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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##?