrons Flashcards
What quantities can be measured in particle physics?
- decay rate of unstable particles
- cross sections of collisions
Fermi’s Golden Rule
-transition rate for an allowed process is:
Γif = 2π |Mfi|² * (phase space)
-where Γif is the transition rate form initial state i to final state f
-Mfi is the transition amplitude, the probability amplitude going from initial to final states
-phase space is the number of allowed final states
Decay Rates
-if particles can decay through multiple channels, then:
Γ = ΣΓi
-where Γ is the total decay rate
-and Γi is the rate of decay to a particular final state
Particle, mass m, decays to n final states through one particular channel with momenta p_k
-golden rule becomes:
dΓi = |Mfi|²/2m d³p1/(2π)³2E1…d³pn/(2π)³2En (2π)^4 δ^4 (po - Σpk)
Cross Section
Definition
- a measure of how often two colliding particles produce a particular final state
- this is an intrinsic property
Cross Section
Experiment
- a typical experiment will accelerate bunches of particles towards each other (most will miss) some collide and their collision energy can produce new particles
- the rate at which this happens depends on experimental parameters e.g. target density, beam flux etc.
- but these parameters can be factored out for the intrinsic cross section
Cross Section
Golden Rule
-total cross section is denoted, σ
dσ = |Mfi|²/[4(√(k1[-k2)²-m1²m2²]]
d³p1/[(2π)³2E1] … d³pn/[(2π)³2En (2π)^4 δ^4 [k1+k1-Σpf]
-so total σ is the integral over this over p1…pn
How to find Mfi?
-Mfi is the amplitude of the probability of going from i to f:
Mfi = ⟨f|Ef|i⟩
-to find Mfi, add up all of the possible Feyman diagrams for the interaction
-every aspect of a Feynman diagram represents an algebraic quantity
-this could be an infinite number of interactions…
Feynman Diagrams
Vertices
-each vertex carries a factor of e for photon ‘interactions’ where:
e = EM coupling
-so when adding up the possible interactions to find Mfi, the more complex Feynman diagrams have more vertices so are proportional to higher powers of e
-since e«1, these terms can be neglected and we can take the simplest form as a good approximation
How to know which vertices are allowed?
-conservation laws
Conservation Laws
Noether’s Theorem
- there are lots of conserved quantities in particle physics
- many of them derive from continuous symmetries via ‘Noether’s Theorem:
- -if you have a continuous symmetry in a physical system, there is a corresponding conserved quantity
Conservation Laws
Symmetries and Conserved Quantities
- spatial translation -> momentum
- time translation -> energy
- rotational -> angular momentum
- boost -> motion of centre of mass
- phase shift -> charge (not just EM charge, any kind of quantum no. can be though of as a charge)
- strong phase invariance -> colour
Conservation Laws
Feynman Diagrams
-any vertex in a Feynman diagram must conserve momentum, energy etc. and also electric charge etc.
How do we find conserved quantities?
-by looking for forbidden processes
Baryon Number
1 for baryons -1 for anti baryons 0 for mesons \+1/3 quarks -1/3 anti quarks
Allowed Interactions of the Standard Model
Fermions and Gauge Bosons
- charger fermion to charged fermion via photon
- quark to quark via gluon
- any fermion to weak partner via W boson
- any fermion to any fermion via Zo
Allowed Interactions of the Standard Model
Gauge Bosons
- W+, W- and photon
- three gluons
- four gluons
- Zo, W+, W-
- there also four point weak interactions (many combinations)
Allowed Interactions of the Standard Model
Higgs Interactions
-any fermion to any fermion via Higgs
Isospin
Description
- experimentally observed hadrons come in groups of similar mass e.g. (p,n), (Σ-,Σo,Σ+)
- know that a spin 1/2 particle has two states and a spin 3/2 has four states etc.
- introduce isospin that behaves in the same way
- i.e. that p,n and Σ-,Σo,Σ+ are different states of one particle which is not correct but the maths leads to useful properties
Isospin
Definition
-isospin is determined by multiplicity: 2I+1 -so for protons and neutrons multiplicity is 2: 2 = 2I+1 => I=1/2 -we say n,p are a multiplet with I=1/2 -for Σ-,Σo,Σ+ : 3 = 21+1 => I=1 -Σ-,Σo,Σ+ are a multiplet with I=1
Isospin
Third Component
- just as with spin, we need a convention for the third component, I3
- the particle from a multiplet with the lowest (or most negative) charge is assigned the lowest I3
- so Σ-,Σo,Σ+ all have I=1 but respectively have I3=-1,0,+1
Strangeness
Description
-certain particles in particle collision experiments are always produced in pairs at high rate BUT long lifetimes / low decay rate
-but:
dΓ ∝ |M|²
dσ ∝ |M|²
-so transition amplitude is large meaning a large coupling constant so must be produced by the strong nuclear force, and decay by the weak nuclear force
-so introduce a property, strangeness, that is conserved in strong interactions and violated by weak interactions
-> i.e two particles are produced together via the strong interaction with opposite strangeness so it is conserved but then they go in different directions and are separate so when they decay they decay separately so strangeness is conserved then
Quantifying Strangeness
-if a particle decays to an ‘ordinary’ particle via one step:
|S|=1
-if by two steps:
|S|=2
Strangeness-Isospin Diagrams
- take all the hadrons with a certain spin
- plot on an axes x=isospin, y=strangeness
- can build any of these representations using the fundamental representation