Conservation Laws in Particle Physics

Conservation laws are the most powerful tools for predicting whether a particle interaction can occur. If any conserved quantity is violated, the interaction is forbidden — it simply cannot happen. This lesson covers the conservation of charge, baryon number, lepton number, and strangeness, including the crucial distinction between when strangeness is and is not conserved. This is assessed in AQA section 3.2.1.

Specification mapping. This lesson corresponds to AQA A-Level Physics (7408) sub-strand 3.2.1.7 (conservation laws in particle interactions: charge, baryon number, lepton number and strangeness). It is the summative lesson of the 3.2.1 particles strand: every quantum number introduced earlier (quarks at order 3, leptons at order 4, the W/Z bosons that change quark flavour at orders 4–5) is brought together here as a checklist for whether a proposed interaction is allowed. Refer to the official AQA 7408 specification document for the authoritative wording. The lesson is sequenced after Feynman diagrams (order 5) so that students can read vertex conservation off a diagram as well as apply it algebraically.

Synoptic links. Three principal synoptic threads run through this content. First, quark model and hadron classification (3.2.1.6, our order 3) — quark composition directly determines all four conserved quantum numbers, so conservation arguments and quark-counting arguments are two sides of the same coin. Second, nuclear physics (course 6, 3.8) — every nuclear decay equation is constrained by these same conservation laws; the mass-energy and momentum conservation that determine decay-product kinematics complement the discrete conservation laws covered here. Third, mechanics — momentum and energy conservation (course 2) — the continuous conservation laws of A-Level mechanics (energy, momentum, angular momentum) parallel the discrete quantum-number conservation laws here, and both ultimately derive from symmetries via Noether's theorem.

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The Conservation Laws

1. Conservation of Charge (Q)

Charge is conserved in all interactions — no exceptions. The total charge before the interaction must equal the total charge after.

2. Conservation of Baryon Number (B)

Baryon number is conserved in all interactions — no exceptions.

Particle type	Baryon number
Baryons (p, n, Σ, Ξ, etc.)	+1
Antibaryons (p̄, n̄, etc.)	−1
Mesons (π, K, etc.)	0
Leptons (e, μ, τ, neutrinos)	0
Photons	0

3. Conservation of Lepton Number (L)

Lepton number is conserved in all interactions. Each generation of lepton has its own lepton number, and each is conserved separately.

Particle	Lₑ	L_μ	L_τ
e⁻	+1	0	0
e⁺	−1	0	0
νₑ	+1	0	0
ν̄ₑ	−1	0	0
μ⁻	0	+1	0
μ⁺	0	−1	0
ν_μ	0	+1	0
ν̄_μ	0	−1	0
τ⁻	0	0	+1
τ⁺	0	0	−1
All hadrons, photons	0	0	0

4. Conservation of Strangeness (S)

Strangeness is conserved in strong and electromagnetic interactions, but can change by ±1 in weak interactions.

Particle	Strangeness
K⁺ (us̄)	+1
K⁰ (ds̄)	+1
K⁻ (ūs)	−1
K̄⁰ (d̄s)	−1
Σ⁺, Σ⁰, Σ⁻	−1
Ξ⁰, Ξ⁻	−2
Ω⁻	−3
p, n, π, e, μ, γ, neutrinos	0

Key Point: Strange particles are always produced in pairs in strong interactions (associated production) — for example, a K⁺ (S = +1) is produced alongside a Σ⁻ (S = −1), so the total strangeness change is zero. However, when strange particles decay, they do so via the weak interaction (one at a time), which is why they have relatively long lifetimes.

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Applying the Conservation Laws — Method

To determine whether an interaction is allowed or forbidden:

1. Add up each conserved quantity on both sides of the equation.
2. Compare the totals.
3. If any quantity is not conserved, the interaction is forbidden.
4. If all quantities are conserved, the interaction is allowed (though it may still be very unlikely or require specific conditions).

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Worked Examples

Worked Example 1 — Is this interaction allowed?

p + p → p + p + π⁰

Quantity	Before	After
Charge	+1 + 1 = +2	+1 + 1 + 0 = +2 ✓
Baryon number	+1 + 1 = +2	+1 + 1 + 0 = +2 ✓
Lepton number	0	0 ✓
Strangeness	0	0 ✓

Allowed ✓ (strong interaction — strangeness conserved, no leptons involved)

Worked Example 2 — Is this interaction allowed?

p → n + e⁺ + νₑ

Quantity	Before	After
Charge	+1	0 + 1 + 0 = +1 ✓
Baryon number	+1	+1 + 0 + 0 = +1 ✓
Lₑ	0	0 + (−1) + 1 = 0 ✓
Strangeness	0	0 ✓

Allowed ✓ (this is beta-plus decay — a weak interaction)

Worked Example 3 — Is this interaction allowed?

p → n + e⁺ + ν̄ₑ

Quantity	Before	After
Charge	+1	0 + 1 + 0 = +1 ✓
Baryon number	+1	+1 + 0 + 0 = +1 ✓
Lₑ	0	0 + (−1) + (−1) = −2 ✗

Forbidden ✗ — electron lepton number is not conserved. The correct beta-plus decay produces νₑ (not ν̄ₑ).

Worked Example 4 — Is this interaction allowed?

π⁺ → μ⁺ + νₑ

Quantity	Before	After
Charge	+1	+1 + 0 = +1 ✓
Baryon number	0	0 + 0 = 0 ✓
Lₑ	0	0 + 1 = +1 ✗

Forbidden ✗ — electron lepton number is not conserved. (The correct pion decay is π⁺ → μ⁺ + ν_μ.)

Worked Example 5 — Is this interaction allowed?

π⁺ → μ⁺ + ν_μ

Quantity	Before	After
Charge	+1	+1 + 0 = +1 ✓
Baryon number	0	0 + 0 = 0 ✓
L_μ	0	−1 + 1 = 0 ✓
Lₑ	0	0 + 0 = 0 ✓
Strangeness	0	0 ✓

Allowed ✓ (weak interaction)

Worked Example 6 — Strangeness in strong interactions

p + π⁻ → K⁰ + Λ⁰

(The Λ⁰ is a baryon with quark composition uds, strangeness −1)

Quantity	Before	After
Charge	+1 + (−1) = 0	0 + 0 = 0 ✓
Baryon number	+1 + 0 = +1	0 + 1 = +1 ✓
Strangeness	0 + 0 = 0	+1 + (−1) = 0 ✓

Allowed ✓ (strong interaction — strangeness conserved, strange particles produced in a pair)

Worked Example 7 — Strangeness violation in decay

Λ⁰ → p + π⁻

Quantity	Before	After
Charge	0	+1 + (−1) = 0 ✓
Baryon number	+1	+1 + 0 = +1 ✓
Strangeness	−1	0 + 0 = 0 ✗

Strangeness changes by +1. This is allowed only via the weak interaction (strangeness can change by ±1 in weak decays). This is indeed observed — the Λ⁰ decays via the weak interaction with a relatively long lifetime (~2.6 × 10⁻¹⁰ s, compared to ~10⁻²³ s for strong decays).

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Summary Table — What Is Conserved When?

Quantity	Strong	Electromagnetic	Weak
Charge (Q)	✓ Always	✓ Always	✓ Always
Baryon number (B)	✓ Always	✓ Always	✓ Always
Lepton number (Lₑ, L_μ, L_τ)	✓ Always	✓ Always	✓ Always
Strangeness (S)	✓ Conserved	✓ Conserved	✗ Can change by ±1

Exam Tip: In exam questions, you are typically given an interaction and asked whether it can occur. Set up a table with charge, baryon number, lepton numbers, and strangeness. Check each quantity systematically. If even one is violated, the interaction is forbidden. If strangeness changes, the interaction can only proceed via the weak force (and the change must be exactly ±1, not ±2 or more in a single step).

Common Misconception: Students sometimes think that if charge is conserved, the interaction must be allowed. This is not true — all four conservation laws must be satisfied simultaneously. An interaction that conserves charge but violates baryon number or lepton number is still forbidden.

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Specimen Exam Question

Specimen question modelled on the AQA paper format (9 marks).

A student claims that the following interactions can occur:

(i) p + p → p + Σ⁺ + K⁰

(ii) K⁻ + p → Ω⁻ + K⁺ + K⁰

(iii) p → e⁺ + νₑ + π⁰

For each interaction:

(a) Apply the four conservation laws (charge, baryon number, lepton number, strangeness) and state whether the interaction is allowed or forbidden. Show your reasoning in a structured table for each case. [6 marks]

(b) For each allowed interaction, name the fundamental force responsible and justify your answer. [3 marks]

AO Breakdown

For this 9-mark item: AO1 (knowledge) carries roughly 2 marks for recall of quantum-number values (the K⁰ has S = +1, the Ω⁻ has S = −3 and B = +1, etc.). AO2 (application) carries roughly 5 marks for systematically applying conservation across all three interactions, including the partial-credit "this interaction conserves Q, B, Lₑ but violates strangeness by −3 so cannot proceed via any interaction" reasoning. AO3 (evaluation) carries roughly 2 marks for naming and justifying the responsible force in each allowed case. Examiners weight the structured table heavily — a candidate who lists Q, B, S, L explicitly in a table is materially harder to mark down than one who writes a paragraph of mixed prose.

Grade-band Model Answers

Grade C response (5 marks out of 9).

(i) p + p → p + Σ⁺ + K⁰. Charge: 1 + 1 = 2 on the left; 1 + 1 + 0 = 2 on the right ✓. Baryon number: 1 + 1 = 2 on the left; 1 + 1 + 0 = 2 on the right ✓. Strangeness: 0 on the left; −1 + 1 = 0 on the right ✓. Lepton number: 0 on the left and right ✓. Allowed.

(ii) K⁻ + p → Ω⁻ + K⁺ + K⁰. Charge: −1 + 1 = 0; −1 + 1 + 0 = 0 ✓. Baryon: 0 + 1 = 1; 1 + 0 + 0 = 1 ✓. Strangeness: −1 + 0 = −1; −3 + 1 + 1 = −1 ✓. Allowed.

(iii) p → e⁺ + νₑ + π⁰. Charge: 1 = 1 + 0 + 0 = 1 ✓. Lepton number: 0 → −1 + 1 + 0 = 0 ✓. Baryon number: 1 → 0 — not conserved. Forbidden.

(b) (i) is strong because no leptons. (ii) is strong because no leptons.