Time-Reversal-Invariance Violation in the $N\!d$ System and Large-$N_C$

A minimal set of five low energy constants (LECs) for time-reversal and parity violating ($\not{T}\!\!\!\not{P}$) nucleon-nucleon ($N\!N$) interactions at low energies ($E\!<\!m_{\pi}^2/M_N$) is given. Using a large-$N_C$ (number of colors in QCD) analysis we show that one linear combination of LECs is $\mathcal{O}(N_C)$, three LECs are $\mathcal{O}(N_C^{0})$, and one linear combination of LECs is $\mathcal{O}(N_C^{-1})$. We also calculate the $\not{T}\!\!\!\not{P}$ observables of neutron spin rotation through a polarized deuteron target and a spin correlation coefficient in nucleon-deuteron scattering using pionless effective field theory. Using the large-$N_C$ analysis we show that the spin correlation coefficient and the neutron spin rotation are predominantly determined by same two LECs in the large-$N_C$ basis.