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Motivated by the two recent observations in the WASA-at-COSY detector, weinvestigate the $\eta^3$He nucleus with the $\eta NNN$ few body method. Weconstruct the effective $s$-wave energy dependent $\eta N$ potential whichreproduce the $\eta N$ subthreshold scattering amplitude in the 2005Green-Wycech model. It gives the $\eta$ separation energy and decay width of0.19 MeV and 1.71 MeV, respectively. We also construct various sets ofeffective $s$-wave energy independent $\eta N$ potentials where thecorresponding complex scattering lengths (a) are within the range given in mosttheoretical models. We obtain the bound $\eta^3$He nucleus with decay width ofabout 5 MeV when a is (1.0 fm, 0.3 fm), and of about 10 MeV when a is (1.0 fm,0.5 fm).

Probing the existence of \eta^3He mesic nucleus with a few-body approach