Tommaso Dorigo reports a series of three different measurements at the Large Hadron Collider that differ with moderate statistical significance from the Standard Model expectation, each of which would be consistent with a modification of the Standard Model in which there are five Higgs bosons instead of just one, a situation called "two Higgs doublet" models.
Lepton Flavor Violation in Higgs boson decays
Most recently, the CMS experiment has seen 2.6 sigma evidence (without considering look elsewhere effects) of lepton flavor violation in Higgs boson decays; specifically decays that seem to involve one muon and one tau.
Baryon number (the number of quarks minus the number of anti-quarks in a system) is perfectly conserved in the Standard Model (with one exceedingly rare and obscure example called a sphaeleron), as is lepton number (the number of leptons minus the number of anti-leptons in a system).
Lepton flavor number conservation means that the number of third generation leptons (taus and tau neutrinos) minus the number of third generation anti-leptons (anti-taus and tau antineutrinos) is constant, and similarly for the second generation (muons and muon neutrinos) and first generation (electrons and electron-neutrinos).
Neutrinos violate lepton flavor number every time they oscillate, and there is no corresponding concept to lepton flavor number of quarks. But, there are no other observed violations of lepton flavor number conservation involving charged leptons (i.e. electrons, muons and taus) to date (for example, in Z boson decays), up to limits of more restrictive than one in a million Z boson decays, when all other Z boson decays have branching fractions on the order of 3.363% (which embodies the "democratic" principle or "universality" principle of weak force decays).
The fact that lepton flavor number violation could be occurring in Higgs boson decays, but not in other circumstances, could indicate that the Higgs sector we observe is not quite the Standard Model Higgs sector.
The muon-tau decays make up about 0.9% of all observed Higgs boson decays, when they shouldn't happen at all, except for extremely rare (i.e. one a billionish) instances with virtual oscillating neutrino loops that indirectly violate lepton flavor conservation.
B meson decays to excited kaons and muon-antimuon pairs
Another experimental anomaly from the LHCb experiment involves the angular distribution and branching fraction of decay products of B mesons (made of a bottom quark and and an anti-up quark) to an excited spin-1 kaon (made of a strange quark and an anti-up quark with aligned spins) called a K* and a muon-antimuon pair (i.e. B-> K*mumu)
These decays deviated from the Standard Model expectation by 3.7 sigma (2.8 sigma with the look elsewhere effect considered) in relatively low energy interactions. Dorigo notes, however, that in general, the assumption that error is distributed in a normal distribution (i.e. Gaussian) overstates the actual rarity of atypical events which actually have a distribution with longer tails of a particular type (known as the Student t test distribution type 10 with coefficient 1/1.11).
You need to be a hard core physicist to figure out that having five Higgs bosons rather than one could produce this effect, but at least one such person thinks that it could.
B meson decays violating lepton universality
A third experimental anomaly from the LHCb involves the ratio of the decay discussed above and a similar one that substitutes electron pairs for muon pairs (B->K*mumu/B->K*ee), which should be almost exactly unity if a principle called "lepton universality" applies. Instead, the decays with muon pairs are about 25% less common than the decays with electron pairs.
This is a 2.6 sigma effect before considering look elsewhere effects.
Standing alone a sub-three sigma deviation from the Standard Model expectation in an experiment is no big deal when you have thousands of scientists making hundreds or thousands of measurements every year.
But, if three different Standard Model deviations of this significance out of the modest number seen at the LHC are all consistent with the same theoretical tweak to the Standard Model, and that tweak wouldn't be expected to modify anything else that has been observed to fit the Standard Model expectation, then the results look more likely to be pointing at New Physics, in this case, the two Higgs doublet model.
Lubos discusses the underlying proposed theoretical physics model with more gusto and enthusiasm than I can muster. One gross oversimplification of what is going on that he mentions when sums up the commonalities between the effects, is that basically, in this model, there is something special about muons that differs from the Standard Model expectation for them. (One might imagine that strange quarks in a parallel position in the Standard Model mass matrix, might have a similar anomaly that is yet to be discovered).
This model would imply a mixing angle of muons and taus of 0.002, and a Z' boson with a mass in the vicinity of 550 GeV to 3,200 GeV, and an additional Higgs field with a vacuum expectation value in the tens of TeVs, in addition to BSM neutrino physics.
It is also worth noting that the LHC exclusions of non-Standard Model Higgs bosons in the absence of other supersymmetric phenomena are less impressive than you might intuitively expect, although aside from these data points, there is really no other evidence pointing positively to the existence of five Higgs bosons which one might think would have a more noticable signature.
Even taken together, these results are still a long shot that is more likely to result from experimental error, or a failure to include something in the theoretical model generating the Standard Model expectation.
For example, the theoretical expectation might inappropriately omit the probability that a Higgs boson will decay to a W boson pair with one W boson decaying to a tau and a tau-antineutrino, and another decaying to a muon and muon antineutrino, where noise in the signal obscures the missing energy from the antineutrinos produces and makes it look as if the Higgs boson decayed directly to a muon and a tau, because the data cuts aren't strict enough to exclude this kind of W boson pair decays from a Higgs boson with a noisy background that makes missing energy hard to notice.
But, these results do leave a rather more promising experimental window for beyond the Standard Model physics than most, although with the downside that the extra four Higgs bosons would have very subtle effects indeed on the phenomenology of the universe with a possible exception at very high energies.
Background on Two Higgs Doublet Models
The Standard Model has one Higgs doublet, which means it should have four Higgs bosons, but the W+, W- and Z boson "eat" three of them, leaving just one Higgs boson. A two Higgs doublet model has (2*4)-3=5 Higgs bosons, adding a positively charged Higgs boson (H+), a negatively charged Higgs boson (H-), a pseudoscalar Higgs boson, i.e. spin-0, odd parity (A), and an extra scalar Higgs boson just like the Standard Model one, but either heavier or lighter in mass, a lighter one (h) and a heavier one (H0).
All supersymmetric models have at least 5 Higgs bosons (two Higgs doublets), and some non-minimal SUSY models have 9 (three Higgs doublets) or 13 (four Higgs doublets) or more, with the Higgs bosons beyond the first five sometimes having exotic properties like a +/-2 electric charge. But, one can have a multiple Higgs doublet model as a stand alone feature without superpartners to the Standard Model particles.