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


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.


  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.


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