$\eta$-nuclear interaction: Optical model versus coupled-channels approach

The existence of possible $\eta$-nuclear bound states is closely related to the corresponding scattering lengths. While the sign of its real part may indicate a bound state, a large (always positive) imaginary part can prevent such a state. Most theoretical calculations for, e.g., ${}^3\text{He}$ predict quite sizable imaginary parts with no bound state. It is shown that a generalization of the conventional phenomenological optical model potential to coupled channels, based otherwise on the same assumptions but treating the pionic channel explicitly, can yield much smaller inelasticity still starting with elementary $\eta N$ interactions giving the same $\eta N$ scattering lengths. As representative examples this decrease is argued by model calculations in the cases of $\eta {}^3\text{He}$ and $\eta {}^{12}\text{C}$.

Figure 1

Nuclear scattering length $a\left(\eta {}^3\text{He}\right)$ as a function of the elementary $\eta N$ scattering length calculated for potentials yielding the same elementary lengths (with the constraint $a_{\mathrm{I}}=a_{\mathrm{R}}/2$). Solid line: single-channel optical model, dashed line: coupled channels. The two values of the $\eta N$ range parameter $b$ are indicated.

Figure 2

Superficial $\eta N$ scattering lengths for an optical potential (solid) vs coupled model (dashed) as a function of strength and range. The elementary $\eta N$ models yielding the same scattering length ($0.55+0.27i$) fm have been modified varying either the strength or range by a multiplicative factor between 0 and 5. The curves starting from 0 vary the strength, whereas those starting discontinuously from factor 1 correspond to varying the range.

Figure 3

$\eta {}^3\text{He}$ scattering length for the coupled model as a function of the elementary $\eta N$ scattering length supplemented by an optical potential to account for the inelasticity to two pions. The solid curves present the real part for two values of the elementary range while the dashed ones are for the imaginary part.

Figure 4

$\eta$-nuclear scattering length on carbon as a function of elementary $\eta N$ scattering length calculated for models yielding the same elementary lengths (with the constraint $a_{\mathrm{I}}=a_{\mathrm{R}}/2$). Solid line: single channel optical model; dashed line: pure coupled channels (the imaginary part multiplied by 10); dotted line: coupled channels plus two-pion inelasticity. The range parameter $b=0.6$ fm is used throughout. Note that the real part of the amplitude is negative.