{"id":"90eee70d-f5ec-4038-a0a9-c96ebf995c68","arxiv_id":"2412.09246","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Randomly displaced sparse grids inside distillation produce an unbiased estimator that makes local tetraquark operators affordable, with sampling noise already negligible at every-eighth-point spacing.","lead":"In lattice QCD, a standard quark-smearing trick called distillation becomes very expensive when the operator is a local tetraquark. The authors show that sampling only a random subset of lattice points removes that bottleneck, and a first test on the Tcc(3875)+ state finds only a small energy shift from such operators.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The variance of the sparse-grid estimator is demonstrated only on a single 32^3 ensemble; if it grows relative to gauge noise at larger volumes, the claimed V^3 scaling advantage for tetraquarks does not materialize.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the volume scaling of the position-sampling variance. I agree with that assessment. Unbiasedness is straightforward to prove and is not the weak point; the paper's one-line sketch in Section 3 can be filled in without difficulty. The genuine risk is that the estimator's variance may not be subdominant to gauge noise across the parameter range where the method is claimed to be advantageous. The numerical tests are genuine evidence: the meson results against Nsep=1 are reassuring, and the tetraquark Nsep=4 vs Nsep=8 comparison is a reasonable first check. But the tetraquark integrand is extremely short-range, so the sparse-grid estimator at Nsep=8 relies on rare near-coincidences; the variance of such an estimator can behave differently as the volume and the number of modes grow. A single-volume, single-lattice-spacing demonstration does not establish the V^3 scaling claim, which is the paper's stated advantage over standard distillation. The Tcc shift result is explicitly preliminary and is not central to the method claim; the method itself is the contribution. The conditional verdict is appropriate. The paper should be published only with a clear statement that the variance scaling with volume is an open question, or better, with an additional test on a larger volume. I do not see grounds to reject: the method is plausible, the implementation seems sound, and the numerical results are consistent with the claims at the tested point.","tokens_in":8448,"tokens_out":25524,"duration_ms":252321,"concrete_test":"Compute the local DD* tetraquark two-point function on the B450 ensemble with Nsep=2 (4096 grid points per time slice) and compare the effective-mass plateau error with the Nsep=8 result. If the errors are statistically consistent, the position-sampling variance is negligible at this volume; if the error decreases significantly, the 'dominated by Monte Carlo error' conclusion fails. To probe the volume scaling directly, repeat this Nsep comparison on a larger CLS ensemble at the same beta (e.g., 48^3 or 64^3) with N scaled proportionally to the volume and a*Nsep fixed; if the ratio of the Nsep=8 to Nsep=2 variance grows with volume, the claimed V^3 advantage is not realized.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that position-space sampling with randomly displaced sparse grids yields an unbiased estimator whose statistical error is dominated by gauge-field Monte Carlo noise, enabling a V^3 cost scaling for local tetraquark operators. The load-bearing premise is that the variance of the sparse-grid estimator remains subdominant relative to gauge noise when Nsep is held fixed in physical units and the number of Laplacian modes N is scaled with volume. The evidence in Section 4 is limited to one 32^3 ensemble at a=0.0762 fm. For tetraquarks, no full-lattice (Nsep=1) reference is available; the only comparison is Nsep=4 versus Nsep=8, which has limited power to detect a moderate position-sampling variance contribution. The tetraquark integrand contains two charm propagators and decays very rapidly with the sampled pair separation (roughly as e^{-2 m_D r} times light-quark factors); at Nsep=8a≈0.61 fm the estimator is supported on rare near-coincidences between the independently displaced sink and source grids. Whether the variance of such rare events remains negligible relative to gauge noise as the volume grows is not demonstrated. If, contrary to the implicit assumption, the position-sampling variance grows faster than the gauge noise, Nsep would need to be reduced at larger volumes, worsening the cost scaling from V^3 back toward V^5 or V^7. The unbiasedness lemma itself is not the issue: for independent uniform offsets, each pair (x',x) is included with probability 1/Nsep^6, so the prefactor makes the estimator unbiased for any fixed gauge field.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a position-space sampling method for evaluating local multiquark operators in the distillation framework. The key idea is to replace the full spatial lattice sums in the smeared-propagator contractions by sums over two independently randomly displaced sparse grids, with normalization factors restoring the full-lattice result. The authors argue that the contraction cost then scales as V^3 (with N proportional to V and fixed physical grid spacing) instead of V^5 or V^7. They test the estimator on a single 32^3 CLS ensemble with N=32 Laplace modes: pion, D, and D* effective masses are independent of Nsep up to Nsep=8 (and Nsep=16 at zero momentum), and two local Tcc tetraquark operators show no reduction in plateau-error when Nsep is lowered from 8 to 4. Adding these local operators to a basis of two bilocal DD* operators in a GEVP produces a downward shift aDeltaE=0.00032(3) in the lowest level, which is small compared with the statistical error of the energy.","tokens_in":8826,"tokens_out":12843,"duration_ms":132262,"significance":"If the volume-scaling assumption is correct, the method is a genuinely useful tool: local tetraquark and hexaquark operators become much cheaper in distillation, and the paper provides a clean numerical demonstration on one ensemble, including an exact Nsep=1 reference for mesons. The estimator is simple and is unbiased by construction for independent uniform offsets, and the authors are careful to label the Tcc result as preliminary and to note that more bilocal operators may remove the shift. The principal open question is the volume dependence of the position-sampling variance, which the paper does not address.","major_comments":[{"comment":"The claim that keeping a fixed physical point separation a Nsep yields a V^3 scaling of the computational cost is conditional on the position-sampling variance remaining negligible relative to the gauge-field Monte Carlo noise as the volume grows. The variance tests in Secs. 4.2 and 4.3 are performed on a single 32^3 ensemble at one lattice spacing, so the volume dependence of this variance is not established. If the position-sampling variance grows with volume, Nsep would need to be reduced and the V^3 scaling would degrade toward V^5 or V^7. Please either provide a theoretical variance bound, add a test on at least one additional volume or lattice spacing, or explicitly restrict the larger-volume claim to the ensemble studied.","section":"Sec. 3, cost scaling"},{"comment":"For the two local tetraquark operators, the conclusion that the statistical error is dominated by the Monte Carlo error is based only on comparing Nsep=4 with Nsep=8, because the Nsep=1 full-lattice reference is not available for these contractions. Since both estimates are computed on the same gauge configurations and the fitted plateau values are not the actual finite-volume energies, this comparison has limited power to detect a moderate position-sampling variance contribution. I recommend adding a quantitative variance estimate, for example from several independent offset samples on fixed gauge configurations, or a Nsep=2 comparison for at least one tetraquark interpolator, or explicitly stating that the tetraquark variance has not yet been established.","section":"Sec. 4.3, tetraquark Nsep comparison"}],"minor_comments":[{"comment":"The definition of the randomly displaced sparse grid should state that the shifted coordinates are taken modulo the spatial lattice extent; as written, an arbitrary offset \\tilde{x}\\in\\Lambda_3 can push some grid points outside the lattice.","section":"Sec. 3, Eq. (11)"},{"comment":"The unbiasedness statement is deferred with 'one can show'; since this is the theoretical foundation of the estimator, a one-sentence proof (average over independent uniform offsets for fixed gauge field) should be included.","section":"Sec. 3, Eq. (10)"},{"comment":"The statement that there is 'no increase in error' up to Nsep=8 or 16 is based on a visual comparison of error bars; please provide the numerical values of the errors or a ratio test to support this conclusion.","section":"Sec. 4.2, Fig. 1"},{"comment":"Please state the plateau fit ranges and the number of effective-mass points used in the fits, since the comparison between Nsep=4 and Nsep=8 depends on this choice.","section":"Sec. 4.3, Fig. 2"},{"comment":"The word 'significant' for the shift a\\Delta E=0.00032(3) should be qualified as statistical significance within the chosen operator basis; the authors' own caveat that additional bilocal operators could produce a similar shift means that the result is not yet evidence for a physical local-tetraquark component.","section":"Sec. 4.4, Fig. 3"},{"comment":"In reference [3], the collaboration name contains a typo ('W ASA-at-COSY Collaborationcollaboration').","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a proceedings-style contribution presenting a promising stochastic estimator. The core numerical results are clean, but the volume-scaling claim, which is a central selling point, is currently supported by only one ensemble. I would not reject the paper; the authors should either add a second-volume or finer-lattice test, or soften the scaling claim and state the single-volume limitation in the abstract/conclusion."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper offers a simple, genuinely new idea: randomly displace sparse grids inside distillation to estimate local multiquark correlators without constructing high-rank tensors. The estimator is unbiased, the cost-scaling improvement is real, and the numerical tests are clean and consistent with the claims. This is not a rehash of the cited sparsening methods or the localized distillation basis; the random-displacement mechanism is new and does the work of restoring full momentum projection.\n\nWhat the paper does well: it demonstrates on the B450 ensemble (32^3, a=0.0762 fm) that meson two-point functions show no error growth up to Nsep=8 (or 16 for the pion at zero momentum), and tetraquark two-point functions show no reduction in error from Nsep=8 to Nsep=4. The energies are consistent with the Nsep=1 reference for mesons. The unbiasedness proof is deferred with 'one can show', but that is a non-issue: for independent uniform offsets, each sink-source pair is included with probability 1/Nsep^6, so the prefactor makes the estimator unbiased configuration-by-configuration. The Tcc study is explicitly preliminary, and the observed downward shift aDeltaE = 0.00032(3) is flagged as small and possibly sensitive to the small bilocal basis.\n\nThe soft spot is the volume-scaling claim. The V^3 cost advantage assumes the position-sampling variance stays subdominant to gauge noise when Nsep is held fixed in physical units and N scales with volume. The evidence is limited to a single ensemble. For tetraquarks there is no full-lattice reference; the Nsep=4 vs 8 comparison has limited power. The tetraquark integrand decays roughly as e^{-2 m_D r}, so at Nsep=8a~0.61 fm the estimator leans on rare near-coincidences between the independent sink and source grids. If that variance grows with volume, Nsep would have to shrink, eroding the scaling advantage. This is an untested assumption, not a demonstrated flaw, and the authors are honest that Nsep choice needs further study.\n\nThis is a proceedings-length paper, but the method is substantive and likely to be used. The right next step is a multi-volume study of the variance, which the authors should be pushed to do. The paper deserves a serious referee and publication after that follow-up.\n\nRecommendation: send to peer review.","headline":"A genuinely new random-displacement sparse-grid estimator for distillation; clean numerical evidence on one ensemble, with an untested volume-scaling assumption worth chasing.","tokens_in":9310,"tokens_out":2043,"would_cite":true,"duration_ms":20598,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A position-space sampling scheme for distillation makes local multiquark interpolators computationally affordable, and a preliminary Tcc study finds a small downward energy shift when local operators are added.","keywords":["lattice QCD","distillation","position-space sampling","sparse grids","stochastic estimator","tetraquark","Tcc(3875)+","variational method"],"falsifier":"Compute the same estimator on a larger spatial volume, e.g. $48^3$ or $64^3$ at the same physical $N_{\\mathrm{sep}}$ and with $N$ scaled with volume, and compare the sampled-correlator error to the full-lattice result at $N_{\\mathrm{sep}}=1$: if the relative error grows with volume, the claimed $V^3$ advantage over standard contractions fails.","tokens_in":8266,"feed_emoji":"⚛️","tokens_out":8661,"duration_ms":77522,"temperature":0.7,"pith_summary":"This paper proposes and tests a way to compute local multiquark operators inside distillation without paying the usual high-rank tensor cost. The idea is to evaluate the smeared quark propagator only on randomly displaced sparse grids, and to rescale the sum so that the result is an unbiased stochastic estimator of the full correlator. On a single $64\\times32^3$ ensemble with 32 Laplacian modes, the statistical error from the sampling is negligible compared to the gauge-field Monte Carlo error once the grid spacing $N_{\\mathrm{sep}}$ is at most 8, for pions, $D$ mesons, and local tetraquark operators. The method drops the contraction cost from scaling like $N^5$ or $N^7$ to $N(N_s/N_{\\mathrm{sep}})^6$, which is a volume scaling of $V^3$ when the physical grid spacing is held fixed. Applying it to $T_{cc}(3875)^+$, adding two local operators to a two-operator bilocal basis lowers the ground-state energy by $a\\Delta E = 0.00032(3)$, small compared with the energy error.","feed_headline":"Sparse-grid estimator keeps tetraquark errors Monte Carlo dominated","feed_subtitle":"An unbiased estimator lets local tetraquark operators be computed with errors dominated by gauge noise.","key_machinery":"The load-bearing object is the randomly displaced sparse grid $\\tilde{\\Lambda}_3 = \\{a\\,\\mathbf{n} + \\tilde{\\mathbf{x}} \\mid n_k = 0, N_{\\mathrm{sep}}, 2N_{\\mathrm{sep}},\\dots,N_s-N_{\\mathrm{sep}}\\}$ with independent random offset $\\tilde{\\mathbf{x}}\\in\\Lambda_3$ chosen separately for sink and source, per configuration and time. Evaluated inside the estimator (Eq. 10), this grid replaces the exact position sums by a Monte Carlo average; the random displacement guarantees that no position is systematically excluded, so the momentum projection is exact and the sparse grid changes only the variance, not the mean. It is this object that converts the expensive high-rank mode tensors into cheap contractions whose cost scales with the grid volume $N(N_s/N_{\\mathrm{sep}})^6$ rather than with $N^5$ or $N^7$.","core_discovery":"In distillation, a local tetraquark correlator normally requires first forming rank-four mode tensors by summing over all positions, making the contraction cost scale as $N^5$ in the number of Laplacian modes, and as $N^7$ for a hexaquark. The paper's claim is that this can be avoided by swapping the summation order: compute $\\bar{D}^{-1}_{\\mathrm{sm},f}(\\mathbf{x}',t';\\mathbf{x},t)$ only for $\\mathbf{x}'$ and $\\mathbf{x}$ on independently chosen, randomly displaced sparse grids, multiply by $|\\Lambda_3|^2/(|\\tilde\\Lambda'_3||\\tilde\\Lambda_3|)$, and average over the random offsets. Because the offsets are chosen fresh per gauge configuration and source time, the momentum projection is exact and the estimator is unbiased; $N_{\\mathrm{sep}}$ controls only the variance. The numerical evidence is that for the pion, $D$, $D^*$, and two local $T_{cc}$ operators, the sampled correlators are statistically equivalent to the full-lattice ones for $N_{\\mathrm{sep}}\\le 8$, with errors dominated by gauge noise. In the variational analysis, the $T_{cc}$ ground state moves down by $a\\Delta E = 0.00032(3)$ when the two local operators are added.","pith_inferences":["If the variance remains gauge-dominated on larger volumes, this estimator could be combined with stochastic sources to reduce $N$ itself, a step the paper does not take.","The observed shift may reflect an incompletely saturated operator basis: adding more bilocal operators could produce a comparable shift, so the local-operator effect should be re-examined with a larger basis before physical conclusions are drawn.","The random-displacement idea may extend beyond position: the same unbiased-sampling logic could be applied to spin, color, or taste contractions where exact summation is costly.","The method's practical range likely depends on momentum: the paper sees error rise at $N_{\\mathrm{sep}}=16$ for nonzero-momentum mesons, suggesting high-momentum local operators need finer grids."],"forward_implications":["Local tetraquark and hexaquark interpolators become usable in larger volumes; the contraction cost scaling improves from $V^5$ or $V^7$ to $V^3$ when $N_{\\mathrm{sep}}$ is held fixed.","Correlators computed this way are unbiased at any $N_{\\mathrm{sep}}$, so the method adds no systematic shift; in practice $N_{\\mathrm{sep}}=8$ already gives errors that are gauge-noise dominated.","The same estimator works for single-meson, heavy-quark meson, and local tetraquark operators, so it is a general replacement for mode doublets and triplets in contractions.","Keeping the same random offsets for all sink times preserves correlations across $t'$, which is necessary for reliable plateaus and GEVP analysis.","For the $T_{cc}$, the ground-state energy shifts downward by $a\\Delta E = 0.00032(3)$ when local operators join the basis, indicating local structure matters, though the shift is small compared with the energy uncertainty."],"supporting_citations":[{"why":"It defines the distillation framework of Laplacian eigenvectors and perambulators that this method modifies.","marker":"[4]"},{"why":"It establishes that the number of modes must scale with physical volume to keep the smearing radius fixed, which underlies the cost-scaling comparison.","marker":"[10]"},{"why":"It supplies the sparse-grid construction to which the random displacement is added, along with earlier sparsening cost benchmarks.","marker":"[11, 12]"},{"why":"It provides the Gamma method used to compute the statistical errors that the sampling error is compared against.","marker":"[13]"},{"why":"It provides the generalized eigenvalue method used to extract the $T_{cc}$ ground-state energy and the shift when local operators are added.","marker":"[17]"}],"fun_headline_variants":["Sparse-grid sampling slashes local tetraquark cost in distillation","Unbiased estimator keeps tetraquark errors at gauge noise floor","Position-space sampling nails tetraquark correlators without N^5 cost","Local tetraquark operators now feasible via sparse-grid averaging","T_cc ground state shifts with local operators in variational basis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the variance of the sparse-grid estimator remaining dominated by gauge-field Monte Carlo noise as the physical volume grows while $N_{\\mathrm{sep}}$ is held fixed and $N$ is scaled with volume; the variance tests are done on one $32^3$ spatial lattice at a single lattice spacing.","fun_headline_variants_meta":{"raw":{"variants":["Sparse-grid sampling slashes local tetraquark cost in distillation","Unbiased estimator keeps tetraquark errors at gauge noise floor","Position-space sampling nails tetraquark correlators without N^5 cost","Local tetraquark operators now feasible via sparse-grid averaging","T_cc ground state shifts with local operators in variational basis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001051,"raw_usage":{"total_tokens":4429,"prompt_tokens":978,"completion_tokens":3451,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":3363}},"tokens_in":594,"tokens_out":3451,"duration_ms":24275,"temperature":1.0,"reasoning_tokens":3363,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:10:51.294374+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same estimator on a larger spatial volume, e.g. $48^3$ or $64^3$ at the same physical $N_{\\mathrm{sep}}$ and with $N$ scaled with volume, and compare the sampled-correlator error to the full-lattice result at $N_{\\mathrm{sep}}=1$: if the relative error grows with volume, the claimed $V^3$ advantage over standard contractions fails.","supporting_citations":[],"review_version":1}