Quantum field theory is the most predictive theory of nature ever tested, yet the scattering amplitudes produced from the standard application of Lagrangians and Feynman rules belie the simplicity of the underlying physics, obscuring the physical answers behind off-shell actions and gauge redundant descriptions. The aim of the modern S-matrix program (or the "amplitudes" subfield) is to reformulate specific field theories and manifest underlying structures in order to make high multiplicity and/or high loop scattering calculations tractable.
Many of the systems amenable to amplitudes techniques are actually intimately related to each other through the double-copy relations. We argue that conformal invariance is common thread linking several of the scalar effective field theories appearing in the double copy. For a derivatively coupled scalar with a quartic O(p⁴) vertex, classical conformal invariance dictates an infinite tower of additional interactions that coincide exactly with Dirac-Born-Infeld theory analytically continued to spacetime dimension D = 0. For the case of a quartic O(p⁶) vertex, classical conformal invariance constrains the theory to be the special Galileon in D = -2 dimensions. We also verify the conformal invariance of these theories by showing that their amplitudes are uniquely fixed by the conformal Ward identities. In these theories, conformal invariance is a much more stringent constraint than scale invariance.
Although many of the theories in the double-copy admit a high degree of space-time symmetry, amplitudes tools can be applied to non-relativistic theories as well. We explore the scattering amplitudes of fluid quanta described by the Navier-Stokes equation and its non-Abelian generalization. These amplitudes exhibit universal infrared structures analogous to the Weinberg soft theorem and the Adler zero. Furthermore, they satisfy on-shell recursion relations which together with the three-point scattering amplitude furnish a pure S-matrix formulation of incompressible fluid mechanics. Remarkably, the amplitudes of the non-Abelian Navier-Stokes equation also exhibit color-kinematics duality as an off-shell symmetry, for which the associated kinematic algebra is literally the algebra of spatial diffeomorphisms. Applying the double copy prescription, we then arrive at a new theory of a tensor bi-fluid. Finally, we present monopole solutions of the non-Abelian and tensor Navier-Stokes equations and observe a classical double copy structure.</p
