Squaring 4-momentum gives wrong result

Question

I‘m doing an exercise where a particle $A$ with mass $m_A$ decays into two particles $B$ and $C$ with mass $m_B$ and $m_C$. The goal is to calculate the absolute value of the momentum of the particles $B$ and $C$. In the solution they start with ($E$ is the energy) $$(m_Ac,0,0,0)=(E_B/c,0,0,p)+ (E_B/c,0,0,-p)$$ and when the square it they get: $$m_A^2c^2=m_B^2c^2+ m_C^2c^2+2(E_BE_C/c^2+|p|^2)$$ But if I calculate the square I get at the end $-|p|^2 $ and not $|p|^2$ because of the minkowski metric. So where is my mistake?

Answer 1

With $E_B = \sqrt{p^2+m_B^2}$ and $E_C = \sqrt{p^2+m_C^2}$:

$$\begin{align}\begin{pmatrix}\sqrt{p^2+m_B^2} + \sqrt{p^2+m_C^2}\\ 0 \\ 0 \\ 0\end{pmatrix}^2 &= \left(\sqrt{p^2+m_B^2} + \sqrt{p^2+m_C^2}\right)^2\\ &=m^2_B + m^2_C + 2p^2 + 2 \underbrace{\sqrt{p^2+m_B^2}}_{E_B}\underbrace{\sqrt{p^2+m_C^2}}_{E_C} \end{align}$$

So

$$ \begin{align} \sqrt{p^2+m_B^2}\sqrt{p^2+m_C^2} &= \frac{m_A^2 - 2p^2 - m^2_B - m^2_C}{2}\\ (p^2+m_B^2)(p^2+m_C^2) &=\frac{\left(m_A^2 - 2p^2 - m^2_B - m^2_C \right)^2}{4}\\ \end{align} $$

From there it follows that

$$p=\frac{\sqrt{m_A^4 - 2 m_A^2 m_B^2 + m_B^4 - 2 m_A^2 m_C^2 - 2 m_B^2 m_C^2 + m_C^4}}{2 m_A}$$

This can be simplifed by using the Källén function

$$\lambda(a,b,c) = a^2+b^2+c^2-2(ab+bc+ca)$$

Then

$$p = \frac{\sqrt{\lambda(m_A^2, m_B^2, m_C^2)}}{2m_A}$$