In 2 consecutive decays, determine max and min energies for a particle

[peguerosdc]

Homework Statement

Considering first that Z at rest decays into $\tau^+ \tau^-$ and then $\tau^-$ decays into $\pi^- \nu_{\tau}$. What are the max and min energies of the $\pi$ in the rest frame of Z?

Relevant Equations

Nothing special. Just the definition of the four-momentum.

Hi!
Instead of just describing my procedure and all my derivations, I really just want to ask if my approach makes sense (actually I have 2 options) to calculate the maximum energy. I am considering c=1 and the problem suggests to consider the neutrino massless:

For the first decay, $Z \rightarrow \tau^+ \tau^-$:
As Z is at rest, there is not much to do in here. $\tau^+$ and $\tau^-$ decay in opposite directions with the same momentum. The energy of the $\tau^+$ (which I'll write it just as E with a subscript "+" from now on. The same for $\tau^-$) is:

$$
E_+ = \frac{ m_z^2 + m_+^2 - m_{-}^2 }{2m_z}
$$

The energy of $\tau^-$ is the same just exchanging + with -.

For the second decay, $\tau^- \rightarrow \pi^- \nu_{\tau}$:
I checked a similar example in Griffiths' problem 3.22 where a particle A at rest decays into particles B,C,D,... and his approach to calculate the minimum energy of a particle (let's say, B) is to suppose that it is at rest and the momentum of A distributes along all the other particles (C,D,...).
Even if it sounds really unlikely, I think this applies to my case as well as it satisfies conservation of energy and momentum and there is nothing preventing it from happening, so in my case it looks reasonable to say that:

$$
E_{min} = m_{\pi}
$$

For the maximum energy is where I run into trouble.

My first approach is to just do the opposite: now the neutrino is at rest and all the momentum of the $\tau^-$ goes to the $\pi^{-}$ such that $\vec{p_{-}} = \vec{p_{\pi}}$. The momentum of the $\tau^-$ is easy calculate (see Griffiths's problem 3.19) and, from squaring the four-momentum $p_{\nu}$, I can arrive at an expression in terms of the energy of the first decay and $\vec{p_{-}}$:

$$
E_{max} = \frac{1}{E_{-}} \left( \frac{m_{-}^2 + m_{\pi}^2}{2} + \vec{p_{-}} \cdot \vec{p_{\pi}} \right)
$$

I am not sure if this is correct mainly because if the neutrino is massless, is it possible for it to be at rest? I haven't worked/read much about neutrinos before (though I know we have realized they are not massless, but anyway), but if they are supposed to be massless it sounds they should move exactly at the speed of light.

So, my second approach is to approximate these 2 decays as just one process $Z \rightarrow \tau^- \pi^- \nu_{\tau}$ and do the same as Griffiths' 3.22: suppose $\tau^-$ and $\nu_{\tau}$ behave as a single "big" particle moving in the opposite direction as $\pi^-$. I don't think this is correct (or at least I don't see it intuitively) as, in the second decay, I don't see a reason why the neutrino moving in the opposite direction of $\pi$ (so it ends up in the same direction as $\tau^+$) yields the maximum energy, but I don't really know what else to try.

Any comments are appreciated.

Thanks!

 

[mfb]

The pion can't be at rest. Consider the decay in the lab frame: You can't have any momentum component orthogonal to the tau momentum if you want a pion a trest. That just gives you two options for the decay, and the momentum will be determined purely by the Z, tau and pion mass in both cases. It would be a remarkable coincidence if the pion can be at rest. And the neutrino can't be at rest either.

You can guarantee that a particle at rest is an option if (a) your decaying particle is at rest and (b) there are at least three particles involved in the decay. Neither of these two conditions is satisfied here.

So, my second approach is to approximate these 2 decays as just one process

That approach doesn't work because you are missing the constraint of pi+nu summing to a particle of the tau mass.

Think of the decay geometry first. Which way can the tau decay to give the pion the largest momentum in the frame of the Z decay? In which direction will the pion fly, relative to the tau?