Physics

Meteorology

Dipole

Pion

Mathematics

Mathematical physics

Lattice QCD

Particle physics

Quantum chromodynamics

Extrapolation

Mathematical analysis

Electric dipole moment

Nucleon

Acoustics

Bar (unit)

Lattice (music)

Quenched approximation

Neutron electric dipole moment

Quantum mechanics

Quark

Chiral perturbation theory

Physical review. C

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 $0.07\phantom{\rule{0.16em}{0ex}}\mathrm{fm}<a<0.11\phantom{\rule{0.16em}{0ex}}\mathrm{fm}$ 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 $\left|\overline{\ensuremath{\theta}}\right|<1.98\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ ($90%$ 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 ${S}_{n}=\ensuremath{-}0.10(43)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{4pt}{0ex}}\overline{\ensuremath{\theta}}\phantom{\rule{4pt}{0ex}}e\phantom{\rule{0.16em}{0ex}}{\text{fm}}^{3}$.

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