Conservation Laws and Symmetries

A guiding maxim of particle physics is that anything not forbidden happens: if a conceivable decay or reaction is never observed, some conservation law forbids it. This lesson catalogues those laws, ties each to an underlying symmetry through Noether's theorem, and shows how the strangeness, parity, and CP rules sort a reaction into the strong, electromagnetic, or weak interaction, or rule it out entirely.

Symmetry and Noether's theorem

Every conservation law follows from a symmetry of the laws of physics, a result proven by Emmy Noether in 1918.¹

As in classical physics, a conservation law is often discovered empirically before its symmetry is identified. Most of the particle-physics laws below are still empirical: no underlying symmetry has been found for them.

Additive quantum numbers

Several conserved quantities are additive: the total over all particles before a reaction must equal the total after.

Baryon number. Every baryon has $B=+1$, every antibaryon $B=-1$, and all other particles $B=0$. Conservation of $B$ requires, for antiproton production $p+p\to p+p+p+\bar{p}$, that a proton accompany every antiproton ($B=2$ before and $B=1+1+1-1=2$ after). Together with energy conservation this makes the proton, the lightest baryon, stable. No known symmetry demands baryon conservation, and searches for proton decay place its lifetime above $10^{32}$ years.

Lepton number. Lepton number is conserved independently for each flavor. The electron and electron neutrino have $L_e=+1$, the positron and electron antineutrino $L_e=-1$; muon and tau numbers $L_{\mu }$, $L_{\tau }$ are assigned likewise. The absence of ${\mu }^{-}\to e^{-}\gamma$, searched for over years, was the first evidence that $L_e$ and $L_{\mu }$ are separately conserved. Neutron decay $n\to p+e^{-}+{\bar{\nu }}_e$ conserves both $B$ and $L_e$, which fixes the emitted particle as an electron _anti_neutrino.

Strangeness, isospin, and hypercharge

Some quantities are conserved only in certain interactions. Strangeness $S$, introduced by Gell-Mann and Nishijima in 1952, is conserved in strong and electromagnetic processes but not weak ones, where it may change by $\Delta S=\pm 1$.

The motivation was the behavior of strange particles. In ${\pi }^{-}+p\to {\Lambda }^0+K^0$ the cross section is large, characteristic of the strong interaction, yet the ${\Lambda }^0$ and $K^0$ decay slowly ($\sim 10^{-10}$ s), characteristic of the weak interaction. Assigning $S=+1$ to the $K^0$ and $S=-1$ to the ${\Lambda }^0$ lets strangeness be conserved in the strong production but forces the subsequent single-particle decays through the weak interaction, where $S$ need not hold. These particles are always produced in pairs of opposite strangeness (associated production).

Isospin. Hadrons cluster into charge multiplets of nearly equal mass, such as the proton-neutron pair. Because the strong force is charge-independent, the members are viewed as different charge states of one particle, described by an isospin vector $I$ in an abstract charge space. Its third component $I_3$ is quantized into $2I+1$ values, and the charge relates to it by

The nucleon has $I=\frac{1}{2}$ with $I_3=+\frac{1}{2}$ (proton) and $I_3=-\frac{1}{2}$ (neutron); the pion is an $I=1$ triplet. Isospin is conserved only when the strong interaction acts alone.

Hypercharge. The combination $Y\equiv B+S+C+\mathcal{B}+T$ (for the light hadrons, $Y=B+S$) is twice the average charge of a multiplet, giving

Plots of $Y$ against $I_3$ reveal the regular patterns that led to the quark model. The eight lightest spin-$\frac{1}{2}$ baryons trace a hexagon with two particles at the center.

Multiplicative quantum numbers and discrete symmetries

Parity. The parity operation reflects the space coordinates through the origin, $\vec{r}\to -\vec{r}$. If the wave function is unchanged, parity is even ($P=+1$); if it changes sign, parity is odd ($P=-1$). Parity is multiplicative, not additive, and for an atomic state $P=(-1{)}^{\ell }$.

Until 1956 parity was assumed conserved everywhere. Lee and Yang noted that the evidence covered only strong and electromagnetic processes and proposed that the weak interaction might violate parity. Wu and Ambler tested this by aligning the spins of ${}^{60}\text{Co}$ nuclei at $0.01$ K and measuring the beta-decay electron directions. More electrons emerged opposite to the nuclear spin than along it: a mirror-asymmetric outcome, so parity is not conserved in the weak interaction.

TCP and CP violation. Any relativistic quantum theory is invariant under the combined operation of time reversal $T$, charge conjugation $C$ (particle $\leftrightarrow$ antiparticle), and parity $P$:

This forces particles and antiparticles to share mass and lifetime. Because the weak interaction gives $P=-1$, at least one of $C$ or $T$ must also break there. In 1964 Christenson and collaborators found that the long-lived neutral kaon $K_L^0$, which normally decays to three pions ($CP=-1$), decays to two pions ($CP=+1$) about once in a thousand: $CP$ is violated. Within the Standard Model, $CP$ violation requires three generations of quarks, and it is a condition for the observed matter-antimatter asymmetry of the universe. With $TCP=+1$ and $CP=-1$, time-reversal symmetry $T$ must also break, establishing an absolute direction for time.

The decisive test between the three interactions is hypercharge (equivalently strangeness), conserved by the strong and electromagnetic forces but not the weak. A decay that changes hypercharge must proceed weakly and so is slow; one that conserves it and involves no leptons can proceed strongly or electromagnetically and is fast.

Resonances

Excited states of hadrons decay via the strong interaction in $\sim 10^{-23}$ s, too fast to leave a track. They are detected as resonances: peaks in the cross section for scattering one hadron on another, at the collision energy matching the excited state's mass. By the uncertainty principle $\Delta E\approx \hslash /\Delta t$, the short lifetime gives a broad energy width.

The regularities in the $Y$-$I_3$ diagrams and the conservation rules were the raw material from which the quark model was built, the subject of the Standard Model lesson.