Production of Z boson - Cross Section

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Homework Statement

Calculate the ratio $R = \frac{\sigma_{had}}{\sigma_{\mu+\mu-}}$ for energy around $10~GeV$.
At sufficiently high energies, the $e^+e^- \rightarrow \mu^+ \mu^-$ reaction can proceed via the $Z^0$ boson. Assuming vertex factors for EM and weak interaction are the same, at what beam energy (below $M_Z$) would the contribution to cross section from $\gamma$ be the same as the contribution from the $Z^0$ exchange?

Homework Equations

The Attempt at a Solution

Since at around $10~GeV$, production of u,d,s,c quarks are possible, $R = 3 \times \left[ (\frac{1}{3})^2 + (\frac{2}{3})^2 + (\frac{1}{3})^2 + (\frac{2}{3})^2 + (\frac{1}{3})^2 \right] = \frac{11}{3}$.

Since we can assume the vertex factor for EM and weak interaction to be the same, thus the cross-section for ($e+e \rightarrow \mu^+\mu^-$) is the same for weak and EM. We have just calculated $R = \frac{11}{3}$.

Since cross-section grows as $\sigma \sim E_{cm}^2$, total contribution from EM processes is $\left(1 + \frac{11}{3}\right) E^2$. Contribution from $Z^0$ process is simply $M_w^2$.

Does this mean that $E = \sqrt{\frac{3}{14}}m_w = 0.46m_W$?

 

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