Conservation Laws in Particle Physics

When particles collide or decay in the subatomic world, they do not do so randomly. Nature imposes strict accounting rules — conservation laws — that determine which reactions can actually happen and which are forbidden. Just as a bank account must balance (money in = money out), certain quantities carried by particles must balance before and after every interaction. These laws are among the most powerful tools in particle physics: if a proposed reaction violates even one conservation law, it simply does not occur, no matter how much energy is available. In classical physics you already know about conservation of energy and momentum. Particle physics adds several new conserved quantities that have no everyday analogue: electric charge, baryon number, lepton number (and its flavours), and strangeness. Each of these is an additive quantum number — you assign a value to every particle, sum the values on the left side of a reaction equation, sum them on the right, and check they are equal. If they are not equal, the reaction is forbidden by that conservation law. Understanding these laws lets physicists both predict the products of collisions and explain the observed lifetimes of unstable particles. A particle that cannot decay without violating a conservation law lives far longer than one whose decay is freely permitted. This interplay between symmetry and conservation — formalised by Noether's theorem — is at the heart of the Standard Model of particle physics.

Key takeaways
  • Conservation of Electric Charge (Q)
  • Conservation of Baryon Number (B)
  • Conservation of Lepton Number (L)
  • Conservation of Strangeness (S)

Conservation of Electric Charge (Q)

Electric charge is conserved in every interaction — electromagnetic, weak, and strong. Each particle carries a charge Q (in units of the elementary charge e = 1.6 × 10⁻¹⁹ C). For example, a proton has Q = +1, an electron Q = −1, a neutron Q = 0, and a photon Q = 0. In any reaction, the total charge on the left must equal the total charge on the right. Consider beta-minus decay: n → p + e⁻ + ν̄ₑ. Left side: Q = 0. Right side: (+1) + (−1) + 0 = 0. ✓ Charge is conserved.

Conservation of Baryon Number (B)

Baryons are particles made of three quarks — protons and neutrons are the most familiar. Every baryon is assigned baryon number B = +1, every antibaryon B = −1, and all mesons, leptons, and photons have B = 0. Baryon number is conserved in all known interactions. This is why a free proton never decays into lighter non-baryonic particles — there is no lighter baryon for it to become, and creating a baryon–antibaryon pair from the proton alone would require an extra baryon on the right, violating conservation. Example: p + p → p + p + p + p̄ has B: 2 → 2. ✓ The reaction p → e⁺ + γ has B: 1 → 0. ✗ Forbidden.

Conservation of Lepton Number (L)

Leptons (electrons, muons, tau particles, and their corresponding neutrinos) carry lepton number. In fact there are three separate lepton family numbers: electron lepton number Lₑ, muon lepton number Lμ, and tau lepton number Lτ, each conserved independently. Each electron or electron-neutrino carries Lₑ = +1; their antiparticles carry Lₑ = −1. The same applies for the muon and tau families. In beta-minus decay n → p + e⁻ + ν̄ₑ: before, Lₑ = 0; after, Lₑ(e⁻) + Lₑ(ν̄ₑ) = (+1) + (−1) = 0. ✓ The antineutrino is essential — without it, Lₑ would not balance.

Conservation of Strangeness (S)

Strangeness is a quantum number assigned to account for the unexpectedly long lifetimes of certain particles like the kaon (K) and the lambda baryon (Λ). These particles are produced quickly by the strong force (in ~10⁻²³ s) but decay slowly via the weak force (~10⁻¹⁰ s). The strange quark s carries S = −1, its antiquark s̄ carries S = +1; all up and down quarks have S = 0. The key rule: strangeness IS conserved in strong and electromagnetic interactions, but IS NOT conserved in weak interactions (it can change by ΔS = ±1). This is why strange particles must be produced in pairs (associated production): π⁻ + p → K⁰ + Λ⁰ gives S: 0 + 0 → (+1) + (−1) = 0. ✓

The Key Equation

∑iQibefore=∑iQiafter,∑iBibefore=∑iBiafter,∑iLibefore=∑iLiafter\sum_i Q_i^{\text{before}} = \sum_i Q_i^{\text{after}}, \quad \sum_i B_i^{\text{before}} = \sum_i B_i^{\text{after}}, \quad \sum_i L_i^{\text{before}} = \sum_i L_i^{\text{after}}
  • Q_i — Electric charge of particle i (multiples of e): Charge in units of elementary charge; can be integer or fractional (for quarks).
  • B_i — Baryon number of particle i (dimensionless): +1 for baryons, −1 for antibaryons, 0 for all other particles.
  • L_i — Lepton number of particle i (dimensionless): +1 for leptons, −1 for antileptons, 0 for all other particles; applies separately to each lepton family.

Worked Example

Problem. Determine whether the following reaction is allowed or forbidden, and state which conservation law(s) it violates if any: μ⁻ → e⁻ + γ

Le:0≠+1    and    Lμ:+1≠0  ⟹  ForbiddenL_e: 0 \neq +1 \;\;\text{and}\;\; L_\mu: +1 \neq 0 \implies \text{Forbidden}

Answer. Forbidden — violates both electron lepton number (Lₑ: 0 ≠ +1) and muon lepton number (Lμ: +1 ≠ 0).

Figure
Diagram
Conservation Laws in Beta-Minus Decay: n → p + e⁻ + ν̄ₑBEFOREnNeutronQ =0B =+1Lₑ =0S =0AFTERpprotone⁻electronν̄ₑantineutrinoQ:+1−10B:+100Lₑ:0+1−1S:000sum=+1sum=0 ✓Each coloured row must balance: Q(0=0)✓ B(+1=+1)✓ Lₑ(0=0)✓ S(0=0)✓
Origins · 1 minMurray Gell-Mann and how this was discovered
MG
Murray Gell-Mann
1929–2019 · American

Gell-Mann introduced the concept of strangeness in 1953 (independently also proposed by Kazuhiko Nishijima) to explain why certain particles produced rapidly by the strong force decayed only slowly. He later proposed the quark model (1964), providing a deeper explanation for all baryon and meson quantum numbers, and received the Nobel Prize in Physics in 1969. Gell-Mann named quarks after a nonsense word in James Joyce's novel 'Finnegans Wake': 'Three quarks for Muster Mark!' He insisted on the unusual pronunciation to rhyme with 'ork' rather than 'ark'.

  • The Standard Model of Particle Physics — Conservation laws are consequences of symmetries encoded in the Standard Model; understanding which particles exist tells you which quantum numbers appear.
  • Quarks and Hadrons — Baryon number and strangeness are fundamentally properties of quarks; knowing quark content lets you calculate these numbers directly.
  • Feynman Diagrams — Feynman diagrams are a visual tool for representing particle interactions; conservation laws must hold at every vertex in these diagrams.
  • Radioactive Decay and Beta Decay — Beta decay is the most accessible example for applying lepton number and baryon number conservation in a quantitative way.