Relating partial width to helicity

[sk1105]

My lecture notes give an example of two decay modes of $K^+$, namely $K^+\rightarrow \mu^+ \nu_\mu$ and $K^+\rightarrow e^+ \nu_e$. Both of these decays are suppressed due to helicity considerations which I understand, and the suppression factors are $\frac{m_\mu c^2}{E_\mu}$ and $\frac{m_ec^2}{E_e}$ respectively.

My notes then say that the ratio of partial widths of these decays is given by $\frac{\Gamma(K^+\rightarrow \mu^+ \nu_\mu)}{\Gamma(K^+\rightarrow e^+ \nu_e)} = \frac{m_\mu^2}{m_e^2}$.

This immediately follows on from the previous discussion, suggesting that there is some link or equivalence between decay amplitude suppression and partial widths, but I can't quite get my head round it. Thank you for your help.

 

[mfb]

$E_\mu \approx E_e$, and probabilities (and branching fractions and partial widths, they are all proportional to each other) are proportional to the amplitude squared.

 

[sk1105]

Ah ok the proportionality makes sense. My notes don't mention that we approximate the energies to be equal. In a high-energy collision it is clear that the difference in rest energy between the electron and the muon is negligible compared to $\sqrt{s}$, but in this case are we saying it is negligible compared to the kaon rest energy?

 

[mfb]

Not negligible if you are interested in precision predictions, but it is a small effect. The 500 MeV from the kaon lead to roughly 250 MeV for the muon (gamma=2.3) and 250 MeV for the neutrino (gamma=very large). To conserve momentum, the muon gets a bit less energy and the neutrino gets a bit more - you can calculate the difference, it is not large. Electron and neutrino get 250 MeV each to a very good approximation.

The ratio is completely dominated by the squared electron to muon mass ratio.

 

[sk1105]

Ah I think that has cleared it up for me; thanks for your help.

 

[Envelope]

but in this case are we saying it is negligible compared to the kaon rest energy?

That's right. This approximation is good to about 10% in the ratio of partial widths, as the ratio of phase space factors is $\frac{(m_K^2-m_e^2)^2}{(m_K^2-m_\mu^2)^2}=1.1$. This is a much worse approximation to make for charge pion decay (ratio of phase space factors $\sim5.6$.)