Electric Dipole Moments of Nucleons, Nuclei, and Atoms: The Standard Model and Beyond

TL;DR

The review builds a comprehensive framework linking CP-violating sources at the Fermi/weak scale to observable EDMs across hadronic, nuclear, atomic, and molecular systems, through a hierarchy of effective operators and scale-by-scale running and matching. It delineates how dimension-four and dimension-six CPV operators generate hadronic and nuclear matrix elements, such as nucleon EDMs, TVPV pion–nucleon couplings ${\bar{g}}_\pi^{(i)}$, and Schiff moments, whose magnitudes determine experimental sensitivities. By aggregating results from HB$\chi$PT, lattice QCD, QCD sum rules, and quark models, the paper provides benchmark values and uncertainties for these matrix elements, and discusses how different CPV sources imprint distinct patterns in light and heavy nuclei, as well as in paramagnetic and diamagnetic atoms and molecules. The authors emphasize that reducing theoretical uncertainties in hadronic and nuclear matrix elements is essential for robustly inferring the nature of CPV from EDM limits, and they survey representative BSM scenarios (SUSY, extended gauge symmetry, extra dimensions) to illustrate the interpretive power of future EDM measurements that could probe scales beyond the LHC. The work thus offers a model-independent EFT framework for EDMs, clarifies current theoretical gaps, and outlines the path toward extracting high-energy CPV information from low-energy precision experiments with broad implications for cosmology and fundamental physics.

Abstract

Searches for the permanent electric dipole moments (EDMs) of molecules, atoms, nucleons and nuclei provide powerful probes of CP violation both within and beyond the Standard Model (BSM). The interpretation of experimental EDM limits requires careful delineation of physics at a wide range of distance scales, from the long-range atomic and molecular scales to the short-distance dynamics of physics at or beyond the Fermi scale. In this review, we provide a framework for disentangling contributions from physics at these disparate scales, building out from the set of dimension four and six effective operators that embody CP violation at the Fermi scale. We survey existing computations of hadronic and nuclear matrix elements associated with Fermi-scale CP violation in systems of experimental interest, and quantify the present level of theoretical uncertainty in these calculations. Using representative BSM scenarios of current interest, we illustrate how the interplay of physics at various scales generates EDMs at a potentially observable level.

Figures (9)

• Figure 1: Electric dipole moments and the interplay of various scales. For purposes of illustration, only the impact of dimension six CPV operators is shown. Below the weak scale, some operators, such as the fermion EDMs and quark chromo EDMs are effectively dimension five, carrying an explicit factor of the Higgs vacuum expectation value $\langle H^0\rangle$. A summary of the operators of interest to this article appears in Table \ref{['tab:effopsum']}. See text for a full discussion.
• Figure 2: Lattice computation of nucleon EDMs induced by the QCD $\theta$-term. The pion mass squared dependence of $d_n$ (left) and $d_p$ (right) obtained using various approaches. Square symbols denote the results in external electric field method in $N_f=2$ clover fermion Shintani:2008nt, and circle symbols denote one in form factor method Shintani:2009 with same gauge configurations. Red bar denotes the bound of EDM in $N_f=2$ domain-wall fermion in Berruto:2005hg, and diamond is a result from EDM form factor of imaginary $\theta$ method quoted in Aoki:2008gv. Note that the error bar of diamond symbol may be an underestimate due to large systematic error associated chiral symmetry breaking of clover fermion. The triangle symbol is model estimate in current algebra.
• Figure 3: Lattice computation of ${\bar{\theta}}$-dependence of $d_n$ using the form factor method methodAoki:2008gv for ${\bar{\theta}}^I=0.2$. Shown is the squared momentum transfer dependence at three mass parameters $K=0.1382$--0.1367 which correspond to $m_\pi^2 = 0.3$--0.85 GeV$^2$. These are results in $N_f=2$ clover fermion configurations.
• Figure 4: Dependence of $\eta(E_\mathrm{ext})$ for YbF Hudson:2002az. Vertical axis gives $E_\mathrm{eff}$ as a function of the applied field $E_\mathrm{ext}$. Scaling $E_\mathrm{eff}$ by twice its asymptotic value gives $\eta(E_\mathrm{ext})$ . Figure reprinted with permission from Phys. Rev. Lett. 89 023003 (2002).
• Figure 5: Illustrative one-loop (left) and two-loop (right) contributions to the fermion EDM and quark CEDM in the MSSM .