{"id":"929c1962-b921-40d1-8431-ae1afd156a06","arxiv_id":"2506.22081","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Non-observation of periodic modulation in tritium beta decay sets a new 95% CL exclusion on ultralight Nelson-Barr scalar dark matter: f below 7.0e9 to 1.4e7 GeV is ruled out for m_phi in 3.4e-23 to 1.7e-20 eV.","lead":"This paper uses the lack of periodic oscillations in tritium decay data to constrain an ultralight scalar dark matter particle that arises from the Nelson-Barr solution to the strong CP problem. It derives how this scalar would slightly change nuclear masses and binding energies, producing tiny time-varying decay rates, and converts a null result into new exclusions on the scalar mass and decay constant.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted direct CKM modulation of |V_ud|^2 dominates the tritium decay residual; the central 95% CL limit on f is off by orders of magnitude.","rationale":"The reader's CONDITIONAL verdict focused on nuclear-model uncertainties and a sign inconsistency, but the more serious problem is a missing leading contribution. The paper's own Eq. (17) states that |V_ud|^2 oscillates with amplitude of order κT_ud ~ 0.08 φ/f. In any weak decay, the rate is proportional to |V_ud|^2, so this modulation enters directly at first order. The paper instead computes only the phase-space modulation caused by nuclear mass and binding-energy shifts, which yields 2.44×10^-5 φ/f. The direct CKM term is roughly 3,000 times larger. Because the likelihood in Eq. (60) depends on the signal amplitude through As, the quoted exclusion limit on f scales linearly with this amplitude; an error of this size changes the central quantitative claim by orders of magnitude. This is not a matter of outside-consensus disagreement or rough nuclear approximations; it follows from the model's own CKM time-dependence and standard weak-interaction physics. The paper could be corrected by adding the omitted term, but as written the headline constraint is not reliable. The reader's identified sign inconsistency is real but secondary, which is why agreement is only partial.","tokens_in":14415,"tokens_out":12743,"duration_ms":124218,"concrete_test":"Add the direct CKM contribution to Eq. (46): replace 2.44×10^-5 φ/f by −2 T_ud κ φ/f + 2.44×10^-5 φ/f (or the exact expression for δ|V_ud|^2/|V_ud|^2 from Eqs. (17)–(18)), propagate this into As in Eq. (59), and recompute the 95% CL exclusion line from Eq. (60). If the resulting bound on f changes by more than an order of magnitude relative to Fig. 1, the headline constraint is not the model's prediction and the paper must be revised.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central amplitude, Eq. (46), takes δΓ/Γ ≈ δI^β/I^β = 2.44×10^-5 φ/f, coming solely from the phase-space shift from δ(M_i−M_f) in Eq. (41). But the model itself predicts a much larger direct effect: Eq. (17) gives |V_ud|^2 ≈ |V^0_ud|^2 (1 − 2 T_ud κ φ/f). With the paper's own inputs T_ud = V^d_ud V^d_cd / |V^0_ud|^2 ~ 0.2 and κ ~ 0.2, the fractional modulation is δ|V_ud|^2/|V_ud|^2 ~ −8×10^-2 φ/f. Since tritium β-decay rate Γ is proportional to |V_ud|^2 × I^β, the full residual is δΓ/Γ ≈ δ|V_ud|^2/|V_ud|^2 + δI^β/I^β. The direct CKM term exceeds the quoted phase-space term by roughly three orders of magnitude. The paper never includes this term; Eqs. (45)–(47) only propagate the phase-space integral. Consequently, the bound on f derived from Eq. (61) is not the model's prediction. Including the direct V_ud term would shift the excluded region by about the same factor, making the constraint far stronger and potentially conflicting with existing CKM-variation limits shown in Fig. 1. This is an internal zero-parameter issue, independent of the nuclear approximations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper considers ultralight scalar dark matter in a Nelson-Barr solution to the strong CP problem, where the pseudo-Goldstone field φ oscillates and induces time-dependent modulations of CKM matrix elements and quark masses. The authors calculate the resulting periodic modulation of the tritium β-decay rate through the neutron-proton mass difference and nuclear binding energies, evaluate the phase-space integral, and apply a frequentist hypothesis test with stochastic-amplitude marginalization to the null JRC tritium data. They obtain a 95% CL exclusion on the decay constant f below 7.0×10^9 to 1.4×10^7 GeV for masses m_φ in the range 3.4×10^-23 to 1.7×10^-20 eV, and they compare this with existing CKM-variation and equivalence-principle constraints.","tokens_in":14706,"tokens_out":5863,"duration_ms":61003,"significance":"If the calculation were correct, the claimed exclusion would be a new and interesting probe of ultralight scalar dark matter in the Nelson-Barr framework, using a genuine null dataset and a statistical pipeline that partly validates itself by reproducing the axion limit of Ref. [14]. The paper is clearly organized and the frequentist/stochastic-amplitude treatment follows standard methods. However, the central decay-rate amplitude in Eq. (46) omits the direct |V_ud|^2 modulation that the same model predicts in Eq. (17); until that term is included, the numerical exclusion region cannot be considered a prediction of the model. This is the main load-bearing issue.","major_comments":[{"comment":"The decay-rate residual in Eq. (46) includes only the phase-space shift δI^β/I^β, but the model itself predicts an oscillating CKM element: Eq. (17) gives |V_ud|^2 ≈ |V^0_ud|^2 (1 − 2 T_ud κ φ/f). With the inputs quoted in §II, κ∼0.2 and T_ud∼0.2, this implies δ|V_ud|^2/|V_ud|^2 ≈ −8×10^-2 φ/f, which is about three orders of magnitude larger than the +2.44×10^-5 φ/f phase-space term. Since the tritium decay rate is proportional to |V_ud|^2 times the phase-space integral, Eq. (46) should be replaced by δΓ/Γ ≈ δ|V_ud|^2/|V_ud|^2 + δI^β/I^β. As written, the amplitude, the coefficient b in Eq. (61), and the exclusion region in Fig. 1 are not the model's prediction. Including the direct CKM term would strengthen the constraint on f by roughly three orders of magnitude, and the resulting region should be checked against the CKM-variation limits shown in Fig. 1.","section":"§III.C, Eqs. (17) and (46)"},{"comment":"There is an internal sign and normalization inconsistency in the derivation of δM_i − δM_f. Eq. (23) predicts a negative coefficient, (m_n − m_p) ≈ (1.55 − 1.75×10^-7 φ/f) MeV. Combining this with Eq. (38) and B_f − B_i ≈ −0.76 MeV in Eq. (39) gives a φ/f coefficient of approximately −1.75×10^-7 − 0.76×(4.55×10^-7/8.1) ≈ −2.18×10^-7 MeV, not +1.33×10^-7 MeV as stated in Eq. (41). The sign flip and the mismatch in magnitude indicate at least one intermediate step is inconsistent. Because the bound is derived from the squared amplitude, the sign error alone would not change the limit, but the magnitude discrepancy and the unexplained sign change should be corrected before the amplitude can be considered reliable.","section":"§III.A–C, Eqs. (23), (40), (41)"}],"minor_comments":[{"comment":"The display of the nonzero couplings lacks commas between g1 ≠ 0 and ˜g2 ≠ 0, and the footnote 'vector-like up-type 1 quark' appears to contain a typographical artifact.","section":"§II and Eq. (5)"},{"comment":"In the sentence before Eq. (53), 'TakingAs = 0' should be 'Taking As = 0', and in Eq. (60) the notational chain 'α = ... ≥ β = ...' is confusing because the two error rates are written as a single continued equality.","section":"§IV.A and Eq. (60)"},{"comment":"The label 'MACROSCOPE' in Fig. 1 should read 'MICROSCOPE'.","section":"Fig. 1"},{"comment":"In the discussion of stochastic amplitudes, the meaning of the ensemble average and the observation-time averaging in Eq. (54) could be clarified; in particular, the relation between the fixed amplitude Φ_DM and the Rayleigh-distributed Φ0 is stated but not formally derived.","section":"§IV.B"}],"recommendation":"major_revision","confidential_remarks":"The missing direct |V_ud|^2 term is the central issue and is fully internal to the model, so the exclusion region in Fig. 1 is not the model's prediction as it stands. If the recomputed constraint, including the direct CKM term, turns out to be already excluded by existing CKM-variation limits, the main novelty of the paper would largely disappear. I recommend asking the authors to recompute the amplitude and the limit, and to fix the sign/normalization inconsistency in §III."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the punchline: the main numerical result is not trustworthy. The authors derive a time-dependent |V_ud|^2 in Eq. (17) with fractional amplitude around -0.09 φ/f, but when they compute the tritium decay rate in Sec. III.C, they only propagate the phase-space shift from the endpoint energy, which is about 2.44e-5 φ/f. Since Γ ∝ |V_ud|^2, the direct CKM term dominates by three orders of magnitude and should be included. I checked the algebra: T_ud ≈ 0.23 and κ ≈ 0.2 give δ|V_ud|^2/|V_ud|^2 ≈ -0.092 φ/f. This is an internal zero-parameter issue, not a question of nuclear modeling. Correcting it changes the 95% CL exclusion by a factor of ~4000 in f, making the bound much stronger and the comparison with the axion limit qualitatively different.\n\nCredit where it's due: the idea of probing Nelson-Barr scalar DM through nuclear decays is novel, the paper is clearly structured, and the frequentist pipeline is verified by reproducing the axion limit from Ref. [14]. The nuclear physics chain (OPE potential, square-well wavefunction, SU(4) scaling) is a reasonable first pass, though the authors don't propagate uncertainties from these approximations.\n\nThe other soft spots are secondary. There is a sign mismatch between Eq. (23) and Eq. (41) that suggests a slip in the intermediate algebra; the bound uses the squared amplitude so the sign cancels, but the magnitude also looks off by about 60%. The parameters κ and log(Λ/v) have order-one freedom that isn't reflected in the error analysis. These are fixable. The missing CKM term is not.\n\nWho is this for? Someone working on ultralight scalar DM in Nelson-Barr models or on nuclear decay as a DM probe would want to see this, but only in a corrected form. As it stands, the paper's central constraint is not a prediction of the model. I'd recommend sending it to review, but with the expectation of major revision: the authors need to include the direct CKM modulation and redo the limit. The idea is worth refereeing, but the current numbers should not be quoted.","headline":"The paper's central constraint is off by three orders of magnitude because it omits the direct CKM modulation of |V_ud|^2 that the model itself predicts.","tokens_in":15286,"tokens_out":9188,"would_cite":false,"duration_ms":84332,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The null observation of periodic variations in tritium beta decay excludes, at 95% confidence, a Nelson-Barr ultralight scalar dark matter with decay constant below 7.0e9 to 1.4e7 GeV for masses between 3.4e-23 and 1.7e-20 eV.","keywords":["ultralight scalar dark matter","Nelson-Barr model","strong CP problem","tritium beta decay","periodic nuclear decay rates","CKM matrix modulation","frequentist hypothesis testing","nuclear binding energy"],"falsifier":"A concrete check is to recompute the amplitude from first principles: the neutron-proton mass-difference coefficient in Eq. (23) and the final decay-rate coefficient in Eq. (41) have opposite signs and different magnitudes, so a corrected calculation would settle the absolute scale of the bound. Experimentally, a tritium dataset with greater statistical power that still shows no sinusoidal residual at the predicted amplitude and frequency would push the excluded $f$ higher, while a residual matching the predicted linear-in-$1/f$ scaling would confirm the Nelson-Barr interpretation.","tokens_in":14171,"feed_emoji":"⌛","tokens_out":14513,"duration_ms":143973,"temperature":0.7,"pith_summary":"The paper tries to establish that the null observation of periodic variations in the tritium $\\beta$-decay rate can be turned into a new exclusion on an ultralight scalar dark-matter candidate in the Nelson-Barr model, a proposal that solves the strong CP problem by spontaneous CP breaking. In that model the dark-matter scalar modulates the CKM matrix and the quark masses; the paper shows how those oscillations propagate through the neutron-proton mass difference and nuclear binding energies into a small oscillating residual in the tritium decay rate. Using twelve years of tritium data in which no statistically significant periodicity appears, it derives the quantitative claim: at 95% confidence the decay constant $f$ is excluded below $7.0\\times 10^9$ to $1.4\\times 10^7$ GeV for masses $m_\\phi$ in the range $3.4\\times 10^{-23}$ to $1.7\\times 10^{-20}$ eV. If correct, this gives a new way to probe a strong-CP solution through ordinary radioactivity, with the bound set by the steadiness of a nuclear clock rather than by a dedicated dark-matter detector.","feed_headline":"Steady tritium decay rules out ultralight Nelson-Barr dark matter","feed_subtitle":"Steady tritium beta decay over 12 years puts a 95% bound on the Nelson-Barr scalar's coupling.","key_machinery":"The load-bearing object is the oscillating residual $I(\\phi)=(\\Gamma(\\phi)-\\langle\\Gamma\\rangle)/\\langle\\Gamma\\rangle = 2.44\\times 10^{-5}\\phi/f$, which converts a dark-matter-induced frequency shift into a measurable fractional change in the decay rate. Two nuclear inputs carry the calculation: the $\\phi$-dependent neutron-proton mass difference, fixed by the chiral relation $4c_5B_0(m_u-m_d)$, and the $\\phi$-dependent tritium binding energy, built from a one-pion-exchange potential, a three-dimensional square-well deuteron wavefunction, and empirical SU(4) scaling from the deuteron to tritium. These enter the $\\beta$-decay phase-space integral, and the resulting residual is tested with a Rice-distributed likelihood whose Type-I and Type-II error rates, with the ultralight-field amplitude treated as Rayleigh-distributed at low mass, produce the exclusion. A key structural identity is that the residual scales linearly with $1/f$, whereas the axion analogue scales as $1/f_a^2$, which is why the Nelson-Barr bound is weaker for a given decay constant.","core_discovery":"In the Nelson-Barr model, spontaneous CP breaking produces a pseudo-Nambu-Goldstone boson $\\phi$; when this field is ultralight and forms the local dark matter, it oscillates at frequency $m_\\phi$ and, through the field-dependent CKM matrix, induces periodic modulations of weak-interaction parameters and quark masses. The paper's central calculation follows these modulations into nuclear physics: the neutron-proton mass difference shifts through the QCD term $4c_5B_0(m_u-m_d)$, and the tritium binding energy shifts through a one-pion-exchange potential evaluated with a square-well deuteron wavefunction and SU(4) scaling. The combined effect is an oscillating residual in the tritium $\\beta$-decay rate, $I(\\phi)=2.44\\times 10^{-5}\\,\\phi/f$, with $\\phi/f = \\sqrt{2\\rho_{DM}}/(f m_\\phi)\\cos(m_\\phi t+\\delta)$. Applying a frequentist hypothesis test to the null tritium data yields the paper's main result: at 95% CL, decay constants below $7.0\\times 10^9$ to $1.4\\times 10^7$ GeV are excluded for $m_\\phi$ between $3.4\\times 10^{-23}$ and $1.7\\times 10^{-20}$ eV.","pith_inferences":["Beyond the paper, the same residual formula could be applied to other beta-emitting isotopes; nuclei with larger $Q$-values would give a larger $\\delta E$/keV lever arm and could push the excluded $f$ further at the same mass.","Beyond the paper, an independent recalculation of the amplitude from first principles would be a useful check: the printed coefficient changes sign between Eq. (23) and Eq. (41), and although the current bound depends on the square of the amplitude and may be insensitive to that sign, a corrected coefficient would shift the absolute scale of the exclusion.","Beyond the paper, the linear-$1/f$ scaling implies that combining several isotopes could help distinguish the Nelson-Barr scalar from axion-like dark matter in future positive detections, since the two models predict different amplitude-versus-frequency relations."],"forward_implications":["A Nelson-Barr scalar with $f$ below roughly $10^7$--$10^{10}$ GeV and mass in the quoted window would imprint a periodic modulation on the tritium decay rate that the twelve-year dataset does not contain, so that parameter region is ruled out.","At the same mass and decay constant, the excluded region is about three orders of magnitude weaker than the axion-model bound because the Nelson-Barr residual carries an extra $(m_q/v)^2$ one-loop suppression and scales as $1/f$.","For masses below $10^{-16}$ eV the scalar-field amplitude must be described as a Rayleigh-distributed stochastic variable; the paper's bound already includes this correction, so it remains valid when the coherence time is longer than the observation time.","The result adds radioactive-decay clocks to the set of probes of the Nelson-Barr parameter space, complementing equivalence-principle tests and searches for CKM-matrix variations."],"supporting_citations":[{"why":"Supplies the Nelson-Barr model with ultralight scalar dark matter and the formulas for $\\phi$-dependent CKM elements and quark masses.","marker":"[30]"},{"why":"Provides the tritium null result, the axion-model residual calculation this work extends, and the experimental parameters ($N=50000$, frequency range) that set the search window.","marker":"[14]"},{"why":"Is the tritium beta-decay dataset from which the null periodic signal and the 0.4% rate uncertainty are taken.","marker":"[51]"},{"why":"Supplies the frequentist Type-I/Type-II error analysis and the Rayleigh-distribution correction for the stochastic ultralight-field amplitude.","marker":"[52]"},{"why":"Gives the NLO chiral formula for the QCD contribution to the neutron-proton mass difference that the scalar field modulates.","marker":"[38]"},{"why":"Provides the first-order perturbation method using a three-dimensional square-well wavefunction to compute the deuteron binding-energy shift.","marker":"[39]"},{"why":"Supplies the one-pion-exchange potential, the square-well wavefunction parameters, and the deuteron binding energy used in the calculation.","marker":"[40]"},{"why":"Relates the pion mass to the light quark masses, carrying the scalar-field dependence into the pion potential.","marker":"[41]"},{"why":"Gives the pion-nucleon coupling $g_{\\pi NN}$ as a function of pion and nucleon masses, used in the potential.","marker":"[42]"},{"why":"Provides the empirical SU(4) scaling that converts the deuteron binding-energy shift into the average tritium binding energy used in Eq. (38).","marker":"[49]"}],"fun_headline_variants":["Tritium decay data tighten Nelson-Barr dark matter limits","Nuclear decay clocks rule out ultralight Nelson-Barr dark matter","Steady tritium decay constrains dark matter from CP solution","Tritium beta decay excludes Nelson-Barr scalar dark matter","Ultralight Nelson-Barr dark matter limited by tritium decay"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result depends on the size of the tiny periodic wiggle the authors compute in the tritium decay rate; that size comes from approximate nuclear calculations, and an error there would move the excluded region.","fun_headline_variants_meta":{"raw":{"variants":["Tritium decay data tighten Nelson-Barr dark matter limits","Nuclear decay clocks rule out ultralight Nelson-Barr dark matter","Steady tritium decay constrains dark matter from CP solution","Tritium beta decay excludes Nelson-Barr scalar dark matter","Ultralight Nelson-Barr dark matter limited by tritium decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1441,"prompt_tokens":956,"completion_tokens":485,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":395}},"tokens_in":572,"tokens_out":485,"duration_ms":5068,"temperature":1.0,"reasoning_tokens":395,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:12:00.842541+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to recompute the amplitude from first principles: the neutron-proton mass-difference coefficient in Eq. (23) and the final decay-rate coefficient in Eq. (41) have opposite signs and different magnitudes, so a corrected calculation would settle the absolute scale of the bound. Experimentally, a tritium dataset with greater statistical power that still shows no sinusoidal residual at the predicted amplitude and frequency would push the excluded $f$ higher, while a residual matching the predicted linear-in-$1/f$ scaling would confirm the Nelson-Barr interpretation.","supporting_citations":[{"cited_title":"Bento, G","cited_arxiv_id":null,"evidence_quote":"Supplies the Nelson-Barr model with ultralight scalar dark matter and the formulas for $\\phi$-dependent CKM elements and quark masses."},{"cited_title":"Pomm ´e and K","cited_arxiv_id":null,"evidence_quote":"Provides the tritium null result, the axion-model residual calculation this work extends, and the experimental parameters ($N=50000$, frequency range) that set the search window."},{"cited_title":"The absolute mass of neutrino and the first unique forbidden beta-decay of 187Re","cited_arxiv_id":"1101.3413","evidence_quote":"Is the tritium beta-decay dataset from which the null periodic signal and the 0.4% rate uncertainty are taken."},{"cited_title":"Pomm ´e et al., Metrologia 54, 19 (2016)","cited_arxiv_id":null,"evidence_quote":"Supplies the frequentist Type-I/Type-II error analysis and the Rayleigh-distribution correction for the stochastic ultralight-field amplitude."},{"cited_title":"Gasser, H","cited_arxiv_id":null,"evidence_quote":"Gives the NLO chiral formula for the QCD contribution to the neutron-proton mass difference that the scalar field modulates."}],"review_version":1}