# 11.2 Particle conservation laws

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By the end of this section, you will be able to:
• Distinguish three conservation laws: baryon number, lepton number, and strangeness
• Use rules to determine the total baryon number, lepton number, and strangeness of particles before and after a reaction
• Use baryon number, lepton number, and strangeness conservation to determine if particle reactions or decays occur

Conservation laws are critical to an understanding of particle physics. Strong evidence exists that energy, momentum, and angular momentum are all conserved in all particle interactions. The annihilation of an electron and positron at rest, for example, cannot produce just one photon because this violates the conservation of linear momentum. As discussed in Relativity , the special theory of relativity modifies definitions of momentum, energy, and other familiar quantities. In particular, the relativistic momentum of a particle differs from its classical momentum by a factor $\gamma =1\text{/}\sqrt{1-{\left(v\text{/}c\right)}^{2}}$ that varies from 1 to $\infty ,$ depending on the speed of the particle.

In previous chapters, we encountered other conservation laws as well. For example, charge is conserved in all electrostatic phenomena. Charge lost in one place is gained in another because charge is carried by particles. No known physical processes violate charge conservation. In the next section, we describe three less-familiar conservation laws: baryon number, lepton number, and strangeness. These are by no means the only conservation laws in particle physics.

## Baryon number conservation

No conservation law considered thus far prevents a neutron from decaying via a reaction such as

$\text{n}\to {\text{e}}^{+}+{\text{e}}^{\text{−}}.$

This process conserves charge, energy, and momentum. However, it does not occur because it violates the law of baryon number conservation. This law requires that the total baryon number of a reaction is the same before and after the reaction occurs. To determine the total baryon number, every elementary particle is assigned a baryon number     B . The baryon number has the value $B=+1$ for baryons, –1 for antibaryons, and 0 for all other particles. Returning to the above case (the decay of the neutron into an electron-positron pair), the neutron has a value $B=+1,$ whereas the electron and the positron each has a value of 0. Thus, the decay does not occur because the total baryon number changes from 1 to 0. However, the proton-antiproton collision process

$\text{p}+\stackrel{\text{−}}{\text{p}}\to \text{p}+\text{p}+\stackrel{\text{−}}{\text{p}}+\stackrel{\text{−}}{\text{p}},$

does satisfy the law of conservation of baryon number because the baryon number is zero before and after the interaction. The baryon number for several common particles is given in [link] .

Conserved properties of particles
Particle name Symbol Lepton number $\left({L}_{e}\right)$ Lepton number $\left({L}_{\mu }\right)$ Lepton number $\left({L}_{\tau }\right)$ Baryon number ( B ) Strange-ness number
Electron ${\text{e}}^{\text{−}}$ 1 0 0 0 0
Electron neutrino ${\upsilon }_{e}$ 1 0 0 0 0
Muon ${\mu }^{\text{−}}$ 0 1 0 0 0
Muon neutrino ${\upsilon }_{\mu }$ 0 1 0 0 0
Tau ${\tau }^{\text{−}}$ 0 0 1 0 0
Tau neutrino ${\upsilon }_{\tau }$ 0 0 1 0 0
Pion ${\pi }^{+}$ 0 0 0 0 0
Positive kaon ${\text{K}}^{+}$ 0 0 0 0 1
Negative kaon ${\text{K}}^{\text{−}}$ 0 0 0 0 –1
Proton p 0 0 0 1 0
Neutron n 0 0 0 1 0
Lambda zero ${\text{Λ}}^{0}$ 0 0 0 1 –1
Positive sigma ${\text{Σ}}^{+}$ 0 0 0 1 –1
Negative sigma ${\text{Σ}}^{\text{−}}$ 0 0 0 1 –1
Xi zero ${\text{Ξ}}^{0}$ 0 0 0 1 –2
Negative xi ${\text{Ξ}}^{\text{−}}$ 0 0 0 1 –2
Omega ${\text{Ω}}^{\text{−}}$ 0 0 0 1 –3

## Baryon number conservation

Based on the law of conservation of baryon number, which of the following reactions can occur?

$\begin{array}{}\\ \\ \left(\text{a}\right)\phantom{\rule{0.2em}{0ex}}{\pi }^{\text{−}}+\text{p}\to {\pi }^{0}+\text{n}+{\pi }^{\text{−}}+{\pi }^{+}\hfill \\ \left(\text{b}\right)\phantom{\rule{0.2em}{0ex}}\text{p}+\stackrel{\text{−}}{\text{p}}\to \text{p}+\text{p}+\stackrel{\text{−}}{\text{p}}\hfill \end{array}$

## Strategy

Determine the total baryon number for the reactants and products, and require that this value does not change in the reaction. Solution

For reaction (a), the net baryon number of the two reactants is $0+1=1$ and the net baryon number of the four products is $0+1+0+0=1.$ Since the net baryon numbers of the reactants and products are equal, this reaction is allowed on the basis of the baryon number conservation law.

For reaction (b), the net baryon number of the reactants is $1+\left(-1\right)=0$ and the net baryon number of the proposed products is $1+1+\left(-1\right)=1.$ Since the net baryon numbers of the reactants and proposed products are not equal, this reaction cannot occur.

## Significance

Baryon number is conserved in the first reaction, but not in the second. Baryon number conservation constrains what reactions can and cannot occur in nature.

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