  We present deep LOFAR observations between 120-181 MHz of the "Toothbrush"
(RX J0603.3+4214), a cluster that contains one of the brightest radio relic
sources known. Our LOFAR observations exploit a new and novel calibration
scheme to probe 10 times deeper than any previous study in this relatively
unexplored part of the spectrum. The LOFAR observations, when combined with
VLA, GMRT, and Chandra X-ray data, provide new information about the nature of
cluster merger shocks and their role in re-accelerating relativistic particles.
We derive a spectral index of $\alpha = -0.8 \pm 0.1$ at the northern edge of
the main radio relic, steepening towards the south to $\alpha \approx - 2$. The
spectral index of the radio halo is remarkably uniform ($\alpha = -1.16$, with
an intrinsic scatter of $\leq 0.04$). The observed radio relic spectral index
gives a Mach number of $\mathcal{M} = 2.8^{+0.5}_{-0.3}$, assuming diffusive
shock acceleration (DSA). However, the gas density jump at the northern edge of
the large radio relic implies a much weaker shock ($\mathcal{M} \approx 1.2$,
with an upper limit of $\mathcal{M} \approx 1.5$). The discrepancy between the
Mach numbers calculated from the radio and X-rays can be explained if either
(i) the relic traces a complex shock surface along the line of sight, or (ii)
if the radio relic emission is produced by a re-accelerated population of
fossil particles from a radio galaxy. Our results highlight the need for
additional theoretical work and numerical simulations of particle acceleration
and re-acceleration at cluster merger shocks.
