  The first direct determination of the ground-state-to-ground-state
${\beta^{-}}$-decay $Q$-value of $^{77}$As to $^{77}$Se was performed by
measuring their atomic mass difference utilizing the double Penning trap mass
spectrometer, JYFLTRAP. The resulting $Q$-value is 684.463(70) keV,
representing a remarkable 24-fold improvement in precision compared to the
value reported in the most recent Atomic Mass Evaluation (AME2020). With the
significant reduction of the uncertainty of the ground-state-to-ground-state
$Q$-value and knowledge of the excitation energies in $^{77}$Se from
$\gamma$-ray spectroscopy, the ground-state-to-excited-state $Q$-value of the
transition $^{77}$As (3/2$^{-}$, ground state) $\rightarrow$ $^{77}$Se$^{*}$
(5/2$^{+}$, 680.1035(17) keV) was refined to be 4.360(70) keV. We confirm that
this potential low $Q$-value ${\beta^{-}}$-decay transition for neutrino mass
determination is energetically allowed at a confidence level of about
60$\sigma$. Nuclear shell-model calculations with two well-established
effective Hamiltonians were used to estimate the partial half-life for the low
$Q$-value transition. The half-life was found to be of the order of 10$^{9}$
years for this first-forbidden non-unique transition, which rules out this
candidate a potential source for rare-event experiments searching for the
electron antineutrino mass.
