Nucleon electric dipole form factor in QCD vacuum

TL;DR

This work investigates nucleon electric dipole moments arising from a small CP-violating angle $\theta$ in QCD by employing the instanton liquid model (ILM) of the QCD vacuum. A key result is a model-independent relation $F^f_3(Q^2)/\theta = (\chi_t / n_{I+A}) F^f_2(Q^2)$ linking the CP-odd electric dipole form factor to the CP-even Pauli form factor, controlled by the topological susceptibility $\chi_t$ and instanton density $n_{I+A}$. Using lattice inputs for $F_2$ and ILM parameters, the authors predict proton and neutron EDMs (at low $Q^2$) that are broadly consistent with recent lattice estimates, e.g., $d_n = -3.81 \times 10^{-3} e\theta\, \mathrm{fm}$ and $d_p = 3.57 \times 10^{-3} e\theta\, \mathrm{fm}$. The results provide a practical bridge between hadronic magnetic moments, topological properties of the QCD vacuum, and lattice QCD data, offering a route to constrain $\theta$ and interpret EDM measurements within a semi-classical topological framework.

Abstract

In the QCD vacuum, the nucleon form factors receive contributions from the underlying ensemble of topological pseudoparticles, which are sensitive to a finite vacuum angle $θ$. We use this observation to derive a novel relationship between the Pauli and electric dipole form factors, for light quark flavors. This relationship allows for an explicit derivation of the proton and neutron electric dipole moments induced by a small CP violating $θ$ angle, in terms of the vacuum topological susceptibity times pertinent magnetic moments. The results compare well with some recent lattice estimates.

Figures (6)

• Figure 1: Visualization of instanton (yellow) and anti-instanton (blue) configurations in the deep-cooled Yang-Mills vacuum with Moran:2008xq.
• Figure 2: The ILM results of the topological charge $\Delta$ in unquenched (red-solid) and quenched (blue-solid) vacuum, compared to the lattice results from the ETMC collaboration Alexandrou:2020mds using twisted mass clover-improved fermions $N_f = 2+ 1+ 1$, in a 4-volume $64^3 \times 128~a^4$ with lattice spacing $a=0.0801(4)$ fm, and physical pion mass $m_\pi=139$ MeV. For the ILM parameters see below.
• Figure 3: Leading pseudoparticle (single instanton) contribution to the quark EM current in the ILM. The dashed lines refer to the exchange induced by a classical instanton and a propagating quark. The details can be found in Appendix \ref{['app:q_prop']} and \ref{['App:EM_inst']}
• Figure 4: The CP-odd electric dipole form factor $F_3(Q^2)$ from the ILM for proton (blue) and neutron (red) is obtained using \ref{['F32F']} with ILM parameters given in Table \ref{['tab:parameters_ILM']} and combining the parameters in \ref{['F32F']} with the measurement Perdrisat:2006hj for $F_2(Q^2)$ of proton and neutron fitted by dispersion analysis Belushkin:2006qa.
• Figure 5: The neutron electric dipole form factor predicted using the ILM relation \ref{['F32F']} combined with $F_2$ lattice results obtained by CSSM and QCDSF/UKQCD Collaborations CSSM:2014knt in a 4-volume $32^3 \times64$ ensemble with pion mass $m_\pi=310$ MeV (red band) and lattice spacing $a=0.074(2)$ fm. The $Q^2$ dependence is compared to the lattice results from $\chi$QCD collaboration Liang:2023jfj with $m_\pi=339$ MeV (blue data). The green band is a lattice linear fit and the yellow band is a lattice square fit with additional $Q^4$ terms.