Let's break down the problem step by step. --- ### **a) Smallest Value of x** **Given:** - Muon energy \( E = 600 \) GeV (incident) - Proton at rest - \( Q^2 = 4 \) GeV\(^2\)/c\(^2\) - Minimum energy of scattered muon \( E' = 0 \) **The Bjorken variable \( x \) is given by:** \[ x = \frac{Q^2}{2 m_p \nu} \] where: - \( Q^2 \) is the four-momentum transfer squared - \( m_p \) is the proton mass (\(\approx 0.938\) GeV/c\(^2\)) - \( \nu = E - E' \) is the energy transfer to the proton (in lab frame) **For smallest \( x \), maximize \( \nu \):** - Maximum \( \nu \) occurs when \( E' = 0 \), so \( \nu_{\text{max}} = E = 600 \) GeV \[ x_{\text{min}} = \frac{Q^2}{2 m_p E} \] **Plug in the values:** \[ x_{\text{min}} = \frac{4}{2 \times 0.938 \times 600} = \frac{4}{1125.6} \approx 0.0036 \] --- ### **b) Number of Partons Resolved** Assuming the distribution is given (not shown in the image), let’s denote the parton distribution as \( f(x) \). If you need to **count the number of partons** with \( x > 0.3 \), \( x > 0.03 \), and all \( x \), you would integrate \( f(x) \) over those ranges: - For \( x > 0.3 \): \( N_{x>0.3} = \int_{0.3}^1 f(x) dx \) - For \( x > 0.03 \): \( N_{x>0.03} = \int_{0.03}^1 f(x) dx \) - For full range: \( N_{\text{full}} = \int_{x_{\min}}^1 f(x) dx \) You'd need the explicit form of \( f(x) \) to compute the actual numbers. --- #### **Summary of Answers:** **a)** \[ x_{\text{min}} \approx 0.0036 \] **b)** - Integrate the given parton distribution over the specified \( x \) ranges to find the number of resolved partons. If you provide the explicit form of the parton distribution, I can help with the actual integration!