Systematic few-body analysis of $\eta d, \eta {}^3\mathrm{He}$, and $\eta {}^4\mathrm{He}$ interaction at low energies

The Alt-Grassberger-Sandhas $N$-body theory is used to study interaction of $\eta$ mesons with $d, {}^3\mathrm{He}$, and ${}^4\mathrm{He}$. Separable expansion of the subamplitudes is adopted to convert the integral equations into the quasi-two-body form. The resulting formalism is applied to fit the existing data for low-energy $\eta$ production of few-nucleon targets. On the basis of this fitting procedure the scattering lengths $a_{\eta d}, a_{\eta {}^3\mathrm{He}}, a_{\eta {}^4\mathrm{He}}$ as well as the subthreshold behavior of the elementary $\eta N$ scattering amplitude are obtained.

Figure 1

The nonzero potentials $Z_{{\alpha }_4,{\beta }_4}^{{\alpha }_3}$. The potentials $Z_{1,2}^2$ and $Z_{1,2}^3$ may be obtained from $Z_{2,1}^2$ and $Z_{2,1}^3$ via mirror rotation. The dashed and the solid lines represent, respectively, $\eta$ mesons and nucleons.

Figure 2

The generalized potentials $Z_{2,1}^2$ and $Z_{2,1}^3$ as defined in Eqs. (11) and (12). Notations as in Fig. 1.

Figure 3

Diagrammatic representation of the potentials $Z_{{\alpha }_3,{\beta }_3}^{{\alpha }_2}$. Notations as in Fig. 1. Other potentials may be obtained from those shown in the figure via mirror rotation.

Figure 4

The potentials $Z_{1,3}^2$ and $Z_{3,3}^2$ as defined in Eqs. (28) and (29). The form factor $g_2^3\left(q\right)$ determines interaction of two nucleons in the presence of the $\eta N$ interacting pair. It differs from the form factor $g_1\left(q\right)$ in $Z_{2,1}^2$ depicted in Fig. 2.

Figure 5

The potentials $Z_{{\alpha }_2,{\beta }_2}$ (32). Notations as in Fig. 1.

Figure 6

Structure of the potential $Z_{4,2}$ as defined by the general formula (33).

Figure 7

Cutoff functions $F_L\left(E\right)$ and $F_R\left(E\right)$ determined by Eqs. (48) and (49) with $a=2\mathrm{MeV}, r=5\mathrm{MeV}, d_L=d_R=-100\mathrm{MeV}$.

Figure 8

Left panels: scattering lengths, calculated with the modified $\eta N$ amplitude $f_{\eta N}{F}_L$ as function of the cutoff parameter $d$. The function $F_L$ is defined in Eq. (48). Right panels: same as in the left panels but calculated with $f_{\eta N}{F}_R$, where $F_R$ is defined by Eq. (49). Calculations are performed with Set I (Table 2) of the $\eta N$ parameters. For $F_{L/R}$ we used the same parameters $a$ and $r$ as in Fig. 7.

Figure 9

The $\eta A$ amplitude squared, $|f_{\eta A}{|}^2$, for $A=d$ and ${}^4\mathrm{He}$. The curve is our fit. Empty and filled circles show the values of $|f_{\eta A}{|}^2$ extracted from the data for $pn\to \eta d$ [15] and $pd\to \eta pd$ [16] (left panel) and $dd\to \eta {}^4\mathrm{He}$ [21, 22] (right panel). The normalization of all results is arbitrary.

Figure 10

The total cross section for $dp\to \eta {}^3\mathrm{He}$ as a function of the excess energy $Q$. The data are taken from [17] (squares), [18] (empty circles), and [19] (filled circles). The curve shows our fit.

Figure 11

The off-shell $\eta N$ scattering amplitude in the $S_{11}$ partial wave calculated with the parameters (Set II in Table 1) adjusted to the data in Fig. 9 as described in Sec. 5. Solid curve: real part. Dashed curve: imaginary part. Crosses and circles represent the results of the $K$-matrix analysis of Ref. [46].

Figure 12

Dependence of the $\eta {}^4\mathrm{He}$ scattering length on the ${}^4\mathrm{He}$ binding energy. Solid curves: real part. Dashed curves: imaginary part. The pairs of curves 1 and 2 are obtained with the $\eta N$ parameters I and II (Table 2), respectively.

Figure 13

Solid lines: Real and imaginary parts of the $\eta N$ local potential $v_{\eta N}(E_{\eta N},r)$ (56) which is equivalent to our separable potential (41). Dashed lines: the $\eta N$ potential $v_{\eta N}(\delta \sqrt{s},r)$ from Ref. [29] calculated at constant value of its energy argument $\delta \sqrt{s}=\delta \sqrt{s_{\mathrm{sc}}}$ for two scale parameters $\mathrm{\Lambda }=2 {\mathrm{fm}}^{-1}$ ($\delta \sqrt{s_{\mathrm{sc}}}=19.48$ MeV) and $\mathrm{\Lambda }=4 {\mathrm{fm}}^{-1}$ ($\delta \sqrt{s_{\mathrm{sc}}}=29.75$ MeV). All potentials are averaged over the ${}^4\mathrm{He}$ density.