Calculating The Proton and Neutron Electric Dipole Moment

The Standard Model assumes, and there are multiple theoretical arguments to support, that there is no charge parity (CP) violation (which is equivalent to dependence upon the direction of time) in the strong force. 

There is an obvious place in the Standard Model equations of the strong force to insert a CP violation parameter, however, which is called the θ term. The θ term is zero if there is no CP violation in the strong force. 

But, while the theory assumes that the θ term is zero, experiments can never directly rule out a very small non-zero value for the θ term. 

A non-zero value for the θ term would have important qualitative implications, especially as a possible source of matter-antimatter asymmetry in the universe. The Standard Model and available observational evidence strongly support that this matter-antimatter asymmetry is an "initial condition" of the universe, contrary to the naive expectation that there should be equal amounts of matter and antimatter at the moment of the Big Bang. As a new paper discussed below explains:

Symmetries and their breaking are essential topics in modern physics, among which the discrete symmetries C (charge conjugation), P (parity), and T (time reversal) are of special importance. This is partially because the violation of the combined C and P symmetries is one of the three Sakharov conditions that are necessary to give rise to the baryon asymmetry of the universe (BAU). However, despite the great success of the standard model (SM), the weak baryogenesis mechanism from the CP violation within the SM contributes negligibly (∼ 16 orders of magnitude smaller than the observed BAU). This poses a hint that, besides the possible θ term in QCD, there could exist beyond-standard-model (BSM) sources of CP violation and thus the study of CP violation plays an important role in the efforts of searching for BSM physics.

One of the main ways to probe the magnitude of the θ term is to measure the electric dipole moments of the proton and the neutron, which are measurements that can be made with exquisite precision. No non-zero electric dipole moment has been observed for either the proton or the neutron. But strict upper bounds on this electromagnetic property of the nucleons have been established and those bounds can be incrementally improved over time.

A new paper uses Lattice quantum chromodynamics (QCD) methods to calculate from first principles the relationship between the observable quantities of the proton and neutron electric dipole moments, and the theoretical Standard Model parameter which is the θ term. 

The paper concludes that the electric dipole moment of the neutron is −0.00148(35)θ¯ e⋅fm and that the electric dipole moment of the proton is 0.0038(14)θ¯ e⋅fm. 

Thus, the θ term is about 675 times the magnitude of the neutron electric dipole moment and about 263 times the magnitude of the proton electric dipole moment, although both are zero if the θ term is zero (except for a weak force contribution about five orders of magnitude smaller than the current experimental limit). This also implies the the ratio of the electron dipole moment of the proton to the electron dipole moment of the neutron should be about -2.6.

The body text of the paper explains that:

The first experimental upper limit on the neutron EDM (nEDM) was given in 1957  as ∼ 10^−20 e·cm. During the past 60 years of experiments, this upper limit has been improved by 6 orders of magnitude. The most recent experimental result of the nEDM is 0.0(1.1)(0.2) × 10^−26 e·cm, which is still around 5 orders of magnitude larger than the contribution that can be offered by the weak CP violating phase. Currently, several experiments are aiming at improving the limit down to 10^−28 e·cm in the next ∼10 years. 

. . .

By using the most recent experimental upper limit of dn, our results indicate that θ¯ < 10^−10. 

This limit is equivalent to less than ± 1.2 x 10^-13 e·fm, which implies that the magnitude of the θ term must be less than about ± 10^-10, a constraint that will improve by about two orders of magnitude in the next decade.

This is too small by more than ten orders of magnitude to make a meaningful dent in the Sakharov conditions. The θ term would have to be roughly on O(1) after running to extremely high energy scales to explain the matter-antimatter asymmetry of the universe. But, the strong force becomes weaker, not stronger, at higher energy scales, so CP violation in the strong force should be less important at these energy scales, not more important.

Personally, I'm confident that the θ term is exactly zero, and that there are no new CP violating physics at higher energies, at least up to about the GUT scale, that explain the matter-antimatter asymmetry of the universe. 

Neither the zero value of the θ term, nor the existence of matter-antimatter asymmetry in the universe at a infinitesimal time after the Big Bang are "problems" in physics to be solved. They are simply descriptive features of our reality.

The paper and its abstract are as follows:

We calculate the nucleon electric dipole moment (EDM) from the θ term with overlap fermions on three domain wall lattices with different sea pion masses at lattice spacing 0.11 fm. Due to the chiral symmetry conserved by the overlap fermions, we have well defined topological charge and chiral limit for the EDM. Thus, the chiral extrapolation can be carried out reliably at nonzero lattice spacings. We use three to four different partially quenched valence pion masses for each sea pion mass and find that the EDM dependence on the valence and sea pion masses behaves oppositely, which can be described by partially quenched chiral perturbation theory. With the help of the cluster decomposition error reduction (CDER) technique, we determine the neutron and proton EDM at the physical pion mass to be dn=−0.00148(14)(31)θ¯ e⋅fm and dp=0.0038(11)(8)θ¯ e⋅fm. This work is a clear demonstration of the advantages of using chiral fermions in the nucleon EDM calculation and paves the road to future precise studies of the strong CP violation effects.

Jian Liang, et al., "Nucleon Electric Dipole Moment from the θ Term with Lattice Chiral Fermions" arXiv:2301.04331 (January 11, 2023).