Weak interaction - half-lifetime of a decay

[Soff]

Hello!
I need help with an exercise about the weak interaction:

How do the mass, the q-value and the vertices depend on the half-lifetime of a decay?

The myon decay has a half-lifetime of 0.0000015 seconds, the neutron decay has a half-lifetime of 607 seconds.

Why is there such a big difference between this two times? Is this because the q-value of a myon is 105MeV while the q-value of a neutron is 0.782MeV? Or has it maybe to do with the involved mas?

Please help me!

 

[arivero]

Hello!
I need help with an exercise about the weak interaction:

How do the mass, the q-value and the vertices depend on the half-lifetime of a decay?

The myon decay has a half-lifetime of 0.0000015 seconds, the neutron decay has a half-lifetime of 607 seconds.

Why is there such a big difference between this two times? Is this because the q-value of a myon is 105MeV while the q-value of a neutron is 0.782MeV? Or has it maybe to do with the involved mas?

Please help me!

Really good exersice. Muon decay is covered in most textbooks, and all the other weak decays except the neutron scale with the quintic power of the mass, in a first approximation. For the neutron, which has a rare decay between two quarks of similar masses in the presence of another one of the same order, the calculation is troublesome and I only know of a textbook covering it, probably Griffith's.

You can argue that the q-value is not the whole history if you consider the decay of the charged pion to muon.

 

[sir-pinski]

The half-lifetime of a decay is a measure of how long it takes for half of the original particles to decay into other particles. In the case of weak interaction, this is dependent on a few factors such as the mass of the particles involved, the q-value (energy released during the decay), and the vertices (points where the decay occurs).

In general, the half-lifetime of a decay is inversely proportional to the q-value. This means that the higher the q-value, the shorter the half-lifetime and vice versa. This can be observed in your example, where the myon decay with a q-value of 105MeV has a much shorter half-lifetime compared to the neutron decay with a q-value of only 0.782MeV.

The mass of the particles involved also plays a role in determining the half-lifetime. Generally, the heavier the particles, the longer the half-lifetime. This is because heavier particles have a stronger binding energy, making it more difficult for them to decay into other particles.

The vertices also have an impact on the half-lifetime. If there are more vertices involved in the decay process, it can lead to a shorter half-lifetime. This is because more vertices mean more opportunities for the decay to occur.

In summary, the half-lifetime of a decay is influenced by the q-value, mass of the particles, and the number of vertices involved. The large difference in half-lifetime between the myon and neutron decay is mainly due to the q-value, but the mass and vertices also play a role. I hope this helps to answer your question.