the renormalization of the LECs. In this section we write the final expression of the $NN$ TRV potential $ V^{(c.m.)}_{TRV}$ as, $$\begin{aligned}
    V^{(c.m.)}_{TRV}&=&
   V^{ ({\rm OPE})}_{TRV} +
   V^{ ({\rm TPE})}_{TRV} +
   V^{ ({\rm 3\pi,0})}_{TRV} +
   V^{ ({\rm 3\pi,1})}_{TRV} \nonumber\\
   &&+  V^{ ({\rm RC})}_{TRV} +
   V^{ ({\rm 3\pi,RC})}_{TRV} +
   V^{ ({\rm CT})}_{TRV} \ ,\label{eq:dec}\end{aligned}$$ namely as a sum of terms due to one-pion exchange (OPE), two-pion exchange (TPE), three-pion exchange at NLO (3$\pi$,0) and at N2LO (3$\pi$,1), relativistic corrections derived from the OPE (RC) and from the $3\pi$-exchange ($3\pi$,RC), and contact contributions (CT). From now on we define also $g_0^*=g_0+g_2/3$ (see Appendix \[app:pot\] for details). The OPE term is the contribution of diagram (a) of order $Q^{-1}$ (LO) and it reads, =0.7pt $$\begin{aligned}
  {V_{TRV}}^{(\rm OPE)}
  &=&\frac{\overline{g}_A\overline{g}_0^*}{2\overline{f}_\pi}({\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2})\frac{i\bmk\cdot
    ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omk^2}\nonumber\\
  &+&
  \frac{\overline{g}_A\overline{g}_1}{4\overline{f}_\pi}\Big[(\tau_{1z}+\tau_{2z})
    \frac{i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omega_k^2}\nonumber\\
    &&\qquad
    +(\tau_{1z}-\tau_{2z})\frac{i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})}{\omega_k^2}\Big]
  \nonumber \\
  &+&\frac{\overline{g}_A\overline{g}_2}{6\overline{f}_{\pi}}
  (3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2})\frac{i\bmk\cdot
    ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omk^2}\ ,\nonumber\\
  \label{eq:isotope}\end{aligned}$$ where there are an isoscalar, an isovector and an isotensor components and $\overline{g}_0^*=\overline{g}_0+\overline{g}_2/3$. The coupling constants $\overline{g}_A/\overline{f}_{\pi}$, $\overline{g}_0^*$, $\overline{g}_1$, $\overline{g}_2$ are renormalized coupling constants, having reabsorbed the various infinites generated by loops and given as combinations of the “bare” LECs entering the Lagrangian. The expression for $\overline{g}_A/\overline{f}_{\pi}$ is the same as reported in Ref. [@viviani14]. The LECs $g_{S1}^{(2)}$, $g_{S2}^{(2)}$, etc. enter 