Weak decay - cross sections

[zakk87]

Weak decay -- cross sections

I have to estimate the ratio between the cross sections of the following weak processes:

$$\nu_{\mu} + e^{-} \Rightarrow \nu_{e} + \mu^{-}$$
$$\bar{\nu_{e}} + e^{-} \Rightarrow \bar{\nu_{\mu}} + \mu^{-}$$

My teacher says that as the first process is $$LL \Rightarrow LL$$ (i.e. chirality of the initial and final states) and the second process is $$LR \Rightarrow LR$$, when can immediately conclude that, the ratio between the cross sections is 3:1 (provided that the incoming neutrinos energies are the same).

I cannot understand why he jumps to that conclusion.

Thank you in advance for any reply!

 

[AndyW]

Dear fellow scientist,

Thank you for bringing up this interesting topic. it is important for us to understand and question the conclusions made by our colleagues. In this case, your teacher's conclusion may be correct, but it is always good to understand the reasoning behind it.

Firstly, let's define cross section. Cross section is a measure of the probability of a particle interaction happening within a given target. In other words, it is a measure of the target's "size" for a specific interaction.

In weak interactions, the chirality (handedness) of the initial and final states plays a crucial role. This is because the weak interaction is chiral, meaning it only interacts with particles of a specific handedness. In your first process, the initial state is LL (left-handed neutrino and left-handed electron) and the final state is also LL (left-handed neutrino and left-handed muon). Similarly, in the second process, the initial state is LR (left-handed anti-neutrino and right-handed electron) and the final state is also LR (left-handed anti-neutrino and right-handed muon).

Now, let's look at the Feynman diagrams for these processes. In the first process, the interaction involves the exchange of a W^- boson, which only interacts with left-handed particles. This means that the strength of the interaction will depend on the number of left-handed particles involved, which in this case is 2 (one left-handed neutrino and one left-handed electron). On the other hand, in the second process, the interaction involves the exchange of a W^+ boson, which only interacts with right-handed particles. This means that the strength of the interaction will depend on the number of right-handed particles involved, which in this case is 2 (one right-handed anti-neutrino and one right-handed electron).

Therefore, the ratio of the cross sections for these two processes will depend on the ratio of left-handed to right-handed particles involved. In this case, for the first process, we have a ratio of 2:0 (left-handed to right-handed particles), while for the second process, we have a ratio of 0:2 (left-handed to right-handed particles). This gives us a ratio of 2:0 for the first process and 0:2 for the second process, which simplifies to 1:0 and 0:1 respectively. This means that the ratio