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$\Delta I =1/2$ process of $K\to\pi\pi$ decay on multiple ensembles with periodic boundary conditions

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Pith's one-line read The paper reports the first periodic-boundary-condition lattice-QCD continuum extrapolation of ε'/ε, finding Re(ε'/ε)=17.5(6.8)(4.9)(5.0)×10^-4, in agreement with experiment.

desk verdict First PBC continuum extrapolation of ε' is a real step, but the new 1.4 GeV point has a postponed on-shell interpolation systematic that makes the central value conditional. read the letter →

arxiv 2501.18077 v1 pith:2QRCBBXY submitted 2025-01-30 hep-lat

Masaaki Tomii This is my paper · ORCID
classification hep-lat
keywords latticeQCDdirectCPviolationkaondecayepsilon-primeDeltaI=1/2amplitudecontinuumextrapolationperiodicboundaryconditionsdomainwallfermions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper reports the first attempt to take the continuum limit of $\varepsilon'$, the measure of direct CP violation in $K\to\pi\pi$ decay, using periodic boundary conditions on the lattice. It computes the $\Delta I = 1/2$ amplitude $A_0$ on two ensembles with inverse lattice spacings $a^{-1}\approx 1.0$ GeV and $1.4$ GeV, the finer ensemble new and the coarser one updated with roughly doubled statistics. Extrapolating $A_0$ linearly in $a^2$, the paper obtains $\mathrm{Re}(\varepsilon'/\varepsilon) = 17.5(6.8)(4.9)(5.0)\times 10^{-4}$, in good agreement with the experimental value $16.6(2.3)\times 10^{-4}$ and with the earlier G-parity calculation. The result is preliminary, and the paper explicitly notes that the $O(a^2)$ scaling-violation estimate is uncertain because only two relatively coarse lattice spacings are used.

What carries the argument

The load-bearing objects are the $\Delta I = 1/2$ amplitude $A_0$ and the ratio $\varepsilon'/\varepsilon$, built from renormalized matrix elements $M_i^{\overline{\mathrm{MS}}}(\mu)$ and three-flavor Wilson coefficients $z_i(\mu)$ and $y_i(\mu)$. The two-pion final state is isolated with a variational GEVP over momentum and $\sigma$ operators, the Lellouch-Lüscher factor converts finite-volume states to infinite-volume matrix elements, linear interpolation in $E_{\pi\pi}/m_K$ reaches the on-shell point, and RI/SMOM step scaling to $\mu = 4.0$ GeV supplies the nonperturbative renormalization. The continuum limit is taken by a two-point linear fit of $A_0$ in $a^2$, with the scaling-violation error estimated from the shift to an $O(a^4)$ fit.

What would settle it

Compute the same $\Delta I = 1/2$ amplitude and $\mathrm{Re}(\varepsilon'/\varepsilon)$ on an ensemble with $a^{-1}\approx 2.7$ GeV using the identical periodic-boundary-condition analysis; if the new point lies off the $O(a^2)$ line by more than the quoted scaling-violation error, the continuum extrapolation and its uncertainty would be wrong.

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Extended reading notes

Core claim

On its own terms, this paper establishes that the $\Delta I = 1/2$ $K\to\pi\pi$ amplitude and $\varepsilon'$ can be computed with periodic boundary conditions at two lattice spacings and extrapolated to the continuum. The central number is $\mathrm{Re}(\varepsilon'/\varepsilon) = 17.5(6.8)(4.9)(5.0)\times 10^{-4}$, which agrees with both the experimental value $16.6(2.3)\times 10^{-4}$ and the earlier G-parity lattice result $21.7(2.6)(6.2)(5.0)\times 10^{-4}$. The paper also reports that the bare matrix elements of the electroweak penguin operators $Q_7$ and $Q_8$ show some tension between the periodic and G-parity calculations at the shared inverse spacing of about 1.4 GeV, while the other operators are consistent. All of these results are labeled preliminary.

Load-bearing premise

The continuum numbers rest on the assumption that discretization errors behave as the square of the lattice spacing across the two spacings used here, so adding a third, finer lattice could move the central value.

Editorial extensions

If this is right

  • The value Re(ε'/ε)=17.5(6.8)(4.9)(5.0)×10^-4 is compatible with the experimental world average, so the Standard Model remains consistent with direct CP violation in kaon decays at current precision.
  • The consistency of most bare matrix elements between the periodic and G-parity calculations at the shared 1.4 GeV spacing supports periodic boundary conditions as a viable route for this process.
  • The continuum extrapolation, though based on two spacings, gives the first estimate of ε' in which a discretization error is assigned rather than quoted as a single-spacing uncertainty.
  • The paper's stated outlook is that adding finer periodic-boundary-condition ensembles with inverse spacings up to 2.7 GeV should reduce the finite-lattice-spacing error, and that the dominant remaining within-isospin error is the ~12% perturbative truncation of the Wilson coefficients.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the linear $a^2$ extrapolation holds on finer lattices, the periodic-boundary-condition route could reach the experimental precision for $\varepsilon'$ using already-generated domain-wall ensembles, avoiding the overhead of G-parity boundary conditions.
  • The reported tension in the $Q_7$ and $Q_8$ bare matrix elements between the two boundary-condition treatments suggests that finite-volume or boundary-condition effects on the left-right penguin operators deserve a dedicated study before the final error budget is closed.
  • A direct test of the claimed order-independence of interpolation and renormalization would be to repeat the earlier procedure, renormalizing before the energy interpolation, on the new 1.4 GeV ensemble and compare the on-shell matrix elements.
  • Once a third spacing is included, the current $O(a^2)$ scaling-violation estimate, based on only two relatively coarse points, is likely to be revised; the fine ensembles already generated provide the data for that test.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This proceedings paper reports preliminary lattice QCD results for the ΔI = 1/2 K→ππ matrix elements and the direct CP-violation parameter ε′/ε, computed with periodic boundary conditions (PBC) on two ensembles with inverse lattice spacings a⁻¹ ≈ 1.0 GeV and 1.4 GeV. The coarser ensemble updates an earlier PBC calculation with roughly doubled statistics, while the finer ensemble is new. The authors perform a GEVP analysis to isolate the first-excited two-pion state, apply Lellouch–Lüscher factors, interpolate bare matrix elements to the on-shell energy, renormalize in RI/SMOM with step scaling, and combine the renormalized amplitudes with external Wilson coefficients. They present the first continuum extrapolation of the ΔI = 1/2 amplitude and of ε′/ε from PBC data, obtaining Re(ε′/ε) = 17.5(6.8)(4.9)(5.0)×10⁻⁴, consistent with experiment and with the earlier GPBC result, and estimate an O(a²) scaling-violation systematic by comparing with an O(a⁴) fit. The paper is explicitly preliminary, with several systematic effects deferred to a later full publication.

Significance. If correct, this work provides the first continuum-extrapolated value of ε′/ε from lattice QCD with periodic boundary conditions, a milestone that could eventually reduce the dominant finite-spacing systematic in this quantity. The analysis follows well-established procedures from the RBC/UKQCD program: the GEVP state isolation, A2A/AMA methods, RI/SMOM renormalization with step scaling, and Lellouch–Lüscher factor treatment are all standard and internally consistent. The paper also honestly acknowledges the limitations of a two-point continuum extrapolation on coarse lattices and the preliminary nature of the result. The main risk is that the central claim — the continuum-extrapolated ε′/ε — depends on a new finer ensemble whose on-shell interpolation systematic is unquantified and explicitly postponed to the full paper. That omission directly affects the headline number and the conclusions drawn from it.

major comments (2)
  1. [Section 3, Figure 3 and text on the 32³ lattice] The on-shell interpolation systematic for the new finer ensemble is left unquantified. The paper states: 'On the 32³ lattice, the n = 1 two-pion energy is about 11% larger than the kaon mass and this could enhance the systematic error, but we postpone the estimation of this effect until the full paper.' This ensemble is the finer point in the continuum extrapolation (Figure 4), so the central result Re(ε′/ε) = 17.5(6.8)(4.9)(5.0)×10⁻⁴ does not include this systematic, despite the interpolation gap being roughly twice as large as on the 1.0 GeV lattice (6%). The omission is load-bearing: if the interpolation bias at 1.4 GeV is not negligible, the continuum extrapolation and thus the quoted ε′/ε would shift by an amount not covered by the quoted errors. This concern is reinforced by the 'some tension' observed for Q₇ and Q₈, which are important for Im A₀, at this same lattice spacing. The authors should either provide an estimate of this systematic (e.g., from a comparison of linear and quadratic interpolation, or from the GPBC data) or explicitly exclude it from the quoted central value and error budget, with a clear statement that the presented continuum result is conditional on this systematic being small.
  2. [Section 3, Figure 4 and continuum limit discussion] The continuum extrapolation uses only two lattice spacings with an assumed O(a²) form, and the scaling-violation systematic is estimated by comparing with an O(a⁴) fit. With only two data points, an O(a⁴) alternative is not a test of the extrapolation; it simply shifts the curve by a fixed amount. The paper acknowledges this ('somewhat uncertain since we use only two lattices with relatively large spacings'), but the quoted 11% scaling-violation error for Im A₀ should be understood as a qualitative measure rather than a robust estimate. The authors should make this limitation more prominent in the abstract and summary, and ideally present the O(a⁴) central value explicitly so that readers can see the sensitivity. This issue is separate from the interpolation systematic of the previous comment but compounds its impact on the final ε′/ε.
minor comments (4)
  1. [Section 2] The phrase 'the244 lattice' appears in the discussion of the coarser ensemble; this is likely a typo for 'the 24³×64 lattice' or 'the 24³ lattice'. Please clarify.
  2. [Section 2, Eq. (6)] In the displayed equation for C³ᵖᵗₙ,ᵢ, the angle brackets are rendered as 'D' and 'E' (apparently a LaTeX/encoding issue). The expectation value notation should be fixed in the final version.
  3. [Section 4] The word 'Möbuis' in the sentence about existing ensembles is a typo for 'Möbius'.
  4. [References] Reference [9] is a PoS proceedings contribution by C. Kelly; it would be helpful to include the full author list and arXiv identifier if available, to improve traceability of the ensemble parameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central ε′ result is a direct lattice computation with external Wilson coefficients and benchmark comparisons.

full rationale

The derivation chain is self-contained: the new ingredient that dominates the ε′ prediction, Im A0, is obtained from lattice three-point functions, GEVP-optimized two-pion operators, Lellouch-Lüscher factors, and RI/SMOM renormalization with step scaling, none of which is defined in terms of the final ε′ value. Wilson coefficients are taken from external references [21,22] and are not fitted here. The ε′ formula (Eq. 8) uses experimental Re A0 and Re A2 and the earlier ΔI=3/2 result for Im A2 [4]; this is a standard input decomposition, not a definition of the target in terms of itself, and the dominant Im A0 term is computed in this paper. The PBC-vs-GPBC comparison is used as a consistency check; the GPBC 2020 result is not an input to the continuum fit. The continuum extrapolation (Figure 4) is a two-point linear fit in a² with a scaling-violation estimate from an O(a⁴) alternative; it is statistically weak, as the paper acknowledges, but not circular. The postponed estimate of the on-shell interpolation systematic on the 32³ lattice (Section 3) and the observed Q7/Q8 tension with GPBC are accuracy and systematic-error concerns, not instances where a prediction reduces by construction to an input. No self-citation chain is load-bearing: prior RBC/UKQCD work supplies methodology, ensembles, and the small Im A2 input, but the central continuum-extrapolated result is independently computed here.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No invented entities; the paper uses established lattice QCD degrees of freedom and introduces no new particles, forces, or conserved quantities. The free parameters listed are the slopes and alternative fits used in the continuum extrapolation and on-shell interpolation, which are determined by the data and influence the central result.

free parameters (3)
  • O(a^2) slope of A0 continuum extrapolation
    Re A0 and Im A0 are extrapolated linearly in a^2 to a=0 using the 1.0 and 1.4 GeV data points (Figure 4). The slopes are fixed by the two ensembles and the intercepts are the continuum values.
  • Linear interpolation coefficient to the kaon mass for finite-volume matrix elements
    Matrix elements are interpolated linearly in Eππ/mK to the on-shell point using the two finite-volume states (Figure 3). The slope of this interpolation is determined by the two data points.
  • O(a^4) alternative extrapolation value used for the scaling-violation systematic
    The O(a^2) scaling violation error is estimated as the difference between the O(a^2) and O(a^4) continuum extrapolations; the O(a^4) curve is an alternative fit to the same two lattice points.
assumptions (5)
  • domain assumption The Lellouch-Lüscher factor converts finite-volume two-pion matrix elements to infinite volume for the two lowest states in the elastic region.
    Used in Section 2 and Figure 3; the 243 and 323 lattices have only the two lowest states in the elastic region where the factor is strictly valid.
  • domain assumption O(a^2) scaling is the leading discretization error for these domain wall fermion ensembles.
    Invoked for the continuum extrapolation in Section 3 and Figure 4. No third lattice spacing is available to justify the assumed form.
  • domain assumption Linear interpolation of matrix elements to the on-shell point introduces a controlled systematic; the 6% offset on the 243 lattice is small and the 11% offset on the 323 lattice is postponed.
    Stated in Section 3. The paper estimates the effect as much smaller than statistical error on the 243 lattice but explicitly postpones the 323 estimate to the full paper.
  • domain assumption RI/SMOM step-scaling to 4 GeV with perturbative conversion to MS is valid.
    Standard nonperturbative renormalization method cited from Ref. [20]. The paper separately notes the 12% perturbative truncation error from Wilson coefficients.
  • domain assumption The GEVP with optimized two-pion operators isolates the first-excited two-pion state.
    Section 2, Eq. (2). The variational method of Lüscher-Wolff is standard, but the reliability depends on the operator basis and the re-basing prescription.

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Cite this review

Pith. "Pith review of $\Delta I =1/2$ process of $K\to\pi\pi$ decay on multiple ensembles with periodic boundary conditions." pith.science (2026). https://pith.science/paper/2QRCBBXY

@misc{pith2026250118077,
  author       = {Pith},
  title        = {Pith review of: $\Delta I =1/2$ process of $K\to\pi\pi$ decay on multiple ensembles with periodic boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2QRCBBXY}},
  note         = {Machine review of arXiv:2501.18077}
}
abstract

We present our preliminary results for the $\Delta I = 1/2$ matrix elements of $K\to\pi\pi$ decay and $\varepsilon'$, the measure of direct $CP$ violation in $K\to\pi\pi$, computed on multiple ensembles with periodic boundary conditions (PBC) at the inverse lattice spacings of $a^{-1} \approx 1.0$~GeV and 1.4~GeV. The finer lattice ensemble is newly introduced as an extension to the first PBC calculation [1], while the calculation on the coarser ensemble is updated with the approximately doubled statistics. Our first attempt to take the continuum limit is also discussed with acknowledging potential significance of the $O(a^2)$ scaling violation on these coarse lattices.

Figures

Figures reproduced from arXiv: 2501.18077 by the authors.

Figure 1
Figure 1. Two-pion effective energies 𝐸 eff 𝑛 (𝑡0 + Δ, 𝑡0, 𝛿𝑡) = − 1 Δ ln 𝜆𝑛 (𝑡0 + Δ, 𝑡0, 𝛿𝑡) for the four low-lying states on the 323 lattice plotted in lattice units. The three low-lying states (𝑛 = 0, 1, 2) are obtained from the GEVP with three re-based operators, while the result for 𝑛 = 3 is obtained from the GEVP with four re-based operators. Here we choose Δ = 2 and 𝛿𝑡 = 10. where 𝑅 𝐾 (𝑡) = √︄ e −𝑚eff 𝐾 (𝑡)𝑡 𝐶𝐾 (𝑡) , (… view at source ↗
Figure 2
Figure 2. Unrenormalized effective matrix elements of 𝑄6 with the 𝑛 = 1 two-pion final state on the 323 lattice plotted in lattice units. Eq. (3) over 𝑡1 using the covariance matrix in 𝑡1 ≥ 4, where no significant dependence on 𝑡1 is observed. The result of various GEVP bases are plotted. ‘3 × 3 GEVP’ corresponds to the GEVP constructed with operators that create pions at rest, pions with spatial momentum ± 2𝜋 𝐿 (0, 0, 1) and… view at source ↗
Figure 3
Figure 3. Interpolation of unrenormalized matrix elements of 𝑄6 (left) and 𝑄8 (right) multiplied by the Lellouch-Lüscher factor to the on-shell point on the 323 lattice plotted in lattice units. The corresponding GPBC results in Ref. [6] are also plotted. Lelloush-Lüscher factor multiplied to the on-shell point. Although they are bare matrix elements, we can compare the results with those from GPBC calculation [6] because the… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Continuum extrapolation of the real (left) and imaginary (right) parts of 𝐴0. The smaller and larger error bars of lattice results represent the statistical and total errors, respectively. Since all systematic errors are mutual for the two lattice spacings with an exce…
Figure 5
Figure 5. Figure 5: History of RBC/UKQCD calculation of Re(𝜀 ′ /𝜀) compared with the world average of experiments. The smaller and larger error bars of lattice results represents the statistical and total errors, respectively. This work represents an important first step towards reducing …

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. From scattering towards multi-hadron weak decays

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    A review of current lattice QCD scattering calculations shows that finite-volume formalisms now enable multi-hadron weak decay studies with direct relevance to flavour physics.

Reference graph

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