Borromean Feshbach resonance in $^{11}$Li studied via $^{11}$Li($p$,$p'$)

A dipole resonance of $^{11}$Li is found by a $^9$Li + $n$ + $n$ three-body model analysis with the complex-scaling method. The resonance can be interpreted as a bound state in the $^{10}$Li + $n$ system, i.e., a Feshbach resonance in the $^9$Li + $n$ + $n$ system. As a characteristic feature of the Feshbach resonance of $^{11}$Li, the $^{10}$Li + $n$ threshold is open above the $^{9}$Li + $n$ + $n$ one, which reflects a distinctive property of the Borromean system. A microscopic four-body reaction calculation for the $^{11}$Li($p$,$p'$) reaction at 6 MeV/nucleon is performed by taking into account the resonant and nonresonant continuum states of the three-body system. The calculation of angular distributions of the elastic and inelastic scattering as well as the energy spectrum reproduced a recent experimental result. Furthermore, the $E1$ strength distribution from a Coulomb dissociation experiment was also reproduced in this framework. This means that the existence of the Borromean Feshbach resonance may consistently answer a long-standing open question of an excited state of $^{11}$Li.