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According to this Wikipedia article on Effective field theory, the effective field theories used in QFT can be seen as an expansion in $1/M$, where $M$ is a characteristic mass scale of a certain involved object in the theory.

I think that this statement requires some clarifications. Since $E \sim M$ due to known relation $E=mc^2$ we are dealing with energy scales and $1/M$ behaves like a distance scale.

Question: The aspect I do not understand is which values/quantities are considered as being able to be expressed as (Taylor)-series expansions in powers of $1/M$ for any fixed effective quantum field theory concretely? Can they be classified? That is which 'objects' precisely allow such expansion within the world of a fixed EFT and which not?

For example, all observables? All amplitudes of physical processes like e.g. scatterings?
Is it known a priori which of all mathematical expressions characteristically involved in a fixed EFT allow such expression as a series in $1/M$?

Short remark: When in context of QFT one says that some expression can be expanded in $1/M$, precisely one means that it is expanded in the ratio $a/M$ with $a \sim E$ depending on context, right?

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    $\begingroup$ All of the above. The truncation in powers of $1/M$ is performed on the action and therefore all quantities you would compute with said action. $\endgroup$ Jul 22, 2021 at 11:55
  • $\begingroup$ so every quantity which involves the action term? $\endgroup$
    – user267839
    Jul 22, 2021 at 12:10
  • $\begingroup$ Additionally: do you really mean that these are expanded in $1/M$ or in $a/M$ where $a \sim M \sim E$ and $a/M$ small, where $a$ depends on context? $\endgroup$
    – user267839
    Jul 22, 2021 at 12:11
  • $\begingroup$ A small parameter you expand in is always a dimensionless ratio, yes. $\endgroup$ Jul 22, 2021 at 12:14
  • $\begingroup$ Can it be said more more precisely which quatities allow such expansion? You wrote, all of the above I mentioned. What I wrote above where just some examples which firstly came into my mind. Can it be answered more conceptionally? So in the sense of a kind of classification or criteria, that is if we fix a EFT and take any expression or term which contains parameters from that theory, and we want to know if it is expressible as such expanstion within the given EFT, then we just have to check if it fullfill the criterion? $\endgroup$
    – user267839
    Jul 22, 2021 at 13:03

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A complete answer to your question is given by the Appelquist Carrazone Decoupling theorem. Under certain mild conditions the effects of a massive particle at low energies decouple (besides finite threshold effects) and are suppressed by powers of the heavy mass scale (1/M). This means ALL observables are subject to this theorem.

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