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R1

From IFPA wiki
  1. From [math]U(y,x)\to e^{i\alpha(y)}U(y,x)e^{-i\alpha(x)}[/math] and [math]n\cdot D\psi=\lim_{\epsilon\to 0}\frac{\psi(x+\epsilon n)-U(x+\epsilon n,x)\psi(x)}{\epsilon}[/math], find the transformation laws of [math]A_\mu[/math], and the formula for [math]D_\mu[/math] as a function of [math]A_\mu[/math] if [math]U(x+\epsilon n,x)=1-i\epsilon n\cdot A[/math].
  2. Write the QCD lagrangian as a function of the coloured fields [math]A_\mu^a[/math], starting from its expression as a function of the fields [math]\mathbb A_\mu[/math] and using [math]{\mathbb A}_\mu=\sum_{a=1}^8 \frac{\lambda_a}{2} A_\mu^a[/math].
  3. Derive the equation of motion of the field [math]A_\mu^a[/math].
  4. Derive the expression of the 3-gluon vertex.
  5. Calculate the colour factors of the following diagrams: Diagrams.png

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