  We explore the double copy of effective field theories (EFTs), in the
recently proposed generalized color-kinematics and Kawai-Lewellen-Tye (KLT)
approaches. In the former, we systematically construct scalar numerators
satisfying the Jacobi identities from simpler numerator "seeds" with trace-like
permutation properties. This construction has the advantage of being easily
applicable to any multiplicity, which we exemplify up to 6-point. It employs
the linear map between color factors formed by single traces of generators and
by products of the structure constants, which also relates the generalized KLT
and color-kinematics formalisms, allowing to produce KLT kernels at arbitrary
order in the EFT expansion. At 4-point, we show that all EFT kernels are
generated and that they only yield double-copy amplitudes which can also be
obtained from the traditional KLT kernel. We perform initial checks suggesting
that the same conclusions also hold at 5-point. We focus on single-trace
massless scalar EFTs which however also control the higher-derivative
corrections to gauge and gravity theories.
