in Eq. (\[eq:isotope\]) through the renormalized constants $\overline{g}_0$, etc. (see Appendix \[app:pot\] for more details). Note that, as mentioned before, the correct expressions of the renormalized constants in term of the bare ones should contain also the contributions of additional diagrams not considered here (see Ref. [@viviani14] for the procedure to be followed for the PC and PV potentials, respectively).

The TPE term comes from the not singular contribution of panels (e)-(h) of Fig.  \[fig:diagNN\], taking also into account the subtracting terms given in Eq. (\[eq:vtrv1\]). As discussed in Appendix \[app:pot\] this term has no isovector component, in agreement with the result reported in [@CM11]. Therefore, as reported in Eqs. (\[eq:pote\]) and (\[eq:box\]), we have, =1.0pt $$\begin{aligned}
  {V_{TRV}}^{(\rm TPE)}&=\frac{\ga{g}_0^*}{\fp\Lx^2}
   {\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\ i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\ L(k)\nonumber\\
  &+\frac{\ga^3 {g}_0^*}{\fp\Lx^2}\ {\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\  i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})
  \ (H(k)-3L(k))\nonumber\\
  &-\frac{\ga {g}_2}{3\fp\Lx^2}
   (3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2})
   \ i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\ L(k)\nonumber\\
   &-\frac{\ga^3 {g}_2}{3\fp\Lx^2}(3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2})\nonumber\\
   &\qquad\qquad\times i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\ 
   (H(k)-3L(k))\ ,\end{aligned}$$ where the loop functions $L(k)$ and $H(k)$ are defined in Eqs. (\[eq:sL\]) and  (\[eq:H\]). In this and the following NLO and N2LO terms the bare coupling constants $\ga$, $\fp$, $g_0^*$, $\ldots$ can be safely replaced by the corresponding physical (renormalized) values.

The $3\pi$-exchange term gives a NLO contribution through the diagram (i) of Fig. \[fig:diagNN\], =1.0pt $$\begin{aligned}
  {V_{TRV}}^{(3\pi,0)}=&-\frac{5\ga^3\deltatre M}{4\fp\Lx^2}\pi
  \Big[(\tau_{1z}+\tau_{2z})
    