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Particle decay formulae

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Problem Statement
Consider the decay of a particle of mass M, at rest, into two particles with masses ##m_1## and ##m_2##, both nonzero. With an appropriate choice of axes, the momentum vectors of the final particle can be written: $$p_1 = (E_1,0,0,k)$$ $$p_2 = (E_2,0,0,-k)$$ with ##E_1^2 = k^2 + m_1^2, E_2^2 = k^2 + m_2^2##.


a) Show that ##k = \dfrac{\sqrt{(M^4 -2M^2(m_1^2+m_2^2)+(m_1^2-m_2^2)^2}}{2M} ##



b) Take the limit ##m_2 \rightarrow 0 ## and show that this reproduces the result for the decay into one massive and one massless particle.

c) Find formulae for ##E_1## and ##E_2## in terms of M, m1, m2.
Relevant Equations
Energy momentum relation: ##E^2 = p^2 + m^2##
Attempt at solution:

By conservation of momentum: $$P = (M,0,0,0) = p_1 + p_2 = (E_1 + E_2, 0, 0,0)$$ thus
$$ M = E_1 + E_2 = 2k^2 + m_1^2 + m_2^2$$
Now $$E_1^2 - E_2^2 = m_1^2 - m_2^2 = (m_1 + m_2)(m_1-m_2)$$
$$ = M(m_1-m_2) = (2k^2+m_1^2+m_2^2)(m_1-m_2)$$
Isolating k: $$ k = \sqrt{\dfrac{M-m_1^2-m_2^2}{2}}$$
 

vela

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##M \ne m_1+m_2##. Also, ##M## can't be equal to ##m_1^2+m_2^2+2k^2##. The units don't work out.
 
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How come?
 

vela

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Mass isn't conserved in special relativity.
 
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You're right. Is it fair to say though that ## P = p_1 + p_2## and ##p_1^2 = m_1^2##, ##p_2^2 = m_2^2##?
 

vela

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You're right. Is it fair to say though that ## P = p_1 + p_2## and ##p_1^2 = m_1^2##, ##p_2^2 = m_2^2##?
Yes. Try squaring ##p_1 = P - p_2##.
 
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Ok, I think I got it. Squaring this term like you suggested gives:

$$ m_1^2 = P^2 - 2P\cdot p_2 + p_2^2$$
$$m_1^2 = M^2 - 2ME_2 + m_2^2$$

Isolating ##E_2##:
$$ E_2 = \dfrac{(M^2-m_1^2-m_2^2)}{2M}$$
Which if we plug it into:
$$ \vec{p_2}^2 = k^2 = E_2^2 - m_2^2 $$

Returns the relation asked, if I didn't mess up the calculation.

Can you help me with this one?
 

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