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The role of the violation of parity in yielding spin polarized muon beams can be directly identified in the rest frame of the pion, again.

We already considered the kinematics of this decay. We can now concentrate on angular momentum conservation, recalling that the $\pi^+$ is a spinless particle. Since, instead, the neutrino has spin one half, like the muon, the angular momenta of the two particles must come out opposite to each other. Furthermore the neutrino is highly relativistic: assuming that it is massless, its spin would be aligned with its linear momentum, i.e. its helicity is equivalent to chirality (handedness) multiplied by $\hbar/2$ .

The violation of parity in this event is equivalent to stating that only neutrinos of negative helicity (spin antiparallel to linear momentum) exist. Positive helicity is conversely the only possibility for antineutrinos.

This is often simplified by saying that, although in the macroscopic world the mirror image of any possible event is also a possible event, this symmetry is broken in the microscopic realm of elementary particles

This statement and the conservation of angular momentum impose that also the muon must have negative helicity. This is illustrated by the sketch on the right.

Therefore if we select by transport a very small solid angle of directions diverging from the primary target, and a momentum of exactly 29.79 MeV/c, the beam will be predominantly of muons from the decays of pions at rest on the surface of the target, and very nearly (*) 100% spin-polarized backwards with respect to linear momentum.

By the same argument negative forward muons would be produced in a similar geometry.

This is only one of the possible schemes for the production of muon beams, the most convenient for the majority of µSR experiments, since it provides fully polarized muons of relatively low energies, which stop in condensed matter within few hundreds micron from the surface (roughly 80 µm for Pb and 1 mm for water). In this way muons are probes of the bulk and require moderate amounts of material.

However higher energy beams can be obtained, simply by selecting the corresponding linear momentum in the transport channel. In this case the transported particles are pions and sufficient length of the transport path must be allowed for most pions to decay (with mean lifetime $\tau_\pi=26.033(5)$ ns.). This is accomplished by superconducting solenoids, on dedicated beamlines, where the helicoidal path is much longer than the physical length of the device. The use of higher energy muons (typically 80 to 100 MeV/c) is mandatory for pressure experiments, where the muons must penetrate a massive pressure cell.

(*) The lower limit on the mass of the muon neutrino (presently $m_\nu\,<\, 2.2\, \mbox{Mev/c}^2$ ) is set also by this argument. A finite mass implies that, notwithstanding parity violation, the spin of the neutrino has a component transverse to momentum, and so does the muon, resulting in a reduction of the spin polarization. Any larger mass than the quoted limit would result in a reduction of the experimental muon spin polarization in this kind of beams, outside the present best experimental error.

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