R1

1. From [math]\displaystyle{ U(y,x)\to e^{i\alpha(y)}U(y,x)e^{-i\alpha(x)} }[/math] and [math]\displaystyle{ 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]\displaystyle{ A_\mu }[/math], and the formula for [math]\displaystyle{ D_\mu }[/math] as a function of [math]\displaystyle{ A_\mu }[/math] if [math]\displaystyle{ 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]\displaystyle{ A_\mu^a }[/math], starting from its expression as a function of the matrix fields [math]\displaystyle{ \mathbb A_\mu }[/math] and using [math]\displaystyle{ {\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]\displaystyle{ A_\mu^a }[/math].
4. Calculate the divergent part of the loop in [math]\displaystyle{ \lambda\Phi^3 }[/math] for the number of spacetime dimensions d=6.
5. Calculate the divergent part of the diagram in [math]\displaystyle{ \lambda\Phi^4 }[/math]. How does it contribute to mass renormalisation?