New insight into the $nd{\to }^3$H$\gamma$ 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 $nd\to$ ${}^3\mathrm{H}\gamma$ 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 $nd\to$ ${}^3\mathrm{H}\gamma$ 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\to$${}^3\mathrm{H}\gamma$ up to $\text{next-to-next-to-leading}\text{order}{\text{(N}}^2\text{LO)}$ with the results of the previous model-dependent theoretical calculations and experimental data. The more acceptable results for cross section ${\sigma }_{\mathrm{tot}}^{\left(2\right)}=0.297\left(\text{LO}\right)+0.124\left(\text{NLO}\right)+0.048\left({\text{N}}^2\text{LO}\right)=[0.469\pm 0.033]\text{mb}$ show order by order convergence and cutoff independence. No three-body currents are needed to renormalize observables up to ${\text{N}}^2\text{LO}$ in this process.

Figure 1

The $M1$ $nd\to$ ${}^3\mathrm{H}\gamma$ diagrams at ${\text{N}}^n\text{LO}$ ($n\le 2$). The “$\left(n\right)$” superscript denotes the contribution up to ${\text{N}}^n\text{LO}$. All possible diagrams in the $M1$ transition of the $nd\to$ ${}^3\mathrm{H}\gamma$ process up to ${\text{N}}^n\text{LO}$ are shown in the first and second lines. The diagrams in the third line are the expanded version of the first diagram in second line. The solid, wavy, and double lines represent a nucleon, a photon, and a dibaryon field, respectively. The nucleon-nucleon-photon ($NN\gamma$) and dibaryon-dibaryon-photon ($dd\gamma$) vertices show the one- and two-body $M1$ interactions. ${\mathcal{D}}^{\left(n\right)}$ is the $2\times 2$ propagator matrix of the dibaryon auxiliary fields and the three-body force is indicated by $\mathcal{H}$. The dashed oval and dashed half oval denote the $Nd$ scattering amplitude and the normalized triton wave function up to ${\text{N}}^n\text{LO}$, respectively.

Figure 2

The $a_4$ diagram in Fig. 1. $k$ and $E_i$ denote the incoming c.m. momentum and the total energy of the initial $nd$ system, respectively. All notation are the same as in Fig. 1.

Figure 3

Comparison between different theoretical results for the total cross section of the $nd\to$ ${}^3\mathrm{H}\gamma$ process. The points from left to right denote the results computed by experiment [15], our LO EFT($\pi /$), our NLO EFT($\pi /$), our ${\text{N}}^2\text{LO}$ EFT($\pi /$), earlier ${\text{N}}^2\text{LO}$ EFT($\pi /$) [3], AV14/VIII(IA+MI+MD)[14], AV18/IX(IA+MI+MD)[14], AV14/VIII(IA+MI+MD+${\Delta }_{PT}$)[14], AV18/IX(IA+MI+MD+${\Delta }_{PT}$)[14], AV14/VIII(IA+MI+MD+$\Delta$)[14], AV18/IX(IA+MI+MD+$\Delta$) [14], AV18/IX(gauge inv.) [27], and AV18/IX(gauge inv.$+3N$-current) [27] methods, respectively. The thin band indicates the error band of the experimental result of the cross section. Two horizontal dashed lines determine the upper and lower limits due to our systematic EFT($\pi /$) error at ${\text{N}}^2\text{LO}$.

Figure 4

The $Nd$ scattering diagrams up to ${\text{N}}^n\text{LO}$ ($n\le 2$). All notations are the same as in Fig. 1.