Understanding the Center of Mass Energy in Fixed Target vs. Collider Experiments

[vbrasic]

Homework Statement

While not explicitly a homework question, I am having some trouble with deriving expressions for the center of mass energy in a fixed target experiment versus a collider experiment. The question is effectively, "Derive an expression for the center of mass energy in a fixed target experiment and compare this to the center of mass energy in a collider experiment."

Homework Equations

The momentum 4-vector. Also, the formula for center of mass energy $\sqrt{s}=\sqrt{(p_1+p_2)^2}$.

The Attempt at a Solution

For a fixed target experiment, we have the two momentum 4-vectors, $(\frac{E_b}{c},p_b)$, and $(m_tc,0)$, for the beam particle and target particle respectively. Then, $$s=\frac{E_b^2}{c^2}+m_t^2c^2+2E_bm_t-p_b^2.$$

We can group the first and last term together to obtain $m_b^2c^2+m_t^2c^2+2E_bm_t$. However, my textbook at this point claims that this is equivalent to $2m^2c^2+2Em$. My question is then, would this not only hold true for $m_b\approx m_t$?

Similarly, for a collider experiment, we have, $s=(\frac{E_A}{c}+\frac{E_B}{c})^2\rightarrow s=\frac{(E_A+E_B)^2}{c^2}$. Again, my textbook claims that this is equivalent to $\frac{4E^2}{c^2}$, which again I think, should only hold true for $E_A\approx E_B$.

If I am not understanding incorrectly, why can these approximations be made?

 

[Orodruin]

They indeed seem to be assuming the masses to be equal.