Particle Conservation Laws, Collisions, and Creation

In every particle reaction, electric charge, energy, momentum, baryon number and each lepton number are conserved (as are angular momentum and, in strong and electromagnetic interactions, strangeness); these rules determine which reactions can occur, how particles are created in collisions (for example pair production and the production of pions) and how they annihilate. In brief: to check a reaction, add up the charge Q, baryon number B and lepton numbers L_e, L_μ, L_τ on both sides; they must be equal. A new particle can be created in a collision only if the available energy in the center-of-mass frame exceeds the total rest energy of the products. Energy is converted into mass according to E = mc². Conservation laws also explain why the proton and electron are stable.

When particles collide at high energies, new particles that did not exist before are created, but not at random: every reaction obeys strict conservation laws. This guide lists the conserved quantities, explains how to test a reaction, discusses pair production, annihilation and thresholds, compares allowed and forbidden reactions in a table, works through numerical examples and corrects common mistakes.

What quantities are conserved?

• Electric charge: the total charge before and after is equal (the charge of quarks is ±1/3 or ±2/3 e).
• Energy (mass-energy) and momentum: the total energy, including rest energy, and the total vector momentum are conserved. Therefore, a single photon cannot produce an electron-positron pair in empty space (momentum could not be conserved); it needs a nucleus nearby to take up momentum.
• Baryon number B: +1 for baryons (+1/3 for quarks), −1 for antibaryons, 0 for others. It explains the stability of the proton.
• Lepton numbers: L_e (electron, electron neutrino +1; positron, antineutrino −1), L_μ and L_τ are separately conserved (except for neutrino oscillations).
• Strangeness S: conserved in strong and electromagnetic interactions, but not in weak ones (kaons and Λ particles are produced in pairs by the strong force, but decay by the weak force: “strange” particles live long compared with the strong-interaction time of 10⁻²³ s).
• Angular momentum (spin), and in some interactions parity and CP are also conserved.

How are reactions tested?

1. Write the reaction with the particles on both sides.
2. For each conserved quantity (Q, B, L_e, L_μ, L_τ, and S for strong reactions), add up the values on the left and right.
3. If any is not equal, the reaction is forbidden.
4. Check the energy: the sum of the rest masses of the products must not exceed the available energy.

Examples: neutron decay n → p + e⁻ + ν̄_e: Q: 0 → +1 − 1 + 0 = 0 ✓; B: 1 → 1 ✓; L_e: 0 → 0 + 1 − 1 = 0 ✓. Proton decay p → e⁺ + π⁰ would conserve charge but violates baryon number (1 → 0) and lepton number, so it has never been observed (the lifetime is longer than 10³⁴ years); grand unified theories predict a tiny probability for it.

How are particles created?

• Pair production: a photon of energy E ≥ 2mc² passing close to a nucleus creates a particle-antiparticle pair: γ → e⁻ + e⁺ (E ≥ 1.022 MeV). The excess energy becomes kinetic energy of the pair.
• Annihilation: e⁻ + e⁺ → 2γ (at rest, each 0.511 MeV), or into other particles at higher energies (the production of Z bosons at LEP, the quark-antiquark pairs, then hadron jets).
• Collision production: kinetic energy of colliding particles can create new ones: p + p → p + p + π⁰, or p + p → p + p + p + p̄ (baryon number: 2 → 2 + 1 − 1 = 2 ✓). In a fixed-target experiment, much of the beam energy is used for the motion of the center of mass, so the threshold energy is much higher than the rest energy of the products; in a collider, the colliding beams have zero total momentum, and all the energy is available.
• Threshold: the minimum kinetic energy of the projectile for a reaction a + b (at rest) → products is T_th = (M_f² − (m_a + m_b)²)c² ÷ (2m_b), where M_f is the sum of the rest masses of the final particles.

Allowed and forbidden reactions compared

Worked example: checks and thresholds

1. Beta-plus decay: p → n + e⁺ + ν_e. Charge: +1 → 0 + 1 + 0 ✓. B: 1 → 1 ✓. L_e: 0 → 0 + (−1) + (+1) = 0 ✓. This cannot happen for a free proton (the products are heavier than the proton), but occurs inside nuclei, where the energy is supplied by the binding energy.
2. Pion decay: π⁻ → μ⁻ + ν̄_μ. Q: −1 → −1 + 0 ✓. L_μ: 0 → +1 − 1 = 0 ✓. The reaction π⁻ → e⁻ + ν̄_μ would violate lepton-flavor numbers (L_e: 0 → +1; L_μ: 0 → −1).
3. Pair production threshold: E_min = 2 × 0.511 MeV = 1.022 MeV, which corresponds to a gamma-ray wavelength of 1,240 eV·nm ÷ 1.022 × 10⁶ eV = 1.2 × 10⁻³ nm. For a photon of 5.0 MeV near a nucleus, the pair shares 5.0 − 1.022 = 3.98 MeV as kinetic energy.
4. Threshold for pion production: p + p → p + p + π⁰ with a target proton at rest: T_th = m_π(4m_p + m_π)c² ÷ (2m_p) = 135.0 × (4 × 938.3 + 135.0) ÷ (2 × 938.3) = 135.0 × 3,888.2 ÷ 1,876.5 = 279.7 MeV, about twice the rest energy of the pion (135 MeV), as the center of mass carries most of the energy of the beam.
5. Antiproton production: p + p → p + p + p + p̄ needs a threshold T_th = 6m_pc² = 5.63 GeV (formula: ((4m)² − (2m)²) ÷ 2m = 6m), found by Segrè and Chamberlain’s team at Berkeley with a 6.2 GeV proton beam in the Bevatron in 1955.
6. Annihilation: an electron and a positron colliding head-on, each with a kinetic energy of 45.6 GeV (LEP), have a total energy of 91.2 GeV in the center-of-mass frame, which is the mass of the Z⁰: the collision can produce a Z boson, which decays into quark pairs or leptons.

How were the laws discovered?

Conservation of charge was established in the 18th century (Franklin, 1747). Baryon number conservation was proposed by Ernst Stückelberg (1938) and Eugene Wigner (1949). Strangeness was introduced in 1953 by Murray Gell-Mann and Kazuhiko Nishijima to explain why strange particles are produced rapidly but decay slowly. Parity violation in weak decays was predicted by Lee and Yang and discovered by Chien-Shiung Wu in 1956, and CP violation in kaon decays by Cronin and Fitch in 1964. Emmy Noether’s theorem (1918) links conservation laws to symmetries: energy to time-translation invariance, momentum to space-translation invariance, and charge to a gauge symmetry.

Which mistakes are common?

Mistake 1: Thinking that mass is conserved in particle reactions

Mass-energy is conserved. The rest mass of the products may be larger or smaller than the initial rest mass, with the difference converted from or into kinetic energy.

Mistake 2: Forgetting the antineutrino in beta decay

Without it, lepton number and energy-momentum balance are violated.

Mistake 3: Using the rest energy of the products as the threshold in a fixed-target experiment

The center of mass has kinetic energy that is not available for creating particles, so the threshold is higher.

Mistake 4: Assuming that every reaction that conserves energy and charge is allowed

Baryon and lepton numbers (and other conserved quantities) must be conserved too.