Amplitude's positivity vs. subluminality: Causality and Unitarity Constraints on dimension 6 & 8 Gluonic operators in the SMEFT

We derive the causality and unitarity constraints on dimension 6 and dimension 8 Gluon field strength operators in the Standard Model Effective Field Theory (SMEFT). In the first part of the paper, we use the 'amplitude analysis' i.e. dispersion relation for $2\rightarrow2$ scattering in the forward limit, to put bounds on the Wilson coefficients. We show that the dimension 6 operators can exist only in the presence of certain dimension 8 operators. It is interesting that the square of the dimension 6 Wilson coefficients can be constrained in this case even at the tree level. In the second part of this work, we successfully rederive all these bounds using the classical causality argument that demands that the speed of fluctuations about any non-trivial background should not exceed the speed of light. We also point out some subtleties in the superluminality analysis regarding whether the low-frequency phase velocity can always be used as the relevant quantity for Causality violation: as an example, we show that, due to these subtleties, if a small pion mass is added in the chiral Lagrangian, it is unclear if any strict positivity bound can be derived on the dimension 8 Wilson coefficient. Finally, we mention an interesting non-relativistic example where the subluminality requirement produces a stronger bound than the 'amplitude analysis'.