The Center of Mass for proton-proton collisions at the LHC

Question

If bunches of protons are being circulated in both directions of the LHC collider with each proton having an energy $E_p=7\ \mathrm{TeV}$, then using the following "Lorentz Invariant Quantity" expression, $s$, for a collider:

$$s=4E_p^2$$

I can then take the square root $s$ to get

$$\sqrt{s}=\sqrt{4\times7^2}\ \mathrm{TeV}=14\ \mathrm{TeV}$$

Which is the center of mass energy for proton-proton collisions at the LHC.

I found the expression, $s$ on the top of page 5 of some Oxford university notes. However, I don't really get where $s$ was derived from, so I am not sure why it has a different form for this type of proton-proton collision.

Answer 1

The definition of $s$ is the following:

$$ s=(p_1+p_2)^2,$$

where $p_1$ and $p_2$ are the 4-momenta of each colliding proton. For head on collision of particles with same energy and momentum (like in LHC) these explicitly read $p_1=(E_p,\vec{p})$ and $p_2=(E_p,-\vec{p})$. Plug this in the definition: \begin{eqnarray} s &=& p_1^2 + p_2^2 +2p_1\cdot p_2\\ &=&(E_p^2-|\vec{p}|^2) + (E_p^2-|\vec{p}|^2)+2 (E_p^2+|\vec{p}|^2)\\ &=& 4E_p^2 \end{eqnarray}