In developing the next-generation frequency standard, the precision of atomic spectroscopy has been considerably improved. For instance, the relative uncertainty of is realized in a clock transition of [1]. The corresponding accuracy of frequency (energy) is of the order of mHz ( eV).

We consider the possibility of constraining or probing a new physics beyond the standard model of particle physics with such high precision spectroscopy. It is practically impossible to calculate transition frequencies of many-electron atomic systems at the above level of accuracy. To overcome this difficulty, a method to use isotope shifts (IS) was proposed [2] and further studied in a quantitative manner [3]. ¹

The typical magnitude of observed IS is the order of GHz ( eV), much smaller than optical transition energies, and such quantities are evaluated by perturbation. If nuclei were infinitely heavy and point-like, the electrons in an atom could not sense the difference of the nuclei among isotopes and no IS would exist. Thus, it is customary to decompose observed IS into the mass shift (MS), which is due to the finite nuclear mass, and the field shift (FS), due to the finite nuclear size [5]. In other words, the nuclear recoil causes the MS and the change of the nuclear Coulomb field causes the FS.

The isotope shift of a transition (labeled by ) between an isotope pair and , , in the leading order perturbation is expressed by (1) where represents the difference of the reduced masses, denotes that of the mean-squared radii of nuclei, and and are electronic factors. The electronic factors solely depend on the transition and are common for isotopes.

The key idea to exploring new physics using IS is King’s linearity [6] and its breakdown by a new interaction [2]. Suppose isotope shifts of several isotope pairs in two distinct transitions () of an atom (or an ion). It is convenient to introduce the modified isotope shift (mIS) so that Eq. (1) is written as (2)

We note that atomic masses are measured with typical precision of or better [7]. ² Eliminating the nuclear factor , we obtain a linear relation between the mIS values of two transitions, (3) where and .

This linear relation is broken by a new force of particle exchange between an electron and a neutron. In the presence of such particle exchange, a particle shift (PS), a new source of IS, is added to the IS expression in Eq. (1), (4)

The linear relation in Eq. (3) is also modified as (5) where (6) and denotes the atomic mass unit. We use to quantify the nonlinearity due to PS for a pair of transitions. We note that values of for different pairs of transitions cannot be directly compared in general.

In the standard model, as pointed out in Ref. [2], a Z or Higgs boson leads to such a nonlinearity. However, the nonlinearities by such heavy particles are strongly suppressed and their measurement is beyond the reach of the present technology [2,8]. If there exists a light boson that couples to the electron and the neutron, the nonlinearity may be detectable depending on its mass and couplings.

Other conceivable sources of nonlinearity within the standard model are discussed in the literature. It is pointed out in Ref. [9] that the subleading FS values in the and states are likely to cause a nonlinearity as well as the FS in the second-order perturbation. A rough estimate of nonlinearities is given in Ref. [10], in which approximate analytic wave functions are used in contrast to numerical ones in the present work.

The effect of the subleading FS, other than the leading contribution shown in Eq. (1) (e.g. a term proportional to as described in Sect. 4), is effectively included in Eq. (5) by replacing with , where represents the FS nonlinearity. The FS nonlinearity is a potential background in the search of new physics with the IS nonlinearity, though it can be removed by using three or more transitions in principle [8].

In this work, we evaluate the PS and FS nonlinearities employing numerical relativistic wave functions and compare the result with that of our previous work with nonrelativistic (NR) wave functions [8]. We find that the sensitivity to a new electron–neutron interaction is enhanced in the mediator mass range of about 1 to 10 MeV in our relativistic evaluation but the FS nonlinearity significantly increases as far as a state is involved in the relevant transitions.

The rest of the paper is organized as follows. We present the formulation to evaluate PS and FS in Sect. 2. In Sect. 3, our method to obtain relativistic wave functions is described. We give formulas to estimate PS and FS nonlinearities with the obtained wave functions in Sect. 4. The current experimental status and future prospects are given in Sect. 5. Section 6 is devoted to our conclusion. We use the natural units, , unless otherwise stated.

As in the previous work, we employ the single-electron approximation to evaluate PS and FS. One valence electron is supposed to change its state (orbital) in a transition while the state of the other electrons is kept intact. We denote the radial densities of the initial and final states of this electron by and , respectively. We account for them in Sect. 3.

A new spin-independent force between an electron and a neutron by exchanging a scalar or vector boson of mass is described by the potential, (7) where is the mediator spin, and and denote the neutron and electron coupling constants, respectively. ³ The PS electronic factor in Eq. (4) is given by (8) where and the radial densities are normalized as . We note that the point-like nucleus is assumed in Eq. (7) so that it is valid for the inverse of the mediator mass larger than the nuclear radius .

We also evaluate the FS with the same electron densities to estimate the magnitude of the FS nonlinearity. The FS is the variation of the transition energy owing to the difference of the Coulomb potential of two nuclei, and . Assuming spherically symmetric nuclear charge distributions, the potential difference between isotopes is also spherically symmetric and represented by (9) where is the nuclear charge, , and the spherical nuclear charge distribution of an isotope denoted by is normalized as . Then, the FS of transition is given by (10)

In order to evaluate the radial electron density in a relativistic manner, we solve the Dirac equation with an effective potential that represents the potential by the nucleus and the electrons other than the one involved in a transition.

The leading contribution in the FS, in Eq. (1), is identified by expanding as . We note that, as illustrated in the inset of Fig. 1, no linear term appears provided that the nuclear charge distribution has no cusp at the origin [9].

Rewriting Eq. (10) as (22) it is straightforward to find that (23) where [16]. Thus in the formula of PS nonlinearity in Eq. (6) is given by , which is solely determined by the wave functions at the origin.

As for the PS, we find (24) and that the leading term in the case of heavy mediator disappears in the expression of the PS nonlinearity in Eq. (6) so that as [8].

In our analysis below, we numerically evaluate the integration in without expanding . The FS is also obtained by numerical integration in Eq. (10) [or (22)] without expansion. We decompose the FS into the leading and subleading contributions as . Then, the FS nonlinearity is given by (25) as in the same manner as the PS nonlinearity. Strictly speaking, evaluated by subtracting the leading FS from the total FS depends on the isotope pair . However, our numerical result shows that the dependence is weak and less than a few per cent among the isotope pairs analyzed below.

We analyze the same experimental data of the singly charged calcium and ytterbium ions as our previous work [8], in which NR wave functions are employed. The IS in the transitions of – (397 nm) and – (866 nm) of are measured with kHz precision for the isotope pairs (40, 42), (40, 44) and (40, 48) [17]. The 397-nm transition is treated as the – transition and the 866-nm as the – in the single-electron approximation. As for , the IS of – (369 nm, –) [18] and – (935 nm, –) [19] are measured for (172, 170), (172, 174) and (172, 176). The experimental precision of the former transition is MHz and that of the latter is MHz. A summary of these data and King’s plots are given in Ref. [8].

It turns out that the data satisfy the linearity within the error. Comparing the data with the IS formula including the nonlinearity, i.e. Eq. (5) with the PS nonlinearity parameter replaced by the experimental nonlinearity parameter , we construct the as described in Appendix E of Ref. [8] and obtain quantitative bounds on the nonlinearity as presented in Table 1. We also present the expected sensitivity of future experiments in Table 1, extrapolating the experimental precision to 1 Hz. We note that IS measurements of 1-Hz precision are conceivable since the IS of the – (729 nm) transition is measured with a precision better than 10 Hz [20].

Our prediction of the PS nonlinearity indicated in Table 1 is that for a representative set of mediator mass and couplings in Eq. (7), and . The PS nonlinearity of for this set of parameters is the same order as the sensitivity expected in experiments of 1-Hz precision. For , the sensitivity of 1-Hz experiments is sufficiently high to probe the PS nonlinearity for the same set of parameters, provided that the FS nonlinearity is suppressed.

Figure 2 shows the current constraints on the mass and couplings as well as those expected with experimental data of 1-Hz precision. The region above the upper lines are excluded by the present data, while the lower lines express the expected sensitivity of future experiments. The red (blue) lines are the case of (). The results of the relativistic calculation in the present work are represented by the solid lines and those of the NR calculation in our previous work [8] are shown by the dash–dotted lines for comparison. We observe that the PS nonlinearity is enhanced and the sensitivity is improved significantly for MeV in the relativistic calculation of compared to the NR case. This is due to the nonvanishing electron density of the state at the origin, shown in Fig. 1. The relativistic effect is less notable for in the depicted mass range.

The peak structure that implies the loss of sensitivity at the corresponding mediator mass is due to the cancellation in the PS electronic factor in Eq. (8). Since the peak position depends on the involved wave functions, it is useful for evading the sensitivity loss to combine two or more linearity tests, e.g. and as shown in Fig. 2.

For the case, the present work employs the same experimental data (of the same transitions) as Ref. [3]. Our single-electron calculation reasonably agrees with the more sophisticated one in Ref. [3]. For example, the peak position of the present constraint in Fig. 1 of Ref. [3] is reproduced well in our calculation.

The current and expected constraints of IS nonlinearity in the and transitions in Fig. 2 are compared with those by other experiments and observations for each case of a scalar and a vector mediators in Fig. 3. The bounds on a scalar (vector) mediator are presented in the top left-hand (top right-hand) panel as well as the IS constraints, and the IS constraints are compared with other bounds on a scalar (vector) mediator in the bottom left-hand (bottom right-hand) panel. The constraints from the IS are identical for both the scalar and vector cases. The grey shaded regions of smaller mass in all the panels are excluded by the fifth force search [21,22]. The shaded regions bounded from below by the orange lines indicate the regions excluded by the electron measurement [23,24], the dark photon search at BaBar [25], and the beam dump experiments [26,27], combined with the neutron scattering experiments [28–30]. ⁵ The BaBar result is relevant in the mass region above 20 MeV for the vector mediator. The beam dump experiments are applied in the region between 100 keV (1 MeV) and MeV for the case of a scalar (vector) mediator assuming no invisible decay modes of the mediator. The electron is used in the rest of the mass region of the orange lines. The combined terrestrial bounds are not identical for the scalar and vector mediators. The regions on the left-hand side of the brown lines are constrained by the stellar cooling [31]. We note that there is an uncertainty that may invalidate these stellar constraints in the relatively strong coupling regions above the brown dotted lines [32]. The black vertical bars in the right-hand panels indicate the coupling range suggested by the 17-MeV Atomki anomaly in the internal conversion [33–36].

In the fifth column of Table 1, we present the FS nonlinearity parameter in Eq. (25) evaluated with the relativistic wave functions. It turns out that the FS nonlinearity is enhanced by about two orders of magnitude compared to the NR calculation, which gives and for and , respectively. This is due to the nonvanishing wave functions at the origin for both and as illustrated in Fig. 1 for the latter.

If is smaller than , the experimental search of new force through the IS nonlinearity becomes difficult. This is the case in the shaded regions below the red () and blue () dashed lines in Fig. 3. These dashed lines represent the expected sensitivities in experiments of 0.3-Hz precision for and 1 kHz for . ⁶ We observe that the extrapolated experiment of the same pair of transitions has the potential sensitivity to probe the parameter space of the mass range from to keV that has not been excluded by other experiments nor observations. As for , the maximal signal of new physics is comparable to or smaller than the background of the FS nonlinearity in the transition pair considered in the present work, although the signal sensitivity (the lower blue solid lines in Fig. 3) does not differ much from that of a different pair of transitions given in Ref. [3].

In our previous work [8], we described a method to remove the FS nonlinearity by generalizing the linearity relation with three or more transitions. Here we consider an alternative way to avoid the large FS nonlinearity.

The inclusion of two distinct states that have nonvanishing wave functions at the origin, such as and in this work, leads to the large FS nonlinearity as argued in Ref. [8]. This is because the FS is governed by the wave functions inside the nucleus. Hence it is possible to suppress the FS and its nonlinearity by replacing the state with a state whose wave function vanishes at the origin, e.g. the state shown in Fig. 1. Our estimation of the magnitudes of FS nonlinearity for the cases of is presented in the last column of Table 1. We observe that the use of instead of reduces the FS nonlinearity to the level of the NR calculation. This is plausible because of the similarity of the and NR wave functions as seen in Fig. 1. We note that the PS nonlinearity in the mediator mass range below MeV is not significantly affected by substituting for . It is possible in principle to employ a state other than as far as it vanishes at the origin.

We have examined the implication of relativistic calculation in the search of new intra-atomic force using the IS nonlinearity. The relativistic wave functions in the single-electron approximation are obtained by numerically solving the Dirac equation. We have employed the effective potential described by the Thomas-Fermi model with the Helm nuclear charge distribution. We have evaluated the PS and FS nonlinearities with these wave functions. The calculated nonlinearities and the current experimental bounds by the and the IS measurements are presented in Table 1 as well as the expectation for future experiments.

In Fig. 2, the bounds on the mass and couplings of the force mediator in the relativistic calculation are compared with those in the NR calculation [8]. We have found that the NR calculation underestimates the magnitude of the PS nonlinearity in the mass range greater than MeV for , while the relativistic effect in the PS is less sizable for . Our results of the single-electron approximation in Ref. [8] and the present work are consistent with the result of the same pair of transitions in Ref. [3].

The current bound and the future sensitivity of IS nonlinearity are also compared with constraints from other experiments and observations in Fig. 3. It is indicated that, although the current bound is weaker than the best other constraints, the IS nonlinearity has a potential sensitivity to probe the unexplored parameter space in future. This observation also agrees with Ref. [3] although they considered different transitions from ours. Future experiments of 1-Hz precision have been supposed in our numerical calculation as an illustration. As mentioned above, IS measurements with the precision of the order of 1 Hz are likely in the near future. When such data of two or more transitions become available, the sensitivity of IS nonlinearity to new physics is expected to be notably improved.

Moreover, it turns out that the FS nonlinearity is significantly enhanced by the relativistic effect and the new force search beyond the IS precision of about 1 kHz is limited for the pair of transitions employed in this work. The case is limited at and beyond about 0.3 Hz, so that the problem is less serious. The tendency of more significant FS nonlinearities for heavier elements is consistent with the result of Ref. [10], though the pairs of transitions treated are not identical.

The large FS nonlinearity is due to the nonvanishing value of the wave function at the origin, unlike the NR p state. Therefore, instead of , we have considered the use of the state, the wave function of which vanishes at the origin like the NR p state. We have found that the FS nonlinearity is suppressed in the case as shown in Table 1.

In conclusion, the experimental search of new physics with the IS nonlinearity has a potential sensitivity beyond the existing terrestrial constraints. A proper selection of transitions is important as well as improvements of experimental precision.