contributions from transition currents at N2LO. Of course, a complete treatment of the EDM up to N2LO needs to take care of them but hereafter we neglect their contribution, showing only the effects of N2LO TRV potential. In future work we plan to include the N2LO current contributions in the calculations.

The nucleon EDM of an $A$ nucleus can be expressed as $$\begin{aligned}
  d^A & = & {\langle}\psi^A_{+} | \hat{D}_{\rm TRV} | \psi^A_{+} {\rangle}+2\, {\langle}\psi^A_{+} | \hat{D}_{\rm PC} | \psi^A_{-} {\rangle}\nonumber\\
  & \equiv & d_{\rm TRV}^A+d_{\rm PC}^A\ ,\end{aligned}$$ where $|\psi^A_{+}{\rangle}$ $(|\psi^A_{-}{\rangle})$ is defined to be the even-parity (odd-parity) component of the wave function. In general, due to the smallness of the LECs, the EDM can be expressed as linear on the TRV LECs, $$\begin{aligned}
  d^A_{\rm TRV}&=&d_pa_p+d_na_n\\
  d^A_{\rm PC}&=&g_0a_0+g_1a_1+g_2a_2+\Delta_3 a_\Delta\nonumber\\
  &+&C_1A_1+C_2A_2+C_3A_3+C_4A_4+C_5A_5\ ,
  \label{eq:da}\end{aligned}$$ where the $a_i$ for $i=0,1,2$, $a_\Delta$, $A_i$ for $i=1,\dots,5$ and $a_p$, $a_n$ are numerical coefficients independent on the LECs values (however, they do depend on the cutoff $\Lambda$ in the PC and TRV chiral potentials).

We evaluate also the theoretical errors associated with the chiral expansion of the nuclear potential. We express the error on the numerical coefficients for the deuteron as, $$\left(\delta a_i\right)^2=\left(\delta a_i^{\text{PC}}\right)^2
  +\left(\delta a_i^{\text{TRV}}\right)^2\ ,
  