{\cal{L}}_{\rm TRV}^{\pi N\,(1)}=\frac{g_V^{(1)}}{2M\fp}\big[\overline{\psi}
    \partial_\mu(\vec{\pi}\times\vec{\tau})_3\partial^\mu\psi+\ h.c.\ \big]\,,
  \label{eq:ltrv1}\end{aligned}$$ while no Lagrangian terms for the pure pionic interaction are allowed. At order $Q^2$ we get six new contributions, =1.0pt $$\begin{aligned}
  {\cal{L}}_{\rm TRV}^{\pi N\,(2)}&=&
   {g_{S1}^{(2)}\over\fp^2}\overline{\psi}\partial_\mu\partial^\mu
  (\pi\cdot\tau)\psi\nonumber\\
  &+&{g_{S2}^{(2)}\over 2M^2\fp^2}\big[\overline{\psi}\partial_\mu\partial_\nu
  (\pi\cdot\tau)\partial^\mu\partial^\nu\psi+{\rm h.c.} \big]\nonumber\\
  &+&{g_{V1}^{(2)}\over\fp^2}\overline{\psi}\partial_\mu\partial^\mu
  \pi_3\psi\nonumber\\
  &+&{g_{V2}^{(2)}\over 2M^2\fp^2}\big[\overline{\psi}\partial_\mu\partial_\nu
  \pi_3\partial^\mu\partial^\nu\psi+{\rm h.c.} \big]\nonumber\\
  &+&{g_{T1}^{(2)}\over\fp^2}\overline{\psi}\partial_\mu\partial^\mu
  \pi_3\tau_3\psi\nonumber\\
  &+&{g_{T2}^{(2)}\over 2M^2\fp^2}\big[\overline{\psi}\partial_\mu\partial_\nu
    \pi_3\tau_3\partial^\mu\partial^\nu\psi+{\rm h.c.} \big]\ ,
  \label{eq:ltrv2}\end{aligned}$$ where $S$, $V$, $T$ stand for isoscalar, isovector and isotensor respectively and $\fp\simeq92$ MeV is the pion decay constant. The three-pion interaction that appears at this level gives contributions of order $Q^2$ to the TRV nuclear potential which is beyond our purpose, therefore we do not consider it. The Lagrangian contains also four-nucleon contact terms ${\cal L}^{\rm CT}_{\rm TRV}$ representing interactions originating from excitation of resonances and exchange of heavy mesons. At lowest order ${\cal L}^{\rm CT}_{\rm TRV}$ contains only five independent four-nucleon contact terms with a single gradient.

In the following we also need the PC Lagrangian up to N2LO: =1.0pt $$\begin{aligned}
  {\cal L}^{PC}&=& {\cal L}_{\pi\pi}^{(2)}+\ldots\nonumber\\
   &+&  {\cal L}_{N\pi}^{(1)}+{\cal L}_{N\pi}^{(2)}+{\cal
    L}_{N\pi}^{(3)}+\ldots+ {\cal L}^{PC}_{CT}\ ,\label{eq:Lpc} \\
  {\cal L}_{\pi\pi}^{(2)}&=& {f_\pi^2\over 4} \langle\nabla_\mu U^\dag
  \nabla^\mu U + \chi^\dag U+\chi U^\dag\rangle\ , \label{eq:Lpc_pipi2}\\
  {\cal L}_{N\pi}^{(1)} &=& \overline{\psi}\Bigl(i\gamma^\mu D_\mu -M
  +{g_A\over 2} \gamma^\mu\gamma^5 u_\mu \Bigr)\psi\
  , \label{eq:Lpc_pin1}\\
   {\cal L}_{N\pi}^{(2)}  &=& 