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REVIEW 3 major objections 6 minor 47 references

2024 Update on $\varepsilon_K$ with lattice QCD inputs

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The standard model, with the exclusive value of $|V_{cb}|$, predicts only about two-thirds of the measured $|\varepsilon_K|$, a $5.1\sigma$ deficit.

desk verdict A transparent, honest update: the 5.1 sigma |epsilon_K| tension is real arithmetic but conditional on exclusive |V_cb|, and the abstract overstates it. read the letter →

arxiv 2501.00215 v2 pith:DQAVWLEE submitted 2024-12-31 hep-lat

Seungyeob Jwa , Jeehun Kim , Sunghee Kim , Sunkyu Lee , Weonjong Lee , Jaehoon Leem , Sungwoo Park This is my paper · ORCID
classification hep-lat
keywords epsilon_KCPviolationkaonmixinglatticeQCDCKMmatrixexclusivevsinclusiveV_cbstandardmodelpredictionlong-distanceeffects
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

This paper updates the standard-model prediction of the CP-violating parameter $|\varepsilon_K|$ in neutral-kaon mixing, using lattice QCD inputs for the hadronic matrix elements and CKM parameters. With the exclusive value of $|V_{cb}|$ from $\bar{B} \to D^*\ell\bar{\nu}$ decays, the prediction is $|\varepsilon_K|_{\rm SM} = 1.453(152) \times 10^{-3}$, compared with the experimental value $2.228(11) \times 10^{-3}$, a $5.1\sigma$ deficit in which the standard model produces only about two-thirds of the observed CP violation. The tension drops to $1.1\sigma$ when the inclusive value of $|V_{cb}|$ from the heavy-quark expansion is used instead, giving $2.050(162) \times 10^{-3}$. The same pattern persists under an alternative treatment of the QCD correction factors, where the exclusive-input tension rises to $5.7\sigma$. The paper's direct message is that the long-standing $\varepsilon_K$ discrepancy now lives almost entirely inside the choice of $|V_{cb}|$, whose exclusive and inclusive determinations still disagree.

What carries the argument

The central object is the standard-model expression for $\varepsilon_K$, which combines the CKM matrix elements, the kaon bag parameter $\hat{B}_K$, the QCD correction factors $\eta_{cc}$, $\eta_{ct}$, and $\eta_{tt}$, and the long-distance parameters $\xi_0$ and $\xi_{\rm LD}$. The load-bearing switch is the value of $|V_{cb}|$: the exclusive determination produces a prediction about 35% below experiment, while the inclusive determination brings the same lattice inputs into agreement. The paper checks robustness by varying two further choices, the scheme used for the QCD corrections (the traditional charm-top unitarity scheme versus the up-top unitarity scheme) and the estimate of the long-distance parameter $\xi_{\rm LD}$, and finds the qualitative pattern unchanged.

What would settle it

Compute the inclusive semileptonic $B$-decay rate directly in lattice QCD, or measure an independent exclusive $|V_{cb}|$ with total uncertainty below $0.3 \times 10^{-3}$: if the value lands near $42 \times 10^{-3}$ rather than $38.4 \times 10^{-3}$, the $\varepsilon_K$ deficit disappears, while a confirmation of the lower value would leave the deficit standing.

Watch

Extended reading notes

Core claim

The central claim is that, with the input set the authors adopt—lattice QCD values for $\hat{B}_K$, $\xi_0$, $\xi_{\rm LD}$, $f_K$, $m_c$, and the CKM parameters, together with the exclusive $|V_{cb}|$ from $\bar{B} \to D^*\ell\bar{\nu}$ semileptonic decays—the standard model predicts $|\varepsilon_K|_{\rm SM} = 1.453(152) \times 10^{-3}$, while the measured value is $2.228(11) \times 10^{-3}$. The gap is $5.1\sigma$, meaning the standard model explains roughly 65% of the experimental value. Replacing the exclusive $|V_{cb}|$ with the inclusive value obtained from the heavy-quark expansion raises the prediction to $2.050(162) \times 10^{-3}$, leaving a $1.1\sigma$ tension. Using the up-top unitarity scheme for the QCD correction factors amplifies the exclusive-input tension to $5.7\sigma$. The paper therefore argues that the $\varepsilon_K$ deficit is real under the exclusive $|V_{cb}|$ input, and that resolving the exclusive-versus-inclusive $|V_{cb}|$ discrepancy is the key to knowing whether the standard model fails in kaon CP violation.

Load-bearing premise

The paper's $5.1\sigma$ tension rests on a single premise: that the exclusive value of the CKM element $|V_{cb}|$ is the correct one; if that value is biased low by a few parts in $10^{-3}$, the discrepancy drops to about $1\sigma$.

Editorial extensions

If this is right

  • If the exclusive $|V_{cb}|$ value is correct, the standard model fails to explain about 35% of the observed CP violation in $K^0$-$\bar{K}^0$ mixing, a gap far larger than the remaining lattice uncertainties.
  • The largest single lever on the prediction is $|V_{cb}|$, contributing about 52% of the error budget in the traditional scheme and 63% in the up-top scheme, so reducing its uncertainty is the highest-value next step.
  • With the inclusive $|V_{cb}|$ input, the same lattice inputs give a prediction within $1.1\sigma$ of experiment, so the reported tension is not a generic lattice-QCD problem but a property of the exclusive $|V_{cb}|$ choice.
  • Under the up-top unitarity treatment of QCD corrections, the exclusive-input tension grows to $5.7\sigma$, indicating the deficit is not an artifact of the traditional correction scheme.

Reading between the lines

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

  • Beyond the paper: if a future lattice calculation of inclusive $B \to X_c \ell\bar{\nu}$ decays, or a new exclusive channel, moves $|V_{cb}|$ up to roughly $41 \times 10^{-3}$, the reported $5.1\sigma$ deficit would largely be an artifact of the exclusive input being biased low.
  • Beyond the paper: because the discrepancy tracks $|V_{cb}|$ so closely, new-physics interpretations of the $\varepsilon_K$ deficit should be postponed until the exclusive-inclusive $|V_{cb}|$ puzzle is settled by independent measurements.
  • Beyond the paper: the stable ordering of the two QCD-correction schemes suggests that a higher-order calculation of the up-top scheme's remaining perturbative uncertainty could sharpen the comparison, but the decisive input will remain $|V_{cb}|$.
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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

3 major / 6 minor

Summary. This proceedings paper updates the SWME collaboration's evaluation of the standard-model prediction for |ε_K| using lattice QCD inputs. The inputs include the FLAG-24 value of R̂_B_K, the angle-only-fit (AOF) Wolfenstein parameters, several exclusive and inclusive determinations of |V_cb|, ξ_0 from the indirect method, ξ_LD, f_K, and m_c. The central result is that with the FNAL/MILC-22 exclusive |V_cb| the SM predicts |ε_K| = 1.453(152)×10^-3, compared with the experimental value 2.228(11)×10^-3, a 5.1σ deficit; with other exclusive inputs the tension ranges between 4.1σ and 5.1σ. With the inclusive kinetic-scheme value (Gambino-21) the prediction rises to 2.050(162)×10^-3 and the tension drops to 1.1σ. An alternative evaluation using the BGS η_i of u-t unitarity gives a 5.4σ tension with the same exclusive input. The paper also presents error budgets in which |V_cb| is the dominant source of uncertainty.

Significance. The calculation is a careful, forward evaluation with a transparent error budget and several valuable cross-checks: the angle-only fit avoids correlation between (ε_K, |V_cb|) and (ρ̄, η̄); multiple exclusive and inclusive |V_cb| inputs are tabulated; and the BGS central-value mismatch is conservatively added as a systematic error. The central values and significances reproduce simple error propagation from the tabulated inputs. The important message of the paper is that the 5σ tension is conditional on the exclusive |V_cb| choice and largely restates the exclusive-inclusive |V_cb| puzzle; this is explicitly acknowledged in the abstract, where the qualifier 'with exclusive |V_cb|' appears and where the disappearance with inclusive |V_cb| is stated. The main weakness is the phrasing 'strong tension ... between the SM theory and experiment', which can encourage an overbroad reading beyond the conditional statement. The paper is a useful update, but its headline claim is only as strong as the exclusive |V_cb| input.

major comments (3)
  1. [Abstract and Section 7.1] The central claim of a 5.1σ tension is explicitly conditional on the exclusive |V_cb| input, and the paper itself shows the tension disappears with inclusive |V_cb| (Table 7(a), rows 5-6; Table 9, rows 5-6). Because |V_cb| contributes about 52% (Table 8(a)) and 63% (Table 8(b)) of the error budget, the reported discrepancy is quantitatively a restatement of the exclusive-inclusive |V_cb| puzzle. The abstract's qualification is present, but the sentence 'represents a strong tension ... between the SM theory and experiment' should be rephrased to state explicitly that the tension is conditional on the exclusive determination being correct, so that the result is not read as an independent kaon-sector anomaly.
  2. [Section 4, Eq. (2)] The indirect method for ξ_0 uses Eq. (2) with the experimental value of |ε_K| (PDG-24, Table 4) as an input. This makes the subsequent 'SM prediction' for |ε_K| mildly circular, since the quantity being predicted is used in the determination of a nuisance parameter. The numerical effect is likely small because ξ_0 enters only as a sub-percent correction, but the paper should state this explicitly and, if space permits, quote the result obtained with the direct method for ξ_0 (the inputs are already given in Table 4). Without such a statement, the phrase 'evaluated directly from the standard model' in the abstract is stronger than the actual procedure.
  3. [Section 7.2, Table 9] The BGS result in Table 9 is obtained after adding the central-value mismatch δ_εK^BGS ≡ |ε_K|_u-t - |ε_K|_c-t to the error in quadrature. The justification is a single sentence ('small and tiny approximations'). Since this is a nonstandard way of combining two theoretical determinations, the paper should explain why the difference is a systematic uncertainty rather than an indication that one of the two methods is missing a contribution. A brief examination of the convergence (e.g., the relative size of the NNLO term) would make the 5.4σ–5.7σ significance in Table 9 more robust.
minor comments (6)
  1. [Section 5, Table 5(b)] The text says that the pole mass M_t is taken from PDG-24, but Table 5(b) lists m_t(m_t) = 162.77 GeV, which is the MS-bar mass, not the pole mass. Please correct the scheme notation.
  2. [Section 7.1] The statement that the SM 'describes only 2/3' of the experimental value is based on central values only, while the prediction carries a 10% uncertainty. Please add 'central value' (e.g., 'the central value of the SM prediction is about 65% of experiment') to avoid implying the uncertainty is negligible.
  3. [Figure 2] The time-evolution plot would be more informative if the points were labeled with the corresponding year and the input set used (e.g., which |V_cb| and R̂_B_K were adopted), since the time evolution mixes several changes of inputs and the reader cannot tell which points correspond to the present analysis.
  4. [Table 4] The source for the experimental value of Re A_0 is marked 'NA'; please provide the exact reference (e.g., PDG or the original RBC-UKQCD papers) so that readers can trace the value.
  5. [General] The paper does not display the master formula for |ε_K|. Since this is an update of Ref. [5], a short equation or a reference to the equation number in the earlier paper would help readers assess the sensitivity to inputs (e.g., the approximate |V_cb|^4 scaling).
  6. [Table 3] The note that the exclusive determinations are consistent within 1σ would benefit from a statement about whether the FLAG and HFLAV averages account for common systematic uncertainties among the lattice calculations.

Circularity Check

2 steps flagged · score 2.0 of 10

Minor self-referential inputs, not load-bearing; the core epsilon_K evaluation is a forward calculation.

  1. self definitional [Section 4, Table 4 and text: 'In the indirect method...']
    "In the indirect method, one determines ξ0 using Eq. (2) with lattice QCD results for ξ2 combined with experimental results for ε′/ε, ε_K, and ω. ... Here we use the results of the indirect method for ξ0 to evaluate ε_K, since the total errors are much smaller than those of the direct method."

    Eq. (2), Re(ε′/ε) = ω/(√2|ε_K|)(ξ2−ξ0), is inverted to obtain ξ0 = ξ2 − √2|ε_K|_exp Re(ε′/ε)/ω. The experimental value of the target observable |ε_K|_exp is therefore an input to the hadronic parameter ξ0, which is then used in the SM evaluation of |ε_K|_SM. This makes the 'prediction' partially dependent on the experimental value it claims to predict. The effect is numerically tiny: ξ0 is about −1.7×10⁻⁴, so its contribution to |ε_K| is at the 0.1% level, far below the reported 5σ tension. It does not drive the central claim.

  2. self definitional [Section 5, Eq. (5)]
    "ξLD = (0 ± 1.6)% of |ε_K|SM."

    The uncertainty assigned to the long-distance parameter ξLD is defined as a fixed percentage of |ε_K|SM, the very quantity being predicted. Because the central value is zero, the prediction's central value is unaffected; only the error bar is self-referential. This is a minor circularity in the error budget, not in the central result.

full rationale

The core calculation is a forward SM evaluation: Wolfenstein parameters are taken from an angle-only fit specifically to avoid correlations with ε_K and |V_cb|; |V_cb| is taken from independent exclusive and inclusive determinations; and the resulting |ε_K|SM is compared with experiment. No parameter is fitted to the target value of |ε_K| in a way that forces the discrepancy. The largest error source, |V_cb|, is external to this paper, and the paper explicitly shows that replacing exclusive |V_cb| with inclusive |V_cb| removes the tension—this is a conditionality, not circularity. The self-citations for η_cc (Ref. [6]) and the SWME B_K point inside the FLAG average are independent inputs, not fits to ε_K. The only genuine self-referential steps are the use of the indirect method for ξ0, which inverts Eq. (2) and thereby uses the experimental |ε_K| to fix a small hadronic parameter, and the RBC-UKQCD ξLD error defined as a percentage of |ε_K|SM. Both are numerically small and do not affect the central 5σ claim. Overall circularity is minor.

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

The paper introduces no new fitting, no new entities, and no ad hoc parameters of its own. All inputs are drawn from external literature. The main modeling choices (AOF for rho-bar and eta-bar, indirect xi_0, RBC-UKQCD or BGI for xi_LD, BGS delta treated as an error) are stated explicitly and are standard practice.

assumptions (5)
  • domain assumption The standard model expression for epsilon_K, including the factorization of QCD corrections into eta_i and hadronic matrix elements, is correct.
    Used throughout Sections 5-7; the formula is from Refs. [5,6] and is not re-derived here.
  • domain assumption The angle-only-fit (AOF) Wolfenstein parameters are independent of epsilon_K and |V_cb| and so are safe inputs.
    Section 2 explicitly states AOF is used to avoid unwanted correlations between (epsilon_K, |V_cb|) and (rho-bar, eta-bar).
  • domain assumption External lattice QCD results (FLAG-24, FNAL/MILC-22, RBC-UKQCD, HPQCD) are accurate within quoted errors.
    Tables 1-5 adopt these results without independent verification; e.g., B_K from FLAG-24 and exclusive |V_cb| from FNAL/MILC-22.
  • domain assumption The indirect method for xi_0 is preferred over the direct method.
    Section 4 states that the indirect method is used because its total errors are much smaller than the direct method.
  • ad hoc to paper The BGS eta_i values for u-t unitarity are valid, and the central-value mismatch with the traditional method can be treated as an additional systematic error.
    Section 7.2: the mismatch delta_BGS is added in quadrature to estimate a total error, which is a reasonable but ad hoc way to handle an internal inconsistency.

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Cite this review

Pith. "Pith review of 2024 Update on $\varepsilon_K$ with lattice QCD inputs." pith.science (2026). https://pith.science/paper/DQAVWLEE

@misc{pith2026250100215,
  author       = {Pith},
  title        = {Pith review of: 2024 Update on $\varepsilon_K$ with lattice QCD inputs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQAVWLEE}},
  note         = {Machine review of arXiv:2501.00215}
}
abstract

We report recent progress on $\varepsilon_K$ evaluated directly from the standard model (SM) with lattice QCD inputs such as $\hat{B}_K$, exclusive $|V_{cb}|$, $|V_{us}|$, $|V_{ud}|$, $\xi_0$, $\xi_2$, $\xi_\text{LD}$, $f_K$, and $m_c$. We find that the standard model with exclusive $|V_{cb}|$ and lattice QCD inputs describes only $2/3 \cong 65\%$ of the experimental value of $|\varepsilon_K|$ and does not explain its remaining 35\%, which represents a strong tension in $|\varepsilon_K|$ at the $5.1\sigma \sim 4.1\sigma$ level between the SM theory and experiment. We also find that this tension disappears when we use the inclusive value of $|V_{cb}|$ obtained using the heavy quark expansion based on the QCD sum rule approach. We also report results for $|\varepsilon_K|$ obtained using the Brod-Gorbahn-Stamou (BGS) method for $\eta_i$ of $u-t$ unitarity, which leads to even a stronger tension of $5.7\sigma \sim 4.2\sigma$ with lattice QCD inputs.

Figures

Figures reproduced from arXiv: 2501.00215 by the authors.

Figure 1
Figure 1. |𝜀𝐾 | SM with exclusive |𝑉𝑐𝑏 | (FNAL/MILC-22) in units of 1.0 × 10−3 : (a) |𝜀𝐾 | SM with 𝜂𝑖 of 𝑐 − 𝑡 unitarity (the traditional method), and (b) |𝜀𝐾 | SM with 𝜂𝑖 of 𝑢 − 𝑡 unitarity (the BGS method). obtained using the BGI estimate for 𝜉LD in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Chronicle of (a) Δ𝜀𝐾 /𝜎Δ𝜀𝐾 , and (b) Δ𝜀𝐾 and 𝜎Δ𝜀𝐾 . to reduce the errors in |𝑉𝑐𝑏| as much as possible. Part of the errors in exclusive |𝑉𝑐𝑏| come from experiments in BELLE, BELLE2, BABAR, and LHCb, which are beyond our control, but will decrease thanks to on-going accumulation of higher statistics in BELLE2 and LHCb. Part of the errors in exclusive |𝑉𝑐𝑏| come from the theory used to evaluate the semi-leptonic form f… view at source ↗

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Reviewed August 10, 2026 · model on record in the stance chip above.