the $NN$ potential $V$ from iterations of it in the Lippmann-Schwinger (LS) equation, $$V+V\, G_0\, V+V\, G_0 \, V\, G_0 \, V+\dots \ ,
\label{eq:lse}$$ where $G_0$ denotes the free two-nucleon propagator. If we assume that, $$\langle N^\prime N^\prime |V|NN\rangle=\sum_n V^{(n)}\ ,$$ with $V^{(n)}$ of order $Q^n$, it is possible to assign to any $T^{(n)}$ the terms in the LS equation that are of the same order. Generally in term like $[V^{(m)}G_0V^{(n)}]$ is of order $Q^{m+n+1}$ because $G_0$ is of order $Q^{-2}$ and the implicit loop integration brings a factor $Q^3$ (for a more detailed discussion, see Ref. [@Pastore11]).

In our case the two nucleons interact via a PC potential plus a very small TRV component. The $\chi$EFT Lagrangian implies the following expansion in powers of $Q$ for $T= T_{PC} + T_{TRV}$: $$\begin{aligned}
 T_{PC}&=&T^{(0)}_{PC}+T^{(1)}_{PC}+T^{(2)}_{PC}+\ldots ,\label{tpc}\\
 T_{TRV}&=&T^{(-1)}_{TRV}+T^{(0)}_{TRV}+T^{(1)}_{TRV}+\ldots .\label{ttrv}\end{aligned}$$ Assuming that the potential $V=V_{PC}+V_{TRV}$ have a similar expansion, $$\begin{aligned}
 V_{PC} & = & V_{PC}^{(0)}+V_{PC}^{(1)}+V_{PC}^{(2)}+\dots \\
 V_{TRV} & = & V_{TRV}^{(-1)}+V_{TRV}^{(0)}+V_{TRV}^{(1)}+\dots ,\end{aligned}$$ we can match order by order the $T$ and the terms in the LS equation obtaining the form definition of the TRV $NN$ potential from the scattering amplitude, $$\begin{aligned}
 V_{TRV}^{(-1)} & = & T_{TRV}^{(-1)}\ ,\label{eq:vtrvml}\\
 V_{TRV}^{(0)}  & = &  T_{TRV}^{(0)}-\left[V_{TRV}^{(-1)}G_0 V_{PC}^{(0)}\right]
                     