New insight into $nd\rightarrow$ $^3H\gamma$ process at thermal energy with pionless effective field theory

We take a new look at the neutron radiative capture by a deuteron at thermal energy with the pionless effective field theory (EFT($\pi\!\!\!/$)) approach. We present in detail the calculation of $nd\rightarrow$ $^3H\gamma$ amplitudes for incoming doublet and quartet channels leading to the formation of a triton fully in the projection method based on the cluster-configuration space approach. In the present work, we consider all possible one-body and two-body photon interaction diagrams. In fact, additional diagrams that make significant changes in the results of the calculation of the total cross section in the $nd\rightarrow$ $^3H\gamma$ process are included in this study. The properly normalized triton wave function is calculated and taken into consideration. We compare the cross section of the dominant magnetic M1-transition of $nd\rightarrow$ $^3H\gamma$ up to next-to-next-to-leading order $\textrm{N}^2\textrm{LO}$ with the results of the previous model-dependent theoretical calculations and experimental data. The more acceptable results for cross section $\sigma^{(2)}_{tot}=0.297\;(\textrm{LO})+0.124\;(\textrm{NLO})+0.048\;(\textrm{N}^2\textrm{LO})=[0.469\pm0.033]\:\textrm{mb}$ show order by order convergence and cutoff independence. No three-body currents are needed to renormalize observables up to $\textrm{N}^2\textrm{LO}$ in this process.