Particle physics - exercises

Uncle_John

1. The problem statement, all variables and given/known data
Calculate the ratio of scattering cross sections for hadron and muon production
[itex] \sigma(e^{+} e^{-} \rightarrow hadrons) / \sigma(e^{+} e^{-} \rightarrow \mu^{+}\mu{-})[/itex],
just underneath and just a bit above the treshold for quark production [itex]t \bar{t}[/itex]
(Note only the exchange of the photons)

2. Relevant equations

Equation for cross section(i think):

[itex]\sigma = \frac{K}{(M_{invariant} - M)^2 c^4 + (\frac{\Gamma}{2})^2} [/itex]
What represents the [itex]\Gamma[/itex] in this equation?
How do i calculate the treshold for above productions

Any help appreciated

Uncle_John

Looking at the [itex]t\overline{t}[/itex] production from:
[itex]\gamma \rightarrow t\overline{t}[/itex]

so minimum [itex]E_{\gamma} = 2m_{t}c^2[/itex]

But still I don't see how can i get data to calculate [itex]\Gamma [/itex] and [itex]M[/itex] in formula for cross section

paco_uk

The equation you have given for the cross section is the Breit-Wigner formula which applies in the region of a resonance (e.g. when the centre of mass energy is just enough to create a charmonium state such as the J/Psi).

I think the ratio you are being asked for is for production away from resonances. In this case the cross section for the photon diagram is:

[tex]
\sigma \sim \frac{4 \pi}{3} (\hbar c^2)^2 C \frac{Q_{f} \alpha^2}{E^2}
[/tex]

Where C is the colour factor and Q_f is the charge of the fermion involved. For hadron production you need to some over the the charges of all the quarks which can be produced at the energy you are considering (hence the difference in cross section above and below the threshold for t).