  A classically scale-invariant 6d analog of the 4d Yang-Mills theory is the
4-derivative $ (\nabla F)^2 + F^3$ gauge theory with two independent couplings.
Motivated by a search for a perturbatively conformal but possibly non-unitary
6d models we compute the one-loop $\beta$-functions in this theory. A
systematic way of doing this using the background field method requires the
expression for the $b_6$ Seeley-DeWitt coefficient for a generic 4-derivative
operator. It was previously unknown and we derive it here. As an application,
we also compute the one-loop $\beta$-function in the (1,0) supersymmetric $
(\nabla F)^2$ 6d gauge theory constructed in hep-th/0505082.
