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I'm trying to better understand the nature of infrared divergencies in YM theory; for now, I'm only interested in soft divergences. The usual explanation one is given about the origin of IR divergences is related to Landau equations and the Coleman-Norton picture, which are supposed to show the "physicality" of IR divergences. Indeed, most examples of IR divergences explicitly require on shell momenta in order to appear: more specifically, when one sets to zero certain momenta, these trigger other internal propagator to go on shell and these provide the extra powers in the denominator which produce the divergence. The typical example is the one loop correction to the vertex in QED, where the integrand, up to factors which are not relevant in determining the divergence, behaves as:

$$\frac{1}{k^2 (p-k)^2 (p'+k)^2}=\frac{1}{k^2 k\cdot(p-k) k\cdot(p'+k)} \approx \frac{1}{k^4} \ \ \text{when} \ k\rightarrow 0.$$

(Here $p$ and $p'$ are the external fermion momentum and k the photon loop momentum). Then the fourth power of the momentum in the denominator cancels partially with the three powers provided by the measure in polar coordinates yielding a logarithmic divergence.

This principle works perfectly as long as the number of loops is low. However, in general, one can think of subdiagrams of a general diagram which could produce a divergence without requiring any internal momentum to be on shell. An example could be:

enter image description here

All propagators are here intended to be internal, and can be set to zero without triggering another momentum to go on shell. The diagram can be though as a subdiagram nested deep in a much larger diagram. Now power-counting for this diagram, when all momenta are sent to zero, yields a degree of divergence

$$D=2 I - N_3 - 3L=1$$

where $I$ is the total number of propagators, $N_3$ the number of 3-vertices (which carry a power of momentum in the numerator) and $L$ the number of loops. Thus this configuration seems to yield an infrared soft divergence.

Is this reasoning correct? Is there a way to eliminate the divergence? I have the impression that either I have made some embarrassing mistake or there is some argument based on the fact that this singularity when thought in complex space is not a pinch singularity and can thus be eliminated with an appropriate choice of contour. In this case, can someone direct me to appropriate literature?

Thank you for any help.

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