Live Fast, Die $\alpha$-Enhanced: The Mass-Metallicity-$\alpha$ Relation
  of the Milky Way's Disrupted Dwarf Galaxies
  The Milky Way's satellite galaxies ("surviving dwarfs") have been studied for
decades as unique probes of chemical evolution in the low-mass regime. Here we
extend such studies to the "disrupted dwarfs", whose debris constitutes the
stellar halo. We present abundances ([Fe/H], [$\alpha$/Fe]) and stellar masses
for nine disrupted dwarfs with $M_{\star}\approx10^{6}-10^{9}M_{\odot}$ from
the H3 Survey (Sagittarius, $Gaia$-Sausage-Enceladus, Helmi Streams, Sequoia,
Wukong/LMS-1, Cetus, Thamnos, I'itoi, Orphan/Chenab). The surviving and
disrupted dwarfs are chemically distinct: at fixed mass, the disrupted dwarfs
are systematically metal-poor and $\alpha$-enhanced. The disrupted dwarfs
define a mass-metallicity relation (MZR) with a similar slope as the $z=0$ MZR
followed by the surviving dwarfs, but offset to lower metallicities by
$\Delta$[Fe/H]$\approx0.3-0.4$ dex. Dwarfs with larger offsets from the $z=0$
MZR are more $\alpha$-enhanced. In simulations as well as observations,
galaxies with higher $\Delta$[Fe/H] formed at higher redshifts -- exploiting
this, we infer the disrupted dwarfs have typical star-formation truncation
redshifts of $z_{\rm{trunc}}{\sim}1-2$. We compare the chemically inferred
$z_{\rm{trunc}}$ with dynamically inferred accretion redshifts and find almost
all dwarfs are quenched only after accretion. The differences between disrupted
and surviving dwarfs are likely because the disrupted dwarfs assembled their
mass rapidly, at higher redshifts, and within denser dark matter halos that
formed closer to the Galaxy. Our results place novel archaeological constraints
on low-mass galaxies inaccessible to direct high-$z$ studies: (i) the redshift
evolution of the MZR along parallel tracks but offset to lower metallicities
extends to $M_{\star}\approx10^{6}-10^{9}M_{\odot}$; (ii) galaxies at
$z\approx2-3$ are $\alpha$-enhanced with [$\alpha$/Fe]$\approx0.4$.
