-\left[V_{PC}^{(0)}G_0 V_{TRV}^{(-1)}\right]\ ,\label{eq:vtrv0}\\
 V_{TRV}^{(1)}  & = &  T_{PV}^{(1)}-\left[V_{TRV}^{(0)}G_0 V_{PC}^{(0)}\right]
               -\left[V_{PC}^{(0)}G_0 V_{TRV}^{(0)}\right]\nonumber\\
          & - & \left[V_{TRV}^{(-1)}G_0 V_{PC}^{(1)}\right]-
                \left[V_{PC}^{(1)}G_0 V_{TRV}^{(-1)}\right]\nonumber\\
          & - & \left[V_{TRV}^{(-1)}G_0 V_{PC}^{(0)}G_0 V_{PC}^{(0)}\right]
             - \left[V_{PC}^{(0)}G_0 V_{TRV}^{(-1)}G_0 V_{PC}^{(0)}\right]
             \nonumber\\
         & - & \left[V_{PC}^{(0)}G_0 V_{PC}^{(0)}G_0 V_{TRV}^{(-1)}\right]
             \ .\label{eq:vtrv1}\end{aligned}$$ The generalization for the $NNN$ TRV potential is straightforward.

The $NN$ TRV potential
----------------------

We define the following momenta, $$\begin{aligned}
  &&\bmK_j=(\bmp_j'+\bmp_j)/2\ , \quad
  \bmk_j=\bmp_j'-\bmp_j\ ,\label{eq:notjb1}\end{aligned}$$ where $\bmp_j$ and $\bmp'_j$ are the initial and final momenta of nucleon $j$. From the overall momentum conservation $\bmp_1+\bmp_2=\bmp_1'+\bmp_2'$, we can define $\bmk=\bmk_1=-\bmk_2$. We also define $\bmK=(\bmK_1-\bmK_2)/2$, $\bmP=\bmp_1+\bmp_2=\bmK_1+\bmK_2$, in this way is possible to write the TRV $NN$ potential as, $$V_{TRV}(\bmk,\bmK_1,\bmK_2)= V_{TRV}^{(c.m.)}(\bmk,\bmK)+ 
   V^{(\bmP)}_{TRV}(\bmk,\bmK)\ ,\label{eq:widetildev2}$$ where the term $ V^{(\bmP)}_{TRV}(\bmk,\bmK)$ represents boost corrections to $  V^{(c.m.)}_{TRV}(\bmk,\bmK)$ [@girlanda10], the potential in the center-of-mass (c.m.) frame. Below we ignore these boost corrections and provide expressions for $ V^{(c.m.)}_{TRV}(\bmk,\bmK)$ only.

![ \[fig:diagNN\] Time-ordered diagrams contributing to the TRV potential (only a single time ordering is shown). Nucleons and pions are denoted by solid and dashed lines, respectively. The open (solid) circle represents a PC (TRV) vertex.](diagNN.eps "fig:").

The diagrams that give contribution to the $NN$ TRV potential are shown in Fig. \[fig:diagNN\]. We do not consider diagrams which give contribution only to 