Methanol isotopologues as a probe for spatial and temporal variations of the electron-to-proton mass ratio

We present results on numerical calculations of the sensitivity coefficients, Qmu, of microwave molecular transitions in (13C)H3OH and CH3(18O)H to the hypothetical variation in the fundamental physical constant mu - the electron-to-proton mass ratio. The invariability of mu in time and space is one of the basic assumptions of the Standard Model of particle physics which can be tested at cosmological scales by means of astronomical observations in the Galaxy and external galaxies. Our calculations show that these two methanol isotopologues can be utilized for such tests since their microwave transitions from the frequency interval 1-100 GHz exhibit a large spread in Qmu values which span a range of -109<Qmu<78. We show that the thermal emission lines of (13C)H3OH observed in the star-forming region NGC6334I constrain the variability of mu at a level of 3x10^-8 (1{\sigma}), which is in line with the most stringent upper limits obtained previously from observations of methanol (CH3OH) and other molecules in the Galaxy.


INTRODUCTION
Physical theories extending the Standard Model (SM) of elementary particle physics allow for possible space-time variations of fundamental physical constants.Modern experimental capabilities make it possible to test such theories with high accuracy both in laboratory experiments with atomic clocks and in studies of astronomical objects.The main research in this area has focused on dimensionless constants such as the fine structure constant α=e 2 / c and the electron-to-proton mass ratio µ=me/mp since they are independent of the choice of unit system.Small changes in α and µ produce offsets in the line positions of atoms and molecules (for a review, see, e.g., Kozlov & Levshakov 2013).
Measurements of fractional changes in µ are based on the facts that (i) the the molecular energy levels are dependent on µ (Thompson 1975), and that (ii) the molecular electron-vibro-rotational transitions have specific dependencies on µ (Varshalovich & Levshakov 1993).The response of a transition to a variation of µ is characterized by its ⋆ E-mail: lev@astro.ioffe.rudimensionless sensitivity coefficient Qµ, which is defined as where the fractional changes in the electron-to-proton mass ratio is given by ∆µ/µ = (µ obs − µ lab )/µ lab . (2) Here µ obs and µ lab are the extraterrestrial and terrestrial values of µ, respectively.In molecules, enhanced sensitivity coefficients are found in tunneling transitions since probability of tunneling depends exponentially on the mass of tunneling particles.The effect was firstly considered for the inversion transition of NH3 by Flambaum & Kozlov (2007) and later on for hindered rotations in the non-rigid tops (Jansen et al. 2011;Levshakov et al. 2011).The tunneling occurs generally in every internal rotor molecule, especially in methanol (CH3OH) where the methyl group CH3 can make torsional vibrations with respect to the hydroxyl group OH.The hydrogen atom in the hydroxyl group can be placed in three possible positions with equal energies, and in order to move from one configuration to another, it must pass through the potential barrier caused by the methyl group.Thus, there is a hindered internal rotation (rotation of the hydrogen atom with respect to the methyl group).This type of molecules are also characterized by a strong interaction between internal (hindered) and overall rotations.
The sensitivity coefficients for such transitions in various molecules were previously calculated in a series of papers.For example, the sensitivity coefficient of the (J, K) = (1, 1) inversion transition of ammonia (NH3) is equal to Kµ = −4.461(Flambaum & Kozlov 2007), in the methyl mercaptan molecule CH3SH the sensitivity coefficients range from Kµ = −14.8 to 12.2 (Jansen et al. 2013), in the methylamine molecule CH3NH2 -from Kµ = −19 to Kµ = 24 (Ilyushin et al. 2012).The calculations of the sensitivity coefficients for methanol CH3OH were performed by two independent methods (Jansen et al. 2011, Levshakov et al. 2011) resulting in the self-consistent values of Qµ ranging from −17 to 43.
Astrophysical limits on ∆µ/µ have been obtained by different methods in observations of Galactic and extragalactic objects.The extragalactic constraint on the variability of µ at a level of 8 × 10 −6 (1σ) at the highest redshift z = 4.22 (look-back time 12.4 Gyr) towards the quasar J1443+2724 is deduced from the analysis of the Lyman and Werner absorption lines of molecular hydrogen H2 (Bagdonaite et al. 2015).In the Milky Way, the most stringent upper limits on µ-variations based on ammonia and methanol observations are, correspondingly, ∆µ/µ < 2 × 10 −8 (Levshakov et al. 2013), and ∆µ/µ < 6 × 10 −8 (Daprà et al. 2017).Both limits are given at a 3σ confidence level.However, it should be noted that these constraints were obtained at the marginal accuracy of modern spectral observations and, therefore, may be affected by systematics of unknown values and caused by inaccessible factors.In order to estimate such kind of systematics, measurements involving different facilities, different objects, and different probes -in our case molecular transitions -are required.
Here we calculate the sensitivity coefficients for methanol isotopologues 13 CH3OH and CH3 18 OH in order to check whether they are suitable to test µ-variations at the above mentioned levels.Section 2 describes the effective Hamiltonian and its parameters, as well as the determination of the sensitivity coefficients.Section 3 presents the results of these calculations.In Section 4, we describe existing detections of isotopic methanol transitions and use observations of the 13 CH3OH thermal emission lines from the young stellar object NGC 6334I to constrain ∆µ/µ.The results obtained are summarized in Section 5.

Quantum-mechanical model
The procedure for calculating sensitivity coefficients is described in details in Levshakov et al. (2011).Here are given only the main aspects.The procedure is based on an approach by Rabli & Flower (2010) which implies a simple and convenient form of the effective Hamiltonian with six spectroscopic constants having clear physical meaning.The dependence of these spectroscopic constants on µ is easily understood within the Born-Oppenheimer approximation.This Hamiltonian is physically transparent and quite accurate for calculations of sensitivity coefficients.
In Rabli & Flower (2010), it is applied a direct method of solving the Schrödinger equation which involves diagonalizing the Hamiltonian, expressed on a basis of torsion-rotation states where 0 ω 2π is the torsion angle of the internal rotation of the CH3 group relative to the OH radical, V (ω) is the torsional potential, F is the torsional constant (see below), D J Km J is a rotation matrix element which is a function of the Euler angles specifying the orientation of the molecule relative to a space-fixed laboratory coordinate system, J is the rotational angular momentum, mJ is the projection of the rotational angular momentum on the space-fixed z-axis, K is its projection on the symmetry axis of the molecule, ρ = 0.8102 for 13 CH3OH (Xu & Lovas 1997), and ρ = 0.809 for CH3 18 OH (Fisher et al. 2007) is the fraction contributed by the CH3 group to the rotational angular momentum of the molecule about its symmetry axis.
For a given value of J, the non-vanishing matrix elements of the methanol Hamiltonian, expressed in the mentioned above basis, are the following (in this model the conversion for the rotational parameters A > B > C were used) where δ i,k is Kronecker's symbol.Diagonalization of the Hamiltonian matrix, formed by these matrix elements, yields the eigen-energies and eigenfunctions of A-and E-methanol simultaneously, for a given J.The torsional angular momentum quantum number, m, satisfies the relation m = 3s + σ, where s is any integer and σ = 0 for A-methanol, σ = ±1 for E-methanol.
Six parameters of the effective Hamiltonian are following: three rotational parameters A, B, C, one parameter D, describing interaction of internal rotation with overall rotation, the kinetic coefficient F , and the depth of the three-fold symmetric torsion potential V3, The values of V3 and F are taken from Xu & Lovas (1997) for 13 CH3OH and Fisher et al. (2007) for CH3 18 OH, parameters A, B, C, D were calculated from the moments of inertia of 13 CH3OH and CH3 18 OH (Lees et al. 1973) according to the formulas: In these equations, = h/2π, Ia, I b , Ic are the moments of inertia, and I ab is the product of inertia about the a-and baxes in the a, b, c-axis system whose a-axis is parallel to the internal rotation axis (assumed to be that of the methyl top), the c-axis perpendicular to the COH plane (for details, see Lees & Baker 1968).The numerical values of the moments of inertia are given in Table 1, while Table 2 lists the calculated spectroscopic parameters for 13 CH3OH, CH3 18 OH, and 12 CH3 16 OH along with the kinetic coefficients F and potential barriers V3.Errors of Ia, I b , Ic, I ab and A, B, C, D are not given in Tables 1 and 2, since, in general, errors in Qµ caused by the model itself are by orders of magnitude larger than uncertainties in the moments of inertia and in the spectroscopic rotational parameters.Therefore, in what follows we consider these values as constants.

Determination of the sensitivity coefficients
Microwave transitions, in our case, torsion-rotation transitions, can be expressed by the quantum numbers of the upper and lower levels -total angular momentum J and its projection K on the axis of the molecule.Hereafter, it will be denoted as Ji and Ki -quantum numbers for the upper level, and J k and K k -quantum numbers for the lower level.
The sensitivity coefficient Q µ,ik for the transition JiKi → J k K k is given by where q ik = qi − q k , qi and q k are the q-factors, individual for each level, which shows a response of the level to a small change of µ, and f is the laboratory transition frequency taken from Anderson et al. (1987Anderson et al. ( , 1990)), Kuriyama et al. (1986), Hughes et al. (1951) for 13 CH3OH, and from Ikeda et al. (1998) for CH3 18 OH.
To determine the sensitivity coefficients Qµ, we first find the dependence of the eigenvalues Ei on ∆µ/µ: where the coefficient qi reveals the response of the level Ei to a small change in µ, i.e., |∆µ/µ| ≪ 1.This is done by diagonalizing the effective Hamiltonian for the three sets of parameters that correspond to µ = µ0 and µ = µ0(1 ± ε), where ε is equal to 0.001 or 0.0001 (for details, see Levshakov et al. 2011).

RESULTS
Molecular transitions with the total angular momentum J from 0 to 13 were selected for calculations, and the quantum numbers K for the upper and lower levels were chosen for |∆K| = 1, since transitions without changing K are purely rotational and have approximately the same sensitivity to µ-variations, Qrot ≈ 1.The ranges of J values chosen are based on the fact that such transitions are described by our model quite well.For J > 13, the model turns out to be insufficiently accurate.Transitions of a wide frequency interval from 1 to 900 GHz are considered.Among them lowfrequency transitions are found to be the most interesting, since they have the highest sensitivities to changes in µ.
The calculated sensitivity coefficients for 13 CH3OH are listed in Table 3 where the quantum numbers J and K of the upper and lower levels, the rest frequency f , and the sensitivity coefficient Qµ are indicated.Table 4 contains the similar information as Table 3 but for CH3 18 OH.Both tables show only transitions with large sensitivity coefficients, |Qµ| ≫ 1.These transitions can be observed under real conditions in the interstellar medium.
Errors of the sensitivity coefficients were calculated as follows.Parameters of the spin-rotational Hamiltonian depend on the mass ratio µ.One can see from Eqs. (9-12) that A, B, C, and D are inversely proportional to the molecular moments of inertia.To a first approximation, the equilibrium internuclear distances do not depend on µ and, therefore, these parameters scale linearly with µ.Within this approximation the kinetic coefficient F in Eq. ( 3) is also proportional to µ, while the potential barrier V3 in Eq. ( 8) is independent of µ.However, the vibrational wave functions of the molecule depend on µ and in the next approximation the internuclear distances also weakly depend on µ.This, as well as centrifugal and other corrections, affects scaling of all parameters with µ.We estimated in Levshakov et al. (2011) that the Hamiltonian parameters are proportional to µ n+ε , where n = 0 for V3 and n = 1 for all other parameters, while |ε| < 0.02.This uncertainty is the dominant source of errors when we calculate sensitivity coefficients Qµ.The resulting errors are determined by summing quadratically the errors caused by changes in the scalings of these six parameters within the uncertainty interval |ε| < 0.02.The errors determined in this way are majorizing estimates of the uncertainties in the values of the sensitivity coefficients Qµ.
We note that our procedure yields Qµ in concordance with values for two lines of CH3 18 OH at 2.604 GHz and 11.629 GHz, and one line of 13 CH3OH at 1.989 GHz found by Jansen et al. (2011), who used a different numerical approach.
Thus, methanol isotopologues 13 CH3OH and CH3 18 OH can provide of a distinct interest in the issues of µ-variations: • there are significantly different sensitivity coefficients of both signs between different transitions with ∆K = ±1; • all transitions with ∆K = 0 have Qµ ≈ 1, and thus can serve as anchors together with highly sensitive lines; • these transitions are in a fairly narrow frequency range which makes them convenient for simultaneous observations; • methanol isotopologues, being observed at various galactocentric distances, can be used for probing the hypothetical coupling of the dark matter with the baryonic matter depending on local environmental conditions.

OBSERVATIONAL CONSTRAINTS ON
µ-VARIATIONS Methanol isotopologues have previously been observed in Galactic and extragalactic sources at different frequencies.
In most cases, high-frequency transitions were observed for both isotopologues (e.g., Nummelin et al. 1998).The most common objects with the methanol isotopologues emission are located in the molecular cloud complex in the Orion (OMC-1) and in the molecular cloud near the center of the Galaxy (Sgr B2).
In addition to these two locations, the emission lines of 13 CH3OH were detected in the young stellar object (YSO) IRAS16293-2422 (Parise et al. 2004;Kuan et al. 2004), in the star-forming region G34.3+0.15 (Macdonald et al. 1996), and in the HII region IRAS18089-1732(Beuither et al. 2004), whereas the absorption lines of 13 CH3OH were previously observed towards the continuum sources Sgr A, Sgr B2, and W33 (Kuiper et al. 1989) and in one of the two lines of sight towards the quasar PKS 1830-211 at redshift z = 0.89 (Muller et al. 2021;Kanekar et al. 2015;Bagdonaite et al. 2013).
In this extragalactic absorber, the detected highfrequency transitions of 13 CH3OH were used to investigate the cosmological invariance of µ at a look-back time of half the present age of the Universe.For this purpose the authors calculated sensitivity coefficients for the transitions 1−1 − 10A at 303.692 GHz, 2−1 − 20A at 304.494 GHz, and 3−1 − 30A at 305.699 GHz (Qµ = 1.904, 1.902, and 1.898, respectively) and estimated the upper limit on |∆µ/µ| < 1.2 × 10 −7 (1σ).However, as noted above, highfrequency transitions do not have large values of Qµ making high-precision measurements of ∆µ/µ difficult.
But a remarkable fact in methanol observations is that its isotopologues have also been observed in the Galaxy at low-frequencies with high Qµ values of different signs which is of great interest for probing spatial and temporal variations of µ.
For instance, a series of thermal emission lines of 13 CH3OH (23-44 GHz) was recently observed in the YSO NGC 6334I by Wu et al. (2023).We selected from their Table 3 the most accurate line positions with known sensitivity coefficients and listed them in our Table 5.It is seen that ∆Qµ in this case can be as large as 20.6.
The fractional changes in µ can be estimated from a pair of molecular transitions (i, j) with different values of Qµ (Levshakov et al. 2022): where Vj and Vi are the local standard of rest radial velocities, VLSR, of molecular transitions with corresponding sensitivity coefficients Qµ,j and Qµ,i, and c is the speed of light.
The sample mean ∆µ/µ and its error based on the total list of n = 8 lines from Table 5 is expected to be more precise than that based on a single pair of lines with the largest difference between Qµ values, e.g., between the 5th and 1st lines.However, the improvement is not as high as 1/ √ n because the individual values of ∆µ/µ are correlated.
The obtained constraint is in line with the most stringent upper limit on |∆µ/µ| < 2 × 10 −8 (1σ) found from observations of Class I methanol (CH3OH) masers distributed in the Milky Way disk over a large range of the galactocentric distances (Levshakov et al. 2022), and from measurements of methanol thermal emission lines towards the dense dark cloud core L1498 (Daprà et al. 2017).
CH3 18 OH was not detected in maser emission, but its low-frequency transitions with high sensitivity coefficients were also observed in interstellar space.The emission line 102 −101E at 34.831 GHz (Qµ = −11) was detected towards NGC 6334I (Wu et al.2023), and the absorption line 20 − 3−1E at 11.629 GHz (Qµ = 33) -at two positions in the direction of Sgr B2 (Gardner et al. 1989).
These observations and detections of isotopic methanol transitions in the low-frequency range from 1 to 40 GHz, where the calculated sensitivity coefficients demonstrate large values of different signs, show great potential for investigations of fundamental physical principles on Galactic and extragalactic scales.

SUMMARY
The numerical calculations discussed in this paper were designed to study response of the microwave molecular line positions in the methanol isotopologues 13 CH3OH and CH3 18 OH on small changes in the electron-to-proton mass ratio, µ.Varying µ would cause shifts in the line positions different for individual lines.This opens up the opportunity to test invariability of µ through comparison of astronomical spectra with laboratory determinations.
That the electron-to-proton mass ratio may not be a constant would imply that the relative strength of the electromagnetic force compared to the strong nuclear force is space-time dependent.No evidences for either spatial or temporal changes in µ have been found yet, however, at a level of a few times 10 −8 .
The improvement in this constraint is currently limited to uncertainties of ∼ 10 kHz in the rest frame frequencies of the methanol isotopologues, corresponding to uncertainties of ∼ 100 m s −1 in the velocity scale for the lowfrequency microwave transitions listed in Tables 3, and 4. The recently measured radial velocities of 13 CH3OH lines (Wu et al. 2023) have similar errors (see Table 5).Future laboratory and astronomical investigations of methanol isotopologues with higher quality spectra can improve by more than 10 times the most stringent limit on µ-variation in the Galaxy enabling these tests to be performed at a level of 10 −9 .
Our main results are as follows.
(i) The spectroscopic rotational parameters A, B, C, and D for methanol isotopologues 13 CH3OH and CH3 18 OH were calculated and presented in Table 2.
(ii) The previously developed procedure for calculating sensitivity coefficients Qµ for methanol lines (Levshakov et al. 2011) was used to analyze torsion-rotation transitions of the methanol isotopologues in the range 1-100 GHz.
(iii) The calculated Qµ coefficients were shown to have significantly different values of both signs which span an interval from −32 to 78 for 13 CH3OH (Table 3), and from −109 to 19 for CH3 18 OH (Table 4).
(iv) For the three previously known Qµ factors in these isotopologues (Jansen et al. 2011), good agreement with our results was obtained.
(v) The constraint on the µ-variability at the 3 × 10 −8 (1σ) level was obtained from the 13 CH3OH thermal emission lines observed recently in the young stellar object NGC 6334I by Wu et al. (2023).
In conclusion, we note that the methanol isotopologues lines were observed in the Milky Way both near the Galactic nucleus and in the disk at different galactocentric distances, which opens new possibilities for testing µ as a function of the gravitational potential on galactic scales.

Table 1 .
Moments of inertia for the main methanol molecule and its isotopologues (in units amu Å2 ).12 CH 316OH a 13 CH 3 OH b CH 3 18 OH b References: a Lees & Baker (1968); b Lees et al. (1973).