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We present the calculation of the $K_{l3}$ form factors with $N_f = 2 + 1$ nonperturbatively $O(a)$-improved Wilson quark action and Iwasaki gauge action at the physical point on a large volume of (10.9 fm)$^3$ at one lattice spacing of $a = 0.085$ fm. We extract the form factors from 3-point functions with three different time separations between the source and sink operators to confirm suppression of excited state contributions. The form factors are calculated in very close to the zero momentum transfer, $q^2 = 0$, thanks to the large volume, so that stable interpolations to $q^2 = 0$ are carried out. Using our form factors, we obtain the form factor at $q^2 = 0$, $f_+(0) = 0.9603(16)(^{+14}_{\ -4})(44)(19)(1)$, where the first, second, and fifth errors are statistical, systematic errors from fit functions and the isospin breaking effect, respectively. The third and fourth errors denote the finite lattice spacing effects estimated from the renormalization factor and contribution beyond the leading order SU(3) chiral perturbation theory (ChPT). The result of $f_+(0)$ yields the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, $|V_{us}| = 0.2255(13)(4)$, where the first error comes from our calculation and the second from the experiment. This value is consistent with the ones determined from the unitarity of the CKM matrix and the $K_{l2}$ decay within one standard deviation, while it is slightly larger than recent lattice calculations by at most 1.5 $\sigma$. Furthermore, we evaluate the shape of the form factors and the phase space integral from our results. We confirm that those results are consistent with the experiment, and also $|V_{us}|$ determined with our phase space integral agrees with the one in the above.

We investigate the finite size effect on the vector meson and the baryon sectors using a subset of the "PACS10" configurations which are generated, keeping the space-time volumes over (10 fm$)^4$ in 2+1 flavor QCD at the physical point. Comparing the results on (5.5 fm$)^4$ and (10.9 fm$)^4$ lattices the ground states of octet baryons , which are stable on the lattice, show no finite size effect within less than 0.5% level of statistical errors. For those of vector mesons, which are unstable on the lattice, we observe that the effective masses are well below the experimental resonance levels both on (5.5 fm$)^4$ and (10.9 fm$)^4$ lattices. For the decuplet baryon sector we have found that the time dependence of the effective mass looks quite similar to that for the vector meson sector including the $\Omega$ baryon channel. We discuss its origin due to a possible mixing with the nearby multihadron states. Since the $\Xi$ baryon mass can be determined with the smallest ambiguity among the vector meson and the baryon masses, we use it together with the pion and kaon masses as the physical inputs to determine the physical point., Comment: 9 pages, 8 figures: published version

We investigate the finite size effect on pseudoscalar meson masses and decay constants using a subset of the "PACS10" configurations which are generated keeping the space-time volumes over (10 fm$)^4$ in 2+1 flavor QCD at the physical point. We have tried two kinds of analyses, fixing $\kappa$ values or measured axial Ward identity quark masses. Comparing the results on (5.4 fm$)^4$ and (10.8 fm$)^4$ lattices, we have found a sizable finite size effect on the pseudoscalar meson sector in the former analysis: a 2.1(8)%, 4.8(1.6)%, and 0.36(31)% finite size effect on $m_\pi$, $m_{\rm ud}$, and $f_\pi$, respectively, on the (5.4 fm$)^4$ lattice. For the latter analysis, the finite size effect on the pseudoscalar meson decay constants is 0.66(33)% for $f_\pi$, 0.26(13)% for $f_K$, and 0.40(32)% for $f_K/f_\pi$. These values with two-sigma error bars are consistent with the predictions from the full one-loop SU(3) chiral perturbation theory, which are 0.20% for $f_\pi$, 0.08% for $f_K$, and 0.13% for $f_K/f_\pi$. The finite size effect on the pseudoscalar meson masses is hardly detected under the current statistical precision., Comment: 7 pages, 10 figures, published version

We present our result for the $K\to\pi\pi$ 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 ${\bf p}=(0,0,2\pi/L)$ 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 $48^3\times 64$ ($La=4.4\,{\rm fm}$) 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 present our result for the $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ decay amplitudes for both the $\mathrm{\ensuremath{\Delta}}I=1/2$ and $3/2$ processes with the improved Wilson fermion action. Expanding on the earlier works by Bernard et al. and by Donini 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 ${m}_{K}\ensuremath{\sim}2{m}_{\ensuremath{\pi}}$. 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 $a=0.091\text{ }\text{ }\mathrm{fm}$, ${m}_{\ensuremath{\pi}}=280\text{ }\text{ }\mathrm{MeV}$, and ${m}_{K}=580\text{ }\text{ }\mathrm{MeV}$ on a ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ ($La=2.9\text{ }\text{ }\mathrm{fm}$) 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 $\mathrm{Re}{A}_{0}=60(36)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}8}\text{ }\text{ }\mathrm{GeV}$ and $\mathrm{Im}{A}_{0}=\ensuremath{-}67(56)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}12}\text{ }\text{ }\mathrm{GeV}$ for a matching scale ${q}^{*}=1/a$. The dependence on the matching scale ${q}^{*}$ for these values is weak.

We present the results of $1+1+1$ flavor $\mathrm{QCD}+\mathrm{QED}$ simulation at the physical point, in which the dynamical quark effects in QED and the up-down quark mass difference are incorporated by the reweighting technique. The physical quark masses together with the lattice spacing are determined with ${m}_{{\ensuremath{\pi}}^{+}}$, ${m}_{{K}^{+}}$, ${m}_{{K}^{0}}$ and ${m}_{{\ensuremath{\Omega}}^{\ensuremath{-}}}$ as physical inputs. Calculations are carried out using a set of $2+1$ flavor QCD configurations near the physical point generated by the nonperturbatively $O(a)$-improved Wilson quark action and the Iwasaki gauge action at $\ensuremath{\beta}=1.9$ on a ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ lattice. We evaluate the values of the up, down and strange quark masses individually with nonperturbative QCD renormalization.

Renormalization factors for bilinear and four-quark operators with the Kogut-Susskind fermion action are perturbatively calculated to one-loop order in the general covariant gauge. Results are presented both for gauge invariant and non-invariant operators. For four-quark operators the full renormalization matrix for a complete set of operators with two types of color contraction structures are worked out and detailed numerical tables are given., Comment: 37 pages, LaTeX with epsf.sty, UTHEP-261/July 1993

We perform a lattice QCD study of the $\ensuremath{\rho}$ meson decay from the ${N}_{f}=2+1$ full QCD configurations generated with a renormalization group improved gauge action and a nonperturbatively $O(a)$-improved Wilson fermion action. The resonance parameters, the effective $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ coupling constant and the resonance mass, are estimated from the $P$-wave scattering phase shift for the isospin $I=1$ two-pion system. The finite size formulas are employed to calculate the phase shift from the energy on the lattice. Our calculations are carried out at two quark masses, ${m}_{\ensuremath{\pi}}=410\text{ }\text{ }\mathrm{MeV}$ (${m}_{\ensuremath{\pi}}/{m}_{\ensuremath{\rho}}=0.46$) and ${m}_{\ensuremath{\pi}}=300\text{ }\text{ }\mathrm{MeV}$ (${m}_{\ensuremath{\pi}}/{m}_{\ensuremath{\rho}}=0.35$), on a ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ ($La=2.9\text{ }\text{ }\mathrm{fm}$) lattice at the lattice spacing $a=0.091\text{ }\text{ }\mathrm{fm}$. We compare our results at these two quark masses with those given in the previous works using ${N}_{f}=2$ full QCD configurations and the experiment.

We investigate the charm quark system using the relativistic heavy quark action on 2+1 flavor PACS-CS configurations previously generated on 323×64 lattice. The dynamical up, down, and strange quark masses are set to the physical values by using the technique of reweighting to shift the quark-hopping parameters from the values employed in the configuration generation. At the physical point, the lattice spacing equals a-1=2.194(10)??GeV and the spatial extent L=2.88(1)??fm. The charm quark mass is determined by the spin-averaged mass of the 1S charmonium state, from which we obtain mcharmMS? (μ=mcharmMS? )=1.260(1)(6)(35)??GeV, where the errors are due to our statistics, scale determination and renormalization factor. An additional systematic error from the heavy quark is of order αs2f(mQa)(aΛQCD), f(mQa)(aΛQCD)2, which are estimated to be a percent level if the factor f(mQa) analytic in mQa is of order unity. Our results for the charmed and charmed-strange meson decay constants are fD=226(6)(1)(5)??MeV, fDs=257(2)(1)(5)??MeV, again up to the heavy quark errors of order αs2f(mQa)(aΛQCD), f(mQa)(aΛQCD)2. Combined with the CLEO values for the leptonic decay widths, these values yield, Vcd, =0.205(6)(1)(5)(9), Vcs, =1.00(1)(1)(3)(3), where the last error is because of the experimental uncertainty of the decay widths.

We present a lattice QCD calculation of the $\rho$ meson decay width via the $P$-wave scattering phase shift for the I=1 two-pion system. Our calculation uses full QCD gauge configurations for $N_f=2$ flavors generated using a renormalization group improved gauge action and an improved Wilson fermion action on a $12^3\times24$ lattice at $m_\pi/m_\rho=0.41$ and the lattice spacing $1/a=0.92 {\rm GeV}$. The phase shift calculated with the use of the finite size formula for the two-pion system in the moving frame shows a behavior consistent with the existence of a resonance at a mass close to the vector meson mass obtained in spectroscopy. The decay width estimated from the phase shift is consistent with the experiment, when the quark mass is scaled to the realistic value., Comment: LaTeX2e, 16 pages, 6 eps figures, uses revtex4 and graphicx

We discuss a possibility that the Neutron Electric Dipole Moment (NEDM) can be calculated in lattice QCD simulations in the presence of the CP violating $\theta$ term. In this paper we measure the energy difference between spin-up and spin-down states of the neutron in the presence of an uniform and static external electric field. We first test this method in quenched QCD with the RG improved gauge action on a $16^3\times 32$ lattice at $a^{-1}\simeq$ 2 GeV, employing two different lattice fermion formulations, the domain-wall fermion and the clover fermion for quarks, at relatively heavy quark mass $(m_{PS}/m_V \simeq 0.85)$. We obtain non-zero values of NEDM from calculations with both fermion formulations. We next consider some systematic uncertainties of our method for NEDM, using $24^3\times 32$ lattice at the same lattice spacing only with the clover fermion. We finally investigate the quark mass dependence of NEDM and observe a non-vanishing behavior of NEDM toward the chiral limit. We interpret this behavior as a manifestation of the pathology in the quenched approximation., Comment: LaTeX2e, 51 pages, 43 figures, uses revtex4 and graphicx, References and comments added, typos corrected, accepted by PRD

We perform a nonperturbative determination of the $O(a)$-improvement coefficient ${c}_{\mathrm{SW}}$ and the critical hopping parameter ${\ensuremath{\kappa}}_{c}$ for ${N}_{f}=3$, 2, and 0 flavor QCD with the (RG) renormalization-group-improved gauge action using the Schr\"odinger functional method. In order to interpolate ${c}_{\mathrm{SW}}$ and ${\ensuremath{\kappa}}_{c}$ as a function of the bare coupling, a wide range of $\ensuremath{\beta}$ from the weak coupling region to the moderately strong coupling points used in large-scale simulations is studied. Corrections at finite lattice size of $O(a/L)$ turned out to be large for the RG-improved gauge action, and hence we make the determination at a size fixed in physical units using a modified improvement condition. This enables us to avoid $O(a)$ scaling violations which would remain in physical observables if ${c}_{\mathrm{SW}}$ determined for a fixed lattice size $L/a$ is used in numerical simulations.

Three-flavor QCD simulation with the Oa� -improved Wilson fermion action is made employing an exact fermion algorithm developed for an odd number of quark flavors. For the plaquette gauge action, an unexpected first-order phase transition is found in the strong coupling regime (� & 5:0) at relatively heavy quark masses (mPS=mV � 0:74-0:87). Strong metastability persists on a large lattice of size 12 3 � 32, which indicates that the transition has a bulk nature. The phase gap becomes smaller toward weaker couplings and vanishes at � ' 5:0, which corresponds to a lattice spacing a ' 0: 1f m. These results imply that realistic simulations of QCD with three flavors of dynamical Wilson-type fermions at lattice spacings in the range a � 0:1-0: 2f mare not possible with the plaquette gauge action. Extending the study to improved gauge actions, we do not observe evidence for first-order phase transition, at least within the � �; � � range we explored. This suggests the possibility that the phase transition either moves away or weakens with improved gauge actions. Possible origins of the phase transition are discussed.

We perform a non-perturbative determination of the O(a)-improvement coefficient c_SW for the Wilson quark action in three-flavor QCD with the plaquette gauge action. Numerical simulations are carried out in a range of \beta=12.0-5.2 on a single lattice size of 8^3x16 employing the Schr\"odinger functional setup of lattice QCD. As our main result, we obtain an interpolation formula for c_SW and the critical hopping parameter K_c as a function of the bare coupling. This enables us to remove O(a) scaling violation from physical observables in future numerical simulation in the wide range of \beta. Our analysis with a perturbatively modified improvement condition for c_SW suggests that finite volume effects in c_SW are not large on the 8^3x16 lattice. We investigate N_f dependence of c_SW by additional simulations for N_f=4, 2 and 0 at \beta=9.6. As a preparatory step for this study, we also determine c_SW in two-flavor QCD at \beta=5.2. At this \beta, several groups carried out large-scale calculations of the hadron spectrum, while no systematic determination of c_SW has been performed.

We present a lattice QCD calculation of phase shift including the chiral and continuum extrapolations in two-flavor QCD. The calculation is carried out for $I=2$ $S$-wave $\ensuremath{\pi}\ensuremath{\pi}$ scattering. The phase shift is evaluated for two momentum systems, the center of mass and laboratory systems, by using the finite-volume method proposed by L\"uscher in the center of mass system and its extension to general systems by Rummukainen and Gottlieb. The measurements are made at three different bare couplings $\ensuremath{\beta}=1.80$, 1.95 and 2.10 using a renormalization group improved gauge and a tadpole improved clover fermion action, and employing a set of configurations generated for hadron spectroscopy in our previous work. The illustrative values we obtain for the phase shift in the continuum limit are $\ensuremath{\delta}(\mathrm{deg} .)=\ensuremath{-}3.50(64)$, $\ensuremath{-}9.5(30)$ and $\ensuremath{-}16.9(64)$ for $\sqrt{s}(\mathrm{G}\mathrm{e}\mathrm{V})=0.4$, 0.6 and 0.8, which are consistent with experiments.

We present an unquenched lattice calculation for the B 0 -B 0 transition amplitude. The calculation, carried out at an inverse lattice spacing l/a = 2.22(4) GeV, incorporates two flavors of dynamical quarks described by theO(a)-improved Wilson fermion action and heavy quarks described by non-relativistic QCD. Particular attention is paid to the uncertainty that arises from the chiral extrapolation, especially the effect of pion loops, for light quarks, which we find could be sizable for the leptonic decay constant, whereas it is small for the B parameters. We obtain f B d = 191(10)( + 1 2 -22) MeV, f B s /f B d = 1.13(3)( + 1 3 -2), B B d (m b ) = 0.836(27)( + 5 6 -62), B B s /B B d = 1.017(16)( + 5 6 -17), and ξ = 1.14(3)( + 1 3 -2), where the first error is statistical, and the second is systematic, including uncertainties due to chiral extrapolation, finite lattice spacing, heavy quark expansion, and perturbative operator matching.

We explore application of the domain wall fermion formalism of lattice QCD to calculate the $K\to\pi\pi$ decay amplitudes in terms of the $K\to\pi$ and $K\to 0$ hadronic matrix elements through relations derived in chiral perturbation theory. Numerical simulations are carried out in quenched QCD using domain-wall fermion action for quarks and an RG-improved gauge action for gluons on a $16^3\times 32\times 16$ and $24^3\times 32\times 16$ lattice at $\beta=2.6$ corresponding to the lattice spacing $1/a\approx 2$GeV. Quark loop contractions which appear in Penguin diagrams are calculated by the random noise method, and the $\Delta I=1/2$ matrix elements which require subtractions with the quark loop contractions are obtained with a statistical accuracy of about 10%. We confirm the chiral properties required of the $K\to\pi$ matrix elements. Matching the lattice matrix elements to those in the continuum at $\mu=1/a$ using the perturbative renormalization factor to one loop order, and running to the scale $\mu=m_c=1.3$ GeV with the renormalization group for $N_f=3$ flavors, we calculate all the matrix elements needed for the decay amplitudes. With these matrix elements, the $\Delta I=3/2$ decay amplitude shows a good agreement with experiment in the chiral limit. The $\Delta I=1/2$ amplitude, on the other hand, is about 50--60% of the experimental one even after chiral extrapolation. In view ofthe insufficient enhancement of the $\Delta I=1/2$ contribution, we employ the experimental values for the real parts of the decay amplitudes in our calculation of $\epsilon'/\epsilon$. We find that the $\Delta I=3/2$ contribution is larger than the $\Delta I=1/2$ contribution so that $\epsilon'/\epsilon$ is negative and has a magnitude of order $10^{-4}$. Possible reasons for these unsatisfactory results are discussed.

We explore applications of the maximum entropy method (MEM) to determine properties of unstable particles using the four-dimensional O(4) $\phi^4$ theory as a laboratory. The spectral function of the correlation function of the unstable $\sigma$ particle is calculated with MEM, and shown to yield reliable results for both the mass of $\sigma$ and the energy of the two-pion state. Calculations are also made for the case in which $\sigma$ is stable. Distinctive differences in the volume dependence of the $\sigma$ mass and two-pion energy for the stable and unstable cases are analyzed in terms of perturbation theory.

~i! The definition of the improved axial current in Eq. ~24! in Sec. III B should be Am 5Am2cA1mP with our definition of the axial current Am5 qg5gmq . The sign of the O(a) improvement term used in the paper was incorrect. This error affects the numerical results of the AWI quark mass and the pseudoscalar decay constant at finite lattice spacings. Our conclusions about these observables, however, are unchanged as we see below. Results on other quantities, such as the hadron spectrum, are unaffected. Table XXI shows the corrected results for the AWI quark mass at simulation points in two-flavour QCD. Recalculated parameters of chiral extrapolations are given in Tables VII and XIV. Physical quark masses, mud and ms , at finite lattice spacings are summarized in Table XVI. Figures 29–32 show separate and combined continuum extrapolations of the quark masses. Combining the statistical and systematic error listed in Table XVII, the corrected results for the light quark masses are given by

We present our results of lattice calculations of $B$ parameters, which parameterize $\Delta B$=2 transition amplitudes together with the leptonic decay constant. Calculations are made in the quenched approximation at $\beta$=5.7, 5.9, 6.0 and 6.1, using NRQCD action for heavy quark and the $O(a)$-improved Wilson action for light quark. The operators are perturbatively renormalized including the correction of $O(\alpha_s/(aM)^m)$ ($m\ge$0). We examine the scaling behavior of $B$ parameters, and discuss the systematic uncertainties based on the results with several different truncations of higher order terms in 1/M and $\alpha_s$ expansions. We find $B_{B_d}(m_b)=0.84(3)(5)$, $B_{B_s}/B_{B_d}=1.020(21)(^{+15}_{-16})(^{+5}_{-0})$ and $B_{S_s}(m_b)=0.85(1)(5)(^{+1}_{-0})$ in the quenched approximation. The errors represent statistical and systematic as well as the uncertainty in the determination of strange quark mass.

Validity of perturbative estimation of renormalization factors in weak matrix element calculations in lattice QCD is examined for the ${\mathit{K}}^{0\mathrm{\ensuremath{-}}}$K${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{0}$ mixing matrix by comparing results for gauge invariant and noninvariant operators. A large disagreement found for uncorrected results for the two cases is shown to be removed by the one-loop renormalization factor. This indicates that the large scaling violation in the mixing matrix previously reported is not due to an artifact of prescription of lattice calculations. Our estimate of B${\mathrm{^}}_{\mathit{K}}$ for the continuum in quenched QCD is 0.61(13)--0.83(7).

We present a lattice QCD calculation of $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\pi}l\ensuremath{\nu}$ semileptonic decay form factors in the small pion recoil momentum region. The calculation is performed on a quenched ${16}^{3}\ifmmode\times\else\texttimes\fi{}48$ lattice at $\ensuremath{\beta}=5.9$ with the nonrelativistic QCD action including the full $1/M$ terms. The form factors ${f}_{1}(v\ensuremath{\cdot}{k}_{\ensuremath{\pi}})$ and ${f}_{2}(v\ensuremath{\cdot}{k}_{\ensuremath{\pi}})$ defined in the heavy quark effective theory for which the heavy quark scaling is manifest are adopted, and we find that the $1/M$ correction to the scaling is small for the B meson. The dependence of the form factors on the light quark mass and on the recoil energy is found to be mild, and we use a global fit of the form factors at various quark masses and recoil energies to obtain model independent results for the physical differential decay rate. We find that the ${B}^{*}$ pole contribution dominates the form factor ${f}^{+}{(q}^{2})$ for small pion recoil energy, and obtain the differential decay rate integrated over the kinematic region ${q}^{2}g18{\mathrm{GeV}}^{2}$ to be $|{V}_{\mathrm{ub}}{|}^{2}\ifmmode\times\else\texttimes\fi{}(1.18\ifmmode\pm\else\textpm\fi{}0.37\ifmmode\pm\else\textpm\fi{}0.08\ifmmode\pm\else\textpm\fi{}0.31){\mathrm{psec}}^{\ensuremath{-}1},$ where the first error is statistical, the second is that from perturbative calculation, and the third is the systematic error from the finite lattice spacing and the chiral extrapolation. We also discuss the systematic errors in the soft pion limit for ${f}^{0}{(q}_{\mathrm{max}}^{2})$ in the present simulation.

We present a calculation for the ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}$ decay amplitude using a quenched simulation of lattice QCD with the Wilson quark action at $\ensuremath{\beta}{=6/g}^{2}=6.1.$ The decay amplitude is extracted from the ratio, the $\stackrel{\ensuremath{\rightarrow}}{K}\ensuremath{\pi}\ensuremath{\pi}$ three-point function divided by either $K$ and $\ensuremath{\pi}$ meson two-point functions or $K$ meson two-point function and $I=2 \ensuremath{\pi}\ensuremath{\pi}$ four-point function; the two different methods yield consistent results. Finite size effects are examined with calculations made on ${24}^{3}\ifmmode\times\else\texttimes\fi{}64$ and ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ lattices, and are shown that they are explained by one-loop effects of chiral perturbation theory. The lattice amplitude is converted to the continuum value by employing a one-loop calculation of chiral perturbation theory, yielding a value in agreement with experiment if extrapolated to the chiral limit. We also report on the $K$ meson $B$ parameter ${B}_{K}$ obtained from the ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}$ amplitude using chiral perturbation theory.