Continuum limit of Bk from 2+1 flavor domain wall QCD

We determine the neutral kaon mixing matrix element B-K in the continuum limit with 2 + 1 flavors of domain wall fermions, using the Iwasaki gauge action at two different lattice spacings. These lattice fermions have near exact chiral symmetry and therefore avoid artificial lattice operator mixing. We introduce a significant improvement to the conventional nonperturbative renormalization (NPR) method in which the bare matrix elements are renormalized nonperturbatively in the regularization invariant momentum scheme (RI-MOM) and are then converted into the (MS) over bar scheme using continuum perturbation theory. In addition to RI-MOM, we introduce and implement four nonexceptional intermediate momentum schemes that suppress infrared nonperturbative uncertainties in the renormalization procedure. We compute the conversion factors relating the matrix elements in this family of regularization invariant symmetric momentum schemes (RI-SMOM) and (MS) over bar at one-loop order. Comparison of the results obtained using these different intermediate schemes allows for a more reliable estimate of the unknown higher-order contributions and hence for a correspondingly more robust estimate of the systematic error. We also apply a recently proposed approach in which twisted boundary conditions are used to control the Symanzik expansion for off-shell vertex functions leading to a better control of the renormalization in the continuum limit. We control chiral extrapolation errors by considering both the next-to-leading order SU(2) chiral effective theory, and an analytic mass expansion. We obtain B-K((MS) over bar)(3 GeV) = 0.529(5)(stat)(15)(chi)(2)(FV)(11)(NPR). This corresponds to (B) over cap ((RGI ) over bar)(K) = 0.749(7)(stat)(21)(chi)(3)(FV)(15)(NPR). Adding all sources of error in quadrature, we obtain (B) over cap ((RGI ) over bar)(K)0.749(27)(combined), with an overall combined error of 3.6%.