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Weinberg in his QFT Volume 1 points out in Chapter 12, section 12.3, near Fig. 12.4 (Is Renormalizability necessary?) that for expansions in EFTs in powers of $k/M$, where $k$ is the energy scale of interest and $M$ is some common large mass scale of the theory, when $k>>M$, the S-matrix elements violate unitarity bounds. I am not sure what he means by this statement.

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Unitarity says that the sum of the probabilities for all possible out states (for a given in state) must be 1. Since the probability is the square of an amplitude, if any individual amplitude has a norm which is greater than 1, unitarity can't be satisfied. In an effective field theory, the amplitudes typically grow as $\sim E/M$ to some positive power, where $E$ is the energy of the process and $M$ is the mass scale. As $E$ grows above the cutoff, eventually this factor will become large enough that the amplitude will have a norm larger than 1.

That's the idea in a nutshell. There are statements like the Froissart bound which make this more precise. There are also some beautiful examples of how effective theories can be "UV completed", and how the UV complete theory "cures" the unitarity problem in the effective field theory by "softening" the amplitude. The role of the massive $W$ and $Z$ bosons is to cure this kind of problem in Fermi's effective theory of the weak interactions, and the Higgs boson cures a related issue due to the longitudinal mode of the massive $W$ and $Z$ particles.

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