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 $\ensuremath{\theta}$ 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 $410\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$, 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 $0.07\phantom{\rule{0.16em}{0ex}}\mathrm{fm}lal0.11\phantom{\rule{0.16em}{0ex}}\mathrm{fm}$ 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 ${d}_{n}=\ensuremath{-}0.00152(71)\phantom{\rule{4pt}{0ex}}\overline{\ensuremath{\theta}}\phantom{\rule{4pt}{0ex}}e\phantom{\rule{0.16em}{0ex}}\text{fm}$ and ${d}_{p}=0.0011(10)\phantom{\rule{4pt}{0ex}}\overline{\ensuremath{\theta}}\phantom{\rule{4pt}{0ex}}e\phantom{\rule{0.16em}{0ex}}\text{fm}$. Assuming the $\ensuremath{\theta}$ term is the only source of CP violation, the experimental bound on the neutron electric dipole moment limits $\left|\overline{\ensuremath{\theta}}\right|l1.98\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ ($90%$ CL). A first attempt at calculating the nucleon Schiff moment in the continuum resulted in ${S}_{p}=0.50(59)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{4pt}{0ex}}\overline{\ensuremath{\theta}}\phantom{\rule{4pt}{0ex}}e\phantom{\rule{0.16em}{0ex}}{\text{fm}}^{3}$ and ${S}_{n}=\ensuremath{-}0.10(43)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{4pt}{0ex}}\overline{\ensuremath{\theta}}\phantom{\rule{4pt}{0ex}}e\phantom{\rule{0.16em}{0ex}}{\text{fm}}^{3}$.