Perturbiner methods for effective field theories and the double copy

Abstract

Perturbiner expansion provides a generating function for all Berends-Giele currents in a given quantum field theory. We apply this method to various effective field theories with and without color degrees of freedom. In the colored case, we study the U(N ) non-linear sigma model of Goldstone bosons (NLSM) in a recent parametrization due to Cheung and Shen, as well as its extension involving a coupling to the bi-adjoint scalar. We propose a Lagrangian and a Cachazo-He-Yuan formula for the latter valid in multi-trace sectors and systematically calculate its amplitudes. Furthermore, we make a similar proposal for a higher-derivative correction to NLSM that agrees with the subleading order of the abelian Z-theory. In the colorless cases, we formulate perturbiner expansions for the special Galileon and Born-Infeld theories. Finally, we study Kawai-Lewellen-Tye-like double-copy relations for Berends-Giele currents between the above colored and colorless theories. We find that they hold up to pure gauge terms, but without the need for further field redefinitions.

L∞-algebras and the perturbiner expansion

“…In this subsection, we very briefly review the perturbiner expansion for Yang-Mills theory in general spacetime dimensions, following the discussion in [20]. The reader might see [21] for a more thorough treatment.…”

“…, a im ). Inserting this expansion back in (2.31), and equating coefficients with the same number of generators T a i on both sides, gives the following recursion relation [20,21]…”

“…To go further, we must say a bit about how to extend the d'Alembertian operator and the binary operation , to E (R 1,d−1 , g ⊗ g ). This will be provided by what in reference [21] is called the colour-dressed version of the perturbiner ansatz. We therefore introduce, for each ordered sequence of positive integers i 1 < i 2 < · · · < i m , the notations…”

“…As a by product, we shall also show that the actual tree-level scattering amplitudes can obtained by plugging in directly the perturbiner expansions into the homotopy Maurer-Cartan actions. This is to be contrasted with the original deduction of these perturbiner expansions [20,21], where they are merely thought of as ansatzs that lead to a series of recursion relations, such that finding a solution to the recurrence leads to finding a solution to the equations of motion. It could therefore appear that by stressing the abstract algebraic L ∞ -structure of the theories in question one gains a deeper conceptual understanding.…”

“…They have also been applied to derive Berends-Giele recursion relations for the bi-adjoint scalar theory [20] and the study of α -deformations of Yang-Mills theory [26]. In addition to all these, Mizera and Skrzypek have constructed perturbiner expansions for effective field theories with and without color, including non-linear sigma models, special Galileon theory, and Born-Infeld theory [21]. Lastly, and even more recently, the perturbiner expansion has been applied to obtain multiparticle super Yang-Mills superfields in the BCJ gauge [27].…”