The recent experimental data of anomalous magnetic moments strongly indicates the existence of new physics beyond the standard model. An energetic µ + beam is a potential option to the µ + rare decay, µSR(the spin rotation, resonance and relaxation) technology, future muon colliders and neutrino factories. It is proposed a prompt µ + acceleration in a nonlinear donut wakefield driven by a shaped steep-rising-front Laguerre-Gaussian (LG) laser pulse. An analytical model is given and shows that a µ + beam can be focused by the electron cylinder at the center line of the donut bubble and accelerated by the front part of the longitudinal wakefield. A shaped LG laser with a shorter rise time can push plasma electrons to generate a higher-density electron sheath at the front of the bubble, which can enlarge the acceleration field. The acceleration field driven by the steep-rising-front shaped LG laser pulse is about four times higher than that driven by a normal LG laser pulse. Our simulation results show that a 300MeV µ + bunch can be accelerated to 2GeV and its transverse size is focused from initial w0 = 5µm to w = 1µm in the donut bubble driven by the steep-rising-front shaped LG laser pulse with the normal intensity a = 22.

The muon plays a key role in the field of particle physics and applied physics. To build the neutrino factories or muon colliders, high-quality muon sources are needed. At present, we can only get the low-flux cosmic-ray muons and low-energy accelerator-generated muons. The key issue about accelerating a low-initial-energy muon beam in the plasma wakefield driven by an electron beam is the phase matching between muons and a wakefield. A plasma density down-ramp is considered as an effective method for accelerating a low-initial-energy muon beam, and the decreasing phase velocity at the back edge of the wakefield can lower the muon trapped energy threshold. A 100 MeV muon beam can be accelerated to 6.21 GeV in the plasma wakefield based on a negative plasma density gradient. The trapping and accelerating process can be controlled by adjusting the parameters of the density down-ramp.