  We study collective excitations in three- and four-band superconductors with
inter-band frustration, which causes neither 0 nor $\pi$ inter-band phases in
the superconducting state. Using a low-energy spin-Hamiltonian originating from
a multi-band tight-binding model, we find that a massless Leggett mode occurs
in a wide parameter region of this four-band system. This massless mode is
related to the fact that the mean-field energy does not depend on a continuous
superconducting phase. In other words, we find a link of the massless mode with
a degeneracy between a time-reversal-symmetry-breaking state (neither 0 nor
$\pi$ phases) and a time-reversal-symmetric state (either 0 or $\pi$ phases).
Therefore, the mass of this collective mode characterizes well the
time-reversal symmetry in frustrated multi-band superconductors.
