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Chapter 43: Problem 50

Which of the following reactions are possible, and by what interaction could they occur? For those forbidden, explain why. (a) \(\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{0}+\mathrm{p}+\pi^{0}\) (b) \(\mathrm{K}^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\pi^{0}\) (c) \(\mathrm{K}^{+}+\mathrm{n} \rightarrow \Sigma^{+}+\pi^{0}+\gamma\) (d) \(\mathrm{K}^{+} \rightarrow \pi^{0}+\pi^{0}+\pi^{+}\) (e) \(\pi^{+} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{e}}\)

Short Answer

Expert verified
(a) Weak interaction possible; (b) Possible via strong interaction; (c) Weak/electromagnetic interaction possible; (d) Forbidden; (e) Possible via weak interaction.

Step by step solution

01

Analyze Reaction (a)

In this reaction, \[\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{0}+\mathrm{p}+\pi^{0}\]we need to check the conservation of baryon number, charge, and strangeness. The initial and final baryon number is +1, so baryon number conservation is satisfied. The initial charge is 0 (\(\pi^{-}\) = -1, \(\mathrm{p}\) = +1), and the final charge is also 0 (\(\mathrm{K}^{0}\) = 0, \(\mathrm{p}\) = +1, \(\pi^{0}\) = 0). Strangeness changes from 0 to +1, which cannot occur in strong interaction but in weak interaction it's possible. Reaction (a) is forbidden in strong interactions due to strangeness conservation but possible via weak interaction.
02

Analyze Reaction (b)

For the reaction \[\mathrm{K}^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\pi^{0}\]we check the conservation of baryon number, charge, and strangeness. The baryon number changes from +1 to +1, so it's conserved. The charge is 0 initially (\(\mathrm{K}^{-} = -1\), \(\mathrm{p} = +1\)) and 0 finally (\(\Lambda^{0} = 0\), \(\pi^{0} = 0\)). Strangeness changes from -1 to -1, which can happen in the strong interaction. Thus, this reaction is possible via strong interaction.
03

Analyze Reaction (c)

For the reaction \[\mathrm{K}^{+}+\mathrm{n} \rightarrow \Sigma^{+}+\pi^{0}+\gamma\]baryon number is +1 initially and +1 finally, so it's conserved. The charge is also conserved (+1 on both sides). The strangeness changes from 0 (\(\mathrm{K}^{+}\)) to +1 (\(\Sigma^{+}\)), which implies this reaction cannot occur via strong interaction due to strangeness violation. The reaction is possible via electromagnetic or weak interaction but less likely to occur naturally.
04

Analyze Reaction (d)

In this decay reaction, \[\mathrm{K}^{+} \rightarrow \pi^{0}+\pi^{0}+\pi^{+}\]we should check the initial and final state conservation laws. The strangeness changes from +1 to 0, which is not possible in any interaction. Thus, this reaction is forbidden.
05

Analyze Reaction (e)

For the decay reaction \[\pi^{+} \rightarrow \mathrm{e}^{+}+u_{\mathrm{e}}\]we have to check conservation laws. Here, the charge is conserved since +1 goes to +1, and lepton number is conserved with 0 turning into +1 (lepton - u_{\mathrm{e}}). This is possible via weak interaction and is a common weak decay for \(\pi^{+}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Baryon Number Conservation
Baryon number is a quantum number assigned to subatomic particles such as protons, neutrons, and their antiparticles. In particle interactions, the total baryon number remains constant. For instance, in the reaction involving \(\pi^{-} + \text{p} \rightarrow \text{K}^{0} + \text{p} + \pi^{0}\), the initial and final baryon numbers are both +1, satisfying the conservation rule.
  • Baryons and Antibaryons: Baryons like protons and neutrons have a baryon number of +1, whereas antibaryons have a baryon number of -1.
  • Conservation Law: The net baryon number before and after any reaction must remain unchanged. This ensures protons and neutrons are never created or destroyed alone, preserving the stability of atoms.
Understanding this concept aids in recognizing allowable particle interactions.
Charge Conservation
Charge conservation is a fundamental principle in physics indicating that the total electric charge remains constant in an isolated system. In particle reactions, the sum of electrical charges before the interaction must equal the sum after. For example, in \(\mathrm{K}^{-} + \mathrm{p} \rightarrow \Lambda^{0} + \pi^{0}\), initial and final charges are both zero, fulfilling this principle.
  • Significance: It ensures that electrical neutrality or overall charge state of particles remains unchanged.
  • Applications: Charge conservation is critical in analyzing reactions, particularly chemical reactions or when evaluating potential candidates for certain processes in particle physics.
The understanding of charge conservation helps predict which particle interactions are possible.
Strangeness Conservation
Strangeness is a property of particles, related to their constituent quarks moving beyond ordinary up and down flavors to include strange quarks. In some interactions, especially strong interactions, strangeness is conserved. For instance, in the reaction \(\mathrm{K}^{+} + \mathrm{n} \rightarrow \Sigma^{+} + \pi^{0} + \gamma\), the initial net strangeness of 0 changes to +1, which is not allowed in strong interactions.
  • Conserved in Strong Interactions: While strong forces, like those binding the atomic nucleus, preserve strangeness, weak forces do not necessarily do so.
  • Practical Implications: Observing strangeness allows scientists to track particle creation, transformation, and decay along these lines.
Recognizing when strangeness is or isn't conserved is crucial for determining the nature of interactions.
Weak Interaction
The weak interaction is one of the fundamental forces of nature, responsible for processes like beta decay in nuclear physics. It allows particles to change their type or flavor and can involve neutrinos. In scenarios like the decay \(\pi^{+} \rightarrow e^{+} + u_{e}\), it plays a vital role.
  • Flavor Changing: Weak interactions are unique for allowing particles to transform from one type to another, such as a neutron converting into a proton.
  • Involvement of Neutrinos: Often involves particles that are hard to detect, like neutrinos, making these interactions less apparent but highly significant in particle physics processes.
Understanding weak interaction broadens comprehension of particle decay and transformation mechanisms.
Strong Interaction
The strong interaction, also known as the strong nuclear force, is the force responsible for holding the nuclei of atoms together. This force binds quarks to form protons, neutrons, and other particles. It's exemplified in reactions like \(\mathrm{K}^{-} + \mathrm{p} \rightarrow \Lambda^{0} + \pi^{0}\), where conservation laws uphold through strong interactions.
  • Strength and Range: The strongest of the fundamental forces, but operates over extremely small distances, typically less than the diameter of a nucleus.
  • Particle Binding: Ensures quarks remain within nucleons and nucleons within atomic nuclei, effectively providing the universe's stability.
The strong interaction is essential for understanding binding forces in subatomic particles.

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Most popular questions from this chapter

What is the quark combination needed to produce a \(\mathrm{D}^{0}\) meson \((Q=B=S=0, c=+1) ?\)

Which of the following reactions are possible, and by what interaction could they occur? For those forbidden, explain why. (a) \(\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{+}+\Sigma^{-}\) (b) \(\pi^{+}+\mathrm{p} \rightarrow \mathrm{K}^{+}+\Sigma^{+}\) (c) \(\pi^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\mathrm{K}^{0}+\pi^{0}\) (d) \(\pi^{+}+\mathrm{p} \rightarrow \Sigma^{0}+\pi^{0}\) (e) \(\pi^{-}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{e}^{-}+\bar{\nu}_{\mathrm{e}}\)

(1) What is the total energy of a proton whose kinetic energy is 4.65 \(\mathrm{GeV}\) ?

(I) The B \(^{-}\) meson is a b\overline{u} \text { quark combination. } ( a ) \text { Show that } this is consistent for all quantum numbers. (b) What are the quark combinations for \(\mathrm{B}^{+}, \mathrm{B}^{0}, \mathrm{B}^{0} ?\)

One decay mode for a \(\pi^{+}\) is \(\pi^{+} \rightarrow \mu^{+}+\nu_{\mu} .\) What would be the equivalent decay for a \(\pi^{-}\) ? Check conservation laws.
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