{"id":"31e2049f-528b-4bea-9d45-85ca8707d6f1","arxiv_id":"1908.04153","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"Complex Langevin simulations, using a twisted-boundary-condition auxiliary-field order parameter, correctly flag spontaneous supersymmetry breaking in several zero-dimensional N=2 models, including new complex-action cases.","lead":"This paper uses complex Langevin simulations to test whether supersymmetry is spontaneously broken in a set of zero-dimensional supersymmetric field theories with complex actions. It reports which toy models break supersymmetry and which preserve it, and concludes that the method captures both cases.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Power-law drift tails and missing independent checks leave the complex-action SUSY-breaking conclusions unverified; a direct thimble-integral comparison is needed.","rationale":"The reader's weakest assumption—that the CL process samples the correct equilibrium despite power-law drift—is exactly the load-bearing point. I independently checked the paper's appendices: Appendix A2 explicitly documents the power-law falloff, and the Fokker-Planck numbers in Tables VIII–X are too noisy to validate correctness. Because the complex-action models have no exact checks, the central claim cannot be accepted as stated. However, the analytical framework (twisted boundary conditions, order parameter) is sound, and the real-action results match known patterns, so rejection is too strong. Conditional acceptance requiring an independent finite-α verification is the right level. My proposed check—direct numerical integration of the zero-dimensional path integral on the thimble—would settle the issue conclusively, since these are one-dimensional integrals that can be computed to high precision. If the CL values match, the paper's complex-action predictions gain real support; if not, the abstract claim must be weakened to only the verified real-action cases.","tokens_in":25036,"tokens_out":14394,"duration_ms":128342,"concrete_test":"Directly evaluate the finite-α partition function and ⟨B⟩_α for the complex models by high-precision numerical integration of Eqs. (36)–(37) along a contour that makes exp(−S_eff) converge (e.g., the Lefschetz thimble or a rotated Gaussian contour). Apply this to W'=ig(φ²+μ²) (g=1,3), W'=igφ(φ²+μ²) (g=1,3), and W'=−ig(iφ)^{1+δ} (δ=1,3), at the same α values as Tables III, V, and VI. If the CL results deviate from the exact integrals by more than the quoted errors, the complex-action conclusions (and the abstract claim) are unsupported; if they agree, the power-law tail is benign for these observables.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that complex Langevin simulations successfully capture the presence or absence of SUSY breaking—extends to complex actions where no exact result is available. Appendix A2 and Figs. 21–22 show that the drift-magnitude distribution P(u) falls off as a power law for all models studied, whereas the correctness criterion of Refs. [42,43] requires exponential suppression; the text defers this to future work. The Fokker-Planck criterion in Appendix A1 is not decisive: Tables VIII–X list values whose statistical errors are comparable to or larger than the central values, so a nonzero deviation cannot be excluded. Consequently, the new predictions (SUSY preserved for W'=ig(φ²+μ²) and for the PT-symmetric δ=1,3 potentials; SUSY broken for W'=igφ(φ²+μ²)) all rest on the unverified premise that the CL process reaches the correct complex equilibrium despite the power-law drift tail. Supporting evidence of CL systematics is the α-dependence of ⟨B⟩_α in Table II for the real double-well, even though the analytic ⟨B⟩_α is exactly α-independent. The abstract-level claim is therefore stronger than the evidence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies complex Langevin dynamics (CL) with stochastic quantization to a class of zero-dimensional N=2 supersymmetric models with real and complex actions. The authors use the expectation value of the auxiliary field under twisted boundary conditions, extrapolated to zero twist, as an order parameter for spontaneous supersymmetry breaking. They first validate the CL method on bosonic correlators for actions derived from -g/N (iφ)^N, then treat a real double-well superpotential and a general real polynomial superpotential, where exact analytic results for SUSY breaking are available. They then turn to complex superpotentials, including W'=ig(φ^2+μ^2), W'=igφ(φ^2+μ^2), and PT-symmetric delta-potential models W'=-ig(iφ)^{1+δ} with δ=1,2,3,4, and conclude that SUSY is preserved in the first and third classes and broken in the second. The paper also contains an appendix discussing two standard CL reliability criteria: the Fokker-Planck correctness criterion and the decay of the drift-magnitude distribution.","tokens_in":25229,"tokens_out":4242,"duration_ms":43711,"significance":"If the complex-action predictions are correct, the paper would demonstrate that complex Langevin dynamics can determine SUSY-breaking patterns in a nontrivial class of zero-dimensional models, including PT-symmetric ones, where exact results are not available. The analytic setup with twisted boundary conditions (Eqs. (33)-(39)) is a useful framework, and the real-action checks (Tables II, IV and the bosonic checks in Table I) are clean and agree with exact results. The authors also deserve credit for explicitly reporting the drift-tail diagnostic and flagging the power-law behavior, rather than hiding it. However, the paper's main new conclusions all concern complex actions for which the manuscript's own reliability diagnostics are inconclusive or fail, so the significance of the claimed results is currently limited by lack of independent verification.","major_comments":[{"comment":"The drift-tail diagnostic fails exactly for the models that produce the new physics. For W'=g(φ^2+μ^2) (Fig. 21) and W'=-ig(iφ)^{1+δ} (Fig. 22), the probability distribution P(u) decays as a power law in u, whereas the criterion of Refs. [42,43] requires exponential suppression before complex Langevin results can be trusted. The text states that 'this needs further investigations, and we save it for future work,' so the manuscript explicitly defers the issue. Every complex-action conclusion—Table III for W'=ig(φ^2+μ^2), Table V for W'=igφ(φ^2+μ^2), and Tables VI-VII for the δ-potentials—rests on the unverified premise that the CL process samples the correct complex-weight equilibrium despite the power-law tail. This is load-bearing: it is precisely the abstract's claim of 'successfully capture the presence or absence of supersymmetry breaking.' An independent check, such as a Lefschetz-thimble evaluation of ⟨B⟩_α for at least one of these actions, is needed before these predictions can be regarded as established.","section":"Appendix A2, Figs. 21-22"},{"comment":"The Fokker-Planck criterion is presented as a correctness check, but the numbers do not rule out a nonzero ⟨~LB⟩. For example, Table VIII gives ⟨~LB⟩_{α→0} = 0.0023(230)+i0.0555(2357) for g=1, and Table IX gives −1.2716(2.421)−i0.1173(2.122) for δ=1; in essentially all rows the statistical error is comparable to or larger than the central value. The criterion is also applied to a single observable B, whereas the cited literature (Ref. [31]) requires a complete set of observables. Thus Appendix A1 provides at best a necessary condition and cannot compensate for the failed drift-tail criterion.","section":"Appendix A1, Tables VIII-X"},{"comment":"For W'=g(φ^2+μ^2), the exact ⟨B⟩_α is independent of α because the α-dependent factor cancels in Eq. (47)/(48). The simulated values in Table II nevertheless drift from −i4.2250(72) at α=0.05 to −i4.1537(74) at α=0.8 for g=1, and the reported α→0 extrapolation −i4.2340(123) differs from the exact value −i4.115 by about 3%. This α-dependence is a systematic CL artifact; it should be quantified and addressed before the same extrapolation procedure is used as the basis for the complex-model conclusions.","section":"Table II and Eq. (48)"}],"minor_comments":[{"comment":"The title 'PT-symmetric models inspiredδ-potentials' should read 'inspired by δ-potentials'.","section":"Section IV.C title"},{"comment":"The caption contains a typo: 'g = 0.5 ad δ = 1, 3' should be 'g = 0.5 and δ = 1, 3'.","section":"Table VI caption"},{"comment":"The caption reads 'The first105 points' and 'every102 steps'; these should be typeset as 10^5 and 10^2.","section":"Fig. 10 caption"},{"comment":"The analytic evaluations for δ=0,2,4 are performed as real integrals over φ, while the CL simulations explore the complexified plane; a sentence explaining the contour/analytic-continuation prescription used in the exact evaluations would improve readability.","section":"Section IV.C"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a real split: the analytic machinery and the real-action simulations are solid, and the authors are commendably transparent about the drift-tail problem. But the paper's headline contribution is the set of complex-action SUSY predictions, and those rest entirely on an assumption that the manuscript's own diagnostics contradict (power-law drift tails) or fail to confirm (inconclusive Fokker-Planck checks). I would not reject outright because the issues are fixable: an independent thimble calculation for one or two of the complex actions, or a substantial downgrade of the abstract and Section V claims, would resolve the main concern. As written, however, the central claim outruns the evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does two things well. First, it gives a clean numerical template for using complex Langevin with a twisted-boundary-condition order parameter to probe SUSY breaking in zero-dimensional N=2 models. Second, it checks itself wherever it can: the bosonic models match exact values, and the real polynomial and double-well superpotentials agree with analytic SUSY-breaking patterns. Those checks are reproducible in principle and give me real confidence in the method's setup.\n\nThe genuinely new content is the set of complex-action results: W'=ig(phi^2+mu^2) preserves SUSY, W'=ig phi(phi^2+mu^2) breaks it, and the PT-symmetric delta-potentials preserve it for delta=1 through 4. I have not seen these in the cited literature. But here is where the paper's central claim outruns its evidence. For these complex cases there is no independent check, and the paper's own reliability diagnostics are not reassuring. The drift-magnitude distribution decays as a power law for every complex model, while the correctness criterion the authors cite demands exponential suppression. The Fokker-Planck criterion in Appendix A1 is inconclusive: the values in Tables VIII–X have statistical errors comparable to or larger than the central values, so a nonzero deviation cannot be excluded. The authors are transparent about this, deferring the issue to future work, but the abstract-level claim that the simulations \"successfully capture\" the presence or absence of SUSY breaking is stronger than the evidence supports.\n\nA smaller but telling warning is the alpha-dependence of <B>_alpha in the real double-well, where the analytic expectation is exactly alpha-independent. The extrapolated value still lands close to the exact result, so this is a minor systematic worry, not a fatal one. Still, it suggests the alpha->0 fits carry uncertainties that are not fully captured by the reported errors.\n\nWho is this for? Lattice and stochastic-quantization people working on the sign problem, and anyone wanting a toy model for CL reliability tests. It is a useful methodological contribution even if the complex-action physics claims remain unproven. The paper deserves a serious referee, not a desk rejection. A referee should push for independent checks where possible, such as a Lefschetz-thimble or direct numerical integration for the complex models, and for a more careful scoping of the conclusions. If the claims are softened and the diagnostics strengthened, this would be a solid contribution.","headline":"Honest and useful complex Langevin study of zero-dimensional SUSY models, but the complex-action conclusions outrun the reliability evidence.","tokens_in":25769,"tokens_out":1624,"would_cite":false,"duration_ms":18453,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that complex Langevin dynamics with twisted boundary conditions correctly identifies spontaneous supersymmetry breaking in zero-dimensional $\\mathcal{N}=2$ models with real and complex actions, recovering known results…","keywords":["complex Langevin dynamics","supersymmetry breaking","stochastic quantization","twisted boundary conditions","zero-dimensional quantum field theory","PT-symmetric potentials","sign problem","auxiliary field order parameter"],"falsifier":"Compute $\\langle B\\rangle_\\alpha$ for $W'=ig(\\phi^2+\\mu^2)$ or a $\\delta$-potential by an independent route, such as direct numerical quadrature of the twisted partition function along a steepest-descent contour or a Lefschetz-thimble sum, and compare the $\\alpha\\to0$ extrapolation with the Langevin values; disagreement would show the complex Langevin equilibrium is not the path integral. A second check is to extend the Langevin time and see whether the high-drift tail of $P(u)$ develops the exponential falloff that the criterion requires.","tokens_in":24788,"feed_emoji":"⚛️","tokens_out":11494,"duration_ms":112250,"temperature":0.7,"pith_summary":"This paper asks whether spontaneous supersymmetry breaking can be detected in zero-dimensional $\\mathcal{N}=2$ supersymmetric field theories when the action is complex, the regime where standard Monte Carlo fails because the probability weight is not positive. The authors use complex Langevin dynamics with real Gaussian noise: the real field is complexified and evolved in a fictitious time by the drift of the effective action, and expectation values are read from the stationary ensemble. To regularize the $0/0$ form of the auxiliary-field expectation value, they give the fermions twisted boundary conditions with phase $e^{i\\alpha}$ and use $\\langle B\\rangle_\\alpha$ as the order parameter, extrapolating $\\alpha\\to0$. The simulations reproduce the known analytic pattern for real polynomial superpotentials (even-degree $W'$ breaks SUSY, odd-degree preserves it) and yield new results for complex actions: $W'=ig(\\phi^2+\\mu^2)$ preserves SUSY, $W'=ig\\phi(\\phi^2+\\mu^2)$ breaks it, and the $PT$-symmetric $\\delta$-potentials preserve SUSY for $\\delta=1,2,3,4$. The paper itself flags that for the complex models the drift-magnitude distribution decays as a power law, while the standard correctness criterion requires exponential suppression, making the new complex-action conclusions conditional on the equilibrium being right.","feed_headline":"Simulations find which toy models break supersymmetry","feed_subtitle":"Complex Langevin dynamics with twisted boundary conditions reproduces known cases and decides new ones with complex actions.","key_machinery":"The load-bearing object is the twisted-boundary-condition order parameter: fermions are assigned boundary phase $e^{i\\alpha}$, which softly breaks supersymmetry and replaces the ill-defined $\\langle B\\rangle=0/0$ with a finite $\\langle B\\rangle_\\alpha$; spontaneous supersymmetry breaking is then read off from whether the $\\alpha\\to0$ extrapolation of $\\langle B\\rangle_\\alpha$ vanishes. The evolution is the complex Langevin update $\\phi(\\tau+\\Delta\\tau)=\\phi(\\tau)-\\Delta\\tau\\,\\partial S^{\\rm eff}_\\alpha/\\partial\\phi+\\sqrt{\\Delta\\tau}\\,\\eta(\\tau)$, where the effective action $S^{\\rm eff}_\\alpha=\\tfrac12 W'^2-\\ln(e^{i\\alpha}-1+W'')$ generates a complex drift and the noise is real Gaussian to tame imaginary excursions. Two diagnostics carry the reliability argument: the Fokker-Planck condition $\\langle\\tilde{L}O\\rangle=0$ on observables, and the probability distribution $P(u)$ of the drift magnitude $u$, whose exponential tail is the cited criterion for correct convergence.","core_discovery":"The central claim is that complex Langevin dynamics with stochastic quantization correctly captures the presence or absence of spontaneous supersymmetry breaking in a class of zero-dimensional $\\mathcal{N}=2$ models with real and complex actions, with the twisted-boundary auxiliary-field expectation value $\\langle B\\rangle_\\alpha$ serving as the order parameter. Where analytic answers exist, the method reproduces them: the real double-well $W'=g(\\phi^2+\\mu^2)$ gives a non-vanishing $\\langle B\\rangle$ at $\\alpha=0$ and hence broken SUSY, while a real even-degree polynomial $W'$ breaks SUSY and an odd-degree one preserves it. For complex actions the paper reports new determinations: $W'=ig(\\phi^2+\\mu^2)$ preserves SUSY, $W'=ig\\phi(\\phi^2+\\mu^2)$ breaks it, and the $PT$-symmetric $\\delta$-potential family $W'=-ig(i\\phi)^{1+\\delta}$ preserves SUSY for $\\delta=1,2,3,4$. The authors state that these simulations successfully capture the pattern, while noting that the complex-action results rest on the unverified reliability of the complex Langevin equilibrium in the face of power-law drift tails.","pith_inferences":["An editor's extension: the same order parameter could be applied at multiple couplings $g$ and masses $\\mu$; a stable $\\alpha\\to0$ intercept across a parameter scan would separate a genuine SUSY phase from a numerical artifact at a single point.","The observed power-law tail in $P(u)$ suggests a testable boundary: fit the tail exponent as a function of $g$, $\\delta$, and $\\alpha$, and compare predictions with a thimble or direct quadrature reference to map where the method can be trusted.","If the complex-action results are confirmed, the natural next probe is the continuous-$\\delta$ extension the authors mention; the $\\langle B\\rangle_\\alpha$ observable could reveal whether SUSY restoration is abrupt or smooth as $\\delta$ moves between integer values.","The method's success on these zero-dimensional models would not automatically transfer to higher-dimensional lattices; the drift-tail diagnostic should be re-checked there, since the power-law behavior may worsen with more degrees of freedom."],"forward_implications":["If the complex Langevin equilibrium is correct, then the same twisted-boundary $\\langle B\\rangle_\\alpha$ order parameter can diagnose spontaneous supersymmetry breaking in any zero-dimensional model where an analytic answer is unavailable.","The even/odd degree rule for real polynomial superpotentials becomes a validated benchmark: even-degree $W'$ breaks supersymmetry and odd-degree preserves it, and the method reproduces this rule numerically.","The complex double-well family splits by parity-like structure: $W'=ig(\\phi^2+\\mu^2)$ preserves supersymmetry while $W'=ig\\phi(\\phi^2+\\mu^2)$ breaks it, showing that complex coefficients do not uniformly destroy or preserve supersymmetry.","The $PT$-symmetric $\\delta$-potentials preserve supersymmetry for $\\delta=1,2,3,4$, matching the perturbative expectation that parity-breaking alone does not break supersymmetry, now at the nonperturbative level.","The method is positioned to be extended to the 1- and 2-dimensional supersymmetric models listed as future work, where the action is genuinely complex and no perturbative or exact result is available."],"supporting_citations":[{"why":"Introduces complex probabilities and the complex Langevin idea underlying all simulations.","marker":"[4]"},{"why":"Supply the Fokker-Planck operator criterion and diagnostics used in Appendix A1 to check that observables satisfy $\\langle\\tilde{L}O\\rangle=0$ at equilibrium.","marker":"[30, 31]"},{"why":"Introduce twisted boundary conditions as a regulator for supersymmetric models, which makes the auxiliary-field order parameter $\\langle B\\rangle_\\alpha$ well-defined.","marker":"[39, 40]"},{"why":"Establish the exponential-suppression condition on the drift distribution that the paper tests in Appendix A2.","marker":"[42, 43]"},{"why":"Defines the parity-broken supersymmetric model and gives the perturbative result that supersymmetry is unbroken, the analytic anchor for the $\\delta$-potential family.","marker":"[36]"},{"why":"Provides the exact one- and two-point correlation functions used as analytic benchmarks for the bosonic complex-action tests in Section II.","marker":"[38]"}],"fun_headline_variants":["Complex Langevin sims capture SUSY breaking pattern","Simulations decide SUSY breaking for complex actions","Langevin dynamics tests SUSY breaking in toy models","New simulations pin down SUSY breaking cases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim stands or falls on the assumption that the complex Langevin process samples the correct equilibrium distribution for the complex actions, even though the measured drift-magnitude distribution $P(u)$ falls off as a power law rather than the exponential tail that the cited correctness criterion demands.","fun_headline_variants_meta":{"raw":{"variants":["Complex Langevin sims capture SUSY breaking pattern","Simulations decide SUSY breaking for complex actions","Langevin dynamics tests SUSY breaking in toy models","New simulations pin down SUSY breaking cases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000532,"raw_usage":{"total_tokens":2517,"prompt_tokens":857,"completion_tokens":1660,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":1599}},"tokens_in":473,"tokens_out":1660,"duration_ms":12815,"temperature":1.0,"reasoning_tokens":1599,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:50:05.884974+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\langle B\\rangle_\\alpha$ for $W'=ig(\\phi^2+\\mu^2)$ or a $\\delta$-potential by an independent route, such as direct numerical quadrature of the twisted partition function along a steepest-descent contour or a Lefschetz-thimble sum, and compare the $\\alpha\\to0$ extrapolation with the Langevin values; disagreement would show the complex Langevin equilibrium is not the path integral. A second check is to extend the Langevin time and see whether the high-drift tail of $P(u)$ develops the exponential falloff that the criterion requires.","supporting_citations":[{"cited_title":"(A2) Once the equilibrium distribution is reached, assuming that it exists, we can remove theτ dependence from the observables","cited_arxiv_id":null,"evidence_quote":"Introduces complex probabilities and the complex Langevin idea underlying all simulations."},{"cited_title":"NUMERICAL INTEGRATION OF MULTIPLICATIVE NOISE STOCHASTIC DIFFERENTIAL EQUATIONS,","cited_arxiv_id":null,"evidence_quote":"Provides the exact one- and two-point correlation functions used as analytic benchmarks for the bosonic complex-action tests in Section II."}],"review_version":1}