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Register a new userCalculation of $K to pipi$ decay amplitudes with improved Wilson fermion
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We present results of our trial calculation of the $K to pipi$ decay amplitudes with the improved Wilson fermion action. Calculations are carried out with $N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson fermion action at $a=0.091,{rm fm}$, $m_pi=280,{rm MeV}$ and $m_K=560,{rm MeV} (sim 2 m_pi)$ on a $32^3times 64$ ($La=2.9 {rm fm}$) lattice.
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We present our result for the $Ktopipi$ decay amplitudes for both the $Delta I=1/2$ and $3/2$ processes with the improved Wilson fermion action. Expanding on the earlier works by Bernard {it et al.} and by Donini {it et al.}, we show that mixings with four-fermion operators with wrong chirality are absent even for the Wilson fermion action for the parity odd process in both channels due to CPS symmetry. Therefore, after subtraction of an effect from the lower dimensional operator, a calculation of the decay amplitudes is possible without complications from operators with wrong chirality, as for the case with chirally symmetric lattice actions. As a first step to verify the possibility of calculations with the Wilson fermion action, we consider the decay amplitudes at an unphysical quark mass $m_K sim 2 m_pi$. Our calculations are carried out with $N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion action at $a=0.091,{rm fm}$, $m_pi=280,{rm MeV}$, and $m_K=580,{rm MeV}$ on a $32^3times 64$ ($La=2.9,{rm fm}$) lattice. For the quark loops in the penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver method work very well for variance reduction. We obtain, for the first time with a Wilson-type fermion action, that ${rm Re}A_0 = 60(36) times10^{ -8},{rm GeV}$ and ${rm Im}A_0 =-67(56) times10^{-12},{rm GeV}$ for a matching scale $q^* =1/a$. The dependence on the matching scale $q^*$ for these values is weak.
We present results for the $Ktopipi$ decay amplitudes for both the $Delta I=1/2$ and $3/2$ channels. This calculation is carried out on 480 gauge configurations in $N_f=2+1$ QCD generated over 12,000 trajectories with the Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson fermion action at $a=0.091,{rm fm}$, $m_pi=280,{rm MeV}$ and $m_K=580,{rm MeV}$ on a $32^3times 64$ ($La=2.9,{rm fm}$) lattice. For the quark loops in the Penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver techniques work very well for variance reduction. We obtain, for the first time with a Wilson-type fermion action, that ${rm Re}A_0 = 60(36) times10^{ -8},{rm GeV}$ and ${rm Im}A_0 =-67(56) times10^{-12},{rm GeV}$ for a matching scale $q^* =1/a$. The dependence on the matching scale is weak.
We present our result for the $Ktopipi$ decay amplitudes for both the $Delta I=1/2$ and $3/2$ processes with the improved Wilson fermion action. In order to realize the physical kinematics, where the pions in the final state have finite momenta, we consider the decay process $K({bf p}) to pi({bf p}) + pi({bf 0})$ in the nonzero momentum frame with momentum ${bf p}=(0,0,2pi/L)$ on the lattice. Our calculations are carried out with $N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion action at $a=0.091,{rm fm}$ ($1/a=2.176,{rm GeV}$), $m_pi=260,{rm MeV}$, and $m_K=570,{rm MeV}$ on a $48^3times 64$ ($La=4.4,{rm fm}$) lattice. For these parameters the energy of the $K$ meson is set at that of two-pion in the final state. We obtain ${rm Re}A_2 = 2.431(19) times10^{ -8},{rm GeV}$, ${rm Re}A_0 = 51(28) times10^{ -8},{rm GeV}$, and $epsilon/epsilon = 1.9(5.7) times10^{-3}$ for a matching scale $q^* =1/a$ where the errors are statistical. The dependence on the matching scale $q^*$ of these values is weak. The systematic error arising from the renormalization factors is expected to be around $1.3%$ for ${rm Re}A_2$ and $11 %$ for ${rm Re}A_0$. Prospects toward calculations with the physical quark mass are discussed.
We report a direct lattice calculation of the $K$ to $pipi$ decay matrix elements for both the $Delta I=1/2$ and 3/2 amplitudes $A_0$ and $A_2$ on 2+1 flavor, domain wall fermion, $16^3times32times16$ lattices. This is a complete calculation in which all contractions for the required ten, four-quark operators are evaluated, including the disconnected graphs in which no quark line connects the initial kaon and final two-pion states. These lattice operators are non-perturbatively renormalized using the Rome-Southampton method and the quadratic divergences are studied and removed. This is an important but notoriously difficult calculation, requiring high statistics on a large volume. In this paper we take a major step towards the computation of the physical $Ktopipi$ amplitudes by performing a complete calculation at unphysical kinematics with pions of mass 422,MeV at rest in the kaon rest frame. With this simplification we are able to resolve Re$(A_0)$ from zero for the first time, with a 25% statistical error and can develop and evaluate methods for computing the complete, complex amplitude $A_0$, a calculation central to understanding the $Delta =1/2$ rule and testing the standard model of CP violation in the kaon system.
We calculate the PCAC mass for $(2+1)$ flavor full QCD with Wilson-type quarks. We adopt the Small Flow-time eXpansion (SFtX) method based on the gradient flow which provides us a general way to compute correctly renormalized observables even if the relevant symmetries for the observable are broken explicitly due to the lattice regularization, such as the Poinc{a}re and chiral symmetries. Our calculation is performed on heavy $u, d$ quarks mass ($m_{pi}/m_{rho}simeq0.63$) and approximately physical $s$ quark mass with fine lattice $a simeq 0.07$~fm. The results are compared with those computed with the Schrodinger functional method.
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