- The 3.7 sigma deviation from standard model predictions in the differential distribution of the B➝K*μ+μ- decay products.
- The 2.6 sigma violation of lepton flavor universality in B+→K+l+l- decays.
The first anomaly is statistically more significant. However, the theoretical error of the standard model prediction is not trivial to estimate and the significance of the anomaly is subject to fierce discussions. Estimates in the literature range from 4.5 sigma to 1 sigma, depending on what is assumed about QCD uncertainties. For this reason, the second anomaly made this story much more intriguing. In that case, LHCb measures the ratio of the decay with muons and with electrons: B+→K+μ+μ- vs B+→K+e+e-. This observable is theoretically clean, as large QCD uncertainties cancel in the ratio. Of course, 2.7 sigma significance is not too impressive; LHCb once had a bigger anomaly (remember CP violation in D meson decays?) that is now long gone. But it's fair to say that the two anomalies together are marginally interesting.
One nice thing is that both anomalies can be explained at the same time by a simple modification of the standard model. Namely, one needs to add the 4-fermion coupling between a b-quark, an s-quark, and two muons:
with Λ of order 30 TeV. Just this one extra coupling greatly improves a fit to the data, though other similar couplings could be simultaneously present. The 4-fermion operators can be an effective description of new heavy particles coupled to quarks and leptons. For example, a leptoquark (scalar particle with a non-zero color charge and lepton number) or a Z' (neutral U(1) vector boson) with mass in a few TeV range have been proposed. These are of course simple models created ad-hoc. Attempts to put these particles in a bigger picture of physics beyond the standard model have not been very convincing so far, which may be one reason why the anomalies are viewed a bit skeptically. The flip side is that, if the anomalies turn out to be real, this will point to unexpected symmetry structures around the corner.
this paper we can see that there are several other 2-sigmish anomalies in B-decays that may possibly have the same origin. More data and measurements in more decay channels should clarify the picture. In particular, violation of lepton flavor universality may come together with lepton flavor violation. Observation of decays forbidden in the standard model, such as B→Keμ or B→Kμτ, would be a spectacular and unequivocal signal of new physics.