Double copy for tree-level form factors. Part II. Generalizations and special topics

Both the Bern, Carrasco, and Johansson (BCJ) and the Kawai, Lewellen, and Tye (KLT) double-copy formalisms have been recently generalized to a class of scattering matrix elements (so-called form factors) that involve local gauge-invariant operators. In this paper, we continue the study of double copy for form factors. First, we generalize the double-copy prescription to form factors of higher-length operators tr(ϕ^m) with m ≥ 3. These higher-length operators introduce new non-trivial color identities, but the double-copy prescription works perfectly well. The closed formulae for the CK-dual numerators are also provided. Next, we discuss the $\overrightarrow{\upsilon }$ \( \overrightarrow{\upsilon} \) vectors which are central ingredients appearing in the factorization relations of both the KLT kernels and the gauge form factors. We present a general construction rule for the $\overrightarrow{\upsilon }$ \( \overrightarrow{\upsilon} \) vectors and discuss their universal properties. Finally, we consider the double copy for the form factor of the tr(F ²) operator in pure Yang-Mills theory. In this case, we propose a new prescription which involves a gauge invariant decomposition for the form factor and a mixture of different CK-dual numerators appearing in the expansion. The new prescription for the more complicated double copy has been verified up to five external gluons.