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!)

 

[sir-pinski]

Hello Phileas,

The angular distributions of Bhabha Scattering and Muon Pair Production differ primarily due to the difference in mass between the particles involved. Bhabha Scattering involves the scattering of two electrons, which have a much smaller mass compared to muons. This results in a more isotropic angular distribution, with the scattered electrons being more evenly distributed around the original direction of the particles.

On the other hand, Muon Pair Production involves the production of two muons, which have a much larger mass compared to electrons. This results in a more forward-peaked angular distribution, with the majority of the produced muons being emitted in the forward direction. This is due to the fact that the higher mass of the muons makes them less likely to scatter at large angles.

Additionally, the energy of the incoming particles also affects the angular distribution. In Bhabha Scattering, the electrons have a lower energy compared to the muons in Muon Pair Production. This leads to a wider angular distribution in Bhabha Scattering, as the lower energy particles are more likely to scatter at larger angles.

In summary, the main differences in the angular distributions of Bhabha Scattering and Muon Pair Production are due to the difference in mass and energy of the particles involved. Both processes, however, can provide valuable insights into the fundamental interactions of particles and play a crucial role in understanding the subatomic world.