Single particle EDMs are the simplest systems because there is no theory involved to evaluate, e.g., shielding or enhancement on an EDM by any internal structure of the investigated system. Any measured EDM can be directly attributed to the fundamental particle itself. Within the group of single particle experiments we can distinguish between to our present knowledge ‘point’ particles such as the leptons electron (e), muon (μ) or tauon (τ) and particles that have known substructure such as the neutron (n) or the proton (p).
These leptonic particles are the cleanest objects to investigate, because they have no known internal structure. An observed particle EDM would be an undoubted property of the particle itself. Among the elementary particles with an established quark substructure are the nucleons p and n and atomic nuclei consisting of them. In these composed systems an EDM can arise in addition to the intrinsic EDMs of the constituents from properties of the interaction between the constituents. As an example, in the deuteron an EDM can arise not only from constituent EDMS, but also from CP-odd parts in the interaction between the proton and the neutron .
Among the presently ongoing EDM searches there are several independent projects aiming towards the discovery of an EDM of the neutron from sources outside of the Standard model (see e.g. [72, 73]). The modern ones exploit the fluxes available for ultra-cold neutrons at various reactors or accelerators worldwide. These competing projects try to achieve improvements over the presently best result [67, 74] largely through the implementation of sophisticated technical developments in their challenging setups.
The latest completed neutron EDM measurement has yielded a precision result in a spin precession experiment . The neutron EDM searches show that now the experiments have reached a level of accuracy where subtle effects can mimic a false EDM, because such effects may be coupled to the electric field and, e.g., reverse if the field is reversed. The final limit in the latest experiment had therefore a need for correction even after publication. It concerns the systematic effects which are caused by the particles moving in the apparatus, in particular also in a partly inhomogeneous magnetic field. The static field in the laboratory is experienced by the moving particle not as a static one. This time dependent field in the eigenframe of the particles can have large consequences, if its frequency comes close to the frequencies of the neglected terms in rotating wave approximation (Bloch-Siegert term). In particular a false EDM can be simulated by Larmor frequency shifts which are introduced by geometric phases [74, 75], as such effects can be proportional to the electric field. At present the bound on the neutron EDM is e cm (90% C.L.).
The neutron experiments show very visibly the sensitivity of precision EDM searches to systematics caused by the environment. At first glance such may not be obvious in precision experiments at the forefront of possibilities. Major improvements of the apparatuses in the ongoing set of many independent experiments concern control of the magnetic field via co-magnetometry by exploiting, e.g., Hg magnetometers or squids which are installed close to the actual measurement volume or a buffer gas co-magnetometer which is discussed . It is considered essential that a co-magnetometer really senses the magnetic field in the fiducial volume, at best it measures simultaneously with the EDM search experiment and it co-exists within the the same volume under observation. This condition can be met, e.g., with two mixed gases one of which serves as a co-magnetometer and the second one is searched for an EDM. Such a setup is possible with 3He as magnetometer with known much lower sensitivity to an EDM and 129Xe as a candidate atom for an atomic EDM (see e.g. ).
The neutron EDM has at present the attention of a number of independent experimental projects which are actively pursued at different laboratories worldwide. This includes experiments which follow the more traditional routes and implement significant refinement, where neutron flux is a central issue. Such projects are under way at the Institute Laue Langevain in Grenoble, France , at the Paul Scherrer Institute (PSI) in Villigen, Switzerland , at the Technical University München, Germany , at the Spallation Neutron Source in Oak Ridge, USA , and a joint project of RCNP KEK and TRIUMF . At the future European Spallation Source in Lund, Sweden, also crystal-diffraction promises to yield competitive limits . These remarkable experimental efforts are accompanied by theoretical activities among which [84, 85].
3.1.2 Charged Particles
In the past a few EDM search experiments have been performed on charged particles exploiting just the properties of these systems themselves directly and without using external electric fields that are applied to the systems in the laboratory. These projects have produced respectable limits.
As an example of the possibilities, already in 1971, a lifetime measurement of the state in the muonic helium ion (4He was carried out. This corresponds to a limit on the muon EDM of e cm .2 A finite EDM would have shortened in that experiment the lifetime of the metastable state. Further, also the good agreement between theory and experiment for the transition frequency of the 22S1/2-22P3/2 transition in (4He had been interpreted in the 1970ies in terms of a limit for the muon EDM of order e cm . The discussion in the following decades on the validity of the reported lifetime of the 2s state of (4He under the high pressure conditions (40 bar)in the experiment [88-90] has primarily been crucial for the interpretation of the laser spectroscopy result in terms of the mean square charge radius of the α-particle the value of which must be still viewed with caution, therefore.
However, this issue is less relevant for pointing out a method to search for an EDM which is different from spin precession experiments. Also, at no time the best limit on the muon EDM has been affected, because of two main reasons: (i) The longevity of the 2s state in (4He has been measured independently . The lifetime of the 22S1/2state has been determined to be ns at a pressure of 200 mbar. This experimental result provides for an independent extraction of a limit on the muon EDM also in the range of 10−15 e cm and demonstrates the feasibility of the method. (ii) A limit on the muon EDM of e cm [91, 92] had been reported already in 1960 from a muon beam experiment at the Nevis cyclotron facility, New York, USA.
This remarkable limit on the muon EDM at the time had been reached by searching for a spin rotation in the motional electric field which muons experience when they pass through a magnetic field. Furthermore, already in 1961 the first muon g-2 experiment at CERN had been modified to establish an even more stringent limit of e cm  by looking for a precession of the muon spin out of the plane of orbit in a magnetic field, which would have been caused by a finite muon EDM interacting with the motional electric field.
The majority of searches for EDMs in charged particles to date have been performed using neutral systems such as atoms, molecules or condensed matter, which contains the objects of interest, such as electrons or nuclei, as constituents. The choices for approaches in experiments are a consequence of the fact that most of the attempted measurements of an EDM require an electric field E. All presently known and ongoing projects follow that route. A charged particle experiences in a static electric field of trivial geometry (such as, e.g., a constant homogeneous field) an acceleration. This has been considered a severe limitation by the majority in the community, because the particle would be expelled from the fiducial volume rather rapidly. However, this argument is only valid for rather straightforward, such as homogeneous, electric field geometries (see e.g. ). There are, however, field topologies in which this argument does not hold. and this fact had been demonstrated already in the early 1960ies in muon experiments (see above).
EDM searches do not necessarily require an external homogeneous electric field at all. This fact had apparently not been recognized widely in the atomic physics based community which has been concerned primarily with low energy high precision experiments to search for an EDM until the late 1990ies. Therefore searches for charged particle EDMs such as the electron or of nucleons and nuclei had been performed exclusively in neutral atoms. This has prohibited setting up dedicated high precision EDM searches which can investigate free charged particles directly. Geometries for experiments where this is possible are, e.g., magnetic storage rings with a radial electric field in the storage region.
The muon has been the first particle that was investigated in a non-trivial field topology, i.e. a limit has been established for the muon EDM parasitically with precise measurements of the muon magnetic anomaly . One example, where this method has been exploited successfully to obtain higher precision already in the 1970ies, is the search for an EDM of the muon in a storage ring  which has been conducted parasitically to the precise measurement of the muon magnetic anomaly . An out of orbit plane precession of the muon spin was searched for along with the data taking for the muon g-2 value; the underlying measurement principle(concerning the origin of the electric field and the detection of the muon spin direction) was the same as the one that been employed already in 1961 .
The most recent muon g-2 experiment at the Brookhaven National Laboratory (BNL), Upton, New York, United States, searched also for a muon EDM in a parasitic measurement along the main data taking [96, 97]. An EDM would have manifested itself in an oscillation of the the decay electron distribution around the muon orbit plane. The so far best limit on the muon EDM has been established to be d e cm (90% C.L.). Because the measurement of the muon magnetic anomaly has yielded a manifest some 3.5 standard deviation difference between experiment and theory  a new g-2 experiment will be set up at Fermilab, Batavia, United States, to verify or refute a discrepancy, which would be a sign of New Physics . This new experiment aims for an improvement in by a factor of 5. At the same time it has a potential to improve the present bound on by one order of magnitude.
Since the wider recognition of the possibility to exploit charged particles directly a number of further systems has been thoroughly considered. Precision experiments both with trapped cold molecular ions in Paul traps  and at high energies using dedicated storage rings muons and nuclei, such as proton, deuteron, triton and others [59, 96, 111-118] are presently under way and aim for most competitive results, i.e. limits on the respective EDMs of order 10−29 e cm or better. The projects include such with a stepwise approach with intermediate accuracy of order 10−24 e cm to 10−25 e cm using to a large part existing storage ring devices such as COSY in Jülich, Germany [112, 114], and such that want to reach the most stringent bounds in one go such as an at the BNL based initiative [57, 115].
As far as the experimental approaches are concerned the situation is characterized by a rapid sequence of proposed concepts which range from primarily magnetic storage to primarily electrostatic storage. Which of the concepts is better suited for a real experiment in future will depend on the system under investigation, where its magnetic anomaly plays a major role for the choice of the type of storage ring and technique, as well as resources and the actual collaboration performing a real experiment at a future time. For magnetic storage experiments with sensitivity at the 10−24e cm level major equipment exists already with a moderate pole diameter (order 1 m) magnet at PSI  and the COSY storage ring at FZ Jülich .
A dedicated storage ring for an EDM experiment was first considered for muons . Longitudinally polarized particles are injected into a magnetic storage ring with a suited radially directed electric field to compensate a spin precession in the plane of orbit which originates from the muon magnetic moment. In such a situation a muon EDM would express itself as a spin rotation around the radius of the particle orbit. This can be observed as a time dependent change of the above/below the plane of orbit counting rate ratio. The time dependence is to first order linear for all realistic achievable fields and experimental geometries) Different from most other EDM searches, in such an experiment the possible muon flux is a major limitation.
For models with nonlinear mass scaling of EDM's such a muon EDM experiment would already be more sensitive to certain New Physics models than the present limit on the electron EDM (see [119-121]). Note, in certain Left-Right or supersymmetric symmetric models a value of as high as 10−22 e cm is possible.
For the muon in particular a limit of 10−24e cm can be potentially reached with a relative moderate cost project that exploits largely existing key equipment and existing surface muon (about 28 MeV/c momentum) facilities for a first dedicated ring EDM experiment  with discovery potential. This value is in the range of a proposed experiment using an existing magnet system at the Paul Scherrer Institut in Villigen, Switzerland . An experiment carried out at a more intense muon source could provide a probe to CP violation for particles in the second generation with correspondingly higher sensitivity. The muon as a particle from the second generation may have different routes to an EDM than the mostly investigated first generation particles. However, the community has not yet really started to exploit this open window of opportunity.3
The deuteron is the simplest known nucleus. Here an EDM could arise not only from a proton or a neutron EDM, but also from CP-odd nuclear forces between the two particles . It has been shown recently  that in certain scenarios the deuteron can be significantly more sensitive than the neutron. The situation is evident for the case of quark chromo-EDMs, where the EDMs induced into deuteron () and neutron(), respectively, are
where and are the chrom -EDM of the down quark and the up quark, respectively. This implies that the deuteron could have a much higher sensitivity to quark chromo-EDMs which arises from the proton-neutron interaction within the deuteron. Because of its rather small magnetic anomaly the deuteron is a particularly interesting candidate for a magnetic ring EDM experiment, as it provides for the possibility to compensate in a magnetic storage ring the magnetic spin precession  with a suited electric field. In such an experiment scattering off a target can be used to observe spin precession, for example. Deuteron polarimeter studies turned out to be very encouraging .
Also heavier nuclei in atomic systems with partly filled electronic shells have been considered [124-127]. Because of its rather low magnetic anomaly the 223Fr nucleus is a particularly interesting further candidate for a ring EDM experiment with sensitivity to a nuclear EDM.
Possible ring EDM experiments are presently being intensively discussed and the favoured approaches are subject to frequent changes, yet. A thorough comparison of the various ambitious approaches, which often aim for many orders of magnitude improvements over presently established EDM limits, is not possible unless experimentally the viability of any of the possible paths will have been demonstrated.