# Cross Section Difference Bhabha Scattering and Muon Pair Production

• Phileas.Fogg
In summary, the angular distributions of Bhabha scattering and muon pair production differ due to an additional Feynman diagram in Bhabha scattering, resulting in different scattering amplitudes and cross-sections. In the center-of-mass frame, the angular dependence is proportional to the squares of the Mandelstam variables t and u, which are related to the scattering angle theta. This is a high-energy approximation that assumes negligible particle masses compared to the beam energy.
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

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!)

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.

## 1. What is the difference between Bhabha scattering and muon pair production in cross section?

Bhabha scattering is a process in which an electron and a positron interact and exchange energy through the emission of a photon. Muon pair production, on the other hand, involves the creation of a muon and an antimuon through the interaction of a high-energy photon with a nucleus. The cross section, or the probability of these processes occurring, is different due to the different energy scales and involved particles.

## 2. How is the cross section for Bhabha scattering and muon pair production calculated?

The cross section for these processes is calculated using quantum field theory and perturbation theory. The calculations involve the interactions of the particles and their probabilities, as well as the energy and momentum of the particles involved. These calculations can be complex and require advanced mathematical techniques.

## 3. What factors affect the cross section for Bhabha scattering and muon pair production?

The cross section for these processes is affected by several factors, including the energy of the incoming particles, the type and properties of the particles involved, the strength of their interactions, and the presence of any external fields. Additionally, the cross section can also be influenced by the theoretical models and assumptions used in the calculations.

## 4. Why is the cross section for muon pair production higher than Bhabha scattering?

The cross section for muon pair production is higher than Bhabha scattering due to the difference in energy and mass scales. Muon pair production involves much higher energies, which results in a higher probability for the interaction to occur. Additionally, the particles involved in muon pair production are heavier than those in Bhabha scattering, which also contributes to a higher cross section.

## 5. How do experiments measure the cross section for Bhabha scattering and muon pair production?

Experiments measure the cross section for these processes by counting the number of events that occur within a given time and with a specific energy range. These events are then compared to the expected number of events based on theoretical predictions, and the ratio between the measured and expected numbers gives the cross section. This process requires precise and sensitive detectors as well as careful data analysis.

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