CP violating moments

Charge and parity symmetry (CP) violating moments include permanent Electric dipole moments (EDM) of electrons, neutrons, and atoms. Searches for these moments is one of the most active frontiers of precision measurement physics, as increased accuracy of null readings allows improved constraints on theoretical CP violation within and beyond the Standard Model.

The nuclear electric dipole moment (nuclear EDM) ${\displaystyle d_{\text{e}}^{N}}$ is an intrinsic property of a nucleus ${\displaystyle N}$ that can be non-zero only if CP violating interactions occur within the atom or the nucleus itself. They have been surmised to exist since at least 1963^[1] in both light^[2] and heavy nuclei.^[3]

CP-violating properties and interactions that can contribute to a nuclear electric dipole moment include the neutron EDM, the proton EDM as well as CP-violating interactions of nucleons with meson or photons. At a more fundamental level, nuclear EDMs can originate from the electroweak sector, a non-zero ${\displaystyle {\bar {\theta }}_{\text{QCD}}}$ or from physical processes that go beyond the standard model of particle physics. A time varying nuclear EDM would be evidence for (and give the mass of) axion-like dark matter.^[4]

The Schiff theorem provides a complication to searches for Nuclear EDMs. The theorem, first proposed by Leonard I. Schiff in 1963, states that a nuclear EDM will be screened by the nucleus's electrons.^[1] This occurs as the electrons arrange themselves so as to cancel an applied external electric field and minimize the electric dipole energy of the nucleus. If the electric charge and dipole distributions of the nucleus are identical (for example if treated as a point dipole and charge) this screening will be complete.

This motivates the use of atoms with mismatches between their nuclear electric charge and dipole distributions. First, Schiff moments (which survive this screening) scale with the nuclear charge ${\displaystyle Z}$, motivating the selection of heavy atoms. Secondly, Schiff moments are enhanced by nuclear levels close in energy and of opposite parity (namely those present in an octupole deformed nuclei).^[5] The Schiff moment may be derived from a Taylor expansion of the classical electrostatic interaction between an electron and a nuclear charge density.^[6]

Experimental searches of subatomic EDMs have nearly always been conducted with a powerful external electric field ${\displaystyle \mathbf {E} }$ collinear with an external magnetic field ${\displaystyle \mathbf {B} }$ that exploits the fact that the potential energy depends on the relative orientation of the electromagnetic fields

${\displaystyle U=-(\mathbf {m} _{N}\cdot \mathbf {B} +\mathbf {d} _{\text{e}}^{N}\cdot \mathbf {E} )}$

where ${\displaystyle \mathbf {m} _{N}}$ is the nuclear magnetic moment. The Larmor frequency is proportional to the potential energy which depends on whether the two external fields are parallel or antiparallel to each other. Flipping the direction of ${\displaystyle \mathbf {E} }$ and subtracting the two corresponding Larmor frequencies leads to a result that is proportional to ${\displaystyle d_{\text{e}}^{N}}$. As this technique relies on large electric fields, it is always applied to neutral systems like the neutron or atoms making it difficult to apply to an electrically charged system like a proton or an ionized atom.

The best upper limit on an atomic EDM was measured on ${\displaystyle ^{199}{\text{Hg}}}$, ${\displaystyle |d_{\text{e}}^{\text{Hg}}|<7.4\times 10^{-30}e\cdot }$cm (95% C.L.) using electric voltages of ${\displaystyle \pm 6~{\text{kV}}}$ and ${\displaystyle \pm 10~{\text{kV}}}$.^[7]

The neutron electric dipole moment (nEDM), denoted d_n, is a measure for the distribution of positive and negative charge inside the neutron. A nonzero electric dipole moment can only exist if the centers of the negative and positive charge distribution inside the particle do not coincide. So far, no neutron EDM has been found. The current best measured limit for d_n is (0.0±1.1)×10⁻²⁶ e⋅cm.^[8]

A permanent electric dipole moment of a fundamental particle violates both parity (P) and time reversal symmetry (T). These violations can be understood by examining the neutron's magnetic dipole moment and hypothetical electric dipole moment. Under time reversal, the magnetic dipole moment changes its direction, whereas the electric dipole moment stays unchanged. Under parity, the electric dipole moment changes its direction but not the magnetic dipole moment. As the resulting system under P and T is not symmetric with respect to the initial system, these symmetries are violated in the case of the existence of an EDM. Having also CPT symmetry, the combined symmetry CP is violated as well.

As it is depicted above, in order to generate a nonzero nEDM one needs processes that violate CP symmetry. CP violation has been observed in weak interactions and is included in the Standard Model of particle physics via the CP-violating phase in the CKM matrix. However, the amount of CP violation is very small and therefore also the contribution to the nEDM: |d_n| ~ 10⁻³¹ e⋅cm.^[9]

From the asymmetry between matter and antimatter in the universe, one suspects that there must be a sizeable amount of CP-violation. Measuring a neutron electric dipole moment at a much higher level than predicted by the Standard Model would therefore directly confirm this suspicion and improve our understanding of CP-violating processes.

As the neutron is built up of quarks, it is also susceptible to CP violation stemming from strong interactions. Quantum chromodynamics – the theoretical description of the strong force – naturally includes a term that breaks CP-symmetry. The strength of this term is characterized by the angle θ. The current limit on the nEDM constrains this angle to be less than 10⁻¹⁰ radians. This fine-tuning of the angle θ, which is naturally expected to be of order 1, is the strong CP problem.

Supersymmetric extensions to the Standard Model, such as the Minimal Supersymmetric Standard Model, generally lead to a large CP-violation. Typical predictions for the neutron EDM arising from the theory range between 10⁻²⁵ e⋅cm and 10⁻²⁸ e⋅cm.^[10][11] As in the case of the strong interaction, the limit on the neutron EDM is already constraining the CP violating phases. The fine-tuning is, however, not as severe yet.

In order to extract the neutron EDM, one measures the Larmor precession of the neutron spin in the presence of parallel and antiparallel magnetic and electric fields. The precession frequency for each of the two cases is given by

${\displaystyle h\nu =2\mu _{\text{n}}B\pm 2d_{\text{n}}E}$,

the addition or subtraction of the frequencies stemming from the precession of the magnetic moment around the magnetic field and the precession of the electric dipole moment around the electric field. From the difference of those two frequencies one readily obtains a measure of the neutron EDM:

${\displaystyle d_{\text{n}}={\frac {h\,\Delta \nu }{4E}}}$

The biggest challenge of the experiment (and at the same time the source of the biggest systematic false effects) is to ensure that the magnetic field does not change during these two measurements.

The first experiments searching for the electric dipole moment of the neutron used beams of thermal (and later cold) neutrons to conduct the measurement. It started with the experiment by James Smith, Purcell, and Ramsey in 1951 (and published in 1957) at ORNL's Graphite Reactor (as the three researchers were from Harvard University, this experiment is called ORNL/Harvard or something similar, see figure in this section), obtaining a limit of |d_n| < 5×10⁻²⁰ e⋅cm .^[12][13] Beams of neutrons were used until 1977 for nEDM experiments. At this point, systematic effects related to the high velocities of the neutrons in the beam became insurmountable. The final limit obtained with a neutron beam amounts to |d_n| < 3×10⁻²⁴ e⋅cm.^[14]

After that, experiments with ultracold neutrons (UCN) took over. It started in 1980 with an experiment at the Leningrad Nuclear Physics Institute (LNPI) obtaining a limit of |d_n| < 1.6×10⁻²⁴ e⋅cm .^[15] This experiment and especially the experiment starting in 1984 at the Institut Laue-Langevin (ILL) pushed the limit down by another two orders of magnitude yielding the best upper limit in 2006, revised in 2015.

During these 70 years of experiments, six orders of magnitude have been covered, thereby putting stringent constraints on theoretical models.^[16]

The latest best limit of |d_n| < 1.8×10⁻²⁶ e⋅cm has been published 2020 by the nEDM collaboration at Paul Scherrer Institute (PSI).^[8]

Currently, there are at least six experiments aiming at improving the current limit (or measuring for the first time) on the neutron EDM with a sensitivity down to 10⁻²⁸ e⋅cm over the next 10 years, thereby covering the range of prediction coming from supersymmetric extensions to the Standard Model.

• n2EDM of the nEDM collaboration^[17] is in commissioning phase at the UCN source at the Paul Scherrer Institute.^[17] In October 2025 the experiment was close to physics data taking with neutrons.^[18] The baseline apparatus is expected to reach a sensitivity of 10⁻²⁷ e⋅cm after 500 days of operation.^[19] An upgrade phase `n2EDMagic` is planned with higher sensitivity running at a higher nominal magnetic field value..
• TUCAN, a UCN nEDM experiment under construction at TRIUMF^[20]
• nEDM@SNS experiment under construction (as of 2022) at the Spallation Neutron Source. In late 2023 DOE and NSF funding for the experiment was stopped, which put the experiment to stall.^[21][22][23]
• PNPI nEDM experiment at the Institut Laue-Langevin.^[24]
• PNPI nEDM experiment at the reactor PIK is expected to reach sensitivity of 10⁻²⁷ e⋅cm per year.^[25]
• PanEDM experiment being built at the Institut Laue-Langevin^[26]
• LANL Electric Dipole Moment (LANL nEDM) at Los Alamos National Laboratory^[27]
• Beam EDM at University of Bern, Switzerland^[28]

The Cryogenic neutron EDM experiment or CryoEDM was under development at the Institut Laue-Langevin but its activities were stopped in 2013/2014.^[29]

The electron electric dipole moment d_e is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field:

${\displaystyle U=-\mathbf {d} _{\rm {e}}\cdot \mathbf {E} .}$

The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin).^[30] Within the Standard Model, such a dipole is predicted to be non-zero but very small, at most 10⁻³⁸ e⋅cm,^[31] where e stands for the elementary charge. The discovery of a substantially larger electron electric dipole moment would imply a violation of both parity invariance and time reversal invariance.^[32][33]

In the Standard Model, the electron EDM arises from the CP-violating components of the CKM matrix. The moment is very small because the CP violation involves quarks, not electrons directly, so it can only arise by quantum processes where virtual quarks are created, interact with the electron, and then are annihilated.^[31][a]

If neutrinos are Majorana particles, a larger EDM (around 10⁻³³ e⋅cm) is possible in the Standard Model.^[31]

Many extensions to the Standard Model have been proposed in the past two decades. These extensions generally predict larger values for the electron EDM. For instance, the various technicolor models predict |d_e| that ranges from 10⁻²⁷ to 10⁻²⁹ e⋅cm.^[34] Some supersymmetric models predict that |d_e| > 10⁻²⁶ e⋅cm^[35] but some other parameter choices or other supersymmetric models lead to smaller predicted values. The present experimental limit therefore eliminates some of these technicolor/supersymmetric theories, but not all. Further improvements, or a positive result,^[36] would place further limits on which theory takes precedence.^[37]

As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that

${\displaystyle \mathbf {d} _{\rm {e}}=\int ({\mathbf {r} }-{\mathbf {r} }_{0})\rho ({\mathbf {r} })d^{3}{\mathbf {r} }}$

depends on the point ${\displaystyle {\mathbf {r} }_{0}}$ about which the moment of the charge distribution ${\displaystyle \rho ({\mathbf {r} })}$ is taken. If we were to choose ${\displaystyle {\mathbf {r} }_{0}}$ to be the center of charge, then ${\displaystyle \mathbf {d} _{\rm {e}}}$ would be identically zero. A more interesting choice would be to take ${\displaystyle {\mathbf {r} }_{0}}$ as the electron's center of mass evaluated in the frame in which the electron is at rest.^[38]

Classical notions such as the center of charge and mass are, however, hard to make precise for a quantum elementary particle. In practice the definition used by experimentalists comes from the form factors ${\displaystyle F_{i}(q^{2})}$ appearing in the matrix element^[39]

${\displaystyle \langle p_{f}|j^{\mu }|p_{i}\rangle ={\bar {u}}(p_{f})\left\{F_{1}(q^{2})\gamma ^{\mu }+{\frac {i\sigma ^{\mu \nu }}{2m_{\rm {e}}}}q_{\nu }F_{2}(q^{2})+i\epsilon ^{\mu \nu \rho \sigma }\sigma _{\rho \sigma }q_{\nu }F_{3}(q^{2})+{\frac {1}{2m_{\rm {e}}}}\left(q^{\mu }-{\frac {q^{2}}{2m_{\text{e}}}}\gamma ^{\mu }\right)\gamma _{5}F_{4}(q^{2})\right\}u(p_{i})}$

of the electromagnetic current operator between two on-shell states with Lorentz invariant phase space normalization in which

${\displaystyle \langle p_{f}\vert p_{i}\rangle =2E(2\pi )^{3}\delta ^{3}({\bf {p}}_{f}-{\bf {p_{i}}}).}$

Here ${\displaystyle u(p_{i})}$ and ${\displaystyle {\bar {u}}(p_{f})}$ are 4-spinor solutions of the Dirac equation normalized so that ${\displaystyle {\bar {u}}u=2m_{\text{e}}}$, and ${\displaystyle q^{\mu }=p_{f}^{\mu }-p_{i}^{\mu }}$ is the momentum transfer from the current to the electron. The ${\displaystyle q^{2}=0}$ form factor ${\displaystyle F_{1}(0)=Q}$ is the electron's charge, ${\displaystyle \mu ={\tfrac {F_{1}(0)\ +\ F_{2}(0)}{2m_{\rm {e}}}}}$ is its static magnetic dipole moment, and ${\displaystyle {\tfrac {-F_{3}(0)}{2m_{\rm {e}}}}}$ provides the formal definition of the electron's electric dipole moment. The remaining form factor ${\displaystyle F_{4}(q^{2})}$ would, if nonzero, be the anapole moment.^[32]

Electron EDMs are usually not measured on free electrons, but instead on bound, unpaired valence electrons inside atoms and molecules. In these, one can observe the effect of ${\displaystyle U=-\mathbf {d} _{\rm {e}}\cdot \mathbf {E} }$ as a slight shift of spectral lines. The sensitivity to ${\displaystyle \mathbf {d} _{\rm {e}}}$ scales approximately with the nuclear charge cubed.^[40] For this reason, electron EDM searches almost always are conducted on systems involving heavy elements.^[35]

To date, no experiment has found a non-zero electron EDM. As of 2020 the Particle Data Group publishes its value as |d_e| < 0.11×10⁻²⁸ e⋅cm. Here is a list of some electron EDM experiments after 2000 with published results:

The ACME collaboration is, as of 2020, developing a further version of the ACME experiment series. The latest experiment is called Advanced ACME or ACME III and it aims to improve the limit on electron EDM by one to two orders of magnitude.^[48][49]

Besides the above groups, electron EDM experiments are being pursued or proposed by the following groups:

• University of Groningen: BaF molecular beam^[50]
• John Doyle (Harvard University), Nicholas Hutzler (California Institute of Technology), and Timothy Steimle (Arizona State University): YbOH molecular trap^[51]
• EDMcubed collaboration, Amar Vutha (University of Toronto), Eric Hessels (York University): oriented polar molecules in an inert gas matrix^[52][53]
• David Weiss (Pennsylvania State University): Cs and Rb atoms trapped inside an optical lattice^[54]
• TRIUMF: Fountain of laser cooled Fr^[55]
• EDMMA collaboration: Cs in an inert gas matrix^[56]

• Electron magnetic moment
• Anomalous electric dipole moment
• Anomalous magnetic dipole moment
• Electric dipole spin resonance
• Parity (physics) § Parity violation
• CP violation
• Charge conjugation
• T-symmetry
• Axion – a hypothetical particle proposed to explain the strong force's unexpected preservation of CP symmetry
• Electric dipole spin resonance
• Nucleon magnetic moment – the corresponding magnetic property, which has been measured