Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 3

In four dimensions the action is dimensionless and $[{\partial }_{\mu }]=1$. The kinetic terms giveso $[\Phi ]=1$ and $[\Psi ]=3/2$.

Up to a constant and total derivatives, the most general renormalizable Lagrangian invariant underisHere $\lambda >0$ is required for stability at large field. The last line is the allowed charge-conserving scalar-fermion Yukawa interaction; its precise two-component expression depends on the chosen spinor notation.

Forwith $\lambda >0$, spontaneous symmetry breaking occurs when $m_{\Phi }^2<0$. ThenSince $\Phi$ has charge two, the transformations preserving a chosen nonzero vacuum satisfy $e^{2i\alpha }=1$. The unbroken subgroup is therefore ${\mathbb{Z}}_2$, generated by $\alpha =\pi$; it acts as $\Psi \mapsto -\Psi$.

With the convention that the fields have charges $q_{\Phi }=2$ and $q_{\Psi }=1$, the Noether current may be writtenup to an overall sign convention for the generator. The field equations, including the invariant Yukawa interaction, give ${\partial }_{\mu }{J}^{\mu }=0$. Subject to vanishing flux at spatial infinity, the Noether chargeis conserved and generates the global $U(1)$ transformations.

Promoting $\alpha$ to a spacetime-dependent parameter makes the scalar kinetic term vary by both terms linear and quadratic in the gauge field. Merely adding $e{J}^{\mu }{A}_{\mu }$ accounts for the linear term but is not invariant because the scalar current itself changes under a local transformation.

Withminimal coupling givesThe missing term is the seagull vertex $4e^2{A}^2|\Phi {|}^2$. This agrees exactly with replacing partial derivatives by gauge covariant derivatives.

The Dirac kinetic term is first order in derivatives:Its expansion is only linear in $A_{\mu }$, so the current coupling already completes the gauge-invariant fermion kinetic term and no $A^2\overline{\Psi }\Psi$ term arises.

The invariant interaction isbecause ${\Phi }^{\ast }$ has charge $-2$ while the fermion pair has charge $+2$. Once $\Phi$ has a vacuum expectation value, it generates a Majorana-type fermion mass and can split the two Majorana components of the original Dirac field.

The operator has dimensionso its coupling is classically marginal in four dimensions. By comparison, $\overline{\Psi }\Psi$ has dimension three and its coefficient is a relevant mass parameter.