### 5.4 Spin polarized torsion balances

Clock comparison experiments constrain the parameter for protons and neutrons. Spin polarized
torsion balances are able to place comparable limits on the electron sector of the mSME [56]. The best
limits on (where is the spatial direction, including that parallel to the earth’s rotation axis) for the
electron come from two balances, one in Washington [170, 141] and one in Taiwan [148]. We detail the
Washington experiment for pedagogical purposes - the two approaches are similar. In the Washington
experiment two different types of magnets (SmCo and Alnico) are arranged in an octagonal shape. Four
SmCo magnets are on one side of the octagon and four Alnico magnets are on the other. The
magnetization of both types of magnets is set to be equal and in the angular direction around
the octagon. This minimizes any magnetic interactions. However, with equal magnetization
the net electron spin of the SmCo and Alnico magnets differs as the SmCo magnets have a
large contribution to their overall magnetization from orbital angular momentum of Sm ions.
Therefore the octagonal pattern of magnets has an overall spin polarization in the octagon’s
plane.
A stack of four of these octagons are suspended from a torsion fiber in a vacuum chamber. The magnets
give an estimated net spin polarization equivalent to approximately aligned electron spins.
The whole apparatus is then mounted on a turntable. As the turntable rotates a bound on
Lorentz violation is obtained in the following manner. Lorentz violation in the mSME gives
rise to an interaction potential for non-relativistic electrons of the form , where
stands for direction and is the electron magnetic moment. As the turntable rotates, since
points in some fixed direction in space, the interaction produces a torque on the torsion
balance. The magnet apparatus therefore twists on the torsion fiber by an amount given by

where is the horizontal component of , is the frequency of rotation, is an initial phase
due to orientation, and is the torsion constant. Since and are known, a measurement of
will give the magnitude of . Since is also known, gives a limit on the size of . The
absence of any extra twist limits all components of for the electron to be less than . The
Taiwan experiment uses a different material () [148]. The bounds from this experiment are of
order for the components of perpendicular to the spin axis and for the
parallel component.
To conclude this section, we note that the torsion balance experiments are actually sensitive enough to
also constrain the dimension 5 operators in Equation (40). Assuming that all lower dimension operators are
absent, the constraint on the dimension five operators is [230].