Rigorous global extraction of particle-level CP-violating parameters from atomic and molecular EDM experiments: state-of-the-art nuclear structure, extended molecular systems, and the overdetermined measurement set

Permanent electric dipole moments (EDMs) of atoms and molecules violate parity (P) and time-reversal (T) symmetry and are among the most sensitive probes of CP violation beyond the Standard Model. Gaul and Berger (JHEP 08 (2024) 100) developed a phenomenological global analysis that selects complementary atomic and molecular systems to minimize the allowed volume in the space of six effective P,T-odd parameters (the electron EDM, an effective nucleon EDM, three nucleon-electron current couplings, and the CP-odd pion-nucleon coupling), but the model treats nuclear structure through crude analytic estimates of nuclear Schiff moments and magnetic quadrupole moments, covers only atoms and rigid linear molecules in Hund's coupling cases (a)-(c) with minimal angular momenta, and assumes equal uncertainties for exactly six experiments. The open problem, stated explicitly in that work, is to convert the now overdetermined set of more than six performed and planned EDM experiments into rigorous bounds on fundamental particle-level CP-violating parameters (the electron EDM, individual quark EDMs, and CP-odd four-fermion couplings) by incorporating state-of-the-art nuclear-structure calculations with quantified uncertainties, extending the molecular model to higher angular momenta and non-rigid polyatomic molecules, and determining how to weight the enlarged measurement set optimally under realistic uncertainties. No published work performs the integrated extraction, so the problem remains open, although its constituent components - a hadronic-scale global fit, ab initio Schiff moments for light nuclei, and operator-level inverse-problem frameworks - have progressed individually.

Background

A permanent electric dipole moment (EDM) of a particle, atom, or molecule is a vector quantity that, in a nondegenerate quantum system, must align with the system's angular momentum. Because the EDM and the angular momentum transform differently under parity (P) and time reversal (T), a permanent EDM violates both symmetries; by the CPT theorem, T violation then implies violation of CP, the combined charge-conjugation and parity symmetry. The CP violation contained in the Standard Model (the phase of the CKM, or Cabibbo–Kobayashi–Maskawa, quark-mixing matrix) generates EDMs orders of magnitude below current detectability, and additional CP violation is in any case needed to explain the cosmic matter-antimatter asymmetry. Searches for permanent EDMs in neutrons, atoms, and molecules therefore probe beyond-the-Standard-Model CP violation at energy scales complementary to high-energy colliders.

In heavy atoms and polar molecules, P,T-odd effects are strongly amplified. The relativistic Sandars enhancement makes the atomic EDM induced by the electron EDM (eEDM) grow roughly with the third power of the nuclear charge number $Z$, and the tiny rotational constants of polar molecules expose the valence electrons to enormous effective electric fields. The current eEDM limit, $|d_e| < 4.1\times 10^{-30}\ e\cdot\mathrm{cm}$, comes from spectroscopy of the molecule ThO. In diamagnetic systems (atoms or molecules with closed electronic shells), Schiff screening - the near-cancellation of the nuclear EDM by the atomic electrons - leaves the nuclear Schiff moment $S$, the leading P,T-odd nuclear moment that survives screening, as the principal carrier of hadronic CP violation. For nuclei with spin $I \geq 1$, a nuclear magnetic quadrupole moment (NMQM) $M$ can also arise; it is a rank-2 P,T-odd magnetic moment that is not screened and can dominate open-shell systems.

At the hadronic level, the effective P,T-odd interaction of an atom or linear molecule can be parameterized by six constants: the eEDM $d_e$; an effective nucleon EDM $d_N$; three CP-odd nucleon-electron current couplings $k_s$, $k_T$, $k_p$ (of scalar-pseudoscalar, tensor-pseudotensor, and pseudoscalar-scalar type); and the CP-odd pion-nucleon coupling $g_{\pi\mathrm{NN}}$, which controls the Schiff moment and NMQM through the nuclear force. The latter two are effective hadronic parameters: translating them into quark-level CP violation (quark EDMs, quark chromo-EDMs, CP-odd four-fermion couplings) requires chiral effective field theory (the systematic low-energy description of nuclear forces as an expansion in powers of pion mass and momentum) and nonperturbative quantum chromodynamics (QCD, the theory of the strong interaction) input that is itself still under development. Each experiment $i$ measures a P,T-odd shift $d_i = \sum_a \alpha_{ia} x_a$, linear in the parameter vector $\vec{x} = (d_e, d_N, k_s, k_T, k_p, g_{\pi\mathrm{NN}})$, so at least six experiments with linearly independent sensitivity rows are needed to bound all six parameters; the allowed region is an ellipsoid whose volume (the coverage volume) quantifies the constraining power of a chosen set of experiments.

Gaul and Berger built a simple analytical electronic-structure model that minimizes this coverage volume over nuclear charge numbers and electronic/nuclear angular momenta, validated by density-functional-theory calculations of all relevant P,T-odd electronic structure parameters for the atoms and molecules of past and currently planned experiments. Their rules of thumb show that complementarity - systems whose effective Hamiltonians contain different P,T-odd terms - matters more than raw enhancement and, contrary to naive expectations from $Z$-scaling of individual enhancement factors, that medium-heavy systems with $Z \leq 54$ can tighten global bounds by orders of magnitude once the number of experiments grows. Global analyses had been performed earlier at the nucleon and pion level (Chupp and Ramsey-Musolf, 2015) and within specific beyond-Standard-Model frameworks such as left-right symmetry (Ramsey-Musolf and Vasquez, 2021); this was the first from the electronic-structure perspective.

The model's simplifications, stated by the authors themselves, define the open question. Nuclear structure enters only through crude analytic estimates of $S$ and $M$ (following standard reviews): the optimization does not distinguish a valence proton from a valence neutron, sets all nuclear magnetic moments to $+1$ nuclear magneton regardless of their actual signs (which differ by isotope and can change the coverage volume), and omits detailed many-body effects such as core polarization, which is known to change Schiff-moment coefficients by large factors and even their sign. The systems treated are atoms and rigid linear molecules in Hund's coupling cases (a)-(c) - the standard schemes describing how electronic orbital and spin angular momenta couple to the internuclear axis - represented with minimal angular-momentum values; vibrational angular momentum (the bending-mode angular momentum that makes linear polyatomic molecules such as YbOH or SrOH degenerate and gives them $l$-type level doubling) is neglected, as is the use of different rotational transitions to separate dipole-type from quadrupole-type (NMQM) contributions. Finally, the optimization assumes exactly six experiments with equal uncertainties, whereas in reality more than six EDM experiments have already been performed (neutron; the diamagnetic atoms $^{199}$Hg and $^{129}$Xe; paramagnetic Cs and Tl; the molecules YbF, ThO, and HfF$^+$), so the actual system of equations is overdetermined.

Problem Statement

With more than six EDM experiments already performed or planned, the system of equations relating measured P,T-odd shifts to the underlying CP-violating parameters is overdetermined. The problem posed by the source is to convert the qualitative global-analysis framework into a rigorous global extraction of fundamental CP-violating parameters, specifically:

(1) Replace the crude analytic estimates of nuclear Schiff moments and nuclear magnetic quadrupole moments by state-of-the-art nuclear-structure calculations with quantified uncertainties - incorporating the distinction between a valence proton and a valence neutron, the signs of nuclear magnetic moments, and detailed nuclear many-body effects - so that the global fit yields rigorous bounds on fundamental particle-level parameters (for example the electron EDM $d_e$, individual quark EDMs, and CP-odd four-fermion couplings) rather than on the effective hadronic parameters $d_N$ and $g_{\pi\mathrm{NN}}$.

(2) Extend the system coverage beyond atoms and rigid linear molecules in Hund's coupling cases (a)-(c) (the standard schemes describing how electronic orbital and spin angular momenta couple to the internuclear axis of a molecule) with minimal angular momenta: to higher electronic and nuclear angular momenta, to polyatomic molecules with vibrational angular momentum, and to the use of different rotational transitions to disentangle the dipole (EDM-type) and quadrupole (NMQM-type) contributions.

(3) For the enlarged, overdetermined set of measurements with realistic experimental and theoretical uncertainties, establish how to weight the experiments optimally - that is, minimize the allowed coverage volume in the six-parameter effective P,T-odd space of the Background, translating the resulting bounds into the fundamental particle-level parameters of item (1) - and thereby refine the guidance for which future experiments are most valuable.

The single objective uniting these components is a rigorous, uncertainty-quantified global determination of the allowed particle-level CP-violating parameters from the full set of atomic and molecular EDM measurements, with a defensible account of how nuclear- and hadronic-theory uncertainties propagate into the bounds. The parts of the Problem Statement above are conjunctive components of this single objective rather than alternative routes to it: item (1) fixes the nuclear-structure input and the fundamental parameters to be bounded, item (2) fixes the molecular coverage, and item (3) fixes the weighting of the enlarged measurement set. Within item (2), the three extensions are independently addressable directions - completing any one of them is substantive progress, whereas the full resolution combines the integrated extraction over the enlarged, optimally weighted measurement set with all three.

References

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