  We investigate the viability of primordial black hole (PBH) formation in the
Standard Model (SM) in a scenario that does not rely on specific inflationary
features or any exotic physics such as phase transitions or non-minimal
coupling to gravity. If the Brout-Englert-Higgs (BEH) field lies exactly at the
transition between metastability and stability, its potential exhibits an
inflexion point due to radiative corrections. The BEH can act like a stochastic
curvaton field, leading to a non-Gaussian tail of large curvature fluctuations
that later collapse into PBHs when they re-enter inside the horizon. This
scenario would require a precise value of the top-quark mas to ensure the Higgs
stability, which is disfavored but still consistent with the most recent
measurements. However, we also find that large curvature fluctuations are also
generated on cosmological scales that are inconsistent with cosmic microwave
background (CMB) observations. We therefore conclude that the SM cannot have
led to the formation of PBHs based on this mechanism. Nevertheless, a variation
of the scenario based on the Palatini formulation of gravity may have provided
the conditions to produce stellar-mass PBHs with an abundance comparable to
dark matter, without producing too large curvature fluctuations on cosmological
scales.
