24 Nov 2024

Particle Physics: Conservation Laws

Baryon number, lepton number, strangeness, isospin, and charge conservation in particle reactions.

msc semester-iii particle-physics conservation-laws strangeness

The conservation laws of energy, momentum, and charge govern all processes. In particle physics, additional empirical conservation laws are also crucial. They are:

1. Conservation of baryon number
2. Conservation of lepton number
3. Conservation of strangeness
4. Conservation of isospin
5. Conservation of electric charge

Baryon Number and Lepton Number

• Experimental results show that whenever a baryon is created in a decay or nuclear reaction, an antibaryon is also created. This scheme can be quantified by assigning every particle a quantum number, the baryon number, as follows: $B=+1$ for all baryons, $B=-1$ for all antibaryons, and $B=0$ for all other particles. The law of conservation of baryon number states that

  Whenever a nuclear reaction or decay occurs, the sum of the baryon numbers before the process must equal the sum of the baryon numbers after the process.
  

• The conservation of lepton number is similar to the conservation of baryon number. The lepton number is defined as $L=+1$ for all leptons, $L=-1$ for all antileptons, and $L=0$ for all other particles. The law of conservation of lepton number states that

  Whenever a nuclear reaction or decay occurs, the sum of the lepton numbers before the process must equal the sum of the lepton numbers after the process.
  

Assignments

1. Use the law of conservation of baryon number to determine whether each of the following reactions can occur:
   • $p + p \rightarrow p + p + \pi^0$
   • $p + p \rightarrow p + p + K^0$
   • $p + p \rightarrow p + p + \bar{K}^0$
   • $p + p \rightarrow p + p + \Lambda^0$
   • $p + p \rightarrow p + p + \Sigma^+$
   • $p + p \rightarrow p + p + \Xi^0$
   • $p + p \rightarrow p + p + \Xi^-$
   • $p + p \rightarrow p + p + \Omega^-$
   • $p + p \rightarrow p + p + n + \bar{n}$
   • $p + p \rightarrow p + p + n + \bar{n} + \bar{\Xi}^0$
   • $p + p \rightarrow p + p + n + \bar{n} + \bar{\Omega}^-$
2. Now use Lepton number conservation to determine whether the following reactions can occur:
   • $\mu^- \rightarrow e^- + \bar{\nu_e} + \nu_{\mu}$
   • $\pi^+ \rightarrow \mu^+ + \nu_{\mu}+ \nu_e $
3. Each of the following reactions is forbidden. Determine what conservation laws are violated for each reaction.
   • $p + \bar{p} \rightarrow \mu^+ + e^-$
   • $\pi^- + p \rightarrow p + \pi^+$
   • $p + p \rightarrow p + p + n$
   • $\gamma + p \rightarrow \eta + \pi^0$
   • $\nu_e + p \rightarrow n + e^+$

---

Strange Particles and Strangeness

Many particles discovered in the 1950s were produced by the interaction of pions with protons and neutrons in the atmosphere. A group of these—the kaon $k$, lambda $\Lambda$ and sigma $\Sigma$ particles—exhibited unusual properties both as they were created and as they decayed; hence, they were called strange particles.

• One unusual property of strange particles is that they are always produced in pairs.

For example, when a pion collides with a proton, a highly probable result is the production of two neutral strange particles:

\[\pi^- + p \rightarrow \Lambda^0 + K^0\]

• The second peculiar feature of strange particles is that although they are produced in reactions involving the strong interaction at a high rate, they do not decay into particles that interact via the strong force at a high rate. Instead, they decay very slowly, which is characteristic of the weak interaction. Their half-lives are in the range $10^{-10}$ s to $10^{-8}$ s, whereas most other particles that interact via the strong force have much shorter lifetimes on the order of $10^{-23}$ s.

To explain these unusual properties of strange particles, a new quantum number S, called strangeness, was introduced, together with a conservation law. The production of strange particles in pairs is handled mathematically by assigning $S = +1$ to one of the particles, $S = -1$ to the other, and $S = 0$ to all non-strange particles. The law of conservation of strangeness states that

In a nuclear reaction or decay that occurs via the strong force, strangeness is conserved; that is, the sum of the strangeness numbers before the process must equal the sum of the strangeness numbers after the process. In processes that occur via the weak interaction, strangeness may not be conserved.

Assignments

1. Whether this reaction can occur: $\pi^- + p \rightarrow p + K^0$

2. Use the law of strangeness conservation to determine whether the reaction $\pi^0+n\rightarrow K^++\Sigma^+$ occurs.

3. Show that the reaction $\pi^-+p\rightarrow \pi^-+\Sigma^+$ does not conserve strangeness.