Cross Section Difference Bhabha Scattering and Muon Pair Production

[Phileas.Fogg]

Hello,
how do the angular distributions of Bhabha Scattering

$$e^+ e^- \Rightarrow e^+ e^-$$

and Muon pair production

e⁺ e^- $$\Rightarrow \mu^+ \mu^-$$

differ?

Regards,
Phileas Fogg

 

[Sideways]

The difference is caused by the fact that there is an additional Feynman diagram in Bhabha scattering. (The electron process can occur by scattering OR pair annihilation/production, whereas the muon process can only occur by the latter.)

This gives a scattering amplitude (in terms of the Mandelstam variables s,t,u) of

M_mu = (u^2 + t^2)/s^2

and

M_el = (s^2 + u^2)/t^2 + 2u^2/(ts) + M_mu

The cross-section is proportional to the squares of these quantities. In the center-of-mass frame (same as the lab frame for a two-beam collider), s has no angular dependence, and

t ~ 1-cos theta

and

u ~ 1+cos theta

So the relative angular dependence isn't hard to work out (though I don't know it offhand).
(These values are all from Quarks and Leptons by Halzen & Martin)

The rest is algebra. Note that this is a high-energy approximation, which assumes that the particle masses are negligible compared to the beam energy.

(Hope this is what you were looking for!)