Particle Interactions
1. Particle Conservation Laws
Learning outcomes
- I can apply conservation of charge.
- I can apply conservation of baryon number.
- I can apply conservation of lepton number.
- I can determine whether particle interactions are possible.
- I can analyze simple particle reactions.
What Is a Conservation Law?
When particles interact, decay, or transform, some properties may change. A neutron can become a proton, particles can be created, and matter and antimatter can annihilate.
However, certain physical quantities must remain balanced.
These rules are called conservation laws.
For the particle reactions in this topic, three especially useful quantities are:
- electric charge
- baryon number
- lepton number
A proposed reaction is allowed only if the relevant conservation laws are satisfied, along with other requirements such as conservation of energy and momentum.
A useful way to think about it is:
Total before reaction = Total after reaction
Conservation of Electric Charge
Electric charge is conserved in every known physical interaction.
This means:
Total charge before = Total charge after
Particles can exchange charge or transform into different particles, but the total electric charge of an isolated system cannot change.
Common Particle Charges
| Particle | Symbol | Charge |
|---|---|---|
| Proton | p | +1 |
| Neutron | n | 0 |
| Electron | e⁻ | −1 |
| Positron | e⁺ | +1 |
| Electron neutrino | νₑ | 0 |
| Electron antineutrino | ν̄ₑ | 0 |
| Photon | γ | 0 |
Quarks have fractional charges:
| Quark | Charge |
|---|---|
| Up | +⅔ |
| Down | −⅓ |
| Charm | +⅔ |
| Strange | −⅓ |
| Top | +⅔ |
| Bottom | −⅓ |
Antiquarks have the opposite electric charge of their corresponding quarks.
Example 1: Checking Charge
Consider beta-minus decay:
n → p + e⁻ + ν̄ₑ
Check the charge.
Before
Neutron:
Q = 0
After
Proton:
+1
Electron:
−1
Antineutrino:
0
Total:
+1 − 1 + 0 = 0
Therefore:
0 = 0
Charge is conserved.
Conservation of Baryon Number
Particles called baryons are assigned a quantity called baryon number.
Ordinary baryons have:
B = +1
Antibaryons have:
B = −1
Particles that are not baryons have:
B = 0
For the processes considered here:
Total baryon number before = Total baryon number after
Common Baryon Numbers
| Particle | Baryon Number |
|---|---|
| Proton | +1 |
| Neutron | +1 |
| Antiproton | −1 |
| Antineutron | −1 |
| Electron | 0 |
| Positron | 0 |
| Neutrino | 0 |
| Photon | 0 |
Protons and neutrons are both baryons because they are made from three valence quarks.
Electrons and neutrinos are leptons, not baryons.
Photons are bosons.
Baryon Number and Quarks
At a deeper level, each quark is assigned:
B = +⅓
Each antiquark has:
B = −⅓
A proton contains:
uud
Therefore:
B = ⅓ + ⅓ + ⅓
B = +1
A neutron contains:
udd
Again:
B = ⅓ + ⅓ + ⅓
B = +1
A meson contains a quark and an antiquark:
B = +⅓ − ⅓ = 0
This is why mesons are hadrons but are not baryons.
Example 2: Checking Baryon Number
Consider again:
n → p + e⁻ + ν̄ₑ
Before:
B = +1
After:
Proton:
B = +1
Electron:
B = 0
Antineutrino:
B = 0
Therefore:
+1 = +1
Baryon number is conserved.
Conservation of Lepton Number
Leptons are assigned a lepton number.
Ordinary leptons have:
L = +1
Antileptons have:
L = −1
Non-leptons have:
L = 0
Therefore:
Total lepton number before = Total lepton number after
Common Lepton Numbers
| Particle | Lepton Number |
|---|---|
| Electron | +1 |
| Electron neutrino | +1 |
| Muon | +1 |
| Muon neutrino | +1 |
| Tau | +1 |
| Tau neutrino | +1 |
| Positron | −1 |
| Antineutrino | −1 |
| Proton | 0 |
| Neutron | 0 |
| Photon | 0 |
Why Is the Antineutrino Needed?
Consider an incomplete version of beta-minus decay:
n → p + e⁻
Check lepton number.
Before:
L = 0
After:
Electron:
L = +1
Therefore:
0 ≠ +1
Lepton number would not be conserved.
The complete reaction includes an electron antineutrino:
n → p + e⁻ + ν̄ₑ
Now:
L = 0 + (+1) + (−1)
L = 0
So lepton number is conserved.
The neutrino or antineutrino is therefore essential when writing many weak-interaction reactions.
Three Checks for a Particle Reaction
For introductory particle reactions, we can perform three quick checks:
1. Charge
Qbefore = Qafter
2. Baryon number
Bbefore = Bafter
3. Lepton number
Lbefore = Lafter
A convenient table is:
| Quantity | Before | After | Conserved? |
|---|---|---|---|
| Charge | ? | ? | Yes/No |
| Baryon number | ? | ? | Yes/No |
| Lepton number | ? | ? | Yes/No |
Example 3: Beta-Minus Decay
Consider:
n → p + e⁻ + ν̄ₑ
Charge
Before:
0
After:
+1 − 1 + 0 = 0
Conserved.
Baryon number
Before:
+1
After:
+1 + 0 + 0 = +1
Conserved.
Lepton number
Before:
0
After:
0 + 1 − 1 = 0
Conserved.
Therefore, this reaction passes all three tests.
Example 4: Beta-Plus Decay
In a suitable nucleus, beta-plus decay can be represented at the nucleon level as:
p → n + e⁺ + νₑ
Check each conservation law.
Charge
Before:
+1
After:
0 + 1 + 0 = +1
Conserved.
Baryon number
Before:
+1
After:
+1 + 0 + 0 = +1
Conserved.
Lepton number
Before:
0
After:
0 − 1 + 1 = 0
Conserved.
Therefore, the particle identities satisfy these three conservation laws.
Note that an isolated free proton cannot spontaneously undergo this decay because the reaction would also fail the energy requirement. Beta-plus decay occurs in suitable proton-rich nuclei where the overall nuclear energy balance permits it.
Conservation Laws Do Not Tell the Whole Story
Passing charge, baryon-number, and lepton-number tests does not automatically prove that a reaction will occur.
Other quantities must also be conserved, including:
- energy
- momentum
- angular momentum
Other quantum rules can also restrict particle interactions.
Therefore:
Failing a conservation law → reaction is forbidden
but:
Passing these three tests → reaction may be possible, but further checks may be required
This is an important distinction.
Example 5: Is This Reaction Possible?
Consider:
n → p + γ
Let's test it.
Charge
Before:
0
After:
+1 + 0 = +1
Charge is not conserved.
Therefore, the reaction is not possible as written.
We do not need to continue checking the other conservation laws because one violation is enough to rule it out.
Example 6: A Missing Particle
Suppose we are given:
n → p + e⁻ + X
Determine particle X.
Charge
Before:
0
Known products:
+1 − 1 = 0
Therefore X must have:
Q = 0
Baryon number
Before:
+1
Known products already contain:
B = +1
Therefore:
BX = 0
Lepton number
Before:
0
The electron contributes:
L = +1
Therefore X must contribute:
L = −1
A neutral particle with:
Q = 0
B = 0
L = −1
is an antineutrino.
More specifically:
X = ν̄ₑ
Therefore:
n → p + e⁻ + ν̄ₑ
Matter and Antimatter
Conservation laws become particularly useful when dealing with antimatter.
An antiparticle has the same mass as its corresponding particle but opposite values of certain quantum numbers.
For example:
| Particle | Charge | B | L |
|---|---|---|---|
| Proton | +1 | +1 | 0 |
| Antiproton | −1 | −1 | 0 |
| Electron | −1 | 0 | +1 |
| Positron | +1 | 0 | −1 |
| Neutrino | 0 | 0 | +1 |
| Antineutrino | 0 | 0 | −1 |
This allows particle-antiparticle pairs to be created or annihilated while preserving the total conserved quantities.
Electron-Positron Annihilation
An electron and positron can annihilate:
e⁻ + e⁺ → γ + γ
Check the conservation laws.
Charge
Before:
−1 + 1 = 0
After:
0 + 0 = 0
Conserved.
Baryon number
Before:
0
After:
0
Conserved.
Lepton number
Before:
+1 − 1 = 0
After:
0
Conserved.
The electron and positron disappear as particles, but their energy and momentum are carried away by photons.
For annihilation at rest in the centre-of-momentum frame, two photons are required so that momentum can also be conserved.
Pair Production
The reverse process can also occur under suitable conditions:
γ → e⁻ + e⁺
However, a single photon cannot produce an electron-positron pair in otherwise empty space while conserving both energy and momentum.
Pair production therefore occurs in the presence of another object, commonly a nucleus:
γ + nucleus → e⁻ + e⁺ + nucleus
The nucleus can recoil and help conserve momentum.
The photon must also have sufficient energy to create the rest mass of both particles.
The minimum rest-energy required for the pair itself is:
2 × 0.511 MeV = 1.022 MeV
Conservation Laws at the Quark Level
Particle transformations can also be examined using quarks.
A neutron has:
udd
A proton has:
uud
During beta-minus decay:
n → p + e⁻ + ν̄ₑ
one down quark changes into an up quark:
d → u + W⁻
The W⁻ then decays:
W⁻ → e⁻ + ν̄ₑ
Charge remains conserved throughout.
Initially:
d charge = −⅓
After the first step:
u = +⅔
W⁻ = −1
Total:
+⅔ − 1 = −⅓
The charge is exactly the same as before.
A More Precise Look at Lepton Number
At an introductory level, we often use one total lepton number L.
Particle physics can also distinguish between different lepton flavours:
- electron lepton number
- muon lepton number
- tau lepton number
For example:
electron and electron neutrino → electron family
muon and muon neutrino → muon family
tau and tau neutrino → tau family
In many basic particle interactions, these family numbers provide another useful way to track reactions.
Neutrino oscillations make the full picture more subtle, so total lepton number is the most useful starting point for this course.
Worked Reaction Analysis
Consider:
p + e⁻ → n + νₑ
This represents electron capture at the particle level.
Charge
Before:
+1 − 1 = 0
After:
0 + 0 = 0
Conserved.
Baryon number
Before:
+1 + 0 = +1
After:
+1 + 0 = +1
Conserved.
Lepton number
Before:
0 + 1 = +1
After:
0 + 1 = +1
Conserved.
Therefore, the reaction passes all three conservation tests.
Try Comparing These Reactions
Consider:
A. n → p + e⁻ + ν̄ₑ
B. n → p + e⁻
C. e⁻ + e⁺ → γ + γ
Reaction A conserves charge, baryon number, and lepton number.
Reaction B conserves charge and baryon number but violates lepton number.
Reaction C conserves all three.
This demonstrates why conservation laws are useful: they allow us to test proposed reactions without needing to observe every reaction experimentally.
A Particle Conservation Table
This table is useful when solving problems:
| Particle | Q | B | L |
|---|---|---|---|
| p | +1 | +1 | 0 |
| n | 0 | +1 | 0 |
| p̄ | −1 | −1 | 0 |
| n̄ | 0 | −1 | 0 |
| e⁻ | −1 | 0 | +1 |
| e⁺ | +1 | 0 | −1 |
| ν | 0 | 0 | +1 |
| ν̄ | 0 | 0 | −1 |
| γ | 0 | 0 | 0 |
For many introductory questions, this table is enough to determine whether a proposed reaction violates one of the major conservation laws.
A Reliable Problem-Solving Method
When analyzing a particle reaction:
Step 1: Write down every particle.
Identify all particles before and after the reaction.
Step 2: Check charge.
Calculate:
Qbefore
and:
Qafter
They must be equal.
Step 3: Check baryon number.
Calculate:
Bbefore
and:
Bafter
They must be equal.
Step 4: Check lepton number.
Calculate:
Lbefore
and:
Lafter
They must be equal.
Step 5: Look for a missing particle.
If one quantity does not balance, determine what properties a missing particle would require.
Step 6: Remember energy and momentum.
Even if Q, B, and L balance, the reaction must also satisfy conservation of energy and momentum and any other relevant quantum rules.
Did You Know?
Conservation laws played an important role in the discovery of the neutrino.
During beta decay, physicists found that the observed electron did not carry all of the expected energy and momentum.
Wolfgang Pauli proposed that another very light, electrically neutral particle must also be produced.
That particle became known as the neutrino.
The neutrino allowed conservation laws to remain satisfied and was later detected experimentally.
This is a powerful example of physicists using conservation laws not only to describe reactions, but also to predict the existence of previously unknown particles.
Connecting the Ideas
Particle conservation problems can be approached as a balancing system:
Particle reaction
↓
Check electric charge
↓
Check baryon number
↓
Check lepton number
↓
Check energy and momentum
↓
Determine whether the reaction is physically possible
Conservation laws therefore act as fundamental rules controlling which particle transformations can occur.
Key Terms
Conservation law – A physical rule stating that a particular quantity remains constant in an isolated system.
Electric charge – A conserved property that determines electromagnetic interactions.
Baryon number – A quantum number used to distinguish baryons and antibaryons.
Lepton number – A quantum number associated with leptons and antileptons.
Baryon – A hadron such as a proton or neutron with baryon number +1.
Lepton – A fundamental particle such as an electron, muon, tau, or neutrino.
Antiparticle – A particle corresponding to another particle but with opposite values of certain quantum numbers.
Annihilation – A process in which a particle and antiparticle disappear and their mass-energy is transformed into other particles.
Pair production – The creation of a particle-antiparticle pair from energy under suitable conditions.
Key Takeaways
- Particle reactions must obey fundamental conservation laws.
- Electric charge is conserved: total charge before equals total charge after.
- Ordinary baryons such as protons and neutrons have B = +1.
- Antibaryons have B = −1, while non-baryons have B = 0.
- Leptons have L = +1 and antileptons have L = −1.
- Beta-minus decay requires an antineutrino to conserve lepton number.
- Conservation laws can be used to identify missing particles.
- Particle-antiparticle annihilation can conserve charge, baryon number, and lepton number.
- A reaction that violates a required conservation law is forbidden.
- A reaction that passes these three tests is not automatically guaranteed to occur; energy, momentum, angular momentum, and other quantum requirements must also be satisfied.
- Conservation laws are powerful tools for analyzing and predicting nuclear and particle interactions.