CP violating moments

A particle (left) with a magnetic moment, μ (blue), has an associated spin. The particle may have a permanent electric dipole moment (EDM), d (red), which arises from separation of positive and negative charges. A parity (P) transformation does not change μ but reverses the EDM sign. A time-reversal (T) transforms does not change the EDM but flips μ. The particle could be a composite particle, such as a neutron, or a fundamental particle, such as an electron.

Searching for the violation of time-reversal symmetry (T) is actively pursued to understand the Standard Model of particle physics and to discover new physics beyond the Standard Model. These searches are highly motivated by problems such as Baryogenesis and the Strong CP problem. Assuming CPT symmetry, T violation is equivalent to CP violation - the combined violation of charge conjugation (C) and parity symmetry (P), thus CP violation and T symmetry violation are often used interchangeably.

A powerful means to discover new sources of CP violation is to search for CP violating moments, such as the permanent Electric dipole moments (EDM) of the electron, neutron, or atoms. Searches for these moments is an active frontier of precision measurement physics, as increased accuracy, constraining the magnitude of EDMs to smaller and smaller values, improves constraints on theoretical CP violation within and beyond the Standard Model, helping to rule out new physics theories.

Theory

The paradigm CP violating moment is an EDM, which violates both P and T symmetries. Under time reversal, a particle's magnetic dipole moment changes its direction, whereas the electric dipole moment is 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 if there is an EDM.

Neutron EDM

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 Template:Val.

Standard Model prediction

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: Template:Nowrap.

Matter–antimatter asymmetry

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.

Strong CP problem

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.

SUSY 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 Template:Nowrap and Template:Nowrap. 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.

Experimental technique

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

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

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:

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

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.

History

The history of neutron EDM limits including the latest best result by the nEDM collaboration at PSI. The prediction stemming from the Standard Model is also indicated.

The first experiments searching for the neutron EDM used thermal neutron beams. It started with the experiment by James Smith, Purcell, and Ramsey in 1951 (published in 1957) at ORNL's Graphite Reactor, obtaining a limit of Template:Nowrap 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 is Template:Nowrap.

Experiments with ultracold neutrons (UCN) then took over. The first was done in 1980 with an experiment at the Template:Ill (LNPI) obtaining a limit of Template:Nowrap. 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, and later revised in 2015.

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

The latest best limit of Template:Nowrap was published in 2020 by the nEDM collaboration at Paul Scherrer Institute (PSI).

Current experiments

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 Template:Nowrap over the next 10 years, thereby covering the range of prediction coming from supersymmetric extensions to the Standard Model.

• n2EDM of the nEDM collaboration is in commissioning phase at the UCN source at the Paul Scherrer Institute. In October 2025 the experiment was close to physics data taking with neutrons. The baseline apparatus is expected to reach a sensitivity of Template:Nowrap after 500 days of operation. 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
• 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.
• PNPI nEDM experiment at the Institut Laue-Langevin.
• PNPI nEDM experiment at the reactor PIK is expected to reach sensitivity of Template:Nowrap per year.
• PanEDM experiment being built at the Institut Laue-Langevin
• LANL Electric Dipole Moment (LANL nEDM) at Los Alamos National Laboratory
• Beam EDM at University of Bern, Switzerland

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

Nuclear EDM

The nuclear electric dipole moment (nuclear EDM) ${𝐝}_{\text{e}}^N$ is an intrinsic property of a nucleus 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 in both light and heavy nuclei.

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 ${\overline{\theta }}_{\text{QCD}}$ 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.

Schiff theorem

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. 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 $Z$, 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). The Schiff moment may be derived from a Taylor expansion of the classical electrostatic interaction between an electron and a nuclear charge density.

Measurements

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

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

where ${𝐦}_N$ 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 $𝐄$ and subtracting the two corresponding Larmor frequencies leads to a result that is proportional to $d_{\text{e}}^N$. 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.

List of Nuclear EDM Results

Year	Location	Principal Investigators	Method	Species	Experimental upper limit on Template:Abs
1989	Yale University	Edward Hinds	Molecular beam	TlF	Template:Val
2016	Argonne National Laboratory	Zheng-Tian Lu, Roy J. Holt, Peter Mueller, Jaideep T. Singh	Atomic beam	225Ra	Template:Val
2016	University of Washington, Seattle	Blayne Heckel	Vapor Cell	199Hg	Template:Val
2019	University of Michigan, Ann Arbor	Timothy E. Chupp, Lars Trahms, Peter Fierlinger	Vapor Cell	129Xe	Template:Val
2022	University of Science and Technology of China	Zheng-Tian Lu, Tian Xia	Magneto-optical trap	171Yb	Template:Val

Electron EDM

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

$U=-{𝐝}_{\mathrm{e}}\cdot 𝐄.$

The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin). Within the Standard Model, such a dipole is predicted to be non-zero but very small, at most Template:Nowrap, 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.

Implications for Standard Model and extensions

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.Template:Efn

If neutrinos are Majorana particles, a larger EDM (around Template:Val) is possible in the Standard Model.

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 Template:Abs that ranges from 10⁻²⁷ to 10⁻²⁹ e⋅cm. Some supersymmetric models predict that Template:Nowrap 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, would place further limits on which theory takes precedence.

Measurements of the electron's electric dipole moment.

Formal definition

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

${𝐝}_{\mathrm{e}}=\int (𝐫-{𝐫}_0)\rho (𝐫)d^3𝐫$

depends on the point ${𝐫}_0$ about which the moment of the charge distribution $\rho (𝐫)$ is taken. If we were to choose ${𝐫}_0$ to be the center of charge, then ${𝐝}_{\mathrm{e}}$ would be identically zero. A more interesting choice would be to take ${𝐫}_0$ as the electron's center of mass evaluated in the frame in which the electron is at rest.

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 $F_i(q^2)$ appearing in the matrix element

$⟨p_f|j^{\mu }|p_i⟩=\overline{u}(p_f)\{F_1(q^2){\gamma }^{\mu }+\frac{i{\sigma }^{\mu \nu }}{2m_{\mathrm{e}}}{q}_{\nu }{F}_2(q^2)+i{ϵ}^{\mu \nu \rho \sigma }{\sigma }_{\rho \sigma }{q}_{\nu }{F}_3(q^2)+\frac{1}{2m_{\mathrm{e}}}(q^{\mu }-\frac{q^2}{2m_{\text{e}}}{\gamma }^{\mu }){\gamma }_5{F}_4(q^2)\}u(p_i)$

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

$⟨p_f|p_i⟩=2E(2\pi {)}^3{\delta }^3({𝐩}_f-{𝐩}_{𝐢}).$

Here $u(p_i)$ and $\overline{u}(p_f)$ are 4-spinor solutions of the Dirac equation normalized so that $\overline{u}u=2m_{\text{e}}$, and $q^{\mu }=p_f^{\mu }-p_i^{\mu }$ is the momentum transfer from the current to the electron. The $q^2=0$ form factor $F_1(0)=Q$ is the electron's charge, $\mu =\frac{F_1(0)+F_2(0)}{2m_{\mathrm{e}}}$ is its static magnetic dipole moment, and $\frac{-F_3(0)}{2m_{\mathrm{e}}}$ provides the formal definition of the electron's electric dipole moment. The remaining form factor $F_4(q^2)$ would, if nonzero, be the anapole moment.

Experimental measurements

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 $U=-{𝐝}_{\mathrm{e}}\cdot 𝐄$ as a slight shift of spectral lines. The sensitivity to ${𝐝}_{\mathrm{e}}$ scales approximately with the nuclear charge cubed. For this reason, electron EDM searches almost always are conducted on systems involving heavy elements.

To date, no experiment has found a non-zero electron EDM. As of 2020 the Particle Data Group publishes its value as Template:Nowrap. Here is a list of some electron EDM experiments after 2000 with published results:

List of Electron EDM Results

Year	Location	Principal Investigators	Method	Species	Experimental upper limit on Template:Abs
1958	Cornell University	Edwin E. Salpeter	Lamb shift	H	Template:Val
1963	Harvard University	Nicolaas Bloembergen	Crystal	Cr:Al2O3	Template:Val
2002	University of California, Berkeley	Eugene Commins, David DeMille	Atomic beam	Tl	Template:Val
2011	Imperial College London	Edward Hinds, Ben Sauer	Molecular beam	YbF	Template:Val
2013	Yale University	David DeMille	Vapor Cell	PbO	Template:Val
2014	Harvard-Yale (ACME I experiment)	David DeMille, John Doyle, Gerald Gabrielse	Molecular beam	ThO	Template:Val
2017	JILA	Eric Cornell, Jun Ye	Ion trap	HfF+	Template:Val
2018	Harvard-Yale (ACME II experiment)	David DeMille, John Doyle, Gerald Gabrielse	Molecular beam	ThO	Template:Val
2022	JILA	Eric Cornell, Jun Ye	Ion trap	HfF+	Template:Val

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.

See also

• Electron magnetic moment
• Anomalous electric dipole moment
• Anomalous magnetic dipole moment
• Electric dipole spin resonance
• Template:Slink
• 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