Relating partial width to helicity

1. May 2, 2015

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.

2. May 2, 2015

Staff: Mentor

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

3. May 3, 2015

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?

4. May 3, 2015

Staff: Mentor

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.

5. May 3, 2015

sk1105

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

6. May 3, 2015

Envelope

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 $\sim5.6$.)