What is $a_\mu$ in particle physics? In terms of a muon; and its magnetic moment? What is, '$a_\mu$ × $10^9$ - 1165900 '? 
What does $a_\mu × 10^9 - 1165900$ mean in the muon $g-2$ experiment at Fermilab?
 A: The spin magnetic moment of a fundamental particle with mass $m$, charge $q$, and spin 1/2 is
$$\vec\mu=g\frac{q}{2m}\vec S$$
where $\vec S$ is its spin vector. The "$g$-factor" is a dimensionless number. The Dirac equation predicts that it should be exactly 2.
However, the complications of quantum field theory cause $g$ to differ slightly from 2, so the small dimensionless number
$$a=\frac{g-2}{2},$$
measuring the fractional departure of $g$ from the Dirac value of 2, is of intense theoretical interest. It is called the "anomalous magnetic moment". For an electron we write $a_e$ and for a muon we write $a_\mu$.
The currently-accepted theoretical prediction for $a_\mu$ is
$$a_\mu=0.001\,165\,918\,04.$$
If you are wondering where this strange value comes from, it is approximately $\alpha/2\pi$, where $\alpha$ is the famous fine- structure constant, which measures the coupling between charged particles and photons. After this famous 1948 calculation by Schwinger, theorists have gone on to calculate corrections involving higher powers of $\alpha$ using perturbation theory in the form of Feynman diagrams with more loops. Each additional power of $\alpha$ is harder and harder to calculate.
Multiplying this value by a billion and subtracting 1165900 gives
$$a_\mu\times 10^9 -1165900 = 18.04,$$ a nicer number to keep in one's head. (After all, the digits before this are not in question.) The $1\sigma$ uncertainty in this number is 0.51.
The fact that, as your chart shows, the experimental number is 20-21 rather than 18 is the current tension between theory and experiment. If the theoretical prediction holds up, and the measurements hold up, then the Standard Model will need to be revised.
But the theoretical prediction is a fiendishly difficult calculation involving something like 12,000 complicated Feynman diagrams, and another group of theorists are claiming it is wrong and that the correct theoretical value is much closer to 20, so it is possible that there is no problem with the Standard Model.
