University Physics IV · Particle Physics · 14.5

Conserved Numbers & Discrete Symmetries

Most reactions you can write down never happen. This is the audit that tells you which: balance charge, baryon number and each lepton family first, then ask which of the three interactions is on duty — because the weak one is allowed to change flavour and to tell left from right, and the other two are not.

Open the lab ↓Practice & worked examples ↓

Build the model

Connect the measurement to the mechanism.

The conservation rules of particle physics are two different kinds of thing, and the skill is telling them apart. A few are theorems: charge, energy-momentum and angular momentum follow from continuous symmetries by Noether's theorem, and CPT invariance follows from Lorentz invariance and locality, which is why a particle and its antiparticle are required to share a mass and a lifetime exactly. The rest is bookkeeping invented to record an absence.

Baryon number was assigned because the proton does not decay; separate electron, muon and tau numbers because μ⁻ → e⁻γ has never been seen; strangeness because kaons are made in pairs by the strong interaction and then live some 10¹⁴ times longer than it took to make them. Each was read off a missing process rather than derived, and each has a limit: the weak interaction changes flavour freely, isospin is only as good as the up–down mass difference is small, and neutrino oscillation has already broken the individual lepton-family numbers. The discrete symmetries cost most.

Parity is exact at the strong and electromagnetic vertices, which are pure vector, and maximally violated by the charged weak current, which is V−A and couples to the left-handed chiral projection of a fermion field and to nothing else. So the ledger's honest output is a veto, not a rate: it says what cannot happen, and which interaction must be the one that does.

Simple definition: A conserved quantum number is an additive label whose total is unchanged across a reaction — but only for the interactions that respect it, which is almost never all three.
Example: Σ⁻ → n + π⁻ takes strangeness from −1 to 0. The strong and electromagnetic interactions conserve strangeness exactly, so only the weak one is left, and the lifetime is 1.5 × 10⁻¹⁰ s rather than the 10⁻²³ s a strong decay takes.

Baryon numberB = (n_q − n_q̄)/3

The proton is the lightest B = 1 state, so B conservation is what makes ordinary matter stable.

Every quark +1/3, every antiquark −1/3; leptons and gauge bosons 0. Additive over the whole final state.

Lepton-family numbersΔLₑ = ΔL_μ = ΔL_τ = 0

Vetoes μ⁻ → e⁻γ. Oscillation breaks it, so strictly only the total L survives.

+1 for ℓ⁻ and ν_ℓ, −1 for ℓ⁺ and ν̄_ℓ, 0 for everything else; one ledger per generation.

Parity of a two-body stateP = P₁ P₂ (−1)^L

Turns parity into an arithmetic test on a reaction, not just a label attached to one state.

Intrinsic parity: quark +1, antiquark −1, photon −1, so P(π) = −1. L is the relative orbital angular momentum.

The V−A charged currentj^μ = ū γ^μ (1 − γ⁵) u = 2 ū γ^μ P_L u

The W has no right-handed field to couple to, so parity violation sits in the vertex, not in a fitted constant.

P_L = (1 − γ⁵)/2 projects the left-handed chiral field; P_L + P_R = 1 and P_L² = P_L.

Beta asymmetry from a polarised sourceW(θ) ∝ 1 + A P (v/c) cos θ, A = −1

cos θ here multiplies a pseudoscalar, so a non-zero coefficient is parity violation measured directly.

θ from the nuclear spin, P the polarisation, v/c the electron speed; A = −1 is maximal violation.

CPT theoremm(X) = m(X̄) and τ(X) = τ(X̄)

So a measured CP violation forces an equal T violation; the K⁰ mass equality tests it below 10⁻¹⁸.

Follows from Lorentz invariance, locality and a Hermitian Hamiltonian — it is derived, not fitted.

Theorems, and rules invented to record an absence

Two different kinds of rule share the word conservation. Charge, energy-momentum and angular momentum are conserved because of Noether's theorem: each follows from a continuous symmetry of the action, and no experiment is expected to break them. CPT invariance is a theorem too, forced by Lorentz invariance, locality and a Hermitian Hamiltonian in any quantum field theory. Everything else on the ledger was invented to explain a missing process. Baryon number exists because the proton is stable — the partial lifetime for p → e⁺π⁰ now exceeds 10³⁴ years, some twenty-four orders of magnitude beyond the age of the universe. Separate electron, muon and tau numbers exist because μ⁻ → e⁻γ has never been seen, down to a branching ratio near 10⁻¹³. Strangeness exists because kaons and hyperons are produced copiously and then decay absurdly slowly. Assigning a quantum number records an absence in arithmetic; it does not explain it.

The audit, in the order that decides fastest

Given a proposed reaction, work down a fixed list and stop at the first failure. Charge is exact and takes one line. Baryon number is exact at every Standard Model vertex: count quarks minus antiquarks and divide by three. Then run the three lepton-family ledgers separately — this is where μ⁻ → e⁻γ dies, with ΔL_μ = −1 and ΔLₑ = +1 even though the total lepton number is untouched. Then energy: a decay needs a positive Q value, a collision needs invariant mass above threshold. Only after all that do you look at flavour, and flavour asks a different question. A change of strangeness, charm or beauty does not veto a process; it routes it, because the strong and electromagnetic interactions conserve flavour and the charged weak current does not. The first four tests answer whether the process can happen at all. The fifth answers who has to do it, and therefore how slowly.

Flavour and isospin are approximate by construction

Strangeness is conserved exactly by the strong and electromagnetic interactions and broken by the weak one, by one unit per W emitted, so a single weak decay has |ΔS| = 1 and ΔS = 2 processes are suppressed far below that. One rule settles the whole strange-particle puzzle of the 1950s. In π⁻ + p → K⁰ + Λ⁰ the strangeness ledger reads 0 → (+1) + (−1) = 0, so the strong interaction is free to make the pair, and it does in about 10⁻²³ s. Neither product can then decay strongly, because that would change S on its own, so the Λ⁰ lives 2.63 × 10⁻¹⁰ s — some 10¹⁴ times its own production time. Isospin is approximate for a different reason: it treats u and d as one doublet, which works only because m_d − mᵤ and α are both small beside the QCD scale. The proton–neutron splitting of 1.3 MeV on 939 MeV sets the error, so isospin predictions are worth about one per cent and no more.

Parity, and what makes an observable pseudoscalar

Parity inverts every spatial coordinate, r → −r. Polar vectors — position, momentum, electric field — change sign under it. Axial vectors, built from a cross product, do not: L = r × p picks up two sign changes, and spin transforms the same way. So a spin dotted into a momentum, ⟨J⟩⋅p̂, changes sign under P where an ordinary scalar does not. It is a pseudoscalar, and if the interaction responsible commuted with P its average would have to vanish exactly. That is what makes the Wu experiment a proof rather than a hint. Cobalt-60 polarised near 0.01 K emits its beta electrons preferentially opposite to the nuclear spin, following 1 − P (v/c) cos θ. With a polarisation near 0.6 and v/c near 0.6 the front-to-back modulation is about 36 per cent, and it reverses when the polarising field reverses, which is what rules out an apparatus artefact.

V−A: why the violation is maximal, and only there

Write the projector P_L = (1 − γ⁵)/2. The charged weak vertex is ū γ^μ (1 − γ⁵) u, which is 2 ū γ^μ P_L u: the W couples to the left-handed chiral field and has no coupling whatever to the right-handed one. In the massless limit chirality equals helicity, so the neutrino from a weak decay is purely left-handed and its mirror image, a right-handed neutrino, has zero amplitude. A mirror-reflected weak process does not merely run at a different rate — the leading term does not run at all. That is what maximal means, and it is the same statement as the vector and axial couplings being equal in size, g_A/g_V = −1 at the quark level, dressed by QCD to −1.27 for the nucleon. The electromagnetic and strong vertices are pure γ^μ, which commutes with parity, so they cannot generate a cos θ term at all. The neutral weak current is the intermediate case: the Z couples to both chiralities with strengths split by sin²θ_W ≈ 0.231, so it violates parity measurably but nothing like maximally.

C, T, CPT, and why CP violation is the prize

Charge conjugation fails as badly as parity: apply C to a left-handed neutrino and you get a left-handed antineutrino, which the weak current does not couple to and which has never been seen. The combination CP maps a left-handed neutrino onto a right-handed antineutrino, which does exist, so CP very nearly survives — and for the seven years after Wu it was taken to be exact. It is not. The long-lived neutral kaon, nominally CP-odd, decays to π⁺π⁻ with a branching ratio of 1.97 × 10⁻³. CPT then forces a T violation of exactly the same size, and the K⁰ ↔ K̄⁰ rate difference has been measured directly. In B mesons the effect is not small at all: sin 2β ≈ 0.70. This matters beyond spectroscopy, because Sakharov showed that a universe ending up with more matter than antimatter needs CP violation. The CKM phase supplies some, and falls short of the observed baryon asymmetry by roughly ten orders of magnitude — one of the clearest places the Standard Model is known to be incomplete.

Change one variable at a time

Make the relationship visible.

Interactive model

Left-handed current fraction h1.00

Nuclear polarisation P0.60

Electron speed β = v/c0.60 c

Set h to 0.5 and every ray lands back on the circle: equal left- and right-handed coupling conserves parity whatever the polarisation. Push h to 1 for pure V−A, then drop β to 0.15 and the pattern rounds out again — a slow electron cannot show its chirality.

ASYMMETRY (2h−1) P β0.36

W AT θ = 0°0.64

W AT θ = 180°1.36

UP / DOWN COUNTS0.471

Live interpretationASYMMETRY (2h−1) P β: 0.36. W AT θ = 0°: 0.64. W AT θ = 180°: 1.36. UP / DOWN COUNTS: 0.471

Catch the common trap

Explain before calculating.

Misconception check

The error here is treating helicity and chirality as the same word. Textbooks say the neutrino is left-handed, students carry that straight over to the electron, and then cannot explain why a mirror-reflected beta decay is suppressed rather than forbidden. Helicity is spin⋅p̂, and for anything with mass it is frame-dependent: overtake a 1 MeV electron and its helicity flips, so helicity cannot be what a Lorentz-invariant vertex couples to. Chirality is the invariant label the projector (1 − γ⁵)/2 selects, and that is what the W actually sees. The two agree only in the massless limit; a fermion of mass m and energy E carries a wrong-chirality admixture of order m/E. Everything quantitative in this topic lives in that gap — the beta asymmetry carries a factor v/c instead of 1, and π⁺ → e⁺ν is suppressed by (mₑ/m_μ)² rather than being forbidden outright.

Cobalt-60 nuclei are polarised at about 0.01 K, and the beta electrons come out preferentially opposite to the nuclear spin, with a modulation proportional to v/c that reverses when the polarising field is reversed. What has the measurement established?

Choose an answer to test the model.

Practice & worked examples

Reason from the model, then test the result.

EasyAudit p → e⁺ + π⁰ and ν̄ₑ + p → n + e⁺ against charge, baryon number and electron-family number. Say which is forbidden, and for the survivor name the interaction that can mediate it and find its threshold. Use mₚ = 938.272, mₙ = 939.565, mₑ = 0.511 MeV/c².

Charge decides neither. For the decay, +1 → +1 + 0 ✓. For the reaction, 0 + 1 → 0 + 1 ✓.
Baryon number kills the first. The proton has B = +1 while a positron and a pion both have B = 0, so ΔB = −1. No Standard Model vertex changes B, and the search agrees: the partial lifetime for p → e⁺π⁰ exceeds 10³⁴ years.
The second passes every ledger. Baryon: 0 + 1 = 1 + 0. Electron family: the ν̄ₑ carries Lₑ = −1 and the e⁺ carries Lₑ = −1, so −1 + 0 = 0 + (−1).
Name the interaction. A u quark in the proton becomes a d, and a neutrino is absorbed. Only the charged weak current does either, so this is W exchange — the inverse beta decay Reines and Cowan used to detect the antineutrino.
Threshold. The mass deficit is mₙ + mₑ − mₚ = 939.565 + 0.511 − 938.272 = 1.804 MeV. Allowing for the neutron recoil, E_ν̄ = [(mₙ + mₑ)² − mₚ²] / (2 mₚ) = 3388.5 / 1876.5 = 1.806 MeV.

Answerp → e⁺π⁰ is vetoed by baryon number, ΔB = −1. ν̄ₑ + p → n + e⁺ balances every ledger and runs on the charged weak current, with a threshold antineutrino energy of 1.806 MeV.

MediumThe Λ⁰ (uds, B = +1, S = −1) is produced in π⁻ + p → K⁰ + Λ⁰ and decays as Λ⁰ → p + π⁻ with τ = 2.63 × 10⁻¹⁰ s. Show that the production conserves strangeness while the decay does not, and use the ratio of the lifetime to the strong timescale to name the interaction behind each.

Production ledger. Charge: −1 + 1 = 0 + 0 ✓. Baryon: 0 + 1 = 0 + 1 ✓. Strangeness: π⁻ and p carry S = 0, while K⁰ (ds̄) has S = +1 and Λ⁰ (uds) has S = −1, so the total is still 0 ✓. Nothing is broken, so the strong interaction may act — which is why strange particles are always made in pairs.
The strong timescale is the time light takes to cross a hadron: 1.0 × 10⁻¹⁵ m ÷ 3.00 × 10⁸ m s⁻¹ = 3.3 × 10⁻²⁴ s.
Decay ledger. Charge and baryon number still balance, but strangeness runs −1 → 0, so ΔS = +1. The strong and electromagnetic interactions conserve S exactly, so both are barred; only the charged weak current changes quark flavour, here s → u with the emitted W⁻ materialising as the π⁻ (dū).
Check the rate against that verdict: 2.63 × 10⁻¹⁰ s ÷ 3.3 × 10⁻²⁴ s ≈ 8 × 10¹³. Kinematics cannot account for fourteen orders of magnitude — the Q value is 1115.68 − 938.27 − 139.57 = 37.8 MeV, perfectly comfortable. It is the ledger forcing a change of interaction.
The decay also violates parity. With P(p) = +1 and P(π⁻) = −1, matching the Λ⁰'s J^P = ½⁺ would allow only the L = 1 amplitude, since P = (+1)(−1)(−1)^L. Both L = 0 and L = 1 are present, and their interference makes a polarised Λ⁰ emit its proton asymmetrically about the spin — the same V−A signature Wu found.

AnswerProduction conserves strangeness, 0 → (+1) + (−1), and runs strong at ~10⁻²³ s. The decay has ΔS = +1, so only the charged weak current can drive it, and it is about 8 × 10¹³ times slower.

HardThe π⁺ has spin 0 and decays through the V−A current. Explain why R = Γ(π⁺ → e⁺νₑ) / Γ(π⁺ → μ⁺ν_μ) equals (mₑ/m_μ)² [(m_π² − mₑ²)/(m_π² − m_μ²)]², evaluate it with m_π = 139.57, m_μ = 105.66 and mₑ = 0.511 MeV/c², and say why a phase-space argument gets the answer backwards. Measured value: 1.233 × 10⁻⁴.

Put the neutrino along +z. The pion has J = 0, so the two spins must cancel: the ν_ℓ is left-handed with S_z = −½, so the ℓ⁺ must carry S_z = +½. But the ℓ⁺ travels along −z, so its spin is antiparallel to its momentum — it is produced in the helicity state the V−A current does not supply.
For a massless fermion that amplitude vanishes exactly; a mass m reopens it at order m/E, so the rate carries a factor m_ℓ². The two-body result is Γ ∝ m_ℓ² (m_π² − m_ℓ²)² / m_π³, and G_F², f_π² and the CKM element all cancel in the ratio.
Squares: m_π² = 139.57² = 19479.8, m_μ² = 105.66² = 11164.0, mₑ² = 0.511² = 0.261, all in MeV².
Phase-space factors: m_π² − mₑ² = 19479.5 and m_π² − m_μ² = 8315.7. Their ratio is 2.3425, and squared it is 5.487.
Helicity factor: (mₑ/m_μ)² = (0.511/105.66)² = (4.836 × 10⁻³)² = 2.339 × 10⁻⁵. So R = 2.339 × 10⁻⁵ × 5.487 = 1.283 × 10⁻⁴.
Phase space alone points the other way. The daughter momentum is (m_π² − m_ℓ²)/(2m_π), which is 69.8 MeV/c for the positron against 29.8 MeV/c for the muon, so a spinless estimate would favour the electron channel. The observed factor of 8000 against it is chirality; the 4 per cent gap between 1.283 and 1.233 × 10⁻⁴ is radiative corrections.

AnswerR = 1.283 × 10⁻⁴ at tree level against 1.233 × 10⁻⁴ measured. The m_ℓ² is wrong-helicity suppression, while phase space, favouring the positron by 69.8 to 29.8 MeV/c, would have predicted the opposite.