Leading low-energy effective action in $6D$, ${\cal N}=(1,1)$ SYM theory

We elaborate on the low-energy effective action of $6D,\,{\cal N}=(1,1)$ supersymmetric Yang-Mills (SYM) theory in the ${\cal N}=(1,0)$ harmonic superspace formulation. The theory is described in terms of analytic ${\cal N}=(1,0)$ gauge superfield $V^{++}$ and analytic $\omega$-hypermultiplet, both in the adjoint representation of gauge group. The effective action is defined in the framework of the background superfield method ensuring the manifest gauge invariance along with manifest ${\cal N}=(1,0)$ supersymmetry. We calculate leading contribution to the one-loop effective action using the on-shell background superfields corresponding to the option when gauge group $SU(N)$ is broken to $SU(N-1)\times U(1)\subset SU(N)$. In the bosonic sector the effective action involves the structure $\sim \frac{F^4}{X^2}$, where $F^4$ is a monomial of the fourth degree in an abelian field strength $F_{MN}$ and $X$ stands for the scalar fields from the $\omega$-hypermultiplet. It is manifestly demonstrated that the expectation values of the hypermultiplet scalar fields play the role of a natural infrared cutoff.