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Both the Bern, Carrasco and Johansson (BCJ) and the Kawai, Lewellen and Tye(KLT) double-copy formalisms have been recently generalized to a class ofscattering matrix elements (so-called form factors) that involve localgauge-invariant operators. In this paper we continue the study of double copyfor form factors. First, we generalize the double-copy prescription to formfactors of higher-length operators ${\rm tr}(\phi^m)$ with $m\geq3$. Thesehigher-length operators introduce new non-trivial color identities, but thedouble-copy prescription works perfectly well. The closed formulae for theCK-dual numerators are also provided. Next, we discuss the $\vec{v}$ vectorswhich are central ingredients appearing in the factorization relations of boththe KLT kernels and the gauge form factors. We present a general constructionrule for the $\vec{v}$ vectors and discuss their universal properties. Finally,we consider the double copy for the form factor of the ${\rm tr}(F^2)$ operatorin pure Yang-Mills theory. In this case, we propose a new prescription thatinvolves a gauge invariant decomposition for the form factor and a combinationof different CK-dual numerators appearing in the expansion.

Double copy for tree-level form factors II: generalizations and special topics