I have this problem:
A particle B is standing still while another one, A, is moving towards it with initial 4-momentum $(E,p,0,0)$. Calculate the change in particle A's 4-momentum as viewed from the particle B's rest frame, in terms of the initial energy E and the scattering angle $\theta$.
I am a bit confused about the 4-momentum conservation. Initially we have $p^i_A=(E,p,0,0)$ and $p^i_B=(m_B,0,0,0)$ finally we should have $p^f_A=(E_f, p_f \cos(\theta),p_f \sin(\theta),0)$ and $p^f_B=(m_B,0,0,0)$. To get the change in momentum I would do $p^f_A-p^i_A$. But the total momentum should be conserved in any frame, but I am not sure how does that work here. In order to conserve it, we would need $E=E_f$ and $\theta=0$ but then the problem would be trivial and also physically you can obviously have angles other than 0. What am I doing wrong?