Lattice calculation of electric dipole moments and form factors of the nucleon

We analyze commonly used expressions for computing the nucleon electric dipole form factors (EDFF) $F_3$ and moments (EDM) on a lattice and find that they lead to spurious contributions from the Pauli form factor $F_2$ due to inadequate definition of these form factors when parity mixing of lattice nucleon fields is involved. Using chirally symmetric domain wall fermions, we calculate the proton and the neutron EDFF induced by the $CP$-violating quark chromo-EDM interaction using the corrected expression. In addition, we calculate the electric dipole moment of the neutron using a background electric field that respects time translation invariance and boundary conditions, and we find that it decidedly agrees with the new formula but not the old formula for $F_3$. Finally, we analyze some selected lattice results for the nucleon EDM and observe that after the correction is applied, they either agree with zero or are substantially reduced in magnitude, thus reconciling their difference from phenomenological estimates of the nucleon EDM.

Figure 1

Nucleon connected three-point function [Eq. (3)] and labeling of the source-sink separation $t_{\mathrm{sep}}$ and operator insertion time $t_{\mathrm{op}}$.

Figure 2

Abelian gauge potential (41) corresponding to a uniform background electric field on a periodic lattice [22]. The horizontal arrows show positive, and the triple vertical arrows show negative Abelian potential $\propto L_z$ on the corresponding lattice links. The resulting circulation of the Abelian potential is the same around each plaquette (shown by the black arrows) and equal to the flux of the electric field $\frac{2\pi }{L_z{L}_t}$.

Figure 3

“Clover” definition of the gauge field strength tensor on a lattice. The arrows correspond to products of gauge links taken around the plaquettes beginning and ending at point $x$.

Figure 4

Quark-connected contractions of the nucleon, quark current (dots), and cEDM operators (crosses).

Figure 5

Quark-disconnected contractions of the nucleon, quark current (dots), and cEDM operators (crosses).

Figure 6

Propagators required for computing quark-connected contractions of the nucleon, quark current, and cEDM operators.

Figure 7

Effective energy plots from the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices, together with two-state fits.

Figure 8

Nucleon vector current form factors from the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices. Disconnected contractions are not included.

Figure 9

Plateau plots for the neutron and proton Pauli form factors: the three smallest $Q^2>0$ points. Results are shown for the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices.

Figure 10

Chiral rotation angle ${\alpha }_5$ of the proton field induced by $u$- and $d$-quark cEDM interactions on the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices. The angles ${\alpha }_5$ for the neutron are related by the $SU(2{)}_f$ symmetry $u\leftrightarrow d$. The chromo-EDM interactions are not renormalized and may contain mixing with other operators.

Figure 11

Nucleon electric dipole form factors $F_3$ induced by $u$- and $d$-quark chromo-EDM interactions from the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices. The chromo-EDM interactions are not renormalized and may include mixing with other operators. Disconnected contractions are not included for either current or cEDM insertion.

Figure 12

Linear $Q^2$ fits to the neutron EDFF $F_3$ (same data as in Fig. 11) including only the three smallest $Q^2>0$ points and source-sink separations $T=8a$, $10a$. Results are shown for the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices.

Figure 13

Plateau plots for the neutron EDFF form factors: the three lowest $Q^2>0$ points. Results are shown for the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices.

Figure 14

The neutron EDM computed from energy shifts with two values of the electric field. The units are dimensionless, and the scale is the same as for $F_3$. The values used in comparison are computed as averages of the $t=6\dots 9$ conservative plateaus common for both ensembles and both cEDM flavors. Results are shown for the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices.

Figure 15

Comparison of the neutron EDFF $F_{3n}(Q^2)$ computed with the conventional (“OLD”) [5, 6, 7, 8, 9, 10, 11] and the “NEW” formula (c12) to the neutron EDM $\zeta$ computed from the energy shift (see Fig. 14). The “OLD” $F_{3n}(Q^3)$ data are extrapolated with the dipole fit, and the “NEW” with the linear fit. Data points are shifted horizontally for legibility. Results are shown for the $24^3\times 64$ (left) and $16^3\times 32$ (right) lattices.

Figure 16

Corrected (filled symbols) and original (open symbols) values for the neutron form factor $F_3$ at a nonzero imaginary $\theta$ angle from Ref. [8]. The linear parts in the limit $\theta \to 0$ are shown in Table 3.