---
title: Conservation Laws and Symmetries
module: Symmetries and Conservation Laws
moduleNumber: 3
lessonNumber: 1
order: 301
summary: >
  Which decays occur is decided by conservation laws, each tied by Noether's
  theorem to a symmetry of physical law. Energy, charge, baryon number, and
  lepton number are conserved universally; strangeness, isospin, and parity hold
  in the strong and electromagnetic interactions but break in the weak one, whose
  parity and CP violation distinguish matter from antimatter.
topics: [Symmetries and Conservation Laws]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 12 — Particle Physics; §12-3 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.[^tl-noether]

> **Theorem (Noether).** Each continuous symmetry of the laws of physics implies
> a conserved quantity. Invariance under a transformation of a coordinate yields
> conservation of the quantity conjugate to it.

| Symmetry (invariance under) | Conserved quantity |
| --- | --- |
| Translation in time | Energy |
| Translation in space | Linear momentum |
| Rotation about a point | Angular momentum |
| Gauge (scale) transformation | Electric charge |

$$
% caption: Noether's correspondence: each continuous symmetry of physical law on
% the left produces the conserved quantity on the right.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  sym/.style={draw, minimum width=34mm, minimum height=8mm, align=center, font=\scriptsize},
  con/.style={draw, minimum width=30mm, minimum height=8mm, align=center, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[sym] (t) at (0,3.0) {time translation};
  \node[sym] (x) at (0,2.0) {space translation};
  \node[sym] (r) at (0,1.0) {rotation};
  \node[sym] (g) at (0,0.0) {gauge transformation};
  \node[con, draw=acc, text=acc] (e) at (6,3.0) {energy};
  \node[con, draw=acc, text=acc] (p) at (6,2.0) {linear momentum};
  \node[con, draw=acc, text=acc] (l) at (6,1.0) {angular momentum};
  \node[con, draw=acc, text=acc] (q) at (6,0.0) {electric charge};
  \draw[->, acc] (t) -- (e);
  \draw[->, acc] (x) -- (p);
  \draw[->, acc] (r) -- (l);
  \draw[->, acc] (g) -- (q);
\end{tikzpicture}
$$

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.

> **Example (Testing a decay).** Consider $n \to p + \pi^-$. Charge is conserved
> ($0 = +1 - 1$) and $B$ is conserved ($1 = 1 + 0$), but the rest energy of the
> products, $m_p c^2 + m_\pi c^2 = 938.3 + 139.6 = 1077.9$ MeV, exceeds the
> neutron rest energy $939.6$ MeV. The decay violates energy conservation and
> does not occur.

## 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**).

$$
% caption: Associated production. The strong reaction conserves strangeness by
% making a pair of opposite S (the kaon +1, the lambda -1); each product then
% decays slowly through the weak interaction, which does not conserve strangeness.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % incoming
  \draw[->, acc, thick] (-2.4,0.6) -- (-0.4,0.2) node[pos=0, left, font=\scriptsize] {pion};
  \draw[->, acc, thick] (-2.4,-0.6) -- (-0.4,-0.2) node[pos=0, left, font=\scriptsize] {$p$};
  \fill[acc] (-0.2,0) circle (2pt);
  \node[font=\scriptsize] at (-0.2,-1.1) {strong: $S$ conserved};
  % outgoing strange particles
  \draw[->, acc, thick] (0,0) -- (2.2,1.1) node[right, font=\scriptsize] {kaon $K^0$};
  \draw[->, acc, thick] (0,0) -- (2.2,-1.1) node[right, font=\scriptsize] {lambda};
  % weak decays downstream (dashed continuation)
  \draw[acc, thick, dashed] (2.2,1.1) -- (4.0,1.6);
  \node[font=\scriptsize, align=center] at (4.9,1.6) {weak decay\\(slow)};
  \draw[acc, thick, dashed] (2.2,-1.1) -- (4.0,-1.6);
  \node[font=\scriptsize, align=center] at (4.9,-1.6) {weak decay\\(slow)};
\end{tikzpicture}
$$

**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

$$
Q = I_3 + \frac{B + S}{2}.
$$

The nucleon has $I = \tfrac{1}{2}$ with $I_3 = +\tfrac{1}{2}$ (proton) and
$I_3 = -\tfrac{1}{2}$ (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

$$
Y = 2(Q - I_3).
$$

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

$$
% caption: The baryon octet plotted in the hypercharge-isospin plane, labeled by
% quark content. Sloping lines are constant charge; horizontal lines are constant
% hypercharge (and strangeness).
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.1]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-2.4,0) -- (2.4,0) node[right, font=\scriptsize] {$I_3$};
  \draw[->, black] (0,-1.9) -- (0,1.9) node[above, font=\scriptsize] {$Y$};
  % top row Y=1
  \fill[acc] (-0.8,1.2) circle (2.2pt); \node[above, font=\scriptsize] at (-0.8,1.3) {$udd$};
  \fill[acc] (0.8,1.2) circle (2.2pt);  \node[above, font=\scriptsize] at (0.8,1.3) {$uud$};
  % middle row Y=0
  \fill[acc] (-1.6,0) circle (2.2pt); \node[left, font=\scriptsize] at (-1.7,0) {$dds$};
  \fill[acc] (0,0) circle (2.2pt);
  \fill[acc] (0.18,0.18) circle (2.2pt); \node[above right, font=\scriptsize] at (0.1,0.15) {$uds$};
  \fill[acc] (1.6,0) circle (2.2pt); \node[right, font=\scriptsize] at (1.7,0) {$uus$};
  % bottom row Y=-1
  \fill[acc] (-0.8,-1.2) circle (2.2pt); \node[below, font=\scriptsize] at (-0.8,-1.3) {$dss$};
  \fill[acc] (0.8,-1.2) circle (2.2pt);  \node[below, font=\scriptsize] at (0.8,-1.3) {$uss$};
  % hexagon outline
  \draw[acc!55] (-0.8,1.2) -- (0.8,1.2) -- (1.6,0) -- (0.8,-1.2) -- (-0.8,-1.2) -- (-1.6,0) -- cycle;
\end{tikzpicture}
$$

## 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**.

$$
% caption: The parity test. A spinning nucleus emits an electron along its spin;
% the mirror reverses the spin sense but not the emission direction, so a
% preferred direction distinguishes the process from its mirror image.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % real process
  \draw[acc, thick] (0,-0.8) circle (0.35);
  \draw[->, acc, thick] (0,-0.35) -- (0,0.6) node[right, font=\scriptsize] {spin};
  \draw[->, acc, very thick] (0,-0.35) .. controls (0.5,0.3) .. (0.7,1.2);
  \node[acc, right, font=\scriptsize] at (0.7,1.2) {electron};
  \node[font=\scriptsize] at (0,-1.6) {process};
  % mirror line
  \draw[black, dashed, thick] (2.3,-2.0) -- (2.3,2.0);
  \node[black, font=\scriptsize, above] at (2.3,2.0) {mirror};
  % mirror image
  \begin{scope}[xshift=4.6cm, xscale=-1]
    \draw[acc, thick] (0,-0.8) circle (0.35);
    \draw[->, acc, thick] (0,-0.35) -- (0,0.6);
    \draw[->, acc, very thick] (0,-0.35) .. controls (0.5,0.3) .. (0.7,1.2);
  \end{scope}
  \node[font=\scriptsize] at (4.6,-1.6) {mirror image};
\end{tikzpicture}
$$

**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 $P$:

$$
TCP\,\Psi(\vec{r}, t) = \Psi(\vec{r}, t), \qquad TCP = +1.
$$

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
$K^0_L$, 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.

| Conserved quantity | Strong | Electromagnetic | Weak |
| --- | --- | --- | --- |
| Energy, momentum, charge $Q$ | Yes | Yes | Yes |
| Baryon number $B$, lepton number $L$ | Yes | Yes | Yes |
| Isospin $I$ | Yes | No | No |
| Strangeness $S$, hypercharge $Y$ | Yes | Yes | No ($\Delta S = \pm 1, 0$) |
| Parity $P$ | Yes | Yes | No |

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.

> **Example (Which interaction?).** For each decay the mass of the parent exceeds
> that of the products (energy conserved), no leptons appear, and charge and
> baryon number balance. The remaining test is hypercharge $Y$.
>
> - $\Sigma^+ \to p + \pi^0$: the $\Sigma^+$ has $Y = 0$, the proton $Y = 1$, the
>   pion $Y = 0$, so $\Delta Y = 1$. Hypercharge changes, so the decay is weak
>   (lifetime $\sim 10^{-10}$ s).
> - $\Sigma^0 \to \Lambda^0 + \gamma$: both have $Y = 0$, so $\Delta Y = 0$. With
>   a photon in the final state the decay is electromagnetic (lifetime
>   $\sim 10^{-20}$ s).
> - $\Xi^0 \to n + \pi^0$: the $\Xi^0$ has $Y = -1$, the neutron $Y = 1$, the pion
>   $Y = 0$, so $\Delta Y = 2$. Hypercharge cannot change by $2$ in a single decay,
>   so this cannot occur at all.

## 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
$\Delta E \approx \hbar/\Delta t$, the short lifetime gives a broad energy width.

$$
% caption: A resonance appears as a peak in the scattering cross section at the
% energy matching the excited-state mass; the peak width is inversely related to
% the state's lifetime.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[right, font=\scriptsize] {collision energy};
  \draw[->, black] (0,0) -- (0,4.0) node[above, font=\scriptsize] {cross section};
  % Lorentzian-like peak centered at x=3.2
  \draw[acc, very thick] plot[domain=0.3:6.1, samples=80] (\x, {0.3 + 3.2/(1 + (\x-3.2)*(\x-3.2)*3.0)});
  % width markers
  \draw[black, dashed] (3.2,0) -- (3.2,3.5);
  \node[black, below, font=\scriptsize] at (3.2,0) {$E_0$};
  \node[acc, right, font=\scriptsize] at (3.3,3.4) {resonance};
\end{tikzpicture}
$$

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](/particle-physics/electroweak-higgs/standard-model) lesson.

[^tl-noether]: Tipler & Llewellyn, §12-3 — Noether's theorem, the additive and multiplicative conservation laws, parity violation in the weak interaction, and TCP invariance with CP violation in the neutral kaon system.