Probing the QCD $\bar θ$ term with paramagnetic molecules

TL;DR

The paper addresses how hadronic CP violation, notably the QCD $\bar{\theta}$ term, can be probed with paramagnetic molecular EDMs. It employs heavy-baryon chiral perturbation theory to derive CP-violating semileptonic electron-nucleus interactions, separating contributions from meson-exchange and two-photon pion-loop processes and renormalizing ultraviolet divergences with a meson-electron counterterm. The authors compute the CP-odd couplings $C_{SP}$ in terms of hadronic CP-odd meson-nucleon interactions, providing explicit expressions and a robust bound $|\bar{\theta}| < 1.5 \times 10^{-8}$ (90% C.L.) from HfF$^+$ data, with a near-term potential to reach the neutron EDM sensitivity by improving paramagnetic measurements by 1–2 orders of magnitude. They also show that, for the $\bar{\theta}$ term, pion-loop diagrams can dominate in heavy systems due to cancellations in the leading ME pieces, and discuss implications for disentangling hadronic CP sources via paramagnetic versus diamagnetic EDMs. Overall, the work offers a solid EFT-based framework to reinterpret paramagnetic EDM experiments in terms of strong CP violation and other hadronic CP-odd operators, highlighting the method’s experimental relevance and guiding future refinements.

Abstract

The experimental search for CP violation in paramagnetic atomic and molecular systems has made impressive progress in recent years. This has led to strong upper limits on the electron electric dipole moment. The same measurements can also be used to constrain hadronic sources of CP violation through CP-violating interactions between the electrons and the nucleus. We employ heavy-baryon chiral perturbation theory to compute such CP-violating semileptonic electron-nucleus interactions arising from the QCD theta term. We sharpen earlier results by determining the relevant short-distance effects and by an explicit two-loop calculation of meson-photon diagrams. We derive a bound of $|\barθ| < 1.5 \cdot 10^{-8}$ at $90\%$ confidence, based on HfF$^+$ experiments at JILA. A further experimental improvement of one to two orders of magnitude would make paramagnetic molecular electric dipole moment experiments competitive with the neutron and diamagnetic atom program in constraining strong CP violation and higher-dimensional CP-odd quark-gluon operators.

Figures (4)

• Figure 1: ME diagrams contributing to $C_\mathrm{SP}$. We denote electrons by single and nucleons by double straight lines, pions and eta mesons by dashed lines, and photons by waving lines. The grey square stands for a CP-violating nucleon-nucleon-meson interaction, the black dot for the WZ photon-meson coupling, and the grey dot for the meson-lepton counterterm, which renormalizes the photon loop.
• Figure 2: PL diagrams contributing to $C_\mathrm{SP}$. We denote electrons by single and nucleons by double straight lines, pions and eta mesons by dashed lines, and photons by waving lines. The grey square stands for a CP-violating nucleon-nucleon-pion interaction, and the orange dot for the nucleon magnetic moment. For each type of diagram only one topology is shown.
• Figure 3: Fit of the parameter integral $I(r)$ over $x$, $\bar{\lambda}$, $a$, $b$ and $c$ in Eq. \ref{['eq:paramintegral']} to a function $C_1 (\log{r}+C_2)$ ($r$ in units of $m_e/m_\pi$). The orange dots show the integral result at various values of $r$, and the blue line shows the best fit result in this range, with $C_1 \simeq -1.57 \simeq - \frac{\pi}{2}$ and $C_2 \simeq -1.77$.
• Figure 4: PL diagrams contributing to $C_\mathrm{SP}$: we show two of the four possible topologies for the only non-zero diagram out of the three classes in Fig. \ref{['fig:fullNLOdiagrams']}. The final two topologies, diagrams C and D, can be obtained by switching the CP-even and CP-odd pion-nucleon vertices in diagrams A and B, respectively. We define $q = p_e' - p_e = p - p'$.