How can we control the resolution in experiments (collider, accelerator, etc.)?

Question

Consider for example we perform the electron-proton scattering experiment.

If we accelerate the incident electron with a certain energy, then it means that we can control the center-of-mass energy of this process. In terms of Mendelstam variables, we know the value of $s$.

But, I wonder how can we control (or measure) the momentum transfer, $Q^2$, which stands for the resolution. If we want to investigate the internal structure of the proton deeply, we need a high $Q^2$ value.

Since $s$ and $t$ are independent variables, it seems that we can change $s$ and $t$ independently.

How can we do that in experiments?

Thank you.

Answer 1

We can't control $t$. We can only measure it, from the energy and direction of the scattered electron. It's not like $s$ which we can choose by arranging an appropriate accelerator and target.

So to study high $Q^2$ we set up an experiment which will measure the (relatively few) large-angle scatters and not get swamped by the more numerous small-angle scatters. From the measured scattered electron momentum $\vec p'$ (and $E'=\sqrt{m_e^2+p'^2}\approx p'$) one has $t=(E-E')^2-(\vec p - \vec p')^2$. So we know it for the events we select and measure, but we don't force the electrons to scatter at any particular angle.