  We discuss a new family of multi-quanta bound states in the Standard Model,
which exist due to the mutual Higgs-based attraction of the heaviest members of
the SM, namely, gauge quanta $W,Z$ and (anti)top quarks, $\bar t, t $. We use a
self-consistent mean-field approximation, up to a rather large particle number
$N$. In this paper we do not focus on weakly-bound, non-relativistic bound
states, but rather on "bags" in which the Higgs VEV is significantly
modified/depleted. The minimal number $N$ above which such states appear
strongly depends on the ratio of the Higgs mass to the masses of $W,Z,\bar{t},
t $: For a light Higgs mass $m_H \sim 50\, GeV$ bound states start from $N\sim
O(10)$, but for a "realistic" Higgs mass, $m_H\sim 100\, GeV$, one finds
metastable/bound
  $W,Z$ bags only for $N\sim O(1000)$. We also found that in the latter case
pure top bags disappear for all N, although top quarks can still be well bound
to the W-bags. Anticipating cosmological applications (discussed in a companion
paper) of these bags as "doorway states" for baryosynthesis, we also consider
the existence of such metastable bags at finite temperatures, when SM
parameters such as Higgs, gauge and top masses are significantly modified.
