  Radio halos and relics are Mpc-scale diffuse radio sources in galaxy
clusters, with a steep spectral index $\alpha>1$ ($S\propto \nu^{-\alpha}$). It
has been proposed that they arise from particle acceleration induced by
turbulence and weak shocks, injected in the intracluster medium (ICM) during
mergers. MACS J1149.5+2223 (MACS J1149) is a high redshift ($z=0.544$) galaxy
cluster possibly hosting a radio halo and a relic. We analysed LOFAR, GMRT, and
JVLA radio data at 144, 323, 1500 MHz, and Chandra X-ray data to characterise
the thermal and non-thermal properties of the cluster. We obtained radio images
at different frequencies to investigate the spectral properties of the radio
halo. We used Chandra X-ray images to constrain the thermal properties of the
cluster. We measured a steep spectrum of the halo, with $\alpha=1.49\pm 0.12$
between 144 and 1500 MHz. The radio surface brightness distribution across the
halo is found to correlate with the X-ray brightness of the ICM, with a
sub-linear slope in the range 0.4 to 0.6. We also report two possible cold
fronts in north-east and north-west, but deeper X-ray observations are required
to firmly constrain the properties of the upstream emission. We show that the
combination of high redshift, steep radio spectrum, and sub-linear radio-X
scaling of the halo rules out hadronic models. An old ($\sim 1 $ Gyr ago) major
merger likely induced the formation of the halo through stochastic
re-acceleration of relativistic electrons. We suggest that the two possible
X-ray discontinuities may actually be part of the same cold front. In this
case, the coolest gas pushed towards the north-west might be associated with
the cool core of a sub-cluster involved in the major merger. The peculiar
orientation of the south-east relic might indicate a different nature of this
source and requires further investigation.
