Higher-dimensional invariants in $6D$ super Yang-Mills theory

We exploit the $6D,\mathcal{N}=(1,0)$ and $\mathcal{N}=(1,1)$ harmonic superspace approaches to construct the full set of the maximally supersymmetric on-shell invariants of the canonical dimension $\mathbf{d}=\mathbf{12}$ in $6D,\mathcal{N}=(1,1)$ supersymmetric Yang-Mills (SYM) theory. Both single- and double-trace invariants are derived. Only four single-trace and two double-trace invariants prove to be independent. The invariants constructed can provide the possible counterterms of $\mathcal{N}=(1,1)$ SYM theory at four-loop order, where the first double-trace divergences are expected to appear. We explicitly exhibit the gauge sector of all invariants in terms of $\mathcal{N}=(1,0)$ gauge superfields and find the absence of $\mathcal{N}=(1,1)$ supercompletion of the $F^6$ term in the abelian limit.