Exploring the physical properties of lensed star-forming clumps at
  $2\lesssim z \lesssim6$
  We study the physical properties (size, stellar mass, luminosity, star
formation rate) and scaling relations for a sample of 166 star-forming clumps
with redshift $z \sim 2-6.2$. They are magnified by the Hubble Frontier Field
galaxy cluster MACS~J0416 and have robust lensing amplification ($2\lesssim \mu
\lesssim 82$) computed by using our high-precision lens model, based on 182
multiple images. Our sample extends by $\sim 3$ times the number of
spectroscopically-confirmed lensed clumps at $z \gtrsim 2$. We identify clumps
in ultraviolet continuum images and find that, whenever the effective spatial
resolution (enhanced by gravitational lensing) increases, they fragment into
smaller entities, likely reflecting the hierarchically-organized nature of star
formation. Kpc-scale clumps, most commonly observed in non-lensed fields, are
not found in our sample. The physical properties of our sample extend the
parameter space typically probed by $z \gtrsim 1$ non-lensed observations and
simulations, by populating the low mass (M$_\star \lesssim 10^7$ M$_\odot$),
low star formation rate (SFR $\lesssim 0.5$ M$_\odot$ yr$^{-1}$), and small
size (R$_\mathrm{eff} \lesssim 100$ pc) regime. The new domain probed by our
study approaches the regime of compact stellar complexes and star clusters. In
the mass-size plane our sample span the region between galaxies and globular
clusters, with a few clumps laying in the region populated by young star
clusters and globular-clusters. For the bulk of our sample, we measure
star-formation rates which are higher than those observed locally in compact
stellar systems, indicating different conditions for star formation at high
redshift and in the local Universe.
