The main goal of hadron colliders it to study quarks and gluon scattering to get insight into the strong force.

💃 Example: here are some example of simple interactions which we could study

💎 Conclusion: Quarks and gluons do not exist freely in nature, thus we scatter hadrons

Structure of proton

💃 Example: we could imagine a proton at a moment in time having this configuration

Of course as time goes by the configuration would change based on the timeframe and energy creating a bag of quarks and gluons.

However we distinguish 2 permanent features

• The valence quarks (black): while they could get annihilated and created the quantum numbers remain the same.
• The sea of quarks and gluons (red): this is in constant fluctuation

Hadron-hadron collisions

Quasi-Feynman diagram:

• The large red dots represent the protons as bags of particles
• This exact interaction her a certain likelihood called the cross section generally or more specifically here the partonic interaction

• To find these probabilities we need the probability of getting (in our example) one $u$ quark and one $\overline u$ quark from our protons, $f_{u/p}$ and $f_{\overline u/p}$ respectively

💼 Case: lets consider the cross section of a proton proton interaction to a final state $X$ ($pp\to X)$

• The formula is

  $$ \sigma_{qq\to X}=\sum_{i,j} \int \int \text dx_1 \,\text dx_2\,\underbrace{f_{i/p}(x_1,Q^2)f_{j/p}(x_2,Q^2)}\text{parton distribution functions}\,\underbrace{\hat \sigma{ij\to x}(x_1,x_2,Q^2)}_\text{partonic cross section} $$

  where $x$ is the fraction of proton momentum that is carried by the quark and $Q^2$ is the momentum scale of a hard scatter

  🗒️ Note: $f_{i/p}$ probability of getting a quark of flavour $i$ from a proton $p$

💎 Conclusion: if we know the parton (quarks, antiquarks and gluons) distribution functions and calculate the partonic cross section we can find the cross section of the interaction

Parton distribution functions

We cant derive these from first principle so instead we measure them

We use deep inelastic scattering, scattering an $e^-$ against a proton mediated by a $\gamma$

🗒️ Note: we could make a similar diagram for the other types of quarks an antiquarks, here $u$

• The cross section is $\propto$ parton distribution of the proton

• We define the 4 momentum transfer

  $$ q^\mu = p^\mu_e - p^\mu_{e'} =(E_q,\vec q) $$

• We can then define the momentum scale $Q^2$ as

  $$ Q^2=-|q^\mu |^2 $$

• We can also define the fraction of the proton momentum carried by the up quark $x$ as

  $$ x=\frac{Q^2}{2(E_qE_p-\vec p \cdot \vec q)} $$

  where $E_p^\mu = (E_p ,\vec p )$ is the four momentum of the initial proton

lets now relate the quantities which we know to the parton distribution functions