For this calculation we employ a single lattice of size 323×64. We utilize 2+1 flavors of Möbius domain wall fermions with Ls=12 and Möbius parameters b+c=32/12 and b-c=1 and light and strange quark masses of 1×10-4 and 0.045, respectively. We use the Iwasaki+DSDR gauge action with β=1.75, corresponding to an inverse lattice spacing of a-1=1.3784(68)  GeV. The dislocation suppressing determinant ratio (DSDR) term [18] is a modification of the gauge action that suppresses the dislocations, or tears in the gauge field that enhance chiral symmetry breaking at coarse lattice spacings. This enables the calculation to be performed with larger lattice spacings, and hence larger physical volumes, at fixed computational cost, ensuring good control over finite-volume systematic errors. We use GPBC in three spatial directions in order to obtain nearly physical kinematics for our K→ππ decays.

The lattice parameters are equal to those of the 32ID ensemble documented in Refs. [19,20] except for the boundary conditions and the fact that we now simulate with a lighter, physical pion mass of 142 MeV versus the 170 MeV pion mass of the 32ID ensemble. This allows the use of existing measurements such as the lattice spacing, and also enables the computation of the nonperturbative renormalization factors in a regime free of the complexities associated with GPBC.

The ensemble used for our 2015 calculation comprised 864 molecular dynamics (MD) time units (after thermalization), upon which 216 measurements were performed separated by 4 MD time units. Subsequent to the calculation, it was discovered [21] that an error existed in the generation of the random numbers used to set the conjugate momentum at the start of each trajectory, which gave rise to small correlations between widely separated lattice sites. While the resulting effects were determined to be 2-to-3 orders of magnitude smaller than our statistical errors, we nevertheless do not include these configurations in the present calculation.

In the period following our previous publication, we have dramatically increased the number of measurements. Configurations were generated on seven independent Markov chains originating from widely separated configurations of our original ensemble. Subsequent algorithmic improvements, particularly the introduction of the exact one-flavor algorithm (EOFA) [22–24] further enhanced our rate of generation such that we have completed over 5000 additional MD time units to date.

Continuing with a measurement separation of 4 MD time units, we can potentially perform almost 1300 measurements in total. For this analysis, we include measurements on ∼60% of the available configurations, totaling 741. We aim to provide updated results containing measurements on the remaining portion in a future publication. For further information on the ensemble properties, generation algorithms, and details of the configurations used for this analysis we refer the reader to Ref. [17].

Aside from the central values of our fit parameters we must also estimate the standard error and the goodness of fit. These are obtained via bootstrap resampling, specifically the nonoverlapping block bootstrap variant [25] which allows us to account for mild autocorrelation effects observed in our data. A block size of 8 is used.

The bootstrap measurement of the goodness of fit is a technique developed specifically for this and our companion work [17], and is detailed in Ref. [26]. To summarize, the goodness of fit is typically parametrized by a p value that represents the likelihood that the data agree with the model, allowing only for statistical fluctuations. The p value is computed by first measuring q2=∑i,j(x¯i-f(p→,i))(cov)ij-1(x¯j-f(p→,j)),(3)where x¯i are the ensemble means of the data at coordinate i, p→ are the fitting parameters, f is the model function, and (cov)ab is the covariance matrix. The value obtained for q2 is then compared to the null distribution that describes how this quantity varies between independent experiments if only statistical fluctuations are allowed around the model. The null distribution is typically assumed to be the χ2 distribution, but this is inappropriate when the fluctuations in the covariance matrix between experiments become significant, as is the case for our ππ measurements [26]. In that work we demonstrate that the null distribution can be estimated directly from the data through a simple bootstrap procedure, allowing for a more reliable p value that is free from assumptions. This procedure also has the benefit of allowing us to neglect the autocorrelations in the determination of the covariance matrix on each bootstrap ensemble, which dramatically improves the statistical error but changes the definition of q2 in a subtle way that cannot be accounted for by traditional methods.

Measurements are performed using the all-to-all (A2A) propagator technique of Ref. [27], whereby the quark propagator is decomposed into an exact low-mode contribution obtained from a set of, in our case 900, predetermined eigenvectors and a stochastic approximation to the high-mode contribution. This allows for the maximal translation of correlation functions in order to take full advantage of each configuration, as well as easy implementation of arbitrarily smeared source and sink operators. We perform full spin, color, flavor, and time dilution such that the stochastic source is required only to produce a delta function in the spatial location.

For all quantities we use smeared meson sources with an exponential (1s hydrogen wave function–like) structure, Θ(|x→-y→|)=exp(-|x→-y→|/α),(4)where α=2 is the smearing radius and x→ and y→ are the spatial coordinates of the two quark operators. Several technicalities must be considered when using G-parity boundary conditions, including limitations on the allowed quark momenta which has implications for the cubic rotational symmetry, the preservation of which is essential for producing an operator that projects onto the rotationally symmetric (s-wave) ππ state. These are detailed in Ref. [17].

More specific details of the various measurements are provided in the following sections.

G-parity boundary conditions mix quark flavor at the boundary, introducing additional Wick contractions in which a quark propagates through the boundary and is annihilated by an operator of the opposite quark flavor. In Ref. [10] we introduced a notation whereby the quark field and its G-parity partner are placed in a two-component vector, ψl=(dCu¯T),(5)where C is the charge conjugation matrix. We will refer to the index of these vectors as a “flavor index.” In this notation the propagator becomes a 2×2 “flavor matrix,” and Pauli matrices inserted appropriately describe the flavor structure. In this notation the Wick contractions assume an almost identical form to those of the periodic case.

The strange quark is introduced into the G-parity framework as a member of an isospin doublet that includes a fictional degenerate partner, s′, into which the strange quark transforms at the boundary. The corresponding field operator is ψh=(sCs¯′T).(6)With the introduction of this extra quark flavor a square root of the s/s′ determinant is required in order to generate a 2+1 flavor ensemble [10].

Following Ref. [10], a stationary (G-parity even) kaonlike state can be constructed as |K˜0⟩=12(|K0⟩+|K′0⟩),(7)where K0 is the physical kaon and K′0 a degenerate partner with quark content s¯′u. This |K˜0⟩ state can be created using the following operator: OK˜0(t)=i2∑x→,y→eip→·(x→-y→)ψ¯l(x→,t)γ5Θ(|x→-y→|)⁢12(1+σ2)ψh(y→,t),(8)where p→=(1,1,1)π2L is the quark momentum and Θ is defined in Eq. (4). Note that in the above equation and for the other operators presented in this document, the projection operators 12(1±σ2) appear; these are necessary to define quark field operators that are eigenstates of translation and hence have definite momentum [10].

The two-point function CK(t1,t2)=⟨0|OK˜0†(t1)OK˜0(t2)|0⟩(9)is measured for all t1 and t2, and subsequently averaged over t2 at fixed t=t1-t2. The data are folded in t; i.e., data with t=t1-t2 are averaged with those with t=LT-(t1-t2), where LT is the lattice temporal extent, to improve statistics. We perform correlated fits to the following function: CK(t)=AK(e-mKt+e-mK(LT-t)),(10)where the second term accounts for the state propagating backwards in time through the lattice temporal boundary. The chosen fit range, the p value, and the results of the fit are given in Table II. In physical units our kaon mass is 490.5(2.4) MeV, which is within 2% of the physical neutral kaon mass.

The isospin triplet of pion states can be constructed from the operators listed in Sec. V.A. of Ref. [10]. Due to the isospin symmetry the resulting two-point functions all have the same Wick contractions and are most conveniently generated with the neutral pion operator, Oπ(p→π,t)=-i2∑x→,y→ei(p→1·x→+p→2·y→)ψ¯l(x→,t)γ5σ3Θ(|x→-y→|)⁢Pp→πψl(y→,t),(11)where p→π=p→1+p→2 is the total pion momentum and Pp→π is a flavor projection operator of the form 12(1±σ2) whose sign depends on the particular choice of the quark momentum, per the discussion in Sec. IV.G. of Ref. [10]. We measure the two-point function with four different momentum orientations related by cubic transformations in order to improve the statistical error: p→π∈{(1,1,1),(-1,1,1),(1,-1,1),(1,1,-1)} in units of π/L. The corresponding choices of quark momentum are given in Ref. [17]. The two-point function Cπ(p→π;t1,t2)=⟨0|Oπ†(p→π,t1)Oπ(p→π,t2)|0⟩(12)is again averaged over all source time slices and also over all four momentum orientations, and the data folded to improve statistics. Correlated fits are performed to the function, Cπ(t)=Aπ(e-Eπt+e-Eπ(LT-t)),(13)where t=t1-t2 as before. The chosen fit range, the p value, and the results of the fit are also given in Table II. In physical units, and assuming the continuum dispersion relation, our pion mass is 142.3(8) MeV, approximately 5% larger than the physical value of 135 MeV. The small effect of this difference on our final results is expected to be negligible in comparison to our other errors.

Details of the strategy for measuring the I=0 ππ two-point function can be found in Ref. [17]. In summary, we construct three operators with the quantum numbers of the I=0 ππ state: The first and second operators, labeled ππ(111) and ππ(311), comprise two single-pion operators carrying equal and opposite momenta separated by Δ=4 time slices in order to reduce the overlap with the vacuum state. The pion momenta in the former reside in the set (±1,±1,±1)π/L, and those of the latter in the set (±3,±1,±1)π/L and permutations thereof. We average over all nonequivalent directions of the pion momentum in order to project onto the rotationally symmetric state. The final, σ operator corresponds to the scalar two-quark operator 12(u¯u+d¯d). As mentioned previously, the pion and σ bilinear operators are smeared with a hydrogen wave function (exponential) smearing function of radius 2 lattice sites in order to improve their overlap with the lowest-energy states.

Two-point correlation functions are constructed from pairs of source and sink operators; thus, Cαβππ(t1,t2)=⟨0|Oα†(t1)Oβ(t2)|0⟩-⟨0|Oα†(t1)|0⟩⟨0|Oβ(t2)|0⟩,(14)where we include an explicit vacuum subtraction. Here t1 specifies the earliest time in which any fermion operator appearing in the annihilation operator Oα† is evaluated, and likewise t2 is the latest time appearing in the creation operator Oβ, such that t=t1-t2 is the time of propagation of the shortest-lived pion state. We average over many t2 at fixed t=t1-t2, and the data are folded to improve statistics as follows: Cαβππ(t)→12[Cαβππ(t)+Cαβππ(LT-Δi-Δj-t)],(15)where Δi=4 for the ππ(111) and ππ(311) operators and zero for the σ operator. To the matrix of correlation functions we perform simultaneous correlated fits to the functions, Cαβππ(t)=∑i=0imaxAαiAβi(e-Eit+e-Ei(LT-Δi-Δj-t)).(16)

We will use the result obtained by uniformly fitting to the temporal range 6–15 with all three ππ source/sink operators and allowing for two intermediate states (imax=1), which represents the “best fit” in Ref. [17]. The results, reproduced from that work are given in the second column of Table III for the convenience of the reader. Note that our ππ and kaon energies differ by 2.2(3)%, where the error is statistical only, and as such our K→ππ calculation is not precisely energy conserving. The effect of this difference is incorporated as a systematic error on our final result, as discussed in Sec. VII B.

It is interesting to compare the statistical errors of our ππ ground-state fit parameters to those of our 2015 analysis, which was performed using a single operator [ππ(111)] and the same tmin=6 as our present analysis. Previously we obtained Aππ(111)0=0.3923(60),E0=0.3606(74).(17)Comparing these to the results of this work in Table III we find that the error on the ground-state amplitude has reduced by a factor of 1.9 and the energy by a factor of 6.7. The former is compatible with the expected 741/216=1.9 reduction in errors due to the increased statistics, but the latter has improved by a far greater amount. In Ref. [17] we demonstrate that this improvement in the errors is a result of the additional operators, in particular the σ operator, which vastly enhance the resolution on the ground-state energy.

The I=0 ππ scattering phase shift is obtained via Lüscher’s method [11,28] and has the value δ0=32.3(1.0)(1.8)°,(18)where the errors are statistical and systematic, respectively. The procedures by which we estimate our errors are detailed in Ref. [17].

Our decision to fit the ππ two-point function with two states limits the number of states that we can include in our K→ππ matrix element analysis. In order to study the possibility of residual contamination from a third state we repeat the analysis of the ππ two-point function with three states, the results of which are given in the third column of Table III. For a stable fit to the ππ data we found it necessary to use tmin=4, which is lower than the tmin=6 used for the primary fit and which exposes the result to enhanced excited state contamination. However, comparing the results between the second and third columns of Table III we find little relative difference in the parameters associated with the ground state, suggesting any such effects on the K→ππ matrix elements are small.

As detailed in Sec. VI A, the Lellouch-Lüscher finite volume correction to the K→ππ matrix elements requires the evaluation of the derivative of the phase shift with respect to the ππ energy evaluated at the kaon mass scale, or more specifically with respect to the variable q=kL/2π where k2=(Eππ/2)2-mπ2 is the square of the interacting pion momentum. This derivative cannot presently be obtained experimentally at this energy scale, and therefore an interpolating ansatz or direct lattice measurement is required.

In Ref. [17], alongside the stationary state examined above, we also compute the ππ energy at several nonzero center-of-mass momenta, allowing us to obtain the phase shift at two values of the rest-frame energy that are lower than the kaon mass as well as a threshold determination of the scattering length. These results are also close to their corresponding dispersive predictions, albeit with somewhat larger excited-state systematic errors. Using these results we can directly measure the derivative of the phase shift with respect to the energy using a finite-difference approximation, for which we obtain dδ0dq=1.76(74)  rad(19)from the difference with the nearest energy to the kaon mass, and dδ0dq=1.33(17)  rad(20)from the next to nearest.

We can also obtain the derivative from the dispersive prediction of Colangelo et al. [16]. The derivative with respect to s=Eππ2, computed at our lattice ππ energy using Eqs. (17.1)–(17.3) of Ref. [16] with mπ=135  MeV, is found to be dδ0ds=3.36(3)×10-6  rad MeV-2,(21)where the error is the statistical error arising from the uncertainty in the lattice spacing and measured lattice ππ energy. Note that this result is obtained at the physical pion mass, which is 5% smaller than our lattice value. In order to estimate the impact of the difference in pion masses on this derivative we use NLO chiral perturbation theory [16,29] (ChPT) to estimate the derivative with respect to energy at k=0.1  GeV, at which ChPT is expected to be reliable. Assuming that the slope with respect to s is roughly constant (which is well motivated by the dispersion theory result; cf. Fig. 7 of Ref. [16]), we estimate the change in dδ0ds evaluated at our lattice ππ energy as 1.2%. This value is small relative to the final systematic error we assign to the derivative in Sec. VII D and can therefore be neglected here. Finally, applying ds/dq=4.18(5)×105  MeV2, where again the errors are statistical, we obtain dδ0dq=1.405(5)  rad.(22)

The near linearity of the dispersive prediction suggests that a linear ansatz, dδ0dEππ≈δ0Eππ-2mπ,(23)may also be appropriate. With this ansatz we find dδ0dq=1.259(36)  rad.(24)

Given that the derivative of the phase shift is a subleading contribution and that the above values are all in reasonable agreement, we expect that the Lellouch-Lüscher factor can be obtained reliably. The variation in these results will be taken into account in our systematic error in Sec. VII D.

In our 2015 work [1] we also considered a linear ansatz in q, dδ0dq≈δ0q,(25)for which we obtain dδ0dq=0.790(22)  rad.(26)This value is not as well motivated as the ansatz in Eq. (24) and is in disagreement with all four of the above results. Given the good agreement between our measured phase shifts and the above estimates of the derivative with the dispersive predictions, we will not include this result in our systematic error estimate.

For use later in this document we define here an optimal operator that maximally projects onto the ππ ground state relative to the first-excited state.

Under the excellent assumption that the backwards-propagating component of the time dependence is small in the fit window, the two-point functions can be described as a sum of exponentials: Cαβππ(t)=∑iAαiAβie-Eit,(27)where again Greek indices denote operators and Roman indices states. We wish to define an optimized operator that projects onto the ground state: Oopt=∑αOαrα,(28)for which Coptππ(t)=⟨0|Oopt†(t)Oopt(0)|0⟩≈[Aopt0]2e-E0t,(29)where the approximate equality indicates that additional exponential terms resulting from excited-state contamination, although suppressed, still exist for an optimal operator composed of a finite number of ππ operators. Expanding the Green’s function, ⟨0|Oopt†(t)Oopt(0)|0⟩=∑αβrα⟨0|Oα†(t)Oβ(0)|0⟩rβ=∑i∑αβrαAαiAβirβe-Eit=∑i[∑αAαirα]2e-Eit.(30)Without loss of generality we can fix Aopt0=1, which alongside Eq. (29) is sufficient to define ri: ∑αAαirα=δi,0.(31)If the number of states is equal to the number of operators, this can be interpreted as a matrix equation, Ar→=0^,(32)where the row index of A is the state index i and the column index the operator index α. Here 0^ is a unit vector in the 0 direction, and as such r→=A-10^,(33)which gives rα=[A-1]α,0;(34)i.e., r→ is the first column of the inverse matrix.

As our ππ fits include only two states, we drop the noisier ππ(311) operator in order to form a square matrix of correlation functions. We then obtain r→T=(5.24(18)×10-7,-2.86(17)×10-4),(35)where the elements are the coefficients of the ππ(111) and σ operators, respectively. In Fig. 1 we compare the effective energy obtained with the optimal operator to that of the ππ(111) and σ operators alone. We observe a marked reduction in the ground-state energy and a noticeable improvement in the length of the plateau region resulting from the removal of excited-state contamination, as well as a significant improvement in the statistical error. This optimal operator will also be used in our matrix element fits in the following section.

On the lattice we measure the following three-point functions: Ci(t,tsepK→snk)=⟨0|Osnk†(tsepK→snk)Qi(t)OK˜0(0)|0⟩,(36)where t denotes the time separation between the kaon and four-quark operators, and tsepK→snk the time separation between the kaon and the ππ “sink” operator, Osnk. As described in Ref. [30], the Wick contractions of these functions fall into four categories based on their topology, as illustrated in Fig. 2.

Note that here and below we take care to differentiate between the G-parity kaon state K˜0, which is a G-parity even eigenstate of the finite-volume Hamiltonian, and the physical kaon K0 that is not an eigenstate of the system. The matrix elements of the physical kaon are related to those of the G-parity kaon by a constant multiplicative factor of 2 that serves as the analogue of the Lellouch-Lüscher finite-volume correction as described in Sec. VI.B. of Ref. [10].

In order to maximize statistics we translate the three-point function over multiple kaon time slices and average the resulting measurements. As the statistical error is dominated by the type3 and type4 diagrams these are measured with kaon sources on every time slice, 0≤tK<LT. The far more precise type1 and type2 contributions are measured every eighth time slice in order to reduce the computational cost. For the remainder of this section we will assume all correlation functions to have been averaged over the kaon time slice where appropriate.

We compute each diagram with five different time separations between the kaon and the ππ sink operators, tsepK→snk∈{10,12,14,16,18}, with the ΔS=1 four-quark operator inserted on all intervening time slices. Note these five time separations specify the time between the kaon operator and the closest single-pion factor in the ππ operator for those cases when the ππ operator is a product of single-pion operators evaluated on different time slices. (This convention of specifying the minimum time separation from those ππ operators which are nonlocal in the time is followed throughout this paper.) As these ππ operators comprise back-to-back moving pions with zero total momentum, we must measure each diagram for all possible orientations of the pion momenta in order to project onto the rotationally symmetric state.

The type3 and type4 diagrams both contain a light or strange quark loop beginning and ending at the operator insertion point that results in a quadratic divergence regulated by the lattice cutoff. This divergence is removed by defining the subtracted operators [30,31], Qi→Qi-αis¯γ5d.(37)We will henceforth denote the unsubtracted operator with a hat notation, Q^i. The coefficients αi in Eq. (37) are defined by imposing the condition, ⟨0|{Q^i(t)-αi(t)[s¯γ5d](t)}OK˜0(0)|0⟩=0,(38)where we have allowed αi to vary with time as this was found to offer a minor statistical improvement. Although the matrix element of this pseudoscalar operator vanishes by the equations of motion for energy-conserving kinematics and is therefore not absolutely necessary for our calculation, the subtraction reduces the systematic error resulting from the small difference between our ππ and kaon energies while simultaneously reducing the statistical error and suppressing excited-state contamination.

Due to having vacuum quantum numbers, the I=0 ππ operators project also onto the vacuum state and this off-shell matrix element dominates the signal unless an explicit vacuum subtraction is performed, Ci(t,tsepK→snk)→Ci(t,tsepK→snk)-⟨0|Osnk†(tsepK→snk)|0⟩⟨0|Qi(t)OK˜0(0)|0⟩.(39)However, due to our definition of the subtraction coefficient αi in Eq. (38), the vacuum matrix elements appearing in the right-hand side vanish making this subtraction unnecessary. In practice this cancellation is not exact in our numerical analysis for the following reason: While the ππ “bubble” ⟨0|Osnk†|0⟩ is formally time-translationally invariant we observed a minor statistical advantage in evaluating this quantity with the ππ operator on the same time slice as it appears in the full disconnected Green’s function that is being subtracted, such that it is maximally correlated. Therefore, for the rightmost term in Eq. (39) we compute 1ntK∑tK∈{tK}⟨0|Osnk†(tK+tsepK→snk)|0⟩⟨0|⁢{Q^i(t+tK)-αi(t)[s¯γ5d](t+tK)}OK˜0(tK)|0⟩,(40)where tK is the kaon time slice and {tK} the set of time slices upon which measurements were performed, i.e., with the product of the K→vacuum matrix element and the ππ bubble performed under the average over the kaon source time slice rather than after. As suggested by the above, the coefficients αi(t) are computed separately from the tK-averaged matrix elements, and therefore the cancellation between the two terms in brackets is exact only up to the degree to which the time translation symmetry is realized at finite statistics. Due to our large statistics we found the difference in the fitted Q6 matrix element obtained with and without the vacuum subtraction to be at the 0.1% level.

We perform measurements with all three two-pion operators described in Sec. III D. For the K→ππ matrix elements of the four-quark operators, the full set of Wick contractions for the ππ(111) and ππ(311) sink operators can be found in Appendixes B.1 and B.2 of Ref. [32], and those of the σ operator in Appendix A of this document. The Wick contractions for the K→ππ matrix elements of the pseudoscalar operator (with all three sink operators) as well as the K→vacuum matrix elements of this and the four-quark operators are provided in Appendix B of this document.

In Fig. 3 we plot the contributions of the four classes of Wick contraction illustrated in Fig. 2 to the three-point functions of the (subtracted) Q2 and Q6 operators with the ππ(111) sink operator. As the individual topologies are not separately interpretable as Green’s functions of the QCD path integral, their time dependence is not necessarily described by the propagation of physical eigenstates of the QCD Hamiltonian. As such we cannot combine our datasets with different tsepK→snk when generating such plots, and instead plot with a single, fixed tsepK→snk=16. Despite the inability to interpret the time dependence physically, we can look at the relative contributions of each topology within the central region of the plot in which the behavior of the combined data is dominated by the kaon and ππ ground states, i.e., the region in which we perform our fits below. Our final choices of cut incorporate data from this set in the range 6≤t≤11 (cf. Sec. IV E 4). In this window we observe that for both the C2 and the C6 correlation functions, the contribution of the noisy, type4 disconnected diagrams is largely consistent with zero, albeit with much larger errors for the former. C2 appears dominated by the type1 and type3 diagrams, which both contribute with the same sign, with a negligible contribution from the type2 diagrams. The contribution of the type1 and type3 diagrams appears to behave similarly for the C6 three-point function; however, here we observe a strong cancellation between those and the type2 diagrams.

The subtraction coefficients αi are computed via Eq. (38) as the following ratio of two-point functions, αi(t)=⟨0|Q^i(t)OK˜0(0)|0⟩⟨0|[s¯γ5d](t)OK˜0(0)|0⟩,(41)where the average of the correlation functions over the kaon source time slice is implicit as above.

The Wick contractions for the ⟨0|Q^i(t)OK˜0(0)|0⟩ two-point functions are identical to the components of the type4 K→ππ diagrams that are connected to the kaon. While these connected components are formally independent of the sink two-pion operator, in practice these quantities were computed using code that was organized differently for the ππ and σ operators. As described in Appendix B of this paper and Appendix B.2 of Ref. [32], the factors entering the type4 diagrams that determine the αi were constructed from two separate bases of functions of the quark propagators, one for the σ and the other for the ππ(⋯) operators, where for each basis γ5 Hermiticity was used in a different way. While γ5 Hermiticity is an exact relation, the fact that we are using a stochastic approximation for the high modes of the all-to-all propagator allows small differences to arise between the values of the αi computed in these two bases. We therefore have separate results for the αi from the ππ and σ three-point functions calculations.

In Fig. 4 we plot the time dependence of the αi for all ten operators. We observe excellent agreement between the results obtained from the two different bases of contractions as expected. For a number of operators we find statistically significant but relatively small excited-state contamination for small t that in all cases appears to die away by t=6. While the effects of this contamination are unlikely to significantly affect our final results, the cuts that we later apply to our fits nevertheless exclude data with t<6.

The K→ππ matrix elements of the pseudoscalar operator s¯γ5d are required to perform the subtraction of the divergent loop contribution. In this section we independently analyze these matrix elements in order to understand their time dependence and the corresponding effect of the subtraction on the amount of excited state contamination in the final K→ππ result.

In the limit of large time separation between the source/sink operators and the four-quark operator, only the lowest-energy ππ and kaon states are present. Since the pseudoscalar matrix elements vanish by the equations of motion when the decay conserves energy and the kaon and ππ ground-state energies in our calculation differ by only 2%, we expect the subtraction to result in only a negligible shift in the central value but a marked improvement in the statistical errors in this limit. However, at finite time separations, the contributions of the excited states may take a long time to die away due to the increasing magnitude of the corresponding matrix elements between initial and final states of different energies. It is this concern that prompts us to study this system more carefully.

The lattice three-point function CP(t,tsepK→snk)=⟨0|Osnk†(tsepK→snk)[s¯γ5d](t)OK˜0(0)|0⟩(42)for a generic sink ππ operator, Osnk, has the following time dependence: CP(t,tsepK→snk)=∑ijAiniAoutjMPijexp(-Einit)exp(-Eoutj(tsepK→snk-t)),(43)where the subscript “in” refers to the incoming kaonic state, “out” refers to the outgoing two-pion state, and MPij is the matrix element for the term involving in and out states i and j, respectively. It is convenient to define an “effective matrix element” by dividing out the ground-state time dependence and operator amplitudes, MPeff,snk(t′,tsepK→snk)=MP00+∑i,j≠0Ain′iAout′jMPijexp(-ΔEini(tsepK→snk-t′))⁢exp(-ΔEoutjt′),(44)where t′=tsepK→snk-t(45)is the separation between the four-quark operator and the sink and (46)Ain/out′i=Ain/outi/Ain/out0,(46a)ΔEin/outi=Ein/outi-Ein/out0.(46b)Note that MPeff,snk is dependent on the sink operator through the terms involving the excited states, in which a ratio of ground and excited state amplitudes appears.

We measure the correlation function Eq. (42) for each of our three two-pion operators. Note that a vacuum subtraction is also required here and is performed in the same way as for the four-quark operators. In Fig. 5 we plot MPeff,snk for the ππ(111) and σ operators for each of the five values of tsepK→snk. The corresponding data for the ππ(311) operator are much noisier and have therefore been excluded. The form of this plot can be explained as follows: As Ein0≈Eout0 we expect MP00 to be small. If we then assume that the dominant excited state contributions come from the term involving the excited kaon state and ground ππ state (i=1, j=0) and the term with the ground kaon state and the first excited ππ state (i=0, j=1), then we expect the data to behave as MPeff,snk(t′,tsepK→snk)≈Ain′1MP10exp(-ΔEin1tsepK→snk)exp(+ΔEin1t′)+Aout′1MP01exp(-ΔEout1t′).(47)This ansatz then implies an exponentially falling contribution from the excited pion state and an exponentially growing piece from the excited kaon state, giving rise to a bowl-like shape assuming that Ain′1 and Aout′1 have the same sign, which appears to be the case here. Furthermore, the exponentially growing piece in t′ is expected to be larger for smaller tsepK→snk, and indeed we observe that the turnover point at which the exponentially growing term begins to dominate occurs sooner for smaller tsepK→snk.

While the effective matrix elements of both sink operators initially trend toward zero, for the more precise ππ(111) data it seems that none of the five datasets are statistically consistent with zero at their maxima, suggesting we do not reach the limit of ground-state dominance. This is not necessarily an issue for our calculation given that the subtraction will heavily suppress these contributions in our final result, and furthermore the inclusion of multiple sink operators will improve our ability to extract the ππ ground-state matrix element. In order to disentangle these two effects it is convenient to examine the three-point function for the optimized sink operator discussed in Sec. III F. The time dependence of MPeff,snk for this operator is also shown in Fig. 5. By definition this operator heavily suppresses Aout′j for j>0, and indeed we find the data to be much flatter in the low-t′ region and also considerably closer to zero. The exponential growth and tsepK→snk dependence that enter due to the excited kaon term are expected to be largely unaffected by this transformation; however, it seems that in several cases the plateaus extend much further into the large-t′ region than previously. It is likely that is due to an accidental cancellation owing to the fact that Aout′1 is positive for the ππ(111) operator and negative for the σ operator (cf. Table III), and hence the exponentially growing terms for these operators have opposite signs.

We conclude by discussing the expected size of the excited-state contamination in the matrix elements of the subtracted four-quark operators arising from the pseudoscalar operator. In the K→ππ calculation, this dimension-3 operator is introduced to remove what in the continuum limit would be a quadratic divergence resulting from the self-contraction between two of the four quark operators appearing in those operators Q^i with a component transforming in the (8,1) or (8,8) representations of SU(3)L×SU(3)R. In our lattice calculation these terms behave as 1/a2 when expressed in physical units. To leading order in a this 1/a2 coefficient does not depend on the external states and is therefore removed from our ⟨0|(ππ)Q^iK˜0|0⟩ amplitude by the subtraction defined above, even though the coefficients αi are determined from the ⟨0|Q^iK˜0|0⟩ matrix element in Eq. (41). Because of the chiral structure of the (8,1) and (8,8) operators, these coefficients have the structure: αi∼ms-mda2+⋯ [8], where the ellipsis represents terms which are not power divergent.

Thus, the s¯γ5d subtraction removes the leading 1/a2 term in the matrix element of Q^i, leaving behind a finite piece of size ∼(ms-md)ΛQCD2s¯γ5d. This remainder is not physical and depends on the condition chosen to define the αi. However, it will contribute to our final result if Eππ≠mK. For the ground-state component (i=0, j=0) this term is thus heavily suppressed by the factor (Eππ0-mK). However, for the excited states we expect this piece to be on the order of the physical contribution from the dimension-six four-quark operator. As such it may result in a modest enhancement of the excited state matrix elements. Providing we are able to demonstrate that we have the excited ππ and kaon states under control through appropriate cuts on our fitting ranges, this should pose no obstacle to our calculation.

For a lattice of sufficiently large time extent that around-the-world terms in which states propagate through the lattice temporal boundary can be neglected, and assuming that the four-quark operator is sufficiently separated from the kaon source that the kaon ground state is dominant, the three-point Green’s functions Ci of the weak effective operators defined in Eq. (36) have the general form, Ci(t,tsepK→snk)=∑j12AKAsnkje-mKtMije-Ej(tsepK→snk-t),(48)where Mij=⟨(ππ)j|Qi|K0⟩ is the matrix element of the four-quark operator Qi with the ππ state j, with Mi0 corresponding to the physical K→ππ matrix elements required to compute A0. The factor of 1/2 relates the matrix element involving the kaon G-parity eigenstate to that of the physical kaon [10]. Here AK is the amplitude of the G-parity kaon operator, Asnkj are the amplitudes of the sink operator with the state j, and Ej is the energy of that state. These parameters are fixed to those obtained from the two-point function fits in Sec. III: AK and mK to the results given in Table II, and Asnkj and Ej to the results obtained from the three-operator, two-state ππ fits given in the second column of Table III.

We perform simultaneous correlated fits over multiple sink operators to the form Eq. (48) in order to determine the matrix elements Mij, allowing for one or more states j. Independent one-state fits are also performed to the optimized sink operator defined in Sec. III F. The fits are performed to each weak effective operator separately, in the ten-operator basis (the relationship between these ten linearly dependent operators serves as a useful cross-check of the fit results) using the strategy outlined in Sec. II B. We apply a cut tmin on the separation t between the kaon and the four-quark operator in order to isolate the ground-state kaon, and also a cut tmin′ on the separation t′=tsepK→snk-t between the four-quark and sink operators. These cuts, the number of sink operators, and the number of excited ππ states included in the fit are varied in order to study systematic effects.

For use below we again define an “effective matrix element” in which the ground-state ππ and kaon amplitudes and time dependence are multiplied out, Mieff,snk(t′)=Ci(t,tsepK→snk)(12AKAsnk0e-mKte-E0(tsepK→snk-t))-1=Mi0+∑jAsnkjAsnk0Mije-(Ej-E0)t′.(49)These effective matrix elements converge exponentially to the ground-state matrix element at large t′. Note that, unlike in Sec. IV C, we are assuming that a cut, tmin, on the separation between the kaon and four-quark operators has been applied that is sufficient to isolate the contribution of the kaon ground state. As a result, these effective matrix elements can be assumed to be independent of tsepK→snk, and a weighted average of our five datasets of different tsepK→snk can be applied to improve the statistical resolution of the data presented in our plots.

In this section we examine the results of fitting various subsets of our data, with the goal of finding an optimal fit window in which systematic errors arising from both excited ππ and kaon states are minimized.

In Figs. 6 and 7 we plot the fitted ground-state matrix elements Mi0 as a function of tmin′ for various choices of tmin, the number of sink operators and the number of states. The three-operator fits are performed using the ππ(311), ππ(111), and σ sink operators; for the two-operator fits we drop the noisier ππ(311) data; and for the one-operator fits we further drop the σ data. The one-operator, one-state fits are equivalent to those performed in our 2015 work, albeit with more statistics and more reliable ππ energies and amplitudes.

The discussion below will be focused on these figures. We will first discuss general features addressing the quality of the data and the reliability of the fits, and we will then concentrate on searching for evidence of systematic effects (or lack thereof) arising from kaon and ππ excited states. Based on those conclusions we will then present our final fit results.