For one complex scalar in the <fundamental representation>, the most general power-counting-renormalizable gauge-invariant Lagrangian is
$$
\boxed{\mathcal L=-\frac14F_{\mu\nu}^aF^{a\mu\nu}
+(D_\mu\Phi)^\dagger D^\mu\Phi-V(\Phi),
\qquad
V(\Phi)=-\mu^2\Phi^\dagger\Phi
+\lambda(\Phi^\dagger\Phi)^2},
$$
apart from a constant and the four-dimensional <Yang-Mills theta term>. Stability requires $\lambda>0$. A cubic candidate $\epsilon_{ijk}\Phi_i\Phi_j\Phi_k$ vanishes because the scalar components commute.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 3 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-305/3/b/ii/solution