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An implicit assumption when working with dimensional regularisation is that the divergences are always of the form $\varepsilon^{-n}$ for some integer $n$ (e.g. refs.1&2). Is there any way to argue that this is always the case? Or can, at least in principle, non-analytic divergences appear somehow? What if the action contains some of the standard "unphysical" ingredients such as non-locality, higher derivatives, kinetic terms with the wrong sign, etc.?

References.

  1. Srednicki's QFT, chapter 28. Free copy here.

  2. https://golem.ph.utexas.edu/category/2008/10/hopf_algebraic_renormalization.html

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