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General conditions ensuring relativistic causality in an effective field theory based on the derivative expansion

We discuss the general conditions ensuring relativistic causality in an effective theory based on the derivative expansion. Relativistic causality implies that the Green function vanishes in the space-like region. It is known that a naive derivative expansion violates causality in some cases such as the first-order relativistic dissipative hydrodynamics. We note that the Lorentz covariance, and the equal order of time and space derivatives do not ensure causality. We derive the general conditions for causality that should be satisfied by any effective theories respecting special relativity. The conditions are the followings: (i) the imaginary part of poles of the Green function is bounded at the large momentum limit, (ii) the front velocity is smaller than the speed of light, and (iii) the coefficient of the highest-order time derivative does not include space derivatives.