10.2 Conservation Laws in Particle Interactions

Conservation Laws for Particle Interactions

Conservation laws determine which particle interactions can occur and which are forbidden. If a proposed reaction violates even one conservation law, it cannot happen. Some of these laws apply universally across all interactions, while others hold only for specific force types.

Fundamental Conservation Principles

A few conservation laws are absolute, meaning they hold in every interaction regardless of which force mediates it:

• Conservation of energy and momentum: Total energy and total momentum of a closed system remain constant before and after any interaction.
• Conservation of electric charge: The total charge of all particles before an interaction must equal the total charge after.
• Conservation of baryon number: The net number of baryons (baryons minus antibaryons) stays constant.
• Conservation of lepton number: The total lepton number for each flavor (electron, muon, tau) is separately conserved.
• Conservation of angular momentum: Total spin angular momentum of the system is unchanged.

Then there are flavor quantum numbers like strangeness and charm. These are conserved in strong and electromagnetic interactions but can be violated in weak interactions. This distinction matters when you're deciding whether a given process is allowed.

Applying Conservation Laws

To check whether a reaction is valid, tally up the relevant quantum numbers on each side. Here's how each one works:

Electric charge conservation — Sum the charges on both sides. They must match.

• Example: $p + e^- \rightarrow n + \nu_e$
• Left side: $+1 + (-1) = 0$. Right side: $0 + 0 = 0$. Charge is conserved.

Baryon number conservation — Assign $B = +1$ to baryons, $B = -1$ to antibaryons, and $B = 0$ to everything else (mesons, leptons, photons).

• Example: $n \rightarrow p + e^- + \bar{\nu}_e$
• Left: $B = +1$. Right: $+1 + 0 + 0 = +1$. Baryon number is conserved.

Lepton number conservation — Assign $L = +1$ to leptons, $L = -1$ to antileptons, and $L = 0$ to non-leptonic particles. Each flavor ($L_e$, $L_\mu$, $L_\tau$) must be conserved separately.

• Example: $\mu^- \rightarrow e^- + \bar{\nu}_e + \nu_\mu$
• $L_\mu$: $+1 = 0 + 0 + 1$ ✓
• $L_e$: $0 = +1 + (-1) + 0$ ✓

Charge, Baryon, and Lepton Conservation

Balancing Equations

The procedure for checking any reaction is straightforward:

1. List every particle on both sides of the reaction.
2. Look up (or recall) each particle's charge $Q$, baryon number $B$, and lepton numbers $L_e, L_\mu, L_\tau$.
3. Sum each quantum number on the left side and on the right side.
4. If any quantum number doesn't match, the reaction is forbidden.

Example — proton-antiproton annihilation:

$p + \bar{p} \rightarrow \pi^+ + \pi^-$

• Charge: $(+1) + (-1) = (+1) + (-1) = 0$ ✓
• Baryon number: $(+1) + (-1) = 0 + 0 = 0$ ✓
• Lepton numbers: all zero on both sides ✓

This reaction is allowed.

Example — charged kaon decay:

$K^+ \rightarrow \pi^+ + \pi^0$

• Charge: $+1 = +1 + 0$ ✓
• Baryon number: $0 = 0 + 0$ ✓
• Strangeness: $+1 \rightarrow 0 + 0$ (changes by 1, so this must proceed via the weak interaction)

Particle Properties and Conservation

When working with antiparticles, remember that every quantum number flips sign. A proton has $Q = +1, B = +1$; an antiproton has $Q = -1, B = -1$.

For more complex reactions involving hyperons or heavier baryons, the same approach applies. Just be careful to assign the correct quantum numbers.

Example:

$\Sigma^+ \rightarrow n + \pi^+$

• Charge: $+1 = 0 + 1$ ✓
• Baryon number: $+1 = +1 + 0$ ✓
• Strangeness: $-1 \rightarrow 0 + 0$ (strangeness changes, so this is a weak decay)

Analyzing Particle Decay Processes

Decay Processes and Conservation Laws

Every particle decay must obey all conservation laws simultaneously. The classic example is neutron beta decay:

$n \rightarrow p + e^- + \bar{\nu}_e$

Check each law:

• Charge: $0 = +1 + (-1) + 0$ ✓
• Baryon number: $+1 = +1 + 0 + 0$ ✓
• Lepton number ($L_e$): $0 = 0 + 1 + (-1)$ ✓

Notice why the antineutrino must be there. Without $\bar{\nu}_e$, lepton number would go from 0 to +1, violating conservation. The conservation law actually predicts which particles appear in the final state.

For decay chains (where a product itself decays), apply conservation laws at each individual vertex. Every step in the chain must independently satisfy all the rules.

Virtual particles can appear as intermediaries in a decay process. They can temporarily violate energy conservation, but only within the limits set by the Heisenberg uncertainty principle ($\Delta E \cdot \Delta t \lesssim \hbar$). The final, observable products must still satisfy all conservation laws.

Visualizing and Analyzing Decays

Feynman diagrams are a useful tool for tracking conservation laws. At every vertex in a Feynman diagram, charge, baryon number, and lepton number must all be conserved.

Example — kaon decay:

$K^+ \rightarrow \mu^+ + \nu_\mu$

• Charge: $+1 = +1 + 0$ ✓
• Baryon number: $0 = 0 + 0$ ✓
• Lepton number ($L_\mu$): $0 = (-1) + (+1) = 0$ ✓

Note that $\mu^+$ is an antimuon, so it carries $L_\mu = -1$, while $\nu_\mu$ carries $L_\mu = +1$. They cancel. Also, the kaon has strangeness $+1$ while the products have strangeness 0, confirming this is a weak decay.

Allowed vs. Forbidden Interactions

Evaluating Particle Interactions

To determine whether a reaction is allowed:

1. Check every conservation law (charge, baryon number, all three lepton flavors, energy-momentum).
2. If all are satisfied, the reaction is allowed.
3. If any one is violated, the reaction is forbidden.
4. If flavor quantum numbers (strangeness, charm) change, the reaction can only proceed through the weak interaction.

Forbidden interaction example — proton decay:

$p \rightarrow e^+ + \pi^0$

• Charge: $+1 = +1 + 0$ ✓
• Baryon number: $+1 \neq 0 + 0$ ✗

Baryon number is violated, so this decay is forbidden in the Standard Model. (Some grand unified theories predict it could happen at extraordinarily low rates, but it has never been observed.)

Hypothetical example — neutrinoless double beta decay:

$^A_Z X \rightarrow ^A_{Z+2} Y + 2e^-$

This would change lepton number by 2 with no neutrinos to compensate. If observed, it would mean lepton number conservation is not absolute. Experiments are actively searching for this process.

Advanced Considerations

Particle generations organize quarks and leptons into three families:

Interactions that stay within a single generation are generally favored. Cross-generation transitions (like $s \rightarrow u$ in kaon decay) are suppressed but still allowed through the weak force.

CPT symmetry (Charge conjugation × Parity × Time reversal) is considered an exact symmetry of nature. Any interaction that violates CPT symmetry is strictly forbidden. Individual symmetries like C, P, or CP can be violated (and are, in weak interactions), but their combined product CPT must hold.

Key forbidden reactions to recognize:

• $\mu^- \rightarrow e^- + \gamma$ — Violates lepton flavor conservation ($L_\mu$ and $L_e$ both change). Forbidden even in weak interactions.
• $n \leftrightarrow \bar{n}$ oscillations — Would violate baryon number by 2 units. Hypothetical and never observed.