Superficial degree of divergence on Weinberg

Reading volume 1 of Weinberg's QFT book, chapter 12, page 505 he says that if you consider a diagram with degree of divergence $D\geq{}0$, its contribution can written as a polynomial of order $D$ in external momenta. As an example he considers the $D=1$ integral

$$\int_0^{\infty}\frac{k\,dk}{k+q}=a+bq+q\ln{q}.$$

where $a$ and $b$ are divergent constants, and we see that we get a polynomial or order 1 in the external momenta $q$. He then says, and I quote

"Now, a polynomial term in external momenta is just what would be produced by adding suitable terms to the Lagrangian, if a graph with $E_f$ external lines of type $f$ (refering to field type) has degree of divergence $D\geq{}0$, then the ultraviolet divergent polynomial is the same as would be producedby adding various interactions $i$ with $n_{if}=E_f$ fields of type $f$ and $d_i\leq{}D$ derivatives."

Can anybody elaborate on this a bit? in particular, how and where does the polynomial arise with the added Lagrangian term?

1. Disclaimer: Renormalization is a huge subject with many facets, such as, e.g. overlapping divergences of subgraphs, regularization, renormalization group, etc. Here we will only elaborate on OP's quote from Ref. 1.

2. Ref. 1 is considering a Feynman diagram $${\cal F}(q_1, \ldots, q_E)$$ in momentum Fourier space, with external 4-momenta $$(q_1, \ldots, q_E)$$, and with internal 4-momenta $$(p_1, \ldots, p_I)$$, which are integrated over. The $$p$$-integrations are assumed to be UV divergent with positive superficial degree of divergence (SDOD) $$D\geq 0$$. Concerning SDOD, see e.g. my related Phys.SE answer here.

3. Here $$E=\sum_f E_f$$ is the total number of external lines, and $$E_f$$ is the number of external lines of field type $$f$$.

4. If we differentiate the Feynman diagram $$D+1$$ times wrt. the external 4-momenta, the integrand becomes UV finite. We conclude that the divergent part of the original Feynman diagram $${\cal F}(q_1, \ldots, q_E)$$ is a polynomial in $$(q_1, \ldots, q_E)$$ of order $$\leq D$$. Note that the coefficients of the polynomial are possibly infinite!

5. We next add new interaction terms to the Lagrangian density $${\cal L}$$ corresponding to $$E$$-vertices with $$E_f$$ fields of field type $$f$$, and possibly a finite number of spacetime derivatives (which in momentum Fourier space becomes a momentum monomial). The new interaction terms are so-called counterterms.

6. Feynman instructs us to sum over all Feynman diagrams with $$E_f$$ external legs of field type $$f$$. In particular, we should also include diagrams consisting of a single $$E$$-vertex coming from the new interaction counterterms. By adjusting possibly infinite coupling constants in front of the new interaction counterterms, the full Feynman diagram can be made finite.

References:

1. S. Weinberg, Quantum Theory of Fields, Vol. 1; Section 12.2, p. 506.

2. M.E. Peskin & D.V. Schroeder, An Intro to QFT, 1995; Section 10.1, p. 319.