Quenched $B_K$-parameter from Osterwalder-Seiler tmQCD quarks and mass-splitting discretization effects

We apply an Osterwalder-Seiler version of twisted mass QCD to a study of the $B_K$ parameter, in which three of the four quark fields making up the relevant $\mathrm{\Delta }S=2$ operator are maximally twisted with the same twist angle, while the fourth one has a twist angle of opposite sign. It is known that this setup ensures automatic improvement of the bare $K^0$-${\bar{K}}^0$ operator matrix element and multiplicative renormalization of the $\mathrm{\Delta }S=2$ operator, at the price of breaking the $K^0$-${\bar{K}}^0$ mass degeneracy by discretization effects. As a result, two dominant systematic errors of the $B_ K$ determination with Wilson fermions are kept under control. With the Clover term included in the fermion action, we perform a feasibility study and find, in the quenched approximation, a significant improvement of the scaling behaviour of $B_K$, compared to earlier standard tmQCD determinations. Moreover, we study in detail the $K^0$-${\bar{K}}^0$ mass splitting that characterizes this approach and confirm that, in the presence of the Clover term, it is greatly reduced in a maximally twisted theory.