Constraining the space of low energy EFTs

Abstract

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). We use the ‘amplitude analysis’ i.e. dispersion relation for $2 \rightarrow2$ scattering in the forward limit, to put bounds on the Wilson coecients. 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 coecients can be constrained in this case even at the tree level. We also 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. We also explore Bell inequality violation for $2 \rightarrow2$ scattering in Effective Field Theories (EFTs) of photons, gluons, and gravitons. Using the CGLMP Bell parameter ($I_2$), we show that, starting from an appropriate initial nonproduct state, the Bell inequality can always be violated in the final state (i.e.,$I_2$ > 2) at least for some scattering angle. For an initial product state, we demonstrate that abelian gauge theories behave qualitatively di↵erently than non-abelian gauge theories (or Gravity) from the point of view of Bell violation in the final state: in the non-abelian case, Bell violation ($I_2$ > 2) is never possible within the validity of EFTs for weakly coupled UV completions. Interestingly, we also find that, for a maximally entangled initial state, scattering can reduce the degree of entanglement only for CP-violating theories. Thus Bell violation in $2 \rightarrow2$ scattering can, in principle, be used to classify CP conserving vs violating theories.