Neutral kaon mixing from 2+1 flavor Domain Wall QCD

TL;DR

This work determines B_K for neutral kaon mixing using lattice QCD with 2+1 dynamical flavors and domain-wall fermions, leveraging good chiral symmetry to enable multiplicative, nonperturbative renormalization of the ΔS=2 operator. The calculation employs large volumes and light pions, uses SU(2)_L×SU(2)_R PQChPT to control the light-quark extrapolation, and achieves a 2+1 flavor result with controlled systematics. The reported values are B_K^{RI}(2 GeV)=0.514(10)(7), B_K^{MS}(2 GeV)=0.524(10)(28), and hat B_K=0.720(13)(37), with a carefully estimated error budget and a comparison to prior lattice determinations. The work demonstrates a substantial reduction in systematic errors via dynamical 2+1 flavors, good chiral symmetry, and nonperturbative renormalization, and outlines needed steps toward finer lattices and NNLO perturbative matching to further reduce discretization and matching uncertainties.

Abstract

We present the first results for neutral kaon mixing using 2+1 flavors of domain wall fermions. A new approach is used to extrapolate to the physical up and down quark masses from our numerical studies with pion masses in the range 240 -- 420 MeV; only $SU(2)_L \times SU(2)_R$ chiral symmetry is assumed and the kaon is not assumed to be light. Our main result is $B_K^{\bar{\rm MS}}(2 \mathrm{GeV}) = 0.524(10)(28)$ where the first error is statistical and the second incorporates estimates for all systematic errors.

Figures (4)

• Figure 1: Typical plateau for the lattice $B$-parameter for the pseudoscalar state made up from quarks of mass $a m^{\rm val}_{x}=0.001$, $a m^{\rm val}_y=0.04$, on the $a m^{\rm sea}_{l}=0.005$, $a m^{\rm sea}_s=0.04$, $24^3$ ensemble.
• Figure 2: Results for $B_P$ together with the NLO partially quenched $SU(2)_L\times SU(2)_R$ ChPT fit to the $24^3$ data plotted versus the light valence quark lattice mass $a m_x$. From top to bottom on the left-hand-side, the three curves are $a m_l$ = 0.01, 0.005 and $a m_x$ respectively. The valence strange quark mass is fixed at its unitary value $a m_y = a m_s = 0.04$. While the statistical errors are large, the growing upward curvature in $m_x$ as the sea quark mass is increased from 0.005 to 0.01 predicted by ChPT is visible. Some $m_x$ values are slightly shifted for clarity.
• Figure 3: A plot of $Z_{B_K}^\mathrm{RGI}(p^2)$ showing that the perturbative running, removed from $Z_{B_K}^\mathrm{RGI}$, accounts for most $p^2$ dependence.
• Figure 4: We compare our 2+1 flavor results with earlier quenchedAliKhan:2001wr and 2 flavor DWFAoki:2004ht as well as 2+1 flavor staggered calculationsGamiz:2006sq. The quenched Iwasaki points show statistical errors only while our point and the staggered point include renormalisation systematics. While our point lies below these Iwasaki results due to our improved chiral limit and flavor content, we expect similar $a^2$ dependence.