Parity-violating photon circular polarization in $p+d\to {}^3\mathrm{He}+\gamma$ with pion-less effective field theory

The nonleptonic weak interaction in the few-nucleon system at very low energies is studied with the calculation of the photon circular polarization in the $pd\to {}^3\mathrm{He}\gamma$ reaction ($P_{\gamma }^{pd}$). For very low energies, one may treat the pion as heavy and integrate it out from the QCD symmetry-based model-independent effective field theory, leaving only short-range strong interactions which are introduced in term of pionless effective field theory formulation. We investigate a complete set of the parity-violating electromagnetic transitions in the $pd\to {}^3\mathrm{He}\gamma$ process. In this paper the interferences of strong, electromagnetic, Coulomb, and parity-violating weak interactions are presented with the calculation of a full set of appropriate diagrams at the leading order. We have obtained energy-dependent $P_{\gamma }^{pd}$ results as a function of the laboratory energy values $0.5\le E_{\mathrm{Lab}}\le 3$ MeV in terms of the low-energy coupling constants. The contributions of the initial $S$- and $P$-wave proton-deuteron systems have been considered to make the final ${}^3\mathrm{He}$ ground state in this energy region.

Figure 1

The PV-scattering diagrams at LO. A single line represents a nucleon and a double line is a dibaryon propagator. The wavy line denotes the exchanged photon. The circle denotes the PV $dNN$ vertex. A dashed oval indicates the PC-scattering amplitudes at LO. The dashed rectangular represents the PV-scattering amplitude. Time-reversed diagrams are not displayed.

Figure 2

The first set of the diagrams for the PV $pd\to {}^3\mathrm{He}\gamma$ process at LO. The dashed rectangular with solid line around it denotes the contribution of all PV $pd$-scattering diagrams depicted in Fig. 1. The $C$ diagram indicates the Coulomb effect that only appears in the $pd\to {}^3\mathrm{He}\gamma$ transition. The dashed half-oval is the normalized ${}^3\mathrm{He}$ wave function at LO. The wavy line which exits from the nucleon denotes the emitted photon and $H$ is the leading-order three-body force.

Figure 3

The second set of the diagrams for the PV $pd\to {}^3\mathrm{He}\gamma$ process at LO. The circle with a cross and wavy line is the PV photon-dibaryon-nucleon-nucleon ($\gamma dNN$) vertex. The dashed rectangular with dashed line around it denotes the PV $pd$ scattering depicted in Fig. 1 without participation of the diagrams which have the full PC $pd$ scattering on the right-hand side.

Figure 4

Variation of the $P_{\gamma }^{pd}$ according to the proton laboratory energies from 0.5 to 3 MeV between $\mathrm{\Lambda }=400$ and 900 MeV based on the $g^{(\overline{x}-\overline{y})}$ presented in the second row of Table 2.

Figure 5

The variation of the $P_{\gamma }^{pd}$ according to photon mass for the proton laboratory energy of 2.65 MeV with $\mathrm{\Lambda }=600$ MeV based on the values of five PV LECs in the second row of Table 2. Dashed line represents the results with extrapolating and the solid line indicates the results without the extrapolation method.