In the one-loop renormalization of ϕ4-theory, only 1PI vertex functions Γ(2) and Γ(4) are regularized and renormalized. But they do not exhaust all the irreducible connected diagrams at one loop. One can have a diagram, for example, with one-loop, 3 vertices and 6 external lines, or with one-loop, 4 vertices and 8 external lines and so on. What about these diagrams? They respectively correspond to Γ(6) and Γ(8). What about these 1PI diagrams with one-loop? Shouldn't they require renormalization as well? In fact these diagrams contribute to the effective potential.
EDIT : arxiv.org/abs/hep-ph/9901312 This might be an useful reference. Please look at the one-loop diagrams in the calculation of the effective potential in ϕ4-theory.
Answer
The naive power counting approach for a d-dimensional theory with coupling constant λ tells us that the amplitude of diagrams with E external lines and V vertices behaves with the cutoff Λ as ∝ΛD with D=d−[λ]V−d−22E where [˙] is the mass dimension. Since ϕ4 in four dimension has a dimensionless coupling, D=4−E and since only diagrams with D≥0 need renormalization, the only diagrams needing it in 4D ϕ4 are those with E≤4. All diagrams with an odd number of external lines vanish due to the ϕ↦−ϕ symmetry, so what's left to renormalize is E=0,2,4, which are the vacuum energy, the propagator, and the 4-vertex, respectively.
The diagrams you ask about exist, but have D<0, and do not need to be renormalized, since they are not diverging when we take the cutoff to infinity.
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