## Abstract

We perform model-independent analyses extracting limits for the electric dipole moment of the electron and the P,T-odd scalar-pseudoscalar (S-PS) nucleon-electron coupling from the most recent measurements with atoms and molecules. The analysis using paramagnetic systems, only, is improved substantially by the inclusion of the recent measurement on HfF^{+} ions, but complicated by the fact that the corresponding constraints are largely aligned, owing to a general relation between the coefficients for the two contributions. Since this same relation does not hold in diamagnetic systems, it is possible to find atoms that provide essentially orthogonal constraints to those from paramagnetic ones. However, the coefficients are suppressed in closed-shell systems and enhancements of P,T-odd effects are only prevalent in the presence of hyperfine interactions. We formulate the hyperfine-induced time-reversal-symmetry breaking S-PS nucleon-electron interaction in general atoms in a mixed perturbative and variational approach, based on electronic Dirac-wavefunctions including the effects of electron correlations. The method is applied to the Hg atom, yielding the first direct calculation of the coefficient of the S-PS nucleon-electron coupling in a diamagnetic system. This results in additionally improved model-independent limits for both the electron EDM and the nucleon-electron coupling from the global fit. Finally we employ this fit to provide indirect limits for several paramagnetic systems under investigation.

## References

- [1]
M.B. Gavela, P. Hernández, J. Orloff and O. Pene,

*Standard model CP-violation and baryon asymmetry*,*Mod. Phys. Lett.***A 9**(1994) 795 [hep-ph/9312215] [INSPIRE]. - [2]
P. Huet and E. Sather,

*Electroweak baryogenesis and standard model CP-violation*,*Phys. Rev.***D 51**(1995) 379 [hep-ph/9404302] [INSPIRE]. - [3]
M.B. Gavela, P. Hernández, J. Orloff, O. Pene and C. Quimbay,

*Standard model CP-violation and baryon asymmetry. Part 2: Finite temperature*,*Nucl. Phys.***B 430**(1994) 382 [hep-ph/9406289] [INSPIRE]. - [4]
J.S.M. Ginges and V.V. Flambaum,

*Violations of fundamental symmetries in atoms and tests of unification theories of elementary particles*,*Phys. Rept.***397**(2004) 63 [physics/0309054] [INSPIRE]. - [5]
M. Pospelov and A. Ritz,

*Electric dipole moments as probes of new physics*,*Annals Phys.***318**(2005) 119 [hep-ph/0504231] [INSPIRE]. - [6]
M. Raidal et al.,

*Flavour physics of leptons and dipole moments*,*Eur. Phys. J.***C 57**(2008) 13 [arXiv:0801.1826] [INSPIRE]. - [7]
T. Fukuyama,

*Searching for New Physics beyond the Standard Model in Electric Dipole Moment*,*Int. J. Mod. Phys.***A 27**(2012) 1230015 [arXiv:1201.4252] [INSPIRE]. - [8]
J. de Vries, E. Mereghetti, R.G.E. Timmermans and U. van Kolck,

*The Effective Chiral Lagrangian From Dimension-Six Parity and Time-Reversal Violation*,*Annals Phys.***338**(2013) 50 [arXiv:1212.0990] [INSPIRE]. - [9]
J. Engel, M.J. Ramsey-Musolf and U. van Kolck,

*Electric Dipole Moments of Nucleons, Nuclei and Atoms: The Standard Model and Beyond*,*Prog. Part. Nucl. Phys.***71**(2013) 21 [arXiv:1303.2371] [INSPIRE]. - [10]
J. Bsaisou, U.-G. Meißner, A. Nogga and A. Wirzba,

*P- and T-Violating Lagrangians in Chiral Effective Field Theory and Nuclear Electric Dipole Moments*,*Annals Phys.***359**(2015) 317 [arXiv:1412.5471] [INSPIRE]. - [11]
T. Chupp, P. Fierlinger, M. Ramsey-Musolf and J. Singh,

*Electric Dipole Moments of the Atoms, Molecules, Nuclei and Particles*, arXiv:1710.02504 [INSPIRE]. - [12]
N. Yamanaka, B.K. Sahoo, N. Yoshinaga, T. Sato, K. Asahi and B.P. Das,

*Probing exotic phenomena at the interface of nuclear and particle physics with the electric dipole moments of diamagnetic atoms: A unique window to hadronic and semi-leptonic CP-violation*,*Eur. Phys. J.***A 53**(2017) 54 [arXiv:1703.01570] [INSPIRE]. - [13]
M. Jung and A. Pich,

*Electric Dipole Moments in Two-Higgs-Doublet Models*,*JHEP***04**(2014) 076 [arXiv:1308.6283] [INSPIRE]. - [14]
B. Graner, Y. Chen, E.G. Lindahl and B.R. Heckel,

*Reduced Limit on the Permanent Electric Dipole Moment of Hg199*,*Phys. Rev. Lett.***116**(2016) 161601 [*Erratum ibid.***119**(2017) 119901] [arXiv:1601.04339] [INSPIRE]. - [15]
V.A. Dzuba, V.V. Flambaum and C. Harabati,

*Relations between matrix elements of different weak interactions and interpretation of the parity-nonconserving and electron electric-dipole-moment measurements in atoms and molecules*,*Phys. Rev.***A 84**(2011) 052108 [*Erratum ibid.***85**(2012) 029901] [arXiv:1109.6082]. - [16]
M. Jung,

*A robust limit for the electric dipole moment of the electron*,*JHEP***05**(2013) 168 [arXiv:1301.1681] [INSPIRE]. - [17]
L.I. Schiff,

*Measurability of Nuclear Electric Dipole Moments*,*Phys. Rev.***132**(1963) 2194 [INSPIRE]. - [18]
I.B. Khriplovich and S.K. Lamoreaux,

*CP Violation Without Strangeness*, Springer, (1997). - [19]
S. Knecht, H.J. Aa. Jensen and T. Fleig,

*Large-Scale Parallel Configuration Interaction. II. Two- and four-component double-group general active space implementation with application to BiH*,*J. Chem. Phys.***132**(2010) 014108. - [20]
V.V. Flambaum and I.B. Khriplovich,

*New Limits on the Electron Dipole Moment and T Nonconserving Electro-Nucleon Interaction*,*Sov. Phys. JETP***62**(1985) 872 [INSPIRE]. - [21]
T. Fleig and M.K. Nayak,

*Electron Electric Dipole Moment and Hyperfine Interaction Constants for ThO*,*J. Molec. Spectrosc.***300**(2014) 16 [arXiv:1401.2284] [INSPIRE]. - [22]
M. Denis et al.,

*Theoretical study on ThF*^{+}*, a prospective system in search of time-reversal violation*,*New J. Phys.***17**(2015) 043005. - [23]
V.A. Dzuba, V.V. Flambaum and S.G. Porsev,

*Calculations of the (P, T)-odd electric dipole moments for the diamagnetic atoms*^{129}*Xe,*^{171}*Yb,*^{199}*Hg,*^{211}*Rn, and*^{225}*Ra*,*Phys. Rev.***A 80**(2009) 032120 [arXiv:0906.5437] [INSPIRE]. - [24]
N.J. Stone,

*Table of nuclear magnetic dipole and electric quadrupole moments*, IAEA Nuclear Data Section Vienna International Centre, Vienna, Austria, (2014), INDC International Nuclear Data Committee. - [25]
K.G. Dyall,

*Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 5d elements Hf-Hg*,*Theoret. Chim. Acta***112**(2004) 403. - [26]
[26] K.G. Dyall and A.S.P. Gomes,

*Revised relativistic basis sets for the 5d elements Hf-Hg*,*Theoret. Chim. Acta***125**(2010) 97. - [27]
A. Kramida, Yu. Ralchenko, J. Reader and and NIST ASD Team,

*NIST Atomic Spectra Database (ver. 5.3)*, http://physics.nist.gov/asd, (2017, February 21), National Institute of Standards and Technology, Gaithersburg, MD., U.S.A., (2015). - [28]
L. Visscher and K.G. Dyall,

*Dirac-Fock Atomic Electronic Structure Calculations using Different Nuclear Charge Distributions*,*Atom. Data Nucl. Data Tabl.***67**(1997) 207. - [29]
M. Denis and T. Fleig,

*In search of discrete symmetry violations beyond the standard model: Thorium monoxide reloaded*,*J. Chem. Phys.***145**(2016) 214307. - [30]
T. Fleig, M.K. Nayak and M.G. Kozlov,

*TaN, a molecular system for probing*P*,*T*-violating hadron physics*,*Phys. Rev.***A 93**(2016) 012505 [arXiv:1512.08729] [INSPIRE]. - [31]
M.G. Kozlov,

*New Limit on the Scalar P, T Odd Electron Nucleus Interaction*,*Phys. Lett.***A 130**(1988) 426 [INSPIRE]. - [32]
A.-M. Martensson-Pendrill,

*Calculation of a P- and T-Nonconserving Weak Interaction in Xe and Hg with Many-Body Perturbation Theory*,*Phys. Rev. Lett.***54**(1985) 1153 [INSPIRE]. - [33]
L. Radžiūtė, G. Gaigalas, P. Jönsson and J. Bieron,

*Electric dipole moments of superheavy elements*—*A case study on copernicium*,*Phys. Rev.***A 93**(2016) 062508 [arXiv:1508.03974] [INSPIRE]. - [34]
K.V.P. Latha, D. Angom, B.P. Das and D. Mukherjee,

*Probing CP-violation with the electric dipole moment of atomic mercury*,*Phys. Rev. Lett.***103**(2009) 083001 [arXiv:0902.4790] [INSPIRE]. - [35]
Y. Singh and B.K. Sahoo,

*Rigorous limits for hadronic and semi-leptonic CP-violating coupling constants from the electric dipole moment of*^{199}*Hg*,*Phys. Rev.***A 91**(2015) 030501 [arXiv:1408.4337] [INSPIRE]. - [36]
B. Sahoo,

*Improved limits on the hadronic and semihadronic CP violating parameters and role of a dark force carrier in the electric dipole moment of*^{199}*Hg*,*Phys. Rev.***D 95**(2017) 013002 [arXiv:1612.09371] [INSPIRE]. - [37]
B.K. Sahoo and B.P. Das,

*Relativistic Normal Coupled-Cluster Theory for Accurate Determination of Electric Dipole Moments of Atoms: First application to*^{199}*Hg atom*,*Phys. Rev. Lett.***120**(2018) 203001 [arXiv:1801.07045]. - [38]
T. Chupp and M. Ramsey-Musolf,

*Electric Dipole Moments: A Global Analysis*,*Phys. Rev.***C 91**(2015) 035502 [arXiv:1407.1064] [INSPIRE]. - [39]
W.B. Cairncross et al.,

*Precision Measurement of the Electron*’*s Electric Dipole Moment Using Trapped Molecular Ions*,*Phys. Rev. Lett.***119**(2017) 153001 [arXiv:1704.07928] [INSPIRE]. - [40]
ACME collaboration, J. Baron et al.,

*Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron*,*Science***343**(2014) 269 [arXiv:1310.7534] [INSPIRE]. - [41]
J. Baron et al.,

*Methods, Analysis, and the Treatment of Systematic Errors for the Electron Electric Dipole Moment Search in Thorium Monoxide*,*New J. Phys.***19**(2017) 073029. - [42]
J.J. Hudson, D.M. Kara, I.J. Smallman, B.E. Sauer, M.R. Tarbutt and E.A. Hinds,

*Improved measurement of the shape of the electron*,*Nature***473**(2011) 493 [INSPIRE]. - [43]
D.M. Kara, I.J. Smallman, J.J. Hudson, B.E. Sauer, M.R. Tarbutt and E.A. Hinds,

*Measurement of the electron*’*s electric dipole moment using YbF molecules: methods and data analysis*,*New J. Phys.***14**(2012) 103051 [arXiv:1208.4507] [INSPIRE]. - [44]
B.C. Regan, E.D. Commins, C.J. Schmidt and D. DeMille,

*New limit on the electron electric dipole moment*,*Phys. Rev. Lett.***88**(2002) 071805 [INSPIRE]. - [45]
P.G.H. Sandars,

*The electric dipole moment of an atom*,*Phys. Lett.***14**(1965) 194. - [46]
P.G.H. Sandars,

*Enhancement factor for the electric dipole moment of the valence electron in an alkali atom*,*Phys. Lett.***22**(1966) 290. - [47]
V.V. Flambaum,

*On enhancement of the electron electric dipole moment in heavy atoms*,*Yad. Fiz.***24**(1976) 383 [INSPIRE]. - [48]
N. Yamanaka, T. Sato and T. Kubota,

*Linear programming analysis of the R-parity violation within EDM-constraints*,*JHEP***12**(2014) 110 [arXiv:1406.3713] [INSPIRE]. - [49]
L.V. Skripnikov,

*Communication: Theoretical study of HfF*^{+}*cation to search for the T,P-odd interactions*,*J. Chem. Phys.***147**(2017) 021101. - [50]
T. Fleig, \( \mathcal{P},\mathcal{T} \)

*-odd and magnetic hyperfine-interaction constants and excited-state lifetime for HfF*^{+},*Phys. Rev.***A 96**(2017) 040502 [arXiv:1706.02893] [INSPIRE]. - [51]
L.V. Skripnikov,

*Combined 4-component and relativistic pseudopotential study of ThO for the electron electric dipole moment search*,*J. Chem. Phys.***145**(2016) 214301. - [52]
M. Abe, G. Gopakumar, M. Hada, B.P. Das, H. Tatewaki and D. Mukherjee,

*Application of relativistic coupled-cluster theory to the effective electric field in YbF*,*Phys. Rev.***A 90**(2014) 022501. - [53]
A. Sunaga, M. Abe, M. Hada and B.P. Das.

*Relativistic coupled-cluster calculation of the electron-nucleus scalar-pseudoscalar interaction constant W*_{S}*in YbF*,*Phys. Rev.***A 93**(2016) 042507. - [54]
B.M. Roberts, V.A. Dzuba and V.V. Flambaum,

*Double-core-polarization contribution to atomic parity-nonconservation and electric-dipole-moment calculations*,*Phys. Rev.***A 88**(2013) 042507 [arXiv:1309.3371] [INSPIRE]. - [55]
D. Mukherjee, B.K. Sahoo, H.S. Nataraj and B.P. Das.

*Relativistic coupled cluster (rcc) computation of the electric dipole moment enhancement factor of francium due to the violation of time reversal symmetry*,*J. Phys. Chem.***A 113**(2009) 12549. - [56]
L.V. Skripnikov, D.E. Maison and N.S. Mosyagin,

*Scalar-pseudoscalar interaction in the francium atom*,*Phys. Rev.***A 95**(2017) 022507 [arXiv:1611.09103] [INSPIRE]. - [57]
H.S. Nataraj, B.K. Sahoo, B.P. Das and D. Mukherjee,

*A Reappraisal of the Electric Dipole Moment Enhancement Factor for Thallium*,*Phys. Rev. Lett.***106**(2011) 200403 [arXiv:1005.1797] [INSPIRE]. - [58]
H.S. Nataraj, B.K. Sahoo, B.P. Das and D. Mukherjee,

*Brief remarks on*“*Electric dipole moment enhancement factor of thallium*”, arXiv:1202.5402. - [59]
B K Sahoo, B P Das, R K Chaudhuri, D Mukherjee and E P Venugopal,

*Atomic electric-dipole moments from Higgs-boson-mediated interactions*,*Phys. Rev.***A 78**(2008) 10501. - [60]
S.G. Porsev, M.S. Safronova and M.G. Kozlov,

*Electric dipole moment enhancement factor of thallium*,*Phys. Rev. Lett.***108**(2012) 173001 [arXiv:1201.5615] [INSPIRE]. - [61]
V.A. Dzuba and V.V. Flambaum,

*Calculation of the (T,P)-odd Electric Dipole Moment of Thallium*,*Phys. Rev.***A 80**(2009) 062509 [arXiv:0909.0308] [INSPIRE]. - [62]
H.S. Nataraj, B.K. Sahoo, B.P. Das and D. Mukherjee,

*Intrinsic Electric Dipole Moments of Paramagnetic Atoms: Rubidium and Cesium*,*Phys. Rev. Lett.***101**(2008) 033002. - [63]
A.M. Mårtensson-Pendrill and P. Öster,

*Calculations of Atomic Electric Dipole Moments*,*Phys. Scripta***36**(1987) 444. - [64]
E.S. Ensberg.

*Experimental upper limit for the permanent electric dipole moment of Rb*^{85}*by optical-pumping techniques*,*Phys. Rev.***153**(1967) 36. - [65]
F.R. Huang-Hellinger Jr.,

*A Search for a Permanent Electric Dipole Moment in Rubidium*, Ph.D. Thesis, University of Washington, Seattle, U.S.A. (1987). - [66]
S.A. Murthy, D. Krause, Z.L. Li and L.R. Hunter,

*New Limits on the Electron Electric Dipole Moment from Cesium*,*Phys. Rev. Lett.***63**(1989) 965 [INSPIRE]. - [67]
J.J. Hudson, B.E. Sauer, M.R. Tarbutt and E.A. Hinds,

*Measurement of the electron electric dipole moment using YbF molecules*,*Phys. Rev. Lett.***89**(2002) 023003 [hep-ex/0202014] [INSPIRE].

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Fleig, T., Jung, M. Model-independent determinations of the electron EDM and the role of diamagnetic atoms.
*J. High Energ. Phys.* **2018, **12 (2018). https://doi.org/10.1007/JHEP07(2018)012

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### Keywords

- Beyond Standard Model
- CP violation