Probing fast oscillating scalar dark matter with atoms and molecules
Dionysios Antypas, Oleg Tretiak, Ke Zhang, Antoine Garcon, Gilad Perez, Mikhail G. Kozlov, Stephan Schiller, Dmitry Budker
PProbing fast oscillating scalar dark matter withatoms and molecules
Dionysios Antypas , Oleg Tretiak , Ke Zhang , AntoineGarcon , Gilad Perez , Mikhail G. Kozlov , , Stephan Schiller ,and Dmitry Budker , Johannes Gutenberg-Universit¨at Mainz, Mainz, Germany, GSI Helmholtzzentrumf¨ur Schwerionenforschung, Darmstadt, Germany, Helmholtz-Institut Mainz, Mainz,Germany Department of Particle Physics and Astrophysics, Weizmann Institute of Science,Rehovot, Israel 7610001 Petersburg Nuclear Physics Institute of NRC “Kurchatov Institute”, Gatchina188300, Russia St. Petersburg Electrotechnical University “LETI”, Prof. Popov Str. 5, 197376St. Petersburg, Russia Institut f¨ur Experimentalphysik, Heinrich-Heine-Universit¨at D¨usseldorf, 40225D¨usseldorf, Germany Department of Physics, University of California, Berkeley, California 94720, USAE-mail: [email protected]
Abstract.
Light scalar Dark Matter with scalar couplings to matter is expectedwithin several scenarios to induce variations in the fundamental constants of nature.Such variations can be searched for, among other ways, via atomic spectroscopy.Sensitive atomic observables arise primarily due to possible changes in the fine-structure constant or the electron mass. Most of the searches to date have focusedon slow variations of the constants (i.e. modulation frequencies ă Keywords : Dark matter, fundamental constant variations, relaxions, atomicspectroscopy, molecular spectroscopy.
1. Introduction
Direct astrophysical observations point to the existence of Dark Matter (DM), whichis estimated to account for «
80% of the total matter in the Universe [1]. This form of a r X i v : . [ phy s i c s . a t o m - ph ] J a n matter interacts with Standard Model (SM) matter gravitationally, but has feeble (ifany) other interactions, making efforts to uncover its origin, composition and propertieschallenging. The prevalent DM candidate scenario assumes that DM consists of WeaklyInteracting Massive Particles (WIMPs), with mass in the 1-1000 GeV range. Howeversearches with accelerators and scintillators have not yielded a clear discovery yet [2].Within another class of motivated scenarios, DM consists of light bosonic particles (ofmass m φ in the 10 ´ ´ ω C “ m φ [3]). Several leadingcandidates, such as the QCD Axion, Axion-Like Particles (ALPs) and others, classifiedaccording to their spin, type of interaction(s) with SM particles and resulting physicalobservable [4], are the focus of a number of completed, ongoing or planned experiments[5]. Within this light bosonic DM landscape, attention is given to the possibility thatDM couples to matter inducing oscillations in the fundamental constants (FC) of nature.Such FC oscillations are expected in cases where DM consists of particles emerging inhigher-dimension theories (such as dilatons) with scalar coupling to the SM particles[6, 7] or relaxions, i.e. ALPs originally introduced to provide a solution to the hierarchyproblem [8]. Within minimal models, the relaxion can account for the observed DM inthe Universe [9]. The related phenomenology arises from a scalar coupling to SM mattervia the relaxion-Higgs mixing [10, 11].Motivated by a number of beyond-SM scenarios, a series of studies involvingastronomical observations and laboratory experiments have focused on probing slowdrifts in the FC (see, for example, [5, 12], and references therein). The possibility thatlight scalar DM with coupling to matter induces FC oscillations, such as for example,oscillations in the fine structure constant α , the electron mass m e or the quantumchromodynamics scale parameter Λ QCD , has motivated searches for such effects as well.A variety of approaches have been proposed to search for light scalar DM-inducedoscillations in the FC. These include use of atomic [6, 13, 14, 15, 16], or nuclear [11] clocksto probe variations of α and detection of oscillations in α , m e via laser interferometry[13, 14], comparison of optical cavities [17] or resonant mass detectors [18]. Effects ofEquivalence-Principle (EP)-violating forces arising due to light scalar DM and meansto detect those were considered in [4, 19]. Analysis of astrophysical data from the earlyUniverse has provided constraints on scalar DM-SM matter couplings [20]. Constraintson such interactions have also been provided via several laboratory results, such as:long-term comparison of Cs and Rb microwave clocks [21, 22], atomic spectroscopy inDy [23], comparison of an optical cavity with itself at different times [24], a comparisonof an ultra-stable Si cavity with a Sr optical clock and H maser [25], a terrestrial networkof optical clocks [26], EP and Fifth-Force (FF) apparatus [27, 28, 29, 30].While the most stringent limits on scalar DM-SM interactions in the low-frequencylimit ( f ă « α and m e are competitive with those of EP experimentswithin part of the explored frequency range (20 kHz – 100 MHz). Here we review theWReSL experiment, discuss experimental progress towards improved searches of rapidFC oscillations with WReSL, and introduce an extension of the technique to molecules,which will allow probing rapid variations of the nucleon mass and thus the Λ QCD scaleparameter. By ‘rapid’, we refer to oscillations at frequencies ą
2. Probing light scalar Dark Matter with WReSL
Let us illustrate the atomic effects arising from a coupling of light scalar DM to SMparticles, and how the WReSL experiment allows to look for observables emerging fromrapidly oscillating FC. In the work reported in [33] the possibly oscillating constantswere α and m e , but the envisioned extension to molecules will focus on the nucleonmass. If DM consists of light bosons of mass m φ the corresponding DM field oscillatescoherently, according to φ p (cid:126)r, t q « ? ρ DM m φ sin p m φ t q , (1)where ρ DM « is the local DM density [34]. The oscillation is coherentwithin time scale associated with the quality factor ω { ∆ ω « π { v « ¨ , where v DM « ´ is the virial velocity of the DM field [35] (coherence is longer within relaxion-halo scenarios [31]). In the case of dilatonic or relaxion DM the constants α and m e acquire a component which oscillates at the frequency m φ : α p (cid:126)r, t q “ α “ ` g γ φ p (cid:126)r, t q ‰ , (2) m e p (cid:126)r, t q “ m e , ” ` g e m e , φ p (cid:126)r, t q ı . (3)The parameters α and m e , refer to the time-averaged values of the constants, and g γ , g e are DM couplings to the photon and the electron, respectively, that the experimentallows to investigate. This investigation exploits the modulation of the frequency of anatomic transition arising from oscillations in α or m e . As the atomic levels have energyproportional to the Rydberg constant R “ p { q m e α , the frequency of an opticaltransition f at acquires an oscillatory component. The resulting fractional modulationin f at is given by δf at f at “ δαα ` δm e m e , “ ´ g γ ` g e m e , ¯ ? ρ DM m φ . (4)The WReSL experiment employs optical spectroscopy of an atomic vapor to proberapid variations in f at . A laser is tuned in frequency to resonantly excite atoms; theexperimental signal contains information about deviations between the laser frequency f L and f at , i.e. the difference f at ´ f L , where the two frequencies are close to eachother (on the optical scale). In the absence of identifiable noise sources, this differenceis to be attributed to variations of FC and hence, to a detection of the couplings g γ and/or g e . However, the sensitivity in detection of g γ and g e from a comparison between f L and f at is not the same over the entire frequency range over which the search isperformed (20 kHz ´
100 MHz) [36]. One has to consider the following two effects, whichare discussed in detail in [32, 33]: i) the sensitivity of the laser frequency itself tooscillations in α and m e , and ii) the response of the atomic signal to oscillations of theconstants, at frequencies comparable to, or larger than the linewidth Γ { π of the opticaltransition. As f L is inversely proportional to the length L of the laser resonator, whichscales as L { αm e , there is in fact modulation in f L with fractional amplitude δf L f L “ δαα ` δm e m e , “ ´ g γ ` g e m e , ¯ ? ρ DM m φ . (5)This modulation occurs at frequencies lower than the cutoff frequency f c1 for soundpropagation in body of the laser resonator, which for the WReSL laser system is f c1 «
50 kHz [33]. A comparison of Eq. (4) and (5) shows that for f ă f c1 there is reducedsensitivity to oscillations of α and there is no sensitivity to oscillations of m e . Anothercutoff is imposed by the decay of the atomic response at frequencies larger than thelinewidth Γ { π . The sensitivity above this cutoff f c2 “ Γ { π (on order of MHz)decays as 1 { f . To summarize, the relation between the measured fractional modulation δ p f at ´ f L q{ f at and the couplings g γ , g e is assumed as follows δ p f at ´ f L q f at “ $’’&’’% g γ ? ρ DM m φ h at p f q , f ď f c1 ´ g γ ` g e m e , ¯ ? ρ DM m φ h at p f q , f ą f c1 . (6)The function h at p f q is the atomic response function, with h at p f q « f ! Γ { π and h at p f q Ñ Γ {p πf q for f " Γ { π . The function h at p f q can be determined by checkingthe response of the atoms to a known frequency modulation imposed on the laser fieldexciting them. From an experimental point of view, it is important to maximize theapparatus fractional frequency sensitivity to obtain optimal sensitivity in detection ofthe couplings g γ , g e . A preliminary WReSL run was carried out in 2019, whose results on scalar DMconstraints were reported in [33]. Optical spectroscopy was done on the D2 line ofcesium (Cs) vapor to search for rapid FC oscillations. A schematic of the setupis shown in Fig. 1. The apparatus implements polarization spectroscopy [37] on the6S { Ñ { transition (natural linewidth Γ { π « . f at and laser frequency f L . In the absence of detection of such oscillations, themeasured power noise spectrum of the signal is used to constrain the fractional variation δ p f at ´ f L q{ f at within the investigated frequency range (20 kHz-100 MHz), and throughEq. (6) constrain the couplings g γ and g e .As no FC oscillations were detected in the first WReSL run [33], limits on scalarDM couplings to the photon and the electron were placed from analysis of the spectrumof the δ p f at ´ f L q{ f at parameter [Eq. (6)], which was constrained to better than 10 ´ inpart of the explored frequency range. Extracted constraints are shown in Fig. (3). Theyare computed for the average galactic DM density ρ DM “ , but also withinthe scenario of a relaxion halo gravitationally bound by Earth [31]. In the latter casethe DM field density ρ DM is greatly enhanced. The resulting DM overdensity that weconsider here is calculated in [31], and its effects on constraining scalar DM interactionsare mostly pronounced in the « g γ , g e are tighter in thevicinity of « PID
Ti:S Laser 852 nm PDPRSpectrum analyzer Lock-in amp. BPD μ -metal shield Cs vapor cell λ /4 PRPBS λ /2 λ /2 WP σ ++ σ - σ + Figure 1: Simplified schematic of the polarization- spectroscopy setup employed inthe 2019 WReSL run [33]. PBS: polarizing beamsplitter, λ /2: half-wave plate, λ /4:quarter-wave plate, PR: partial reflector, WP: Wollaston prism, PD: photodetector,BPD: balanced photodetector. Adapted from [33]. - 2 0 0 - 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 5 Balanced PD output (V)
L a s e r f r e q u e n c y - 3 5 1 , 7 3 0 , 8 7 0 ( M H z )
F = 4
852 nm
F = 3 - F ' = 2 f e a t u r e
Figure 2: Polarization spectroscopy on the 6S { F “ Ñ { F “ , , δ p f at ´ f L q{ f at . Shown in the inset are the hyperfine levelsof the ground 6S { state and excited 6P { state. Adapted from [33]. - 3 2 - 2 8 - 2 4 - 2 0 - 1 6 E P e x p e r i m e n t s
N a t . L = 3 T e V g g (eV)-1 - 1 0 - 9 - 8 - 7 - 2 3 - 2 0 - 1 7 - 1 4 - 1 1 m f ( e V ) E P e x p e r i m e n t s
N a t . L = 3 T e V g e F r e q u e n c y ( H z ) - 3 2 - 2 8 - 2 4 - 2 0 - 1 6 g g (eV)-1 - 1 0 - 9 - 8 - 7 - 2 7 - 2 4 - 2 1 - 1 8 - 1 5 - 1 2 - 9 N a t . L = 3 T e V N a t . L = 3 T e V E P e x p e r i m e n t sE P e x p e r i m e n t s g e m f ( e V ) F r e q u e n c y ( H z )
Figure 3: Constraints in the couplings g γ and g e [shown at the 95% confidence level(CL)], extracted from the earlier WReSL run [33]. Left: Limits considering the averagegalactic DM density. The constraint in green comes from the requirement to maintainNaturalness [18, 4]. Right: Extracted limits (shown at the 95% CL) within the scenarioof a relaxion Earth halo. The shaded area in the plots indicates a region around the laserresonator cutoff frequency f c1 «
50 kHz, where careful consideration of the resonatorresponse is needed to accurately determine the transition in sensitivity of probing g γ , g e . The limits from EP experiments are from [40, 41]. Constraints of [33] in the range40-100 MHz are scaled up here by factor in the range 1-2, to account for improvedapparatus calibration. Adapted from [33].
3. Towards an improved search for scalar DM with WReSL
Substantial technical improvements compared to the earlier WReSL setup [33] haveyielded a greatly enhanced experimental sensitivity. The primary improvement hasbeen a dramatic increase in the measurement duty cycle. The search for FC oscillationsis done via spectral analysis of the polarization spectroscopy signal (Fig. 2). In thefirst WReSL run the analysis was carried out with a commercial, swept-frequencyspectrum analyzer, which performed analysis in a sub-optimal way, resulting in anlow effective duty cycle (of order 10 ´ ), thus necessitating a long acquisition time fora given target sensitivity. Spectral analysis is now done with a homebuilt, computer-based Fast-Fourier-Transform (FFT) spectrometer, whose implementation is based on[42]. The new spectrometer incorporates a 250 MSa/s data-acquisition card and anefficient graphics-processing unit, and performs real-time FFT analysis in our 20 kHz-100 MHz range of interest with « « ˆ ˆ δ p f at ´ f L q{ f at with theupgraded apparatus. Data acquisition of duration «
72 min results in up to «
100 greatersensitivity compared to the previous results (also shown), for which data were acquiredfor 66 hrs. With longer integration we anticipate an additional factor in excess of ˆ δ p f at ´ f L q{ f at variations. A complication in interpreting the spectrumarises due to the enhanced sensitivity; far more spurious peaks due to technical noisesources appear, that must be properly investigated to exploit the full power of themethod to detect, or constrain FC oscillations. There are various ways that some peakscan be rejected, for example, based on the fact that they appear in both signals, fromthe atoms and from the optical cavity, while the latter cannot respond to fast-oscillatingDM field. Such peaks thus likely originate from frequency modulation of the laser light.Additionally, some peaks appear in multiplets with fixed separation (of exactly 20 Hz).These are likely related to electromagnetic pick-up from some device, for example, aswitching power supply in or near the building. Some of the peaks have linewidthsinconsistent with models of galactic DM. This investigation is ongoing.
4. Extension to molecules: probing oscillations of the nuclear mass
While atomic systems are primarily sensitive to variations in α and m e , molecules areadditionally sensitive to variations of the nuclear mass M . This can be exploitedto search for DM-induced oscillations in M , or to leading order, in the Λ QCD scaleparameter, as this parameter predominantly determines the masses of protons andneutrons [43]. Additional contributions can arise due to the couplings to the quarks.For a survey of the proposed, ongoing and completed activities involving searches forFC variations using molecules we refer the reader to [5, 44]. We note that searches forDM with molecular spectroscopy that involve signatures other than FC variations, havealso been proposed (see for example Refs. [45, 46]).In this section we analyse the molecular sensitivity to variations in M . This analysisis useful in guiding future molecule-based searches for FC oscillations, slow or rapid. In - 1 7 - 1 6 - 1 5 - 1 4 - 1 3 - 1 2 - 1 0 - 9 - 8 - 7 E a r l i e r 6 6 h r r u n ( 2 0 1 9 ) 1 . 2 h r r u n ( n e w s e t u p ) d (f at - fL)/ f at m (cid:1) ( e V ) F r e q u e n c y ( H z )
Figure 4: Comparison of constraints on δ p f at ´ f L q{ f at shown for the earlier 2019 run [33]and the new, ongoing experiment with improved sensitivity apparatus. Constraints areshown at the 95% CL. The 2019 data are scaled up by factor in the range 1-2 between40-100 MHz, to account for improved apparatus calibration. A series of technical noisepeaks in the new data are being investigated.their presence, the nuclear mass can be expressed as M p (cid:126)r, t q “ M ” ` g n M φ p (cid:126)r, t q ı , (7)where M is the time-averaged mass, and g n is the coupling of the DM field φ to thenucleons.In molecules, in addition to the electronic levels, there are vibrational and rotationallevels, with the total energy E mol of the system given by E mol “ E el ` E vib ` E rot . (8)The terms in Eq. (8) have generally different sensitivities to a change in the nuclearmass. To examine the sensitivity of these terms to changes in M we will write themsuch that the dependence on M is explicit. As we explain below, it is the vibrationalenergy that is most sensitive to M variations.The energy of a given electronic level E el is proportional to the Rydberg constant R ,so that E el “ C el hcR , where C el is a constant of order unity, independent of FC inthe non-relativistic approximation. In the electronic energy, C el acquires dependence on α when relativistic effects are considered: C el “ C ` C p αZ eff q ` .... (9)The coefficients C i are of order unity. In the electronic transitions considered here(accessible by conventional laser spectroscopy), it is the weakly bound electrons that0transition between orbitals. For these, the effective nuclear charge Z eff »
1. Thus, forthe purpose of this discussion, the dependence of C el on FC can be neglected.Going beyond the approximation of an infinite nuclear mass necessitates replacing m e with the reduced mass m e M {p m e ` M q , so that E el “ C el m e Mm e ` M α c » C el m e α c ´ ´ m e M ¯ . (10)The electronic energy change δE p M q el with a change δM is given by δE p M q el “ C el hcR ´ m e , M ¯ δMM “ E el ´ m e , M ¯ δMM , (11)where the superscript “ p M q ” denotes that only the sensitivity to M is being considered.Note that this sensitivity estimate is accurate up to a factor of order unity.The vibrational energy is given by E vib “ ω e ´ υ ` ¯ ´ ω e χ e ´ υ ` ¯ , (12)where ω e is the fundamental vibrational transition energy, χ e is an anharmonicityconstant, and anharmonic terms of higher order in the vibrational quantum number υ are omitted.According to the Born-Oppenheimer theory, the vibrational energy ω e is given by ω e “ C vib hcR ´ m e M r ¯ { , (13)where M r M is the reduced nuclear mass of the molecule and C vib is a constant of orderunity. The lowest-order corrections to C vib are proportional to p αZ eff q , similar to C el in Eq. (9), due to relativistic effects. They may be neglected here. For simplicity, weneglect the dependence of the anharmonicity constant on FC (the constant χ e scales as p m e { M q { [47]).A change δM induces a change in E vib δE p M q vib “ E vib ´ ´ δMM ¯ . (14)The rotational energy is given by E rot “ BJ p J ` q , (15)where B is the rotational constant, J is the rotational quantum number of the level,and higher-order contributions are omitted. The rotational constant is ¯ h {p I q , where I is the moment of inertia. For a diatomic molecule, it takes the form I “ M r d , with d being the distance between the nuclei. Since d is proportional to the Bohr radius withproportionality constant of order unity, we can reexpress the rotational energy as E rot “ C rot hcR ´ m e M r ¯ J p J ` q , (16)1where C rot is a constant of order unity.The dependence of δE p M q rot on δM is δE p M q rot “ E rot ´ ´ δMM ¯ . (17)One can make use of eqs. (11), (14), and (17) to evaluate the contributions to thevariation of a transition frequency f mol “ f el ` f vib ` f rot , considering optical excitationfrom the ground to an excited electronic level, with p ν, J q Ñ p ν , J q . The rotationalquantum number selection rule for a one-photon, electric-dipole allowed electronictransition in a diatomic molecule is ∆ J “ J ´ J “ , ˘
1. The respective frequencychanges δf p M q due to the nuclear mass only are: hδf p M q el “ p E el ´ E el q ´ m e , M ¯ δMM , (18) hδf p M q vib “ "” ω e ´ υ ` ¯ ´ p ω e χ e q ´ υ ` ¯ ı ´ ” ω e ´ υ ` ¯ ´ ω e χ e ´ υ ` ¯ ı*´ ´ δMM ¯ , (19)and hδf p M q rot “ “ B J p J ` q ´ BJ p J ` q ‰´ ´ δMM ¯ . (20)Let us estimate the relative scaling of the variations δf p M q el , δf p M q vib and δf p M q rot . We considera specific system: iodine I , a homonuclear diatomic molecule with M «
127 u, inwhich electric-dipole transitions in the visible range between the ground X Σ g andexcited B Π u electronic level are conveniently accessible. The constants related to themolecular energy are E el ´ E el “ p h c q ´ , ω e “ p h c q ´ , p ω e χ e q “p h c q .
764 cm ´ , B “ p h c q .
029 cm ´ , ω e “ p h c q ´ , ω e χ e “ p h c q .
615 cm ´ , B “ p h c q . ´ [48].Taking as an example a transition at λ “ .
409 nm [48] with p υ “ , J “ q Ñp υ “ , J “ q we obtain: δf p M q el » . δMM , (21) δf p M q vib » ´
30 THz δMM , (22)and δf p M q rot » . δMM . (23)The relative scaling of the above quantities is | δf p M q vib | : | δf p M q rot | : | δf p M q el | „ . ˆ : ˆ :
1. We see that the vibrational part of this molecular transition is the mostsensitive to M variations, with a smaller effect in the rotational and the electronic levels.2Primarily because transitions involving large change ∆ υ are allowed whereas thechange in J follows the selection rule | ∆ J | ď
1, it is the choice of vibrational levelsthat has to be made optimally for a high-sensitivity search for variations of M . Thereis in fact an optimal ∆ υ value that maximizes the frequency deviations. Due to theanharmonicity of interatomic potential at large number of vibrational quanta, and theexpected atom-like behavior of the system near the dissociation limit, the sensitivityto δM decreases with large ∆ υ . Optimal detection of frequency deviations occurs fortransition between υ “ υ such that the excited state vibrational energy is afraction of the dissociation energy [49]. Selection of such an optimal transition requiressensitivity calculations that should include the sensitivity of the anharmonicity constant χ e [50]. The precise dependences of the energies of a molecule on FC can in principlebe computed ab initio using quantum chemistry techniques. The most complete suchcalculations are possible for the one-electron molecules, i.e. the molecular hydrogenions [51, 52, 53]. However, we do not expect that precise calculations are needed inorder to interpret the experimental data, unless one deals with a (undesired) situationof measuring the differential frequency fluctuations between two systems having closesensitivities.We stress that all molecular energy contributions are proportional to the Rydbergconstant. Furthermore, there is an additional sensitivity to the electron mass in allmolecular energy contributions discussed above. In total, δf mol f mol “ δαα ` δm e m e , ` f mol ´ f el m e , M ´ f vib ´ f rot ¯´ δMM ´ δm e m e , ¯ . (24)As we have seen in many cases, Eq. (24) approximates to δf mol f mol « δαα ` ´ ` f vib f mol ¯ δm e m e , ´ f vib f mol δMM . (25)This shows that molecules have simultaneous sensitivity to electron mass, nuclear massand fine structure constant. Thus, they are general detectors of FC oscillations. Thefractional sensitivity to nuclear mass is determined by the ratio f vib { f mol . It amounts to0.06 in the example above. For the well-known iodine transition R56(32-0) at 532 nm,the difference υ ´ υ “
32 is larger than in the above, so that the ratio increases toapproximately 0.084. We note that a more accurate sensitivity estimate can be obtainedthrough consideration of the mass dependences of the Dunham parameters [54], as isdone, for example, in [50].The sensitivity of a molecular system to these constants is present not only ifone probes an electronic-vibrational transition (as discussed above) but also if oneprobes a purely vibrational transition (without change of electronic state). In thiscase, f mol » f vib in Eq. (25). The ratio relevant to nuclear mass sensitivity increasessensibly to close to ´ .
5. The frequencies of such transitions lie in the infrared spectralrange [55].Let us state the sensitivities of a comparison between a molecular transitionfrequency and an optical cavity resonance frequency. The comparison consists in3measuring the frequency ratio f mol { f L . For simplicity, we assume the realistic casethat the two frequencies are nearly equal, denoted by f mol . Recalling the transition insensitivity to FC oscillations around the acoustic cut-off frequency f c1 of the opticalcavity (see section 2), we combine Eq. (25) with Eq. (5) to obtain for the fractionalvariation of f mol { f L : δ p f mol ´ f L q f mol “ $’’&’’% δαα ` f vib f mol δm e m e , ´ f vib f mol δMM , f ď f c1 δαα ` ´ ` f vib f mol ¯ δm e m e , ´ f vib f mol δMM , f ą f c1 . (26)We can further express Eq. (26) in a manner analogous to Eq. (6) to illustrate thesensitivity in detection of the couplings of the DM field to the photon, electron andnucleons: δ p f mol ´ f L q f mol “ $’’&’’%” g γ ` f vib f mol g e m e , ´ f vib f mol g n M ı ? ρ DM m φ h mol p f q , f ď f c1 ” g γ ` ´ ` f vib f mol ¯ g e m e , ´ f vib f mol g n M ı ? ρ DM m φ h mol p f q , f ą f c1 . (27)The molecular response function h mol p f q is analogous to the function h at p f q of Eq. (6).If we instead compare a molecular transition with an atomic transition that satisfiesEq. (4), the sensitivity to α is lost. It can be regained if an atomic transition is employedfor which there is a strong contribution from relativistic effects [56, 57].
5. Conclusions
We have reviewed the motivations for extending direct searches for fundamental constantvariations to the radio-frequency band, and have described the Weekend Relaxion-SearchLaboratory, an experiment designed to probe rapid oscillations of the fine structureconstant and the electron mass in the 20 kHz-100 MHz range, via atomic spectroscopy.Apparatus upgrades in the original setup, which was used to provide competitive con-straints on scalar Dark Matter within scenarios of relaxion Earth-halos, are expectedto enable an improved search for rapid oscillations of the constants, with up to « Acknowledgements
We are grateful to R. Ozeri for insightful discussions. Thiswork was supported by the Cluster of Excellence “Precision Physics, FundamentalInteractions, and Structure of Matter” (PRISMA+ EXC 2118/1) funded by theGerman Research Foundation (DFG) within the German Excellence Strategy (ProjectID 39083149), by the European Research Council (ERC) under the European Union4Horizon 2020 research and innovation program (project Dark-OST, grant agreement No695405), by the DFG Reinhart Koselleck project and by Internal University ResearchFunding of Johannes Gutenberg-University Mainz. The work of MGK was supportedby the Russian Science Foundation (RSF) grant No 19-12-00157. The work of GP issupported by grants from BSF-NSF, Friedrich Wilhelm Bessel research award, GIF, ISF,Minerva, Yeda-Sela-SABRA-WRC, and the Segre Research Award.
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