Solving Muon Decay Calc: Need Help!

In summary, the conversation is about finding a way to calculate the differential of momentum for a specific decay process involving muons. The equations (0), (1), and (2) are discussed, and the person is struggling to continue with the differentiation process. They are considering different methods and asking for suggestions.
  • #1
ChrisVer
Gold Member
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Please, I'd need some help. Although I am not sure if this is again the correct thread, but since it concerns muon decay I bring it here. So...

I am trying to find out why the differential below, in spherical cordinates becomes:
[itex]d^{3}p_{\bar{v_{e}}}=-\frac{E_{\bar{v_{e}}} E_{v_{μ}}}{E_{e}} dE_{\bar{v_{e}}} dE_{v_{μ}} dφ (0)[/itex]

I already have derived the equation:
[itex] E_{v_{μ}}^{2}= E_{\bar{v_{e}}}^{2}+E_{e}^{2}+2E_{\bar{v_{e}}}E_{e}cosθ (1)[/itex]
I also have the conservation of energy due to delta function:
[itex] E_{v_{μ}}= m_{μ}-E_{\bar{v_{e}}}-E_{e} (2)[/itex]

I stop in a very bad position not knowing how to continue:
[itex]d^{3}p_{\bar{v_{e}}}= p_{\bar{v_{e}}}^{2} dp_{\bar{v_{e}}} dcosθ dφ=E_{\bar{v_{e}}}^{2} dE_{\bar{v_{e}}} dcosθ dφ [/itex]
How would you recommend I continue? I would try to differentiate the [itex](1)[/itex] but it has also cosθ and generally a mess is happening. I also could try to differentiate [itex](2)[/itex] but I would get weird results not coinciding with [itex](0)[/itex]
Any suggestion?
(the mass of muon only exists, in the game, so the electron and neutrinos' masses are neglected, and thus their energies are equal to their momentum's magnitudes)
 
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  • #2
Start with
[tex]d^3p_1d^3p_2d^3p_e\delta^4()/E_1E_2E_e[/tex].
[tex]\rightarrow d^3p_1d^3p_2\delta(E_1+E_2+E_e-M)/E_1E_2E_e[/tex],
[tex]\rightarrow 8\pi^2p_1dE_1p_2dE_2d\cos(\theta)\delta(E_1+E_2+E_e-M)/E_e[/tex],
with [itex]E_e=\sqrt{m^2+p^2_1+p^2_2+2p_1p_2\cos(\theta)}[/itex].
The delta function integration over d\theta gives
[tex]8\pi^2dE_1dE_2.[/tex]
 
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1. What is "muon decay" and why is it important in scientific research?

Muon decay is a process in which a muon particle, a type of elementary particle similar to an electron, decays into other particles. This process is important in scientific research because it allows us to study the fundamental properties of particles and gain a better understanding of the laws of physics.

2. What is a "muon decay calculator" and how does it work?

A muon decay calculator is a tool that uses mathematical equations and experimental data to predict the decay rate of muon particles. It takes into account factors such as the mass and energy of the particles, as well as any external forces acting on them, to calculate the probability of decay at a given time.

3. What is the purpose of solving a muon decay calculation?

The purpose of solving a muon decay calculation is to better understand the behavior of muon particles and their interactions with other particles. This can help us make predictions about the behavior of matter and energy on a microscopic level, which has important implications for various fields of science, including particle physics and cosmology.

4. What are some common challenges when solving a muon decay calculation?

Some common challenges when solving a muon decay calculation include accurately measuring the properties of the particles involved, accounting for any uncertainties in the data, and dealing with complex mathematical equations. It may also be challenging to determine which factors are most important to consider in the calculation.

5. Are there any real-world applications of muon decay calculations?

Yes, there are several real-world applications of muon decay calculations. For example, they are used in medical imaging techniques such as positron emission tomography (PET) scans, and in studies of nuclear reactions and radioactive materials. They are also important in the development of new technologies, such as particle accelerators and nuclear power plants.

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