In scalar-field extensions of the Standard Model that violate the Einstein equivalence principle, the proton-electron mass ratio μ=mp/me\mu = m_p/m_e may depend on the local gravitational potential ϕ=GM/(Rc2)\phi = GM/(Rc^2), parameterized as Δμ/μ=kμ(1)Δϕ+kμ(2)(Δϕ)2\Delta\mu/\mu = k_{\mu}^{(1)}\Delta\phi + k_{\mu}^{(2)}(\Delta\phi)^2. Earth-bound clock and laser comparisons probe only seasonal potential changes of order 10−1010^{-10} and constrain the linear term; the quadratic term is measurable only in strong fields. The best strong-field constraint (Δϕ∼10−4\Delta\phi \sim 10^{-4}, about 10410^4 times the Sun-dominated gravitational potential at Earth's position) comes from H2_2 Lyman-band spectra in the photospheres of the white dwarfs GD 133 and G29-38: ∣Δμ/μ∣≲5×10−5|\Delta\mu/\mu| \lesssim 5\times10^{-5}, equivalent to kμ(2)<1×103k_{\mu}^{(2)} < 1\times10^3. The open problem is to tighten this bound through additional or improved white-dwarf molecular spectroscopy with a full statistical and systematic error budget. No credible constraint tighter than kμ(2)∼103k_{\mu}^{(2)} \sim 10^3 exists, and the problem is open.

Background

The proton-electron mass ratio μ=mp/me\mu = m_p/m_e is one of the dimensionless constants that set the scale of low-energy physics. Theories with new light scalar fields, extra dimensions, or a non-unique vacuum can make such constants vary in space and time, and couplings between the scalar field and ordinary matter can make them depend on the local environment - the ambient matter density or gravitational field - in violation of the Einstein equivalence principle (the statement that local nongravitational experiments give the same results regardless of where and when they are performed). This environmental dependence is parameterized through the dimensionless gravitational potential ϕ=GM/(Rc2)\phi = GM/(Rc^2) by a Taylor expansion Δμ/μ=kμ(1)Δϕ+kμ(2)(Δϕ)2\Delta\mu/\mu = k_{\mu}^{(1)}\Delta\phi + k_{\mu}^{(2)}(\Delta\phi)^2, where Δϕ\Delta\phi is the potential difference between the measurement environment and Earth. Earth-based spectroscopy and atomic-clock comparisons exploiting the Δϕ∼3×10−10\Delta\phi \sim 3\times10^{-10} seasonal variation from the eccentricity of Earth's orbit constrain the linear coupling kμ(1)k_{\mu}^{(1)} at the 10−410^{-4} level (kμ(1)<4×10−4k_{\mu}^{(1)} < 4\times10^{-4} from laser spectroscopy), but the quadratic term, suppressed by (Δϕ)2(\Delta\phi)^2, is unmeasurably small in any terrestrial setting; probing it requires potentials thousands of times stronger. White-dwarf photospheres provide ϕ∼10−4\phi \sim 10^{-4}, roughly 10410^4 times the gravitational potential at Earth's position (itself dominated by the Sun's contribution, ϕ⊕≈10−8\phi_\oplus \approx 10^{-8}), together with resolvable absorption spectra - the strongest gravitational fields in which molecular lines can currently be measured.

Molecular hydrogen was discovered in the photospheres of the hot white dwarfs GD 133 (WD 1116+026) and G29-38 (WD 2326+049) in Hubble Space Telescope Cosmic Origins Spectrograph (HST/COS) ultraviolet spectra (Xu et al., 2013). Because the vibrational and rotational spacings of a molecule scale with its nuclear reduced mass, a change in μ\mu shifts each electronic-vibrational-rotational transition by a transition-dependent amount: Δλi/λi=Ki Δμ/μ\Delta\lambda_i/\lambda_i = K_i\,\Delta\mu/\mu, where the sensitivity coefficients KiK_i (of order 0.1 for the relevant lines) are computed from laboratory level energies of the Lyman (B1Σu+−X1Σg+B^1\Sigma_u^+ - X^1\Sigma_g^+) and Werner (C1Πu−X1Σg+C^1\Pi_u - X^1\Sigma_g^+) bands. At photospheric temperatures of about 12,000-14,500 K many excited rotational and vibrational levels are populated (with weights given by a Boltzmann partition function), so roughly 870 weak, mutually blended Lyman-band lines appear between 1298 and 1444 A at signal-to-noise ratio of only about 15; comparing their observed centroids with laboratory wavelengths constrains Δμ/μ\Delta\mu/\mu.

Bagdonaite et al. (Phys. Rev. Lett. 113, 123002 (2014)) fitted both white-dwarf spectra with a simultaneous Voigt-profile model (a Voigt profile combines thermal Doppler and natural Lorentzian broadening, convolved with the instrumental line-spread function) and obtained Δμ/μ=(−2.7±4.7stat±0.2sys)×10−5\Delta\mu/\mu = (-2.7 \pm 4.7_{\rm stat} \pm 0.2_{\rm sys})\times10^{-5} for GD 133 and (−5.8±3.8stat±0.3sys)×10−5(-5.8 \pm 3.8_{\rm stat} \pm 0.3_{\rm sys})\times10^{-5} for G29-38: no significant variation, but, interpreted as a quadratic coupling at Δϕ≈1.2×10−4\Delta\phi \approx 1.2\times10^{-4} to 1.9×10−41.9\times10^{-4}, a bound kμ(2)<1×103k_{\mu}^{(2)} < 1\times10^3, several orders of magnitude beyond what Earth-based experiments can reach. (A parallel program constrains the fine-structure constant α\alpha from Fe V and Ni V lines in the white dwarf G191-B2B.) The paper's own stated limitations define the open problem: only two white dwarfs have adequate H2_2 spectra; the molecular line data were heterogeneous; Stark- and Zeeman-shift parameters of the molecular Lyman bands have never been measured in the laboratory; and G29-38 is a pulsating (ZZ Ceti-type) star whose few-hundred-second pulsations smear the line profiles during long exposures.

Problem Statement

Does the proton-electron mass ratio μ=mp/me\mu = m_p/m_e depend on the strength of the local gravitational field, and if so how weakly? The problem is to test this and to tighten the strong-field constraint through precision spectroscopy of molecules in white-dwarf photospheres. One concrete route, motivated by the limitations stated in Bagdonaite et al. (2014) and going beyond the existing analysis of two white dwarfs, would be: (i) to obtain spectra of H2_2 in additional white-dwarf photospheres, and/or higher signal-to-noise, better wavelength-calibrated observations of GD 133 and G29-38, sufficient to reduce the statistical uncertainty on Δμ/μ\Delta\mu/\mu below the present ∼4×10−5\sim4\times10^{-5}; (ii) to assess each systematic contribution - wavelength-calibration velocity distortions, Stark and Zeeman shifts of the molecular Lyman bands (for which no laboratory measurements exist), pulsational smearing, and photospheric temperature and partition-function modeling - within a full error budget; and (iii) to translate the measurement into a bound on the gravitational coupling coefficients of μ\mu tighter than the current kμ(2)<1×103k_{\mu}^{(2)} < 1\times10^3, using measurements spanning different photospheric potentials ϕ\phi to separate the linear and quadratic terms where possible. This route is not exclusive. Extension to other molecular or atomic species with independent sensitivity coefficients is desirable as a test of model dependence but is not itself the completion boundary. Completion is a published constraint, with a full statistical and systematic error budget, that demonstrably tightens the strong-field bound on the gravitational dependence of μ\mu.

References

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