Scattering Amplitudes and the Double-Copy Landscape

Abstract

The double copy is a well-known relation between gauge and gravitational theories at the level of scattering amplitudes, allowing color-ordered amplitudes from two gauge theories to be combined via a double-copy kernel to yield the amplitude of a third gravitational theory. This is a surprising result, as straightforwardly multiplying two amplitudes could violate the factorization property of scattering amplitudes. Consistency of double copy demands that the multiplicative rule incorporate an identity model, which is realized in field-theory double copy by the bi-adjoint scalar theory (BAS). In this work, we explore the generalization of the double copy in the context of effective field theories (EFTs). Previous studies demonstrated that for any legitimate generalized double copy, the identity model corresponds to a bi-adjoint scalar EFT (BAS EFT) whose amplitude must exhibit "minimal rank." We bootstrap these BAS EFT identity models by imposing the minimal rank condition and discover that, at 6 points, compatibility with factorization significantly restricts the solutions to a form resembling string theory. Specifically, we introduce a class of solutions termed "hybrid models," which satisfy the field-theory Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations for one of the two color orderings. Remarkably, the other color ordering is then found to exhibit an emergent set of linear relations, as verified at 4 points to 36-derivative order and 5 points to 12-derivative order, precisely the string monodromy relations. Furthermore, we demonstrate that these hybrid models are nothing but a slight generalization of the Z-theories, if one treats the multiple zeta values (MZVs) in the low-energy expansion of the Z-theory amplitude as free parameters. We propose that general BAS EFT identity models can be written as a textit{field-theory} double copy of two hybrid models, and we provide numerical evidence at 4, 5, and 6 points. Our study suggests that the only consistent generalization of double copy must manifest string-theory-like structures, highlighting a close connection to MZVs and number theory.