Consider a complex scalar field (see problems 3.5, 5.1, and 8.7) with L=L0+L1, where
L0L1Lct=−∂μφ†∂μφ−m2φ†φ,=−41Zλλ(φ†φ)2+Lct,=−(Zφ−1)∂μφ†∂μφ−(Zm−1)m2φ†φ.
This theory has two kinds of sources, J and J†, and so we need a way to tell which is which when we draw the diagrams. Rather than labeling the source blobs with a J or J†, we will indicate which is which by putting an arrow on the attached propagator that points towards the source if it is a J†, and away from the source if it is a J.
a) What kind of vertex appears in the diagrams for this theory, and what is the associated vertex factor? Hint: your answer should involve those arrows!
b) Ignoring the counterterms, draw all the connected diagrams with 1≤E≤4 and 0≤V≤2, and find their symmetry factors. Hint: the arrows are important!