  Two next generation muon $g-2$ experiments at Fermilab in the US and at
J-PARC in Japan have been designed to reach a four times better precision from
0.54 ppm to 0.14 ppm and the challenge for the theory side is to keep up in
precision as far as possible. This has triggered a lot of new research
activities. The main motivation is the persisting 3 to 4 $\sigma$ deviation
between standard theory and experiment. As Standard Model predictions almost
without exception match perfectly all other experimental information, the
deviation in one of the most precisely measured quantities in particle physics
remains a mystery and inspires the imagination of model builders. Plenty of
speculations are aiming to explain what beyond the Standard Model effects could
fill what seems to be missing. Here very high precision experiments are
competing with searches for new physics at the high energy frontier lead by the
Large Hadron Collider at CERN. Actually, the tension is increasing steadily as
no new states are found which could accommodate the $g_\mu-2$ discrepancy. With
the new muon $g-2$ experiments this discrepancy would go up at least to 6
$\sigma$, in case the central values do not move, up to 10 $\sigma$ could be
reached if the present theory error could be reduced by a factor of two.
Interestingly, the new $\alpha$ from Berkeley by R. H. Parker et al. Science
360, 191 (2018): $\alpha^{-1}({\rm Cs18})=137.035999046(27)$ gives an $a_e$
prediction $a_e=0.00115965218157(23)$ such that $a_e^{\rm exp}-a_e^{\rm
the}=(-84\pm36)\times 10^{-14}$ shows a $- 2.3~ \sigma$ deviation now.

High Energy Physics - Phenomenology