💼 Case: lets consider a general interaction $AB\to A'B'$ with the following Feynman diagram
Here $X$ is mediating the interaction and is a force carrier
Looking closer at the first node we see that we have a particle $A$ which emits $A'X$ ie $A\to A'X$
💼 Case: lets consider the rest frame of the initial state
The momentum of the initial state ie $A$ is
$$ \begin{aligned} P^\mu_A(m_A,\;\vec 0)
\end{aligned} $$
The momentum of the final state ie $A'+X$ is
$$
P^\mu_{A'+X}=(\sqrt{p_f^2+m_A^2},\;\vec p_f)+(\sqrt{p_f^2+m_X^2},\;-\vec p_f) $$
1️⃣ Lets look at the conservation of energy
The difference in energy is defined as
$$ \boxed{\Delta E_{fi}=\sum E_f -\sum E_i} $$
For out system we get $\Delta E_{fi}=\sqrt{p_f^2+m_A^2}+\sqrt{p_f^2+m^2_X}-m_A$
2️⃣ Lets consider a Massless force carrier ie $m_X=0$
If the recoil momentum is much smaller than the mass so $p_f\ll m_A$ thus
$$ \Delta E_{fi}\approx\underbrace{\sqrt{p^2_f+m_A^2}}{\approx m_A}+\underbrace{\sqrt{p^2_f+m_X^2}}{=p_f}-m_A\approx p_f $$
If the recoil momentum is much larger than the mass so $p_f\gg m_A$ thus
$$ \Delta E_{fi}=\underbrace{\sqrt{p^2_f+m_A^2}}{\approx p_f}+\underbrace{\sqrt{p^2_f+m_X^2}}{=p_f}-m_A\approx2p_f-m_A\approx 2p_f $$
2️⃣ Lets now consider a massive force carrier $m_X \ne 0$
If $p_f\ll m_A,m_X$ we have
$$ \Delta E_{fi}\approx\underbrace{\sqrt{p^2_f+m_A^2}}{\approx m_A}+\underbrace{\sqrt{p^2_f+m_X^2}}{\approx M_X}-m_A\approx M_X $$
if $p_f \gg m_A,m_X$ we have
$$ \Delta E_{fi}=\underbrace{\sqrt{p^2_f+m_A^2}}{\approx p_f}+\underbrace{\sqrt{p^2_f+m_X^2}}{\approx p_f}-m_A\approx2p_f-m_A\approx 2p_f $$
💎 Conclusion: no matter the assumptions energy is not conserved, in a single vertex
To fix this we say that the particles are created over a very short amount of time which allows spontaneous breaking of energy conservation through $\Delta E\Delta t \ge \hbar/2$
💎 Conclusion: the vertex we studied above $A\to A'X$ is not a real process
💼 Case: lets now consider the real process ie $AB\to A'B'$
Here energy is conserved so $m_A+m_B=m_{A'}+m_{B'}$