construction of a TRV potential up to N2LO considering all possible TRV interaction terms in the $\chi$EFT without making any assumption for the chiral order of the LECs. In this way all possible sources of TRV can be studied just setting the LECs to their estimated values and turning on and off the various terms in the Lagrangian. The second is the study of the EDM of $\det$, $\tri$ and $\hel$, providing a suitable framework for the future determination of the LECs. In our calculation the contribution of the TRV three-body force is found to be sizably larger than reported in Ref. [@JB151].

Finally, it is worthy to mention that exists a different approach to the derivation of the TRV nuclear forces based on meson-exchange model [@CP04]. This model includes pion and vector-meson exchange with 10 unknown meson constants. Such a theory, which has a wider energy range of validity but it is less systematic and with no direct connection with the fundamental Lagrangian, has been used to study the EDM of light nuclei [@CP04; @IS08; @NY15; @YH13] and the neutron spin rotation $\vec{n}-\vec{p}$ [@CP06] and $\vec{n}-\vec{d}$  [@CP04; @YH11] scattering.

The present paper is organized as follows. In Sec. \[sec:trvlag\] 