(Electric Dipole Moment of the Electron in Diatomic Molecules)
Principal Investigator: Timo Fleig,
Collaborators: Malaya K. Nayak (Mumbai), Trond Saue (Toulouse), Mikhail G. Kozlov (St. Petersburg)
Students: Malika Denis

The Search for Physics Beyond the Standard Model 
State of the Art
The Standard Model (SM) of elementary particle physics is not the final microscopic theory of the universe.
Its success being beyond question, the SM exhibits striking shortcomings. Apart from the fact
that it does not unify all four forces of nature in one common framework, the SM fails to explain
the very origin of matter. According to the SM, both particles and their antiparticles should have
been produced in (nearly) equal amounts in the early stages of the universe. Yet, we observe an enormous
surplus of denominational matter over antimatter, the socalled Baryon Asymmetry of the Universe
(BAU). This is one of the great unsolved problems in physics.
Electric Dipole Moments (EDMs) as Source of P and T Violation
The existence of electric dipole moments of elementary particles (in particular of leptons)
would indicate P.and.Tviolation and by virtue of the CPT theorem, (CP)violation.
This is one of Sakharov's three conditions which have to be met for explaining the BAU.

If an electron, for instance, had a nonvanishing electric dipole moment, the ensuing interaction
with an external electric field would be Parity (P ) and Timereversal (T )
violating. Such a dipole moment is assumed to be induced by fundamental interactions with other
particles.

Within the framework of the SM, the EDM of the electron could have a nonzero value. However, since the only
known CPviolating mechanism in the SM involves mesons with nonzero strangeness or bottomness,
higherorder loop diagrams
with virtual quarkantiquark pairs are required for producing an electron EDM. Its value is in this
context therefore immeasurably small, on the order of 10 ^{38} e cm.
Extensions to the SM such as Grand Unified Theories (GUT) and SuperSymmetric models comprise new sources of (CP)violation which entails
many orders of magnitude larger values of particle EDMs. In Minimal Supersymmetric Extensions to the SM a host of new particles
is assumed to exist which would allow for oneloop induced EDMs, boosting the electron's value to an
order of 10 ^{27} e cm. Such a value reaches into the measurable range of today's modern
experiments.
Electric dipole moments are unparalleled probes for new (CP)violation with a sensitivity beyond
the reach of the Large Hadron Collider (LHC) for direct detection!
What are we heading for ?
The search for EDMs of elementary particles is, therefore, also the search for New Physics (NP) beyond
the SM. However, not only the spectacular goal of detecting an EDM is of interest. NOT detecting an
EDM in a certain range of values is one important of many guides on how possible extensions to the
SM, consistent NP theories, should be conceived.
Other symmetryviolating interactions accessible by our methods are the scalarpseudoscalar
electron nucleon interaction (ENSPS, P.and.Todd), the nuclear magnetic quadrupole moment  electronic magnetic
field interaction (P.and.Todd), and the nuclear anapole moment (Podd).




Theory Project
The EDM of the electron cannot be measured directly. In all modern experiments an energy shift or a
transition energy in an atom or molecule due to an EDM is measured, from which the particle's EDM
can be interpreted by means of a manybody calculation. The determination of this factor which relates
the atomic physics to the particle physics scale is one of the major objectives of our work.
Thus, the knowledge of the internal electric field interacting with the electron in the
manyparticle system is required. We have recently implemented the respective P,T odd
interaction Hamiltonian into our
KramersRestricted Configuration Interaction method. This method allows for the computation of
P,T odd matrix elements as expectation values over correlated fourcomponent relativistic
CI wavefunctions.
The new method has been and is currently being applied to various diatomic molecules which have been
identified as promising candidates in the search for the electron EDM. Among them are HfF^{+},
ThO, ThF^{+}, and WC. The currently most constraining upper bound to the electron EDM
has been obtained from experimental and theoretical studies on ThO (ACME collaboration and own work).
The measure for the P,T odd interaction is the effective electric field at the position
of an unpaired electron in the molecule (for an electron pair in a closed shell the energy shifts cancel
out).
The studies on heavyelement molecules are challenging and typically require an accurate
description of molecular electronically excited states, including the effects of special relativity.
Our approaches take all atomic electrons into account explicitly, and limit the number of explicit
electrons for correlation calculations to valence and outer core shells.

If the experimental upper bound to the molecular EDM is interpreted in terms of the electron EDM by
dividing this upper bound by the calculated effective electric field, the electron EDM upper bound
results. It currently (January 2019) has the value d_{e} < 1.3 x 10^{29} e cm.
The figure to the left shows the ensuing constraint on theories going beyond the SM (click on the
image).

Strictly speaking,
only theories to the right of the eEDM bar are still valid possibilities as extensions to the SM.
However, accidental cancellations of for instance eEDM and ENSPS effects may lead to a too low upper
bound. Moreover, NP theories are susceptible to varying predictions of EDMs based on freedom in their
parameter spaces.


News Section 
(122016) TaO^{+} proposed as promising candidate for New Physics searches
(112016) New electron EDM and neSPS interaction constants for ThO determined.
(042015) TaN spectrum and P,Todd constants determined
(062013) ANR grants financing for EDMeDM as one of 15 projects and the only theory
project on the principal list (A list).


Recent Publications 
Modelindependent determinations of the electron EDM and the role of diamagnetic atoms
Timo Fleig and Martin Jung
J High Energy Phys (JHEP) 07 (2018) 012
arXiv:1802.02171 (please refer to full JHEP paper, above)
P,TOdd and Magnetic Hyperfine Interaction Constants and ExcitedState Lifetime for HfF^{+}
Timo Fleig
Phys Rev A 96 (2017) 040502(R)
arXiv:1706.02893 (please refer to full PRA paper, above)
TaO^{+}: A candidate molecular ion for searches of physics beyond the standard model
Timo Fleig
Phys Rev A 95 (2017) 022504
arXiv:1611.08741 (please refer to full PRA paper, above)


Topic Related Bibliography 
The Electric dipole moment of the Electron.
E. D. Commins and D. DeMille
, Advanced Series on Directions in HighEnergy Physics  Vol. 20, Chapter 14 (2009)
Electric dipole moments in the {MSSM} reloaded.
J. Ellis, J. S. Lee, and A. Pilaftsis, J. High Energy Phys. 10 (2008) 49
Electric dipole moments as probes of new physics.
M. Pospelov and A. Ritz, Ann. Phys. 318 (2005) 119
Electric Dipole Moments of Leptons.
E. D. Commins, Adv. Mol. Opt. Phys. 40 (1999) 1
CP Violation Without Strangeness.
I. B. Khriplovich and S. K. Lamoreaux, Springer (Berlin, Heidelberg), 1997


Topic Related Links 
Advanced Cold Molecule Electron EDM (ACME) Collaboration.
DeMille (Yale), Doyle (Harvard), Gabrielse (Harvard)
JILA electron EDM search using trapped molecular ions.
Cornell (Boulder)


