Determination of the $\eta {}^3\mathrm{He}$ threshold structure from the low energy $pd\to \eta {}^3\mathrm{He}$ reaction

We analyze the data on cross sections and asymmetries for the $pd\to \eta {}^3\mathrm{He}$ reaction close to threshold and look for bound states of the $\eta {}^3\mathrm{He}$ system. Rather than parameterizing the scattering matrix, as is usually done, we develop a framework in which the $\eta {}^3\mathrm{He}$ optical potential is the key ingredient, and its strength, together with some production parameters, are fitted to the available experimental data. The relationship of the scattering matrix to the optical potential is established using the Bethe-Salpeter equation and the $\eta {}^3\mathrm{He}$ loop function incorporates the range of the interaction given by the empirical ${}^3\mathrm{He}$ density. We find a local Breit-Wigner form of the $\eta {}^3\mathrm{He}$ amplitude $T$ below threshold with a clear peak in ${\left|T\right|}^2$, which corresponds to an $\eta {}^3\mathrm{He}$ binding of about 0.3 MeV and a width of about 3 MeV. By fitting the potential we can also evaluate the $\eta {}^3\mathrm{He}$ scattering length, including its sign, thus resolving the ambiguity in the former analyses.

Figure 1

The process $pd\to \eta {}^3\mathrm{He}$ considering explicitly the $\eta {}^3\mathrm{He}$ rescattering. The square box in the first diagram indicates the full transition amplitude, while the circle in the second diagram stands for the bare transition amplitude prior to the $\eta {}^3\mathrm{He}$ final state interaction. It contains all diagrams that do not have $\eta {}^3\mathrm{He}$ as intermediate state. The oval stands for the $\eta {}^3\mathrm{He}$ optical potential.

Figure 2

Diagrammatic representation of the $\eta {}^3\mathrm{He}$ scattering matrix.

Figure 3

Real (solid line) and imaginary (dashed line) parts of the $G$ of Eq. (10) as functions of the excess energy $Q$.

Figure 4

The fitted $dp\to \eta {}^3\mathrm{He}$ total cross sections compared with experimental data [25, 39].

Figure 5

The fitted $dp\to \eta {}^3\mathrm{He}$ total cross sections below $Q=2\mathrm{MeV}$ compared with experimental data [25, 39].

Figure 6

The fit in the model to the asymmetry parameter $\alpha$ as a function of the center-of-mass $\eta$ momentum $p_{\eta }$ compared with the experimental data [25, 39].

Figure 7

Square of the absolute value of the $\eta {}^3\mathrm{He}\to \eta {}^3\mathrm{He}$ scattering amplitude.

Figure 8

Real and imaginary parts of the $\eta {}^3\mathrm{He}\to \eta {}^3\mathrm{He}$ amplitude $T$ as a function of the excess energy $Q$. Red-solid and red-dashed curves are the real and imaginary parts of the Breit-Wigner structures of Eq. (15) fitted to the calculated amplitude.

Figure 9

Ratio of the real and imaginary parts of the $\eta {}^3\mathrm{He}\to \eta {}^3\mathrm{He}$ amplitude $T$ as a function of the excess energy $Q$.