  We theoretically analyze phonon-assisted tunneling transport in a quantum dot
side connected to a Majorana bound state in a topological superconducting
nanowire. We investigate the behavior of the current through the dot, for a
range of experimentally relevant parameters, in the presence of one long-wave
optical phonon mode. We consider the current-gate voltage, the current-bias
voltage and the current-dot-Majorana coupling characteristics under the
influence of the electron-phonon coupling. In the absence of electron-phonon
interaction, the Majorana bound states suppress the current when the gate
voltage matches the Fermi level, but the increase in the bias voltage
counteracts this effect. In the presence of electron-phonon coupling, the
current behaves similarly as a function of the renormalized gate voltage. As an
added feature at large bias voltages, it presents a dip or a plateau, depending
on the size of the dot-Majorana coupling. Lastly, we show that the currents are
most sensitive to, and depend non-trivially on the parameters of the Majorana
circuit element, in the regime of low temperatures combined with low voltages.
Our results provide insights into the complex physics of quantum dot devices
used to probe Majorana bound states.
