How to Determine Neutrino Energy in Pion Decay?

[XCBRA]

Homework Statement

Pions can decay via the reaction π+ → μ+ νμ. Show that the energy of the neu- trino in the rest frame of the pion is given by

$$E_v = \frac{m^2_∏-m^2_μ}{2m_∏}$$



Pions with energy Eπ in the laboratory frame (Eπ >> mπc2) decay via the above reaction. Show that the maximum energy of the neutrinos in the laboratory frame is given by


$$E^{max}_v = E_∏\frac{m^2_∏-m^2_μ}{m^2_∏} c^2$$

Homework Equations

Lorentz Transform

$$\left(\begin{array}{cc}E'\\p'c\end{array}\right) = \left(\begin{array}{cc}\gamma&\beta\gamma\\-\beta\gamma&\gamma\end{array}\right) * \left(\begin{array}{cc}E\\pc\end{array}\right)$$

The Attempt at a Solution

I am able to get the first result, the problem I am having is changing frames from the first result to the second result. I take the first result and use the Lorentz transform ont he energy, where prime denotes the lab frame and unprimed is the rest frame of the pion.
this give me:


$$E_v' = \frac{m^2_∏-m^2_μ}{2m_∏} \gamma(1-\beta) c^2$$

i then use the fact that E'_∏ = gamma mc^2 to find gamma, giving me:


$$E_v' = \frac{m^2_∏-m^2_μ}{2m^2_∏} E_∏(1-\beta) c^2$$

However I have done something wrong as this means I need beta =-1 to give me the right result. Any help would be greatly appreciated.

 

[Muphrid]

You've done something wrong with your LT matrix. Both $\beta \gamma$ terms should have the same sign.

 

[XCBRA]

Ahh yes, however that was only a mistake I made when typing the problem up to pf, my calculations had both with the -ve sign to get the result below it.