Fundamental Symmetry Frontiers

Status: parity violation in weak interactions, atomic parity violation, and the use of reversals to isolate symmetry-sensitive observables are established. Searches for permanent electric dipole moments, nuclear Schiff and magnetic quadrupole moments, improved nuclear-spin-dependent parity violation, and new boson-mediated interactions are active. No permanent electric dipole moment, molecular weak parity-violating energy difference, nuclear Schiff moment, nuclear magnetic quadrupole moment, or fifth force has been established. Particular beyond-Standard-Model interpretations of present null results are model dependent; claims that one AMO bound identifies the origin of the cosmic matter excess are conjectural.

Last reviewed: 26 July 2026. Numerical limits, confidence levels, theory response coefficients, publication status, and projected sensitivities are date-sensitive. A tighter exclusion curve is not a discovery, and an observed instrumental nonlinearity is not evidence for a new interaction until Standard Model and apparatus explanations have been bounded.

Core Question

Which violations of parity, time reversal, and related low-energy symmetries can atomic, molecular, optical, nuclear-spin, and storage-ring measurements resolve, and what evidence is required to distinguish a fundamental interaction from an ordinary field, a nuclear-structure effect, or a model-dependent reinterpretation?

The frontier is not summarized by one smallest number. It asks whether a measurement can:

define a symmetry-sensitive observable;
project that observable into a calibrated reversal channel;
reject ordinary effects with the same channel parity;
calculate the system response with an uncertainty and convention;
separate several plausible microscopic operators;
construct intervals with demonstrated statistical coverage; and
reproduce a candidate effect in systems with complementary responses.

A trustworthy result preserves the distinction among four statements:

measurement: a fitted phase, frequency, asymmetry, force, or isotope shift;
system-level inference: an atomic or molecular energy coefficient, particle moment, or effective field;
low-energy interpretation: a bound on one or more effective couplings; and
ultraviolet interpretation: a constraint on a particle model, mediator mass, phase, or symmetry-breaking scale.

The first statement can be experimentally direct. Each later statement adds response theory and assumptions.

Why It Matters

Symmetry tests can expose interactions hidden at high energy

Small low-energy observables can be sensitive to virtual heavy particles, light weakly coupled bosons, or interactions suppressed by several powers of a high scale. Long coherence, repeated trials, large internal molecular fields, near-degenerate opposite-parity states, and accurately calculable response coefficients can compensate for tiny couplings.

This complementarity is real but conditional. A schematic dimension-six contribution may scale as

\Delta E \sim \frac{ C\,\mathcal R }{ \Lambda^2 },

where $\mathcal R$ is a system-dependent response, $C$ contains couplings and phases, and $\Lambda$ is a heavy scale. A null result constrains $C/\Lambda^2$ , not $\Lambda$ alone. Loop order, cancellations, flavour structure, renormalization-group running, and operator mixing can all change the inferred scale.

Electric dipole moments probe additional CP violation

A permanent spin-aligned electric dipole moment of a nondegenerate stationary system is odd under parity $P$ and time reversal $T$ . Under the usual assumptions of local relativistic quantum field theory and CPT invariance, such $T$ violation corresponds to CP violation.

The Standard Model contains CP violation, but its predicted flavour-diagonal electron, neutron, atomic, and molecular EDM signals are far below present sensitivity. The observed cosmic matter–antimatter asymmetry motivates additional CP-violating sources. EDM experiments strongly constrain many such sources, but there is no one-to-one implication:

a null EDM does not rule out all baryogenesis mechanisms;
a nonzero EDM would not identify one operator by itself;
several microscopic sources can cancel in one system; and
a successful baryogenesis model must satisfy cosmological and collider constraints in addition to EDM bounds.

Parity violation is both signal and calibration

Weak-interaction parity violation is established. Atomic parity-violation experiments therefore test the magnitude, isotope dependence, spin dependence, and many-body interpretation of a known small amplitude. They can constrain weak charges, nuclear anapole moments, neutron distributions, and new parity-violating interactions.

This differs logically from an EDM null test. The parity-violating amplitude is expected to be nonzero, so agreement with the Standard Model can validate both electroweak and atomic-structure calculations. A discrepancy must still survive radiative, nuclear, and many-electron corrections before it is assigned to a new boson.

Nuclear moments connect several theory layers

Atoms and molecules can translate hadronic CP violation into an electronic or spectroscopic observable. The path may include:

\begin{gathered} \text{quark and gluon operators} \longrightarrow \text{nucleon and pion couplings} \\ \longrightarrow \text{nuclear Schiff or magnetic quadrupole moment} \longrightarrow \text{atomic or molecular energy} \\ \longrightarrow \text{measured phase or frequency}. \end{gathered}

Each arrow has a convention, a calculation, and an uncertainty. Octupole deformation and close opposite-parity nuclear levels may greatly enhance a response, but enhancement does not remove nuclear-model dependence.

New bosons need broad mass and operator coverage

A boson with mass $m_b$ mediates an interaction with characteristic range

\lambda_b = \frac{\hbar}{m_b c}.

Sub-micrometre spectroscopy, centimetre-scale spin sensors, terrestrial baselines, and astronomical systems therefore probe different mediator-mass ranges and coupling combinations. Scalar, pseudoscalar, vector, and axial-vector couplings generate different spin, velocity, parity, and source dependences. No single “fifth-force strength” represents all of them.

Canonical Boundary

This page owns the dated cross-program frontier assessment: current experimental status, emerging platforms, disputed interpretations, evidence standards, and representative literature. It does not repeat the settled derivations in their canonical homes.

Tests of Fundamental Symmetries owns EDM and atomic-parity-violation Hamiltonians, switch-parity estimators, Schiff screening, systematics, representative system-level bounds, and the full observable-to-operator inference chain.
Precision Molecular Measurements owns body-to-laboratory orientation, molecular effective fields, parity-doublet sensors, electron- and nuclear-response coefficients, chiral comparisons, and molecular calibration.
Precision Measurement and Metrology owns measurands, covariance, blinding, calibration, uncertainty budgets, and traceability.
Magnetometry owns spin transduction, comagnetometers, shielding, transfer functions, gradients, and magnetic systematic effects.
Variation of Constants Searches owns drift, oscillation, and transient searches for dimensionless constants and ultralight scalar-field signal models.
Parity and Time Reversal own the operators and transformation laws. From Discrete Symmetries to CPT owns the assumptions relating $T$ , CP, and CPT.
Cold Molecule Frontiers owns the dated production, cooling, trapping, qubit, and dipolar-matter status of molecular platforms.
Precision AMO Frontiers owns cross-platform sensor and new-force comparisons. Optical Clock Frontiers owns clock-network and clock-based ultralight-field status.

This page may quote a record limit to orient the frontier. The canonical method page retains the derivation, sign convention, detailed systematic ledger, and reusable response formula.

Minimal Background

The five-layer inference chain

Let $\mathbf z$ denote detector-level data, $\widehat{\mathbf y}$ fitted system observables, $\mathbf c$ low-energy coefficients, and $\boldsymbol\theta$ parameters of a more fundamental model. A compact inference chain is

\mathbf z \xrightarrow{ \text{calibration and fit} } \widehat{\mathbf y} \xrightarrow{ \text{response theory} } \mathbf c \xrightarrow{ \text{matching and running} } \boldsymbol\theta.

Examples of $\widehat{\mathbf y}$ include:

an electric-field-odd Ramsey frequency;
an APV interference amplitude;
a storage-ring spin-axis tilt;
a hyperfine parity-mixing matrix element;
a force at the source-modulation frequency; or
an isotope-shift residual after mass and field shifts are fitted.

The observable model can be written

\widehat{\mathbf y} = K\mathbf c + B\boldsymbol\eta + \boldsymbol\epsilon,

where $K$ is the signal-response matrix, $B$ maps nuisance parameters $\boldsymbol\eta$ into measured channels, and $\boldsymbol\epsilon$ is residual noise. The matrix $K$ contains atomic, molecular, nuclear, or source-geometry calculations. It is not generally exact or diagonal.

Symmetry signatures are operator specific

Observable or interaction	Leading $P$ character	Leading $T$ character	Typical AMO route
permanent spin-aligned EDM	odd	odd	electric-field-correlated spin precession
nuclear Schiff moment	odd	odd	diamagnetic atom or polar molecule
nuclear magnetic quadrupole moment	odd	odd	open-shell molecule with nuclear spin
nuclear-spin-independent APV	odd	even	weak–Stark or weak–M1 interference
nuclear anapole contribution	odd	even	hyperfine-dependent APV
scalar Yukawa interaction	even	even	spectroscopy, force, or equivalence test
pseudoscalar dipole–dipole interaction	model dependent as a full exchange potential	model dependent	polarized source and spin sensor
axial-vector–vector interaction	often parity odd	coupling and kinematics dependent	APV or spin-dependent force search

The last rows cannot be classified by the mediator name alone. The nonrelativistic potential, source polarization, relative velocity, and chosen couplings determine the laboratory signature.

An EDM channel

For a spin degree of freedom in parallel electric and magnetic fields, a schematic Hamiltonian is

H = -\boldsymbol\mu\cdot\mathbf B -\mathbf d\cdot\mathbf E.

For two spin projections and a calibrated effective electric field $E_{\mathrm{eff}}$ , a common molecular convention gives

\omega_{\mathrm{EDM}} = \frac{ 2d\,\mathcal P E_{\mathrm{eff}} }{ \hbar },

where $\mathcal P$ is the laboratory polarization or orientation factor. The exact factor of two and orientation sign depend on the experiment’s state definitions.

An electric-field reversal is not the antiunitary $T$ transformation. It is a laboratory switch used to project the coefficient with the same sign pattern as the EDM term. Leakage-current fields, motional fields, geometric phases, correlated gradients, and readout nonlinearities can share that sign pattern.

Switch-parity projection

For binary switches $s_q=\pm1$ , write a measured frequency as

f(\mathbf s) = \sum_S f^S \prod_{q\in S}s_q.

The desired coefficient $f^{S_\star}$ is isolated by a balanced projection,

\widehat f^{S_\star} = \frac{1}{2^n} \sum_{\mathbf s} \left( \prod_{q\in S_\star}s_q \right) f(\mathbf s).

Imbalance, drift, missing switch states, and switch-correlated changes make different parity channels nonorthogonal. A mature analysis fits the full design matrix, reports channel covariance, and tests injected signals.

Atomic parity-violation interference

The nuclear-spin-independent weak charge is approximately

Q_W \simeq -N + Z \left( 1-4\sin^2\theta_W \right),

before radiative and nuclear corrections. A weak interaction mixes opposite-parity electronic states and produces a small amplitude $M_{\mathrm{PV}}$ . Interference with a controlled parity-conserving amplitude $M_{\mathrm{PC}}$ gives

\mathcal A_{\mathrm{PV}} \simeq \frac{ 2\operatorname{Re} \left( M_{\mathrm{PC}}^*M_{\mathrm{PV}} \right) }{ |M_{\mathrm{PC}}|^2 }.

Extracting $Q_W$ requires atomic many-body theory, electroweak radiative corrections, and nuclear inputs. Isotope ratios can cancel some electronic factors but introduce neutron-skin and isotope-dependent nuclear structure.

Nuclear moments as response coefficients

A molecular energy can contain several symmetry-violating terms,

\Delta E_i = K_{ie}d_e + K_{iS}\mathcal S + K_{iM}\mathcal M + \sum_a K_{ia}C_a.

Here $\mathcal S$ is a nuclear Schiff moment, $\mathcal M$ a nuclear magnetic quadrupole moment, and $C_a$ other electron–nucleus or hadronic coefficients. The $K$ values depend on the state and convention. An enhancement quoted for one term does not guarantee a well-conditioned fit to all terms.

King plots and new bosons

For isotope pair $A,A'$ , a leading isotope-shift model for transition $i$ is

\delta\nu_i^{AA'} = K_i\mu^{AA'} + F_i\delta\langle r^2\rangle^{AA'} + X_i h^{AA'} + \cdots,

where $\mu^{AA'}$ is a known mass factor, $F_i$ is a field-shift coefficient, and $X_i h^{AA'}$ represents a possible new electron–neutron interaction in one chosen model. Eliminating the leading nuclear-radius term between two transitions produces a linear King relation.

Nonlinearity can also arise from:

second-order mass and field shifts;
higher nuclear moments;
nuclear deformation and polarization;
imperfect nuclear masses;
unresolved line structure; and
correlated spectroscopic systematics.

A nonlinear King plot is therefore an anomaly in a reduced model, not by itself a fifth-force detection.

Static forces and oscillating fields are different searches

A static boson-exchange potential depends on source geometry and range. A generic spin-independent form is

V_\phi(r) = \frac{ \hbar c\,g_1g_2 }{ 4\pi } \frac{e^{-r/\lambda_b}}{r},

with sign and normalization fixed by the chosen interaction Lagrangian. Spin-dependent potentials add combinations of $\boldsymbol\sigma_1$ , $\boldsymbol\sigma_2$ , $\widehat{\mathbf r}$ , and relative velocity.

An ultralight dark-matter field instead supplies a coherent time-dependent background, schematically

\phi(t,\mathbf x) \simeq \phi_0 \cos \left( \omega_\phi t -\mathbf k_\phi\cdot\mathbf x +\varphi \right).

The static-force and dark-field analyses use different source assumptions, coherence models, scan penalties, and nuisance rejection. Limits from one should not be relabelled as limits from the other.

Discovery requires complementary response directions

For two candidate coefficients $c_1,c_2$ and two experiments,

\begin{pmatrix} y_1\\ y_2 \end{pmatrix} = \begin{pmatrix} K_{11} & K_{12}\\ K_{21} & K_{22} \end{pmatrix} \begin{pmatrix} c_1\\ c_2 \end{pmatrix}.

If the response rows are nearly parallel, the determinant is small and the inverse problem is ill conditioned even when each $y_i$ is precise. Complementary atoms, molecules, nuclei, and source geometries identify operators by rotating the response direction, not merely by repeating the same sensitivity.

What Is Known

Parity violation is established; these experiments test its detailed form

The charged-current and neutral-current weak interactions violate parity. Atomic parity violation (APV) is therefore not a search for whether the weak interaction exists. It is a precision test of the weak charge, nuclear-spin dependence, isotope dependence, and possible additional parity-violating interactions.

The cesium measurement by Wood et al. remains a benchmark because cesium combines a precise parity-violating transition amplitude with mature many-body theory. The comparison constrains the nuclear weak charge and electroweak physics at low momentum transfer. It also produced evidence for the nuclear anapole contribution, although extracting hadronic weak couplings from that contribution remains substantially more theory dependent than extracting a spin-independent weak charge.

In ytterbium, Antypas et al. measured APV across four isotopes. The parity-violating amplitude changed with neutron number as expected from the Standard Model within the achieved precision, with approximately $0.5\%$ precision for three isotopes. That isotope-chain result is important for two distinct reasons:

the variation is a within-element differential test, so much electronic structure is common;
neutron-distribution and other isotope-dependent nuclear effects do not vanish and must be included before attributing a residual to new physics.

For a transition with a dominant parity-conserving amplitude $A_{\mathrm{PC}}$ and a weak amplitude $A_{\mathrm{PV}}$ , the leading observable is an interference term,

\left|A_{\mathrm{PC}}+A_{\mathrm{PV}}\right|^2 \simeq \left|A_{\mathrm{PC}}\right|^2 +2\operatorname{Re} \left( A_{\mathrm{PC}}^*A_{\mathrm{PV}} \right).

Its sign under field and polarization reversals is part of the observable. A nonzero fitted interference channel is persuasive only after ordinary amplitudes with the same reversal signature have been measured or bounded.

Established conclusion: atomic parity-violating amplitudes have been observed, and benchmark APV measurements are consistent with the Standard Model at their stated precision. This conclusion does not imply that every nuclear-spin-dependent contribution or isotope-dependent correction has already been isolated.

No permanent electric dipole moment has been established

A permanent electric dipole moment (EDM) of a nondegenerate stationary state aligned with its angular momentum violates parity and time reversal. Subject to the usual assumptions of local relativistic quantum field theory, the CPT connection then relates it to CP violation.

As of 26 July 2026, all mature EDM programs report results consistent with zero. Representative published results are:

Sector and system	Published result	Confidence level	Correct interpretation
paramagnetic molecule, trapped $^{180}\mathrm{Hf}^{19}\mathrm{F}^+$	$d_e=(-1.3\pm2.0_{\rm stat}\pm0.6_{\rm syst})\times10^{-30}\ e\,\mathrm{cm}$ ; $\lvert d_e\rvert<4.1\times10^{-30}\ e\,\mathrm{cm}$	$90\%$	Leading published electron-EDM limit under the stated one-source interpretation
paramagnetic molecule, ThO beam	$\lvert d_e\rvert<1.1\times10^{-29}\ e\,\mathrm{cm}$	$90\%$	Independent molecular-beam constraint with different apparatus and systematics
ultracold neutron	$d_n=(0.0\pm1.1_{\rm stat}\pm0.2_{\rm syst})\times10^{-26}\ e\,\mathrm{cm}$ ; $\lvert d_n\rvert<1.8\times10^{-26}\ e\,\mathrm{cm}$	$90\%$	Direct free-neutron EDM constraint
diamagnetic $^{199}\mathrm{Hg}$ atom	$\lvert d_{\rm Hg}\rvert<7.4\times10^{-30}\ e\,\mathrm{cm}$	$95\%$	Atomic EDM limit requiring atomic and nuclear response theory for microscopic interpretation
radioactive $^{225}\mathrm{Ra}$ atom	$\lvert d_{\rm Ra}\rvert<1.4\times10^{-23}\ e\,\mathrm{cm}$	$95\%$	Direct atomic result in an octupole-deformed-nucleus program; statistically limited
stored deuteron	$\lvert d^d\rvert<2.5\times10^{-17}\ e\,\mathrm{cm}$	$95\%$	First direct deuteron EDM limit and a storage-ring feasibility milestone, not yet competitive with indirect hadronic constraints

These numbers must not be ordered by their powers of ten as if they measured the same quantity. An atomic EDM, a free-neutron EDM, a molecular frequency, and a stored-deuteron spin rotation have different dimensions only after conventions are applied, different response coefficients, and sensitivity to different combinations of low-energy operators.

The current electron-EDM benchmark is conditional

Roussy et al. used trapped HfF $^+$ ions to obtain the 2023 result in the table. Its central value is statistically consistent with zero. The quoted $d_e$ interval assumes that the measured paramagnetic signal is attributed to the electron EDM rather than to a simultaneous scalar–pseudoscalar electron–nucleon interaction or another operator.

For a paramagnetic species $a$ , a more complete leading description is

\Delta E_a = W_{d,a}\,d_e +W_{S,a}\,C_S +\sum_k W_{k,a}\,c_k.

Here $W_{d,a}$ is the electron-EDM response, $W_{S,a}$ is the scalar–pseudoscalar response, and the omitted coefficients represent further operators allowed by the analysis. Reporting a one-dimensional bound on $d_e$ corresponds to setting the other $c_k$ to zero or constraining them externally. That projection is useful and conventional, but it is not an operator-independent measurement of $d_e$ .

The ThO result from the ACME collaboration is weaker numerically but remains essential evidence. It uses a distinct molecule, source, state preparation, interaction region, reversal implementation, and systematic-error budget. If a nonzero HfF $^+$ -like signal were to appear, agreement with ThO or another paramagnetic system after response matching would be a central discovery test.

Null results are not measurements of zero

An interval containing zero does not prove that the true moment vanishes. It states what parameter values are compatible with a specified likelihood, nuisance model, and interval construction. A frequentist confidence procedure is calibrated through repeated-sampling coverage,

\Pr_{\theta} \left[ \theta\in C(X) \right] \geq 1-\alpha,

for parameter values $\theta$ in the procedure’s stated domain. This is not the posterior probability that a particular realized interval contains $\theta$ .

The distinction matters near a physical boundary such as $\lvert d\rvert\geq0$ , when combining statistical and systematic uncertainties, and when a collaboration blinds the symmetry channel. A frontier comparison should preserve:

the signed central estimate when available;
statistical and systematic components;
the confidence or credible level;
the interval construction;
the operator assumptions; and
whether theory-response uncertainty is included.

Neutron, atomic, and nuclear systems probe different CP-odd directions

The neutron EDM is a direct hadronic observable, but its interpretation in terms of quark EDMs, chromo-EDMs, the QCD $\bar\theta$ parameter, or four-fermion operators requires strong-interaction theory. Diamagnetic atoms add atomic screening and nuclear response. A schematic chain is

\mathbf c_{\rm UV} \longrightarrow \mathbf c_{\rm had} \longrightarrow \begin{pmatrix} d_n\\ \bar g_0\\ \bar g_1\\ \bar g_2\\ \vdots \end{pmatrix} \longrightarrow \begin{pmatrix} S_A\\ M_A\\ \vdots \end{pmatrix} \longrightarrow \Delta E_{\rm atom}.

The CP-odd pion–nucleon couplings $\bar g_i$ , a nuclear Schiff moment $S_A$ , and a nuclear magnetic quadrupole moment $M_A$ are intermediate response quantities, not directly interchangeable observables.

The 2020 ultracold-neutron result by Abel et al. remains a leading direct free-neutron constraint. The 2016 mercury result by Graner et al. remains a leading diamagnetic-atom constraint. Their response vectors are complementary: a model that suppresses one observable need not suppress the other.

For a set of measured observables $\mathbf y$ , the correct joint problem is

\mathbf y = \mathbf K_{\rm atom} \mathbf K_{\rm nuc} \mathbf K_{\rm had} \mathbf c +\boldsymbol\epsilon,

with correlated experimental and theory covariance. A table of separate one-operator limits is not equivalent to a simultaneous fit of $\mathbf c$ .

The direct deuteron result is a milestone, not the strongest hadronic bound

In 2026, Andres et al. reported the first direct deuteron EDM search in a storage ring. The experiment constrained

\lvert d^d\rvert < 2.5\times10^{-17}\ e\,\mathrm{cm} \qquad (95\%\ {\rm confidence}).

The result demonstrated spin preparation, long storage, polarimetry, field control, and an EDM-sensitive spin observable in a charged-particle ring. Spin-axis tilts were a dominant systematic issue. The scientific milestone is therefore direct access to a charged hadron’s EDM in a storage ring and the validation of the experimental architecture.

It would be misleading to describe this first result as superseding neutron, atomic, or molecular constraints on hadronic CP violation. Its numerical sensitivity is many orders of magnitude above the long-term target and above indirect constraints in common effective-field-theory scenarios. Future storage-ring reach depends on frozen-spin control, radial electric fields, magnetic-field rejection, beam dynamics, polarimetry, and counter-rotating-beam comparisons.

Enhanced nuclear moments remain targets rather than detections

Octupole-deformed nuclei and certain molecular electronic states can enhance response to CP-odd nuclear moments. Enhancement is a property of the susceptibility,

\Delta E = W_S S +W_M M +\cdots,

not evidence that either $S$ or $M$ is nonzero.

The $^{225}$ Ra atomic EDM search is a direct result in an octupole-deformed nucleus. Its present limit is statistically limited. Radioactive molecules such as RaF and AcF may combine nuclear enhancement with large molecular polarization and internal-state reversals, but no Schiff moment or nuclear magnetic quadrupole moment has been observed.

Two recent molecular achievements sharpen the inputs without changing that null conclusion:

Wilkins et al. observed the nuclear magnetization-distribution, or Bohr–Weisskopf, effect in RaF in 2025. This calibrates nuclear and molecular structure; it is not a CP-violating signal.
Athanasakis-Kaklamanakis et al. produced and spectroscopically characterized gas-phase $^{227}$ AcF in 2025. Calculated enhancement factors motivate future searches; the work did not measure a CP-odd moment.

This distinction is especially important in radioactive systems, where scarce yields make every spectroscopy milestone scientifically substantial. Production, identification, state assignment, laser cycling, polarization, coherence, and a symmetry measurement are separate rungs on the evidence ladder.

No molecular weak parity-violating energy difference has been observed

Nuclear-spin-independent weak interactions can produce an exceedingly small energy difference between enantiomers of a chiral molecule. Calculations and spectroscopic strategies are active, but no molecular parity-violating enantiomeric energy difference has been established.

Nuclear-spin-dependent parity violation is a different target. It includes contributions from electron–nucleon neutral currents, nuclear anapole moments, and the interplay of nuclear-spin-independent weak interactions with hyperfine structure. Near-degenerate opposite-parity molecular levels can enhance the measurable mixing.

Altuntaş et al. demonstrated a Stark-interference method in BaF and constrained an NSD-PV-sensitive matrix element without claiming a nonzero weak signal. Laser cooling of $^{137}$ BaF by Kogel et al. in 2025 prepared up to 112 addressed internal levels and advanced the platform. A 2024 Penning-trap proposal by Karthein et al. described enhanced sensitivity in a single molecular ion. The first is a method result, the second is a control milestone, and the third is a proposal. None is an observation of molecular NSD-PV.

Fifth-force and new-boson searches have set limits, not found a force

AMO experiments constrain new bosons through several observables:

isotope shifts and King-plot nonlinearities;
spin-dependent energy shifts;
comagnetometer and spin-amplifier signals;
parity-violating amplitudes;
spectroscopy at several internuclear separations; and
time-dependent responses to an ultralight background.

No new boson-mediated fifth force has been established in these channels.

Calcium isotope-shift nonlinearity is compatible with known physics

Wilzewski et al. compared isotope shifts in Ca $^{14+}$ and Ca $^+$ and used precise nuclear mass ratios to test King linearity. They observed a nonlinearity, but its size and structure were compatible with higher-order Standard Model and nuclear-polarization effects. After accounting for those effects, the data strengthened constraints on candidate bosons over an approximate mass range

10\ {\rm eV}/c^2 \lesssim m_\phi \lesssim 10^7\ {\rm eV}/c^2.

The correct headline is therefore improved constraints after resolving a Standard Model-sized nonlinearity, not evidence for a fifth force.

For transitions $i$ and isotopes $A,A'$ , a useful residual is

r_i^{AA'} = m\mu_i^{AA'} -F_{ij}\,m\mu_j^{AA'} -K_{ij},

where $m\mu_i^{AA'}$ denotes a chosen modified isotope shift. A nonzero $r_i^{AA'}$ only says that the two-parameter linear model is insufficient. Candidate causes include higher-order field shifts, nuclear polarization, mass uncertainties, line-shape bias, isotope-dependent experimental effects, and a new interaction.

Spin-dependent searches constrain several interaction classes

Polarized sources and spin sensors can test monopole–dipole, dipole–dipole, spin–velocity, and other potentials. The mapping from a measured field-like signal to a coupling product depends on source geometry, spin density, shielding, mediator range, velocity distribution, and the chosen potential convention.

Recent examples include:

magnetic amplification in a polarized gas used by Su et al. to constrain axion-mediated spin interactions;
a SmCo $_5$ polarized source and self-compensating comagnetometer used by Xu et al., with exceptionally strong magnetic rejection, to improve constraints; and
reinterpretation of BaF and cesium parity-violation data by Gaul, Cong, and Budker as constraints on a vector boson.

All are exclusion results. None reported a new interaction.

Complementarity is already experimentally useful

Even in the absence of a discovery, combining systems prevents a misleading one-parameter story. Suppose two observables depend on the electron EDM and a scalar–pseudoscalar coefficient:

\begin{pmatrix} y_a\\ y_b \end{pmatrix} = \begin{pmatrix} W_{d,a} & W_{S,a}\\ W_{d,b} & W_{S,b} \end{pmatrix} \begin{pmatrix} d_e\\ C_S \end{pmatrix} +\begin{pmatrix} \epsilon_a\\ \epsilon_b \end{pmatrix}.

The determinant

D = W_{d,a}W_{S,b} -W_{S,a}W_{d,b}

measures response complementarity. Better precision shrinks uncertainty along each measured direction; a larger $\lvert D\rvert$ resolves the two coefficient directions. A new species is most valuable when it adds both precision and a sufficiently different response ratio.

The same logic governs joint neutron–nucleus analyses, combinations of spin-dependent force geometries, and isotope-shift programs. The frontier is therefore moving from isolated “best limits” toward shared likelihoods, response matrices, and global low-energy fits.

What Is Actively Developing

Next-generation electron-EDM experiments are becoming coherence experiments

The immediate electron-EDM frontier is not only larger effective electric field. It is the simultaneous optimization of polarization, coherence time, state readout, count rate, reversal completeness, and response complementarity.

For $N$ detected particles with contrast $C$ , interrogation time $\tau$ , and effective EDM field $\mathcal E_{\rm eff}$ , a projection-noise scale is

\delta d_e \sim \frac{\hbar} {2C\mathcal E_{\rm eff}\tau\sqrt N}.

This expression is a design scale, not a promised confidence interval. It omits dead time, technical phase noise, imperfect polarization, state leakage, correlations among shots, and systematics that do not average as $N^{-1/2}$ .

Ultracold YbF beam interferometry

Jenkins et al. reported a complete spin-interferometry sequence in a beam of ultracold YbF molecules in 2026. The work combines preparation, coherent spin evolution, field control, and readout in the apparatus intended for a future electron-EDM search. Slower molecules can increase the interaction time while preserving a beam geometry with spatially separated preparation and detection.

The relevant phase is schematically

\phi = \frac{\tau}{\hbar} \left( 2d_e\mathcal E_{\rm eff} +2\mu B_{\rm eff} +\Delta_{\rm geo} +\Delta_{\rm ctrl} \right).

The EDM channel is selected by molecular orientation and laboratory-field reversals. The 2026 paper demonstrated the interferometer; it did not report a new electron-EDM limit. Future reach depends on molecular flux, interaction time, magnetic gradients, geometric phases, state-dependent trajectories, and the full switch correlation matrix.

Engineered clock transitions in YbOH

Takahashi et al. demonstrated engineered molecular clock transitions in YbOH in 2026. At selected avoided crossings, the transition frequency was made at least about 700 times less sensitive to electric-field noise and about 200 times less sensitive to magnetic-field noise while retaining electron-EDM sensitivity.

For a control parameter $x$ , a clock point satisfies

\left. \frac{\partial\omega_{ab}}{\partial x} \right|_{x=x_0} \simeq 0,

whereas the desired symmetry response remains

\left. \frac{\partial\omega_{ab}}{\partial d_e} \right|_{x=x_0} \neq 0.

This is a response-engineering result: the nuisance tangent direction is suppressed without eliminating the signal direction. The demonstrated suppression factors improve the control architecture, but they are not an EDM measurement or limit.

Polyatomic molecules also provide closely spaced parity-doublet structures, multiple internal co-magnetometer channels, and prospects for laser cooling and trapping. Their denser spectra create additional responsibilities: state assignment, off-resonant coupling, tensor shifts, avoided-crossing tracking, and leakage diagnostics must be incorporated into the analysis.

Trapped molecular ions and renewed beam experiments

Trapped molecular ions offer long interrogation and repeated state preparation, while neutral beams offer large ensembles and clean spatial separation of stages. Neither architecture is universally superior.

An experiment-level figure of merit is better written as a vector,

\mathcal F = \left( \mathcal E_{\rm eff}, \tau, N, C, \eta_{\rm pol}, \eta_{\rm read}, \mathbf s_{\rm rev}, \mathbf u_{\rm sys} \right),

where $\eta_{\rm pol}$ and $\eta_{\rm read}$ are polarization and readout efficiencies, $\mathbf s_{\rm rev}$ summarizes reversal leverage, and $\mathbf u_{\rm sys}$ summarizes irreducible nuisance uncertainty. Maximizing $\mathcal E_{\rm eff}\tau\sqrt N$ alone can select an apparatus that is difficult to validate.

The HfF $^+$ and ThO programs continue to motivate improved trapped-ion, beam, and laser-coolable species. A future nonzero result should be reported both in a system-level frequency basis and in a shared operator basis so that species with different $W_d/W_S$ ratios can test it.

Charged-particle storage rings are opening a direct hadronic channel

The first direct deuteron result established a measurement pathway that had previously been dominated by projected designs. The next frontier is to control a stored spin so that an EDM produces a steadily accumulating out-of-plane component rather than a small modulation beneath rapid magnetic-moment precession.

For spin $\mathbf S$ in electric and magnetic fields, the measured evolution contains magnetic-dipole and EDM terms,

\frac{d\mathbf S}{dt} = \boldsymbol\Omega_{\rm MDM} \times\mathbf S +\boldsymbol\Omega_{\rm EDM} \times\mathbf S.

A frozen-spin setting seeks

\boldsymbol\Omega_{\rm MDM} \simeq \mathbf 0

in the co-moving frame while retaining an EDM rotation proportional to an effective radial electric field. In practice, residual radial magnetic fields, vertical orbit shifts, geometric phases, beam-position correlations, polarimeter asymmetries, and clockwise–counterclockwise differences all enter.

Active storage-ring questions include:

whether counter-rotating beams can share sufficiently identical fields;
how beam-based magnetometry constrains radial magnetic fields;
how spin coherence time scales with momentum spread and lattice optics;
whether geometric phases factorize from the EDM channel;
how polarimeter drifts are measured independently; and
how a first proton or improved deuteron result will be combined with neutron and nuclear data.

The decisive milestone is not a projected $10^{-29}\ e\,\mathrm{cm}$ target. It is a demonstrated sequence of increasingly sensitive blinded searches whose magnetic-field and beam-dynamics controls scale with statistical reach.

Radioactive molecules are moving from production to calibrated sensors

Radioactive molecules can concentrate three advantages in one system:

enhanced nuclear response from deformation or near-degenerate nuclear states;
strong molecular polarization in laboratory fields; and
internal molecular levels that reverse the symmetry-sensitive interaction without reversing every laboratory condition.

For a candidate nuclear moment $q_A$ in molecule $m$ , the inferred frequency is

\omega_m = \frac{1}{\hbar} W_{mA}\,q_A,

and the microscopic interpretation is

q_A = \sum_j R_{Aj}\,c_j.

Both $W_{mA}$ and $R_{Aj}$ require uncertainties. A spectacular enhancement in $R_{Aj}$ does not compensate for an unknown sign, uncontrolled state mixing, or a poorly validated $W_{mA}$ .

RaF as a nuclear-structure calibrator

The observation of the Bohr–Weisskopf effect in RaF gives direct access to the finite spatial distribution of nuclear magnetization through molecular hyperfine structure. Such data can test nuclear models used in symmetry-response calculations. The frontier is to turn radioactive-molecule spectroscopy into a closed calibration loop:

\text{spectroscopic data} \longrightarrow \text{nuclear and electronic model test} \longrightarrow \text{response coefficient} \longrightarrow \text{symmetry inference}.

This route is slower than quoting a calculated enhancement factor, but it is more valuable for a future claim.

AcF as a newly accessed actinide molecule

The first gas-phase production and spectroscopy of $^{227}$ AcF established that an actinide fluoride with predicted sensitivity to CP-odd nuclear physics can be created and identified. Immediate tasks include improved term assignments, branching ratios, hyperfine structure, production yield, optical cycling pathways, polarization, and coherence.

A realistic readiness sequence is

\begin{aligned} &\text{produce} \to \text{identify} \to \text{assign} \to \text{prepare} \to \text{polarize}\\ &\qquad \to \text{interrogate} \to \text{reverse} \to \text{bound systematics} \to \text{unblind}. \end{aligned}

AcF was near the beginning of this sequence in 2025. Calling it an EDM experiment without this qualification would collapse years of necessary instrument and theory work into one label.

Nuclear magnetic quadrupole moments

Molecules containing deformed nuclei may have enhanced sensitivity to a CP-odd nuclear magnetic quadrupole moment. Candidate species with favorable electronic structure and laser-cooling pathways are under study. The active questions are:

which isotope offers a usable lifetime and production rate;
which electronic state gives a large, calculable $W_M$ ;
how nuclear deformation and CP-odd nuclear forces set $M$ ;
which internal reversals separate $M$ from electron-EDM and magnetic effects; and
which independent species can resolve operator degeneracies.

No nuclear magnetic quadrupole moment has yet been observed.

Molecular and atomic parity violation are gaining new control handles

Laser-cooled BaF

Laser cooling of $^{137}$ BaF brings longer interaction times and better state control to a nuclear-spin-dependent parity-violation candidate. Addressing up to 112 internal levels illustrates both the opportunity and the complexity: the same hyperfine and rotational structure that supplies useful near-degeneracies also creates dark states, leakage pathways, and control-dependent line shapes.

For opposite-parity levels $\lvert+\rangle$ and $\lvert-\rangle$ with detuning $\Delta$ , a minimal mixing model is

H = \begin{pmatrix} \Delta/2 & dE+iW_{\rm PV}\\ dE-iW_{\rm PV} & -\Delta/2 \end{pmatrix}.

The Stark term $dE$ supplies a controlled parity-conserving amplitude; the weak matrix element $W_{\rm PV}$ is selected through interference. Operating near $\Delta=0$ can enhance the response, but the interpretation requires calibrated detuning, electric-field phase, state purity, and line-shape models. The full derivation belongs in tests of fundamental symmetries.

Single molecular ions

Penning-trap and other trapped-ion concepts aim to exploit long coherence, single-particle readout, and tunable near-degeneracies. Their projected enhancement is active design work, not a measured sensitivity. Before a proposal becomes a search, it must demonstrate state preparation, resolved readout, field calibration, a parity-sensitive interference observable, and reversal-odd systematic diagnostics in the target ion.

Isotope chains and neutron skins

Improved isotope-chain APV can reduce common electronic uncertainty, but nuclear distributions then become a central observable rather than a small afterthought. A schematic isotope difference is

\Delta A_{\rm PV}^{AA'} \simeq K_{\rm at} \left[ \Delta Q_W^{AA'} +\Delta Q_{\rm skin}^{AA'} +\Delta Q_{\rm new}^{AA'} \right].

The common atomic factor $K_{\rm at}$ is useful only to the extent that isotope-dependent electronic corrections are controlled. Nuclear radii, neutron skins, deformation, and isotope-dependent many-body effects can imitate or obscure a new interaction.

Multitransition isotope shifts are becoming overconstrained tests

A two-transition King plot has a simple geometric interpretation but limited diagnostic power: one observed curvature can be distributed among several candidate causes. Adding transitions and isotope pairs turns the problem into a higher-rank consistency test.

Let $M_{iA}$ be a matrix of modified isotope shifts. Under a model with $r$ independent isotope-dependent factors,

\operatorname{rank}M \leq r

up to noise and specified corrections. A new interaction adds a factor only if its isotope and transition dependence is linearly independent of the included Standard Model terms.

The active program therefore includes:

transitions with different electronic overlap at the nucleus;
highly charged ions with complementary relativistic response;
improved Penning-trap nuclear mass ratios;
nuclear-polarization and higher-order field-shift calculations;
blind injection tests for line-center extraction;
covariance-aware low-rank tests; and
simultaneous fits across elements.

The calcium work shows why this overconstrained strategy is necessary. Resolving a nonlinearity into known higher-order effects strengthens the method by identifying the nuisance basis that future searches must include.

Spin-dependent force searches are integrating sources and quantum sensors

Spin-dependent interactions are tested by moving, rotating, modulating, or reorienting a polarized source while a nearby sensor searches for the predicted spatial, temporal, and directional response. The signal template is an integral over the source,

s(t;\lambda) = \int_{\mathcal V} d^3r\, \rho_s(\mathbf r,t)\, \mathcal K \left( \mathbf r-\mathbf r_d, \mathbf v, \boldsymbol\sigma_s, \boldsymbol\sigma_d; \lambda \right),

where $\rho_s$ is the calibrated spin density, $\mathbf r_d$ is the detector location, $\lambda$ is the interaction range, and $\mathcal K$ encodes the chosen potential.

The sensor does not measure a coupling directly. It measures a voltage, optical rotation, precession phase, or frequency that passes through a transfer function,

V(\omega) = G(\omega) \left[ s(\omega) +b(\omega) \right] +n(\omega).

Here $b$ denotes coherent backgrounds and $n$ stochastic noise. Active improvements include:

self-compensating comagnetometers;
spin amplifiers with resonant gain;
dense polarized electron-spin sources;
better source metrology and finite-element field maps;
mechanical monitors at the source modulation frequency;
multiple detector orientations;
sidebands and deliberately detuned null channels; and
searches across mediator range rather than at one nominal geometry.

Large magnetic-shielding or amplification factors are useful only if their frequency dependence and coupling to systematic channels are calibrated in the same operating state used for the search.

Static forces and oscillating dark fields are being analyzed jointly but distinctly

An exchanged boson sourced by laboratory matter produces a spatial potential. An ultralight dark field produces a coherent or partially coherent time series determined by the local field distribution. The same microscopic coupling can enter both analyses, but the likelihoods are different.

For a scan over angular frequencies $\omega_j$ , a dark-field analysis may use

q(\omega_j) = 2\ln \frac{ \mathcal L \left( \widehat A_j,\widehat\varphi_j, \widehat{\boldsymbol\eta}_j \right) }{ \mathcal L \left( A_j=0, \widehat{\boldsymbol\eta}_{0j} \right) },

with nuisance parameters $\boldsymbol\eta$ . A local excess must then be corrected for the scan and tested against the assumed coherence model. For a static source modulation, the phase relation to source position and the spatial range dependence supply different diagnostics.

The active frontier is a common operator vocabulary with separate, experiment-appropriate statistical models. Combining incompatible coherence or source assumptions would produce an apparently stronger but physically meaningless bound.

Shared likelihoods and response databases are becoming essential

A mature global analysis needs more than a table of published upper limits. For experiment $a$ , it ideally needs

\mathcal L_a \left( \mathbf y_a \mid \mathbf K_a\mathbf c, \boldsymbol\eta_a \right),

together with:

the sign and normalization convention for every coefficient;
response coefficients and their covariance;
experimental nuisance correlations;
the exact confidence construction;
the validity range in mediator mass or frequency;
enough information to reproduce the published projection; and
versioned updates when theory coefficients change.

This infrastructure is active research. It can change which operator combination is best constrained even when no underlying experimental datum changes. A reanalysis must therefore identify whether an improvement came from new data, revised response theory, a different prior, a different one-operator assumption, or a corrected convention.

What Is Debated or Uncertain

How directly do EDM bounds constrain the origin of cosmic matter?

Additional CP violation is required in many explanations of the observed baryon asymmetry, and EDMs strongly constrain broad classes of new CP-odd physics. The connection is important but not one-to-one.

A baryogenesis mechanism depends on finite-temperature dynamics, out-of-equilibrium evolution, flavor structure, transport, and the spectrum of new states. A laboratory EDM depends on zero-temperature low-energy operators after matching and running. Schematically,

\mathbf p_{\rm baryo} \xrightarrow[\text{thermal history}]{} Y_B, \qquad \mathbf p_{\rm baryo} \xrightarrow[\text{matching and running}]{} \mathbf c_{\rm EDM} \xrightarrow[\text{response}]{} \mathbf y.

The two arrows can be correlated in a specified model, but an EDM null result does not exclude every baryogenesis mechanism. Conversely, a future EDM discovery would establish new CP violation without by itself proving that the same interaction generated the cosmic asymmetry.

Claims of an “energy-scale reach” are similarly conditional. If $d\sim e\,m\,\sin\phi/(16\pi^2\Lambda^2)$ , solving for $\Lambda$ assumes a loop order, coupling strength, phase, mass insertion, and absence of cancellations. The scale is a model illustration, not an experimentally measured collision energy.

Theory response is part of the uncertainty budget

Electronic-structure calculations provide molecular effective fields, APV matrix elements, and responses to electron–nucleon operators. Nuclear theory provides Schiff moments, magnetic quadrupole moments, anapole moments, and hadronic matching. Lattice QCD, chiral effective field theory, nuclear many-body methods, and relativistic electronic-structure methods each enter different links.

If an observable is

y = \sum_i K_i c_i,

then response uncertainty contributes

\operatorname{Var}(y) = \mathbf c^{\mathsf T} \Sigma_K \mathbf c +\mathbf K \Sigma_c \mathbf K^{\mathsf T} +\cdots.

This dependence on the unknown coefficients makes theory uncertainty more subtle than adding one fixed percentage to an experimental error bar. Correlations also matter when several species use related electronic or nuclear methods.

Disagreement among high-level calculations is not automatically a reason to discard a platform, but it should remain visible. Useful validation includes hyperfine constants, dipole moments, transition energies, isotope shifts, and other observables that test the same wavefunction regions as the symmetry response.

A nonlinearity or modulation is not uniquely new physics

King-plot curvature, a line at a sidereal frequency, a source-synchronous sensor response, or a reversal-odd phase can be an excellent candidate observable. None has a unique interpretation before alternatives are tested.

For hypotheses $H_k$ , the relevant comparison is not

H_0:\text{perfectly linear or exactly zero}

against

H_1:\text{new boson}.

It is a model set such as

\left\{ \begin{array}{l} H_{\rm SM,\,higher\ order},\\ H_{\rm nuclear},\\ H_{\rm apparatus},\\ H_{\rm environmental},\\ H_{\rm analysis},\\ H_{\rm new\ interaction}. \end{array} \right.

The calcium isotope-shift result is a constructive example: a measured nonlinearity survived the simplest model but was compatible with known higher-order effects. A reliable search improves when such effects are measured and added to the nuisance model.

Laboratory and astrophysical bounds answer different questions

Astrophysical cooling, cosmology, equivalence-principle tests, colliders, and AMO experiments often constrain overlapping boson parameter space. The strongest numerical bound can depend on assumptions about:

particle abundance and production;
stellar or supernova modeling;
screening or environmental dependence;
cosmological initial conditions;
mediator stability and decay channels;
couplings to other sectors; and
whether one coupling or several are nonzero.

Laboratory bounds are generally more controlled and reproducible but may be weaker in a minimal model. Astrophysical bounds may be stronger but less robust to nonminimal particle physics or environmental effects. They should be plotted with their assumptions rather than collapsed into a single unqualified exclusion frontier.

One-operator limits can hide cancellations and flat directions

Suppose the likelihood depends on

y_a = \sum_i K_{ai}c_i.

Setting all but one $c_i$ to zero yields a useful conditional bound. Allowing several coefficients can produce correlations, cancellations, and unconstrained directions. A near-null observable may result from small coefficients or from cancellation:

K_{a1}c_1 \simeq -K_{a2}c_2.

Neither possibility can be separated with experiment $a$ alone.

There is no universally correct number of simultaneous operators. The fit should match the scientific claim. A model-independent low-energy analysis usually needs several coefficients and complementary systems; a named ultraviolet model may justify a lower-dimensional parameter surface. Results should state which case is being shown.

Chiral molecular parity violation is theoretically clean in concept but hard in practice

The weak interaction predicts an enantiomer-dependent energy difference, but the expected shifts are extremely small and depend strongly on molecular structure, heavy-atom placement, vibrational averaging, and state choice. Spectroscopy must also distinguish weak parity violation from:

ordinary enantiomer-dependent chemical environments;
Stark and Zeeman shifts;
pressure and collisional shifts;
line pulling and unresolved structure;
isotopic composition;
differential light shifts; and
frequency-reference drift.

An enantiomer comparison alone is not enough if the two samples experience different ordinary environments. Strong programs require racemic controls, isotopologues, field reversals, multiple transitions, and a calculation with validated uncertainty.

Nuclear enhancement can amplify theory dependence as well as signal

Octupole deformation and near-degeneracy can produce large Schiff or magnetic quadrupole responses. The same collective nuclear structure can make the mapping from fundamental operators more model dependent.

S_A = \sum_i R_{Ai}^{(S)}c_i, \qquad M_A = \sum_i R_{Ai}^{(M)}c_i,

then a large $R_{Ai}$ is useful only when its sign, uncertainty, and correlations are sufficiently controlled. Spectroscopic calibration can test parts of the electronic and nuclear model, but it does not directly validate every CP-odd matrix element.

Radioactive-molecule proposals should therefore report both projected statistical sensitivity to $S_A$ or $M_A$ and the current uncertainty in mapping those quantities to microscopic coefficients.

Static fifth forces and ultralight dark matter should not be conflated

A Yukawa force sourced by nearby matter and an oscillating ambient field may involve the same boson but assume different occupation, production, and coherence. In particular, a dark-matter interpretation assumes that the field supplies some fraction of the local dark-matter density. A static force bound does not require that assumption.

For an ultralight scalar constituting density $\rho_\phi$ ,

\phi_0 \sim \frac{\sqrt{2\rho_\phi}}{m_\phi},

so the inferred coupling scales with an assumed $\rho_\phi$ and coherence model. A laboratory source experiment instead scales with measured source mass or spin density and a spatial Green function.

A combined plot is legitimate only when the axes, source assumption, and interaction normalization are explicit.

Discovery thresholds cannot be reduced to five sigma

A small tail probability under one noise model is neither necessary nor sufficient for a fundamental-symmetry discovery. The candidate must survive tests that are physically targeted to the interaction.

A persuasive protocol would require:

a prespecified primary symmetry channel and analysis;
successful blinding and injection recovery;
stability under reasonable nuisance models;
null-channel behavior consistent with calibration;
the predicted scaling with field, coherence time, isotope, source geometry, or mediator range;
persistence under independent hardware or analysis changes;
a second system with a complementary response coefficient; and
a shared operator fit with consistent sign and magnitude.

The final requirement is especially powerful. An ordinary systematic often tracks apparatus details; a fundamental interaction should track the calculated response of a different atom, molecule, nucleus, or source geometry.

Key Experimental Platforms

No platform spans all symmetry sectors. The useful comparison is its target, present evidence stage, dominant leverage, and discovery-critical control.

Platform	Principal target	Status in July 2026	Distinctive leverage	Discovery-critical control
polar paramagnetic molecular beams	electron EDM and semileptonic CP-odd operators	mature null searches; ThO benchmark, YbF interferometer advancing	large internal effective field, large ensembles, multiple switches	motional and leakage fields, geometric phase, trajectory–switch correlations
trapped molecular ions and laser-coolable polyatomics	electron EDM, semileptonic operators, NSD-PV	HfF $^+$ leading electron-EDM projection; YbOH clock-transition control demonstrated	long coherence, internal co-magnetometers, engineered avoided crossings	state assignment, trap-field correlation, micromotion, readout bias
heavy atoms and atomic beams	weak charge, anapole contribution, atomic EDM	APV established in Cs and Yb; Hg and Ra EDM searches null	mature spectroscopy and many-body benchmarks	atomic theory, neutron distributions, stray-field reversal parity
ultracold neutrons	neutron EDM	mature direct null search	direct neutral-hadron observable, long storage	magnetic gradients, comagnetometer response, geometric phase, wall effects
charged-particle storage rings	deuteron and proton EDM	first direct deuteron limit published; next-generation rings developing	direct charged-hadron access and counter-rotating beams	radial magnetic field, orbit and spin coherence, polarimeter drift
radioactive atoms and molecules	Schiff moment, magnetic quadrupole moment, NSD-PV	Ra limit established; RaF and AcF spectroscopy advancing; no CP-odd moment observed	nuclear deformation plus molecular polarization	isotope production, structure calibration, coherence, validated response theory
isotope-shift networks	new spin-independent bosons and nuclear structure	calcium multitransition limits; observed nonlinearity compatible with known effects	differential precision across isotopes and transitions	nuclear mass, polarization, higher-order field shifts, rank sufficiency
comagnetometers and spin amplifiers	spin-dependent bosons, Lorentz violation, ultralight fields	mature null searches with rapidly improving source integration	common-mode magnetic rejection and resonant gain	source magnetism, gain calibration, mechanical pickup, scan trials
chiral-molecule spectroscopy	weak parity-violating enantiomeric energy shift	active spectroscopy and theory; no observation	direct comparison of opposite molecular handedness	chemical equivalence, line pulling, light shifts, vibrational averaging

Platform readiness should be reported as evidence, not adjectives

Terms such as “promising,” “next generation,” and “discovery-ready” are too elastic for comparison. A more useful readiness vector is

\mathbf R = \left( R_{\rm source}, R_{\rm state}, R_{\rm coherence}, R_{\rm readout}, R_{\rm reversal}, R_{\rm response}, R_{\rm blind} \right),

with each component supported by a measured result.

Examples of different readiness components are:

AcF production and term identification establish source and spectroscopy readiness, not reversal readiness.
YbF spin interferometry establishes coherent-control readiness, not a completed EDM interval.
YbOH clock transitions establish nuisance-response engineering, not the final statistical reach.
the direct deuteron search establishes an end-to-end ring measurement at its published sensitivity, while also exposing field and tilt controls that must improve.
the HfF $^+$ result establishes a complete blinded null search and currently supplies the leading electron-EDM projection.

Cross-platform replication is part of the apparatus design

A collaboration should identify its replication partner before a candidate appears. Useful pairs have different systematics and response ratios:

Candidate channel	High-value independent check
paramagnetic-molecule EDM phase	second molecule or architecture with different $W_d/W_S$ and switch implementation
neutron EDM	proton or deuteron ring plus diamagnetic atom, interpreted in a common hadronic basis
isotope-shift curvature	additional transitions, isotope pairs, or element with controlled higher-order nuclear terms
source-synchronous spin force	changed source geometry and a sensor with a different magnetic transfer function
APV isotope trend	independent element and neutron-distribution information
radioactive-molecule CP-odd shift	stable-isotope control, second nuclear structure, and independent electronic calculation

The most informative replication changes the nuisance model while preserving the predicted low-energy coefficient.

Key Theoretical Tools

Reversal algebra and channel projection

Let each switch $s_j=\pm1$ encode a controlled reversal. Any measured scalar can be expanded on the Boolean basis,

y(\mathbf s) = \sum_{S\subseteq\{1,\ldots,n\}} \beta_S \prod_{j\in S}s_j.

For a balanced switch set, the coefficient of channel $S$ is

\widehat\beta_S = \frac{1}{2^n} \sum_{\mathbf s} \left( \prod_{j\in S}s_j \right) y(\mathbf s).

The EDM or PV signal occupies a prespecified channel. Other coefficients are not disposable: they diagnose leakage, imperfect reversals, and correlated backgrounds. With imbalance or missing switch states, the coefficients should be estimated through a design matrix and covariance, not by applying the balanced formula blindly.

The canonical derivation and experiment examples live in tests of fundamental symmetries.

Response matrices with conventions and covariance

Theory should publish a response object, not only a favored final bound:

\left\{ \mathbf K(\mu), \Sigma_K, \mathcal C, \mathcal V \right\}.

Here $\mu$ is the renormalization scale when relevant, $\Sigma_K$ is the response covariance, $\mathcal C$ records sign and normalization conventions, and $\mathcal V$ states the domain of validity. The observable model is then

\mathbf y = \mathbf K(\mu)\mathbf c(\mu) +\boldsymbol\epsilon.

Changing basis or scale must transform $\mathbf K$ and $\mathbf c$ consistently. A sign copied from one molecular convention into another can reverse an inferred correlation without changing any experimental datum.

Useful numerical diagnostics include:

\kappa \left( \mathbf K^{\mathsf T} \Sigma_y^{-1} \mathbf K \right),

the condition number of the information matrix, and its singular vectors. They reveal flat operator directions that a list of marginal bounds hides.

Atomic, molecular, nuclear, and hadronic validation ladders

Response calculations should be checked against observables sensitive to the same parts of the wavefunction or nuclear structure:

Response layer	Useful validation observables
relativistic electronic structure	excitation energies, hyperfine constants, dipole moments, $g$ factors, isotope shifts
molecular polarization and mixing	Stark curves, avoided crossings, parity splittings, transition strengths
nuclear magnetization and charge	isotope shifts, hyperfine anomalies, charge radii, magnetic moments
collective nuclear response	spectroscopy of parity doublets, deformation observables, neighboring isotopes
hadronic matching	lattice-QCD matrix elements, chiral-EFT consistency, scale and scheme checks

Agreement on a convenient observable does not prove the symmetry coefficient correct, but systematic disagreement identifies a model that should not be used without inflated uncertainty.

Precision molecular measurements develops the molecule-as-sensor response in detail.

Effective-field-theory matching and running

The safe interpretation path is layered:

\mathcal L_{\rm UV} \xrightarrow[\mu_{\rm high}\to\mu_{\rm had}]{\rm match+run} \mathcal L_{\rm had} \xrightarrow{\rm nuclear} \mathcal H_{\rm nuc} \xrightarrow{\rm atomic/molecular} \mathcal H_{\rm lab}.

At each arrow, record:

operator basis and normalization;
matching scale and renormalization scheme;
retained and neglected orders;
theory covariance;
assumptions about flavor and CP phases; and
whether cancellations are permitted.

This bookkeeping prevents a published laboratory interval from acquiring a false universality when translated into a named particle model.

A minimum experiment likelihood separates the signal channel from calibrated nuisance channels,

\mathcal L = \mathcal L \left( \mathbf y_{\rm sig}, \mathbf y_{\rm null}, \mathbf y_{\rm cal} \mid \mathbf c, \boldsymbol\eta \right).

The analysis should specify:

which parameters were fixed before unblinding;
how hidden offsets or sign flips were injected;
how calibration data constrain $\boldsymbol\eta$ ;
how non-Gaussian tails and data cuts were tested;
how trial factors enter scans;
how interval coverage was checked; and
which alternate analyses were prespecified.

A blind offset prevents tuning to zero, but it does not by itself prevent selection bias, incorrect uncertainty models, or leakage from correlated channels. Injection recovery and simulated coverage remain necessary.

For covariance and information-matrix tools, see variance and covariance and Fisher information.

Source geometry and transfer-function modeling

For a force search, the coupling is inferred through a spatial integral and an instrument response. Both should be calibrated with geometry uncertainty:

\mu_k(\mathbf g) = g_a g_b \int d^3r\, \rho(\mathbf r) \mathcal K_k \left( \mathbf r;\lambda \right),

\mathbf y(\omega) = \mathbf G(\omega) \boldsymbol\mu(\omega) +\mathbf b(\omega) +\mathbf n(\omega).

The geometry parameters $\mathbf g$ include source position, shape, spin density, orientation, and motion. A Monte Carlo over $\mathbf g$ is useful only if its prior or calibration distribution is experimentally justified. Transfer functions should be measured at the search frequency and operating point, especially for resonant amplifiers and self-compensating sensors.

Reproducible result packages

A durable symmetry result should preserve:

signed channel estimates before one-operator projection;
covariance among signal, null, and calibration channels;
switch definitions and sign conventions;
response coefficients with versions and uncertainty;
source geometry or field maps when relevant;
likelihood or an accurate likelihood surrogate;
interval and trial-factor construction;
machine-readable exclusion data;
a script that reproduces principal tables and curves; and
an immutable record of corrections and reinterpretations.

This package lets future theory improvements update the operator inference without reverse engineering a plotted curve.

Representative Papers and Reviews

The annotations below identify why each source is included. Numerical values in the page are taken from the version-of-record papers rather than press summaries or projected-sensitivity documents.

Broad reviews and interpretation

M. S. Safronova et al., “Search for New Physics with Atoms and Molecules,” Reviews of Modern Physics 90, 025008 (2018), doi:10.1103/RevModPhys.90.025008 — broad AMO review spanning EDMs, APV, constants, dark matter, and new forces.
T. E. Chupp et al., “Electric Dipole Moments of Atoms, Molecules, Nuclei, and Particles,” Reviews of Modern Physics 91, 015001 (2019), doi:10.1103/RevModPhys.91.015001 — experimental and theoretical EDM landscape.
W. B. Cairncross and J. Ye, “Atoms and Molecules in the Search for Time-Reversal Symmetry Violation,” Nature Reviews Physics 1, 510–521 (2019), doi:10.1038/s42254-019-0087-9 — molecular and atomic time-reversal tests.
D. DeMille, N. R. Hutzler, A. M. Rey, and T. Zelevinsky, “Quantum Sensing and Metrology for Fundamental Physics with Molecules,” Nature Physics 20, 741–749 (2024), doi:10.1038/s41567-024-02499-9 — molecular response, coherence, and quantum-sensing opportunities.
B. M. Roberts, V. A. Dzuba, and V. V. Flambaum, “Parity and Time-Reversal Violation in Atomic Systems,” Annual Review of Nuclear and Particle Science 65, 63–86 (2015), doi:10.1146/annurev-nucl-102014-022331 — atomic many-body theory and symmetry observables.
G. Arrowsmith-Kron et al., “Opportunities for Fundamental Physics Research with Radioactive Molecules,” Reports on Progress in Physics 87, 084301 (2024), doi:10.1088/1361-6633/ad1e39 — interdisciplinary roadmap for production, structure, and symmetry measurements.
J. C. Berengut and C. Delaunay, “Precision Isotope-Shift Spectroscopy for New Physics Searches and Nuclear Insights,” Nature Reviews Physics 7, 119–125 (2025), doi:10.1038/s42254-024-00793-2 — King plots, Standard Model nonlinearities, and new-force inference.
L. Cong et al., “Spin-Dependent Exotic Interactions,” Reviews of Modern Physics 97, 025005 (2025), doi:10.1103/RevModPhys.97.025005 — potential conventions, experiments, and constraints for spin-dependent forces.
A. Jadbabaie et al., “Radioactive Molecules as Laboratories of Fundamental Physics,” Nature Reviews Physics 8, 521–530 (2026), doi:10.1038/s42254-026-00950-9 — current radioactive-molecule status and prospects.

Electric dipole moments

T. S. Roussy et al., “An Improved Bound on the Electron’s Electric Dipole Moment,” Science 381, 46–50 (2023), doi:10.1126/science.adg4084 — leading published HfF $^+$ electron-EDM projection.
V. Andreev et al. (ACME Collaboration), “Improved Limit on the Electric Dipole Moment of the Electron,” Nature 562, 355–360 (2018), doi:10.1038/s41586-018-0599-8 — benchmark ThO molecular-beam result.
C. Abel et al., “Measurement of the Permanent Electric Dipole Moment of the Neutron,” Physical Review Letters 124, 081803 (2020), doi:10.1103/PhysRevLett.124.081803 — leading direct ultracold-neutron result.
B. Graner et al., “Reduced Limit on the Permanent Electric Dipole Moment of $^{199}$ Hg,” Physical Review Letters 116, 161601 (2016); Erratum 119, 119901 (2017), doi:10.1103/PhysRevLett.116.161601 — diamagnetic-atom constraint; use the corrected record.
M. Bishof et al., “Improved Limit on the $^{225}$ Ra Electric Dipole Moment,” Physical Review C 94, 025501 (2016), doi:10.1103/PhysRevC.94.025501 — radioactive octupole-deformed atomic result.
A. Andres et al. (JEDI Collaboration), “First Experimental Limit on the Permanent Electric Dipole Moment of the Deuteron,” Physical Review Letters 136, 241801 (2026), doi:10.1103/ns3s-ld4k — first direct storage-ring deuteron limit.
R. A. Jenkins et al., “Spin Interferometry in a Beam of Ultracold Molecules,” Physical Review Letters 136, 253401 (2026), doi:10.1103/6g7x-y15q — complete YbF interferometry sequence for a future electron-EDM measurement, not a new bound.
Y. Takahashi et al., “Engineered Molecular Clock Transitions for Precision Measurements,” Physical Review X 16, 031011 (2026), doi:10.1103/4t7q-d58r — YbOH electric- and magnetic-noise suppression while retaining electron-EDM response.

Parity violation and radioactive molecules

C. S. Wood et al., “Measurement of Parity Nonconservation and an Anapole Moment in Cesium,” Science 275, 1759–1763 (1997), doi:10.1126/science.275.5307.1759 — benchmark cesium APV and nuclear-spin-dependent result.
D. Antypas et al., “Isotopic Variation of Parity Violation in Atomic Ytterbium,” Nature Physics 15, 120–123 (2019), doi:10.1038/s41567-018-0312-8 — weak-charge scaling across four isotopes.
E. Altuntaş et al., “Demonstration of a Sensitive Method to Measure Nuclear-Spin-Dependent Parity Violation,” Physical Review Letters 120, 142501 (2018), doi:10.1103/PhysRevLett.120.142501 — BaF Stark-interference method and null constraint.
F. Kogel et al., “Laser-Cooled $^{137}$ BaF Molecules for Measuring Nuclear-Spin-Dependent Parity Violation,” Physical Review Research 7, L022041 (2025), doi:10.1103/PhysRevResearch.7.L022041 — laser-cooling control milestone.
J. Karthein et al., “Electroweak Nuclear Properties from Single Molecular Ions in a Penning Trap,” Physical Review Letters 133, 033003 (2024), doi:10.1103/PhysRevLett.133.033003 — proposed near-degeneracy enhancement, not an observation.
S. G. Wilkins et al., “Observation of the Distribution of Nuclear Magnetization in a Molecule,” Science 390, 386–389 (2025), doi:10.1126/science.adm7717 — molecular observation of the Bohr–Weisskopf effect in RaF.
M. Athanasakis-Kaklamanakis et al., “Laser Spectroscopy and CP-Violation Sensitivity of Actinium Monofluoride,” Nature 648, 562–568 (2025), doi:10.1038/s41586-025-09814-1 — first gas-phase $^{227}$ AcF production and spectroscopy with calculated CP-odd sensitivities.

New bosons, fifth forces, and cosmic fields

A. Wilzewski et al., “Nonlinear Calcium King Plot Constrains New Bosons and Nuclear Properties,” Physical Review Letters 134, 233002 (2025), doi:10.1103/PhysRevLett.134.233002 — multitransition isotope shifts, precise masses, nuclear polarization, and improved new-boson limits.
H. Su et al., “New Constraints on Axion-Mediated Spin Interactions Using Magnetic Amplification,” Physical Review Letters 133, 191801 (2024), doi:10.1103/PhysRevLett.133.191801 — polarized-gas source and resonant amplification.
Z. Xu et al., “Constraints on Axion Mediated Dipole-Dipole Interactions,” Physical Review Letters 134, 181801 (2025), doi:10.1103/PhysRevLett.134.181801 — SmCo $_5$ spin source and self-compensating comagnetometer.
K. Gaul, L. Cong, and D. Budker, “Constraints on New Vector Boson-Mediated Electron–Nucleus Interactions from Spectroscopy,” Physical Review Letters 136, 181805 (2026), doi:10.1103/d19m-s856 — reinterpretation of BaF and cesium PV data as vector-boson limits.
S. Lahs, D. Comparat, F. Kirk, and B. Roberts, “Atomic Observables Induced by Cosmic Fields,” Physical Review D 113, 015041 (2026), doi:10.1103/j7s3-wjg6 — operator and symmetry map for scalar, pseudoscalar, vector, axial, and tensor cosmic fields.

Common Misconceptions

“The smallest quoted EDM number is the strongest experiment”

False. Electron, neutron, atomic, nuclear, and deuteron EDM results constrain different observables and operator combinations. Compare them only after mapping to a common basis with stated response uncertainties.

“The HfF⁺ result directly measured the electron EDM”

The experiment measured a symmetry-selected molecular frequency compatible with zero. The quoted $d_e$ limit is the leading projection when other contributing operators are fixed or constrained.

“A null result proves the symmetry is exact”

A null result excludes a parameter region under a likelihood, confidence construction, response model, and operator assumptions. It cannot prove that the true coefficient is mathematically zero.

“The first direct deuteron limit is already the best hadronic constraint”

It is a major feasibility result for charged-particle storage rings, but its present numerical reach is not competitive with mature neutron and diamagnetic-system inferences in common models.

“Enhanced RaF or AcF sensitivity means CP violation was observed”

RaF measured nuclear magnetization structure, and AcF was produced and spectroscopically characterized. Both advance response calibration and platform readiness; neither detected a CP-odd moment.

“A nonlinear King plot is evidence for a fifth force”

Nonlinearity rejects a low-rank leading-order model. Higher-order mass and field shifts, nuclear polarization, mass error, and apparatus effects must be tested before a new boson interpretation. The 2025 calcium result improved limits while retaining a viable nuclear explanation for its nonlinearity.

“Magnetic shielding removes magnetic systematics”

Shielding and self-compensation suppress selected magnetic transfer paths. Leakage from source motion, gradients, finite bandwidth, sensor cross-coupling, and calibration fields can remain. The transfer function must be measured in the search configuration.

“Five sigma would be enough for discovery”

A tail probability under one model does not establish a fundamental cause. Predicted reversal parity, scaling, null channels, hardware perturbations, trial corrections, and independent response-matched replication are also required.

“An EDM discovery would by itself explain baryogenesis”

It would establish new CP violation. Connecting it to the cosmic baryon asymmetry requires a specified particle model and finite-temperature, out-of-equilibrium dynamics.

“Laboratory and astrophysical exclusions can be merged without qualification”

They can be compared only with their source, abundance, environmental, screening, and coupling assumptions visible. A numerically stronger astrophysical curve may be less robust to nonminimal models.

Exercises

Exercise 1: Symmetry of a permanent EDM

For a nondegenerate spin-polarized state, consider $H_d=-d\,\mathbf S\cdot\mathbf E/S$ . Determine how this interaction transforms under parity and time reversal. Why does a nonzero permanent EDM aligned with spin violate both symmetries?

Solution

Spin is an axial vector. Under parity,

\mathbf S \xrightarrow{P} \mathbf S, \qquad \mathbf E \xrightarrow{P} -\mathbf E.

Therefore $\mathbf S\cdot\mathbf E$ is parity odd. Under time reversal,

\mathbf S \xrightarrow{T} -\mathbf S, \qquad \mathbf E \xrightarrow{T} \mathbf E,

so the product is also time-reversal odd. A nonzero scalar coefficient $d$ in a stationary nondegenerate state therefore produces a Hamiltonian term that changes sign under either transformation.

The qualification about the state matters. An induced electric dipole in an applied field need not violate either symmetry, and degenerate systems can support oriented superpositions whose interpretation requires more care.

Exercise 2: Phase scale versus published limit

Take $\mathcal E_{\rm eff}=23\ {\rm GV/cm}$ , $\tau=1.0\ {\rm s}$ , and $d_e=10^{-30}\ e\,{\rm cm}$ . Estimate the relative phase $\phi=2d_e\mathcal E_{\rm eff}\tau/\hbar$ . Then estimate the single-shot projection-noise scale for $C=0.80$ and $N=10^6$ . Explain why neither number is a published limit.

Solution

Convert the units:

d_e = 1.602\times10^{-51}\ {\rm C\,m}, \qquad \mathcal E_{\rm eff} = 2.3\times10^{12}\ {\rm V/m}.

The phase is

\phi = \frac{ 2(1.602\times10^{-51}) (2.3\times10^{12}) (1.0) }{ 1.055\times10^{-34} } \simeq 7.0\times10^{-5}\ {\rm rad}.

The idealized projection-noise scale is

\delta d_e \sim \frac{\hbar} {2C\mathcal E_{\rm eff}\tau\sqrt N} \simeq 1.8\times10^{-29}\ e\,{\rm cm}.

The first number is the signal phase for an assumed EDM. The second is an idealized uncertainty for one ensemble under a simplified noise model. A published limit requires repeated data, duty cycle, calibrated contrast, reversal projection, systematic uncertainties, response uncertainty, and a specified interval construction.

Exercise 3: Complementarity and conditioning

Two experiments constrain coefficients $c_1,c_2$ with response matrix

K = \begin{pmatrix} 1 & 1\\ 1 & 1.01 \end{pmatrix}.

Compute its determinant. If the second row is changed to $(1,-1)$ , how does the determinant change? Explain the experimental meaning.

Solution

For the first matrix,

\det K = 1.01-1 = 0.01.

The rows are nearly parallel. Inverting the matrix amplifies small experimental or theory errors, so the combination $c_1-c_2$ is poorly identified even if both measured observables are precise.

With second row $(1,-1)$ ,

\det K = -1-1 = -2.

The rows now probe nearly orthogonal combinations, $c_1+c_2$ and $c_1-c_2$ . The second experiment adds operator discrimination rather than only repeating the first response direction. In a real program the entries have units and covariance, so singular values of the whitened response matrix are more useful than the raw determinant.

Exercise 4: Parity-violation interference channel

Let a controlled Stark amplitude be $A_S=sA_0$ with switch $s=\pm1$ , and let a much smaller parity-violating amplitude $A_{\rm PV}$ be independent of $s$ . To first order in $A_{\rm PV}$ , find $R(+)-R(-)$ for $R(s)=\lvert A_S+A_{\rm PV}\rvert^2$ . Name one ordinary effect that could enter the same fitted switch channel.

Solution

To first order,

R(s) \simeq \lvert A_0\rvert^2 +2s\operatorname{Re} \left( A_0^*A_{\rm PV} \right).

Therefore

R(+)-R(-) = 4\operatorname{Re} \left( A_0^*A_{\rm PV} \right).

The reversal isolates interference, which is linear in the weak amplitude. It does not identify the cause automatically. A switch-correlated stray electric field, polarization change, detuning shift, leakage amplitude, or detector-gain change could occupy the same channel. Independent monitors and additional reversals must constrain those alternatives.

Exercise 5: A King-plot false positive

Suppose an isotope shift contains mass and field terms plus a quadratic field contribution:

\delta\nu_i^{AA'} = K_i\mu^{AA'} +F_i\delta r_{AA'}^2 +G_i \left( \delta r_{AA'}^2 \right)^2.

Why can two modified isotope shifts form a nonlinear King plot even when no new boson exists? Give two experimental strategies for distinguishing the quadratic term from a new interaction.

Solution

The leading King relation assumes only two isotope-dependent factors, $\mu^{AA'}$ and $\delta r_{AA'}^2$ . The quadratic term adds a third factor $(\delta r_{AA'}^2)^2$ . Unless its transition coefficient $G_i$ happens to be aligned with the leading response, the isotope-shift matrix gains rank and the two-transition plot curves.

Useful discriminants include:

add transitions with different ratios of $F_i$ , $G_i$ , and candidate new-boson response;
add isotope pairs so the predicted nuclear-radius scaling is overconstrained;
improve independent charge-radius and nuclear-polarization information;
repeat in another element, where a new boson and nuclear structure scale differently.

A nonlinearity is evidence that the leading model is incomplete, not a unique particle identification.

Exercise 6: Reading the deuteron milestone

The first direct result gives $\lvert d^d\rvert<2.5\times10^{-17}\ e\,{\rm cm}$ at $95\%$ confidence, whereas the neutron result gives $\lvert d_n\rvert<1.8\times10^{-26}\ e\,{\rm cm}$ at $90\%$ confidence. Compute the ratio of the quoted magnitudes. Why is that ratio not a complete ranking of hadronic CP sensitivity, and what was established by the deuteron measurement?

Solution

The numerical ratio is

\frac{ 2.5\times10^{-17} }{ 1.8\times10^{-26} } \simeq 1.4\times10^9.

The direct deuteron interval is therefore about nine orders of magnitude larger as a moment. But $d_n$ and $d^d$ are different hadronic responses, their microscopic operator maps differ, and the intervals use different confidence levels and systematics. A full comparison requires a common hadronic effective theory.

The deuteron experiment established an end-to-end direct storage-ring EDM measurement for a charged stable hadron, including spin-axis determination and a published systematic assessment. It did not establish competitive microscopic CP bounds. Its value is experimental feasibility and a measured starting point for dedicated frozen-spin facilities.

Exercise 7: Design a source-synchronous force null test

A polarized source rotates at angular frequency $\Omega$ , and a comagnetometer sees a line at $\Omega$ . Design a minimum set of changes that would distinguish a spin-dependent interaction from magnetic leakage and mechanical pickup.

Solution

A credible test should alter predicted signal and nuisance responses differently. A minimum set includes:

reverse source spin while preserving mass motion; a spin interaction changes sign, while ordinary mechanical pickup need not;
replace the polarized source with a magnetically matched unpolarized dummy to measure motion and drive leakage;
change source–sensor separation and compare with the mediator-range template;
rotate the detector sensitivity axis and test the predicted tensor angular dependence;
monitor vibration, temperature, current, and ordinary magnetic field at $\Omega$ and its harmonics;
detune the sensor from resonant amplification and verify the measured transfer function;
change $\Omega$ while preserving geometry; and
repeat with a second sensor or source geometry.

The primary likelihood should fit the signal template and monitored nuisance templates jointly. A single disappearing line after one intervention would diagnose that intervention, not establish the full interaction model.

Exercise 8: Build a discovery protocol

A trapped-molecule experiment finds a blinded, reversal-odd phase inconsistent with zero. List the evidence needed before interpreting it as new CP violation, and explain why repeating the same apparatus for more time is not sufficient.

Solution

Within the original apparatus, require:

valid unblinding and injected-signal recovery;
correct sign under every predicted internal and laboratory reversal;
the expected scaling with coherence time, polarization, and effective field;
null-channel consistency;
stability across field magnitude, detuning, trap settings, data subsets, and analysis variants;
direct bounds on correlated magnetic, electric, geometric, and readout effects; and
a signed system-level result with likelihood and response covariance.

Outside the apparatus, require:

an independent experiment using a different species or architecture;
a response matrix that tests whether the two signals agree as $d_e$ , $C_S$ , or another operator combination;
independent response calculations with convention checks; and
consistency with neutron, diamagnetic, collider, and other relevant constraints in any claimed ultraviolet model.

More data in one apparatus can reduce statistical uncertainty but does not rotate an operator degeneracy or expose an apparatus-specific systematic. Independent response directions and nuisance structures are what turn an anomaly into a fundamental interpretation.

Connections to Other Volumes

Parity and time reversal define the discrete transformations used to classify EDM and APV Hamiltonians.
Discrete symmetries and CPT states the assumptions behind translating time-reversal violation into a CP statement.
Tests of fundamental symmetries is the canonical home for EDM and APV Hamiltonians, switch algebra, Schiff screening, and systematic-error derivations.
Precision molecular measurements develops effective fields, molecular polarization, and electronic and nuclear response coefficients.
Magnetometry supplies sensor dynamics, calibration, and magnetic-noise tools used in spin-dependent searches.
Variation-of-constants searches treats adjacent clock and spectroscopy searches for static drifts and oscillating scalar backgrounds.
Cold-molecule frontiers tracks the production and control capabilities behind YbF, YbOH, BaF, RaF, AcF, and related sensors.
Precision AMO frontiers compares symmetry tests with clocks, constants, and other precision programs.
Optical-clock frontiers develops clock networks and isotope-shift capabilities relevant to ultralight fields and new-boson searches.
Variance and covariance and Fisher information provide the mathematical language for correlated response fits and identifiability.

What Changed This Year

This ledger records version-of-record developments through 26 July 2026.

29 January: Lahs et al. published a systematic operator map from scalar, pseudoscalar, vector, axial-vector, and tensor cosmic fields to atomic observables. This is theory infrastructure, not a signal.
6 May: Gaul, Cong, and Budker published vector-boson constraints from reinterpretations of BaF and cesium parity-violation data. The work added limits without adding a new measured anomaly.
16 June: the JEDI Collaboration published the first experimental deuteron EDM limit. It established direct storage-ring feasibility; the measured spin-axis tilts were dominated by systematic effects.
22 June: Jadbabaie et al. published a current perspective on radioactive molecules, clarifying the production, control, theory, and symmetry-measurement ladder.
23 June: Jenkins et al. published the complete YbF ultracold-beam spin-interferometry sequence and a sensitivity assessment. No new electron-EDM interval was reported.
20 July: Takahashi et al. published engineered YbOH clock transitions with strong electric- and magnetic-sensitivity suppression while retaining electron-EDM response. This was a control milestone, not an EDM result.

No 2026 version-of-record paper reviewed here established a permanent EDM, nuclear Schiff moment, nuclear magnetic quadrupole moment, molecular parity-violating energy difference, or fifth force. The leading published electron-EDM projection remains the 2023 HfF $^+$ result.

Fundamental Symmetry Frontiers

Core Question

Why It Matters

Symmetry tests can expose interactions hidden at high energy

Electric dipole moments probe additional CP violation

Parity violation is both signal and calibration

Nuclear moments connect several theory layers

New bosons need broad mass and operator coverage

Canonical Boundary

Minimal Background

The five-layer inference chain

Symmetry signatures are operator specific

An EDM channel

Switch-parity projection

Atomic parity-violation interference

Nuclear moments as response coefficients

King plots and new bosons

Static forces and oscillating fields are different searches

Discovery requires complementary response directions

What Is Known

Parity violation is established; these experiments test its detailed form

No permanent electric dipole moment has been established

The current electron-EDM benchmark is conditional

Null results are not measurements of zero

Neutron, atomic, and nuclear systems probe different CP-odd directions

The direct deuteron result is a milestone, not the strongest hadronic bound

Enhanced nuclear moments remain targets rather than detections

No molecular weak parity-violating energy difference has been observed

Fifth-force and new-boson searches have set limits, not found a force

Calcium isotope-shift nonlinearity is compatible with known physics

Spin-dependent searches constrain several interaction classes

Complementarity is already experimentally useful

What Is Actively Developing

Next-generation electron-EDM experiments are becoming coherence experiments

Ultracold YbF beam interferometry

Engineered clock transitions in YbOH

Trapped molecular ions and renewed beam experiments

Charged-particle storage rings are opening a direct hadronic channel

Radioactive molecules are moving from production to calibrated sensors

RaF as a nuclear-structure calibrator

AcF as a newly accessed actinide molecule

Nuclear magnetic quadrupole moments

Molecular and atomic parity violation are gaining new control handles

Laser-cooled BaF

Single molecular ions

Isotope chains and neutron skins

Multitransition isotope shifts are becoming overconstrained tests

Spin-dependent force searches are integrating sources and quantum sensors

Static forces and oscillating dark fields are being analyzed jointly but distinctly

Shared likelihoods and response databases are becoming essential

What Is Debated or Uncertain

How directly do EDM bounds constrain the origin of cosmic matter?

Theory response is part of the uncertainty budget

A nonlinearity or modulation is not uniquely new physics

Laboratory and astrophysical bounds answer different questions

One-operator limits can hide cancellations and flat directions

Chiral molecular parity violation is theoretically clean in concept but hard in practice

Nuclear enhancement can amplify theory dependence as well as signal

Static fifth forces and ultralight dark matter should not be conflated

Discovery thresholds cannot be reduced to five sigma

Key Experimental Platforms

Platform readiness should be reported as evidence, not adjectives

Cross-platform replication is part of the apparatus design

Key Theoretical Tools

Reversal algebra and channel projection

Response matrices with conventions and covariance

Atomic, molecular, nuclear, and hadronic validation ladders

Effective-field-theory matching and running

Likelihoods, coverage, and blind analysis

Source geometry and transfer-function modeling

Reproducible result packages

Representative Papers and Reviews

Broad reviews and interpretation

Electric dipole moments

Parity violation and radioactive molecules

New bosons, fifth forces, and cosmic fields

Common Misconceptions

“The smallest quoted EDM number is the strongest experiment”

“The HfF⁺ result directly measured the electron EDM”

“A null result proves the symmetry is exact”