  Indirect CP violation in K \rightarrow {\pi}{\pi} decays plays a central role
in constraining the flavor structure of the Standard Model (SM) and in the
search for new physics. For many years the leading uncertainty in the SM
prediction of this phenomenon was the one associated with the nonperturbative
strong interaction dynamics in this process. Here we present a fully controlled
lattice QCD calculation of these effects, which are described by the neutral
kaon mixing parameter B_K . We use a two step HEX smeared clover-improved
Wilson action, with four lattice spacings from a\approx0.054 fm to
a\approx0.093 fm and pion masses at and even below the physical value.
Nonperturbative renormalization is performed in the RI-MOM scheme, where we
find that operator mixing induced by chiral symmetry breaking is very small.
Using fully nonperturbative continuum running, we obtain our main result
B_K^{RI}(3.5GeV)=0.531(6)_{stat}(2)_{sys}. A perturbative 2-loop conversion
yields B_K^{MSbar-NDR}(2GeV)=0.564(6)_{stat}(3)_{sys}(6)_{PT}, which is in good
agreement with current results from fits to experimental data.
