c_1 \overline{\psi}\langle \chi_+\rangle
   \psi  - \frac{c_2}{8M^2}\big[\overline{\psi}\langle u_\mu u_\nu \rangle
     D^\mu D^\nu \psi +\  {\rm h.c.}\big]\nonumber\\
   &+&c_3\overline{\psi}\frac{1}{2}\langle u_\mu u^\mu \rangle \psi+
   c_4 \overline{\psi}\frac{i}{4}[u_\mu,u_\nu]\sigma^{\mu\nu}\psi+
   \ldots \ ,\label{eq:Lpc_pin2}\\
  {\cal L}_{N\pi}^{(3)}  &=& d_{16} \overline{\psi}{1\over 2}
  \gamma^\mu\gamma^5 u_\mu \langle \chi_+\rangle \psi\nonumber \\
  &+&  d_{18} \overline{\psi}{i\over 2}
  \gamma^\mu\gamma^5 [D_\mu, \chi_-] \psi+\ldots\ ,\label{eq:Lpc_pin3}\end{aligned}$$ where the building blocks $U$, $u_\mu$, $\chi_+$, $\chi_-$ and the covariant derivative $D_\mu$ are defined in Ref. [@viviani14]. We have omitted all the terms not relevant in the present work (for the complete expression for Lagrangian $ {\cal L}_{N\pi}^{(2)}$ and ${\cal L}_{N\pi}^{(3)}$ in Ref. [@NF00]). Four-nucleon contact terms (see, for example, Refs.[@EE09; @RM11]) are lumped into ${\cal L}^{PC}_{CT}$. The parameters $c_1$, $c_2$, $c_3$, $c_4$, $d_{16}$, and $d_{18}$ are LECs entering the PC Lagrangian. All the constants entering the terms discussed in this section have to be considered as “bare” parameters (i.e. unrenormalized).

The TRV potential up to order Q {#sec:trvpot}
===============================

In this section, we discuss the derivation of the TRV $NN$ and $NNN$ potential at N2LO. We provide, order by order in power counting the formal expressions for it in terms of the time-ordered perturbation theory (TOPT) amplitudes, following the scheme presented in Ref. [@viviani14]. Then, the various diagrams associated with these amplitudes 