# Deriving threshold energy for a reaction

I'm looking at this wikipedia derivation (link) for the threshold energy of a reaction of the form $$1+2 \rightarrow a + b + c$$

but I get to a different result because wikipedia says

$$E_1 = \frac{E_{CM}^2-(m_1+m_2)^2}{2m_2}$$ but I get

$$E_1 = \frac{E_{CM}^2-(m_1^2+m_2^2)}{2m_2}$$

who's right and who's wrong?

I get the exact same results for beta and gamma shown in the page but the last step seems to give me a different result. These are my steps $$\gamma(E_1+m_2-\beta p_1) = E_{CM}$$ $$\gamma (E_1 - \frac{p_1^2}{E_1+m_2}+m_2) = E_{CM}$$ $$\gamma \frac{\left[ E_1(E_1+m_2) - p_1^2 + m_2(E_1+m_2)\right]}{E_1+m_2} = E_{CM}$$ $$\frac{\gamma}{E_1+m_2}\left[ E_1^2-p_1^2 +m_2^2+ 2E_1 m_2\right] = E_{CM}$$ $$m_1^2+m_2^2 + 2m_2E_1 = E_{CM}^2$$ $$E_1 = \frac{E_{CM}^2-m_1^2-m_2^2}{2m_2}$$

I think your computation is not optimal. It is far easier to see that many of the steps are really just internal consistency constraints of $$\begin{pmatrix} \gamma & -\gamma\beta \\ -\gamma\beta & \gamma \end {pmatrix} \left [ \begin{pmatrix} E \\ p \end {pmatrix} + \begin{pmatrix} m_2 \\ 0 \end {pmatrix} \right ] = \begin{pmatrix} m_a + m_b + m_c \\ 0 \end {pmatrix}$$ The cheapest way to get the answer is really to compute the invariant rest energy $$(E+m_2)^2 - p^2 = (m_a+m_b+m_c)^2\\ E^2 - p^2 + m_2^2 + 2 m_2 E = (m_a+m_b+m_c)^2\\ m_1^2 + m_2^2 + 2 m_2 E = (m_a+m_b+m_c)^2\\ \therefore \qquad E = \frac{(m_a+m_b+m_c)^2-(m_1^2+m_2^2)}{2m_2}$$ Wiki's current version had, instead $$E = \frac{(m_a+m_b+m_c)^2-(m_1+m_2)^1}{2m_2}$$ which is just plain wrong.