ordering cancel out (see Appendix \[app:pot\]). The final expression for the NLO of the $NNN$ TRV potential is, $$\begin{aligned}
  {V_{TRV}}^{\rm{NNN}}&=&\frac{\deltatre g_A^3 M}{4\fp^3}({\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\ \tau_{3z}+
  {\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_3}\ \tau_{2z}+{\boldsymbol{\tau}_2}\cdot{\boldsymbol{\tau}_3}\ \tau_{1z})\nonumber\\
  &&\times\frac{(i\bmk_1\cdot{\boldsymbol{\sigma}_1})\ (i\bmk_2\cdot{\boldsymbol{\sigma}_2})\ (i\bmk_3\cdot{\boldsymbol{\sigma}_3})}
                       {\omega^2_{k_1}\omega^2_{k_2}\omega^2_{k_3}}\ ,
                       \label{eq:NNNpot}\end{aligned}$$ which is in agreement with the expression of Ref. [@JV13; @JB150].

Also in this case we need the Fourier transform of the potential. Using the overall momentum conservation $\bmp_1+\bmp_2+\bmp_3=\bmp_1'+\bmp_2'+\bmp_3'$, which give us that $\bmk_3=-\bmk_1-\bmk_2$, and defining $\bmQ=\bmk_1+\bmk_2$ and $\bmq=\bmk_1-\bmk_2$ the Fourier transform becomes, $$\begin{aligned}
 \langle \bmr_1'\bmr_2'\bmr_3'|& V|\bmr_1\bmr_2\bmr_3\rangle=
 \delta (\bmr_1-\bmr_1')\delta (\bmr_2-\bmr_2')\delta (\bmr_3-\bmr_3')
 \nonumber\\ 
 &\times\int {d^3q\over (2\pi)^3} {d^3Q\over (2\pi)^3}
 V(\bmq,\bmQ)
    e^{-i(\bmq/2)\cdot\bmx_2}\,e^{-i\bmQ\cdot\bmx_1}\ ,
    \label{eq:vrnnn}\nonumber\\\end{aligned}$$ where $\bmx_1=\bmr_2-\bmr_1$ and $\bmx_2=\bmr_3-(\bmr_1+\bmr_2)/2$. Note that with this choice of $\bmQ$ and $\bmq$, the form of the Jacobi vectors appear automatically. We use the regularization function reported in Eq. (\[eq:cutoff\]), where we replace $k$ with $Q$, namely, $$C_\Lambda(Q)={\rm e}^{-(Q/\Lambda)^4}\ ,\label{eq:cutoffnnn}$$ which gives a local form of the $NNN$ TRV potential. A complete description of how to carry out the integration in the Fourier transform is given in Appendix \[sec:NNNr\].

Results {#sec:res}
=======

In this section, we report results for the EDM of $\det$, $\tri$ and $\hel$. Hereafter, we do not use the barred notation for the renormalized LECs but all the coupling constant must be considered as 