Let's answer each question step by step, with explanations: --- ### **Question 1** #### π⁻ → μ⁻ + ν̅_μ: Dependence of decay rate on Cabibbo angle - **Step 1:** The π⁻ (pion) is made of a \( \bar{u} \) and \( d \) quark. The decay process is: \( d \rightarrow u + W^- \rightarrow u + \mu^- + \bar{\nu}_\mu \) - **Step 2:** The transition \( d \rightarrow u \) is a weak interaction, and its strength is proportional to \( \cos \theta_c \) (Cabibbo angle). - **Step 3:** The decay rate depends on the square of the amplitude, so it's proportional to \( \cos^2 \theta_c \). **Final Answer:** **c. cos² θ_c** *The decay rate is proportional to \( \cos^2 \theta_c \) for this process.* --- ### **Question 2** #### Comparing decay probabilities: τ⁻ → ν_τ + e⁻ + ν̅_e τ⁻ → ν_τ + μ⁻ + ν̅_μ - **Step 1:** Both decays are leptonic, and the weak interaction does not distinguish between electrons and muons (ignoring small phase space differences due to mass). - **Step 2:** The only difference is the mass of the muon vs. electron, but the phase space suppression for the muon channel is small for the τ lepton. **Final Answer:** **The two decays have approximately equal probabilities** *Both processes are mediated by the same weak interaction and are nearly equally probable.* --- ### **Question 3** #### Which quark is most likely to appear in s quark decays? - **Step 1:** The s quark (strange) decays via the weak interaction, predominantly to a u quark (since s → u is CKM suppressed, but s → d is dominant). - **Step 2:** In reality, the dominant decay is \( s \rightarrow u + W^- \), but since the answer choices are about which appears most likely, we look at which quark is produced directly via weak decay. - **Step 3:** The s quark most commonly decays into a u quark (Cabibbo-favored). **Final Answer:** **u** *The u quark is most likely to appear in s quark decays.* --- ### **Question 4** #### Which quark is most likely to be produced in c quark decays? - **Step 1:** The c (charm) quark decays via the weak interaction, predominantly to a s quark. - **Step 2:** This is due to the CKM matrix elements: \( c \rightarrow s \) is the most favored. **Final Answer:** **s** *The s quark is most likely to be produced in c quark decays.* --- ### **Question 5** #### μ⁺ → e⁺ + ν̅_μ + ν_e: Dependence of decay rate on Cabibbo angle - **Step 1:** This decay is purely leptonic, involving only leptons (no quarks). - **Step 2:** The Cabibbo angle only affects weak decays involving quark flavor transitions, not purely leptonic decays. **Final Answer:** **No dependence** *The decay rate does not depend on the Cabibbo angle for purely leptonic decays.* --- ### **Question 6** #### Lorentz structure of the weak charged interaction current - **Step 1:** The weak charged current is described by the V–A (Vector minus Axial Vector) structure in the Standard Model. - **Step 2:** This means the current is a combination: \( \text{Vector} - \text{Axial Vector} \). **Final Answer:** **Vector minus axial vector** *The Lorentz structure is V–A (Vector minus Axial Vector).* --- ## **Summary of Answers** 1. **c. cos² θ_c** 2. **The two decays have approximately equal probabilities** 3. **u** 4. **s** 5. **No dependence** 6. **Vector minus axial vector**