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To investigate the light scalar tetraquark candidate $a_0(980)$ (quantumnumbers $J^P =0^+$), a correlation matrix including a variety of two- andfour-quark interpolating operators has to be computed. We discuss efficienttechniques to compute the elements of this correlation matrix, in particulardiagrams with closed quark loops. Furthermore, we present evidence that suchdiagrams are not negligible given our precision, and their contribution isessential to obtain physically meaningful results. In particular, we findindications of the existence of an "additional" state around the two-particlethresholds of $K+ \bar K$ and $\eta +\pi$, which could correspond to the$a_0(980)$ meson.

Importance of closed quark loops for lattice QCD studies of tetraquarks