Dual is Different: EFTs, Axions and Nonpropagating Form Fields in Cosmology

Scalar fields in 4D are known to have equivalent dual descriptions in terms of form-field gauge potentials, but this is often regarded as an arcane fact. Why use more complicated formulations when simpler scalar descriptions exist and are equivalent? We describe three ways in which scalars that arise as duals can differ from their garden-variety counterparts. Two of these -- the interchange of weak and strong couplings and utility in bringing topological information to the low-energy theory -- are relatively well-known, but to these we add a third: dualities that map derivative interactions to non-derivative interactions seem not to commute with the general power-counting arguments that quantify control over the low-energy approximation within any EFT involving gravity. Since both sides of the duality must agree on their low-energy limit, the non-derivative interactions on the scalar side of the duality turn out to be suppressed relative to what would generically be assumed. They are nonetheless technically natural, as is particularly clear in the dual formulation. We argue that scalar fields arising as duals (such as the universal axion $a$ from string vacua) that have the commonly assumed $J^\mu \partial_\mu a$ interaction with matter also have $J^\mu J_\mu$ contact interactions among the respective currents, some of whose implications we explore. We also emphasize the non-trivial role and cosmological implications of non-propagating 3-form gauge potentials and clarify confusing statements recently made in the literature regarding the validity of duality for massive form fields.