Lattice QCD calculation of the pion-nucleon coupling $\bar{g}_0$ induced by the QCD $\Theta$-term
We present lattice QCD results for the CP violating pion-nucleon coupling $\bar{g}_0$ induced by the QCD $\overline\Theta$ term from the analysis of three 2+1+1-flavor ensembles generated with highly improved staggered quarks (HISQ) by the MILC collaboration. These ensembles are at lattice spacing $a\approx 0.09~\text{fm}$ and pion masses of 313, 226 and 138 MeV, respectively. The coupling $\bar{g}_0$ is extracted in two ways. First, from the matrix element of the correlation between the pesudoscalar current and the topological charge evaluated between the nucleon ground state. The data for correlation functions with both the pseudoscalar and axial vector exhibit large contamination from the $N\pi$ excited state. We show that these can be controlled at the leading order using chiral perturbation theory ($\chi$PT) and the axial Ward identity (AWI, also called the partially conserved axial current (PCAC) relation). The result after removing the $N\pi$ contamination and extrapolating to the physical pion mass is, however, noisy: $\bar{g}_0/(2F_\pi)=-7(63)\times 10^{-3} \,{\overline{\Theta}}$. The more precise result $\bar{g}_0/(2F_\pi)=17.4(1.9)\times 10^{-3} \,{\overline{\Theta}}$ is obtained using low energy effective field theory methods or equivalently the AWI. Since contamination from the $N\pi$ excited states arises in the calculation of many nucleon matrix elements, we give an extended discussion on them and the use of the AWI for controlling them in the calculation of $\bar g_0$.