Strong force interacts with all particles with colour charge so quarks and gluons

Interaction vertices

• For quarks and gluons we have

  

• Three and four-point interactions of gluons

  

💎 Conclusion: quark flavour is conserved at each interaction vertex!

Colour states of quarks and gluons

• Quarks carry 3 colour charges ($r,g,b)$ and antiquarks carry $(\overline r,\overline g,\overline b)$

  $$ \begin{aligned} &\text{🔴 red}\,(r) \\ &\text{🟥 anti-red}\,(\overline r) \end{aligned} \qquad\begin{aligned} &\text{🟢 green}\,(g) \\ &\text{🟩 anti-green}\,(\overline g) \end{aligned} \qquad \begin{aligned} &\text{🔵 blue} \,(b) \\ &\text{🟦 anti-blue} \,(\overline b)\end{aligned} $$

  🗒️ Note: the anti-colour charges act like negative electric charge, ie $r$ and $\overline r$ is neutral

• Gluons are composed of a colour and anti-colour charge and have a net charge, there are only 8 combinations allowed

  $$ r\overline g \quad r\overline b \quad b\overline r \quad b\overline g \quad g\overline r \quad g\overline b \quad ; \quad {\scriptsize🔴🟩 \quad 🔴🟦 \quad 🔵🟥 \quad 🔵🟩 \quad 🟢🟥 \quad 🟢🟦} $$

  These are the the trivial ones, then last 2 are

  $$ \tfrac {1}{2} (r\overline r - g\overline g ) \quad \tfrac{1}{\sqrt{6}} (r\overline r + g \overline g- 2 b \overline b) \quad ;\quad ({\scriptsize 🔴🟥} - {\scriptsize🟢🟩}) \quad ({\scriptsize 🔴🟥}+{\scriptsize🟢🟩}-2{\scriptsize🔵🟦}) $$

• These were thrown a bit out of the blue so expand for more details (non examinable)

• We cannot have a colour-singlet gluon so the state

  $$

  \cancel{\tfrac{1}{\sqrt{3}} (r\overline r + b \overline b + g \overline g)} \quad ;\quad \cancel{({\scriptsize 🔴🟥}+{\scriptsize🔵🟦}+{\scriptsize🟢🟩})} $$

Colour-flow diagrams

🎯 Feynman diagram

Here we have the interaction $u\to u$ with a $g$ emitted

🎨 Colour flow diagram

here we have an $r$, $u$ quark which emits a $r\overline g$ gluon turning the quark into a $g$, $u$ quark

💎 Conclusion: Colour is conserved in QCD interactions

Strength of QCD interactions

💼 Case: consider our previous interaction

here the vertex has a coupling strength

$$ \text{coupling strength}\equiv g_\text{s} $$

We define the dimensionless coupling $\alpha_S$

$$ \alpha_\text{s}=\frac{g_\text{s}^2}{4\pi} \quad \Rightarrow \quad \text{what we measure experimentally} $$

🗒️ Note: here we used our general formula in natural units

If we compare the 2 forces we have that $\alpha_\text{s}\gg \alpha_\text{em}$

Running of strong coupling

Here we have 2 competing effects caused by quantum fluctuations