Every quantum number met so far — charge, baryon number, lepton number, strangeness — exists for one practical purpose: deciding which particle interactions can actually happen. This lesson sets out beta-plus decay alongside the beta-minus decay from the last lesson, then walks through the systematic method for checking any proposed interaction against every conservation law at once.
9 min read
Beta-minus and beta-plus decay, at the quark level
The previous lesson showed that neutron decay, $n \rightarrow p + e^- + \bar\nu_e$, is really a single down quark converting into an up quark. Written purely in terms of that quark transition, beta-minus decay is:
$d \rightarrow u + e^- + \bar\nu_e$
A down quark (charge $-\tfrac13$) becomes an up quark (charge $+\tfrac13$), and the extra charge of $-1$ that's disappeared from the quark is carried away by the emitted electron, with the antineutrino balancing lepton number.
The reverse process also occurs: a proton-rich nucleus can undergo beta-plus decay, emitting a positron and a neutrino as one of its up quarks converts into a down quark:
$u \rightarrow d + e^+ + \nu_e$
Here the quark's charge falls from $+\tfrac23$ to $-\tfrac13$, a drop of $1$, carried away by the emitted positron. At the whole-nucleon level, this underlies $p \rightarrow n + e^+ + \nu_e$ inside a nucleus — a free proton can't decay this way, but a proton bound inside a proton-rich nucleus sometimes can. Between them, $d\rightarrow u$ and $u \rightarrow d$ account for every beta decay you'll meet.
The systematic method: checking a proposed interaction
Given any proposed particle interaction, you can decide whether it's allowed without knowing anything about the mechanism. Go through each conserved quantity, add it up on both sides, and check the totals match. If even one conservation law fails, the interaction cannot occur.
MethodChecking whether a proposed interaction is allowed
1
Check charge. Add up the total electric charge of everything on the left-hand side, and everything on the right-hand side. These totals must be equal.
Charge conservation holds in every interaction, without exception, at every level from nucleus to quark.
2
Check baryon number. Add up $B$ for every particle on each side ($+1$ per baryon, $-1$ per antibaryon, $0$ otherwise). The totals must match.
This must hold in every interaction, strong, electromagnetic or weak.
3
Check lepton number — but as two separate totals, not one. Add up $L_e$ on each side, and separately add up $L_\mu$ on each side. Both totals must match independently.
An interaction that conserves the combined lepton count but shuffles electron-type leptons into muon-type ones is still forbidden.
4
Check strangeness — but only if the interaction is a strong or electromagnetic one. Add up $s$ on each side; the totals must be equal for a strong or electromagnetic process.
Strangeness need not be conserved in a weak interaction, where it's allowed to change by 0, +1 or −1.
5
If every relevant total matches, the interaction is allowed by these conservation laws. If any one total fails to match, the interaction is forbidden — checking the remaining quantities becomes unnecessary.
A single broken conservation law is enough to rule an interaction out completely.
Energy and momentum still apply
Charge, baryon number, lepton number and strangeness are the conservation laws specific to particle physics, but energy and momentum are conserved in every particle interaction too. Momentum conservation is why annihilation and pair production only occur under specific geometric conditions; energy conservation is why beta decay needed a third particle, the neutrino — without it, the electron alone couldn't carry a fixed, single energy, and the observed spread of electron energies would violate energy conservation.
Checking charge and baryon number is usually quick. Lepton number needs care, since electron-type and muon-type must be checked as two separate sums. Strangeness is the one most students forget: always ask first whether the interaction is strong/electromagnetic or weak, since that decides whether it needs to balance exactly or is allowed to shift.
Worked exampleIs $K^+ \rightarrow \mu^+ + \nu_\mu$ allowed?
1
Charge. Left: $K^+$ has charge $+1$. Right: $\mu^+$ has charge $+1$, $\nu_\mu$ has charge $0$, total $+1$. Charge matches: $+1 = +1$. ✓
2
Baryon number. The kaon is a meson, $B=0$; the muon and muon-neutrino are leptons, $B=0$ each. Total on both sides: $0 = 0$. ✓
Neither side contains a baryon or antibaryon, so this check is trivially satisfied.
3
Lepton number. $L_e$: zero on both sides (no electron-type leptons appear anywhere). $L_\mu$: left is $0$ (the kaon is not a lepton); right has $\mu^+$ with $L_\mu = -1$ and $\nu_\mu$ with $L_\mu = +1$, summing to $0$. Both totals match: $0=0$ for $L_e$, $0=0$ for $L_\mu$. ✓
$\mu^+$ is the antiparticle of the muon, so it carries $L_\mu=-1$, not $+1$ — a common slip to check carefully.
4
Strangeness. This decay proceeds via the weak interaction (kaon decays are always weak), so strangeness is not required to balance. The kaon carries $s=+1$; the products carry $s=0$, a change of $-1$, which is allowed for a weak process.
Since the interaction is weak, this check doesn't even need to match — it's included here only to show explicitly that the "violation" is permitted, not to be mistaken for a problem.
5
Every relevant conservation law checks out (and the one that doesn't match — strangeness — isn't required to, since this is a weak interaction). The decay $K^+ \rightarrow \mu^+ + \nu_\mu$ is allowed, and is in fact one of the dominant decay modes of the kaon.
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Worked exampleWhy $\Lambda^0 \rightarrow p + \pi^-$ is forbidden as a strong interaction
1
The $\Lambda^0$ baryon has quark content $uds$, so it carries strangeness $s=-1$ (from its single strange quark), charge $0$, and baryon number $+1$.
2
Charge. Left: $0$. Right: proton $+1$, $\pi^-$ has charge $-1$, total $0$. Charge matches: $0=0$. ✓
3
Baryon number. Left: $\Lambda^0$ is a baryon, $B=+1$. Right: the proton is a baryon, $B=+1$; the pion is a meson, $B=0$. Total $+1$. Baryon number matches: $+1=+1$. ✓
4
Lepton number. No leptons appear on either side, so $L_e=0$ and $L_\mu=0$ trivially on both sides. ✓
5
Strangeness. Left: $\Lambda^0$ has $s=-1$. Right: the proton (uud) and $\pi^-$ ($d\bar u$) both carry $s=0$, total $0$. So strangeness goes from $-1$ to $0$, a change of $+1$.
If this decay is proposed to happen via the strong interaction, strangeness must be exactly conserved — no change at all is permitted.
6
Charge, baryon number and lepton number all balance perfectly, but strangeness does not, and a strong interaction cannot change strangeness by any amount. This interaction is therefore forbidden as a strong process.
This is exactly why the $\Lambda^0$ can't decay this way strongly, even though it looks perfectly reasonable by every other conservation law — and it's also why, when the $\Lambda^0$ does eventually decay to $p+\pi^-$, it must do so via the much slower weak interaction instead, where a strangeness change of $\pm1$ is allowed.
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The lesson of the forbidden example
Four out of five conservation laws checking out perfectly still isn't enough — a single failed law is enough to forbid an interaction outright. This is why strangeness is worth checking every time: it's the one most often skipped by mistake, and the one most likely to be the single quantity that fails.
Key takeaways
Beta-minus decay at the quark level: $d \rightarrow u + e^- + \bar\nu_e$ — a down quark becomes an up quark, emitting an electron and an antineutrino.
Beta-plus decay at the quark level: $u \rightarrow d + e^+ + \nu_e$ — an up quark becomes a down quark, emitting a positron and a neutrino.
To check whether a proposed interaction is allowed, sum charge, baryon number, $L_e$, $L_\mu$ and (for strong/electromagnetic interactions) strangeness separately on each side.
If any one of these totals fails to match between the two sides, the interaction is forbidden — a single violated conservation law is enough to rule it out.
Strangeness only has to be conserved in strong and electromagnetic interactions; in a weak interaction it may change by 0, +1 or −1.
Worked check: $K^+ \rightarrow \mu^+ + \nu_\mu$ conserves charge, baryon number and both lepton numbers, and its strangeness change ($-1\rightarrow0$) is permitted because it's a weak decay — the interaction is allowed.
Worked check: $\Lambda^0 \rightarrow p + \pi^-$ conserves charge, baryon number and lepton number, but its strangeness change ($-1\rightarrow0$) is not permitted for a strong interaction — the interaction is forbidden as a strong process, and can only proceed weakly.
Energy and momentum remain conserved in every particle interaction alongside these particle-specific quantum numbers — this is why the neutrino had to exist, and why annihilation/pair production only happen under the right conditions.