The maximum pole of a diagram with $V$ vertices and $L$ loops is at most $1/\varepsilon^{m(V,L)}$, where $m(V,L)=\min (V-1,L).$ The result holds in dimensional regularization, where $\varepsilon = d-D$, $d$ is the physical dimension and $D$ the continued one. Moreover, vertices are counted treating mass terms and the other non-dominant quadratic terms as “two-leg vertices”.

**Proof**

We prove the statement inductively in $V$ and, for fixed $V$, inductively in $L$. The diagrams with $V=1$ and arbitrary $L$ are tadpoles, which vanish identically and therefore satisfy the theorem. Suppose that the statement is true for $V<\overline{V}$, $\overline{V}>1$, and arbitrary $L$. Consider diagrams with $\overline{V}$ vertices. Clearly for $L=1$ the maximal divergence is $1/\varepsilon $, so the theorem is satisfied. Proceed inductively in $L$, i.e. suppose that the theorem is satisfied by the diagrams with $\overline{V}$ vertices and $L<\overline{L}$ loops, and consider the diagrams $G_{\overline{V},\overline{L}}$ that have $\overline{V}$ vertices and $\overline{L}$ loops. If $G_{\overline{V},\overline{L}}$ has no subdivergence, its divergence is at most a simple pole. Higher-order poles are related to the subdivergences of $G_{\overline{V},\overline{L}}$ and can be classified replacing the subdiagrams with their counterterms. Consider the subdiagrams $\gamma _{v,l}$ of $G_{\overline{V},\overline{L}}$, with $l$ loops and $v$ vertices. Clearly, $1\leq l<\overline{L}$ and $1\leq v\leq \overline{V}$. By the inductive hypothesis, the maximal divergence of $\gamma _{v,l}$ is a pole of order $m(v,l)$. Contract the subdiagram $\gamma _{v,l}$ to a point and multiply by $1/\varepsilon^{m(v,l)}$. A diagram with $\overline{V}-v+1\leq $ $\overline{V}$ vertices and $\overline{L}-l<\overline{L}$ loops is obtained, whose maximal divergence, taking into account of the factor $1/\varepsilon ^{m(v,l)}$, is at most a pole of order $m(v,l)+m(\overline{V}-v+1,\overline{L}-l)$. The inequality
\[
m(v,l)+m(\overline{V}-v+1,\overline{L}-l)\leq m(\overline{V},\overline{L}),
\]
which can be derived case-by-case, proves that the maximal divergence of $G_{\overline{V},\overline{L}}$ associated with $\gamma _{v,l}$ satisfies the theorem. Since this is true for every subdiagram $\gamma _{v,l}$, the theorem follows for $G_{\overline{V},\overline{L}}$. By induction, the theorem follows for every diagram. $\Box$
This theorem was proved in
D. Anselmi, *Renormalization of a class of non-renormalizable theories*,

05A1 Renorm

JHEP 0507 (2005) 077 | DOI: 10.1088/1126-6708/2005/07/077

and arXiv:hep-th/0502237.

**Note** In the proof mass terms, if present, are treated as two-leg vertices, the propagator being just the massless one. Any other dimensionful parameters multiplying quadratic terms must be treated in a similar way. It is consistent to do so, since we are interested only in the UV divergences of the quantum theory, and their renormalization, which are polynomial in the masses and the other parameters that multiply quadratic terms. To avoid IR problems in intermediate steps, it is convenient to calculate the UV divergences of Feynman diagrams with deformed propagators having $k^{2}$ replaced by $k^{2}+\delta ^{2}$ in their denominators, and let $\delta $ tend to zero at the end. The limit is smooth, since the theorem of locality of counterterms ensures that the divergent parts of Feynman diagrams depend polynomially on $\delta $. Tadpoles are loops with a single vertex and vanish identically. However, loops with at least two vertices are not tadpoles (even if one of the vertices is a two-leg “mass” vertex) and do give a non-trivial divergent contribution, which can be calculated at $\delta \neq 0$.

**Example **The two-loop scalar self-energy of the massless $\phi^4$-theory ($L=2$, $V=2$) has only a simple pole instead of a double pole.