Planck far-infrared detection of Hyper Suprime-Cam protoclusters at $\bf
  z\sim4$: hidden AGN and star formation activity
  We perform a stacking analysis of {\it Planck}, {\it AKARI}, Infrared
Astronomical Satellite ($IRAS$), Wide-field Infrared Survey Eplorer ($WISE$),
and {\it Herschel} images of the largest number of (candidate) protoclusters at
$z\sim3.8$ selected from the Hyper Suprime-Cam Subaru Strategic Program
(HSC-SSP). Stacking the images of the $179$ candidate protoclusters, the
combined infrared (IR) emission of the protocluster galaxies in the observed
$12-850~\mu$m wavelength range is successfully detected with $>5\sigma$
significance (at $Planck$). This is the first time that the average IR spectral
energy distribution (SED) of a protocluster has been constrained at $z\sim4$.
The observed IR SEDs of the protoclusters exhibit significant excess emission
in the mid-IR compared to that expected from typical star-forming galaxies
(SFGs). They are reproduced well using SED models of intense starburst galaxies
with warm/hot dust heated by young stars, or by a population of active galactic
nuclei (AGN)/SFG composites. For the pure star-forming model, a total IR (from
8 to 1000 $\mu$m) luminosity of $19.3_{-4.2}^{+0.6}\times10^{13}~L_{\odot}$ and
a star formation rate (SFR) of $16.3_{-7.8}^{+1.0}\times10^3~M_{\odot}$
yr$^{-1}$ are found whereas for the AGN/SFG composite model,
$5.1_{-2.5}^{+2.5}\times10^{13}~L_{\odot}$ and
$2.1^{+6.3}_{-1.7}\times10^3~M_{\odot}$ yr$^{-1}$ are found. Uncertainty
remaining in the total SFRs; however, the IR luminosities of the most massive
protoclusters are likely to continue increasing up to $z\sim4$. Meanwhile, no
significant IR flux excess is observed around optically selected QSOs at
similar redshifts, which confirms previous results. Our results suggest that
the $z\sim4$ protoclusters trace dense, intensely star-forming environments
that may also host obscured AGNs missed by the selection in the optical.
