DaveC426913 said:
A fifth fundamental force would be pretty significant, if true.
In their defense, Einsteins' relativity and then quantum mechanics both "turned physics on its head" - even for (especially for) the layperson. It overturned how we see our universe.
Suppose that there really is a 4.2 sigma discrepancy and the BMW calculation is wrong.
One thing we can say with certainty is that a measurement that is 4.2 sigma from a theory prediction for muon g-2 does not tell us "what" is causing the discrepancy.
It only tells us "how big" the discrepancy is, and even then, only partially. A discrepancy could be due to a tiny tweak to something central to calculating muon g-2, or it could be a huge tweak to something that only makes a small contribution to the overall result, or it could be something in between, or a bit of all of those explanations.
But, since the calculation of muon g-2 receives contributions from all three Standard Model forces and most of the Standard Model fundamental particles (in addition to any new physics contributions), it is a very global measure of the consistency of the Standard Model with experiment.
Lots of plausible explanations wouldn't involve a "fifth force", just one or more new particles. For example, while no one is proposing this particular explanation of the muon g-2 anomaly, as proof of concept, if there were a fourth generation of Standard Model fermions (t', b', tau', tau neutrino'), this would change the value of muon g-2 a little, without changing any of the forces of the Standard Model.
To give a more "real" example, one of the big differences between the prediction that says there is a 4.2 sigma distinction between experiment and prediction, and the one that says that there is only a 1.6 sigma distinction, is that the second prediction treats up and down quarks as having different masses, while the first one uses only the average mass of the up and down quarks. This slight tweak in the assumed masses of two Standard Model quarks makes a quite significant impact on the predicted discrepancy between theory and experiment, even though both the up quark and down quark masses are tiny (about 2.5% and 5% respectively, of the muon mass).
Also, keep in mind that the discrepancy, even if it is highly statistically significant, is still tiny. It is on the order of 2 parts per billion.
The same can be said of other anomalies that are out there.
For example, there are several kinds of decays of B mesons (composite particles with a valence quark and anti-quark, one of which is a b quark or anti-b quark) which seem to produce decays that generate more electrons than muons for reasons beyond those attributable to their mass differences even though in the Standard Model, this shouldn't happen. This isn't seen in any other kind of decay process.
But guess what. There are almost no processes of engineering importance, or importance in the post-Big Bang natural world, even in extreme circumstances like the inner structure of neutron stars and supernova, in which the ratio of lepton flavors produced in B meson decays play an important part. It is intellectually interesting and could even lead to a tweak of the Standard Model, but it is not important in any practical sense. Nobody even predicted that b-quarks existed until 1973 and no one in the entire history of life on Earth had knowingly observed one until 1977. B mesons are so ephemeral that they have a mean lifetime on the order of a trillionth of a second, and have only ever been produced in the lab in a handful high energy particle colliders.
So, while there may be a crack or two in the Standard Model that doesn't yet have a full explanation, it is still an incredibly precise and accurate description of the real world, and the cracks that are present are either tiny, or in highly exotic phenomena produced only in the most rarified laboratory conditions.