contribution to the at N2LO, both from the RC of the vertices, and from NLO in the pion propagators (see Appendix \[app:pot\]). The final expression we obtain is $$\begin{aligned}
  {V_{TRV}}&^{(\rm 3\pi,RC)}=-\frac{5\ga^3\deltatre}{16\fp\Lx^2}
  \Big[(\tau_{1z}+\tau_{2z})i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\
  &+(\tau_{1z}-\tau_{2z})i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\Big]\nonumber\\
  &\times\Big(\frac{25}{6}L(k)-\frac{7}{2}\frac{\mp^2}{\omega_k^2}L(k)
  +2\frac{\mp^2}{\omega_k^2}H(k)\Big)\nonumber\\
  &-\frac{25\ga^3\deltatre}{12\fp\Lx^2}\frac{1}
  {\omega_k^2}\ \Big\{(\tau_{1z}+\tau_{2z})
    \Big[\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}\nonumber\\
    &+\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]
  +(\tau_{1z}+\tau_{2z})\nonumber\\
  &\times\Big[\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\Big\}\label{eq:rc3p}\ ,
  \nonumber\\\end{aligned}$$ where in Eqs. (\[eq:rco2\]) and (\[eq:rc3p\]) the TRV coupling constants are renormalized to order $Q^2$.

Last, the potential $V^{ ({\rm CT})}_{TRV}$, derived from the $NN$ contact diagrams (b) of Fig. \[fig:diagNN\], reads $$\begin{aligned}
  V^{\text{(CT)}}_{TRV}&=&\frac{1}{\Lambda_\chi^2f_\pi}\big\{\overline{C}_1\ i\bmk\cdot
      \left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\nonumber\\
      &+&\overline{C}_2\ i\bmk\cdot
      \left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right){\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\nonumber\\
      &+&{\overline{C}_3\over 2}\ 
      \big[i\bmk\cdot\left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\left(\tau_{z1}+\tau_{z2}\right)
        \nonumber\\
        &&-i\bmk\cdot\left({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2}\right)
        \left(\tau_{z1}-\tau_{z2}\right)\big]\nonumber\\
      &+&{\overline{C}_4\over 2}\ 
      \big[i\bmk\cdot\left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\left(\tau_{z1}+\tau_{z2}\right)
        \nonumber\\
        &&+i\bmk\cdot\left({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2}\right)
        \left(\tau_{z1}-\tau_{z2}\right)\big]\nonumber\\
      &+&\overline{C}_5\ i\bmk\cdot
      \left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\left(3\tau_{z1}\tau_{z2}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\right)
      \big\}\label{eq:ct}\ .\end{aligned}$$ Above $\overline{C}_1$, $\overline{C}_2$, $\overline{C}_3$, $\overline{C}_4$ and $\overline{C}_5$ are renormalized LECs since they have reabsorbed various singular terms coming from the TPE diagrams, 3$\pi$ diagrams and the relativistic corrections. Note that it is possible to write ten operators which can enter $V_{TRV}^{(\text{CT})}$ at order $Q$ but only five of them are independent. In this work we have chosen to write the operators in term of $\bmk$, so that the $r$-space version of $V_{TRV}^{(\text{CT})}$ will assume a simple local form with no gradients. We want also to remark that the $C_2$, $C_4$ and $C_5$ LECs are needed in order to reabsorb 