Longitudinal distribution for a neutrino beam

[lLehner95]

Homework Statement

Hello, i'm trying to solve this exercise: to produce neutrino beams, $\pi$ and $K$ meson decays are used. Mesons are produced with a ratio $n_{K}/n_{\pi}=1/10$. Mesons momentum is $p_{0}=100 GeV/c$. Calculate the longitudinal neutrino momentum distribution.

Relevant Equations

In the lab frame of reference:
$p_{T}=p^{*}sin\theta^{*}$
$p_{L}=\gamma p^{*}cos\theta^{*}+\beta\gamma E^{*}=\frac{E}{M}p^{*}cos\theta^{*}+\frac{p}{M}E^{*}$

In the center of mass frame of reference i found that $p^{*}=\frac{[(M^{2}-m_{\nu}^{2}-m_{K}^{2})^{2}-4m_{\nu}^{2}m_{K}^{2})]^{1/2}}{2M}$.
I don't know how to find the momentum distribution $p_{L}(\theta)$ considering that i have 2 different mesons with a specific number ratio $n_{K}/n_{\pi}=1/10$. How can i combine them?

 

[mfb]

Just calculate the distributions separately and then add them. The kaon distribution gets a prefactor of 0.1. Normalize the sum if you like.