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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$\Delta S =2$ operator are maximally twisted with the same twist angle, whilethe fourth one has a twist angle of opposite sign. It is known that this setupensures automatic improvement of the bare $K^0$-$\overline K^0$ operator matrixelement and multiplicative renormalization of the $\Delta S =2$ operator, atthe price of breaking the $K^0$-$\overline K^0$ mass degeneracy bydiscretization effects. As a result, two dominant systematic errors of the $B_K$ determination with Wilson fermions are kept under control. With the Cloverterm included in the fermion action, we perform a feasibility study and find,in the quenched approximation, a significant improvement of the scalingbehaviour of $B_K$, compared to earlier standard tmQCD determinations.Moreover, we study in detail the $K^0$-$\overline K^0$ mass splitting thatcharacterizes this approach and confirm that, in the presence of the Cloverterm, it is greatly reduced in a maximally twisted theory.

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