  Collective excitations reveal fundamental properties and potential
applications of superconducting states. We theoretically study macroscopic
quantum tunneling (MQT) in a Josephson junction composed of multi-band
superconductors, focusing on a phase mode induced by inter-band fluctuations:
the Josephson-Leggett (JL) collective excitation mode. Using the imaginary-time
path-integral method, we derive a formula for the MQT escape rate for
low-temperature switching events. We clarify that the JL mode has two major
effects on the MQT: (i) the zero-point fluctuations enhance the escape rate,
and (ii) the quantum dissipation induced by the couplings to the
gauge-invariant phase difference suppresses the quantum tunneling. We show that
the enhancement exceeds the suppression for a wide range of junction
parameters. This enhancement originates from the single-mode interaction
between the tunneling variable and the inter-band fluctuations.
