In the one-loop renormalization of $\phi^4$-theory, only 1PI vertex functions $\Gamma^{(2)}$ and $\Gamma^{(4)}$ 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 $\Gamma^{(6)}$ and $\Gamma^{(8)}$. 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 $ϕ^4$-theory.
Answer
The naive power counting approach for a $d$-dimensional theory with coupling constant $\lambda$ tells us that the amplitude of diagrams with $E$ external lines and $V$ vertices behaves with the cutoff $\Lambda$ as $\propto\Lambda^D$ with $$ D = d - [\lambda]V - \frac{d-2}{2}E$$ where $[\dot{}]$ is the mass dimension. Since $\phi^4$ in four dimension has a dimensionless coupling, $$ D = 4-E $$ and since only diagrams with $D \geq 0$ need renormalization, the only diagrams needing it in 4D $\phi^4$ are those with $E \leq 4$. All diagrams with an odd number of external lines vanish due to the $\phi\mapsto-\phi$ 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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