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See e.g. Schwartz, QFT and the Standard Model, Eqs. (22.2), (22.3), (22.6), (33.5).

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They are derived from local causal Lagrangians, so their physical effect should not break causality. Typically the effective theory of this kind contains an infinite number of terms with higher and higher derivatives. If you truncate the series by retaining only a finite number of terms, then you may get a violation of causality at high energies, but this may be irrelevant when you apply the effective theory at low energies only. In any case, if you do a resummation of the whole series, the acausality problems should go away.As I have read, the effective Lagrangian are non local sometimes, does that mean they break causlaity ?

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One should also be aware that the series of perturbation theory (including this series of powers of energy-momentum scales of effective theories) are usually asymptotic series (with a convergence radius of 0!).

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Generally, an infinite series in higher and higher derivatives is an indication of possible non-locality. And even without infinite derivatives, an equation of motion with higher than second derivatives may lead to serious problems http://www.scholarpedia.org/article/Ostrogradsky's_theorem_on_Hamiltonian_instabilityAnd what this has to do with "non-locality"?

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For example, some non-renormalizable EFT's will violate unitary (froissard bound) at high energies, simply because the theory is being extrapolated beyond its range of validity.

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