$p+d\to {}^3\mathrm{He}+\gamma$ reaction with pionless effective field theory

We study the proton radiative capture by a deuteron with the pionless effective field theory [EFT($\pi /$)] formalism. The calculation of the $pd\to {}^3\mathrm{He}\gamma$ amplitude is considered for the incoming doublet and quartet channels leading to the formation of a ${}^3\mathrm{He}$. The strong and Coulomb scattering amplitudes for the proton-deuteron ($pd$) scattering are included in this study. In this calculation, the properly normalized ${}^3\mathrm{He}$ wave function has been used at each order. We evaluate both $M1$ and $E1$ transitions in the $pd\to {}^3\mathrm{He}\gamma$ process up to NLO. We calculate the total cross section for the $pd\to {}^3\mathrm{He}\gamma$ process based on the cluster-configuration space and compare it with the experimental data. The cross section results are presented for the incoming proton with the energy $0.5\le E\le 3$ MeV where the lower and upper limits are chosen for the treatment of Coulomb effects perturbatively and the EFT($\pi /$) breakdown scale, respectively. No three-body force is needed to renormalize observables up to NLO other than those we have introduced in the $pd$ scattering amplitudes.

Figure 1

The Faddeev equation of the nucleon-dibaryon scattering in the cluster-configuration space up to the next-to-leading order. The superscript “$n$” denotes the contribution up to ${\mathrm{N}}^n\mathrm{LO}$ ($n\le 1$). The dashed oval is the nucleon-dibaryon scattering amplitude. The solid line indicates a nucleon. The Coulomb internal wavy line with coupling to nucleon from both sides represents an exchanged photon. The double solid line denotes a dibaryon auxiliary field with the propagator ${\mathcal{D}}^{\left(n\right)}$. The $K_s$ and $H$ depict the propagator of the exchanged nucleon and the three-body force, respectively. The $K_c, K_{\mathrm{box}}, K_{\mathrm{tri}}^{\mathrm{in}}$, and $K_{\mathrm{tri}}^{\mathrm{out}}$ denote the kernels with the Coulomb interaction $\sim \alpha$. The diagram with direct Coulomb between the nucleon and deuteron enters only at the next-to-leading order.

Figure 2

The Feynman diagrams which contribute in the $E1$ and $M1 pd\to {}^3\mathrm{He}\gamma$ transition amplitudes up to NLO. The first and second lines show the diagrams which contribute in the $M1$ and $E1$ transitions of the $pd\to {}^3\mathrm{He}\gamma$ process up to NLO. For the $E1$ we do not calculate $s_5$ and the fifth diagram in the second line that are related to the $s_5$ diagram, because it will appear in a higher order than NLO. The third line presents the diagrams which enter only for the $pd\to {}^3\mathrm{He}\gamma$ transition (Coulomb effect). They are depicted by the rectangle with an outgoing wavy line. The wavy line which exits from the nucleon denotes the emitted photon. The dashed half-oval is the normalized ${}^3\mathrm{He}$ wave function. All remaining notations are the same as in Fig. 1.

Figure 3

The neglected diagrams which can be contributed in the final amplitude of the $E1$ and $M1pd\to {}^3\mathrm{He}\gamma$ transition. All notations are the same as previous figures.

Figure 4

Comparison between our results for $\mathrm{\Lambda }=600$ and experimental data. The triangular points denote the experimental data. Our results for total cross section, $E1$, and $M1$ cross section of the $pd\to {}^3\mathrm{He}\gamma$ process at LO and NLO, have been shown.

Figure 5

Curves of cutoff variation between $\mathrm{\Lambda }=200$ and $\mathrm{\Lambda }=600$ MeV for the cross section up to ${\mathrm{N}}^n\mathrm{LO}$ ($n\le 1$) are plotted as a function of laboratory system energy. The dashed line and solid line correspond to our result up to LO and NLO, respectively.