A proton beam with a momentum of \(p=100\text{GeV}\) hits a fixed Hydrogen target. (Discussion beneath).

## a) What is the centre of mass energy \(\sqrt{s}\) for this interaction?

In such an interaction, obviously only a fraction of the momentum carried by the incoming proton will be accessible to the interaction.

Let’s calculate the momentum 4-vectors (in natural units) for this interaction:

\begin{eqnarray}

p_1^\mu &=& (E_p, \vec {p}_1) \\

p_2^\mu &=& (m_p, \vec 0)

\end{eqnarray}

The centre of mass energy \(\sqrt s\) is therefore^{1}

\begin{eqnarray}

s &=& (p_1^\mu + p_2^\mu)^2 \nonumber \\

&=& \big[(E_p, \vec p_1) + (m_p, 0)\big]^2 \nonumber \\

&=& (E_p+m_p)^2-\vec p^2 \nonumber \\

&=& E_p^2 + 2E_pm_p + m_p^2 – \vec p^2 \nonumber \\

&=& 2m_p^2 + 2E_pm_p \nonumber \\

\sqrt s &=& \sqrt{2m_p^2 + 2E_pm_p} \nonumber \\

\label{eq:rg} &\approx& \sqrt{2E_pm_p}

\end{eqnarray}

where \eqref{eq:rg} holds since the incoming proton’s energy is so much higher than the proton rest mass. Plugging in the values yields

$$\sqrt s = 14.2\text{GeV}$$

## b) What beam energy would be required to achieve the same \(\sqrt s\) using a \(pp\) collider like the LHC?

For a \(pp\) collider, almost all of the momentum of the particles is available for the interaction, and so only half the centre of mass energy would be required per beam (7.1 GeV)).

## What are asymmetric colliders used for then? Why not just use a more powerful, more efficient symmetric collider?

Choosing whether to use a fixed-target collider, a slightly asymmetric collider (such as LHCb), or a fully symmetric collider (like ATLAS) depends primarily on your intention and what you’re interested in. At experiments such as ATLAS, the main intention is creating and detecting high-mass particles like the Higgs boson, and less about measuring their various properties so precisely.

However, at LHCb, the collisions are asymmetric, as you can see from the asymmetry of the detector. LHCb deals mainly with lighter particles such as B mesons and kaons and therefore isn’t so interested in the highest possible energy. In fact, the asymmetry of the collisions additionally leads to a moving centre of mass frame after the collision: this relative motion gives the resultant particles a Lorentz boost and enables their decays to be observed with greater ease.

Fixed-target colliders benefit from the obvious advantage of being *easier and cheaper* to construct. They also have a higher luminosity and interaction cross-section.

- For a bit of background if you’re not sure about what exactly is going on, see the Particle Data Group’s kinematics reference. These same equations appear there from [43.3] onwards (page 2 ish). [↩]