In the scenario where an evolving ultralight scalar field (dark matter or dark energy) couples to gluons, the QCD interaction scale Λ_QCD becomes time-dependent while the fine-structure constant α and the electron mass stay fixed. Flambaum and Mansour (JHEP 11 (2025) 086) translated atomic-clock, Oklo, quasar, and Big Bang Nucleosynthesis data into conditional bounds on δΛ_QCD/Λ_QCD within this minimal gluonic scenario and showed that the 8.4 eV 229Th nuclear clock transition would amplify a Λ_QCD drift by four orders of magnitude. The open problem is to convert these large but conditional sensitivities into robust constraints: reduce the nuclear/hadronic theory uncertainties (BBN binding-energy response, Oklo resonance near-cancellation, 229Th sensitivity coefficient, ~10%-level proton mass and g-factor scalings), generalize beyond the gluon-only coupling assumption, and obtain direct 229Th-based limits on Λ_QCD variation. No later work delivering such robust, model-generalized bounds is known to date. The first continuously operating solid-state 229Th nuclear clock (2026) has produced a first direct drift limit on Λ_QCD and new oscillation bounds on strong-force and quark couplings, but over a one-day baseline and with the 229Th sensitivity coefficients still unknown within order-of-magnitude bands; lattice-QCD inputs for the nuclear responses are likewise advancing but incomplete.

Background

Tests of whether the 'constants' of nature vary are only model-independent for dimensionless quantities such as the fine-structure constant α; a dimensionful parameter can meaningfully be said to vary only inside a specified framework [Uzan 2011]. The quantity at issue here is Λ_QCD, the intrinsic energy scale of quantum chromodynamics (of order 200–300 MeV) generated by dimensional transmutation (the emergence of an energy scale from the running of the dimensionless QCD coupling), which sets the nucleon mass, nucleon magnetic moments, and nuclear binding energies. Nearly all searches for varying constants — atomic clock comparisons, quasar absorption spectra, the Oklo natural reactor, Big Bang Nucleosynthesis (BBN) — had targeted α or the dimensionless mass ratios Xe,q=me,q/ΛQCDX_{e,q}=m_{e,q}/\Lambda_{\rm QCD}, treating Λ_QCD as fixed. A well-motivated beyond-Standard-Model framework in which Λ_QCD itself drifts is an ultralight scalar field φ (a dark-matter or dark-energy candidate) coupled to gluons through an operator ϕnGμνaGaμν\phi^n G^a_{\mu\nu}G^{a\mu\nu}: this shifts Λ_QCD in time while leaving α and the electron mass approximately untouched. Flambaum and Mansour (JHEP 11 (2025) 086) analyzed this minimal gluonic scenario and derived: (i) from the Yb+/Cs atomic clock comparison, Λ˙QCD/ΛQCD=(3.2±3.5)×10−17 yr−1\dot{\Lambda}_{\rm QCD}/\Lambda_{\rm QCD}=(3.2\pm3.5)\times10^{-17}\,{\rm yr}^{-1}, using the nucleon-mass scaling δmN/mN≈0.91 δΛQCD/ΛQCD\delta m_N/m_N\approx0.91\,\delta\Lambda_{\rm QCD}/\Lambda_{\rm QCD}; (ii) from the 1.8-Gyr-old Oklo natural fission reactor, whose 149Sm neutron-capture resonance at Er=0.1E_r=0.1 eV arises from a near-cancellation of ~50 MeV nuclear energy contributions, ∣δΛQCD/ΛQCD∣<2×10−9|\delta\Lambda_{\rm QCD}/\Lambda_{\rm QCD}|<2\times10^{-9} over that time span (<1×10−18 yr−1<1\times10^{-18}\,{\rm yr}^{-1} assuming linear drift); (iii) from quasar absorption spectra, constraints via composite parameters (μ=mp/me\mu=m_p/m_e and proton gg-factor combinations) whose translation to Λ_QCD rests on scaling relations accurate only at the ~10% level; (iv) from BBN, δΛQCD/ΛQCD=K(−2.5±0.4)×10−3\delta\Lambda_{\rm QCD}/\Lambda_{\rm QCD}=K(-2.5\pm0.4)\times10^{-3} with K∼1K\sim1 a factor-of-two theory placeholder, whose non-zero central value reflects the cosmological 7Li problem; and (v) that the 8.4 eV isomeric transition in 229Th (a nuclear excited state of uniquely low energy, first laser-excited in 2024 and measured against the 87Sr atomic clock) responds as δω/ω≈Kα(δα/α−7 δΛQCD/ΛQCD+1.5 δmq/mq)\delta\omega/\omega\approx K_\alpha(\delta\alpha/\alpha-7\,\delta\Lambda_{\rm QCD}/\Lambda_{\rm QCD}+1.5\,\delta m_q/m_q) with Kα≈0.6×104K_\alpha\approx0.6\times10^4, so a Λ_QCD drift would be amplified by four orders of magnitude relative to present atomic clocks. The authors stress that all these bounds are conditional: the nuclear-physics inputs carry factor-of-two uncertainties; the results hold only in the pure-gluonic scenario and would change if φ also coupled significantly to photons, electrons, or quarks; and comparing gigayear-lookback measurements (Oklo, quasars, BBN) with laboratory drift rates assumes linear variation in time, 'which has no justification over such large time intervals'. In addition, the proton mass and gg-factor scaling relations entering the quasar interpretation are accurate only at the ~10% level, and improving them would require further work in lattice QCD (the first-principles numerical formulation of QCD on a discrete space-time grid) and hadronic effective field theory.

Problem Statement

In the framework where an evolving ultralight scalar field φ (dark matter or dark energy) couples to gluons and renders the QCD scale Λ_QCD time-dependent, establish a robust bound on — or a detection of — the temporal variation of Λ_QCD with a controlled theory error budget and explicitly characterized model dependence. Relative to the source analysis (Flambaum and Mansour, JHEP 11 (2025) 086), whose bounds are conditional on the minimal gluon-only coupling scenario and carry large nuclear-theory uncertainties, this requires the components addressing the limitations the authors explicitly acknowledge, all serving the single objective above: (1) reduce the nuclear and hadronic theoretical uncertainties that currently limit the translation of observations into δΛQCD/ΛQCD\delta\Lambda_{\rm QCD}/\Lambda_{\rm QCD} — the factor-of-two (K∼1K\sim1) binding-energy response entering the BBN interpretation, the near-cancellation analysis of the Oklo 149Sm resonance, and the sensitivity coefficient of the 8.4 eV 229Th isomeric transition — so that the large nominal sensitivities become robust, theory-error-controlled constraints; (2) improve the proton mass and proton gg-factor scaling relations with Λ_QCD beyond their current ~10% accuracy (for example by lattice QCD and hadronic effective field theory), so quasar absorption-spectrum bounds can be reinterpreted with confidence; (3) generalize the constraints beyond the pure-gluonic scenario to models in which φ also couples significantly to photons, electrons, or quarks, where the Λ_QCD-only interpretation no longer holds; (4) obtain direct constraints on Λ_QCD variation (secular drift and oscillatory ultralight-dark-matter-induced modulation) from the 229Th nuclear clock transition, whose sensitivity to δΛQCD/ΛQCD\delta\Lambda_{\rm QCD}/\Lambda_{\rm QCD} is enhanced by roughly four orders of magnitude relative to present atomic clocks — a first one-day-baseline drift limit and first oscillation bounds now exist, but with sensitivity coefficients still uncomputed, so the required controlled-coefficient limits remain outstanding; and (5) clarify the status of the linear-drift assumption — or replace it with a physically motivated evolution model — when comparing laboratory limits with Oklo, quasar, and BBN constraints spanning lookback times up to ~13 Gyr. A complete answer states the resulting bound (or measurement) together with its uncertainty budget and the coupling scenario in which it holds, and requires the five components above to be addressed jointly, since each removes one uncontrolled nuclear-theory error or one unexamined model dependence; a partial answer that addresses a single component (for example, only one sensitivity coefficient, only the model generalization, or only the linear-drift question) is nevertheless a valid partial result, and must be judged by how much it reduces the theory error or narrows the model dependence in the final constraint.

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