Problem 28 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Which of the following reactions and decays are possible? For those forbidden, explain what laws are violated. (a) \(\pi^{-}+\mathrm{p} \rightarrow \mathrm{n}+\eta^{0}\) (b) \(\pi^{+}+\mathrm{p} \rightarrow \mathrm{n}+\pi^{0}\) (c) \(\pi^{+}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{e}^{+}\) (d) \(\mathrm{p} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{c}}\) \((e) \mu^{+} \rightarrow \mathrm{e}^{+}+\overline{v}_{\mu}\) (f) \(\mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{+}+\nu_{\mathrm{e}}\)

Step-by-Step Solution

Answer

Possible: (a); Forbidden: (b), (c), (d), (e), (f).

Step 1: Checking Charge Conservation

For a reaction or decay to be possible, charge must be conserved. Let's verify each given reaction:(a) \( \pi^{-} + \mathrm{p} \rightarrow \mathrm{n} + \eta^{0} \): On the left, charge is \(-1 + 1 = 0\); on the right, charge is \(0 + 0 = 0\). Charge is conserved.(b) \( \pi^{+} + \mathrm{p} \rightarrow \mathrm{n} + \pi^{0} \): On the left, charge is \(+1 + 1 = +2\); on the right, charge is \(0 + 0 = 0\). Charge is not conserved — forbidden.(c) \( \pi^{+} + \mathrm{p} \rightarrow \mathrm{p} + \mathrm{e}^{+} \): On the left, charge is \(+1 + 1 = +2\); on the right, charge is \(+1 + 1 = +2\). Charge is conserved.(d) \( \mathrm{p} \rightarrow \mathrm{e}^{+} + u_{c} \): On the left, charge is \(+1\); on the right, charge is \(+1 + 0 = +1\). Charge is conserved, but neutrino type leaves doubt.(e) \( \mu^{+} \rightarrow \mathrm{e}^{+} + \overline{v}_{\mu} \): On the left, charge is \(+1\); on the right, charge is \(+1 + 0 = +1\). Charge is conserved.(f) \( \mathrm{p} \rightarrow \mathrm{n} + \mathrm{e}^{+} + u_{e} \): On the left, charge is \(+1\); on the right, charge is \(0 + 1 + 0 = +1\). Charge is conserved.

Step 2: Checking Baryon Number Conservation

The baryon number (B) must be conserved in all reactions/decays.(a) \( \pi^{-} + \mathrm{p} \rightarrow \mathrm{n} + \eta^{0} \): Left: \(B = 0 + 1 = 1\); Right: \(1 + 0 = 1\). Baryon number is conserved.(b) \( \pi^{+} + \mathrm{p} \rightarrow \mathrm{n} + \pi^{0} \): Not applicable as it's forbidden by charge.(c) \( \pi^{+} + \mathrm{p} \rightarrow \mathrm{p} + \mathrm{e}^{+} \): Left: \(0 + 1 = 1\); Right: \(1 + 0 = 1\). Baryon number is conserved.(d) \( \mathrm{p} \rightarrow \mathrm{e}^{+} + u_{c} \): Left: \(1\); Right: \(0 + 0 = 0\). Baryon number is not conserved — forbidden.(e) \( \mu^{+} \rightarrow \mathrm{e}^{+} + \overline{v}_{\mu} \): No baryons involved.(f) \( \mathrm{p} \rightarrow \mathrm{n} + \mathrm{e}^{+} + u_{e} \): Left: \(1\); Right: \(1 + 0 + 0 = 1\). Baryon number is conserved.

Step 3: Checking Lepton Number Conservation

Lepton number conservation is important for these reactions/decays.(a) \( \pi^{-} + \mathrm{p} \rightarrow \mathrm{n} + \eta^{0} \): No leptons involved so lepton number is balanced.(c) \( \pi^{+} + \mathrm{p} \rightarrow \mathrm{p} + \mathrm{e}^{+} \): Electrons appear, lepton numbers (+1) differ without counterparts. Not conserved — forbidden.(d) \( \mathrm{p} \rightarrow \mathrm{e}^{+} + u_{c} \): Lepton numbers: Left: 0; Right: \(0 (for u_c) + 1 (for \mathrm{e}^{+})\). Not conserved — forbidden.(e) \( \mu^{+} \rightarrow \mathrm{e}^{+} + \overline{v}_{\mu} \): Left: Muon lepton number \((-1)\). Right: Electron lepton \((+1)\); Antimuon neutrino \(0\). Not conserved — forbidden.(f) \( \mathrm{p} \rightarrow \mathrm{n} + \mathrm{e}^{+} + u_{e} \): Left: 0; Right: \(1 (\mathrm{e}^{+}) + 1 (u_{e})\). Electron lepton not opposed; thus, not conserved — forbidden.

Key Concepts

Charge Conservation

Charge conservation is a fundamental principle in physics that dictates that the total electric charge in an isolated system remains constant over time. This means that in any reaction or decay, the sum of charges before the event must equal the sum of charges after. For instance, consider reaction

• (a) \( \pi^{-} + \mathrm{p} \rightarrow \mathrm{n} + \eta^{0} \): Initially, the charge is \(-1 + 1 = 0\), and it remains \(0 + 0 = 0\) after. Charge is conserved.
• (b) \( \pi^{+} + \mathrm{p} \rightarrow \mathrm{n} + \pi^{0} \): Here, the initial charge is \(+1 + 1 = +2\) which does not match the final charge of \(0 + 0 = 0\), resulting in a forbidden reaction.

This principle ensures that particles do not appear or disappear spontaneously without observing charge balance.

Baryon Number Conservation

Baryon number conservation is another key rule applied in particle physics. Each nucleon, such as protons and neutrons, has a baryon number of +1. Particles like anti-protons have a baryon number of -1. In a valid reaction or decay, the total baryon number before and after the process must be equal.

• (a) \( \pi^{-} + \mathrm{p} \rightarrow \mathrm{n} + \eta^{0} \): Initially, the baryon number is \(0 + 1 = 1\) and remains \(1 + 0 = 1\) afterward, showing conservation of baryon number.
• (d) \( \mathrm{p} \rightarrow \mathrm{e}^{+} + u_{c} \): This transition would result in a change from \(1\) initial baryon number to \(0 + 0 = 0\) afterward, violating baryon number conservation, thus making the reaction forbidden.

Understanding baryon number conservation helps physicists confirm whether reactions respect the symmetry and laws observed in strong and electromagnetic interactions.

Lepton Number Conservation

Lepton number conservation is a principle that asserts that the total number of leptons minus the number of anti-leptons remains constant. Each lepton is assigned a lepton number of +1, while its associated antilepton is assigned -1.

• (e) \( \mu^{+} \rightarrow \mathrm{e}^{+} + \overline{v}_{\mu} \): Initially, the muon lepton number is \(-1\). The products have an electron lepton number of \(+1\) and antimuon neutrino adds zero to it, disrupting lepton number consistency and rendering the process forbidden.
• (f) \( \mathrm{p} \rightarrow \mathrm{n} + \mathrm{e}^{+} + u_{\mathrm{e}} \): Starts with total lepton number \(0\) and ends with lepton numbers of \(+1\) for \(\mathrm{e}^{+}\) and \(1\) for \(u_{\mathrm{e}}\), causing inequality and therefore not conserving lepton number.

This principle helps ensure that interactions under the umbrella of weak force, like beta decay, respect an unchanging lepton count.