{"id":"bdd4877b-14fb-4ec0-8ae5-9d7d0e7280c7","arxiv_id":"2504.13259","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Tritium bound to graphene decays with a beta spectrum whose endpoint shifts by tens of eV and acquires bound-state features, depending on the assumed final-state potential for helium.","lead":"This paper computes how the electron spectrum of tritium beta decay changes when the tritium is attached to a graphene sheet, using density functional theory together with a quantum treatment of the decaying nucleus. The predicted spectral shifts are tens of electronvolts, huge compared with the 100 meV resolution planned for the PTOLEMY experiment, which matters for how that experiment extracts the neutrino mass.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The spectrum is built on a single-nuclear-coordinate factorization that the paper itself says lacks a timescale/mass hierarchy; if substrate modes participate, the discrete lines and 29 eV endpoint in Fig. 5 are not established.","rationale":"The paper is careful and honest: it identifies this approximation explicitly and treats the sudden/semi-sudden/adiabatic spread as a measure of theory systematics. It also performs genuine DFT calculations and uses a transparent Fermi-golden-rule framework. The central qualitative observation, that a substrate can shift the endpoint by tens of eV relative to a bare nucleus, is plausible and does not rest solely on the one-dimensional overlap; Eq. (16) is dominated by potential minima. However, the claim of discrete, PTOLEMY-resolvable bound-state features does rest on the single-particle factorization, and the authors' own Section VII states there is no mass or timescale hierarchy to justify it. Section VI shows the carbon scaffold recoils with about an eV of energy, so substrate modes are not passive spectators. A quasi-harmonic Franck-Condon calculation is the natural, affordable check: it would diagnose whether the no-phonon lines survive or are washed out by substrate vibrational final states. This is the same weakness the reader identified, and it supports, rather than overturns, the conditional verdict. The paper should not be rejected, but the quantitative spectrum should not be treated as a finished prediction until this check is performed.","tokens_in":30404,"tokens_out":17444,"duration_ms":193014,"concrete_test":"Perform a quasi-harmonic (or 3N-dimensional harmonic) nuclear calculation: build the 3N x 3N Hessians of the DFT potential-energy surface at the initial T minimum and at the final sudden-He minimum, diagonalize to get the phonon modes, and compute the beta spectrum as a Franck-Condon sum over final vibrational states of the full 32-C plus 31-T/He supercell. If the no-phonon line keeps most of the weight and the discrete peaks in Fig. 5 shift by less than about 1 eV, the single-coordinate reduction is adequate; if sidebands carry comparable weight or broaden the features beyond PTOLEMY's 100 meV resolution, Eq. (12) is not a valid basis for the claimed spectrum.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section V solves Eq. (12) for one nucleus moving in U(r,z), with all other nuclei frozen at their initial equilibrium positions. The decay matrix element, Eq. (5), is thereby a one-dimensional overlap between psi_T00 and psi_He,nl. But the physical initial and final states are many-body nuclear wavefunctions; the T-to-He charge change displaces neighboring C and other T/He coordinates (visible in the recoil dynamics of Section VI), so the true spectrum contains Franck-Condon sums over substrate vibrational excitations. Section VII explicitly concedes that there is \"no truly large hierarchy between masses and/or time scales to rigorously justify this separation.\" The endpoint formula (16) uses only potential minima and so may survive, but the discrete bound-state structure claimed for PTOLEMY is a statement about no-phonon lines: if the no-phonon Franck-Condon weight is small or the phonon sidebands are broad, the qualitative spectrum in Fig. 5 is not what an experiment would see. The three sudden/semi-sudden/adiabatic schemes already bracket the electronic uncertainty; the nuclear many-body uncertainty is a separate, unquantified channel.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a combined DFT and single-particle nuclear quantum mechanics framework to compute the β-decay electron spectrum of tritium chemisorbed on graphene. The authors compute the tritium binding potential and, within sudden, semi-sudden, and adiabatic schemes, the potential felt by the helium daughter nucleus immediately after decay. They solve a single-particle Schrödinger equation for the decaying tritium and the final helium, compute decay matrix elements, and predict that the endpoint is shifted by roughly 29 eV relative to vacuum and that the spectrum contains discrete features from helium bound states. They also perform Born-Oppenheimer molecular dynamics to study the post-decay fate of the helium and the substrate. The central claim is that these effects are large compared with the 100 meV target resolution of PTOLEMY and thus constitute a measurable signature of the condensed-matter environment.","tokens_in":1455,"tokens_out":1775,"duration_ms":65660,"significance":"If the quantitative predictions hold, this work would be an important contribution to the planning and interpretation of future tritium-on-substrate neutrino experiments, particularly PTOLEMY, by showing that solid-state effects cannot be ignored at the stated energy resolution. The paper's strength lies in its explicit multi-scheme treatment of the electronic response (sudden, semi-sudden, adiabatic), its systematic exploration of tritium loading and magnetization states, and its unusually candid discussion of the limitations of the approximations. The prediction of a tens-of-eV endpoint shift is a concrete, falsifiable statement that can be tested once a tritiated-graphene source is operated. However, the quantitative spectrum and the discrete-line structure in particular rest on a single-nuclear-coordinate reduction whose validity is not established, as discussed in the major comments.","major_comments":[{"comment":"The central spectral predictions in Fig. 5 are computed from single-particle wave functions for one tritium/helium nucleus with all other nuclei frozen. The paper explicitly concedes in Section VII that there is no large hierarchy of masses or time scales that rigorously justifies this reduction. This matters because the discrete bound-state features shown in Fig. 5 are, in a many-body treatment, zero-phonon lines: the T-to-He charge change will, as shown by the paper's own Born-Oppenheimer molecular dynamics in Fig. 6, displace the carbon and neighboring tritium/helium coordinates, producing Franck-Condon sums over substrate vibrational excitations. If the zero-phonon weight is small or the phonon sidebands are broad, the discrete structure claimed to be visible with PTOLEMY's resolution is not what an experiment would observe. The authors should either provide an estimate of the no-phonon Franck-Condon factor within a tractable many-body model, or substantially soften the claim that the discrete spectral features are robust predictions.","section":"Section VII, Eq. (5), Eq. (12)"},{"comment":"The 29 eV endpoint shift, which is the paper's headline quantitative result, follows from the difference U_T^0 - U_He^0 obtained by aligning the tritium and helium potentials using a displaced-point-charge model with parameters (notably the 0.7 Å displacement between the carbon nucleus and the bond charge) that are not varied and carry no quoted uncertainty. Because the comparison target is a 100 meV energy resolution, even a few eV of systematic uncertainty in the alignment would not destroy the qualitative claim, but an unquantified alignment error is material to the precise value. The authors should provide a sensitivity study or a conservative estimate of the alignment uncertainty, for example by varying the Coulomb model parameters within physically reasonable ranges or by cross-checking with an independent alignment method.","section":"Section V, Eq. (16), SI Section SI.I.F"},{"comment":"The effective potential U(r,z) is fitted to DFT data at only three values of z (1.11, 1.61, 1.95 Å) and then extrapolated throughout the region where the nuclear wave functions are localized. The bound-state spectrum, the wave-function overlaps, and hence the discrete line separations in Fig. 5 depend sensitively on the shape of the potential near its minimum, especially for the narrow sudden potential. No convergence test with respect to the number of z grid points or a direct comparison against additional DFT points is reported. The authors should demonstrate that the discrete spectral features are not artifacts of the three-point interpolation.","section":"SI Section SI.II, Eq. (SI.6), Table SI.4"}],"minor_comments":[{"comment":"The phrase 'for large enough distances (≳ 10−15 Å)' appears to contain a typo; 10−15 Å is one femtometer and cannot be the regime where the Coulomb tail is valid. The intended scale is presumably a few Å or more.","section":"SI Section SI.I.F"},{"comment":"The neglect of the Coulomb distortion of the outgoing electron wave function is stated but not quantified. Since the paper emphasizes a 100 meV resolution, a brief estimate of the size of this effect near the endpoint would help the reader judge whether it is negligible at the target precision.","section":"Section II, footnote 2"},{"comment":"The spectrum in Fig. 5 is computed only for the 100% loading case with a specific relaxation prescription, while Section III documents a strong dependence of the potential parameters on loading and configuration. A sentence clarifying how the curves in Fig. 5 would change under the other prescriptions, or why the chosen case is representative, would improve the readability.","section":"Section V"}],"recommendation":"major_revision","confidential_remarks":"The paper is transparent and the authors are appropriately cautious in Section VII, but the load-bearing quantitative claims—especially the discrete spectral structure—are not yet supported by the single-nuclear-coordinate treatment. A major revision that provides even a rough estimate of Franck-Condon/sideband effects, or that explicitly reframes the discrete-line claims as illustrative of a framework rather than as quantitative predictions, would bring the paper into line with its stated significance. The alignment uncertainty should also be quantified. These are fixable within the scope of the manuscript, so major revision is appropriate rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nYou should know two things about this paper. First, it is the first calculation I know that takes the difference between tritium and helium interaction with graphene seriously in the beta-decay spectrum, and the claimed ~29 eV endpoint shift relative to vacuum is a real effect if the DFT potentials are right. Second, the discrete spectral features that would actually change PTOLEMY's analysis are built on a single-nucleus factorization that the authors themselves admit has no rigorous timescale or mass hierarchy.\n\nWhat is genuinely new: the sudden and semi-sudden treatments of the final helium potential, and the explicit revision of the earlier PTOLEMY assumption that T and He see the same potential. The paper is also unusually honest. Section VII lists the limitations, including missing vibrational and electronic excitations, the single-particle reduction, and the lack of a rigorous non-adiabatic treatment. That candor is real credit.\n\nThe endpoint shift itself is probably robust. It comes from the difference between potential minima, which DFT should get to within an eV or so at this level. The 29 eV number dwarfs PTOLEMY's 100 meV resolution, so the conclusion that a substrate-specific template is needed is solid. The structure of the discrete lines, however, is loaded. The wavefunction overlaps are computed with a single nuclear coordinate and all other nuclei frozen. The stress-test worry is on point: after the decay the charge change displaces neighboring atoms, so phonon sidebands and Franck-Condon sums over substrate modes will redistribute weight. The no-phonon lines in Fig. 5 could be strongly suppressed. The paper's own MD shows significant helium recoil and substrate excitation, which suggests the frozen-lattice assumption for the final state is questionable.\n\nThe interpolation from three z values and the fitted P, k, S, W parameters also mean the potential shape is not controlled beyond the minimum region. And there are no error bars anywhere. So the quantitative spectrum is a first estimate, not a prediction to hand to experimentalists.\n\nMy bottom line: the central qualitative claim, that the substrate shifts the endpoint by tens of eV, holds up well enough to warrant serious referee time. The discrete-line structure should be treated as a motivation for a proper many-body nuclear treatment, not as a finished template. I'd send this to a good referee and ask for heavy revision, and I'd want code/data released if it is to be used by PTOLEMY. I'd take it to the reading group because it makes you think about where the approximations actually bite.\n\nBest,","headline":"The ~29 eV substrate endpoint shift is a robust qualitative result worth taking seriously; the discrete bound-state lines in the spectrum are not yet established.","tokens_in":31198,"tokens_out":2081,"would_cite":true,"duration_ms":21118,"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":"Tens of eV shift tritium's beta endpoint on graphene and replace the smooth vacuum spectrum with discrete helium-state peaks.","keywords":["tritium beta decay","neutrino mass endpoint","graphene substrate","sudden approximation","density functional theory","final-state interactions","helium bound states"],"falsifier":"A measurement of the beta-electron spectrum from tritium chemisorbed on graphene, with resolution of a few eV near the endpoint, would settle the claim: if no discrete helium-bound-state peaks and no endpoint shift of tens of eV appear, the predicted spectrum is wrong. A complementary check is the charge state of the emitted helium: the sudden prescription predicts He++, the semi-sudden predicts He+, and the adiabatic predicts neutral He, so measuring that branching would discriminate among the schemes.","tokens_in":30181,"feed_emoji":"⚛️","tokens_out":9710,"duration_ms":94546,"temperature":0.7,"pith_summary":"The paper argues that when tritium is chemisorbed on graphene, the beta-decay electron spectrum is no longer the smooth vacuum spectrum that neutrino-mass experiments usually fit. Because the daughter helium nucleus, immediately after decay, feels a deep potential created by the nearly unchanged electronic structure, the endpoint shifts by tens of electronvolts and splits into discrete peaks, one for each bound state of helium on the substrate. The paper builds a computational scheme combining density-functional theory for the tritium and helium interaction potentials with a full quantum treatment of the decaying nucleus, using sudden and semi-sudden approximations for the electronic response. A near-endpoint measurement with roughly 0.1 eV resolution would see these shifts, so the work has direct consequences for planned tritium-on-graphene neutrino-mass experiments.","feed_headline":"Tens of eV shift tritium's beta-decay endpoint on graphene","feed_subtitle":"The substrate is not a spectator: discrete helium peaks would replace the vacuum endpoint curve at planned resolutions.","key_machinery":"The load-bearing machinery is a single-particle Schrödinger equation for the decaying nucleus moving in a potential obtained from density-functional theory, combined with an electron-response model for the daughter helium. The sudden approximation freezes the pre-decay electron density when computing the helium potential, producing a deep attractive well; the semi-sudden approximation keeps the density frozen but allows one electron to follow the helium, producing a shallower well; the adiabatic approximation uses the relaxed ground-state density, producing a repulsive potential. The decay rate is then computed from Fermi's golden rule as an overlap integral of the initial tritium wave function, the final helium wave function, and the outgoing electron and neutrino plane waves, and a small-momentum expansion makes the near-endpoint rate analytically tractable.","core_discovery":"The central discovery is that the substrate cannot be treated as a spectator: the final-state interaction between the daughter helium and graphene reshapes the near-endpoint beta spectrum. In the sudden approximation, where the electronic density is frozen from before the decay, the helium potential well is about 58.5 eV deep and its minimum is roughly 29 eV below the tritium minimum; the endpoint for a helium bound final state therefore shifts by tens of eV relative to vacuum, and each helium bound state contributes its own partial endpoint. The paper also finds that the semi-sudden scheme, in which one electron follows the helium, moves the continuum onset close to the endpoint, while the fully adiabatic, relaxed-electron potential is repulsive and gives no bound final states. All three schemes yield spectra whose shapes differ from the vacuum decay, and the paper quantifies those differences as a first measure of theory systematics.","pith_inferences":["Extending beyond the paper: if the discrete helium peaks are real, their measured positions and intensities could be inverted to extract the substrate's electron-relaxation time scale, turning a neutrino experiment into a condensed-matter probe.","Extending beyond the paper: the sudden-to-adiabatic family of potentials suggests a continuous electron-response parameter; a non-adiabatic calculation could show how the peaks broaden into a continuum and shrink the current spread between schemes.","Extending beyond the paper: the same DFT-plus-nuclear-Schrödinger machinery should transfer to tritium on other two-dimensional hosts, where the endpoint shift would encode the host's electronic structure and could in principle be engineered.","Extending beyond the paper: measuring the helium charge-state branching after decay, He++, He+, or neutral He, would be a comparatively cheap decisive test of the three response schemes before a full high-resolution spectrum is built."],"forward_implications":["An experiment resolving about 100 meV near the endpoint should detect an endpoint tens of eV away from the vacuum value, so the substrate spectrum, not the vacuum one, is the relevant template for neutrino-mass extraction.","The near-endpoint spectrum contains discrete helium-bound-state features with separations set by the shape of the sudden well; the number and spacing of these peaks carry information about the electronic response.","The continuum onset depends on the electron-response scheme: with one electron following the helium it sits close to the endpoint, whereas with fully frozen electrons it lies far away, changing where a smooth tail begins.","After the decay the helium is expelled from the graphene within femtoseconds with roughly 1-4 eV of kinetic energy, and periodic releases heat the layer by a few eV; the simulations find no structural damage except in rare head-on collisions at weak bonds."],"supporting_citations":[{"why":"Defines the tritium-on-graphene experimental concept and its expected sensitivity, which motivates the need for a realistic spectrum.","marker":"[15]"},{"why":"Earlier theory update that assumed tritium and helium interact with graphene essentially the same; this is the assumption the paper revises.","marker":"[17]"},{"why":"Establishes nearly 1:1 hydrogen stoichiometry and the band-gap opening used to justify the Born-Oppenheimer treatment.","marker":"[24]"},{"why":"Supplies the density-functional methodology for hydrogenated graphene potentials and binding energies.","marker":"[34]"},{"why":"Provides the generalized-gradient exchange-correlation functional used in all DFT potentials.","marker":"[40]"},{"why":"Supplies the sudden-approximation treatment of final-state interactions in tritium beta decay.","marker":"[56]"},{"why":"Provides the molecular-tritium measurement of final-state charge distributions that the paper cites as the analogue needed for graphene.","marker":"[63]"}],"fun_headline_variants":["Graphene shifts tritium beta-decay endpoint by tens of eV","Substrate reshapes tritium beta spectrum: helium-graphene well","Tritium decay on graphene: tens of eV endpoint shift","Graphene alters tritium beta spectrum via helium interaction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The single most load-bearing premise is that the entire many-body decay problem can be reduced to one nucleus moving in a fixed single-particle potential; the paper itself notes there is no genuinely large hierarchy of masses or time scales that rigorously justifies that separation.","fun_headline_variants_meta":{"raw":{"variants":["Graphene shifts tritium beta-decay endpoint by tens of eV","Substrate reshapes tritium beta spectrum: helium-graphene well","Tritium decay on graphene: tens of eV endpoint shift","Graphene alters tritium beta spectrum via helium interaction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"total_tokens":1239,"prompt_tokens":895,"completion_tokens":344,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":267}},"tokens_in":511,"tokens_out":344,"duration_ms":3583,"temperature":1.0,"reasoning_tokens":267,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:11:57.120427+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A measurement of the beta-electron spectrum from tritium chemisorbed on graphene, with resolution of a few eV near the endpoint, would settle the claim: if no discrete helium-bound-state peaks and no endpoint shift of tens of eV appear, the predicted spectrum is wrong. A complementary check is the charge state of the emitted helium: the sudden prescription predicts He++, the semi-sudden predicts He+, and the adiabatic predicts neutral He, so measuring that branching would discriminate among the schemes.","supporting_citations":[{"cited_title":"Gap open- ing in double-sided highly hydrogenated free-standing graphene,","cited_arxiv_id":null,"evidence_quote":"Establishes nearly 1:1 hydrogen stoichiometry and the band-gap opening used to justify the Born-Oppenheimer treatment."},{"cited_title":"Nano-scale corrugations in graphene: A density functional theory study of structure, electronic properties and hydrogena- tion,","cited_arxiv_id":null,"evidence_quote":"Supplies the density-functional methodology for hydrogenated graphene potentials and binding energies."}],"review_version":2}