# Intuitive explanation of superficial degree of divergence

Consider $$\varphi^p$$ theory in dimension $$D$$. For a Feyman diagram $$\Gamma$$ one can introduce the superficial degree of divergence $$deg(\Gamma)$$. It is defined as $$DL-2I$$ where $$I$$ is the number of internal edges of $$\Gamma$$ and $$L$$ is the loop number. This value can be easily understood by examining the power of the momenta under the integral corresponding to $$\Gamma$$. By a simple combinatorial argument one can show that $$\deg(\Gamma)=(D-\frac{2p}{p-2})L+\frac{2}{p-2}(p-N)$$ where $$N$$ is the number of external edges of $$\Gamma$$. For example for $$p=4$$ one gets $$deg(\Gamma)=(D-4)L+4-N$$. In particular this degree decreases when $$N$$ increases and increases when $$D$$ increases: but the most interesting phenomenon is related to the behaviour wih respect to $$L$$: in small dimension many loops implies smaller degree of divergence while in higher dimension increasing the number of loops causes problems with convergence. There is also this critical value of dimension $$\frac{2p}{p-2}$$ in which the loop number does not make a difference. The only values $$p$$ for which this critical dimension appears to be an integer are $$p=3,4,6$$ with corresponding $$D=6,4,3$$.

Is there any physical reason (not just mere power counting+simple combinatorics) for which one should expect such behaviour of this degree of divergence with respect to $$N,D$$ and in particular $$L$$? By a physical reason I mean some heuristic (not necessarilly rigorous) argument for finiteness/divergence, not involving Feynman rules and power counting.

I am aware that my question is rather vaque nevertheless I hope that my point is clear and I will be grateful for any explanation.

• Those are the values for which the theory is scale-invariant (cf. this PSE post). Aug 20, 2019 at 12:09
• Aug 21, 2019 at 16:15
• Thank you for the links: however the explanations given in these discussions still heavily relies on power counting and combinatorics and is not exactly what I had in mind Aug 23, 2019 at 2:32