[T]he magnetic moment of the proton is 660 times smaller than that of the electron, which means that it is considerably harder to detect. . . . The magnetic moment of the anti-proton is currently only known to three decimal places.
A five year old experiment in Germany has just managed, for the first time, to directly measure the spin of a single proton, and provides a mechanism to measure the magnetic moment of individual protons and anti-protons with much greater precision in the near future at a tiny fraction of the cost of big budget experiments like the LHC. These experiments allow more accurate comparisons of the proton and anti-proton constants, to determine if they truly observe CP symmetry and offer a different methodology than prior experiments which have relied on statistical analysis of the behavior of large groups of protons or anti-protons.
Nobody really expects any surprises in these measurements. There is every reason to believe that the magnetic moments of the proton and anti-proton will be identical, and the values currently known to three significant digits are unlikely to be seriously nudged. But, greater precision in a measurement of one of the most common parts of the background that must be subtracted away to determine what more interesting quantum physics are going on in experiments will make it easier to observe new physics elsewhere with the same experiments (or even when looking at past experimental results). These constants help to set constants of QCD that have wide applicability, and the inaccuracies in these measurements limit the accuracy of theoretical QCD predictions generally.
More accurate measurements may also narrow the range of beyond the Standard Model physics, because a great many otherwise plausible extensions of the Standard Model incorrectly model proton behavior. The most obvious defect in these theories is that a large class of them generically predict proton decay, something that has never been observed, at rates far greater than are consistent with experimental limits on its frequency which are consistent with a half-life with an order of magnitude that is approaching the fourteen billion year age of the universe. But, if precision magnetic moment data were available, that would provide another way to identify extensions of the Standard Model that must be incorrect because they incorrectly model the proton, which is the most common form of baryonic matter, for which it is usually relatively easy to derive a prediction directly from the theory.