### Feynman Diagrams

```Seeing Interactions
Drawing Feynman Diagrams
S
Feynman Diagrams
Richard Feynman created a pictorial representation of particle interactions which now bear
his name. (These were doodles to begin with but now are a powerful tool in particle physics.)
The Feynman diagram is focused on the vertex (where two lines meet).
This is the symbol for a photon
The vertices all have the same basic structure. A quark or lepton enters a
vertex, emits or absorbs an exchange particle, and a related quark or
lepton then emerges from the vertex.
The process is either quark– vertex–quark or lepton–vertex–lepton.
Some of the Rules
These diagrams are NOT the tracks of the particles. The vertices show time ordering only.
To the left of the vertex shows the particles before the interaction and
to the right of the vertex shows the particles after.
Each vertex has a arrowhead pointing towards the vertex and an arrowhead leaving it.
This represents a same generation lepton-lepton transition or a same or
neighboring-generation quark-quark transition.
This shows the electron-electron repulsion.
e
-
e
-
From a distance, this looks like the electrons
are repelling each other.
However, the Feynman diagram shows that
an electron emits a photon and is pushed back
to conserve momentum.
The second electron also changed direction
since it absorbed the photon.
photon
e
e
-
-
The EM force is mediated through a photon!
Virtual Particles
The photon that carries the information in the last example was emitted from a free electron
This certainly violates the conservation of mass-energy (inertia).
However, Heinsenberg’s Uncertainty Principle states
Measurements of the energy of a particle or of an energy level are subject to an uncertainty.
The energy measurement must be completed within a certain time interval, Δt.
The uncertainty in the energy, ΔE is related to Δt through:
h
DE × Dt ³
4p
So the photon can exist, energy can be created (!),
as long as it does not last longer than
h
Dt »
4pDE
Virtual Particles
Since the photon is very quickly absorbed by the other electron,
the energy “borrowed” is returned, (in time).
Particles like this photon are called virtual particles.
Virtual particles are particles that cannot be observed because they are absorbed
as soon as they are created. They may violate conservations laws.
Antimatter in Feynman
diagrams
We have seen evidence, through the Bubble Chamber, that a photon can pairproduce an electron and a positron.
What does this look like in a Feynman diagram?
e
+
e
-
WRONG!
Since every vertex has a
toward it and one
leaving
Antimatter in Feynman
diagrams
We have seen evidence, through the Bubble Chamber, that a photon can pairproduce an electron and a positron.
What does this look like in a Feynman diagram?
e
+
e
Any antiparticle, will
the opposite direction.
-
What’s Wrong?
What’s wrong with each of these Feynman diagrams?
e
e
-
e
+
In both cases electric charge is not conserved.
e
-
-
Weak Interactions
The weak interaction involves the emission or absorption of the W and Z bosons.
The W particle carries either a positive or a negative electric charge between
particles whereas the Z particle carries no electric charge.
The W particle is a quark-changing or a lepton-changing particle.
Gluons and photons carry no charge and do not change the particle flavour.
Weak Interactions
What boson is this? Why?
d quark
u quark
The charge before is -1/3. Afterward it is +2/3.
-1/3 = +2/3 + x
X= -1
It must be the W-
Weak Interactions
Don’t confuse this with:
Wgluon
d quark
u quark
d quark
What particle is leaving the vertex?
A d quark of a different color.
This is not a weak interaction
Weak Interactions
W-
e-
Here is another example of a weak interaction.
νe
Beta Decay
What is the nucleus produced when calcium-46 decays through the emission of a beta
particle?
46
20
Ca ® Sc + e + n
46
21
0
-1
Beta negative decay is the decay of a neutron into a proton emitting
an electron and an antineutrino.
Let’s see it: fun with quarks!
Free neutrons will decay in this manner.
A closer look at decays we have
The Ω- decays to produce a neutral xion (Ξ0) and a negative pion
What is the quark makeup of the omega-minus?
sss
And the quark makeup of the neutral xion?
s
s
s
uss
What type of boson is this?
s
s
Ξ0
u
Look at the charges at that vertex.
Coming in: -1/3
d
u
π-
Leaving: +2/3
So the boson is
negative: W-
A closer look at decays we have
A neutral xion (Ξ0) decays, by weak interaction, into a neutral lamba (Λ0) and a neutral
pion (π0).
The xion (Ξ0) is comprised of uss and the lamba (Λ0) is uds. The pion is (uu)
s
u
s
u
What is the boson?
d
s
u
u
W-
Putting it all together
A negative kaon (K-) hits a stationary proton.
ee+
πP-+
+
ee-
γ
Λ
γ0
KΞ
π00
+
KP
+
Ω-
V
K
-
This creates a K+ and an omega-minus
particle (Ω-).
(A neutral kaon (K0) is produced as well –
as seen with the conservation of
momentum.)
The Ω- decays at W to produce a
neutral xion (Ξ0) and a negative pion
The neutral xion decays into a
neutral lamba (Λ0) and a neutral pion.
This pion decays very quickly into two
photons.
The two photons decay into electronpositron pairs while the lamba decays
into a proton and a negative pion.
Strangeness
The Σ- particle has a half-life in the order of 10-10 s.
The Σ0 particle has a half-life that is in the order of 10-20 s.
That is 10 billion times shorter than the Σ- particle!
They were called strange particles.
A new quantum number called strangeness was used to make sense of their unusual
properties.
A hadron is assigned one positive unit of strangeness for every antistrange quark it contains
and one negative unit of strangeness for every strange quark it contains.
In the weak interactions previously discussed, is strangeness conserved?
Weak and Strange
The Ω- decays to produce a neutral xion (Ξ0) and a negative pion
What is the quark makeup of the omega-minus?
sss
And the quark makeup of the neutral xion?
Before
After
S = -3
S= -2
Strangeness is not conserved
in this interaction
uss
A neutral xion (Ξ0) decays, by weak interaction, into a neutral lamba (Λ0) and a neutral
pion (π0).
The xion (Ξ0) is comprised of uss and the lamba (Λ0) is uds. The pion is (uu)
Before
After
S = -2
S= -1
Strangeness is not conserved
in this interaction
Conservation of Strangeness
Δ+  p +
p-
Δ+ is comprised of uud.
Before
After
S=0
S= 0
Strangeness is conserved in
this interaction
Strangeness is conserved ONLY in electromagnetic and strong interactions
but is violated in weak interactions.
Strangeness
Σ- n + π-
Half-life: 10-10 s
Σ- = dds
Σ0 Λ0 +γ
Half-life: 10-20 s
Σ0 = uds
n = udd
Λ0 = uds
π- = du
Before
After
Before
After
S = -1
S=0
S = -1
S= -1
This is a weak interaction
since the strangeness is
not conserved. It is therefore slow.
This is not a weak interaction
since strangeness is conserved.
This decay takes place through the
electromagnetic interaction and is therefore fast.
The Range of Interaction
The range of an interaction can depend on the mass of the virtual particle.
The fastest a virtual particle can travel is ‘c’.
The range of the interaction will be R.
The virtual particle will reach the second particle in a time no smaller than R/c.
The energy exchanged will be in the order of mc2.
The uncertainties can be estimated to be: ΔE ≈mc2 andΔt ≈R/c
Heisenberg's Uncertainty Principle produces:
R h
mc ´ »
c 4p
2
So
h
R»
4p mc
The Range of Interaction
R»
h
4p mc
Why does the electromagnetic interaction have an infinite range?
It is mediated by a massless photon!
The weak interaction has a short range because it involves the massive W and Z bosons.
The W boson has a mass of about 80 GeVc-2 it follows that
R = 10-18m
Note: the gluons are massless but the strong force is short ranged
and it has to do with the fact that the gluons interact with each other.
Probability
Feynman diagrams are not just pictures. It represents a very definite mathematical
expression called the amplitude of the process.
The square of the amplitude gives the probability of the process actually taking place.
We will learn to assign to each vertex a quantity called the strength of the interaction.
This has the symbol
a
For the electromagnetic interaction, the basic vertex is assigned
a EM
1
»
137
a EM
To find the amplitude of a process, multiple the amplitudes for each
vertex.
This means that the more vertices, the less likely the process is to occur.
Real Particles
Anti Particles
Bosons
Particles
e
p
e+ g
-
-
+
p n0
p- p+ p0
ne nt np
W
-
W
+










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