Constraining New Physics Models with Isotope Shift Spectroscopy
Abstract
Isotope shifts of transition frequencies in atoms constrain generic long and intermediaterange interactions. We focus on new physics scenarios that can be most strongly constrained by King Linearity Violation such as models with vector bosons, Higgs portal and chameleon. With the anticipated precision, King Linearity Violation has the potential to set the strongest laboratory bounds on these models in some regions of parameter space. Furthermore, we show that this method can probe the couplings relevant for the protophobic interpretation of the recently reported Be anomaly. We extend the formalism to include an arbitrary number of transitions and isotope pairs and fit the new physics coupling to the currently available isotope shift measurements.
I Introduction
The standard model of particle physics (SM) is one of the most successful scientific theories. Yet, it cannot be a complete description of nature because, for example, it does not provide a viable dark matter candidate and cannot account for the observed baryon asymmetry of the universe. Yet, the energy scale associated with new physics (NP) is unknown and therefore the experimental program for physics beyond the SM should be as broad as possible. Colliders are one of the main tools to study elementary particles and the LHC is pushing forward the energy frontier. A complementary and vital role is played by lowenergy, precision and/or highintensity experiments, which require a joint effort of the particle, atomic, and nuclear physics communities. The MeVGeV scale is already efficiently probed by a variety of highintensity experiments Alexander:2016aln ; Essig:2009nc ; Essig:2010xa ; Batell:2009yf ; Batell:2009di ; Izaguirre:2014dua ; Dobrescu:2014ita ; Coloma:2015pih ; Frugiuele:2017zvx ; Flacke:2016szy . The existence of new subMeV degrees of freedom can instead be probed by both astrophysical observations and precision experiments. In this context, atomic physics observables play an important role. For example, atomic precision measurements can be used to constrain interactions beyond the SM (BSM), see e.g. Refs. Jaeckel:2010xx ; Ficek:2016qwp . In a broader context, atomic physics probes have been used to test the violation of fundamental laws such as parity (see e.g. Refs. Porsev:2009pr ; Tsigutkin:2009zz ; PhysRevA.81.032114 ; Leefer:2014tga ; Versolato:2011zza ; Dzuba:2012kx ; Roberts:2014bka ; Jungmann:2014kia ), Lorentz symmetry Dzuba:2015xva ; Flambaum:2015rwt and even the time variation of fundamental constants of nature Godun:2014naa ; Rosenband1808 .
A new proposal to constrain NP using isotope shift measurements was presented in Ref. Delaunay:2016brc . In Ref. Berengut:2017zuo this proposal was detailed and it was shown that it can constrain new light degrees of freedom mediating long and intermediaterange spinindependent interactions between electrons and neutrons. These new interactions cause a frequency shift that is factorized to a high degree into a product of a function that solely depends on the electronic degrees of freedom and a function that depends on the nuclear physics ones. Within the validity of this factorization, a linear relation between isotope shifts of different transitions is obtained. This is known as King linearity King:63 ; King:13 . New interactions mediated by light mediators that couple electrons to neutrons generally lead to a nonlinear relation Delaunay:2016brc . We shall denote such an effect as King Linearity Violation (KLV). The absence of a deviation from linearity allows to constrain the New Physics contributions.
Existing measurements of isotope shifts cannot probe so far unconstrained regions of parameter space, but the projected sensitivity allows to explore regions presently left unconstrained. The goal of this work is to investigate how this statement applies to specific models where couplings to other SM particles (in addition to the electron and neutron couplings) and thereby additional constraints become relevant.
The paper is organized as follows. In Sec. II we introduce the notation and provide a brief summary of Ref. Berengut:2017zuo . In Sec. III we generalize the formalism and provide a fit of NP interactions to the available data sets of Ca and Yb. As an application, we identify models for which KLV constraints are relevant, such as a new gauge boson , the Higgs portal, chameleon models and protophobic models in the context of the recently observed Be anomaly, and we discuss the implications before concluding in Sec. V.
Ii Probing new physics via isotope shift measurements
Consider a narrow atomic transition between two atomic states and two even isotopes and . The isotope shift (IS) is defined as the difference of the transition frequencies, . The leading contributions to the IS stem from two sources: the mass shift (MS) and field shift (FS). The former arises from the mass difference of the isotopes and . It can be factorized into an electronic coefficient , which only depends on the transition , and the isotopedependent reduced mass given by
(1) 
which is measured at high precision. The FS originates from the different volumes of the two isotopes. At leading order it also factorizes into the electronic, isotope independent coefficient and the charge radius variance, , where is the nuclear charge radius of isotope . In contrast to , is subject to large experimental uncertainties. The composition of the IS in terms of products of purely electronic and purely nuclear quantities is referred to as factorization King:63 . As a result, these two leading contributions amount to the total IS as
(2) 
where the first term represents the MS and the second one the FS King:63 ; King:13 . The dots denote possible higher order corrections and NP contributions which we will discuss below. It is useful to normalize the frequency shifts by the reduced mass to obtain the socalled modified isotope shifts, , which we will use in the following. As a consequence, the mass shift is reduced to the electronic factor whereas the FS factor is multiplied by the modified charge radius variance, .
When considering several pairs of isotopes, the modified Eq. (2) can be written in vectorial form as
(3) 
where each line corresponds to one set of isotopes. For the example of four isotopes combined to three isotope pairs , where and is the reference isotope, the IS vector of transition is given by , and accordingly. The mass shift vector is denoted by .
With measurements of two transitions the unknown charge radius distribution can be replaced by measured quantities. Solving Eq. (3) with for and replacing it in the equation with leads to
(4) 
with and . Hence, this replacement gives rise to a linear dependence between the two sets of modified frequency shifts , referred to as King linearity King:63 .
In order to quantify the observed linearity, we define a measure of nonlinearity Berengut:2017zuo ,
(5) 
which corresponds to the volume of the parallelepiped spanned by the vectors , and (for illustration see Ref. Berengut:2017zuo ). King linearity is considered to hold if the measure NL is smaller than its uncertainty ^{1}^{1}1At the present level of experimental accuracy, the uncertainties on the isotope masses are smaller by several orders of magnitude than those of the frequency shifts (e.g. smaller for Yb masses Myers2013107 with the present IS accuracy of MHz PhysRevA.94.052511 ; PhysRevA.20.239 ). Therefore we will neglect them in our numerical evaluation. Once the uncertainties of IS measurements will be significantly reduced, the mass uncertainties will have to be taken into account.. In several atoms and ions, King linearity has indeed been established within the experimental uncertainty of on the IS, see e.g. Refs. PhysRevLett.115.053003 ; Shi2016 ; PhysRevA.94.052511 ; PhysRevA.20.239 ; PhysRevA.59.3513 .
In Ref. Berengut:2017zuo it was shown that new physics contributions from light bosons interacting with electrons and neutrons can lead to a deviation from the linear relation in Eq. (4). Thereby, the observation of linearity allows to set bounds on the mass and coupling of a possible new force mediator.
To be specific, a new physics contribution is added as a third term to Eq. (2)
(6) 
where , are the couplings of a new boson to electrons and neutrons, respectively. Furthermore, we have introduced the electronic NP factor and the reduced isotope dependence . Both of them are modeldependent; a specific expression is given below. Proceeding as in the SM case, one can express as a function of , yielding
(7) 
Thus, NP can break King linearity. For unit coupling, the NP contribution to NL is given by the projection of onto the normal vector of the King plane,
(8) 
vanishes if

NP mediates a shortrange interaction, shorter than the nuclear size. In this case the electronic parameter becomes proportional to the electronic parameter of the FS, namely so that the bracket in Eq. (7) vanishes or

the isotopedependent NP contribution is proportional either to or to the reduced charge radius , such that the NP contribution can be absorbed in a redefinition of or , respectively.
Finally, solving the set of equations in Eq. (7) determines the central value of needed to yield a particular data set {, , },
(9) 
The interval of can be obtained via error propagation of the uncertainties on the involved quantities. In case of linearity, is compatible with zero and the method reaches its maximal sensitivity, whereas if nonlinearity is found a bound can be set with the experimental resolution at which nonlinearity emerges. In the following we will adopt the same approach as in Ref. Berengut:2017zuo based on the bestcase projection where linearity holds up to the experimentally achievable precision.
Indeed, nonlinearity cannot only arise from NP, but also from SM higherorder contributions. The dominant effects are expected as corrections to the FS operator (see Refs. PalmerStacey:1982 ; PhysRev.188.1916 ; SmExp:1981 ; 002237002015015 ; PhysRevA.31.2038 for relevant discussions). However, these estimates are not tailored to the most promising elements and transitions. Thus, in order to fully exploit the KLV potential to probe NP interactions, a significant improvement of the atomic theory input will be crucial to match the experimental precision.
In the remainder of the paper, we will consider NP interactions that couple linearly to the SM fermions. Hence the isotopedependent NP part takes the form
(10) 
where in the last step we approximated and amu denotes the atomic mass unit. Therefore, in this approximation, can be written as with . We will further assume a Yukawalike potential of the NP interaction, mediated by a boson of mass ,
(11) 
In the massless limit, where is the Bohr radius and the ionization number, the electronic NP factors can be expressed as Berengut:2017zuo
(12) 
with being the fine structure constant. For the transition between the energy levels and , the effective nuclear charges account for the partial shielding of the nuclear charge for a valence electron at the states , respectively. For later use, we define
(13)  
(14) 
In the numerical evaluation in Sec. III we will use this approximation, see Ref. Berengut:2017zuo for more details.
Iii Fit of the NP coupling
The formalism proposed in Ref. Berengut:2017zuo and summarized in Sec. II is constructed for the minimal case of two transitions and three isotope pairs. However, at present there are already more measurements at comparable accuracy available in some systems, such as three transitions in Ca PhysRevLett.115.053003 ; Shi2016 and four independent isotope pairs in Yb PhysRevA.94.052511 ; PhysRevA.20.239 . Hence the system is overconstrained and a fit of the NP coupling is necessary. In the following we will therefore perform a fit to the data to obtain a limit on the NP coupling .
element  transition  [MHz]  Ref.  
Ca  (D1)  397  0.1  3  PhysRevLett.115.053003 
866  0.1  3  PhysRevLett.115.053003  
(D2)  393  0.1  3  Shi2016  
Yb  399  0.5  4  PhysRevA.94.052511  
555.65  0.5  4  PhysRevA.20.239  
404  10  3  PhysRevA.59.3513  
408  2  3  PhysRevA.59.3513 
element  omitting  
Ca  397 nm  0.00  
866 nm  0.00  
393 nm  0.00  
–  0.04  
Yb  168  0.00  
170  0.00  
172  0.00  
176  0.00  
–  0.34 
Under the assumptions made in Eq. (10) and Eq. (12), Eq. (7) can be written for any two transitions and as
(15) 
where is given in Eq. (14). This equation defines a family of parallel lines whose intercept depends on the new physics couplings and the isotope pair via the third term. The lines live in the isotope shift space where each dimension corresponds to an isotope pair. When combining transitions, we obtain for each isotope pair a line in the dimensional space
(16) 
that can be fitted to the measured isotope shifts. For later convenience we write this equation as
(17) 
where the components of the above vectors follow from Eq. (16).
For the fit we construct a function and marginalize over the entries of the vectors and . Since the measured isotope shifts exhibit similar uncertainties in all transitions, a multidimensional that includes the uncertainties of all transitions needs to be constructed (see e.g. Ref. Hogg:2010yz ). For simplicity we assume that the uncertainties of the measurement are not correlated. In this case the contribution of the pair to the function is given by
(18) 
where the sum runs over all transitions. Here, is the uncertainty of the respective IS measurement, and is the th component of the vector connecting the measured point to the line defined by Eq. (16),
(19) 
with and . The full function is given by summing over all isotope pairs.
By construction, the IS of the transition that appears in the righthand side of Eq. (16) seemingly has a special role. This is, however, not the case as the distance between a point and a line in an dimensional space is invariant under the permutation of coordinates. When minimizing the we have checked the consistency of our computation and its numerical stability by verifying that all permutations of the isotope shifts yield comparable results. We obtain the 95% confidence level limits on shown in Table 2. The near degeneracy between two of the transitions in Ca is reflected in the extremely weak limit in the case of omitting the nondegenerate transition of . Moreover this explains why the limit hardly becomes more stringent when including all three transitions. For Yb the limits get in general weaker by a factor to when one isotope shift measurement is dropped. The removal of leads to the weakest bound.
Omitting one transition of Ca or one isotope pair of Yb leads to the minimal case of transitions and isotope pairs where Eq. (15) is exactly solvable. Hence, the theory parameters , and can be chosen such that the theory predictions of the modified isotope shifts reproduce exactly the measured ones. This is reflected by the vanishing in Tab. 2.
Iv Implications for BSM models
In Ref. Berengut:2017zuo the sensitivity of KLV was compared to other measurements in a modelindependent way. In the following we will translate the KLV bounds into bounds on the parameters of various BSM models and compare them to existing constraints. In addition we will explore the sensitivity of the nearfuture KLV projections with Ca Dstates, Sr, Sr/Sr, and Yb as reported in Ref. Berengut:2017zuo . We focus on those models that can be best probed by KLV measurements. While we discuss in detail the various constraints on the model in the full mass range relevant for KLV, we highlight promising mass values in Higgs portal and chameleon models. Most bounds on the model can be translated also to these models by rescaling. Furthermore, we present updated bounds on the protophobic interpretation of the Be anomaly.
iv.1 Z vector boson from U(1)
One of the frequently studied abelian extensions of the SM gauge group is gauging the difference of baryon and lepton number, . Under this additional interaction all quarks therefore have the same charge and all leptons . The group is made anomalyfree by introducing a righthanded neutrino for each family. In this model, the coupling of the new vector boson is purely vectorial and of equal strength for electrons and neutrons, hence KLV is a promising method to probe this kind of NP interaction.
In Fig. 1 we compare the KLV bounds and projections from different atoms and ions to existing constraints in the mass range of 10 eV to a few MeV. For other overviews collecting bounds on this model and related models see e.g. Refs. Jaeckel:2010ni ; Jaeckel:2010xx ; Harnik:2012ni ; Heeck:2014zfa .
iv.1.1 Laboratory bounds
The existence of a fifth force is severely constrained for a mass by experiments testing the Casimir effect Bordag:2001qi ; bordag2009advances .
In contrast to KLV, other atomic precision measurements such as energy level shifts in Rydberg states PhysRevLett.61.2285 ; PhysRevLett.61.2285 ; Karshenboim:2010cg ; Karshenboim:2010ck and in  and states of atomic H and hydrogenlike He Jaeckel:2010xx provide bounds on where is the proton coupling. In the massless limit, , the NP potential probed by these observables simplifies to a Coulomb potential. In this case the NP interaction is absorbed by a redefinition of the finestructure constant , resulting in a weakening of the bounds. Due to its sensitivity to , KLV is not affected by this redefinition so that its bound remains constant in the massless limit and is the strongest among the atomic spectroscopy bounds for mediator masses below . The intersection of the Ca and Rydberg bound was determined following Ref. Karshenboim:2010ck and lies below the mass range shown in Fig. 1. Yet, one needs to keep in mind that for also other constraints apply, such as from the Casimir effect mentioned above or from tests for a deviation from the Coulomb force, see e.g. Ref. Jaeckel:2010ni .
Neutron scattering is a powerful probe of the interaction between new bosons and neutrons over a wide mass range. Among the neutron scattering experiments, neutron optics Leeb:1992qf provides the strongest constraint on , in this model equivalent to , in the mass range of . For , the comparison of the total to the forward scattering cross section Nesvizhevsky:2007by is most sensitive. Above , the neutronlead (Pb) scattering Pokotilovski:2006up sets the strongest bound. This method is based on the proposal by Ref. Barbieri:1975xy whose bounds are superseded by the ones reported in Refs. Pokotilovski:2006up ; Leeb:1992qf . The collection of the various bounds is shown in Fig. 2, the limit presented in Fig. 1 shows the best bound for each mass.
When comparing to KLV, the considerable uncertainties on the neutron scattering bounds need to be kept in mind Wissmann:1998ta ; Nesvizhevsky:2007by ; Antoniadis:2011zza ; TuckerSmith:2010ra . In particular, the uncertainties related to the electronneutron scattering length, various nuclear inputs, and the missing higherorder terms in the neutronscattering cross section, are not easy to determine precisely. Similarly, the bounds inferred for masses far higher than the neutron energies of Leeb:1992qf are also subject to large uncertainties. In addition, in the derivation of the various neutron bounds it is assumed that the NP contribution to the neutronelectron interaction is either absorbed in the corresponding measurement of the neutronelectron scattering length, , or negligible Nesvizhevsky:2007by .
For masses above a few keV, the neutron bounds are exceeded by constraints that arise from measurements of neutrinoelectron () scattering. Measurements from Borexino Bellini:2011rx and GEMMA Beda:2009kx have been interpreted in the model in Ref. Harnik:2012ni and are the strongest laboratory bounds between and .
The limit on via from the bound on the anomalous magnetic moment of the electron, , is less constraining than the neutron scattering bounds in the same mass range PhysRevLett.100.120801 ; thus we omit it in Fig. 1.
Above the electron threshold, i.e. , and up to , highintensity electron and proton beam dump experiments Essig:2009nc ; Essig:2010xa ; Andreas:2012mt provide the strongest bound on . For a review see Ref. Alexander:2016aln and references therein.
iv.1.2 Astrophysical and cosmological bounds
A large part of the parameter space of a new light boson interacting with SM fermions is constrained by astrophysical probes. In particular, bounds on the cooling rate in horizontal branch stars limit the coupling of as long as its mass is within the thermal reach () Yao:2006px ; Grifols:1986fc ; Grifols:1988fv ; Hardy:2016kme ; Redondo:2013lna . Here we omit the corresponding bound from sun cooling since its excluded region is also covered by the horizontal branch stars. Furthermore, the energy loss in the core of the supernova SN 1987A constrains the neutron coupling for masses of Yao:2006px ; Raffelt:2012sp , though with large uncertainties Blum:2016afe .
The coupling is also severely constrained by the effective number of neutrinos Ahlgren:2013wba ; Heeck:2014zfa . If a light mediator couples to neutrinos, it can thermalize via the inverse decay or , thus contributing to the energy density of the Universe. The first process dominates for , and the second one for . Under the requirement that the NP contribution to , , fulfills at , a large parameter region is excluded.
Although the cosmological and astrophysical bounds exceed all KLV projections, a complementary laboratory probe of this region will nevertheless be valuable, in particular because isotope shifts are a very clean observable and the derived bounds are based on less model assumptions. E.g. the constraint from can be avoided by charging only the righthanded electrons under the new gauge group. However, in such a model other strong constraints arise, for instance, from the cancellation of a gauge anomaly in the case of an anomalous gauge group Dobrescu:2014fca . For examples of UVcomplete models of axial couplings with such phenomenology see Refs. Kozaczuk:2016nma ; Kahn:2016vjr . Moreover, in this class of models the axial coupling In presence of an axial coupling to electrons and vectorial couplings to quarks, atomic parity violation (APV) Bouchiat:2004sp is a more sensitive probe than KLV. For instance for gauge bosons with , the limit from APV Bouchiat:2004sp is several orders of magnitude stronger than the Yb projection. The same conclusion is obtained for constraints on the mixing of a new light gauge boson with the SM Davoudiasl:2012ag .
iv.1.3 KLV bounds and projections
Due to the simple coupling structure of the boson, the limits from KLV are straightforwardly obtained from Ref. Berengut:2017zuo by identifying . The existing bound from Ca is shown by the solid red line, the projections are shown by dashed lines. While the existing bound does not compete with other laboratory bounds, the projected bounds will extend the reach of laboratory bounds in the mass range of , in particular using Sr/Sr and Yb. Hence, KLV has the potential put the regions that are currently only probed by astrophysical and cosmological observables under scrutiny.
iv.2 Higgs portal
Another example for a new light mediator which gives rise to a spinindependent interaction between electrons and neutrons is a scalar singlet mixed with the Higgs boson Patt:2006fw ; OConnell:2006rsp . This model can be linked to the solution of the hierarchy problem via the relaxion mechanism, where the relaxion takes the role of the new scalar that mixes with the Higgs boson Flacke:2016szy ; Choi:2016luu .
We evaluate the KLV sensitivity to this class of models for a very light scalar with mass . In this case the KLV bound on the mixing angle between the singlet and the Higgs boson is given by
(20) 
where we assumed the strongest KLV projections for this mass given in Ref. Berengut:2017zuo , i.e. Yb with a precision at the level. The quark and gluon contribution to the neutron Yukawa coupling can be obtained using Refs. Belanger:2008sj ; Belanger:2013oya ; Shifman:1978zn . For SM Yukawa couplings, the limit from neutron scattering is stronger by one order of magnitude since the KLV observable depends on the electron Yukawa coupling that is much smaller than the neutron Yukawa coupling. Yet, for models where the electron (neutron) coupling is enhanced (suppressed) by at least one order of magnitude with respect to its SM value, KLV could set a stronger bound than neutron scattering. For assumptions regarding the bound from neutron scattering see Sect. IV.1.1. Under these assumptions and if the electron (neutron) coupling is enhanced (suppressed) by a factor of 10, KLV sets a stronger bound than neutron scattering for and the strongest bound of all in the region . Below this range a stronger bound arises from horizontal branch stars Yao:2006px ; Grifols:1986fc ; Grifols:1988fv ; Hardy:2016kme ; Redondo:2013lna and above 1 MeV beam dump experiments are more constraining.
A suppressed can arise for example in less minimal Higgs portal models, such as the leptonic Higgs portal Batell:2016ove . In this model, the singlet couples to leptons with a similar strength as the Higgs boson, i.e. , while its coupling to the quarks is suppressed compared to normal Higgs portal scenarios, hence neutron scattering experiments loose sensitivity.
iv.3 Chameleon models
The chameleon is a scalar field with an effective potential that depends on the density of the environment Khoury:2003aq ; Khoury:2003rn
(21) 
with being a mass scale characterizing the coupling of the chameleon to matter, and is chosen such that the mass of increases with increasing . As a result the mass of the scalar is heavy in a dense environment and light otherwise, which leads to a screening effect in test masses. Therefore it can mediate a longrange force on cosmological scales but avoid constraints from fifthforce experiments.
On atomic scales the chameleon can alter the energy levels of the electrons. The relevant part of the NP perturbation of the Hamiltonian that can be probed by KLV is given by Brax:2010jk ; Brax:2010gp
(22) 
where is the electron mass, the contribution of the neutrons to the nucleus mass, and the distance to the nucleus. In this expression we omit a possible screening of the nucleus as it depends not only on the parameter space of the model but also on the experimental setup Burrage:2014oza .
Assuming the massless case, Eq. (22) can be matched to Eq. (11) and the bounds on can be easily translated into bounds on
(23) 
where is the reduced Planck mass and we used the Yb KLV projection. This is stronger by more than an order of magnitude than the current best bound from measurements of energy levels in hydrogen and helium atoms Jaeckel:2010xx ; Schwob:1999zz ; Simon:1980hu , which leads to a bound of Burrage:2016bwy . Depending on the parameter space, this can even be the strongest bound on . The current bound of is stronger than the present KLV bound from Ca and Yb.
iv.4 Beryllium anomaly
Recently, a anomaly was reported in rare nuclear decays of Be Krasznahorkay:2015iga . The anomaly arises in the isoscalar transition as a bump in the distribution of the opening angle of the emitted electronpositron pairs. This observation can be explained by the emission of a particle with mass in the process , which subsequently decays into an electron positron pair. The best agreement with observations is obtained for being a vector with either axial or vectorial couplings to quarks and electrons Feng:2016jff ; Feng:2016ysn ; Kozaczuk:2016nma ; Kahn:2016vjr . It was noted in Refs. Feng:2016jff ; Feng:2016ysn that the vectorlike interpretation necessitates protophobic couplings to quarks, or else it would be in conflict with other observables. Therefore KLV can provide the necessary method to confirm or reject this hypothesis.
In Fig. 3 we present KLV projections and compare them to existing bounds on and for a fixed mass of . The gray shaded area corresponds to the range of the couplings that explains the observed excess and is not in conflict with other measurements.
The upper bound on comes from measurements. In contrast to Refs. Feng:2016jff ; Feng:2016ysn the plotted bound represents the 95% CL instead of the limit. Another upper bound is provided by KLOE2 Anastasi:2015qla . However even this 90% CL bound is weaker than the 95% CL bound from the magnetic moment and therefore not shown. The lower bound on stems from beam dump experiments requiring that the new particle decays before it leaves the detector. The strongest bound for is provided by the E141 experiment Riordan:1987aw and interpreted in Ref. Andreas:2012mt . The latter corrected Ref. Essig:2009nc that was used in Refs. Feng:2016jff ; Feng:2016ysn .
An upper bound on is set by neutronPb scattering. The strongest constraint for is provided by Ref. Leeb:1992qf which is stronger than the older derived in Ref. Barbieri:1975xy and that was used in Refs. Feng:2016jff ; Feng:2016ysn . The bound shown in Fig. 3 is weaker by a factor of than the one presented in Fig. 2 due to the protophobic nature of the coupling. In contrast to the neutron scattering, KLV does not loose sensitivity in the protophobic case.
The dashed lines show the projected upper bounds of KLV on the product of the couplings , assuming linearity. While Sr/Sr will not suffice to probe the couplings relevant for the Be anomaly, Yb has the potential to exclude or support the existence of a new vector with a mass of . With a precision of Yb will become sensitive to the relevant Be coupling space and with the anticipated precision of the whole region can be covered.
V Conclusions
In this work we extended the proposal of Ref. Berengut:2017zuo to constrain New Physics (NP) by means of isotope shift spectroscopy to enable the inclusion of larger data sets with an arbitrary number of atomic transitions and isotope pairs. As an application of the King linearity violation (KLV) observable to bound NP couplings, we evaluated the constraints resulting from existing data sets of two different atomic systems (Ca and Yb).
We compare the existing KLV bounds and nearfuture projections to present constraints in various models that can potentially be probed by isotope shifts.

BL: The  space is already largely constrained by astrophysical and cosmological bounds. Complementary laboratory probes, however, are not yet able to confirm those bounds in certain areas of the parameter space. Here KLV has the potential to become the strongest laboratory bound for .

Higgs portal: While KLV bounds on standard Higgs portals are weaker than existing laboratory bounds, KLV can supersede them in the case of an enhanced electron or suppressed neutron coupling. For an enhancement (suppression) by a factor of 10, KLV could even set the strongest of all bounds in the range . Such a scenario can be realized e.g. in the leptonic Higgs portal.

Chameleon: KLV will be able to set the strongest lower bound on the interaction scale of the chameleon with matter.

Be anomaly: With the anticipated precision, KLV will fully explore the coupling range of a protophobic vector boson with mass needed to reproduce the observed anomaly in Be decays.
Acknowledgments
We thank Cedric Delaunay and Yotam Soreq for useful discussions and for carefully reading our draft.
The work of GP is supported by grants from the BSF, ERC, ISF, Minerva, and the Weizmann UK Making Connections Programme.
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