^{1}

^{2,3}

^{4,5}

^{6}

^{1}

^{3,2}

^{4}

^{4}

^{7,4}

^{1}

^{8}

^{9}

^{4}

^{2}

^{1}

^{3}.

We present a lattice QCD calculation of the

A key ingredient to explaining the dominance of matter over antimatter in the observable universe is the breaking of the combination of charge-conjugation and parity (

Direct

Lattice QCD is the only known technique for determining the properties of low-energy QCD from first principles with systematically improvable errors. In this regime the high-energy physics is precisely captured by the

For an isospin-symmetric lattice calculation it is convenient to formulate the

In 2015 the RBC and UKQCD Collaborations published

In order to obtain on-shell kinematics, i.e., to ensure that

In our calculation of

The technique of using APBC on the down quark naturally breaks the isospin symmetry. For the

Note that due to the

Among the necessary ingredients in the lattice calculation of the

The observation of a discrepancy from the predicted phase shift increased our motivation to extend the earlier calculation by increasing the statistics and using more sophisticated methods to better analyze the

In Ref.

In this analysis we also include an improved nonperturbative determination of the renormalization factors relating the bare matrix elements in the lattice discretization to those of operators renormalized in the RI-SMOM scheme (see Sec.

Throughout this document results are presented in lattice units unless otherwise stated.

While the current paper is intended to be self-contained, it should be viewed as the third in a series of three closely related papers. The first of these is Ref.

For the convenience of the reader we summarize the primary results of this work in Table

A summary of the primary results of this work. The values in parentheses give the statistical and systematic errors, respectively. For the last entry the systematic error associated with electromagnetism and isospin breaking is listed separately as a third error contribution.

The layout of the remainder of this paper is as follows: In Sec.

In this section we provide an overview of the calculation, including information on the ensemble and the measurement techniques.

For this calculation we employ a single lattice of size

The lattice parameters are equal to those of the 32ID ensemble documented in Refs.

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

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)

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

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

The bootstrap measurement of the goodness of fit is a technique developed specifically for this and our companion work

Measurements are performed using the all-to-all (A2A) propagator technique of Ref.

For all quantities we use smeared meson sources with an exponential (

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

In order to compute the

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.

The strange quark is introduced into the G-parity framework as a member of an isospin doublet that includes a fictional degenerate partner,

Following Ref.

The two-point function

Fit results in lattice units, fit ranges, and

The isospin triplet of pion states can be constructed from the operators listed in Sec. V.A. of Ref.

Details of the strategy for measuring the

Two-point correlation functions are constructed from pairs of source and sink operators; thus,

We will use the result obtained by uniformly fitting to the temporal range 6–15 with all three

Fit parameters in lattice units and the

It is interesting to compare the statistical errors of our

The

Our decision to fit the

As detailed in Sec.

In Ref.

We can also obtain the derivative from the dispersive prediction of Colangelo

The near linearity of the dispersive prediction suggests that a linear ansatz,

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.

In our 2015 work

For use later in this document we define here an optimal operator that maximally projects onto the

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:

As our

A comparison of the effective ground-state energy obtained from the optimal operator [i.e., the optimal combination of the

In this section we detail the measurement and fitting of the

On the lattice we measure the following three-point functions:

The four classes of

Note that here and below we take care to differentiate between the G-parity kaon state

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

We compute each diagram with five different time separations between the kaon and the

The

Due to having vacuum quantum numbers, the

We perform measurements with all three two-pion operators described in Sec.

In Fig.

The contributions of the four Wick contraction topologies

The subtraction coefficients

The Wick contractions for the

In Fig.

The pseudoscalar subtraction coefficient

The

In the limit of large time separation between the source/sink operators and the four-quark operator, only the lowest-energy

The lattice three-point function

We measure the correlation function Eq.

The effective pseudoscalar matrix element

While the effective matrix elements of both sink operators initially trend toward zero, for the more precise

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

Thus, the

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

We perform simultaneous correlated fits over multiple sink operators to the form Eq.

For use below we again define an “effective matrix element” in which the ground-state

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

In Figs.

Fit results in lattice units for the

The extension of Fig.

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

We will first comment on the fits to the optimal operator, labeled opt. in the figures. This approach is outwardly advantageous in that the fits are performed to a single state and the covariance matrix is considerably smaller. In Fig.

The effective matrix elements

In Figs.

Given the above, an interesting question we can ask is whether the models we obtain from our fits with

The

For

The

We now address excited kaon state effects. Because the data rapidly become noisier as we move the four-quark operator closer to the kaon operator and thus further away from the

We can also test for excited kaon effects by examining the data near the kaon operator in more detail, alongside looking for trends in the five different

The

We therefore conclude that excited kaon effects in our results are negligible.

The dominant fit systematic error is expected to be due to excited

We begin by comparing the multioperator fits to the one-operator [

In general we observe excellent agreement between two- and three-operator fits with two states. Unfortunately, as mentioned above, the

In order to study the possibility of residual contamination from a third state we perform three-operator, three-state fits to the matrix elements using the

A further test for excited-state contamination is to study the agreement of the fit curves with the data outside of the fit region. To this end in Fig.

The effective matrix element

For our final result we choose to focus upon the three-operator, two-state fits. While the majority of the corresponding curves in Figs.

As discussed above we choose the three-operator, two-state fit with

Final

We conclude this section with a comparison of the statistical errors of the matrix elements

The Wilson coefficients are conventionally computed in the

In our 2015 calculation we computed the renormalization matrix at a somewhat low renormalization scale of

Due to operator mixing, the renormalization factors take the form of a matrix. This is most conveniently expressed in the seven-operator chiral basis in which the operators are linearly independent and transform in specific representations of the

In the RI-SMOM scheme the renormalized operators are generally defined thus,

These Green’s functions are not gauge-invariant; hence the procedure must be performed using gauge-fixed configurations, for which we employ Landau gauge fixing. The use of momentum-space Green’s functions introduces contact terms that prevent the use of the equations of motion so that additional operators, beyond those needed to determine on-shell matrix elements, must be introduced if all possible operator mixings are to be included, as is required if the RI-SMOM scheme is to have a continuum limit. These are discussed below.

Note that the Wick contractions of Eq.

The quark field renormalization

The independent projection matrices

Similar choices of

The seven weak effective operators mix with several dimension-three and dimension-four bilinear operators. For the parity-odd components these are

Mixing also occurs with the dimension-five chromomagnetic penguin operator and a similar electric dipole operator, conventionally labeled

In addition to the lower-dimension operators there is also mixing with both gauge-invariant and gauge-noninvariant dimension-six two-quark operators. These operators enter at next-to-leading order and above, and are therefore naturally small provided we perform our renormalization at large energy scales.

The gauge-noninvariant dimension-six operators vanish due to gauge symmetry and in many cases also by the equations of motion, and therefore do not contribute to on-shell matrix elements

Of the gauge-invariant dimension-six operators,

Step scaling

We use the step-scaling procedure to obtain the renormalization matrix at a scale of

Due to the presence of disconnected diagrams in our calculation, the choices of quark momenta are restricted to the discrete values allowed by the finite volume. The closest match between allowed momenta on the 32ID and 32Ifine ensembles that can be chosen as an intermediate scale is

We obtain the quark field renormalization for the 32Ifine ensemble via the vector current operator as described in Sec.

On the 32ID ensemble we extend the calculation at

The elements of the

For the measurement of the step-scaling matrix on the 32Ifine ensemble we likewise use

The results for the step-scaling matrix

The elements of the

The elements of the

As mentioned previously, we will also utilize step-scaled renormalization matrices computed at

The elements of the

The elements of the

In this section we combine our lattice measurements with experimental inputs to obtain

Standard Model and other experimental inputs required to determine

As previously mentioned, the Wilson coefficients that incorporate the short distance physics “integrated out” from the Standard Model are known in perturbation theory in the ten-operator basis to NLO in the

For consistency with the NLO determination of the Wilson coefficients we follow Ref.

The Lellouch-Lüscher factor

The calculation of the Lellouch-Lüscher factor requires the derivative of the phase shift with respect to interacting pion momentum, or correspondingly the

We find

The infinite-volume matrix elements of the seven chiral-basis operators

The ten conventional, linearly dependent operators

In our lattice calculation we have evaluated the matrix elements of all ten linearly dependent operators

Since the Fierz identities are not obeyed exactly by the data in Table

The

The bare lattice matrix elements in the seven-operator chiral basis (second column) that minimize the correlated

The results for the seven operators converted to the

Physical, infinite-volume matrix elements in the

The

The

We can now obtain

The

The contributions of each of the ten operators to the real and imaginary parts of

The contributions of each of the ten four-quark operators to

The real and imaginary parts of

We can then use Eq.

In Table

Values of

We could instead choose the parameter

Writing

For our primary result we use the more precise experimental values of

It is illustrative to break the value of

We can also compute a purely lattice value of

In this section we provide further insight into the origin of the significant change between our 2015 result of

The increase in the minimum time separation between the four-quark operator and the sink

The change in the procedure for determining the derivative with respect to energy of the

The increase in statistics from 216 to 741 configurations.

The addition of the

The use of step scaling to raise the renormalization scale from 1.53 GeV to 4.01 GeV.

The change in the value of the experimental inputs, notably that of the CKM ratio

We first note that repeating the

We next repeat the analysis of the

Repeating the above but with the

The result in Eq.

We therefore conclude that the difference in

In this section we describe the procedure used to estimate the systematic errors on our results. We will quote the values as representative percentage errors either on the matrix elements or on

In Sec.

As for the contribution of excited

In order to assign a numerical error to the contamination from excited

Relative differences between the ground-state elements obtained by fitting to three operators and three states with

As our values of

Another means of estimating this systematic error is to vary the subtraction coefficients

Relative differences in the unrenormalized lattice matrix elements of

We use the value provided in Ref.

The three

As described in Sec.

The Schenk parametrization

A linear approximation in the

A direct lattice calculation of the phase shift at energies close to and including the kaon mass.

We expect the remaining finite volume corrections to our matrix elements to be dominated by the (exponentially suppressed) interactions between the final state pions that are not accounted for by the Lüscher and Lellouch-Lüscher prescriptions. In Refs.

In the calculation of our step-scaled nonperturbative renormalization factors with scale

As discussed in Ref.

The relative difference in

The most important systematic errors in determining the renormalization matrix

In Table

The nonzero elements of the matrix

As expected there is a general trend toward smaller values as we increase the scale that appears consistent with the factor of 3 decrease in

We propagate the parametric uncertainties shown in Table

As mentioned previously we compare the NLO and LO determinations of the Wilson coefficients in order to estimate the systematic error arising due to missing higher-order terms. More specifically, we compare

Additionally we consider the same difference of LO vs NLO predictions for

In conclusion, we assign a 12% systematic error on both

We divide the systematic errors into those that affect the calculation of the matrix elements of the

Relative systematic errors on the infinite-volume matrix elements of the

Relative systematic errors on

In this section we collect our final results including systematic errors and discuss the implications of our results. For consistency with our previous work we will use the

The renormalized, infinite-volume matrix elements in the RI and

Physical, infinite-volume matrix elements in the

For the real part of the decay amplitude we take the value from Eq.

The “

In the past

In order to obtain a quantitative, first-principles result for

For earlier theoretical papers on the

For

While for most quantities these corrections enter at the 1% level or below, for

The effects on

Since a careful discussion of these corrections is beyond the scope of this paper we choose to treat these effects of isospin breaking as a systematic error whose size is given by the effect of including

Our first-principles calculation of

The horizontal-band constraint on the CKM matrix unitarity triangle in the

We have described in detail a calculation which substantially enhances our 2015 lattice calculation of

The new calculation reported here is based on an increase by a factor of 3.4 in the number of Monte Carlo samples and includes two additional

We have also included in this new calculation, an improved renormalization technique. As discussed in Sec.

Unfortunately raising the renormalization scale does not result in a similar improvement for the Wilson coefficients as their error is dominated by the use of perturbation theory at the scale of the charm quark mass to match the effective weak interaction theory between three and four flavors. We are presently working

Finally in the current calculation we have adopted a new bootstrap method

Finite lattice spacing effects remain a significant source of systematic error as at present we have computed

A second important systematic error, which we plan to reduce in future work, comes from the effects of electromagnetic and light quark mass isospin breaking. As discussed in Sec.

For our final result we obtain

We believe that

We thank our RBC and UKQCD collaborators for their ideas and support. We are also pleased to acknowledge Vincenzo Cirigliano for helpful discussion and explanation of isospin breaking effects. The generation of the gauge configurations used in this work was primarily performed using the IBM BlueGene/Q (BG/Q) installation at BNL (supported by the RIKEN BNL Research Center and BNL), the Mira computer at the ALCF (as part of the Incite program), Japan’s KEKSC 1540 computer, and the STFC DiRAC machine at the University of Edinburgh (STFC Grants No. ST/R00238X/1, No. ST/S002537/1, and No. ST/R001006/1), with additional generation performed using the NCSA Blue Waters machine at the University of Illinois. The majority of the measurements and analysis, including the nonperturbative renormalization calculations, were performed using the Cori supercomputer at NERSC, with contributions also from the Hokusai machine at ACCC RIKEN and the BG/Q machines at BNL. T. B. and M. T. were supported by U.S. DOE Grant No. DE-SC0010339, and N. H. C., R. D. M., M. T., and T. W. by U.S. DOE Grant No. DE-SC0011941. D. H. was supported by U.S. DOE Grant No. DE-SC0010339. The work of C. J. and A. S. was supported in part by U.S. DOE Contract No. DE-SC0012704. C. K. was supported by the Intel Corporation. C. L. was in part supported by U.S. DOE Contract No. DE-SC0012704. D. J. M. was supported in part by U.S. DOE Grant No. DE-SC0011090. C. T. S. was partially supported by STFC (UK) Grant No. ST/P000711/1 and by an Emeritus Fellowship from the Leverhulme Trust. P.B is a Wolfson Fellow WM/60035 and is supported by STFC Grants No. ST/L000458/1 and No. ST/P000630/1.

In this Appendix we provide the expressions for the Wick contraction required to compute the

For this Appendix we will utilize the notation described in Sec.

The Wick contractions of the

As in Ref.

We will first write down the expressions for the correlation functions

For simplicity, in Eqs.

As described in Sec.

It is easy to see that the

In our notation the pseudoscalar operator becomes

The

The result for