Tau lepton decay problem - weird answer?

[jeebs]

Hi,
I've got a tauon weak-decaying into a charged pion and a neutrino. The tauon has total energy [tex]\ E_{\tau} = 2.5GeV[/tex], rest mass [tex]\ m_{\tau} = 1.777GeV/c^2[/tex], and the pion has rest mass [tex]\ m_{\pi} = 0.1396GeV/c^2[/tex].
This is everything I am told.
I have to find the maximum energy the pion can have after the decay.

So, what I said was that the pion will get the most energy by retaining all of the tauon's momentum, ie.

[tex]\ p_{\tau} = \sqrt{E_{\tau}^2 - m_{\tau}^2} = p_{\pi} = 1.758GeV/c[/tex]

and so [tex]\ E_{\pi} = \sqrt{p_{\pi}^2 + m_{\pi}^2} [/tex]

This gives me the result that [tex]\ E_{\pi} = 1.764GeV [/tex].
However, the neutrino that is also produced has a tiny mass and I have assumed it to have zero momentum, so where has most of the 2.5GeV tauon energy gone?

I had 2.5GeV before the decay, but now I have not much more than 1.764GeV after the decay, so where has the rest of this energy magically disappeared to?

Thanks.

 

[NateD]

The rest of the energy is carried away by the neutrino. Since the neutrino has such a small mass, almost all of its energy is in its kinetic energy. The neutrino's momentum is equal to the difference between the initial tauon's momentum and the final pion's momentum, so it can take away quite a bit of energy.