- #1

Aleolomorfo

- 73

- 4

- Homework Statement
- Consider the decay ##K \rightarrow \mu + \nu##. Find the maximum ##\mu## momentum in a frame in which the ##K## momentum is 5 GeV

- Relevant Equations
- ## P = (E, p_x, p_y, p_z)##

## P^2 = m^2##

My first idea was to find an expression of the muon momentum as a function of the angle and then maximaze the expression. But my attempts were not succesful. I report here my attempt.

Set up (referring to the attached file "bettini.pdf"):

$$ p = (\sqrt{p_k^2+m_k^2},0,0,p_k)$$

$$ k_1 = (\sqrt{p_{\mu}^2+m_{\mu}^2},0,p_{\mu}\sin{\theta}, p_{\mu}\cos{\theta})$$

$$ k_2 = (p_{\nu},0,-p_{\nu}\sin{\phi}, p_{\nu}\cos{\phi})$$

My attempt:

$$ p = k_1 + k_2 $$

$$ k_2 = p - k_1$$

Squaring

$$ k_2^2 = p^2 + k_1^2 -2pk_1$$

$$ 0 = m_k^2 + m_{\mu}^2 - 2\sqrt{(p_k^2+m_k^2)(p_{\mu}^2+m_{\mu}^2)} + 2p_kp_{\mu}\cos{\theta}$$

Continuing from here the computation to obtain ##p_{\mu}## as a function of ##\cos{\theta}## is quite demanding.

My second idea was to think intuitively that the case in which the muon momentum is maximum is when it flies forward and the neutrino backward. And in this particular case the computation is easier.

To sum up, my two questions are: Is there a way to perform my first idea? Is my second idea right?

Thanks in advance!

Set up (referring to the attached file "bettini.pdf"):

$$ p = (\sqrt{p_k^2+m_k^2},0,0,p_k)$$

$$ k_1 = (\sqrt{p_{\mu}^2+m_{\mu}^2},0,p_{\mu}\sin{\theta}, p_{\mu}\cos{\theta})$$

$$ k_2 = (p_{\nu},0,-p_{\nu}\sin{\phi}, p_{\nu}\cos{\phi})$$

My attempt:

$$ p = k_1 + k_2 $$

$$ k_2 = p - k_1$$

Squaring

$$ k_2^2 = p^2 + k_1^2 -2pk_1$$

$$ 0 = m_k^2 + m_{\mu}^2 - 2\sqrt{(p_k^2+m_k^2)(p_{\mu}^2+m_{\mu}^2)} + 2p_kp_{\mu}\cos{\theta}$$

Continuing from here the computation to obtain ##p_{\mu}## as a function of ##\cos{\theta}## is quite demanding.

My second idea was to think intuitively that the case in which the muon momentum is maximum is when it flies forward and the neutrino backward. And in this particular case the computation is easier.

To sum up, my two questions are: Is there a way to perform my first idea? Is my second idea right?

Thanks in advance!