Constraints on the Variation of Physical Constants, Equivalence Principle Violation, and a Fifth Force from Atomic Experiments

The existence of life is consistent with a narrow possible range of values for fundamental physical constants, see e.g. the review [1]. This precise fitting of constants for life may be explained by the variation of these constants in space; life emerged in regions of the universe where the fundamental constants have suitable values. Potential evidence for such spatial variation in the fine structure constant $\alpha$ has been observed in quasar absorption spectra  [2, 3]. Based on this astrophysical data, it is suggested that we are moving in the direction of increasing $\alpha$, which could lead to a slow drift of fundamental constants observable in laboratory experiments  [4, 5].

There are a number of models which predict the space-time variation of fundamental physical constants, which may be related to dynamical dark energy models, dark matter models, the variation of the unification scale and string theory models - see e.g. the review [1]. For example, the space-time variation of fundamental constants may be due to an interaction with a slowly evolving scalar dark energy field or an oscillating dark matter field. The dark matter candidate particles in this class are the pseudoscalar axion (and axion like particles) and the dilaton-like scalar particle [6, 7, 8]. If the mass of the cold dark matter is very light ($m_{\text{DM}}\ll 1\ \text{eV}$), it may be considered to be a classical field oscillating harmonically at every particular point in space. For axions, we may write this as

where $\varphi$ is a (position-dependent) phase and $m_{a}$ is the mass of the axion. Assuming that axions saturate the entire dark matter density, the amplitude $a_{0}$ may be expressed in terms of the local dark matter density $\rho_{\text{DM}}\approx 0.4\ \text{GeV/cm}^{3}$, see e.g. Ref. [9],

Similar expressions are used to describe the case of a scalar field dark matter $\phi$.

The effects of the interaction between the scalar field and fermions may be presented as the apparent variation of fermion masses. This immediately follows from a comparison of the interaction of a fermion with the scalar field $-g_{f}M_{f}\phi^{n}\bar{\psi}\psi$ and the fermion mass term in the Lagrangian $-M_{f}\bar{\psi}\psi$. Adding these terms gives $M^{\prime}_{f}=M_{f}(1+g_{f}\phi^{n})$, with $n=1,2$. Similarly, the interaction of the scalar field with the electromagnetic field may be interpreted as a variable fine structure constant $\alpha^{\prime}=\alpha(1+g_{\gamma}\phi^{n})$. These variations may manifest themselves as a slow drift or oscillations of frequencies of atomic clocks [10, 11, 12]. If the interaction is quadratic in $\phi$, the scalar field becomes interchangeable with the pseudoscalar (axion) field as $\phi^{2}$ always has positive parity [11]. The corresponding theory has been developed in Ref. [13], in which limits on the axion interaction from atomic spectroscopy experiments were obtained (see also Ref. [14, 15]).

In this paper we consider the slow drift of the physical constants, which may be due to an interaction with an evolving non-oscillating dark energy field or the variation of the density of the dark matter field $\rho_{\text{DM}}$ in the case where $n=2$, after averaging over fast oscillations of $\cos^{2}(\omega t+\varphi)=(1+\cos 2(\omega t+\varphi))/2$. Such slow variation also appears in other models [1]. In then present paper we do not assume any specific model predicting the variation of the fundamental constants.

The dependence of atomic transition frequencies on $\alpha$ and the quark masses has been calculated in Refs. [16, 17, 18, 19, 20, 21]. These calculations and results measuring the time dependence of atomic transition frequencies have previously been used to place improved limits on the interaction strength of the low mass scalar field dark matter $\phi$ with photons, electrons and quarks by up to 15 orders in magnitude [11, 22]. The experimental results have been obtained by measuring the oscillating frequency ratios of electron transitions in a range of systems, including Dy/Cs [23], Rb/Cs [24], Yb/Cs [25], Sr/H/Si cavity [26], Cs/cavity [27], Cs/H [28], Al⁺/Hg⁺ [29] and Yb⁺/Yb⁺/Sr [30, 31].

Scalar particles can mediate Yukawa-type interactions between Standard Model particles. A range of experimental methods have been used to constrain these interactions, some of which include Equivalence Principle tests via Torsion Pendulum experiments [32, 33], Lunar Laser Ranging [34] and Atom Interferometry [35]. Experimental limits on Equivalence Principle violation may be expressed in terms of the Eotvos parameter $\eta$

where $\vec{a}_{A}$ and $\vec{a}_{B}$ are the accelerations of two test bodies $A$ and $B$ (which are composed of different materials) towards a third body $C$. Most recently, direct fifth force measurements have placed constraints on $\eta$. Specifically, the MISCROSCOPE mission monitored the difference in acceleration of two freely falling test masses (composed of Pt and Ti) as they orbited the Earth, constraining the Eotvos paramater to be $\eta=(-1\pm 27)\times 10^{-15}$ at a 2-$\sigma$ confidence level [36, 37]. Constraints on this parameter may be repurposed into individual limits on the couplings of the scalar particle to standard model particles.

Another method of constraining these interactions is via atomic spectroscopy measurements. The scalar field is produced by massive bodies and causes the variation of fundamental constants, leading to a variation in the ratio of transition frequencies. Such calculations and measurements have been performed in Refs. [38, 39].

This paper is divided into three parts. In the first part, we seek to obtain limits on the variation of physical constants. The variation of nuclear parameters affects electronic transitions, and the results of the measurements of the variation in the ratio of transition frequencies in Cs/H [28] and the ratio of optical clocks transitions in Al⁺/Hg⁺ [29] allow us to place constraints on the variation of the proton mass $m_{p}$, the variation of the average quark mass $m_{q}=(m_{u}+m_{d})/2$ and the variation of the nuclear charge radius $r_{N}$ and nuclear mass $m_{N}$, all of which may be used to place limits on the variation of the QCD parameter $\theta$. The idea that the dependence of the electronic atomic transition frequencies on the nuclear radius (and subsequently on the hadronic parameters above) may be used in the search for dark matter fields in optical transitions was first proposed in Ref. [30], while the effect of the variation of the nuclear radius (and hadronic parameters) in hyperfine transitions was earlier calculated in Ref. [40]. In this paper, we employ a similar method to Ref. [41], in which the sensitivity of the optical clock transitions in Yb⁺ to the variation in the nuclear radius was used to place constraints on the variation of the hadron and quark masses, and the QCD parameter $\theta$.

In the second part we consider the beyond-standard-model effects of gravity, such as the Einstein Equivalence Principle (EEP) violating term ($c_{00}$) in the Standard Model Extension (SME) Lagrangian [42] and the dependence of fundamental constants on the gravitational potential.

In the third part we consider the effects produced by the interaction between hypothetical scalar particles and standard model particles in the presence of massive bodies creating a Yukawa-type potential mediated by scalar particles. Such effects have previously been considered in Ref. [38]. The scalar field, produced by the Sun and the Moon, affects the fine structure constant $\alpha$ and the fermion mass $m_{f}$. We perform relevant calculations and use the measurement of the variation in the ratio of clock transition frequencies Al⁺/Hg⁺ in Ref. [29], Yb⁺/Cs in Ref. [43] and Yb⁺(E2)/Yb⁺(E3) in Ref. [31] to determine limits on the interactions between this scalar field and standard model particles for a range of scalar particle masses. These constraints are compared to those obtained from the results of the MICROSCOPE mission [36, 37].

Note that when discussing the variation of dimensionful parameters, we must be mindful of the units they are measured in, as these units may also vary. In other words, we should consider the variation of dimensionless parameters which have no dependence on any measurement units. Nuclear properties depend on the quark mass and $\Lambda_{\text{QCD}}$. In this work, we keep $\Lambda_{\text{QCD}}$ constant, meaning our calculations are related to the measurement of the variation of a dimensionless parameter $X_{q}\equiv m_{q}/\Lambda_{\text{QCD}}$. As a result, we measure the quark mass in units of $\Lambda_{\text{QCD}}$ - see Refs. [44, 45, 19]. A similar choice of units is assumed for the variation of hadron masses.

In this paper, we assume natural units $\hbar=c=1$ if $\hbar$ and $c$ are not explicitly presented.

The total electronic energy $E_{\text{tot}}$ of an atomic state contains the energies associated with the finite mass of the nucleus $m_{N}$ (mass shift, MS) and the non-zero nuclear charge radius $r_{N}$ (field shift, FS). Both effects contribute to the isotope shifts of atomic transition frequencies. These are parametrised as [46]

where $K_{\text{MS}}$ and $K_{\text{FS}}$ are the mass and field shift coefficients respectively and $m_{A}$ is the mass of an atom with atomic mass number $A$, which is largely determined by the nuclear mass $m_{N}$. The variation of the total electronic energy associated with the nuclear degrees of freedom can be written as [47]

The mass shift term dominates for light nuclei whilst the field shift term dominates for heavy nuclei. In general, comparing the electronic transition frequencies $\nu_{a}$ and $\nu_{b}$ of two different atomic species, we obtain

where

In this section, we use the above theory along with experimental observations of the drift in atomic transition frequencies to place limits on the variation of quark and hadronic parameters. The approach in this section is similar to that presented in Ref. [41].

Let us first consider the variation of fundamental constants due to the variation of the ratio of transition frequencies in Cs/H. Ref. [28] compared results of the variation in the $\ket{1^{2}S_{1/2},F=1,m_{F}=\pm 1}\rightarrow\ket{2^{2}S_{1/2},F^{\prime}=1,m_{F}^{\prime}=\pm 1}$ two-photon transition in atomic hydrogen to results from clocks based on ¹³³Cs in order to deduce limits on the fractional time variation of the fine structure constant $\alpha$. The comparison of the transition in H against the ground state hyperfine transition in ¹³³Cs gives the following fractional time variation [28]

The variation of this ratio may be related to the variation of fundamental constants - see Ref. [19]

The relative variation of the electron to proton mass ratio can be described as [48]

where $m_{s}$ is the mass of the strange quark. For brevity, we may assume that $\delta m_{s}/m_{s}=\delta m_{q}/m_{q}$. Combining these expressions gives

Therefore, using this expression along with the limit presented in Eq. (11), we may obtain a limit on the variation of the quark mass $m_{q}$ assuming that there is no variation of $\alpha$ and $m_{e}$

When considering the variation of the QCD parameter $\theta$, it is convenient to consider the problem at the hadron level, rather than the quark level. Using the calculations presented in Ref. [19] and the limit (11) we obtain

Using this value, along with the following result from Ref. [13]

we may subsequently place limits on the drift of the proton mass due to the drift of the pion mass

Finally, we may use the relation between the variation of the pion mass and the QCD parameter $\theta$, presented in Ref. [49], to place limits on the linear drift of $\theta$. The pion mass has dependence on $\theta$, and the shift of the pion mass due to a small $\theta$ relative to the pion mass for $\theta=0$ is given by [49]:

Thus, substituting the value for the drift of the pion mass from Equation (16) yields

We may also perform a similar calculation relevant to measurements of a different ratio of transition frequencies. The drift of the ratio of the optical clock transition frequencies of aluminium and mercury has been measured in Ref. [29]. This drift was used to place limits on the temporal variation of the fine structure constant. Using a similar method to that above, this result may be repurposed to extract limits on the variation of the nuclear radius and hadronic parameters. The rate of change in the ratio of the transition frequencies of ${}^{1}S_{0}\rightarrow\ ^{3}P_{0}$ transitions in Al⁺ and ${}^{2}S_{1/2}\rightarrow\ ^{2}D_{5/2}$ transitions in Hg⁺ was found to be [29]

Once again we note the fact that mass shift dominates the isotope shift effects in light elements, while field (volume) shift dominates in heavy elements. As such, the isotopic shift in $\text{Al}^{+}$ is dominated by the mass shift component, while the isotopic shift in $\text{Hg}^{+}$ is dominated by the field shift component.

The mass shift is divided into two components: the normal mass shift (NMS) and the specific mass shift (SMS). The NMS results from a change in the reduced electron mass, and its parameter is easily calculated from the transition frequency using the non-relativistic virial theorem stating that in the case of the Coulomb interaction, the electron’s total energy change corresponds to the change in it’s kinetic energy, with a negative sign:

where the factor in the denominator refers to the ratio of the atomic mass unit to the electron mass. Using the transition frequency from the experimental observation of the ${}^{1}S_{0}\rightarrow\ ^{3}P_{0}$ transition in ²⁷Al⁺ of Ref. [50], we calculate the normal mass shift factor in Al⁺ to be $K_{\text{NMS}}^{\nu_{\text{Al}^{+}}}=-615$ GHz amu. Using the calculated ratio of the specific to normal mass shifts from Ref. [51], we yield a total mass shift parameter of $K_{\text{MS}}^{\nu_{\text{Al}^{+}}}=-1530$ GHz amu.

Now we must consider the effects from the field shift in Hg⁺. We have performed relativistic many-body calculations of the shift of atomic transition frequencies due to the variation of the nuclear radius in Hg⁺. The wave functions and energies in the zeroth-order approximation have been calculated using the Dirac-Hartree-Fock method, including Breit corrections. The correlation corrections were calculated using the CIPT method (configuration interaction with perturbation theory [52]), which allows us to deal with open electronic shells. This approach is similar to that described in Ref. [53]. We have found the field shift parameter for the ${}^{2}S_{1/2}\rightarrow\ ^{2}D_{5/2}$ transition in Hg⁺ to be $85.5$ GHz/fm². Similar calculations are performed to find isotope shifts in atomic transitions. Our value for the isotope shift in the ${}^{2}S_{1/2}\rightarrow\ ^{2}D_{5/2}$ transition of Hg⁺ is in agreement with the calculations of the isotope shift in Hg⁺ presented in Ref. [54]. Thus, using the MS and FS parameters along with Eq. (6), we obtain

Here, we have used the fact that the nuclear radii in all nuclei may be quite accurately related to the internucleon distance $r_{0}$ by the universal formula $r_{N}=A^{1/3}r_{0}$. This implies that these quantities have equivalent fractional variations. Presenting limits on the variation of $r_{0}$ is more useful as it allows one to compare the results of measurements in different nuclei. Noting that $r_{N,\text{Hg}^{+}}^{2}=5.4474$ fm in ¹⁹⁹Hg [55] and $m_{N,\text{Al}^{+}}\approx 27m_{p}$ in ²⁷Al, we yield the following limits

Considering these effects as separate sources of the variation allows us to derive independent limits on the variation of quark and hadronic parameters.

Firstly, using a similar method to the one presented in Ref. [41], we consider the effects arising from a variation in the internucleon distance $r_{0}$. Calculations of the dependence of nuclear energy levels and nuclear radii on fundamental constants were performed in Refs. [44, 56, 45]. Specifically, in Table VI of Ref. [45], the sensitivity coefficients of nuclear radii to the variation of hadron masses for several light nuclei have been presented. These results may be extended to all nuclei due to the relation $r_{N}=A^{1/3}r_{0}$. This relation follows from the constancy of nuclear density and reasonably describes the nuclear radius for nuclei with mass number $A>2$. Therefore, by calculating the variation in $r_{0}$, a fundamental parameter, we can generalize the results for light nuclei to include all nuclei. The sensitivity coefficients are defined by the relation

The sum over hadrons in Refs. [56, 45] includes contributions from $\pi$, nucleon, $\Delta$ and vector mesons (these hadron masses are parameters of the kinetic energy and nucleon interaction operators used in Refs. [56, 45]). The sensitivity to the pion mass is given by the coefficient $K_{\pi}=1.8$ and the sensitivity to the nucleon mass is given by $K_{n}=-4.8$. We neglect contributions from $\Delta$ and vector mesons as they are of a similar magnitude with opposing sign, meaning their resulting contribution is small and unstable. Note that the estimate in Eq. (26) is model dependent and may have an error O(1).

Subsequently, the variation of hadron masses may be related to variation of the quark mass, see e.g. Ref. [57]:

where $m_{q}=(m_{u}+m_{d})/2$ corresponds to the average light quark mass. The sensitivity coefficient for the pion mass is an order of magnitude bigger than that for other hadrons since the pion mass vanishes for zero quark mass ($m_{\pi}\propto m_{q}^{1/2}$) while other hadron masses remain finite. Indeed, according to Refs. [58, 59] $K_{\pi,q}=0.498$ for the pion, while $K_{n,q}=0.06$ for nucleons. The sensitivity coefficients to the quark mass have been calculated for light nuclei in Ref. [45]. The average value is given by

We note that here there are partial cancellations of different contributions, so the sensitivity is smaller than that following from pion mass alone. Refs. [56, 45] have also presented calculations of the dependence of the nuclear energies and radii on the variation of the fine structure constant $\alpha$. Applying Equation (28) to the limit on the variation of the internucleon distance presented in (24), we determine limits on the variation of the quark mass to be

Once again, in order to place limits on the variation of the proton mass and the QCD parameter $\theta$, it is convenient to consider the problem at the hadron level, rather than the quark level. In Ref. [45], the sensitivity of the nuclear radius to the masses of the pion, nucleon, vector meson and delta has been calculated. In the following estimate, we do not include contributions from the vector meson and delta as their contributions are smaller. These contributions also have opposing signs, meaning they partially cancel each other out making their contribution less reliable. The variation of the nuclear radius may be written in terms of the pion and nucleon mass as

where in the last equality we have applied Equation (17). Thus, once again applying Equation (24), we obtain a limit on the drift of the pion mass

Finally, we apply Equation (19) in order to place constraints on the linear drift of the QCD parameter $\theta$

These limits are approximately 44 times weaker than the limits imposed from the sensitivity of the optical clock transitions in Yb⁺ obtained in Ref. [41]. This difference corresponds exactly to the difference in the accuracy of the experimental measurements of the variation in clock frequency ratios, showing that these systems have equivalent sensitivities to the variation of the internucleon distance. We however note that the limits from the Al⁺/Hg⁺ clock ratio are $\sim 2-4$ times stronger than those calculated for the Cs/H system, despite the fact that the accuracy of the Al⁺/Hg⁺ measurements is $\sim 60$ times better. This implies that the sensitivity of the Cs/H ratio to changes in the hadron constants is higher than that of the Al⁺/Hg⁺ ratio.

Let us now consider the effects arising from a variation of the nuclear mass $m_{N}$. The nuclear mass may be related to the proton mass $m_{p}$ by the following relation $m_{N}\approx A\ m_{p}$. As such, these parameters have equivalent fractional variations, meaning the limit from Eq. (25) applies. Using this result, along with the relation between the pion and the proton mass presented in Equation (17), we may place limits on the drift of the pion mass

Using the calculations presented in Ref. [19], we may use this result to place limits on the variation of the quark mass $m_{q}$

We also use the relation between the variation of the pion mass and the QCD parameter $\theta$ presented in Equation (19) to place constraints on the linear drift of $\theta^{2}$

As expected, the limits obtained upon considering the variation in the ratio of frequencies (21) as being due to the drift of the nuclear (and hence nucleon) mass are weaker than the limits from the variation of the internucleon distance.

In some theoretical models, atomic transition frequencies and fundamental constants may depend on the gravitational potential. In Ref. [60] limits on the gravity related variation of fundamental constants are derived from measurements of the drift of atomic clock frequency ratios. In a similar way, we may obtain limits on the gravity related variation of the nuclear radius and nuclear mass using the measurement of the variation in the ratio of transition frequencies in $\text{Al}^{+}/\text{Hg}^{+}$ from Refs. [61, 29]. We will also calculate limits on gravity related variation of these quantities for other systems of interest, Yb⁺/Cs [43] and Yb⁺/Yb⁺ [31].

Noting the dependence of the frequency shift on the nuclear mass and radius from Eq. (6), we introduce the parameters $\kappa_{r,N}$ as follows: (see Ref. [60])

It is instructive to link these parameters $\kappa_{r}$ and $\kappa_{N}$ to the Einstein Equivalence Principle (EEP) violating term in the Standard Model Extension Hamiltonian  [42]. This term may be presented as a correction to the kinetic energy which in non-relativistic form is equal to (see e.g. [62])

where $c_{00}$ is a parameter of the Standard Model Extension (SME) Lagrangian [42], $U$ is the gravitational potential, $p$ is the electron momentum operator and $m_{e}$ is the electron mass. Limits on $c_{00}$ have been found by monitoring the drift of atomic clock frequencies for different transitions (see e.g. [61, 62]). Ref. [61] subsequently used the limits on this parameter to determine a limit on the gravity-related variation of the fine structure constant $\kappa_{\alpha}$

where in the case of a relative change of two transition frequencies in different atomic species $\delta\omega_{a}/\omega_{a}=K_{\alpha a}(\delta\alpha/\alpha)$ and $\delta\omega_{b}/\omega_{b}=K_{\alpha b}(\delta\alpha/\alpha)$, the parameter $\kappa_{\alpha}$ is equal to

where $R_{a}$ and $R_{b}$ represent the relativistic factors of atoms $a$ and $b$ respectively, which describe the deviation from the expectation value of the kinetic energy of a relativistic atomic electron from the value given by the non-relativistic virial theorem [61]

where $\Delta E_{i}$ is the energy shift of the state $i$ due to the relativistic kinetic energy operator. In the non-relativistic limit, $R=1$. Thus, we employ a similar method and derive expressions for the parameters $\kappa_{r}$ and $\kappa_{N}$ describing the variation of the nuclear radius and nuclear mass:

where $K_{1,2,3,4}$ are defined in Eq. (6). The value of $R$ has been calculated for a number of clock transitions in Ref. [61], see Table 1. Ref. [61] also calculated a limit on the value of the SME parameter $c_{00}$ due to the change in the Sun’s gravitational potential for the ratio of these transition frequencies $\nu_{\text{Al}^{+}}/\nu_{\text{Hg}^{+}}$. As such, we may substitute all known quantities into Eqs. (42, 43) and determine a limit on the gravity related variation of the nuclear radius and nuclear mass. We also use the limits on the coupling to gravity of the fine structure constant $\alpha$, presented in Refs. [43] (Yb⁺/Cs) and [31] (Yb⁺/Yb⁺) in order to place improved limits on gravity’s coupling to the nuclear radius $\kappa_{r}$. In performing these calculations, we make use of the following result for the difference in field shift parameters in the electric octupole (E3) and electric quadrupole (E2) transitions in Yb⁺: $K_{1}-K_{2}=2.4\times 10^{-3}$ [30], and the sensitivity of variations in this ratio to the fine structure constant $\alpha$, $k_{\alpha}(E3)-k_{\alpha}(E2)=-6.95$ [18]. The results are presented in Table 2.

In this section, we consider the Yukawa-type interaction mediated by the scalar field produced by massive bodies. Let us first provide an introduction to the phenomenology of this scalar field, following Ref. [38]. A scalar field $\phi$ interacts with the standard model sector via the Yukawa-type Lagrangian:

Here, the first term represents the coupling to fermion fields $f$, with mass $m_{f}$ and $\bar{f}=f^{\dagger}\gamma_{0}$, while the second term represents the coupling to the photon field. $\Lambda_{f}$ and $\Lambda_{\gamma}$ are new-physics energy scales which determine the strength of these couplings. Adding these interaction terms to the relevant terms in the standard model Lagrangian

we observe that we may present the effects of the interaction terms in the form of a variable fermion mass $m_{f}$ and electromagnetic fine-structure constant $\alpha$ (see e.g  [12])

Adding the kinetic term to the interaction Lagrangian (44) gives the following equations of motion for the field $\phi$:

where $m_{\phi}$ is the mass of the scalar particle. This implies that in the presence of the interaction (44), the standard model fermion and photon fields act as sources of the scalar field. Massive bodies such as stars or galaxies, which are composed of atoms, may act as these sources, producing a scalar field mediating Yukawa-type interactions. This field may produce a local variation of fundamental constants in the presence of a massive body with a varying distance to the laboratory. As such, we may investigate the influence of the scalar particle $\phi$ on atomic spectroscopy experiments, see Ref. [38]. In the following subsections we place constraints on the scalar field’s interaction with the Standard Model particles. Specifically, we consider the effects produced by a variation in the scalar field due to both the semi-annual variation in the Sun-Earth distance, and the approximately semi-monthly variation in the Moon-Earth distance. These constraints are compared to those obtained from tests of the equivalence principle by the MICROSCOPE mission [36, 37].

Atomic spectroscopy measurements are used to search for the potential space-time variation of physical constants. In particular, we relate the proton mass $m_{p}$ and the quark mass $m_{q}$ variation to measurements in the variation of two frequency ratios: the comparison of the two-photon transition in atomic Hydrogen to the results from clocks based on ¹³³Cs in Ref. [28], and the variation in the ratio of the two optical clock frequencies in Al⁺ and Hg⁺ in Ref. [29]. We used this data to place new limits on the variation of the proton mass $m_{p}$, as well as independent limits on the variation of the quark mass $m_{q}$, both of which may be used to place limits on the variation of the QCD parameter $\theta$.

In the second part of this paper we considered the beyond-standard-model effects of gravity, such as the Einstein Equivalence Principle (EEP) violating term ($c_{00}$) in the Standard Model Extension (SME) Lagrangian [42] and the dependence of fundamental constants on the gravitational potential, based on the measurements of the dependence of the ratio of atomic transition frequencies Al⁺/Hg⁺  [61, 29], Yb⁺/ Cs [43] and Yb⁺/Yb⁺ [31] to the Sun-Earth distance.

In the third part of this paper we considered the scalar field produced by massive bodies. We determine limits on the interactions of the scalar particle with photons, nucleons and electrons for a wide range of scalar particle masses, basing on the measurements of dependence of the ratio of atomic transition frequencies Al⁺/Hg⁺  [61, 29], Yb⁺/ Cs [43] and Yb⁺/Yb⁺ [31] on the Sun-Earth and Moon-Earth distances. If the scalar particle’s Compton wave length $\hbar/m_{\phi}c$ is bigger than the distance to the source, we place the following limits on the coupling constants of the interaction of the scalar field with photons, nucleons and electrons: $\Lambda_{\gamma}\gtrsim 1\times 10^{21}\ \text{GeV}$, $\Lambda_{n}\gtrsim 5\times 10^{21}\ \text{GeV}$ and $\Lambda_{e}\gtrsim 1\times 10^{20}\ \text{GeV}$.

The work was supported by the Australian Research Council Grants No. DP230101058 and No. DP200100150.