Calculating E+e- -> μ+ μ- Cross Section

In summary: , so in summary, summing over just spin up and spin down and dividing by 2 is equivalent to considering all possible combinations.
  • #1
Silviu
624
11
Hello! When calculating the matrix element for (let's say) ##e^+e^- \to \mu^+ \mu^-##we have to average over initial spins and sum over final spins. I understand the motivation of this, but when the calculation is done, the sum is done for 2 cases: spin up and spin down, so you have to add 8 terms (2 for each particle) and divide by 4 (the 2 initial incoming particles). Why is this summation enough? Shouldn't one integrate over all possible values of spin? It is not like the particle will come with either spin up or spin down on a give axis, they can be a linear combination of these.
 
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  • #2
Ehm, the fact that you have a linear combination of both spin up and down is that you have to sum over them... If the particle was in a known spin state, you wouldn't need to make any summation.
 
  • #3
ChrisVer said:
Ehm, the fact that you have a linear combination of both spin up and down is that you have to sum over them... If the particle was in a known spin state, you wouldn't need to make any summation.
I understand this. My question is why you sum over just the spin up and spin down and divide by 2, and not integrate over all the possible linear combinations? You don't know the spin so it can be anything, not just up or down along the z axis.
 
  • #4
Silviu said:
My question is why you sum over just the spin up and spin down and divide by 2, and not integrate over all the possible linear combinations?
What you are doing is equivalent to considering all possible combinations. You have to consider is the ensemble of the in-state, which corresponds to a density matrix that is proportional to unity.
 
  • #5
Silviu said:
I understand this. My question is why you sum over just the spin up and spin down and divide by 2, and not integrate over all the possible linear combinations? You don't know the spin so it can be anything, not just up or down along the z axis.
is there any other possible state for spin 1/2 ? as far as I know the dim is 2s+1=2.
 
  • #6
ChrisVer said:
is there any other possible state for spin 1/2 ? as far as I know the dim is 2s+1=2.
My point was that the spin doesn't have to be along the z axis, it can be along any other axis. But Orodruin made it clear to me.
 
  • #7
in any axis it may be though, when written in the basis of z you are still having 2. ok
 

1. What is the purpose of calculating the E+e- -> μ+ μ- cross section?

The cross section is a measure of the likelihood of a specific particle interaction occurring. By calculating the cross section for E+e- -> μ+ μ-, we can better understand and predict the behavior of particles in this type of collision.

2. How is the E+e- -> μ+ μ- cross section calculated?

The E+e- -> μ+ μ- cross section is calculated using theoretical models and experimental data. Theoretical models use mathematical equations to predict the likelihood of the interaction, while experimental data is collected from particle accelerators and used to validate the theoretical calculations.

3. What factors affect the E+e- -> μ+ μ- cross section?

The E+e- -> μ+ μ- cross section is affected by the energy of the colliding particles, the angle at which they collide, and the properties of the particles themselves. The cross section may also be influenced by external factors such as the presence of other particles in the collision.

4. How is the E+e- -> μ+ μ- cross section measured in experiments?

In experiments, the E+e- -> μ+ μ- cross section is measured by counting the number of interactions that occur and comparing it to the total number of collisions. This ratio is then used to calculate the cross section. To improve accuracy, multiple experiments are typically conducted and the results are averaged.

5. What are the units of the E+e- -> μ+ μ- cross section?

The E+e- -> μ+ μ- cross section is typically measured in units of picobarns (pb), which is equal to 10^-36 square meters. This unit is commonly used in particle physics to describe the tiny cross sections of particle interactions.

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