Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 6 / b

Define the free Feynman propagators byThe interaction preserves the global phase symmetry $\phi \mapsto e^{i\alpha }\phi$, with the real $\varphi$ neutral. Therefore the charge-two correlator vanishes in a symmetry-invariant perturbative vacuum:In Wick contractions, a complex field contracts only with its conjugate; one vertex has one of each and cannot absorb both external $\phi$ fields. The symmetry shows this vanishing holds to all perturbative orders, not only first order.

The neutral connected correlator has one allowed contraction to each external point at order $g$:The propagators are even functions of their separation; their orientations encode the conjugate-field contractions. Equivalently, with every external momentum Fourier transformed as incoming,No factor of two accompanies the vertex: $\phi$ and ${\phi }^{†}$ are distinct fields.

For the full correlator as literally written, a disconnected scalar three-point tadpole is also possible. Contract the two external complex fields with each other and contract the complex pair at the vertex into a loop. Unless normal ordering is imposed on the interaction or a tadpole counterterm fixes $⟨\varphi ⟩=0$, addThe coincident propagator is regulator dependent; in dimensional regularization the massless scaleless tadpole diagram vanishes. More generally this is $⟨T{\phi }^{†}(x)\phi (y){⟩}_0⟨\varphi (z)⟩$ to this order and is removed by taking the connected correlator or imposing the stated vacuum-renormalization condition. Normalizing by vacuum bubbles alone does not remove an external-leg disconnected tadpole diagram. The charged correlator remains zero in every one of these conventions.^[ref]