Conservation Laws in Particle Physics Revision Notes | A Level AQA Physics

1. Why Conservation Laws Are the Referee of Particle Physics

Particle physicists cannot watch individual quarks collide under a microscope. Instead, they use conservation laws as a set of strict rules that every interaction must obey — if any rule is broken, the interaction cannot happen, no matter how much energy is available.

Think of it like a currency exchange: you can swap your money into different denominations, but the total value must stay the same. In particle interactions, the 'currencies' are charge, baryon number, lepton number, and strangeness — and the total of each must balance on both sides of the interaction equation.

The AQA specification requires you to apply all four conservation laws to any given interaction and state clearly whether it is allowed or forbidden. This section builds that skill from the ground up.

2. The Four Conservation Laws: What They Are and Why They Matter

Before applying the laws, you need to know exactly what each one says. The table below gives you the key quantum numbers for the particles you will encounter most often at A Level.

For most particles, antiparticles carry the opposite sign for every quantum number compared with their particle partner. This is a useful pattern to remember when looking up antiparticle values quickly. Note, however, that some particles are their own antiparticle — for example, the neutral pion ($\pi^0$) and the photon — and these have all quantum numbers equal to zero.

Charge conservation is the most familiar — you already applied it at GCSE when balancing ionic equations. At A Level, you extend the same logic to particle reactions, counting charge in units of $e$ rather than in coulombs.

Baryon number conservation explains why a lone proton never spontaneously decays into lighter particles — there is no lighter baryon for the baryon number to transfer to, so the proton is stable. This is a profound result that follows directly from the conservation law.

The separate conservation of electron and muon lepton numbers is a subtlety that catches many students out. An electron neutrino ($ u_e$) contributes to $L_e$ but not to $L_\mu$, and a muon neutrino ($ u_\mu$) contributes to $L_\mu$ but not to $L_e$. You must check each family independently.

Strangeness is the most nuanced of the four laws because it is not universally conserved — weak interactions are allowed to violate it by one unit. This is why strange particles (like kaons and the lambda baryon) are produced in pairs via the strong force but decay individually via the weak force, often into non-strange products.

The phenomenon of strange particles being produced in pairs is called associated production, and it is direct experimental evidence that strangeness is conserved in strong interactions. When a kaon ($K^+$, $S = +1$) is produced alongside a lambda ($\Lambda^0$, $S = -1$), the total strangeness remains zero — exactly as required.

3. Applying the Laws: A Systematic Method

Every conservation-law question in the AQA exam follows the same structure: you are given an interaction equation and asked whether it can occur. The method is always the same — check each conservation law in turn and state your conclusion.

Here is the four-step method to use every time. Apply it in order, and never skip a step — examiners award marks for each law checked, not just for the final verdict.

• Step 1 — Charge: Sum the charge on the left-hand side (LHS) and the right-hand side (RHS). They must be equal.
• Step 2 — Baryon number: Sum B on LHS and RHS. They must be equal.
• Step 3 — Lepton number: Sum $L_e$ on LHS and RHS (must be equal), then repeat for $L_\mu$.
• Step 4 — Strangeness: Sum S on LHS and RHS. If they are equal, the interaction is allowed by the strong or electromagnetic force. If they differ by ±1, it may proceed via the weak force only. If they differ by more than 1, the interaction is forbidden.

Worked example: Checking Whether a Neutron Decay Is Allowed

Determine whether the following interaction is allowed: $n \rightarrow p + e^- + \bar{ u}_e$

1. LHS: $Q = 0$ RHS: $Q = (+1) + (-1) + (0) = 0$ $0 = 0$ ✓ — Charge is conserved. The proton's +1 and the electron's -1 cancel to give zero, matching the neutron's charge.
2. LHS: $B = +1$ RHS: $B = (+1) + (0) + (0) = +1$ $+1 = +1$ ✓ — Baryon number is conserved. Only the proton carries B = +1; the electron and antineutrino are leptons with B = 0.
3. LHS: $L_e = 0$ RHS: $L_e = (0) + (+1) + (-1) = 0$ $0 = 0$ ✓ — The electron contributes $L_e = +1$ and the electron antineutrino contributes $L_e = -1$, summing to zero — matching the neutron.
4. LHS: $S = 0$ RHS: $S = 0 + 0 + 0 = 0$ $0 = 0$ ✓ — No strange particles are involved, so strangeness is straightforward conserved.

Answer: All four conservation laws are satisfied. This interaction is allowed — it is beta-minus decay of the neutron, which proceeds via the weak interaction.

Notice that the electron antineutrino ($\bar{ u}_e$) is essential in beta-minus decay. Without it, lepton number on the RHS would be $+1$ (from the electron alone) while the LHS is $0$ — a clear violation. The antineutrino was actually predicted to exist before it was detected, purely because lepton number had to be conserved.

4. Strangeness in Detail: Strong, Weak, and Forbidden

Strangeness is the conservation law that requires the most careful thought, because its rules depend on which fundamental force drives the interaction. The strong force conserves strangeness exactly; the weak force allows a change of ±1; no interaction ever changes strangeness by more than 1 in a single step.

This has a powerful practical consequence: if you see an interaction where $\Delta S = 0$, it could be strong or weak. If $\Delta S = \pm 1$, it must be weak. If $\Delta S = \pm 2$ or more, the interaction is forbidden outright — no force can make it happen.

Worked example: Analysing a Strange Particle Interaction

Determine whether the following interaction is allowed, and if so, state which force is responsible: $\pi^- + p \rightarrow K^0 + \Lambda^0$

1. LHS: $Q = (-1) + (+1) = 0$ RHS: $Q = (0) + (0) = 0$ $0 = 0$ ✓ — Charge is conserved.
2. LHS: $B = (0) + (+1) = +1$ RHS: $B = (0) + (+1) = +1$ $+1 = +1$ ✓ — The pion is a meson (B = 0), the proton is a baryon (B = +1). On the RHS, the kaon is a meson (B = 0) and the lambda is a baryon (B = +1). Baryon number is conserved.
3. LHS: $L_e = 0$, $L_\mu = 0$ RHS: $L_e = 0$, $L_\mu = 0$ Both conserved ✓ — No leptons are involved, so lepton number is straightforward zero on both sides.
4. LHS: $S = (0) + (0) = 0$ RHS: $S = (+1) + (-1) = 0$ $0 = 0$ ✓ — The $K^0$ carries $S = +1$ and the $\Lambda^0$ carries $S = -1$. They are produced together so that total strangeness remains zero — this is associated production.
5. $\Delta S = 0$, so the strong force can drive this interaction. — Because strangeness is conserved exactly ($\Delta S = 0$), the strong interaction is responsible. This is consistent with the short timescale of such production events observed experimentally.

Answer: The interaction is allowed. All four conservation laws are satisfied. Strangeness is conserved ($\Delta S = 0$), so the strong force is responsible. This is an example of associated production of strange particles.

5. Identifying Forbidden Interactions

The real exam skill is not just confirming that an interaction works — it is spotting exactly which law is broken when an interaction is forbidden. Examiners expect you to name the violated law and show the numerical check that proves the violation.

A common trap is to stop checking once you find one violation. Always check all four laws and report every violation — some interactions break more than one simultaneously, and the mark scheme may award a mark for each one identified.

Worked example: Identifying a Forbidden Interaction

Show that the following interaction is forbidden and identify all violated conservation laws: $p + p \rightarrow p + n + \mu^+$

1. LHS: $Q = (+1) + (+1) = +2$ RHS: $Q = (+1) + (0) + (+1) = +2$ $+2 = +2$ ✓ — Charge is conserved. The antimuon ($\mu^+$) carries charge +1.
2. LHS: $B = (+1) + (+1) = +2$ RHS: $B = (+1) + (+1) + (0) = +2$ $+2 = +2$ ✓ — Baryon number is conserved. The antimuon is a lepton, so it carries B = 0.
3. LHS: $L_\mu = 0 + 0 = 0$ RHS: $L_\mu = 0 + 0 + (-1) = -1$ $0 eq -1$ ✗ — The antimuon carries $L_\mu = -1$. There is no muon or muon neutrino on the LHS to balance this. Muon lepton number is violated.
4. LHS: $S = 0 + 0 = 0$ RHS: $S = 0 + 0 + 0 = 0$ $0 = 0$ ✓ — No strange particles are involved, so strangeness is conserved.

Answer: The interaction is forbidden. Muon lepton number is violated: LHS gives $L_\mu = 0$ but RHS gives $L_\mu = -1$. All other conservation laws are satisfied. To make this interaction allowed, a muon neutrino ($ u_\mu$, $L_\mu = +1$) would need to appear on the LHS, or a muon antineutrino ($\bar{ u}_\mu$, $L_\mu = -1$) on the RHS.

6. Energy and Momentum Conservation

Beyond the four quantum-number laws, the AQA specification also requires you to recognise that energy and momentum are conserved in every particle interaction. These are not separate rules to memorise — they are the same fundamental conservation principles from mechanics, now applied at the subatomic scale.

Energy conservation at this level connects to Einstein's mass-energy equivalence: $E = mc^2$. When particles are created in a collision, the kinetic energy of the incoming particles is converted into the rest-mass energy of the new particles. This is why high-energy particle accelerators are needed to produce heavy particles — you must supply enough energy to account for the rest mass of everything produced.

Momentum conservation means the total momentum of all particles before the interaction equals the total momentum after. In practice, AQA questions at this level ask you to recognise these principles rather than perform relativistic momentum calculations — but you must be able to state them clearly and apply them qualitatively.

• Energy is conserved: the total energy (including rest-mass energy) before an interaction equals the total energy after.
• Momentum is conserved: the total momentum of all particles before an interaction equals the total momentum after.
• These two principles, combined with the four quantum-number conservation laws, fully constrain which interactions are possible.
• A reaction that conserves all quantum numbers but would require more energy than is available still cannot occur — energy conservation is a real constraint.

7. Exam Technique and Common Mistakes

Conservation-law questions are among the most reliably mark-able in the AQA paper — the method is fixed, the marks are predictable, and the same mistakes appear year after year. The guidance below targets the errors that most frequently cost students marks.

A reliable exam strategy is to draw a small table with rows for Q, B, $L_e$, $L_\mu$, and S, and columns for LHS and RHS. Fill in each cell from the data table provided, sum each column, and compare. This structured layout makes it almost impossible to miss a law and gives the examiner a clear audit trail for your marks.

Mastering conservation laws means you can tackle any particle interaction the examiner presents — even with unfamiliar particles — because the method never changes: check charge, baryon number, lepton number (both families), and strangeness in turn, show your working numerically, and state a clear conclusion. That systematic approach is what separates full-mark answers from partial credit.