egative-parity nucleon resonances $N^*(1535)$ and $N^*(1650)$ in $\frac12^-$channel. We then have to construct an optimal operator which dominantly couples to $N^*(1535)$ or $N^*(1650)$. We employ the following two independent nucleon fields, $
N_1(x)\equiv \varepsilon_{\rm abc}u^a(x)(u^b(x)C\gamma_5 d^c(x))
$ and $
N_2(x)\equiv \varepsilon_{\rm abc}\gamma_5 u^a(x)(u^b(x)C d^c(x)),
$ in order to construct correlation matrices and to separate signals of $N^*(1535)$ and $N^*(1650)$. (Here, $u(x)$ and $d(x)$ are Dirac spinor for u- and d- quark, respectively, and $a,b,c$ denote the color indices.) Even after the successful signal separations, there still remain several signal contaminations mainly because lattices employed in actual calculations are finite systems: Signal contaminations ([*a*]{}) [*by scattering states*]{}, ([*b*]{}) [*by wraparound effects*]{}.

[Comment to ([*a*]{}) :]{} Since our gauge configurations are unquenched ones, the negative parity nucleon states could decay to $\pi$ and N, and their scattering states could come into the spectrum. The sum of the pion mass $M_\pi$ and the nucleon mass $M_N$ is however in our setups heavier than the masses of the lowest two states (would-be $N^*(1535)$ and $N^*(1650)$) in the negative parity channel. We then do not suffer from any scattering-state signals.

[Comment to ([*b*]{}) :]{} The other possible contamination is wraparound effects [@Takahashi:2005uk]. Let us consider a two-point baryonic correlator $\langle N^*(t_{\rm snk}) \bar N^*(t_{\rm src})\rangle$ in a Euclidean space-time. Here, the operators $N^*(t)$ and $\bar N^*(t)$ have nonzero matrix elements, $\langle 0|N^*(t)|N^*\rangle$ and $\langle N^*|\bar N^*(t)|0\rangle$, and couple to the state $|N^*\rangle$. Since we perform unquenched calculations, the excited nucleon $N^*$ can decay into $N$ and $\pi$, and even when we have no scattering state $|N+\pi\rangle$, we could have another “scattering states”. The correlator $\langle N^*(t_{\rm snk}) \bar N^*(t_{\rm src})\rangle$ can still accomm