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REVIEW 3 major objections 6 minor 24 references

Proton decay matrix elements on PACS configurations

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

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Pith's one-line read Two independent lattice QCD actions now give consistent proton decay matrix elements at the physical point.

desk verdict A clean, honest preliminary Wilson-clover calculation of proton decay matrix elements; the consistency with RBC2022 is suggestive but cannot be a precision cross-check until a second lattice spacing is included. read the letter →

arxiv 2501.13429 v2 pith:ADANCSL2 submitted 2025-01-23 hep-lat

Ryutaro Tsuji , Yasumichi Aoki , Yoshinobu Kuramashi , Eigo Shintani This is my paper · ORCID
classification hep-lat
keywords protondecaylatticeQCDmatrixelementsWilson-cloverfermionsphysicalpionmassrenormalizationRome-Southamptonschemegrandunification
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 a first calculation of proton decay matrix elements with the Wilson-clover action directly at the physical pion mass, on a $64^{4}$ lattice with a = 0.085 fm. These matrix elements convert a measured proton lifetime into constraints on grand-unified-theory parameters, so a second independent lattice result is a useful cross-check. The paper's twelve renormalized form factors W0 at physical kinematics are mostly consistent with the recent continuum physical-pion result obtained with domain-wall fermions, while they visibly deviate from the older chiral-extrapolated result. The paper describes the work as preliminary and notes that the discretization effect has not yet been included in the quoted uncertainties.

What carries the argument

The central objects are the renormalized three-quark matrix elements $W_0$ for proton-to-pseudoscalar transitions through baryon-number-violating operators, evaluated at physical kinematics. They are extracted from three-point functions at several source-sink separations around 1.7 fm to control excited-state contamination, then renormalized nonperturbatively via the Rome-Southampton method in both the MOM3q and SYM3q schemes (three-quark RI schemes with equal momenta, or with equal-magnitude momenta summing to zero), converted to the MS scheme at 2 GeV using RI/SMOM intermediate schemes. All-mode averaging with a deflated Schwarz-preconditioned solver suppresses statistical noise. This chain turns raw lattice correlators into $W_0$ values that can be compared across different lattice fermion actions and inserted into the proton partial-width formula.

What would settle it

Repeat the calculation on one or more finer ensembles, for example a = 0.06 fm with the same action, and take the continuum limit; if the extrapolated values move outside the reference continuum band by more than the combined quoted errors, the reported agreement is an artifact of the single lattice spacing.

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Extended reading notes

Core claim

The central claim is that Wilson-clover fermions on $64^{4}$ lattices at a = 0.085 fm can produce renormalized proton decay matrix elements $W_0^{RL}$ and $W_0^{LL}$ for the twelve relevant modes at $q^2 \simeq 0$, and that these are consistent with the recent continuum domain-wall result. The paper attributes the tension with the older chiral-extrapolated band to nonlinear chiral behavior near the physical pion mass. Explicitly, the updated renormalization constants are $Z_{RL}^{\rm MS}(2\,\mathrm{GeV}) = 1.016(5)(41)$ and $Z_{LL}^{\rm MS}(2\,\mathrm{GeV}) = 1.018(6)(37)$, with the second bracketed error collecting lattice artifacts, infrared effects, perturbative conversion, and scheme dependence.

Load-bearing premise

The result stands on the assumption that discretization effects at the single lattice spacing of 0.085 fm are small compared with the quoted errors, since no continuum extrapolation is made and the discretization effect is explicitly left out of the reported error budget.

Editorial extensions

If this is right

  • A second independent lattice discretization now backs the hadronic matrix elements used to turn proton-lifetime limits into GUT parameter exclusions.
  • Computing at the physical point removes the need for chiral extrapolation in this quantity, so nonlinear chiral behavior near the physical pion mass can be seen directly.
  • The reported $W_0$ values can be inserted into the partial-width formula to update model-specific proton lifetime predictions.
  • The planned continuum extrapolation on finer lattices will decide whether the agreement with the domain-wall result persists at the percent level.
  • The renormalization constants evaluated with two RI schemes provide a cross-check for any Wilson-clover calculation of baryon-number-violating operators.

Reading between the lines

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

  • A continuum extrapolation at finer lattice spacings could shift $W_0$ by an amount comparable to the quoted errors, so the consistency with the earlier continuum result should be treated as provisional until that extrapolation exists.
  • The agreement between Wilson-clover and domain-wall actions at the physical point suggests that discretization artifacts may be subdominant in current error budgets, but it also implies that future precision will be limited by renormalization-scale and excited-state systematics rather than statistics.
  • If the disagreement with the older chiral-extrapolated band is truly nonlinear chiral behavior, the same effect should appear in other physical-point baryon observables that were previously obtained by chiral extrapolation; testing those correlations could confirm the interpretation.
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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 reports preliminary lattice QCD results for the twelve proton decay matrix elements W0 at physical kinematics, computed on PACS 2+1 flavor Wilson-clover ensembles at a single lattice spacing a≈0.085 fm (a^{-1}=2.3162(44) GeV). The authors study excited-state contamination by varying the source-sink separation, determine the renormalization constants of the three-quark operators in the RI/MOM scheme (MOM3q and SYM3q) with four intermediate schemes, and quote Z_RL=1.016(5)(41) and Z_LL=1.018(6)(37) in MSbar at 2 GeV. The resulting W0 values are compared with the RBC/UKQCD continuum results of Ref. [7] and are stated to be mostly consistent, while a discrepancy is seen against the older chiral-extrapolated RBC results. The paper explicitly labels the results as preliminary and notes that discretization effects have not yet been accounted for.

Significance. If the results are correct, they provide a second independent lattice action (Wilson-clover physical-point, as opposed to domain-wall) supporting the hadronic matrix elements used to convert proton lifetime bounds into constraints on GUT and SUSY-GUT parameters. The paper's strengths include a clear demonstration of plateaus in the source-sink separation at t_sep≳20, a systematic renormalization-scale study with explicit intermediate-scheme dependence, and an honest acknowledgment that the computation is preliminary and that the discretization error is not yet included. The use of physical pion masses avoids the chiral extrapolation that was a major uncertainty in earlier work, and the AMA technique provides reasonable statistical precision. The comparison with RBC/UKQCD, even at the qualitative level, is a useful cross-check for the field.

major comments (3)
  1. [§5, Figure 5, and Summary] The central cross-check against the RBC continuum result [7] rests on the assumption that discretization effects at this single lattice spacing are small compared with the quoted uncertainties, yet the paper gives no estimate of their size and explicitly excludes them from the error budget. The conclusion that the results are "mostly consistent" (Fig. 5 and Summary) is therefore not yet a quantitative cross-check. Please either provide a quantitative estimate of the O(a^2) effect (for example, by comparing with the RBC point at a^{-1}=1.8 GeV shown in Fig. 5, or with any other available lattice-spacing dependence) or rephrase the consistency claim to make clear that it is provisional pending the planned continuum extrapolation.
  2. [§5, Figure 4] The interpolation to the physical kinematics (q^2≈0) is not documented: the fitting function, the number and range of the data points used, and the associated systematic uncertainty are not stated. Since the reported W0(0) values in Fig. 5 are the final quantities, please specify the interpolation procedure and, if possible, the size of the interpolation systematic error.
  3. [§4, Figure 2] The quoted systematic error on the renormalization constant, e.g., 0.041 for Z_RL, is hard to reconcile with the visible spread among the four intermediate schemes in Fig. 2, particularly at low matching scales. Please define the central value (the scale (pa)^2 at which Z is read) and describe how the RMS sum of the systematic contributions (lattice artifacts, IR divergence, perturbative matching, intermediate-scheme dependence) is assembled, so that the quoted total systematic error is reproducible.
minor comments (6)
  1. [Section 3, first paragraph] The text states that the configurations at β=1.82 correspond to a lattice spacing of 0.09 fm, but Table 1 and the abstract give a=0.085 fm (a^{-1}=2.3162(44) GeV); please correct the inconsistency.
  2. [Section 1] The word "nonberturbative" in the Introduction should be "nonperturbative."
  3. [Section 2] The phrase "read into" preceding Eq. (4) should be "reads."
  4. [Eq. (5)] The decay-width formula as displayed appears dimensionally inconsistent if W0 has mass dimension 2 (see Fig. 4's axis label); please verify that the expression matches the convention in Refs. [6,7,9], including any missing mass factor or square on the phase-space bracket that may have been lost in the PDF conversion.
  5. [Figure 3, Table 1] The plateau study is shown for the Gaussian source at m_pi=139 MeV; please clarify whether the same excited-state systematic was checked for the exponential source at 135 MeV, and if so, where the corresponding plot or result is.
  6. [Section 5, last paragraph] The sentence "those are mostly consistent with ours" is ambiguous; please rephrase to "with the PACS results" or "with ours (the present work)" for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proton decay matrix elements are computed from QCD correlation functions with independently evaluated RI/MOM renormalization and compared against an external RBC/UKQCD continuum result.

full rationale

The derivation chain is: (1) compute bare three-point and two-point correlation functions on PACS ensembles; (2) extract W0 via Eq. (4) at several momenta and source-sink separations; (3) renormalize with Z_MS obtained from an independent RI/MOM (Rome-Southampton) evaluation with perturbative matching via Eq. (6); (4) interpolate to physical kinematics and compare with RBC/UKQCD [7]. None of these steps defines the target W0 in terms of the RBC values or fits a parameter to RBC data. The renormalization constants in Eq. (7) are new numbers from this calculation, not imported from [7]. Citations to [7,9,10,19,20] supply conventions (MOM3q/SYM3q schemes, smearing parameters, systematic-error methodology) rather than the reported matrix elements; although some references share authors (Y. Aoki, E. Shintani), the RBC2022 comparison is an external, independently generated lattice result used only as a benchmark. The paper's explicit caveat that discretization effects are not yet included (Section 5) weakens the physical interpretation of the agreement, but that is an uncertainty/correctness limitation, not a circular reduction. No equation in the paper equals its input by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard lattice QCD practice plus the specific choices listed: interpolation to q^2=0, hand-tuned smearing parameters, and the untested single-spacing assumption. No new particles, forces, or dimensions are introduced.

free parameters (2)
  • q^2 interpolation coefficients for W0(q^2) = not quoted
    The physical-kinematics values W0(0) are obtained by fitting or interpolating W0(q^2) over a small momentum range; the fit parameters and functional form are not given, and the reported values depend on this fit.
  • source smearing parameters (A,B) and (W,N) = Exp: (1.2, 0.33); Gauss: (10, 600)
    These are chosen by hand to suppress excited states, as stated in the footnote in Section 3. The choice affects the excited-state contamination estimate and the plateau quality in Figure 3.
assumptions (4)
  • domain assumption Proton decay amplitude factorizes into Wilson coefficients C_I and hadronic matrix elements W0 via the operator product expansion.
    Used in Equations (4) and (5), following Refs. [5, 6]. The paper does not rederive this factorization.
  • domain assumption Perturbative matching from RI/SMOM and RI/SMOM_gamma_mu schemes to MS at 2 GeV is accurate at the matching scales used.
    Invoked in Equation (6) for the renormalization constants. Residual scale dependence is treated as a systematic error in Eq. (7), but the perturbative truncation is not independently verified in this paper.
  • domain assumption Discretization errors at a=0.085 fm are assumed to be smaller than the quoted statistical and systematic errors.
    The paper presents results at a single lattice spacing and states in Section 6 that continuum extrapolation is future work; the comparison to RBC therefore implicitly assumes O(a^2) effects are subdominant.
  • domain assumption The chosen smearing operators and t_sep values are sufficient to suppress excited-state contamination in the three-point functions.
    Section 5 argues from observed plateaus in Figure 3 that t_sep approximately 20 (1.7 fm) is enough, but no two-state fit or alternative operator analysis is shown to quantify residual contamination.

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Cite this review

Pith. "Pith review of Proton decay matrix elements on PACS configurations." pith.science (2026). https://pith.science/paper/ADANCSL2

@misc{pith2026250113429,
  author       = {Pith},
  title        = {Pith review of: Proton decay matrix elements on PACS configurations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADANCSL2}},
  note         = {Machine review of arXiv:2501.13429}
}
abstract

We report the preliminary results of lattice computation for the proton decay matrix elements in $N_f=2+1$ physical point with Wilson-clover fermion. We perform it on the PACS configurations of $64^4$ lattice volume with lattice spacing $a=0.085$ fm, and carefully estimate the systematic uncertainties, especially for the excited state contamination and associated error of the renormalization constant with Regularization Independent (RI, Rome-Southampton) scheme. Our preliminary results of the twelve relevant transition modes in proton decay matrix element and comparison with other lattice results are presented.

Figures

Figures reproduced from arXiv: 2501.13429 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Renormalization constant in MS scheme at 2 GeV with 𝑂 = 𝑅𝐿 via MOM3q (left) and SYM3q (right) with four different intermediate RI/SMOM(𝛾𝜇 ) matching schemes (see, e.g. Ref [19] and references therein.) as a function of matching scale squared. intermediate scheme dependence (see Ref. [19] in detail), in which the total systematic errors are evaluated as the root-mean-squared sum of those systematic uncertainties. 5. … view at source ↗
Figure 3
Figure 3. Time separation dependence of 𝑊0 for 𝑝 → 𝜋 0 mode with Gaussian source in 139 MeV pion. (Left) 𝑊𝑅𝐿 0 , (Right) 𝑊𝐿𝐿 0 . The horizontal axis shows the shifted time-slice of the operator 𝑡 such that 𝑡 = 0 always locates at the central position between the source and sink. our smearing parameter can enough suppress the excited state contamination effect when 𝑡𝑠 ≃ 20 corresponding to 1.7 fm separation [PITH_FULL_IMAGE:f… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Transfer momentum dependence of 𝑊𝑅𝐿 0 (Left) and 𝑊𝐿𝐿 0 (Right) for 12 relevant transition modes with different masses in 139 MeV and 135 MeV pions. Colored bands represent the previous results of chiral extrapolation from the heavy pion mass by RBC [9]. 0 0.05 0.1 0.15…
Figure 5
Figure 5. Figure 5: Summary of matrix elements of proton decay and comparison with other lattice results. Acknowledgement We would like to thank members of the PACS collaboration for useful discussions. Nu￾merical calculations in this work were performed on Oakforest-PACS in Joint Center …

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