Neutron electric dipole moment from lattice QCD

We carry out a feasibility study for the lattice QCD calculation of the neutron electric dipole moment (NEDM) in the presence of the $\theta$ term. We develop the strategy to obtain the nucleon EDM from the $CP$-odd electromagnetic form factor $F_3$ at small $\theta$, in which NEDM is given by $\underset{q^2\to 0}{lim}\theta F_3(q^2)/(2m_N)$, where $q$ is the momentum transfer and $m_N$ is the nucleon mass. We first derive a formula which relates $F_3$, a matrix element of the electromagnetic current between nucleon states, with vacuum expectation values of nucleons and/or the current. In the expansion of $\theta$, the parity-odd part of the nucleon-current-nucleon three-point function contains contributions not only from the parity-odd form factors but also from the parity-even form factors multiplied by the parity-odd part of the nucleon two-point function, and, therefore, the latter contribution must be subtracted to extract $F_3$. We then perform an explicit lattice calculation employing the domain-wall quark action with the renormalization group improved gauge action in quenched QCD at $a^{-1}\simeq 2\mathrm{GeV}$ on a ${16}^3\times 32\times 16$ lattice. At the quark mass $m_{f}a=0.03$, corresponding to $m_{\pi }/m_{\rho }\simeq 0.63$, we accumulate 730 configurations, which allow us to extract the parity-odd part in both two- and three-point functions. Employing two different Dirac $\gamma$ matrix projections, we show that a consistent value for $F_3$ cannot be obtained without the subtraction described above. We obtain $F_3(q^2\simeq 0.58{\mathrm{GeV}}^2)/(2m_N)=-0.024(5)e·\mathrm{fm}$ for the neutron and $F_3(q^2\simeq 0.58{\mathrm{GeV}}^2)/(2m_N)=0.021(6)e·\mathrm{fm}$ for the proton.