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I was trying to derive transition rate for a two-body decay process. In one of the reference I'm following, it consider $a\rightarrow1+2$ decay, and said the daughter particles in center-of-mass reference frame have three-momenta $p*$ and $-p*$. After some mathematical steps, it says:

Therefore, \begin{equation} \Gamma_{fi} = \frac{1}{8\pi^2m_a} \int \left| \mathcal{M_{fi}}\right|^2 g(p_1) \delta(f(p_1)) dp_1 d\Omega \end{equation} with \begin{align} g(p_1)&= \frac{p_1}{4E_1E_2},\\ f(p_1) &= m_a - E_1 - E_2\\ &= m_a - \sqrt{m_1^2+p_1^2} - \sqrt{m_2^2+p_1^2}.\end{align} The Dirac delta-function $\delta(f(p_1))$ imposes energy conservation and is only non-zero for $p_1=p_*$, where $p_*$ is the solution of $f(p_*)=0$ and is the momentum of daughter particle in center-of-mass frame.

I'm confused, why the center-of-mass momentum of daughter particle also made the energy conservation holds?

Notes : I'm following along mark Thomson's Modern Particle Physics, and this is on page 68.

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    $\begingroup$ Greetings! I'm having trouble parsing the sentence which begins with "I'm confused" and ends with a question mark. Have you perhaps omitted a word? $\endgroup$
    – rob
    Dec 4, 2020 at 5:24

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This happens because of four momentum algebra. Given the invariant mass of the particles once the momentum is fixed the energy is also fixed.

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