Current trends in searches for new physics using measurements of parity violation and electric dipole moments in atoms and molecules
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Current trends in searches for new physics using measurements of parity violation andelectric dipole moments in atoms and molecules
V. A. Dzuba ∗ School of Physics, University of New South Wales, Sydney 2052, Australia
V. V. Flambaum † School of Physics, University of New South Wales,Sydney 2052, AustraliaandEuropean Centre for Theoretical Studies in Nuclear Physics (ECT),Strada della Tabarelle 286,I-38123, Villazzano (Trento), Italy
We review current status of the study of parity and time invariance phenomena in atoms, nucleiand molecules. We focus on three most promising areas of research: (i) parity non-conservation ina chain of isotopes, (ii) search for nuclear anapole moments, and (iii) search for permanent electricdipole moments (EDM) of atoms and molecules which are caused by either, electron EDM or nuclear
T, P -odd moments such as nuclear EDM and nuclear Schiff moment.
PACS numbers: 11.30.Er; 12.15.Ji; 31.15.A-
I. INTRODUCTION
The study of the parity and time invariance viola-tion in atoms, molecules and nuclei is a low-energy, rel-atively inexpensive alternative to high-energy search fornew physics beyond the standard model (see, e.g. a re-view [1]). Until very recently the accurate measurementsof the parity non-conservation (PNC) in atoms was oneof the most promising ways of exploring this path. Itculminated in very precise measurements of the PNC incesium [2]. There were even indications that these mea-surements show some disagreements with the standardmodel and might indeed lead to new physics [3]. It tookconsiderable effort of several groups of theorists to im-prove the interpretation of the measurements and resolvethe disagreement in favor of the standard model. Thedisagreement were removed when Breit [4] and quantumelectrodynamic corrections [5] were included and the ac-curacy of the treatment of atomic correlations were im-proved [6, 7].In is unlikely that any new measurements of the PNCin atoms can compete with the cesium experiment inaccuracy of the measurements and interpretation (theheavy alkaline atoms like Fr and Ra + , where the PNCeffect is 20 times larger than in Cs, may be exceptions).Therefore, the main interest in the area has shiftedmostly to three other important subjects: (i) the PNCmeasurements for a chain of isotopes; (ii) the measure-ments of nuclear anapole moments; and (iii) the measure-ments of the P,T-odd permanent electric dipole momentsof atoms and molecules. Below we will briefly review eachof these subjects. ∗ E-mail address: [email protected] † E-mail address: v.fl[email protected]
II. CHAIN OF ISOTOPES
The values measured in atomic PNC-experiments canbe presented in a form E P NC = k P NC Q W , (1)where k P NC is an electron structure factor which comesfrom atomic calculations, and Q W is a weak nuclearcharge. Very sophisticated calculations are needed foraccurate interpretation of the measurements as has beendiscussed in introduction for the case of Cs. An alter-native approach was suggested in Ref. [8]. If the samePNC effect is measured for at least two different isotopesof the same atom than the ratio R = E ′ P NC E P NC = Q ′ W Q W (2)of the PNC signals for the two isotopes does not dependon electron structure factor. It was pointed out howeverin Ref. [9] that possible constrains on the new physiccoming from isotope ratio measurements are sensitive tothe uncertainties in the neutron distribution which aresufficiently large to be a strong limitation factor on thevalue of such measurements. The problem was addressedin Ref. [10] and more recently in Ref. [11]. The authors ofRef. [10] argue that experimental data on neutron distri-bution, such as, e.g. the data from the experiments withantiprotonic atoms[12], can be used to reduce the un-certainty. In more general approach of Ref. [11] nuclearcalculations are used to demonstrate that the neutrondistributions are correlated for different isotopes whichleads to cancelation of the relevant uncertainties in theratio (2).The parameter F of the sensitivity of the ratio (2) tonew physics can be presented in the form F = h p h = (cid:18) RR − (cid:19) N N ′ Z ∆ N , (3)where h p is the new physics coupling to protons(∆ Q new = Zh p + N h n ), h comes from the SM, R is theratio (2) assuming that each isotope has the same protonand neutron distribution (no neutron skin), N and N ′ arethe numbers of neutrons in two isotopes, Z is the numberof protons and ∆ N = N ′ − N . The constrains on newphysics parameter h p are affected by the experimentalerror δ R exp and uncertainties in R due to unsufficientknowledge of nuclear distributions. The later, as is ar-gued in Ref. [11], are correlated and strongly cancel eachother. Estimations of Ref. [11] show that correspondingcontribution to δ F is in the range 10 − ÷ − which isabout an order of magnitude smaller than the uncorre-lated ones. In the end, the isotope-chain measurementsare more sensitive to new physics than current parity-violating electron scattering measurements [13] (by a fac-tor of 10 for such atoms as Cs, Ba and Dy).Experiments on isotope chains are in progress at Berke-ley for Dy and Yb atoms [14, 15], at TRIUMF for Fratoms [16], at Los Alamos for Yb + ions [17], and atGroningen (KVI) for Ra + ions [70]. III. ANAPOLE MOMENT
The notion of the anapole moment was introduced byB. Ya. Zeldovich [19]. Nuclear anapole moment (AM)is the magnetic P and C -odd, T -even nuclear momentcaused by the P -odd weak nuclear forces. Interaction ofelectrons with AM magnetic field (which may be calledthe PNC hyperfine interaction) dominates the nuclear-spin-dependent contribution to the atomic or molecularPNC effect.First calculations of nuclear AM and proposals for ex-perimental measurements were presented in Ref. [20–23].Corrections to the AM interaction with electrons due tofinite nuclear size were considered in Ref. [26]. The au-thors of [22, 23] (see also [24]) note in particular that theeffect of AM is strongly enhanced in diatomic moleculesdue to mixing of the close rotational states of oppositeparity including mixing of Λ or Ω doublets. The PNCeffects produced by the weak charge are not enhanced.Therefore the AM effect dominates PNC in molecules.This greatly simplifies the detection of AM in diatomicmolecules compared to atoms. In atoms the AM effectis 50 times smaller than the weak charge effect; AM ef-fect is separated as a small difference of the PNC effectsin different hyperfine transitions. A review of the par-ity and time invariance violation in diatomic molecules(including the AM effect) can be found in Ref. [25].The idea of the AM contribution enhancement may beexplained as follows. After the averaging over electronwave function the effective operator acting on the angu-lar variables may contain three vectors: the direction of molecular axis N , the electron angular momentum J andnuclear spin I . Scalar products NI and NJ are T -oddand P -odd. Therefore, they are produced by the T, P -odd interactions discussed in the next section. P -odd, T -even operator V P must be proportional to N [ J × I ]. Itcontains nuclear spin I , therefore, the weak charge doesnot contribute. The nuclear AM is directed along the nu-clear spin I , therefore, it contributes to V P . The matrixelements of N between molecular rotational states arewell-known, they produce rotational electric dipole tran-sitions in polar molecules. Therefore, V P (induced by themagnetic interaction of the nuclear AM with molecularelectrons) mixes close rotational-hyperfine states of oppo-site parity. The interval between these levels is five ordersof magnitude smaller than the interval between the oppo-site parity levels in atoms (by the factor m e /M where m e and M are the electron and reduced molecular masses),therefore PNC effects are five orders of magnitude larger.Further enhancement may be achieved by a reduction ofthe intervals by an external magnetic field [23].The effect is further enhanced for heavy molecules.It grows with nuclear charge as Z A / R ( Zα ), where R ( Zα ) is the relativistic factor which grows from R = 1at low Z to R ∼
10 for
Z >
80. Good candidates forthe measurements include the molecules and molecularions with Σ / or Π / electronic ground state [22, 23],for example, YbF, BaF, HgF, PbF, LaO, LuO, LaS, LuS,BiO, BiS, YbO+, PbO+, BaO+, HgO+, etc. Molecularexperiments are currently in progress at Yale [27] andGroningen KVI [28]. An interesting idea of studying AMcontribution to the NMR spectra of chiral molecules werediscussed in Ref. [29].So far the only nuclear AM which has been measuredis the AM of the Cs nucleus. It is done by compar-ing PNC amplitudes between different hyperfine struc-ture sublevels in the same PNC experiment which is dis-cussed in the introduction [2]. Interpretation of the mea-surements [30] indicates some problems. For example,the value of Cs AM is inconsistent with the limit on theAM of Tl [31].To resolve the inconsistencies and obtain valuable in-formation about P-odd nuclear forces it would be ex-tremely important to measure anapole moments for othernuclei. In particular, it is important to measure AM fora nucleus with an unpaired neutron (Cs and Tl have un-paired protons). AM of such nucleus depend on differentcombination of the weak interaction constants providingimportant cross-check. Good candidates for such mea-surements include odd isotopes of Ra, Dy, Pb, Ba, La,Lu and Yb. The Ra atom has an extra advantage be-cause of strong enhancement of the spin-dependent PNCeffect in the S - D transition due to proximity of theopposite-parity state P o (∆ E = 5 cm − ) [32].Experimental work is in progress for Rb and Fr at TRI-UMF [16, 33], and for Dy and Yb at Berkeley [14, 15]. IV. ELECTRIC DIPOLE MOMENT
Permanent electric dipole moment (EDM) of a neu-tron, atom or molecule would violate both P and T in-variance. Under conditions of the CP T -theorem thiswould also mean a CP -violation. The Kobayashi-Maskawa mechanism of the SM leads to extremely smallvalues of the EDMs of the particles. It is also too weakto explain the matter-antimatter asymmetry of the Uni-verse. On the other hand, most of the popular extensionsto the SM predict much larger EDMs which are within ex-perimental reach. The EDM of an atom or a molecule ismostly due to either electron EDM and T,P-odd electron-nucleon interactions in paramagnetic systems (with non-zero total momentum J ) or to the T, P -odd nuclear forcesin diamagnetic systems ( J = 0; nuclear-spin-dependente-N interaction contributes here too). The existence of T, P -odd nuclear forces leads to the
T, P -odd nuclear mo-ments in the expansion of the nuclear potential over pow-ers of distance R from the center of the nucleus. Thelowest-order term in the expansion, the nuclear EDM, isunobservable in neutral atoms due to total screening ofthe external electric field by atomic electrons. It mightbe possible however to observe the nuclear EDM in ions(see below). The first non-vanishing term which survivesthe screening in neutral systems is the so called Schiffmoment. Below we discuss the effects of nuclear andelectron EDM and the Schiff moment. A. Nuclear EDM
It was widely believed that one needs neutral parti-cles (e.g., neutron, neutral atom or molecule) to studyEDMs. This is because the EDM is expected to be verysmall and it would be very hard to see the effect of itsinteraction with external electric field on the backgroundof the much stronger interaction with the electric charge.On the other hand, the EDM of neutral systems is verymuch suppressed by the effect of screening of the ex-ternal electric field by electrons (Schiff theorem). TheSchiff theorem may be violated by the relativistic effect(which dominates in the case of the electron EDM), hy-perfine interaction and finite size effect. For example,the lowest-order
T, P -odd nuclear moment, the nuclearEDM is practically unobservable in the neutral systems(except for a small contribution due to the hyperfine in-teraction). First observable
T, P -odd nuclear moment,the Schiff moment, is non-zero due to finite nuclear size.It is important therefore to explore the possibilityof studying EDMs of charged particles (e.g. muons oratomic ions). There are realistic suggestions of this kindin Refs. [34–36] based on the use of ion storage rings.The external electric field is not totally screened on thenucleus of an ion. Its value is E N = Z i Z E , (4) where E is external electric field, E N is electric field atthe nucleus, Ze is nuclear charge, Z i e is the charge of theion, e is proton charge. The formula (4) can be obtainedin a very simple way. The second Newton law for the ionand its nucleus in the electric filed reads M i a i = Z i eE ,M N a N = ZeE N , where M i is the ion’s mass, a i is its acceleration, M N is nuclear mass ( M N ≈ M i ), and a N is its acceleration.Since the ion and its nucleus move together, the acceler-ations must be the same ( a i = a N ), therefore E N = Z i Z E M N M i ≈ Z i Z E . (5)Different derivation of this formula can be found inRef. [37, 38]. Numerical calculations of the screened elec-tric field inside an atomic ion were performed in a numberof our works (see, e.g. Ref. [37]).The Hamiltonian of the nuclear EDM ( d N ) interactionwith the electric field is given byˆ H d = d N E N = d N Z i Z E . (6)Screening is stronger for diatomic molecules where wehave an additional suppression factor in eq. (5), M N /M i = M / ( M + M ), where M and M are themasses of the first and second nucleus. For the averageelectric field acting on the first nucleus we obtain E N = Z i Z M M + M E . (7)Note that the screeing factor here contains both nu-clear masses. This indicates that the nuclear motion cannot be ignored and the screening problem is more com-plicated than in atoms. For example, in a naive ionicmodel of a neutral polar molecule A + B − , both ions A + and B − should be located in the area of zero (totallyscreened) electric field since their avarage acceleration iszero. This could make A + and B − EDM unobservableeven if they are produced by a nuclear Schiff momentor electron EDM. In a more realistic molecular calcula-tions the Schiff moment and electron EDM effects arenot zero, however, they may be significantly suppressed(in comparison with a naive estimate of ionic EDM in avery strong field of another ion) and the results of thecalculations may be unstable.In the case of monochromatic external electric fieldits frequency can be chosen to be in resonance with theatomic electron excitation energy. Then for the effectiveHamiltonian we can haveˆ H d = d N E N ( t ) ≫ d N E . (8) B. Electron EDM
Paramagnetic atoms and molecules which have an un-paired electron are most sensitive to the electron EDM.The EDM of such systems can be expressed in the form d = Kd e , (9)where d is the EDM of an atom or molecule, d e is electronEDM, and K is electron structure factor which comesfrom atomic calculations. The factor K increases withnuclear charge Z as Z [39] times large relativistic factor R ( Zα ) [40] which may exceed the value of 3 in heavyatoms. A rough estimate of the enhancement factor inheavy atoms with external s / or p / electron is K ∼ Z α R ( Zα ) ∼ − [39, 40].Several orders of magnitude larger K ∼ − ex-ist in molecules due to the mixing of the close rotationallevels of opposite parity (including Λ-doublets) [22]. Fol-lowing Sandars this enhancement factor is usually pre-sented as a ratio of a very large internal molecular fieldto the external electric field which polarizes the molecule.The best current limit on electron EDM comes fromthe measurements of the thallium EDM [41] and reads d e = (6 . ± . × − e cm . (10)Here the value K = −
585 [42] were used for the interpre-tation of the measurements. The value of K for Tl is verysensitive to the inter-electron correlations but two mostcomplete calculations [42, 43] give very close results.In contrast to paramagnetic atoms the diamagnetic(closed shell) atoms are much less sensitive to electronEDM. This is because the only possible direction of theatomic EDM in this case is along nuclear spin and hyper-fine structure interaction must be involved to link elec-tron EDM to nuclear spin. For example, for the mercuryatom K ∼ − [44, 45]. However, due to very strongconstrain on the mercury EDM [46] the limit on electronEDM extracted from these measurements is competitiveto the Tl result (10) [46] d e < × − e cm . (11)New experiments are in progress to measure the elec-tron EDM in Cs [47], Fr [48], YbF [49], ThO [50],PbO [51] and in solid-state experiments [52]. C. Schiff moment
Schiff moment is the lowest-order
T, P -odd nuclear mo-ment which appears in the expansion of the nuclear po-tential when screening of the external electric field byatomic electrons is taken into account. This potentialcan be written as (see the derivation, e.g. in [53, 54]) φ ( R ) = Z eρ ( r ) | R − r | d r + 1 Z ( d · ∇ ) Z ρ ( r ) | R − r | d r, (12) where ρ ( r ) is nuclear charge density normalized to Z , and d is nuclear EDM. The second term in (12) is screening.Taking into account finite nuclear size the lowest-orderterm in the expansion of (12) in powers of R can be writ-ten as [55] ψ ( R ) = − S · R B ρ ( R ) , (13)where B = R ρ ( R ) R dr and S = e (cid:20) h r r i − Z h r ih r i (cid:21) (14)is Schiff moment. The expression (13) has no singular-ities and can be used in relativistic calculations. TheSchiff moment is caused by the T, P -odd nuclear forces.The dominant mechanism is believed to be
T, P -oddnucleon-nucleon interaction. Another important contri-bution comes from the EDMs of protons and neutrons.Schiff moment is the dominant nuclear contributionto the EDM of diamagnetic atoms and molecules. Thebest limit on the EDM of diamagnetic atoms comes fromthe measurements of the EDM of mercury performed inSeattle [46] | d ( Hg) | < . × − | e | cm . (15)Interpretation of the measurements requires atomic andnuclear calculations. Atomic calculations link the EDMof the atom to its nuclear Schiff moment. Nuclear calcula-tions relate Schiff moment to the parameters of the T, P -odd nuclear interactions. Summary of atomic [56–58] andnuclear [59, 60] calculations for diamagnetic atoms of ex-perimental interest is presented in Table I. To comparethe EDM of different atoms we present only the resultsof our nuclear calculations which all were performed bythe same method. For Hg and Ra there are several recentnuclear many-body calculations available (see referencesin the most recent calculation [61]) and new calculationsare in progress.The dimensionless constant η characterizes thestrength of the P, T -odd nucleon-nucleon interactionwhich is to be determined from the EDM measurements.Using (15) and the data from the Table one can get S ( Hg) = ( − . ± . ± . × − e cm (16)and for the T, P -odd neutron-proton interaction η np = (1 ± ± × − . (17) a. Nuclear enhancement. It was pointed out inRef. [60] that Schiff moment of nuclei with octupole de-formation can be strongly enhanced. This can be ex-plained in a very simple way. Nuclear deformation cre-ates an intrinsic Schiff moment in the nuclear referenceframe S intr ≈ eZR N β β π √ , (18) TABLE I: EDMs of diamagnetic atoms of experimental inter-est. Z Atom [ S/ ( e fm )] η e cm Experiment × − e cm × − He 8 × − × − Xe 0.38 0.7 Seattle [63], Ann Arbor [64]Princeton [65]70
Yb -1.9 3 Bangalore [66], Kyoto [67]80
Hg -2.8 4 Seattle [46]86
Rn 3.3 3300 TRIUMF [68]88
Ra -8.2 2500 Argonne [69], KVI [70]88
Ra -8.2 3400 where R N is nuclear radius, β ≈ . β ≈ . T or P invariance and, if no T, P -oddinteraction present, it averages to zero in the laboratoryreference frame due to nuclear rotation. However, when
T, P -odd interaction is included, it can mix close rota- tional states of opposite parity. Small energy intervalbetween these states leads to strong enhancement of thenuclear Schiff moment in the laboratory reference frame S lab ∼ h + | H P T |−i E + − E − S intr ∼ (19)0 . eβ β ZA / ηr eV E + − E − ∼ × − ηe fm , where r = 1 . | E + − E − | ∼
50 keV. The estimate (19) is about 500 timeslarger than the Schiff moment of a spherical nucleus likeHg.It was pointed out in Ref. [62] that octupole deforma-tion doesn’t need to be static. Soft octupole vibrationslead to similar enhancement. Large values of the Schiffmoment for Ra and Rn (see Table I) are due to nuclearoctupole deformation.
Acknowledgments.
This work was supported in partby the Australian Research Council and ECT. [1] J. S. M. Ginges and V. V. Flambaum, Phys. Rep. ,63 (2004).[2] C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J.L. Roberst, C. E. Tanner, C. E. Wieman, Science ,1759 (1997).[3] S.C. Bennett and C.E. Wieman, Phys. Rev. Lett. ,2484 (1999); Phys. Rev. Lett. , 4153(E) (1999); Phys.Rev. Lett. , 889(E) (1999).[4] A. Derevianko, Phys. Rev. Lett. , 1618 (2000); V.A.Dzuba, C. Harabati, W.R. Johnson, and M.S. Safronova,Phys. Rev. A , 044103 (2001); M.G. Kozlov, S.G. Por-sev, and I.I. Tupitsyn, Phys. Rev. Lett. , 3260 (2001);V. A. Dzuba, V. V. Flambaum, M. S. Safronova, Phys.Rev. A, , 022108(2002); W.R. Johnson, I. Bednyakov, and G. Soff, Phys.Rev. Lett. , 233001 (2001); Phys. Rev. Lett. ,079903(E) (2002); M.Yu. Kuchiev and V.V. Flambaum,Phys. Rev. Lett. , 283002 (2002); A.I. Milstein, O.P.Sushkov, and I.S. Terekhov, Phys. Rev. Lett. , 283003(2002); M.Yu. Kuchiev, J. Phys. B , 4101 (2002); A.I.Milstein, O.P. Sushkov, and I.S. Terekhov, Phys. Rev.A , 062103 (2003); J. Sapirstein, K. Pachucki, A.Veitia, and K.T. Cheng, Phys. Rev. A , 052110 (2003);M.Yu. Kuchiev and V.V. Flambaum, J. Phys. B , R191(2003); V. M. Shabaev, K. Pachucki, I. I. Tupitsyn, andV. A. Yerokhin, Phys. Rev. Lett. , 213002 (2005); V.V. Flambaum and J. S. M Ginges, Phys. Rev. A ,052115 (2005).[6] V. A. Dzuba, V. V. Flambaum, and J. S. M. Ginges,Phys. Rev. D , 076013 (2002).[7] S. G. Porsev, K. Beloy, and A. Derevianko Phys. Rev.Lett. , 181601 (2009).[8] V. A. Dzuba, V. V. Flambaum, and I.B. Khriplovich, Z.Phys. D , 243 (1986).[9] E. N. Fortson, Y. Pang, and L. Wilets, Phys. Rev. Lett. , 2857 (1990).[10] A. Derevianko and S. G. Porsev, Phys. Rev. A , 052115(2002).[11] B. A. Brown, A. Derevianko, and V. V. Flambaum, Phys.Rev. C , 035501 (2009).[12] A. Trzcinska, J. Jastrzebski, P. Lubinski, F.J. Hartmann,R. Schmidt, T. von Egidy, and B. Klos, Phys. Rev. Lett. , 082501 (2001).[13] R. D. Young, R. D. Carlini, A. W. Thomas, and J. Roche,Phys. Rev. Lett. , 122003 (2007).[14] A. T. Nguyen, D. Budker D, D. DeMille, and M. Zolo-torev, Phys. Rev. A , 3453 (1997).[15] K. Tsigutkin, D. Dounas-Frazer, A. Family, J. E. Stal-naker, V. V. Yashchuk, and D. Budker, Phys. Rev. Lett. , 071601 (2009).[16] E. Gomez, L. A. Orozco, and G. D. Sprouse, Rep. Prog.Phys. , 79 (2006).[17] J. Torgerson, private communication (2010).[18] L. W. Wansbeek, B. K. Sahoo, R. G. E. Timmermans,K. Jungmann, B. P. Das, and D. Mukherjee, Phys. Rev.A , 050501 (2008).[19] Ya.B. Zeldovich, Zh. Eksp. Teor. Fiz. , 1531 (1957)[Sov. Phys. JETP , 1184 (1958)].[20] V. V. Flambaum and I.B. Khriplovich. ZhETF , 1656(1980) [Sov. Phys. JETP , 835 (1980)].[21] V. V. Flambaum, I. B. Khriplovich, and O. P. Sushkov,Phys. Lett. B. , 367 (1984).[22] O. P. Sushkov and V. V. Flambaum, ZhETF , 1208(1978) [Sov. Phys. JETP , 608 (1978)].[23] V. V. Flambaum and I. B. Khriplovich. Phys. Lett. A , 121 (1985).[24] L. N. Labzovsky, Zh. Eksp. Teor. Fiz. , 856 (1978)[Sov. Phys. JETP , 434 (1978)].[25] M. G. Kozlov and L. N. Labzowsky, J. Phys. B , 1933(1995).[26] V. V. Flambaum and C. Hanhart, Phys. Rev. C , 1329 (1993).[27] D. DeMille, S. B. Cahn, D. Murphree, D. A. Rahmlow,and M. G. Kozlov, Phys. Rev. Lett. , 023003 (2008).[28] T. A. Isaev, S. Hoekstra, and R. Berger,arXiv:1007.1788v2 (2010).[29] S Nahrwold and R. Berger, J. Phys. Chem. , 214101(2009); G. Laubender and R. Berger, ChemCemChem ,395 (2003).[30] V. V. Flambaum and D. W. Murray, Phys. Rev. C ,1641 (1997).[31] P. A. Vetter, D. M. Meekhof, P. K. Majumder, S. K.Lamoreaux, and E. N. Fortson, Phys. Rev. Lett. , 2658(1995).[32] V. V. Flambaum, Phys. Rev. A , R2661 (1999); V.A. Dzuba, V. V. Flambaum, and J. S. M. Ginges, Phys.Rev. A , 062509 (2000).[33] D. Sheng, L. A. Orozco, and E. Gomez, J. Phys. B ,074004 (2010).[34] F. J. M. Farley et al , Phys. Rev. Lett. , 052001 (2004).[35] Y. F. Orlov, W. M. Morse, and Y. K. Semertzidis, Phys.Rev. Lett. , 214802 (2006).[36] V. G. Baryshevsky, J. Phys. G , 035102 (2008).[37] V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, O. P.Sushkov, Phys. Lett. A, , 461-465 (1988).[38] S. Oshima, Phys. Rev. C , 038501 (2010).[39] P. G. H. Sandars, Phys. Lett. , 194 (1965); P. G. H.Sandars, Phys. Lett. , 290 (1966).[40] V. V. Flambaum, Yad. Fiz. , 383 (1976) [Sov. J. Nucl.Phys. , 199 (1976)].[41] B. C. Regan, E. D. Commins, C. J. Schmidt, and D.DeMille, Phys. Rev. Lett. , 071805 (2002).[42] Z.W. Liu and H.P. Kelly, Phys. Rev. A , R4210 (1992).[43] V. A. Dzuba and V. V. Flambaum, Phys. Rev. A ,062509 (2009).[44] V. V. Flambaum and I. B. Khriplovich, Zh. Eksp. Teor.Fiz. , 1505 (1985) [Sov. Phys. JETP , 872 (1985)].[45] A.-M. M˚artensson-Pendrill and P. ¨Oster, Phys. Scripta , 444 (1987).[46] W. C. Griffith, M. D. Swallows, T. H. Loftus, M. V.Romalis, B. R. Heckel, and E. N. Fortson, Phys. Rev.Lett. , 101601 (2009).[47] J. M. Amini, C. T. Munger, and H. Gould, Phys. Rev. A , 063416 (2007); H. Gould, Int. J. Mod. Phys. D ,2337 (2007).[48] H. Kawamura et al , unpublished; Y. Sakemi, unpub-lished; S. Sanguinetti et al , Opt. Lett. , 893 (2009).[49] M. R. Tarbutt, J. J. Hudson, B. E. Sauer, and E. A.Hinds, Faraday Discussions , 37 (2009).[50] A. C. Vutha et al , J. Phys. B , 074007 (2010).[51] A. V. Titov, N. S. Mosyagin, A. N. Petrov, T. A. Isaev, and D. P. DeMille, Progr. Theor. Chem. Phys. , 253(2006).[52] B. J. Heidenreich et al , Phys. Rev. Lett. , 253004(2005); D. Budker, S. K. Lamoreaux, A. O. Sushkov,and O. P. Sushkov, Phys. Rev. A , 1357 (1997).[54] R. A. Sen’kov, N. Auerbach, V. V. Flambaum, and V.G. Zelevinsky, Phys. Rev. A ,032113 (2002).[56] V. A. Dzuba, V. V. Flambaum, J. S. M. Ginges, and M.G. Kozlov, Phys. Rev. A , 012111 (2002).[57] V. A. Dzuba, V. V. Flambaum, and J. S. M. Ginges,Phys. Rev. A , 034501 (2007).[58] V. A. Dzuba, V. V. Flambaum, and S. G. Porsev, Phys.Rev. A , 032120 (2009).[59] O. P. Sushkov, V. V. Flambaum, and I. B. Khriplovich,ZhETF , 1521 (1984) [JETP , 873 (1984)]; V. V.Flambaum, I. B. Khripovich, and O. P. Sushkov, Nucl.Phys. A ,750 (1986).[60] N. Auerbach, V. V. Flambaum, and V. Spevak, Phys.Rev. Lett. , 4316 (1996); V. Spevak, N. Auerbach, andV. V. Flambaum, Phys. Rev. C , 1357 (1997).[61] S. Ban, J. Dobaczewski, J. Engel, A. Shukla, Phys. Rev.C ,015501 (2010).[62] J. Engel, J. L. Friar, and A. C. Hayes, Phys. Rev. C , 035502 (2000); V. V. Flambaum and V. G. Zelevin-sky, Phys. Rev. C , 035502 (2003); N. Auerbach, V.F. Dmitriev, V. V. Flambaum, A. Lisetskiy A, R. A.Sen’kov, and V. G. Zelevinsky, Phys. Rev. C , 025502(2006); N. Auerbach, V. F. Dmitriev, V. V. Flambaum,A. Lisetskiy A, R. A. Sen’kov, and V. G. Zelevinsky,Phys. At. Nuc. , 1654 (2007).[63] T. G. Vold, F. J. Raab, B. Heckel, and E. N. Fortson,Phys. Rev. Lett. , 2229 (1984).[64] M. A. Rosenberry and T. E. Chupp, Phys. Rev. Lett. ,22 (2001).[65] M. V. Romalis and M. P. Ledbetter, Phys. Rev. Lett. ,067601 (2001).[66] V. Natarajan, Eur. Phys. J. D , 33 (2005).[67] T. Takano, M. Fuyama, H. Yamamoto, and Y. Takahashi(unpublished).[68] E. R. Tardiff et al , Nucl. Instr. Meth. Phys. Res. D et al , Pramana J. Phys.75