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# Baryon

**Based on Euclidean notation (not the same as the $\gamma_5$ notation used in the code).**

The first baryon interpolation operator for proton is:

$$
\begin{align}
N_i(x) =& \epsilon_{a,b,c} u_{i,a}(x) \big( u_{j,b}(x) (C\gamma_5)_{j,k} d_{k,c}(x)\big)
\end{align}
$$

where summing over repeated indices is assumed, where

$$
\begin{align}
C =& \gamma_t \gamma_y =
i
\left(
\begin{array}{cc}
\sigma_y & 0 \\
0 & -\sigma_y\\
\end{array}
\right)
\\
C^\dagger =& -C
\\
C^* =& C
\\
C^T =& -C
\\
C \gamma_5 =& \gamma_t \gamma_y \gamma_5 =
-i
\left(
\begin{array}{cc}
\sigma_y & 0 \\
0 & \sigma_y\\
\end{array}
\right)
\\
\epsilon_{0,1,2} =& 1
\end{align}
$$

$$
\begin{align}
\bar{N}_i(x) =& -\epsilon_{a,b,c} \bar u_{i,a}(x) \big( \bar u_{j,b}(x) (C\gamma_5)_{j,k} \bar d_{k,c}(x)\big)
\end{align}
$$