Electric Dipole Moment of the Neutron from $2+1$ Flavor Lattice QCD

We compute the electric dipole moment $d_n$ of the neutron from a fully dynamical simulation of lattice QCD with $2+1$ flavors of clover fermions and nonvanishing $\theta$ term. The latter is rotated into a pseudoscalar density in the fermionic action using the axial anomaly. To make the action real, the vacuum angle $\theta$ is taken to be purely imaginary. The physical value of $d_n$ is obtained by analytic continuation. We find $d_n=-3.9(2)(9)\times {10}^{-16}\theta e\mathrm{cm}$, which, when combined with the experimental limit on $d_n$, leads to the upper bound $|\theta |\lesssim 7.4\times {10}^{-11}$.

Figure 1

The topological susceptibility on the SU(3) symmetric line $m_u=m_d=m_s$ as a function of $m_{\pi }^2$ in units of $t_0$.

Figure 2

The topological charge distribution $p(Q)$ [with $\sum _{Q}p(Q)=1$] of ensemble 4 at ${\kappa }_{\ell }=0.12104$, ${\kappa }_s=0.12062$, and $\lambda =0.005$, together with a Gaussian fit.

Figure 3

The average charge $⟨Q⟩$ as a function of $\theta$ for ensembles 1 and 2 (filled circle) and ensembles 3 and 4 (filled square), together with linear plus cubic fits.

Figure 4

Disconnected insertion of the pseudoscalar density to lowest order. Gluon lines are omitted.

Figure 5

The phase factor $\overline{\alpha }(\overline{\theta })$ as a function of $\overline{\theta }$ for our two sets of quark masses.

Figure 6

The ratio of form factors ${\overline{F}}_3^{\overline{\theta },n}/F_1^{\overline{\theta },p}$ for ${\kappa }_{\ell }={\kappa }_s=0.12090$ and $\lambda =0.005$.

Figure 7

The renormalized form factor ${\overline{F}}_3^{\overline{\theta },nR}(0)$ as a function of $\overline{\theta }$, together with a linear plus cubic extrapolation, ${\overline{F}}_3^{\overline{\theta },nR}(0)=A\overline{\theta }+B{\overline{\theta }}^3$, to $\overline{\theta }=0$.

Figure 8

The dipole moment of the neutron extrapolated to the physical point along the path $\overline{m}=\text{const}$.