First of all, what are two-higgs doublet models and why you should care about them ? For a start, I assume that you do know that we (i.e. CMS and ATLAS, the two giant experiments at the CERN Large Hadron Collider) have discovered, two years ago, a new particle which we tried very hard for a while not to call "Higgs boson" (the first papers in fact claim they found a new boson "while they sought for the Higgs"), but cannot now abstain from calling it such.
The fact is, the Higgs particle was predicted as far back as 50 years ago by a handful of theorists as the only way to explain a number of things. The CERN finding of 2012 fit the bill quite nicely from the start, and by now we are extremely confident that it is a Higgs boson. "A" Higgs boson, mind you: not "the" Higgs boson. The difference is important.
In fact, 50 years ago the existence of the Higgs was postulated as the simple way to make the theory consistent at a mathematical level with phenomena inferred and later verified, such as the existence of very massive weak bosons (those discovered by Rubbia with the UA1 experiment in 1983, the W and the Z particles). But it was not the only way: it was just the simplest. Simplicity is a very highly regarded quality in the organization of the physical world, but there might be reasons why Nature (the bitch, not the magazine) chose something a bit more complex.
To describe the properties of a physical system, even a macroscopic one, physicists use an equation which is called "Lagrangian" (from the Italian mathematician Giuseppe Ludovico Lagrangia) and which specifies the dynamics. The Lagrangian says ALL that there is to know about the possible behaviour of the system, and is thus a very compact and elegant formulations.
In the Standard Model, THE Higgs boson (the only one) is the result of a symmetry breaking mechanism that magically converts the nice and symmetric electroweak Lagrangian, which may include a doublet of complex scalar fields (four degrees of freedom), into a uglier version where a single physical particle with no spin (a "scalar" boson) emerges from the doublet; the three remaining degrees of freedom are used to generate mass terms for the W and Z bosons. That magic is remarkable, but if one had started with TWO doublets of complex scalar fields in the equation (for a total of eight degrees of freedom) the symmetry breaking mechanism would make you end up with FIVE Higgs bosons, as the W and Z still only require three of those.
Two-Higgs doublet models (2DHM) are arguably less economical than the standard model proper. However, they may allow for extra features of the theory, which enable its extension into something eventually more powerful. For instance, Supersymmetry in its simplest manifestations is a two-higgs doublet model: in SUSY there are always at least five Higgs bosons. But 2DHMs do not limit to SUSY, so it is a more general formalism which is quite useful to study in detail.
So far we found a Higgs boson at 125 GeV. It is neutral, it is scalar (zero spin), and it behaves as the only physical state emerging from the breaking of a model with just one doublet. But it is entirely plausible that it is only the first one state we discover of those five. The paper by Passera and collaborators considers the input (there is a light neutral Higgs at 125 GeV) and takes into account all other observed features of electroweak phenomenology - measured value of Z boson properties, anomalous magnetic moment of the muon, B physics observables, direct constraints on the existence of other Higgs states- to try and figure out just how much of 2DHMs is left on the table, not yet excluded directly or indirectly.
The paper is a bit technical so I will stop here, but I will summarize its conclusions. One important observation is that the discrepancy observed in the magnetic moment of the muon with Standard Model predictions, if interpreted in the context of 2DHMs, points in a very specific direction: we can explain that 3-sigma discrepancy if we take a specific kind of 2DHMs, the one called "type X", or "lepton-specific". I won't explain what those are, but they are a specific subset of the 2DHMs one can conceive. They accommodate the observed muon anomaly together with the observed constraints in a particular decay mode of the b-quark, which includes s-quarks and photons. Hence the authors can conclude as follows:
The parameter space favourable for the muon g-2 in type X models is quite limited inThis is a very definite prediction which is worth following with attention. Of course ATLAS and CMS are going to investigate these final states in Run 2, which is due to start in 2015; but the paper adds interest to these signatures.
mass ranges for the heavy neutral and charged scalar: M(H0),M(H+)<200 GeV (with small MA and large tan(β)). These bosons can be searched for in forthcoming collider experiments, even if this parameter region could be elusive because the productions of the additional Higgs bosons A, H, and H+ are suppressed either by 1/ tan^2 β (in single productions, e.g. through gluon fusion) or by cos(β-α) (associated productions of Vφ and hφ). The leading search channels for the extra bosons would then be pair or associated productions through pp -> HH,HA [...] followed by [leptonic decays], which can be readily tested at the next run of the LHC.
Finally, I am quite happy to notice that I have been named in the Acknowledgements section of the paper... Thanks to the authors for the honour!