  An energetic muon beam is an attractive key to unlock new physics beyond the
Standard Model: the lepton flavor violation or the anomalous magnetic moment,
and also is a competitive candidate for the expected neutrino factory. Lots of
the muon scientific applications are limited by low flux cosmic-ray muons, low
energy muon sources or extremely expensive muon accelerators. An prompt
acceleration of the low-energy muon beam is found in the beam-driven plasma
wakefield up to $\mathrm{TV/m}$. The muon beam is accelerated from
$275\mathrm{MeV}$ to more than $10\mathrm{GeV}$ within $22.5\mathrm{ps}$.
Choosing the injection time of the muon beam in a proper range, the
longitudinal spatial distribution and the energy distribution of the
accelerated muon beam are compressed. The efficiency of the energy transfer
from the driven electron beam to the muon beam can reach $20\%$. The prompt
acceleration scheme is a promising avenue to bring the expected neutrino
factory and the muon collider into reality and to catch new physics beyond the
Standard Model.
