Confirming the existence of the strong CP problem in lattice QCD with the gradient flow

We calculate the electric dipole moment of the nucleon induced by the quantum chromodynamics $\theta$ term. We use the gradient flow to define the topological charge and use $N_f=2+1$ flavors of dynamical quarks corresponding to pion masses of 700, 570, and $410\mathrm{MeV}$, and perform an extrapolation to the physical point based on chiral perturbation theory. We perform calculations at three different lattice spacings in the range of $0.07\mathrm{fm}<a<0.11\mathrm{fm}$ at a single value of the pion mass, to enable control on discretization effects. We also investigate finite-size effects using two different volumes. A novel technique is applied to improve the signal-to-noise ratio in the form factor calculations. The very mild discretization effects observed suggest a continuumlike behavior of the nucleon electric dipole moment toward the chiral limit. Under this assumption our results read $d_n=-0.00152\left(71\right)\overline{\theta }e\text{fm}$ and $d_p=0.0011\left(10\right)\overline{\theta }e\text{fm}$. Assuming the $\theta$ term is the only source of CP violation, the experimental bound on the neutron electric dipole moment limits $\left|\overline{\theta }\right|<1.98\times {10}^{-10}$ ($90\%$ CL). A first attempt at calculating the nucleon Schiff moment in the continuum resulted in $S_p=0.50\left(59\right)\times {10}^{-4}\overline{\theta }e{\text{fm}}^3$ and $S_n=-0.10\left(43\right)\times {10}^{-4}\overline{\theta }e{\text{fm}}^3$.