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

We exploit the $6D, {\cal N}=(1,0)$ and ${\cal N}=(1,1)$ harmonic superspace approaches to construct the full set of the maximally supersymmetric on-shell invariants of the canonical dimension ${\bf d=12}$ in $6D, \,{\cal N}=(1,1)$ 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 ${\cal N}=(1,1)$ 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 ${\cal N} = (1, 0)$ gauge superfields and find the absence of ${\cal N}=(1,1)$ supercompletion of the $F^6$ term in the abelian limit.