nuclear and particle physics junior honours: particle physics

Nuclear and Particle Physics Junior Honours:

Particle PhysicsLecture 3: Quantum Electro-Dynamics & Feynman Diagrams

February 15th 2007

AntimatterVirtual Particles

Quantum description of electromagnetismQED vertex

Feynman Diagram and Feynman RulesExamples

Schrödinger and Klein Gordon

E = i! ∂

∂t!p = −i!!∇

ΨE(!r, t) = ψE(!r) exp −iEt/!

−! ∂2

∂t2Ψ("r, t) = −!2c2∇2Ψ("r, t) + m2c4Ψ("r, t)

• However for particles near the speed of light E2=p2c2+m2c4 ⇒

• Solutions with a definite energy, Ep=+(p2c2+m2c4)½, and momentum, p:

Ψ(!r, t) = N exp i(!p · !r − Ept)/!• Also solutions with a energy, Ep=−(p2c2+m2c4)½, and momentum, −p:

• Negative energy solutions are a direct result of E2=p2c2+m2c4.

• We interpret these as anti-particles

Ψ∗(!r, t) = N∗ exp i(−!p · !r + Ept)/!

Klein Gordon Equation

• Quantum mechanics describes momentum and energy in terms of operators:

• E=p2/2m gives time-dependent Schödinger:

• Solution with a definite energy, E:

− !2m∇2Ψ(!r, t) = i! ∂

∂tΨ(!r, t)

Klein Gordon equation is non-

examinable

Antimatter

Dirac Interpretation:

• The vacuum is composed of E<0 energy levels. Each level is filled with two electrons of opposite spin: the Dirac sea.

• A ‘hole’ in the sea of −ve charge and E<0 appears as a state with +ve charge and E>0.

• This idea lead Dirac to predict the positron, discovered in 1931.

Feynman-Stueckelberg

• negative energy particles moving backwards in space and time correspond to

• positive energy antiparticles moving forward in space and time

Nuclear and Particle Physics Franz Muheim 14

AntiparticlesAntiparticles

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exp −i ((−E)(−t)− (−!p ) · (−!r )) /!= exp −i (Et− !p · !r ) /!

Quantum Electrodynamics (QED)

Classical electromagnetism:

• Force between charged particle arise from the electric field

• act instantaneously at a distance

Quantum Picture:

• Force between charged particle described by exchange of virtual “field quanta” - photons.

• Strength of interaction is related to charge of particle.

Matrix ElementFull derivation in 2nd order perturbation theory.

Gives propagator term: for exchange boson


Quantum ElectrodynamicsQuantum Electrodynamics

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QED is the quantum theory of electromagnetic interactions.

charge of electron

charge of proton

propagator

!E =q r

4πε0r2

1/(q2 −m2γ)

Mfi =e2

(q2 −m2γ)

Virtual Particles

• Virtual Particles

• do not have same rest mass as physical particles:

• We say X is “off-mass shell”

• Heisenberg Uncertainty Principle: can borrow energy to create particle if energy (∆E=mc2) repaid within time (∆t), where ∆E∆t≈ħ

• In electromagnetic interactions mass of photon propagator is non-zero.

m2X != E2

X − !p 2X

Force between two charged particles described by exchange of virtual photons.


Virtual ParticlesVirtual Particles

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Example: electron-electron scattering

p1

p2

p3

p4

• Momentum-squared transferred by the photon is:

• The photon is virtual as

q2 = (p3 − p1)2 = (p2 − p4)2

q2 != m2γ

• Only intermediate photons may be virtual. Final state ones must be real!

Feynman Diagrams

• The probability amplitudes for all processes - scattering, decay, absorption, emission can be described by Feynman Diagrams.

• Feynman diagrams are a very useful and powerful tool in particle physics. We will use them a lot in this course. We use them a lot in our research!

“A Feynman diagram is a pictorial representation of a process corresponding to a particular transition amplitude”

Richard Feynman receiving the 1967 Noble prize in

physics for his invention of this technique.


Feynman DiagramsFeynman Diagrams

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• Conventions• Time flows from left to right (occasionally upwards)

• Fermions are solid lines with arrows• Bosons are wavy (or dashed) lines

• Feynman Rules: allow quantitative results for the matrix elements at different orders in perturbation theory.Nuclear and Particle Physics Franz Muheim 4

Feynman DiagramsFeynman Diagrams

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μ+

Electromagnetic VertexBasic electromagnetic process:

• Initial state fermion

• Absorption or emission of a photon

• Final state fermion

Examples: e−→e−γ, e−γ→e−

All electromagnetic interactions are described by the vertex and a photon propagator

√α

QED Conservation Laws

• Momentum, energy and charge is conserved at all vertices

• Fermion flavour (e, µ, τ, u, d ...) is conserved: e.g. u→uγ allowed, c→uγ forbidden

• Parity, π, is conserved.

Coupling strengthMatrix element is proportional to the fermion charge:

The rate of the interaction is proportional to

Use the fine structure constant, α

The strength of the coupling at the vertex is

Mfi ∝ e

|Mfi|2 ∝ e2

√α

α =e2

4πε0!c=

e2

4π≈ 1

137

Basic QED Processes

• The vertex can be twisted: describing many processes. NB: electron going backwards in time is equivalent to a positron.

• None of above processes is real! Violate energy-momentum conservation:

• Join two together to get a real processes

p2γ = (pe1 − pe2)

2 "= m2γ

Each line and vertex in a Feynman diagram corresponds to a term in the matrix element.

• Initial and final fermions are described by a wave function:

• Initial and final photons described by a photon wave function:

• Internal virtual photons described by photon propagator

• At every vertex a coupling constant (and the vertex operator )

• Also internal fermions: descried by fermion propagator(Some technical differences between the way we treat fermions and bosons)

Feynman Rules for QED

Example electron - elastic muon scattering: eµ→eµMatrix element:

• muon wave function• photon propagator• electron wave function

√α

√α

ψe(pµ)

√α

ψγ(pµ)1/(q2 −m2

γ) = 1/q2

∼ 1/q2

q2 = (p1 − p3)2 = (p4 − p2)2

OV

∼ 〈ψe(p1)| e OV |ψe(p3)〉

∼ 〈ψµ(p2)| e OV |ψµ(p4)〉

∝ e2

q2 rate ∝ |Mfi|2 ∝α2

q4

〈ψe(p1)| e OV |ψe(p3)〉1q2

〈ψµ(p2)| e OV |ψµ(p4)〉

• Elastic electron-proton scattering: e−p→e−p

• Momentum transferred to photon from e−:

• Rutherford scattering: e− Au→e− Au, can neglect recoil of the gold atoms: E=Ei=Ef

QED Scattering Examples


ElectronElectron--Proton ScatteringProton Scattering

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q2=

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q2

dσ

dΩ∝ |M|2 ∝ e4

q4=

16π2α2

q4

q2 = (pf − pi)2 = p2f + p2

i − 2pfpi

= 2m2e − 2(EfEi − |!pf ||!pi| cos θ)

≈ −4EfEi sin2(θ/2)

dσ

dΩ∝ Z2π2α2

E4 sin4(θ/2) LAB Frame CoM Frame

• Inelastic e−e+→µ+µ−

• Momentum transferred by photon:

• Use Fermi’s golden rule, including density of final states:

q2 = (pe+ + pe−)2 = s

σ =16πE2

3|M|2 =

4πα2

3s

• QED is formulated from time dependent perturbation theory.

• Perturbation series: break up the problem into a piece we can solve exactly plus a small correction.

• e.g. for e+e−→µ+µ− scattering.

• Many more diagrams have to be considered for a accurate prediction of σ.

• As α is small the lowest order in the expansion dominates, and the series quickly converges!

Higher Orders

23

Higher Orders

So far only considered lowest order term in the

perturbation series. Higher order terms also

contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....

Second order suppressed by relative to first

order. Provided is small, i.e. perturbation is

small, lowest order dominates.

Dr M.A. Thomson Lent 2004

Lowest Order

|M|2 ∝ α2

23

Higher Orders



contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....





2nd Order

+ ...|M|2 ∝ α4

23

Higher Orders



contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....





3rd Order

+ ...

|M|2 ∝ α6

Relativistic quantum mechanics predicts negative energy particles: antiparticles. Two interpretations:

• a positive energy particle travelling backwards in time.• a ‘hole’ in a vacuum filled with negative energy states.

Quantum Electro Dynamics (QED) is the quantum mechanical description of the electromagnetic force.Electromagnetic force propagated by “off-mass shell” photon:

Feynman diagrams can be used to illustrate and calculate a QED matrix elements.Feynman rules are used to calculate the matrix element. Cross section is proportional to the matrix element squared.

All QED interactions are described by a fermion-fermion-photon vertex:• Strength of the vertex is the charge of the fermion.• Fermion flavour and energy-momentum are conserved at vertex.

The photon propagator ~1/q2 where q is the 4-momentum transferred by the photon.

Summary

q2 != m2γ

nuclear and particle physics junior honours: particle physics

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