Continuum limit of $B_K$ 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 $\overline{\mathrm{MS}}$ 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 $\overline{\mathrm{MS}}$ 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 $B_K^{\overline{\mathrm{MS}}}(3\mathrm{GeV})=0.529(5{)}_{\mathrm{stat}}(15{)}_{\chi }(2{)}_{\mathrm{FV}}(11{)}_{\mathrm{NPR}}$. This corresponds to ${\widehat{B}}_K^{\overline{\mathrm{RGI}}}=0.749(7{)}_{\mathrm{stat}}(21{)}_{\chi }(3{)}_{\mathrm{FV}}(15{)}_{\mathrm{NPR}}$. Adding all sources of error in quadrature, we obtain ${\widehat{B}}_K^{\overline{\mathrm{RGI}}}=0.749(27{)}_{\mathrm{combined}}$, with an overall combined error of 3.6%.

Figure 1

Effective mass plateau of the lightest unitary simulated pion ($m_h=0.03$, $m_x=m_y=m_l=0.004$) on the $\mathbf{1}$ ensembles. Here the plateau is obtained from the wall-local pseudoscalar-pseudoscalar correlation function (PP) correlator, but the fit displayed is to all pseudoscalar correlators.

Figure 2

Effective mass plateau of the lightest unitary simulated pion ($m_h=0.04$, $m_x=m_y=m_l=0.005$) on the $\mathbf{2}$ ensembles. Here the plateau is obtained from the wall-local PP correlator, but the fit displayed is to all pseudoscalar correlators.

Figure 3

Effective mass plateau of the heaviest simulated eta ($m_x=m_y=m_h=0.03$, $m_l=0.008$) on the $\mathbf{1}$ ensembles. Here the plateau is obtained from the wall-local PP correlator, but the fit displayed is to all pseudoscalar correlators.

Figure 4

Effective mass plateau of the heaviest simulated eta ($m_x=m_y=m_h=0.04$, $m_l=0.01$) on the $\mathbf{2}$ ensembles. Here the plateau is obtained from the wall-local PP correlator, but the fit displayed is to all pseudoscalar correlators.

Figure 5

A typical $B_K^{\mathrm{lat}}$ matrix element correlator ($m_y=m_h=0.03$, $m_x=m_l=0.004$) on the $\mathbf{1}$ ensembles.

Figure 6

A typical $B_K^{\mathrm{lat}}$ matrix element correlator ($m_y=m_h=0.04$, $m_x=m_l=0.005$) on the $\mathbf{2}$ ensembles.

Figure 7

An overlay of a typical $B_K^{\mathrm{lat}}$ matrix element ($m_y=0.03$, $m_x=m_l=0.004$) on the $\mathbf{1}$ ensembles at two values of the sea strange-quark mass: $m_h=0.03$ (red) and $m_h=0.027$ (blue). The latter is at our closest reweight to the physical strange mass.

Figure 8

The $m_h$ dependence of a typical $B_K^{\mathrm{lat}}$ matrix element ($m_y=0.03$, $m_x=m_l=0.004$) on the $\mathbf{1}$ ensembles.

Figure 9

The four lowest order diagrams. Each circle represents the insertion of the current $\overline{s}{\gamma }_L^{\mu }d$. The $d$ or $\overline{d}$ ($s$ or $\overline{s}$) quarks have momenta $\pm p_1$ ($\pm p_2$) and the flow of fermion number is denoted by the arrow.

Figure 10

The lowest order diagrams, with spinor and color labels exhibited. The notation is as in Fig. 9.

Figure 11

The two independent one-loop Feynman diagrams to be evaluated.

Figure 12

Four one-loop diagrams whose Feynman integrals are given by that of diagram (a1) in Fig. 11.

Figure 13

Four one-loop diagrams whose Feynman integrals are related to that of diagram (b1) in Fig. 11.

Figure 14

We can use the perturbative running to convert the chiral limit of the ratio (91) to $\overline{\mathrm{MS}}$ at 2 GeV for each $p^2$ using $Z_{B_K}^{\mathrm{S}}(p^2)\times \frac{{\omega }_{\mathrm{NDR}}(\mu =2\mathrm{GeV},nf=3)}{{\omega }_{\mathrm{S}}({\mu }^2=p^2,nf=3)}$. This is displayed for all five intermediate MOM schemes $S$ on the $\mathbf{2}$ ensemble set (${24}^3$, $a^{-1}=1.73\mathrm{GeV}$ lattice). The top two panels correspond to the original RI-MOM as the intermediate scheme and the other four rows correspond to the schemes of Sec. 4b. The left-hand panels show the data with the momenta of Table 8 and the right-hand panels show the data using the momenta in Table 9 accessible with the use of twisted boundary conditions. The scatter due to the $O(4)$ symmetry breaking in the left-hand panels is absent in the right-hand panels. For this reason, we use the data with twisted boundary conditions for our analysis.

Figure 15

We can use the perturbative running to convert the chiral limit of the ratio (91) to $\overline{\mathrm{MS}}$ at 2 GeV for each $p^2$ using $Z_{B_K}^S(p^2)\times \frac{{\omega }_{\mathrm{NDR}}(\mu =2\mathrm{GeV},nf=3)}{{\omega }_{\mathrm{S}}({\mu }^2=p^2,nf=3)}$. This is displayed for all five intermediate MOM schemes on the $\mathbf{1}$ ensemble set (${32}^3$, $a^{-1}=2.28\mathrm{GeV}$ lattice). The top two panels correspond to the original RI-MOM as the intermediate scheme and the other four rows correspond to the schemes of Sec. 4b. The left-hand panels show the data with the momenta of Table 8 and the right-hand panels show the data using the momenta in Table 9 accessible with the use of twisted boundary conditions. The scatter due to the breaking of $O(4)$ symmetry is smaller on this finer lattice.

Figure 16

Mixing coefficient at $\beta =2.25$ for $O_1=O_{VV+AA}$ and the operators $O_2=O_{VV-AA}$, $O_3=O_{SS-PP}$, $O_4=O_{SS+PP}$, and $O_5=O_{TT}$. The data shown has been extrapolated to the chiral limit.

Figure 17

Continuum limit step scaling functions for all four SMOM schemes (blue) compared with one-loop perturbation theory (black). The continuum limit is a simple linear extrapolation in $a^2$. The right, $s=1$, point corresponds to 3 GeV.

Figure 18

Partially quenched light valence mass dependence of $B_K$ for the three (${32}^3$) $\mathbf{1}$ ensembles (left panel) and two (${24}^3$) $\mathbf{2}$ ensembles (right panel) at a valence strange-quark mass fixed to be the physical strange mass, and after reweighting in the heavier sea-quark mass to the physical strange mass. The overlaid curves are the partially quenched SU(2) chiral perturbation theory expressions used in our fits.

Figure 19

Partially quenched light valence mass dependence of $B_K$ for the three (${32}^3$) $\mathbf{1}$ ensembles (left panel) and two (${24}^3$) $\mathbf{2}$ ensembles (right panel) at a valence strange-quark mass fixed to be the physical strange mass, and after reweighting in the heavier sea-quark mass to the physical strange mass. The overlaid lines represent analytic fits to this data.

Figure 20

The continuum limit chiral extrapolation obtained from our global fits using NLO $\mathrm{SU}(2)$ PQChPT and LO analytic fits. The data is shown corrected to the continuum limit using the $\mathcal{O}(a^2)$ corrections obtained from both fit forms.

Figure 21

The continuum limit chiral extrapolation obtained from our global fits using NLO $\mathrm{SU}(2)$ PQChPT and LO analytic fits. As opposed to Fig. 20, the data plotted here has not been corrected to the continuum limit. The fit curves plotted are those performed to the continuum data as before.