Semi-abelian Z-theory: NLSM+phi^3 from the open string

TL;DR

The paper shows that semi-abelian Z-theory amplitudes reproduce the leading ${\alpha'}$-order of the extended NLSM$+\phi^3$ theory, clarifying how NLSM pions couple to bi-adjoint scalars via partial symmetrizations and monodromy relations. It introduces a compact ${\cal W}_{\alpha'}$-matrix framework to express semi-abelian disk integrals in terms of $(n-2)!$ non-abelian orderings, derives BCJ numerators of the NLSM from the abelian limit, and demonstrates how these results map to CHY integrands and to a field-theory double-copy structure yielding DBI-Volkov-Akulov-type predictions when combined with sYM. The work provides explicit low-energy expansions, including leading $\alpha'^{2}$, $\alpha'^{4}$ terms, and traces of nested commutators, highlighting the role of sine factors and MZVs in organizing higher-derivative interactions. Overall, the findings offer a robust, all-orders color-kinematics framework for semi-abelian string-inspired theories and illuminate connections to CHY formulations and the broader web of double-copy relations.

Abstract

We continue our investigation of Z-theory, the second double-copy component of open-string tree-level interactions besides super-Yang-Mills (sYM). We show that the amplitudes of the extended non-linear sigma model (NLSM) recently considered by Cachazo, Cha, and Mizera are reproduced by the leading alpha'-order of Z-theory amplitudes in the semi-abelian case. The extension refers to a coupling of NLSM pions to bi-adjoint scalars, and the semi-abelian case involves to a partial symmetrization over one of the color orderings that characterize the Z-theory amplitudes. Alternatively, the partial symmetrization corresponds to a mixed interaction among abelian and non-abelian states in the underlying open-superstring amplitude. We simplify these permutation sums via monodromy relations which greatly increase the efficiency in extracting the alpha'-expansion of these amplitudes. Their alpha'-corrections encode higher-derivative interactions between NLSM pions and bi-colored scalars all of which obey the duality between color and kinematics. Through double-copy, these results can be used to generate the predictions of supersymmetric Dirac-Born-Infeld-Volkov-Akulov theory coupled with sYM as well as a complete tower of higher-order alpha'-corrections.