💼 Case: lets consider $e^+ e^- \to q\overline q$

Feynman diagram description

Picture in COM frame

Now we notice a problem directly, because the quark and antiquark are going away from each other but we can never have a free quark even for an arbitrarily small amount of time, thus either they combine and form a meson or they create quark antiquark pairs and produce 2 mesons.

Continuing on the same interaction lets now make one of the quarks emit a photon

Here we draw a diagram of the interaction

• $\theta$ is the angle of the emitted gluon
• $E$ is the energy of the gluon
• $\vec k$ is the momentum of the gluon

• If we use QCD to find the probability of this emission is

  $$ \text{Prob} \simeq \frac{2\alpha_\text{s} C_f}{\pi} \frac{\text dE}{\text E} \frac{\text d \theta}{\theta} $$

  We notice as $E\to 0$ or $\theta\to 0$ we get a divergence

  🗒️ Note: as $E,\theta\to 0$ then $\alpha_s$ would become very large so this equation derived from perturbation theory wouldn't work anymore

  🗒️ Note: the probability here means, if an interaction with a quark where to happen there is $\text{Prob}$ probability of it being a $g$ emission

💼 Case: using this new information lets look back at our $e^+e^-\to q\overline q$ in more detail

what actually happens is as follows

🧽 Assume: here we only care about the quark $q$

• At high energies we expect many gluons to be created

• These gluons then have a probability of creating more gluons

  or of creating quark and antiquark pairs

• This cascade of events is called a parton shower

• The final result is many hadrons such as $p$ and $\pi$

🗒️ Note: in experiment we need to be able to map a jet (the parton shower) back to its interaction

• To achieve this we use parton-hadron duality:

  $$ \sum^\text{hadrons}{i} \vec p_i\sim \vec p\text{quark} $$

  i.e. the momentum of the quark which initiate the shower is $\sim$ the momentum of all the hadrons

• 🗒️ Note: we can use this to determine the spin of quarks

More complicated interaction

💼 Case: lets now consider $e^+e^- \to q\overline q g$

Feynman diagram description

where the jets are represented by the streaks

Geometrical picture in COM frame

the $e^+ e^-$ beam line is in red into the page

🗒️ Note: the angles of each jet will depend on the momentum split between $q\overline qg$

• The ratio of the 3 jet against 2 jet event cross section is

  $$ \frac{\sigma_\text{2jet} + \sigma_\text{3jet}}{\sigma_\text{2jet}}= \left ( 1 + \frac{\alpha_\text{s}}{\pi}\right ) $$

• 🗒️ Note: we can use the angles to find that gluons are spin-$1$ particle

How to define $N$-jet events