\[tab:cecd\].

  PC interactions    $c_1$   $c_2$   $c_3$
  ----------------- ------- ------- -------
  N2LO               -0.74     -     -3.61
  N3LO               -1.07   3.20    -5.32
  N4LO               -1.10   3.57    -5.54

  :  \[tab:cecd\] Values of the coefficients and $c_1$, $c_2$, $c_3$ in unit of GeV$^{-1}$ taken from Ref. [@DR15].

In the following the values $g_A=1.267$ and $\fp=92.4$ MeV are adopted.

This section is organized as follows. In Sec. \[sec:dedm\], we present the general expression for the EDM operator and the results for the deuteron EDM, while in Sec. \[sec:3Hedm\] we present the calculation of the $\tri$ and $\hel$ EDMs.

Deuteron EDM {#sec:dedm}
------------

The EDM operator ${\hat{\boldsymbol{D}}}$ is composed by two parts, $${\hat{\boldsymbol{D}}}={\hat{\boldsymbol{D}}}_{\rm PC}+{\hat{\boldsymbol{D}}}_{\rm TRV}.$$ The ${\hat{\boldsymbol{D}}}_{\rm PC}$ receives contribution at LO from the nuclear EDM polarization operator $${\hat{\boldsymbol{D}}}_{\rm PC}=e\sum_i\frac{1+\tau_z(i)}{2}\boldsymbol{r}_i,
  \label{eq:dpc}$$ where $e>0$ is the electric unit charge, $\tau_z(i)$ and $\bmr_i$ are the $z$ component of the isospin and the position of the i-th particle. The ${\hat{\boldsymbol{D}}}_{\rm TRV}$ LO contribution comes from the intrinsic nucleon EDM, $${\hat{\boldsymbol{D}}}_{\rm TRV}=\frac{1}{2}\sum_i\left[(d_p+d_n)+(d_p-d_n)\tau_z(i)
    \right]\boldsymbol{\sigma}(i)\ ,
  \label{eq:dtrv}$$ where $d_p$ and $d_n$ are the EDM of proton and neutron respectively and $\boldsymbol{\sigma}(i)$ is the spin operator which act on the i-th particle. As discussed in Refs. [@JV11b; @JB13] both the ${\hat{\boldsymbol{D}}}_{\rm PC}$ and ${\hat{\boldsymbol{D}}}_{\rm TRV}$ receive 