Lorentz Boost of the Lab Momentums

Question

Lets assume we have a two body scattering like;

$$a + b = c+d$$

In the lab frame particle $a$ is moving with a certain beam energy $E_a$, and hits stationary particle $b$. Before the collision, the four momentum components for particle $a$ is $(E_a, 0,0, P_z)$ and for particle $b$ are $(m_b,0,0,0 )$.

I have all four momentum components (after the collision in lab frame) for particle $c$, which is $(E_c, p_{xc}, p_{yc}, P_{zc})$. How can I get the four momentum components of $c$ in the CM frame?

I do know how to write down the total four vector of $(a+b)$ in the CM and Lab frames and transfer them. But knowing all the four vector components of particle $c$, how can I get the four vector of particle $c$ in CM frame?

I want to do the computation in C++, so any mathematical trick how to get four momentum of $c$ particle in CM frame would be very helpful.

Any suggestions will be appreciated.

EDIT:

1. I said the decay assuming the c particle could decay, but if I do know all the momentum components, I can explain that thing, so please ignore that
2. Actually I know the four vector of (a+b) in CM frame calculation and also I also know all the all initial four vector of particle c in LAB frame, my concern is to get the four vector of c particle in CM frame (not lab frame, coz I know and I have the numerical value of the four components too) in terms of LAB frame.

Answer 1

Let $\tilde p_{com}=\tilde p_a+ \tilde p_b$, which you say you know.
Assuming that $\tilde p_{com}$ is future-timelike [and assuming signature $(+,-,-,-)$],
the 4-velocity along $\tilde p_{com}$ is this unit-vector $$\hat t_{com}=\frac{1}{\sqrt{\tilde p_{com}\cdot\tilde p_{com}}}\tilde p_{com}.$$

Given any 4-vector $\tilde Q$,
its t-component in the com-frame is $$Q_{t,com}=\tilde Q\cdot \hat t_{com}.$$

For simplicity, let's assume that all y- and z-components are zero.

To find the unit vector of the x-axis in the com-frame, you can use this trick:
if $\hat t_{com}=A \hat t + B \hat x$, then $\hat x_{com}=B \hat t + A \hat x$. [Check that $\hat x_{com}$ is unit and is orthogonal to $\hat t_{com}$.]

Thus, the x-component of $\tilde Q$ in the com-frame is $$Q_{x,com}=-\tilde Q\cdot \hat x_{com}.$$

Both dot-products $Q_{t,com}$ and $Q_{x,com}$ can be evaluated using components in the lab frame.

(I'll leave all the algebra and C++-coding for you to do.)

UPDATE:
You could also find the spatial velocity $\beta_{com}$ of $\tilde p_{com}$ [e.g. $\beta_{com}=\frac{p_{com,x}}{p_{com,t}}$ in the simple case], then write down a Lorentz transformation matrix with $\beta_{com}$. Then apply that transformation matrix to $\tilde Q$.

Answer 2

You are leaving some degrees of freedom on the table, since you know the masses of the initial particles:

$$ E_a = \sqrt{P_Z^2 + M_a^2} $$

and of course,

$$ E_b = M_b $$.

Hence the initial 4 momentum in the lab is:

$$ p_{\mu} = (M_b + \sqrt{P_Z^2 + M_a^2}, 0, 0, P_Z) $$

Now find the boost that zeros the momentum and apply it to:

$$ p_C = (E_C, \vec p_c) $$