What Is the Ratio of Rates for e+ e- -> mu+ mu- at sqrt(s)=5 GeV?

[Ai52487963]

Homework Statement

For $$e^+ e^-$$ collisions at $$\sqrt{s}=5$$ GeV, estimate the ratio of the rates at which interactions produce hadrons and $$\mu^+ \mu^-$$

Homework Equations

$$\sqrt{s} = 2E = E_{cm}$$

$$\Gamma = \frac{S |p|}{8 \pi \hbar m_1^2 c} |M|^2$$ where M is the matrix element

$$\frac{d \sigma}{d \Omega} = \left(\frac{h c}{8 \pi} \right)^2 \frac{S |M|^2}{(E_1 + E_2)^2} \frac{|p_i|}{|p_f|}$$

The Attempt at a Solution

So I know that at $$\sqrt{s}=5$$ GeV, the propogator has to be a $$\gamma$$ and the ratio of rates should favor the muon production as opposed to the hadrons, but I don't know how to calculate the rates. Likewise, I also know the propogator for $$e^+ e^- \rightarrow q \bar{q}$$ has to be a $$Z^0$$.Is the $$e^+ e^- \rightarrow \mu^+ \mu^-$$ considered a two-body scattering or what?

Basically, I'm torn as to calculating the rate. Do I need to explicitly find the matrix element for each case, or does that divide out?

 

[fzero]

The electromagnetic couplings at the vertices are basically the same and the photon propagator in the middle has the same form, so the matrix elements are very similar. ($$e^+ e^- \rightarrow q \bar{q}$$ is dominated by intermediate photons, especially at c.o.m. energies far below the Z mass.) When you square the matrix element, you have to do sums over the incoming and outgoing spin states, but even this part is similar for the two cases. So the rates have the same form when expressed in terms of momenta and masses for muons or hadrons in the final state. You should check to make sure. The difference in rates is related to the large difference in masses between the muon and available hadronic states.