# 1-loop renormalization of general scalar in general dimension

I'm interested in studying the theory $$S=\int d^dx\left(\frac{1}{2}(\partial\phi)^2-\frac{g}{(2k)!}\phi^{2k}\right)$$ in $$d=2, 3, 4$$ and for $$k=2, 3, 4\dots$$ and I'm having trouble with the the 1-loop renormalization of the said theory. The questions are boldfaced.

I know that for the theory to be renormalizable the mass dimension of the coupling constant must be $$\left[g\right]\geq 0$$, and therefore for a said dimension only some values of $$k$$ are worth studying:

$$d=4\quad\Rightarrow\quad k=2$$
$$d=3\quad\Rightarrow\quad k=2,3$$
$$d=2\quad\Rightarrow\quad k=2,3,4\dots$$

I'm also pretty confident that the Feynman rules are $$k$$-independent: $$\frac{i}{p^2}$$ for the propagator and $$-ig$$ for the vertex (with the appropriate number of external legs, which is obviously $$k$$-dependent).
The doubts come with the loops: my idea was to draw all the possible 1-loop diagrams starting with 1 vertex, and then two, and so on, with the appropriate number of external legs coming out of the vertices depending on $$k$$. The only diagrams that I could draw are the "tadpoles", the "spiders" and the various polygons:
First question: are these the only 1-loop diagrams there are, or am I missing something?
The divergence of the loop integrals are obviously $$d$$-dependent, and in $$d\leq4$$ only the tadpoles and the spiders can be divergent (by simple power-counting), so let's focus on these two.
I know that for a massless scalar theory with $$k=2$$ in $$d=4$$ the tadpoles diagrams can't be regularized and are therefore just considered as an infinite constant shift of the theory. Can one do this with general $$k$$ and $$d$$?
Assuming this is correct, we're left with the spiders. Is there a general formula to find the number of diagrams with the same number of vertices, and external lines? With this I don't mean the symmetry factor (that should be $$\frac{1}{2}$$ for all these spider diagrams), I mean for example the three diagrams in the $$k=2$$ case:
I couldn't find a general formula so I decided to move on and leave this problem for later. Now for the real question: these spider diagrams, what 0-loop diagrams are they "correcting" for $$k\geq3$$? I mean, in the $$k=2$$ case, the diagram is the quantum correction to the vertex, which has 4 external legs. But for the $$k=3$$ case, for example, there isn't a 0-loop diagram with 8 external legs! The first correction to the 6-legged vertex is a 2-loop diagram, so what is this loop correcting?
My attempt to answer this is that for $$k>2$$ some diagrams are just not there and there is no problem with this, but it seems a bit odd:
(the diagrams in brackets are some 2-loop diagrams that I'm not interested in).

• Every spider has degree of divergence $d - 4$. So the $k \geq 3$ ones converge and therefore don't need a 0-loop vertex to correct. Nov 20 '21 at 23:11
• @ConnorBehan so among all the 1-loop diagrams in a general scalar theory, with general (even) coupling and general dimension, the only one that actually matters is the $k=2$ spider? Nov 21 '21 at 0:15
• Assuming finite diagrams don't matter for what you're doing, yes. Nov 21 '21 at 1:00
• Suggestion: Consider to replace the word tadpole with self-loop for clarity. Nov 21 '21 at 11:50