  Decay mode of the $2_1^+$ resonant state of $^6$He populated by the $^6$He
breakup reaction by $^{12}$C at 240 MeV/nucleon is investigated. The
continuum-discretized coupled-channels method is adopted to describe the
formation of the $2_1^+$ state, whereas its decay is described by the
complex-scaled solutions of the Lippmann-Schwinger equation. From analysis of
invariant mass spectra with respect to the $\alpha$-$n$ and $n$-$n$ subsystems,
coexistence of two decay modes is found. One is the simultaneous decay of two
neutrons correlating with each other and the other is the emission of two
neutrons to the opposite directions. The latter is found to be free from the
final state interaction and suggests existence of a di-neutron in the $2_1^+$
state of $^6$He.
