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