Fundamental Particles and Interactions
3. Leptons
Learning Outcomes
- I can identify the members of the lepton family.
- I can describe the properties of electrons and neutrinos.
- I can explain the role of leptons in particle interactions.
- I can compare leptons and quarks.
- I can explain lepton conservation in simple interactions.
- Solutions to the Schrödinger equation for the hydrogen atom
- Quantization of energy levels and orbitals
Leptons
Leptons are one of the two major families of elementary matter particles in the Standard Model.
The other major family is:
quarks
Unlike quarks, leptons do not experience the strong interaction. As far as current experiments show, leptons have no smaller internal structure and are therefore considered:
elementary particles
The six leptons are:
- electron
- electron neutrino
- muon
- muon neutrino
- tau
- tau neutrino
The Six Leptons
Leptons occur in three generations.
| Generation | Charged Lepton | Neutrino |
|---|---|---|
| First | electron (e⁻) | electron neutrino (νₑ) |
| Second | muon (μ⁻) | muon neutrino (νμ) |
| Third | tau (τ⁻) | tau neutrino (ντ) |
Each generation therefore contains:
one charged lepton + one neutrino
The three charged leptons have charge:
−e
The three neutrinos have:
zero electric charge
First Generation
The first generation contains:
electron + electron neutrino
These are the lightest members of the lepton family.
The electron is stable and is a fundamental component of ordinary atoms.
Electron neutrinos are produced in processes including:
- beta decay
- nuclear fusion
- nuclear reactors
- interactions inside stars
First-generation particles dominate ordinary stable matter.
Second Generation
The second generation contains:
muon + muon neutrino
The muon resembles a heavier version of the electron.
It has:
- charge −e
- spin 1/2
- much greater mass than the electron
- no known internal structure
Unlike the electron, the muon is:
unstable
It eventually decays into lighter particles.
Third Generation
The third generation contains:
tau + tau neutrino
The tau has:
- charge −e
- spin 1/2
- much greater mass than the electron
- a very short lifetime
It is also unstable and decays into lighter particles.
The pattern is therefore:
electron → muon → tau
with increasing mass.
Charged Leptons
The three charged leptons are:
e⁻, μ⁻, τ⁻
They have several important properties in common.
All three:
- are elementary
- have charge −e
- have spin 1/2
- participate in the weak interaction
- participate in electromagnetic interactions
- do not participate in the strong interaction
Their major difference is:
mass
Comparing Charged Leptons
| Particle | Symbol | Charge | Approximate Rest Energy |
|---|---|---|---|
| Electron | e⁻ | −1e | 0.511 MeV |
| Muon | μ⁻ | −1e | 105.7 MeV |
| Tau | τ⁻ | −1e | 1777 MeV |
The muon is more than 200 times as massive as the electron.
The tau is even more massive.
The Electron
The electron is the most familiar lepton.
Important properties include:
- symbol: e⁻
- charge: −1e
- elementary particle
- spin: 1/2
- stable
- relatively low mass
Electrons are essential to:
- atomic structure
- electricity
- chemical bonding
- electronics
- electromagnetic radiation interactions
Electrons in Atoms
Electrons occupy quantum states around atomic nuclei.
They are described using:
- wavefunctions
- atomic orbitals
- probability distributions
Electrons are not made from quarks.
Therefore:
electron = lepton
not:
electron = hadron
Neutrinos
The other three leptons are:
- electron neutrino
- muon neutrino
- tau neutrino
Collectively, they are called:
neutrinos
Neutrinos are unusual because they:
- have no electric charge
- have extremely small masses
- do not experience the strong interaction
- do not experience electromagnetic interactions
- interact primarily through the weak interaction and gravity
As a result, neutrinos can pass through enormous amounts of matter without interacting.
Neutrinos Passing Through Matter
Huge numbers of neutrinos pass through Earth continuously.
Most travel straight through because the probability of interaction is extremely small.
This makes neutrinos:
very difficult to detect
Scientists often build enormous detectors containing large amounts of material to increase the chance that a neutrino will interact.
Sources of Neutrinos
Neutrinos are produced in many processes.
Important sources include:
- the Sun
- other stars
- supernovae
- radioactive beta decay
- nuclear reactors
- cosmic-ray interactions
- particle accelerators
Neutrinos can therefore provide information about physical processes that are otherwise difficult to observe.
Solar Neutrinos
The Sun produces energy through:
nuclear fusion
Fusion reactions ultimately produce enormous numbers of:
electron neutrinos
Because neutrinos interact so weakly, they can escape from the Sun very quickly.
Studying them allows scientists to investigate processes occurring deep inside the solar interior.
Neutrino Flavours
There are three neutrino flavours:
νₑ = electron neutrino
νμ = muon neutrino
ντ = tau neutrino
Each corresponds to one of the three charged leptons.
Their antiparticles are:
ν̄ₑ
ν̄μ
ν̄τ
Neutrino Oscillation
Experiments discovered that neutrinos can change from one flavour into another as they travel.
This phenomenon is called:
neutrino oscillation
For example, an electron neutrino may later be detected as a:
- muon neutrino
- tau neutrino
This discovery was extremely important because neutrino oscillation implies that neutrinos have:
non-zero mass
This requires physics beyond the simplest original formulation of the Standard Model.
Muons
A muon is essentially a heavier charged lepton with many properties resembling those of an electron.
Muons are commonly produced when high-energy cosmic rays strike Earth's atmosphere.
They are unstable.
A common muon decay is:
μ⁻ → e⁻ + ν̄ₑ + νμ
This interaction provides an excellent example of lepton conservation.
Tau Leptons
The tau is even more massive than the muon.
Symbol:
τ⁻
It is extremely short-lived.
Tau particles can decay through several different processes involving:
- lighter leptons
- neutrinos
- hadrons
For example:
τ⁻ → e⁻ + ν̄ₑ + ντ
The tau lepton number is preserved by the tau neutrino.
Antileptons
Each lepton has a corresponding:
antilepton
| Lepton | Antilepton |
|---|---|
| electron e⁻ | positron e⁺ |
| electron neutrino νₑ | electron antineutrino ν̄ₑ |
| muon μ⁻ | antimuon μ⁺ |
| muon neutrino νμ | muon antineutrino ν̄μ |
| tau τ⁻ | antitau τ⁺ |
| tau neutrino ντ | tau antineutrino ν̄τ |
Antileptons have opposite values of:
lepton number
The Positron
The antiparticle of the electron is the:
positron
Symbol:
e⁺
The electron and positron have:
- identical mass
- opposite electric charge
Electron:
Q = −e
Positron:
Q = +e
When an electron and positron meet, they can undergo:
annihilation
Electron-Positron Annihilation
A common reaction is:
e⁻ + e⁺ → γ + γ
The electron and positron disappear and their energy is converted into photons.
If they annihilate essentially at rest, each photon commonly has energy:
511 keV
Energy, momentum, electric charge, and other conserved quantities must remain balanced.
Leptons and the Weak Interaction
All leptons participate in the:
weak interaction
The weak interaction is associated with:
- W⁺ bosons
- W⁻ bosons
- Z bosons
It is responsible for processes including:
- beta decay
- muon decay
- neutrino interactions
- many particle transformations
The weak interaction can change one type of particle into another while obeying conservation laws.
Beta-Minus Decay
Beta-minus decay is an important example involving leptons.
The overall reaction is:
n → p + e⁻ + ν̄ₑ
The particles produced include:
- proton
- electron
- electron antineutrino
At the quark level, one down quark changes into an up quark.
But the emitted electron and antineutrino are:
leptons
Why the Antineutrino Is Necessary
Historically, beta decay created a major puzzle.
Experiments showed that the emitted electron did not always carry the same energy.
It initially appeared that energy and momentum might not be conserved.
Wolfgang Pauli proposed that another undetected particle carried away some of the missing:
- energy
- momentum
That particle became known as the:
neutrino
In beta-minus decay, the particle emitted with the electron is specifically an:
electron antineutrino
Lepton Number
To help analyze particle interactions, physicists use a quantum number called:
lepton number
In a simplified treatment:
lepton:
L = +1
antilepton:
L = −1
non-lepton:
L = 0
For many simple interactions:
total lepton number before = total lepton number after
This is called:
lepton number conservation
Basic Lepton Numbers
| Particle | Lepton Number |
|---|---|
| electron | +1 |
| electron neutrino | +1 |
| positron | −1 |
| electron antineutrino | −1 |
| muon | +1 |
| muon neutrino | +1 |
| antimuon | −1 |
| muon antineutrino | −1 |
| proton | 0 |
| neutron | 0 |
| photon | 0 |
Lepton Conservation in Beta Decay
Consider:
n → p + e⁻ + ν̄ₑ
Before:
neutron:
L = 0
Total before:
L = 0
After:
proton:
L = 0
electron:
L = +1
electron antineutrino:
L = −1
Total:
0 + 1 − 1 = 0
Therefore:
lepton number is conserved
Why n → p + e⁻ Is Incomplete
Suppose we incorrectly wrote:
n → p + e⁻
Before:
L = 0
After:
electron gives:
L = +1
Therefore:
0 ≠ +1
Lepton number would not be conserved.
The correct reaction includes:
ν̄ₑ
so that:
+1 + (−1) = 0
Lepton Conservation in Muon Decay
Consider:
μ⁻ → e⁻ + ν̄ₑ + νμ
Using total lepton number:
Before:
muon:
L = +1
After:
electron:
+1
electron antineutrino:
−1
muon neutrino:
+1
Total:
+1 − 1 + 1 = +1
Therefore:
lepton number is conserved
Lepton Flavour Conservation
At an introductory level, it is also useful to track separate lepton-family numbers:
- electron lepton number
- muon lepton number
- tau lepton number
For example:
μ⁻ → e⁻ + ν̄ₑ + νμ
Electron family:
e⁻ gives +1
ν̄ₑ gives −1
Total:
0
Muon family:
μ⁻ initially gives +1
νμ afterward gives +1
Therefore, the family numbers balance in this simple decay.
Neutrino oscillations make the full modern treatment more subtle, but family-number bookkeeping remains useful for introductory particle interactions.
Worked Example: Is This Reaction Allowed?
Consider:
n → p + e⁻
Check electric charge.
Before:
neutron = 0
After:
proton = +1
electron = −1
Total = 0
So electric charge is conserved.
But check lepton number.
Before:
0
After:
+1
Therefore:
lepton number is not conserved
The reaction is incomplete.
Add an electron antineutrino:
n → p + e⁻ + ν̄ₑ
Now lepton number balances.
Worked Example: Electron-Positron Annihilation
Consider:
e⁻ + e⁺ → γ + γ
Before:
electron:
L = +1
positron:
L = −1
Total:
L = 0
After:
photons:
L = 0
Therefore:
lepton number is conserved
Worked Example: Pair Production
Consider:
γ → e⁻ + e⁺
For lepton number:
Before:
photon:
L = 0
After:
electron:
+1
positron:
−1
Total:
0
Therefore lepton number is conserved.
In a real pair-production process, another object such as a nucleus is generally involved so that energy and momentum can both be conserved.
Leptons and Electromagnetism
Charged leptons experience the:
electromagnetic interaction
Therefore:
- electrons interact electromagnetically
- muons interact electromagnetically
- taus interact electromagnetically
Neutrinos have no electric charge.
Therefore, they do not experience ordinary electromagnetic interactions.
This is one reason neutrinos pass through matter so easily.
Leptons and the Strong Interaction
Leptons do not carry colour charge.
Therefore they do not experience the:
strong interaction
This is one of the major differences between:
leptons and quarks
Leptons vs Quarks
| Property | Leptons | Quarks |
|---|---|---|
| Elementary | Yes | Yes |
| Number of types | 6 | 6 |
| Generations | 3 | 3 |
| Strong interaction | No | Yes |
| Colour charge | No | Yes |
| Fractional electric charge | No | Yes |
| Form hadrons | No | Yes |
| Examples | electron, neutrino | up, down |
Both groups are elementary:
fermions
and have:
spin 1/2
Electric Charges: Leptons vs Quarks
Charged leptons have:
−1e
Neutrinos have:
0
Quarks have fractional charges:
+2/3 e
or:
−1/3 e
Therefore:
fractional electric charge is a characteristic feature of quarks, not leptons.
Leptons Do Not Form Hadrons
Quarks combine through the strong interaction to form:
- baryons
- mesons
Leptons do not experience the strong interaction.
Therefore:
leptons do not form hadrons
An electron remains an elementary lepton rather than becoming part of a proton or neutron.
Ordinary Matter
Ordinary atoms contain:
protons + neutrons + electrons
Protons and neutrons contain:
up and down quarks
Electrons are:
leptons
Therefore, ordinary matter is built primarily from three elementary matter particles:
up quarks + down quarks + electrons
This combination forms almost all familiar atomic matter.
Fermions
Both leptons and quarks are:
fermions
Elementary matter fermions have:
spin 1/2
Fermions obey the:
Pauli exclusion principle
This is extremely important for electrons in atoms because it prevents identical electrons from occupying exactly the same quantum state.
The exclusion principle helps explain:
- electron configurations
- atomic structure
- periodic table patterns
- properties of matter
Leptons in Particle Interactions
Leptons frequently appear in particle reactions.
Examples include:
Beta decay
n → p + e⁻ + ν̄ₑ
Muon decay
μ⁻ → e⁻ + ν̄ₑ + νμ
Electron-positron annihilation
e⁻ + e⁺ → γ + γ
Pair production
energy → e⁻ + e⁺
Conservation laws help determine which interactions are physically possible.
Conservation Laws
When examining particle interactions, scientists check several quantities.
These include:
- energy
- momentum
- electric charge
- lepton number
- baryon number
A proposed interaction that violates a required conservation law cannot occur in the form written.
This makes conservation laws powerful tools for analyzing particle reactions.
Example: Charge and Lepton Number Together
Consider:
n → p + e⁻ + ν̄ₑ
Electric charge
Before:
0
After:
+1 − 1 + 0 = 0
Charge conserved.
Lepton number
Before:
0
After:
0 + 1 − 1 = 0
Lepton number conserved.
Both tests work.
Why Classifying Leptons Matters
Grouping particles into the lepton family reveals important patterns.
For example:
electron pairs with electron neutrino
muon pairs with muon neutrino
tau pairs with tau neutrino
This repeated structure helped scientists organize the Standard Model.
The classification also allows physicists to predict:
- possible interactions
- decay products
- conservation rules
- relationships between particles
Detecting Neutrinos
Because neutrinos interact so weakly, scientists need very large detectors.
Many neutrino detectors use enormous volumes of:
- water
- ice
- liquid scintillator
- other detector materials
Very occasionally, a neutrino interacts with matter and produces detectable secondary particles.
By collecting many rare events, scientists can study neutrino properties.
Why Neutrinos Are Important
Neutrinos are useful because they can escape from environments that light cannot easily escape from.
Scientists use neutrinos to investigate:
- the Sun's core
- supernova explosions
- nuclear reactions
- cosmic-ray processes
- fundamental particle properties
They can act as unique messengers from otherwise hidden physical environments.
Neutrinos and the Sun
Photons produced deep inside the Sun interact repeatedly with matter before eventually escaping.
Neutrinos interact much less.
They can escape from the solar interior relatively directly.
Therefore, detecting solar neutrinos gives scientists information about:
nuclear fusion occurring inside the Sun
Common Mistakes
Mistake: Saying electrons are made from quarks.
Electrons are elementary leptons.
Mistake: Saying neutrinos have electric charge.
Neutrinos are electrically neutral.
Mistake: Saying neutrinos have exactly zero mass.
Neutrino oscillation demonstrates that neutrinos have non-zero masses.
Mistake: Saying leptons experience the strong interaction.
They do not carry colour charge and do not participate in the strong interaction.
Mistake: Saying only electrons are leptons.
There are six leptons.
Mistake: Forgetting that muons and taus have charge −e.
All three charged leptons have the same electric charge but different masses.
Mistake: Saying neutrinos never interact with matter.
They interact very weakly, mainly through the weak interaction.
Mistake: Giving an antineutrino lepton number +1.
Antileptons have lepton number −1 in the simplified convention.
Mistake: Writing beta-minus decay without an antineutrino.
The electron antineutrino is necessary for the complete reaction.
Mistake: Assuming a reaction is allowed just because charge is conserved.
Other quantities such as energy, momentum and lepton number must also be considered.
Key Terms
Lepton — elementary fermion that does not participate in the strong interaction.
Electron — stable first-generation charged lepton found in atoms.
Muon — unstable second-generation charged lepton with greater mass than the electron.
Tau — unstable third-generation charged lepton with greater mass than the muon.
Neutrino — electrically neutral, extremely low-mass lepton that interacts mainly through the weak interaction.
Electron neutrino (νₑ) — neutrino associated with the electron family.
Muon neutrino (νμ) — neutrino associated with the muon family.
Tau neutrino (ντ) — neutrino associated with the tau family.
Antilepton — antiparticle corresponding to a lepton.
Positron — positively charged antiparticle of the electron.
Antineutrino — antiparticle corresponding to a neutrino.
Lepton number — quantum number used to track leptons and antileptons in particle interactions.
Lepton conservation — principle that total lepton number is conserved in the simple interactions considered here.
Weak interaction — fundamental interaction responsible for beta decay, neutrino interactions and many particle transformations.
Neutrino oscillation — phenomenon in which neutrinos change flavour while travelling.
Fermion — particle with half-integer spin; quarks and leptons are elementary fermions.
Key Takeaways
- Leptons are elementary matter particles.
- There are six leptons arranged into three generations.
- The first generation contains the electron and electron neutrino.
- The second contains the muon and muon neutrino.
- The third contains the tau and tau neutrino.
- The electron, muon and tau all have charge −e.
- Neutrinos have zero electric charge.
- Electrons are stable and are fundamental components of atoms.
- Muons and taus are heavier and unstable.
- Neutrinos have extremely small but non-zero masses.
- Neutrinos interact very weakly with ordinary matter.
- Leptons do not participate in the strong interaction.
- Charged leptons participate in electromagnetic interactions.
- All leptons participate in weak interactions.
- Leptons are not made from quarks and do not form hadrons.
- Quarks and leptons are both elementary fermions.
- Quarks carry colour charge; leptons do not.
- Each lepton has a corresponding antilepton.
- The positron is the antiparticle of the electron.
- In the simplified convention, leptons have L = +1 and antileptons have L = −1.
- Many simple particle reactions conserve total lepton number.
- Beta-minus decay is n → p + e⁻ + ν̄ₑ.
- The electron and electron antineutrino in beta decay have opposite lepton numbers.
- Muon decay also demonstrates lepton conservation.
- Conservation laws allow scientists to determine whether proposed particle interactions are possible.
- Together with quarks, leptons form the elementary matter particles of the Standard Model.
Check Your Understanding
- What is a lepton?
- Name the six leptons.
- How are the six leptons organized into generations?
- What electric charge does an electron have?
- What electric charge does a neutrino have?
- Compare the electron, muon and tau.
- Why are neutrinos difficult to detect?
- Name three sources of neutrinos.
- What is neutrino oscillation?
- What does neutrino oscillation tell us about neutrino mass?
- What is the antiparticle of an electron?
- What is the lepton number of an electron?
- What is the lepton number of a positron?
- What is the lepton number of an electron antineutrino?
- Show that lepton number is conserved in n → p + e⁻ + ν̄ₑ.
- Explain why n → p + e⁻ is incomplete.
- Show that lepton number is conserved in μ⁻ → e⁻ + ν̄ₑ + νμ.
- Compare leptons and quarks in terms of electric charge and the strong interaction.
- Explain why electrons do not form hadrons.
- Explain how lepton classification and lepton conservation help scientists analyze particle interactions.