Electronic structure of ytterbium monohydroxide molecule to search for axionlike particles
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec Study of electronic structure of ytterbium monohydroxide molecule to search foraxionlike particles
D.E. Maison , , ∗ , V.V. Flambaum , , N.R. Hutzler , L.V. Skripnikov , Petersburg Nuclear Physics Institute named by B.P. Konstantinov of NationalResearch Center “Kurchatov Institute” (NRC “Kurchatov Institute” - PNPI),1 Orlova roscha mcr., Gatchina, 188300 Leningrad region, Russia Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia ∗ School of Physics, The University of New South Wales, Sydney NSW 2052, Australia Johannes Gutenberg-Universit¨at Mainz, 55099 Mainz, Germany and Division of Physics, Mathematics, and Astronomy,California Institute of Technology, Pasadena, CA 91125, United States
Recently the YbOH molecule has been suggested as a candidate to search for the electron electricdipole moment (eEDM) which violates spatial parity (P) and time-reversal (T) symmetries [Phys.Rev. Lett. , 133002 (2017)]. In the present paper we show that the same system can be usedto measure coupling constants of the interaction of electrons and nucleus with axionlike particles.The electron-nucleus interaction produced by the axion exchange induces T,P-violating a EDM ofthe whole molecular system. We express the corresponding T,P-violating energy shift produced bythis effect in terms of the axion mass and product of the axion-nucleus and axion-electron couplingconstants.
I. INTRODUCTION
Verification of the Standard Model (SM) and its exten-sions is of key importance for modern theoretical physicsand cosmology [1]. Despite numerous experimental con-firmations of the SM, several observed phenomena can-not be explained in its frames. Among them are thestrong CP-problem of quantum chromodynamics (QCD),the unknown nature of dark matter and dark energy andthe problem of the baryogenesis [2].A possible solution of the strong CP-problem wassuggested by Peccei and Quinn [3] via modification ofthe QCD Lagrangian. Weinberg and Wilczek also in-dependently noticed that the spontaneous Peccei-Quinn U PQ (1) symmetry violation demands the existence of thepseudoscalar Goldstone boson, afterwards called the ax-ion [4, 5]. Later it was realized that the axion is a suitablecandidate to be the dark matter component [6–8]. There-fore, it also solves another aforementioned problem of theSM. In the QCD axion case there are relations connectingaxion mass and axion-fermion interaction strength. In amore general case of axionlike particles no such connec-tion is assumed. Numerous experimental investigationsled to strong constraints on the axionlike particle prop-erties [9–11] (see also Fig. 2 in Ref. [12]). For brevityaxionlike particles are often called axions, without as-suming the QCD axion properties.It is known that the search for T,P-violating effectssuch as the electron electric dipole moment (eEDM) inthe low-energy regime can be successfully performed withparamagnetic heavy atoms and small molecules contain-ing such atoms [1, 13, 14]. One of the benefits of such ∗ molecules is the existence of closely lying opposite par-ity levels enhancing eEDM effect [15]. For instance, thestrongest current constraints on the eEDM value d e is ob-tained in an experiment using a thorium monoxide molec-ular beam [16]. Other molecular experiments [17, 18] alsosurpass sensitivity of atomic EDM experiments [19]. Theaxion-induced T,P-violating effects can also be measuredin these molecular experiments [12, 20].It was recently suggested to perform experiments tosearch for the electron electric dipole moment using lin-ear triatomic molecules containing heavy atoms. Thesemolecules have a very small energy gap between oppositeparity levels due to l -doubling effect [21–23]. This fea-ture makes it possible to polarize them by relatively weakelectric field. Besides, these molecules can be cooled andslowed by the laser-cooling technique to extremely lowtemperatures [24]. This is the way to increase the co-herence time and, thus, to improve the sensitivity of theexperiment as the uncertainty of the measured energycharacteristic is inversely proportional to the coherencetime. For this reason, the YbOH molecule has been in-tensively considered for the T,P-violating effect search byseveral theoretical groups [25–28].In Ref. [12] the first estimations of the axion-inducedinteractions in a number of diatomic molecules have beenperformed. These estimations have been based on atomiccalculations and scaling. In the present paper we intro-duce an explicit molecular approach to study the axion-mediated T,P-violating interaction based on the relativis-tic Fock-space coupled cluster theory. The dependenceof the corresponding energy shift on the axionlike parti-cle mass is considered in the ytterbium monohydroxide(YbOH) molecule. It is shown that the expected sensi-tivity [22] of the experiment on this molecule will enableto set limits on the axionlike particle coupling constantssurpassing current limits by several orders of magnitude. II. THEORY
The coupling of an axionlike particle a with the SMfermions ψ can be written in the following form: L int = a X ψ ¯ ψ (cid:16) g sψ + ig pψ γ (cid:17) ψ . (1)The coupling constants g sψ and g pψ characterize the scalarand pseudoscalar interactions in the Lagrangian (1). Thismixed scalar and pseudoscalar interaction leads to T,P-violating effects.The electron pseudoscalar and nucleon scalar interac-tions in Eq. (1) with the intermediate boson a of themass m a lead to the T,P-violating Yukawa-type interac-tion potential [12]: V eN ( r ) = + i g sN g pe π e − m a | r − R | | r − R | γ γ . (2)Here r and R are positions of the electron and nucleusunder consideration, respectively; γ –matrices are Diracmatrices defined according to [29] and refer to the elec-tron; g sN and g pe are the coupling constants of the axion-like particle with the nucleus and the electron, respec-tively. This interaction has a similar form as the T,P-violating contact nucleus-electron scalar-pseudoscalar in-teraction [13]: H T,P = + i G F √ Zk T,P n ( r ) γ γ , (3)where G F is the Fermi-coupling constant, Z is the nuclearcharge, k T,P is the coupling constant and n ( r ) is the nu-clear density normalized to unity. It should be noted thatthis form of the interaction (2) corresponds to the inter-action of axion with electron spin. Interaction of axionwith nucleon spin is considered in Ref. [30].Inclusion of the interaction (2) into the electronicHamiltonian leads to T,P-violating energy shifts of elec-tronic states in a manner analogous the shifts createdby the nucleus-electron scalar-pseudoscalar interaction.This shift is proportional to g sN g pe and can be charac-terised by the molecular constant W (eN)ax which dependson the axionlike particle mass m a : W (eN)ax ( m a ) = 1 g sN g pe h Ψ | N X i =1 V eN ( r i ) | Ψ i . (4)Here index i runs over all electrons in the molecule, andΨ is the electronic wavefunction. The characteristic T,P-violating energy shift of the electronic level can be ex-pressed as [31]: δE = g sN g pe · W (eN)ax ( m a ) . (5)The value of W (eN)ax is required for interpretation of theexperimental data in terms of the product of interaction constants. This molecular constant is the analog of theeffective electric field and the molecular constant W T,P that characterizes molecular parameter of the scalar-pseudoscalar nucleus-electron interaction in the electronelectric dipole moment search area (see e.g. [27, 28, 32–37]). Note, however, that in the present case W (eN)ax depends on the axion mass. The typical radius of theinteraction Eq. (2) can be estimated as R Yu ( m a ) ≃ m a / eV) a B , where a B is the Bohr radius. As it is shownbelow, in the limiting case of large R Yu the W (eN)ax con-stant is almost independent on m a and in the oppositelimiting case of high-mass axionlike particles the factor-ization of the W (eN)ax is possible.In the present paper we calculate molecular constant W (eN)ax ( m a ) for the YbOH molecule over a wide rangeof m a values. The molecule is considered in its ground Σ / electronic state.The equilibrium geometry parameters of the moleculeare R (Yb-O) = 2 . R (O-H) = 0 . W (eN)ax parameter, the calculations wereperformed within four basis sets, which are described inTable I. These basis sets are ordered by its quality, i.e.the basD is the best considered basis set. For Yb wehave used Dyall’s family of all-electron uncontracted ba-sis sets. TABLE I. Notation and composition of the basis sets used.Basis set basis on basis onnotation Yb [40] O and H [41–43] ∗ basA AE2Z aug-cc-pVDZ-DK[24 s ,19 p ,13 d ,8 f ,2 g ] [10 s ,5 p ,2 d ] and [5 s ,2 p ]basB AE3Z aug-cc-pVDZ-DK[30 s ,24 p ,16 d ,11 f ,4 g ,2 h ] [10 s ,5 p ,2 d ] and [5 s ,2 p ]basC AE3Z aug-cc-pVTZ-DK[30 s ,24 p ,16 d ,11 f ,4 g ,2 h ] [11 s ,6 p ,3 d ,2 f ] and [6 s ,3 p ,2 d ]basD AE4Z aug-cc-pVTZ-DK[35 s ,30 p ,19 d ,14 f ,8 g ,5 h ,2 i ] [11 s ,6 p ,3 d ,2 f ] and [6 s ,3 p ,2 d ] ∗ The basis sets on O and H were uncontracted.
Electronic correlation effects have been taken into ac-count using the relativistic Fock-space coupled clusterapproach with single and double cluster amplitudes (FS-CCSD) [44] within the finite field approach. Fock-spacesector (0,0) corresponds to the YbOH + cation in itsground electronic state, and open-shell electronic calcu-lations were performed in sector (0,1). All electrons ofYbOH have been included in correlation calculations. InRefs. [37, 45] it has been shown that high energy cut-off is important to ensure including functions that de-scribe spin-polarization and correlation effects for coreelectrons. In the present paper all virtual orbitals havebeen included in correlation treatment. The effect of theGaunt interaction of the electrons was estimated withinthe Dirac-Hartree-Fock-Gaunt approach using the basDbasis set. Its relative contribution reaches the maximalvalue of − m a = 10 eV and does not exceed 1%by absolute value for other presented m a . It was shownin [46], that electronic correlation can affect the Gauntcontribution to the properties of triatomic molecules, butthe absolute value contribution is not significant. We donot include the Gaunt contribution to the values in Ta-ble III.Both the Dirac-Hartree-Fock and the Fock-space cal-culations were performed using the local version of the dirac15 code [47]. The code to calculate matrix ele-ments of the electron-nucleus interaction Eq. (2) was de-veloped in the present paper. III. RESULTS AND DISCUSSION
Tables II and III give the calculated dependence of the W (eN)ax value on the mass of the axionlike particle m a us-ing different basis sets and methods. The final values aregiven in the last column of Table III. The uncertaintyof the final values arises mainly from higher-order corre-lations and can be estimated to be less than 10% [25]. TABLE II. The values of the W (eN)ax constant for the groundelectronic state of YbOH (in units of m e c/ ¯ h ) for various m a using the Dirac-Hartree-Fock method and different basis sets. m a , eV basA basB basC basD10 +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +9 . · − +9 . · − +9 . · − +9 . · − +1 . · − +1 . · − +1 . · − +1 . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − As one can see from Tables II and III, results for low-mass axionlike particles are weakly dependent on the ba-sis set size. However, according to our findings, basisfunctions with small radii should be included in the ba-sis set to describe correctly the electronic wavefunctionasymptotic in the vicinity of a heavy nucleus in the heavyaxion case. It can be seen from comparison of Tables IIand III that the role of correlation effects increases forhigh m a values. Note that in the region from 10 eV to10 eV the W (eN)ax constant changes its sign. This can beexplained by the fact that the effects for low-mass and TABLE III. The values of the W (eN)ax constant for the groundelectronic state of YbOH (in units of m e c/ ¯ h ) for various m a using the relativistic FS-CCSD approach and different basissets. m a , eV basA basB basC basD (Final)10 +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − +1 . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − − . · − high-mass axions arise from different distances [12]. Be-low we discuss separately the low- and the high-mass ax-ion cases. In the latter case the potential (2) is spatiallylocalized in the vicinity of the nucleus and in this casethe W (eN)ax parameter belongs to the class of the “atom ina compound” (AiC) characteristics [48–50]. This meansthat it should have similar correlation trends to othercharacteristics such as the effective electric field actingon the electron electric dipole moment, etc. Low-mass limit.
As one should expect from the po-tential form (2) and as it can be seen from Table III,for m a ≪ eV the W (eN)ax parameter is almost inde-pendent on m a . Thus, for the low-mass axion case wecan explicitly obtain the link between the g sN g pe prod-uct and the energy shift δE defined by Eq. (5) employ-ing the electronic structure calculation. The mass of theaxionlike particle becomes unimportant here. The cur-rent limitation for these interaction constants product | g sN g pe | / (¯ hc ) < ∼ − [12] can be obtained from the in-terpretation of the second generation of the ThO exper-iment [51]. Note, that this constraint is the strongestfor m a > ∼ − eV (see Fig. 2 in Ref. [12]). The corre-sponding energy shift in the YbOH molecule would be δE ≃ µ Hz. This value has the same order of mag-nitude as the sensitivity to T,P-violating effects alreadyachieved in the ThO experiment [16]. In Ref. [22] it wassuggested that using the YbOH molecule the sensitivityto the electron EDM could be increased by up to 4 ordersof magnitude above the that obtained in [52] (and, con-sequently, 3 orders with respect to Ref [16]). Therefore,the expected sensitivity of the YbOH experiment will ul-timately be enough to set orders of magnitude strongerlimitations on | g sN g pe | . High-mass limit.
For axions with mass m a ≫ eVthe approximate dependence is W (eN)ax ≃ f W · m − a , wherethe f W value does not depend on m a . In this case, themeasured energy shift can be parameterized as δE ≈ g sN g pe m a f W , (6)and the constraint for the combination g sN g pe /m a canbe derived from the experimental data. The constraint | g sN g pe | / (¯ hcm a ) < ∼ − GeV − can be obtained from theinterpretation of the experiment with the ThO molecule[12, 16]. The energy shift in the YbOH molecule cor-responding to this constraint would be ∼ µ Hz. Asit has been noted above, the ultimate sensitivity of theYbOH experiment is several orders higher. Thus, in thehigh-mass limit it also will be possible to obtain new con-straints.
ACKNOWLEDGMENTS
Electronic structure calculations have been carried outusing computing resources of the federal collective usage center Complex for Simulation and Data Processing forMega-science Facilities at NRC “Kurchatov Institute”,http://ckp.nrcki.ru/.Molecular coupled cluster electronic structure cal-culations have been supported by the Russian ScienceFoundation Grant No. 19-72-10019. Calculations of the W (eN)ax matrix elements were supported by the foundationfor the advancement of theoretical physics and mathe-matics “BASIS” grant according to the research projectsNo. 20-1-5-76-1 and No. 18-1-3-55-1. Calculation of theGaunt contribution has been supported by RFBR grantNo. 20-32-70177. V.F. acknowledges support by the Aus-tralian Research Council grants No. DP190100974 andDP20010015. N. R. H. acknowledges support by the Gor-don and Betty Moore Foundation (grant 7947) and theAlfred P. Sloan Foundation (grant G-2019-12502). [1] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,A. 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