Torsion-Rotational Transitions in Methanol as a Probe of Fundamental Physical Constants—Electron and Proton Masses

We report on the using of torsion-rotational transitions in the CH₃OH and ¹³CH₃OH molecules to evaluate possible variations of the electron-to-proton mass ratio μ = m_e/m_p from spectral observations of emission lines detected in the microwave range towards the dense molecular cloud Orion-KL. An estimate of the upper limit on the relative changes in μ is obtained by two independent ways with ¹³CH₃OH lines and with the combination of ¹³CH₃OH and CH₃OH lines. The calculated upper limit Δμ/μ < 1.1 × 10^–8(1σ) is in line with the most stringent constraints on the variability of fundamental physical constants established by other astrophysical methods.

The existence of Dark Matter in the Universe follows from a number of observational facts, including flat rotation curves of galaxies at large galactocentric distances, gravitational lensing of cosmologically distant objects and the large-scale structure of the spatial distribution of galaxies [1]. To explain the nature of Dark Matter, various models have been considered, including phenomena extending the Standard Model of particle physics (SM). Several of these models assume the existence of hypothetical scalar fields interacting with the baryonic component of an ordinary substance [2–4]. The result of such interactions can lead to the space-time variations of dimensionless physical constants, such as the fine structure constant α and the electron-to-proton mass ratio μ [5–8].

Therefore, the new theories can be tested experimentally using the differential measurements of μ = m_e/m_p [9]. It should be noted that the mass of the electron m_e is directly related to Higgs-like scalar fields, while the main contribution to the mass of the proton m_p comes from the binding energy of quarks.

Electron-vibro-rotational transitions in molecular spectra have a specific dependence on μ, individual for each transition [10, 11]. The reaction of the transition to a change in μ is characterized by a dimensionless sensitivity coefficient Q_μ, which is defined as

$${{Q}_{\mu }} = \frac{{df{\text{/}}f}}{{d\mu {\text{/}}\mu }},$$

(1)

here df/f is the relative frequency offset, and dμ/μ is given by

$$\frac{{\Delta \mu }}{\mu } = \frac{{{{\mu }_{{{\text{obs}}}}} - {{\mu }_{{{\text{lab}}}}}}}{{{{\mu }_{{{\text{lab}}}}}}},$$

(2)

where μ_obs, μ_lab are astronomical and laboratory values of μ, correspondingly. At the same time, the sensitivity coefficients can take different signs, which leads to an increase or decrease in the observed frequency compared to its laboratory value.

The most stringent limits on $\mu$-variations at large redshifts z were obtained from extragalactic observations towards the quasar J1443+2724 (z = 4.22). From the analysis of the Lyman and Werner absorption lines of molecular hydrogen H₂ the upper limit on Δμ/μ < 8 × 10^–6 was obtained [12]. In the Milky Way, the most stringent upper limits were established from observations of the 23 GHz inversion transition in ammonia NH₃, which has a sensitivity coefficient Q_μ = 4.46 [13], compared to the pure rotational transitions in HC₃N, HC₅N, and HC₇N, which have Q_μ = 1: Δμ/μ < 7 × 10^‒9 [14]. Independent estimates from observations of the thermal emission lines of methanol CH₃OH in the core of the molecular cloud L1498 lead to a value of Δμ/μ < 2 × 10^–8 [15]. Similar constraints follow from the measurements of the radial velocities of methanol maser lines: Δμ/μ < 2 × 10^–8 [16] and Δμ/μ < 2.7 × 10^–8 [17]. All estimates of Δμ/μ are given at a 1σ confidence level.

It should be noted that in previous works methanol isotopologues have not been used extensively. The first estimates of the upper limit on μ-variations were obtained from the TMRT 65-m telescope [18] observations of the thermal emission lines of ¹³CH₃OH in the star-forming region NGC 6334I: Δμ/μ < 3 × 10^–8 [19].

In the molecules with hindered internal motion, enhanced sensitivity coefficients Q_μ are inherent for tunneling transitions, since the tunneling probability depends exponentially on the masses of tunneling particles [13, 20, 21]. The most promising molecule for these studies is methanol (CH₃OH), where the methyl group CH₃ can make torsional vibrations relative to the hydroxyl group OH. In this case, the hydrogen atom of the hydroxyl group can be located in three possible positions with equal energies, and to move from one configuration to another, it must pass through a potential barrier caused by the three hydrogen atoms of the methyl group. So, there is an internal hindered motion of the hydrogen atom relative to the methyl group.

The sensitivity coefficients Q_μ for methanol were calculated by two independent methods in 2011 [20, 21]. The results show that low-frequency (in the range of 1–50 GHz) transitions have high values of Q_μ with different signs: –17 ≤ Q_μ ≤ +43, which, compared to the sensitivity coefficients of the molecular hydrogen H₂ lines (|Q_μ| ~ 10^–2) give a gain in the limiting estimates of Δμ/μ by a factor of more than 1000.

In our previous work [19], a list of molecules with the enhanced sensitivity coefficients was extended due to the methanol isotopologues—¹³CH₃OH with ‒32 ≤ Q_μ ≤ +78, and CH₃¹⁸OH with –109 ≤ Q_μ ≤ +33.

Turning to the practical measurements of Δμ/μ, we note that to estimate this value, pairs of molecular lines with different coefficients Q_μ,1 and Q_μ,2 are used [20]:

$$\frac{{\Delta \mu }}{\mu } = \frac{{{{V}_{1}} - {{V}_{2}}}}{{c({{Q}_{{\mu ,2}}} - {{Q}_{{\mu ,1}}})}},$$

(3)

where V₁ and V₂ are the measured radial velocities of these lines, and c is the speed of light. The conversion from the frequency scale f to the velocity scale V is made by the radio astronomical definition V/c = (f_lab – f_obs)/f_lab.

The accuracy of the Δμ/μ measurements is due to the influence of various factors. Uncertainties in laboratory frequencies and line centers in astronomical spectra are the main sources of errors. In addition, there are systematic errors which can be estimated from observations of different objects in different molecular transitions. Transitions in methanol isotopologues have high sensitivity coefficients of both signs, i.e., they are the most suitable candidates for such studies.

High precision measurements can be carried out from high-resolution spectra, which were recently obtained for the Orion-KL molecular cloud [22]. The published spectra contain lines of methanol CH₃OH and its two isotopologues—¹³CH₃OH and CH₃¹⁸OH. The CH₃¹⁸OH lines are found to be quite weak, and have large errors in radial velocities. However, the ¹³CH₃OH emission spectra display lines with higher intensities, and their positions are determined quite accurately (with errors of 100 m s^–1, which are acceptable for our purposes). Thus, it becomes possible to estimate Δμ/μ independently – with only the lines of ¹³CH₃OH and in combination with the transition ${{J}_{{{{K}_{u}}}}}{-} {{J}_{{{{K}_{l}}}}}$ = 15₂–15₁E in CH₃OH (see Table 1 below).

Table 1. Selected ¹³CH₃OH and CH₃OH transitions in Orion-KL [22]

We selected from the published spectra the pairs of molecular transitions with approximately equal values of the Doppler widths Δ${{{v}}_{{\text{D}}}}$, such that the difference in the sensitivity coefficients ΔQ_μ is maximized. The selected lines and their parameters are given in Table 1. The first column lists the transition described by two quantum numbers—the total angular momentum J and its projection K on the principal axis of the molecule—for the upper (u) and lower (l) levels, the second column displays the transition frequency, the third and fourth columns show the Doppler width and the Local Standard of Rest radial velocity, V_LSR, respectively. The fifth column contains the sensitivity coefficient Q_μ, taken from [19]. The calculation of Q_μ for methanol CH₃OH was performed in this work using our previously developed method [20]. As can be seen from Table 1, the difference in the sensitivity coefficients is ΔQ_μ ~ 30, which provides a confident estimate on the μ-variance.

Using Eq. (3) for the transitions in ¹³CH₃OH, we obtain Δμ/μ = (–1.1 ± 1.5) × 10^–8, which corresponds to the upper limit on Δμ/μ = 1.5 × 10^–8. A similar calculation for the combination of the 6₂–5₃A^– line of ¹³CH₃OH and the 15₂–15₁E line of CH₃OH gives Δμ/μ = (–3.4 ± 1.6) × 10^–8, and the upper limit on Δμ/μ = 1.6 × 10^–8. This yields the average value of 〈Δμ/μ〉 = (–2.3 ± 1.1) × 10^–8, and the corresponding upper limit on the changes in μ – Δμ/μ = 1.1 × 10^—8. This upper limit agrees well with previously obtained constraints from Galactic CH₃OH [15, 16] and ¹³CH₃OH [19] observations.

The results of these investigations do not indicate any significant systematic errors in the estimates of Δμ/μ. It follows that the assumed effects of Higgs-like scalar fields on the masses of elementary particles do not exceed the 10^–8 level in the disk of the Galaxy. This upper limit, 10^–8, also coincides with a limit on the influence of the hypothetical fifth force on hadronic interactions [23], so it can be considered as the most robust at the current time.