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Trying to answer my old question myself: The cited piece is talking about an infrared cutoff, which basically amounts to giving the propagating particle a mass which appears squared in a propagator. So I think, what is meant is considering $$\lim_{\Lambda\to 0}\int \frac{d^4p}{p^2+\Lambda}$$ and this $\Lambda$ of course has a mass dimension of 2.

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