Simulation studies for upgrading the surface muon beamline $\mu$E4 at PSI

The $\mu$E4-LEM beamline at Paul Scherrer Institute (PSI, Switzerland) is a special muon beamline combining the hyprid type surface muon beamline $\mu$E4 with the low energy muon facility (LEM) and delivers ${\mu}^{+}$ with tunable energy up to 30 keV for low-energy muon spin rotation experiments (LE-$\mu$SR). We investigate a possible upgrade scenario for the surface muon beamline $\mu$E4 by replacing the last set of quadrupole triplet with a special solenoid to obtain about 1.4 times original beam intensity on the LEM muon moderator target. In order to avoid the muon beam intensity loss at the LEM spectrometer due to the stray magnetic field of the solenoid, three kinds of solenoid models have been explored and the stray field of the solenoid at the LEM facility is finally reduced to the magnitude of the geomagnetic field. A more radical design, "Super-$\mu$E4", has also been investigated for further increasing the brightness of the low energy muon beam, where we make use of the current $\mu$E4 channel and all sets of quadrupole triplets are replaced by large aperture solenoids. Together with the new slanted muon target E, about 2.2 times the original low energy muon beam intensity can be expected in the Super-$\mu$E4 beamline.