Feynman diagrams

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Review Interaction Gravitational Weak EM Strong

Electroweak Fundamental Composite

Acts on Mass - energy Flavour Electric charge Colour change

On what particles

All

Quarks, leptons

Electrically charged particles

Quarks, gluons

Hadrons

Exchange particle

graviton 𝑊+, 𝑊−, 𝑍0

𝛾 gluons Mesons

Relative strength

10−38 10−5

10−2

1 n/a

Range ∞ ~ 10−18m ∞ ~ 10−15m ©cg

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Graphical visualization of interactions b/w particles

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Feynman diagrams • They are a mathematically representation of the interaction

b/w particles

• Space – time diagrams - time axis is going upwards - space / position axis to the right

• Be aware that some books and papers switch the axes

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Meaning of diagram components

• Straight lines – particles

• Upward arrow = particle moving forward in time

• Downward arrow = antiparticle, also moving forward in time

• Wavy or broken lines with no arrow = exchange particles

• Points where lines come together = vertices, at each vertex Q, L, B must be applied

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Electromagnetic and weak

exchange Strong exchange

Particle

FYI

The ―bubble of ignorance‖ is the actual place in the plot

that exchange particles do their thing.

Ingoing and outgoing particles are labeled.

Consider two electrons approaching one-another from

the top and the bottom of the page…

A purely spatial sketch of this

interaction would look like this:

But if we also apply a time axis,

the sketch would look like this:

The Time axis allows us to draw

the reaction in a spread-out way

to make it clearer.

Feynman diagrams

e-

e-

The bubble of

ignorance

SP

AC

E

TIME

e- e-

e- e-

Basic rules

1. You can only draw straight lines or wavy/ broken lines in any direction

2. You can only connect these lines if you have two lines with arrows meeting a single wavy / broken line Orientation of arrows is important: You must have exactly one arrow going into the vertex and one arrow coming out

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Basic rules continued

3. Your diagram should only contain connected pieces – every line must connect at least to one vertex

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Basic rules continued

4. Each line should be as straight as possible, but wavy lines stay wavy

• Why are these lines not allowed?

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Basic rules continued Diagrams are read from left to right

• If you have up and down lines shift them slightly so they point in either direction

• Matter particles pointing from left to right with arrows are electrons, arrows pointing in opposite direction are positrons

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• For every incoming ray there must be an outgoing one to preserve Q

• If the diagram is rotated, a different interaction is described

What is the story?

𝑒− emits photon, keeps going

𝑒+ absorbs photon, keeps going

𝑒− and 𝑒+annihilate into a photon

Photon produces pair 𝑒− and 𝑒+

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More tips • Left side: incoming particles

Right side: outgoing particles, after interaction

• Internal lines are virtual particles that are never directly observed - they only allow a given set of interactions to occur - allow incoming particle to turn into outgoing particle

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What is the story now?

𝑒− and 𝑒+ annihilate into a photon. Photon produces another 𝑒− and 𝑒+ pair

𝑒− tosses photon to nearby 𝑒+ w/o ever touching it

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Feynman diagrams to represent interactions

EM force b/w two 𝑒−

• Two 𝑒− moving closer

• Interact by exchange of virtual photon

• Moving apart

Weak nuclear force

• Radioactive decay by 𝛽 emission

• 𝛽− - neutron decays into proton - a W boson is exchanged as a quark changes from down to up - W boson immediately decays into 𝑒− and 𝑣 𝑒

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Strong nuclear force b/w quarks b/w nucleons

• Exchange particle is 𝜋+, 𝜋−, 𝜋0 - use dashed line to draw

• b/w a proton and a neutron

• b/w quarks

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Note:

• The shorter the range of the exchange force the more massive the exchange particle

• Exchange particles for gravitation and EM force (range = ∞) must have zero rest mass

• Shortest range in weak interaction, hence heaviest boson

• Strong interaction has exchange particle of intermediate mass

• Strong interaction increases with separation, requires high energy

• High energy allows the creation of hadrons (they have two quarks!)

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Pair annihilation • Matter and antimatter collides such as 𝑒− and 𝑒+

• Both particles are annihilated and 2 gamma rays with same energy but at direction of 1800 to each other are produced

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Pair production

• If sufficient energy is available and gamma rays pass by a nucleus, the annihilation can be reversed and produces an 𝑒− and 𝑒+ pair

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Additional examples

𝒗𝝁 interacts with photon to become 𝒆−

• s quark emits exchange particle and become an u quark

• This is an example for a flavour change

• Quark transforms into a member of another generation

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More examples

𝝁− emits 𝑾− and becomes a 𝒗𝝁 𝝅+ decays into 𝝁+ and 𝒗𝝁

• u quark and 𝑑 quark annihilate

• They produce 𝑊+ particle

• 𝑊+ decays into 𝜇+ and 𝑣𝜇

• 𝑊− changes to particle – antiparticle pair of 𝑒− and 𝑣 𝑒

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More complicated interactions

• EM interactions that led to photon scattering

• Particles in the loop are 𝑒− and 𝑒+

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Collision

• When two particles collide, energy is given off.

• The total energy of the combined system will be zero

• All the energy of the two particles becomes available as collision energy

• The total available energy is given by

• 𝐸𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒2 = 2𝑀𝑐2𝐸 + 𝑀𝑐2 2 + 𝑚𝑐2 2

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Half life and Range • According to Heissenberg’s uncertain principle, the uncertainty in

Energy and time is given by ∆𝐸∆𝑡 ≈ℎ

4𝜋

• The creation of an exchange particle can therefore not last any

longer than ∆𝑡 ≈ℎ

4𝜋∆𝐸=

ℎ

4𝜋𝑚𝑐2

• Distance = speed x time

• The furthest the particle can move or its range is therefore given by 𝑅 ≈ 𝑐∆𝑡

• 𝑅 ≈ℎ

4𝜋𝑚0𝑐

• The range of an interaction is inversely proportional to the rest mass of the virtual exchange particle

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Higgs boson • Elementary particle proposed by Peter Higgs in 1964

• Answers questions how two particles such as W and Z have large mass

• Particle’s mass calculated by Homer Simpson in 1998??????

• Did Homer Simpson Discover the Higgs Boson 14 Years Ago.mp4

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• Higg’s proposed that particles acquire mass as result of interactions with a hypothetical extra electroweak force field called the Higg’s field

• This particle H has mass, but no other conservation characteristics

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Imagine a room full of physicists. The Higgs field.

Suddenly Einstein enters and attempts to cross the

room, but the star-struck physicists cluster around him

and impede his movements, effectively increasing his

mass. High-mass particle.

Now imagine that I enter the room. Nobody wants to

interact with me, so I pass through the physicists

relatively unimpeded—no effective mass for me! Low-

mass (or massless) particle.

Lastly, imagine that somebody whispers a rumor,

causing the physicists to cluster together excitedly on

their own. Field disturbance – Higgs boson.

-Burton DeWilde

Higgs boson analogy

The Higgs boson analogy

Higgs field Einstein Me Big mass Small mass

The “God” particle

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• When the particle H is brought close to another particle, i.e. a proton, the H can interact with the proton b/c there is a force b/w them

• If H and the proton interact, H must be a boson

• Particles created in an accelerator arise from fields that are spread out in space and time

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Discovery of the Higgs particle would be evidence that

the standard model is correct.

Without the Higgs particle, the standard model will not

extend into the realm of general relativity.

String theory could be an alternative to the standard

model.

Here is the expected structure of the Higgs field:

The Higgs boson

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• Using quantum mechanics Mathematics Higgs showed, if H was in lowest energy state of a field – empty space – the field would not be zero

• Hence the H particle interacts with other particles that can gain mass as a result of the interaction

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March 2013 • The particle has

been proven to behave, interact and decay as predicted by the standard model

• H: charge +1 and spin 0

• These are properties of a boson

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Summary of weak force interaction 𝑾+

• Boson involved in interaction b/w neutrino and neutron creating a proton and an electron

𝑾−

• Decay of neutron into proton, electron and neutrino

A d quark changes into a u quark udd (neutron) uud (proton) 𝑊−emitted, decays into 𝑒− and 𝑣 𝑒

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𝑍0

• In collissions b/w particles, where there is no transfer of charges

𝜸 − 𝒑𝒉𝒐𝒕𝒐𝒏

• Virtual photon

• In the interaction b/w electrically charged particles

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Example

• Draw Feynman diagrams to show the following interactions:

a) Positive beta (positron) decay p n + 𝑒+ + 𝑣𝑒

• Proton decays into neutron

• Weak force: 𝑊+ mediator

• Emits positron and electron neutrino

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Example

Proton – electron collision p + 𝑒− n + 𝑣𝑒

• Proton and electron collide

• Produce a neutrino and electron neutrino

• Interaction mediated by 𝑊− boson

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Example • The two types of neutron

– electron neutrino collisions n + 𝑣𝑒 𝑣𝑒 + n

• 𝑍0 mediates force • Neutral current

interaction

• n + 𝑣𝑒 p + 𝑒−

• 𝑊+ interaction • Neutron changes into

proton • Charged current

interaction

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Feynman diagrams

Exam example

A virtual particle is a particle that has a very

short range of influence.

Look at charge…

-1/3 -1

0

The particle must be a W-.