  Recent low-frequency radio observations at 140 MHz discovered a 3 Mpc-long
bridge of diffuse emission connecting the galaxy clusters Abell 0399 and Abell
0401. We present follow-up observations at 60 MHz to constrain the spectral
index of the bridge, which so far has only been detected at 140 and 144 MHz. We
analysed deep (~18 hours) LOw Frequency ARray (LOFAR) Low Band Antenna (LBA)
data at 60 MHz to detect the bridge at very low frequencies. We then conducted
a multi-frequency study with LOFAR HBA data at 144 MHz and uGMRT data at 400
MHz. Assuming second-order Fermi mechanisms for the re-acceleration of
relativistic electrons driven by turbulence in the radio bridge regions, we
compare the observed radio spectrum with theoretical synchrotron models. The
bridge is detected in the 75'' resolution LOFAR image at 60 MHz and its
emission fully connects the region between the two galaxy clusters. Between 60
MHz and 144 MHz we found an integrated spectral index value of -1.44 +\- 0.16
for the bridge emission. For the first time, we produced spectral index and
related uncertainties maps for a radio bridge. We produce a radio spectrum,
which show significant steepening between 144 and 400 MHz. This detection at
low frequencies provides important information on the models of particle
acceleration and magnetic field structure on very extended scales. The spectral
index gives important clues to the origin of inter-cluster diffuse emission.
The steepening of the spectrum above 144 MHz can be explained in a turbulent
re-acceleration framework, assuming that the acceleration timescales are longer
than ~200 Myr.
