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The scalar potential of the $N$-Higgs-doublet model (NHDM) is best analyzed not in the space of $N$ complex doublets ${\ensuremath{\phi}}_{a}$ but in the ${N}^{2}$-dimensional space of real-valued bilinears constructed of ${\ensuremath{\phi}}_{a}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\phi}}_{b}$. In particular, many insights have been gained into $CP$ violation in the 2HDM and 3HDM by studying how generalized $CP$ transformations (GCPs) act in this bilinear space. These insights relied on the fact that GCPs, which involved an odd number of mirror reflection, could be clearly distinguished from Higgs family transformations by the sign of the determinant of the transformation matrix. It was recently pointed out that this criterion fails starting from 4HDM, where the reflection/rotation dichotomy does not exist anymore. In this paper, we restore intuition by finding a different quantity which faithfully discriminates between GCPs and Higgs family transformations in the bilinear space for any number of Higgs doublets. We also establish the necessary and sufficient conditions for an orthogonal transformation in the bilinear space to represent a viable transformation back in the space of $N$ doublets, which is helpful if one prefers to build an NHDM directly in the bilinear space.

Establishing if multi-Higgs potentials are bounded from below (BFB) can be rather challenging, and it may impede efficient investigation of all phenomenological consequences of such models. In this paper, we find the necessary and sufficient BFB conditions for the three-Higgs-doublet model (3HDM) with the global symmetry group $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$. We observed an important role played by charge-breaking directions in the Higgs space, even for situations when a good-looking neutral minimum exists. This remark is not limited to the particular model we consider but represents a rather general feature of elaborate multi-Higgs potentials which must be carefully dealt with. Also, applying this method to Weinberg's model (the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetric 3HDM) turned out to be more challenging than was believed in the literature. In particular, we have found that the approach taken in a paper from 2009 does not lead to the necessary and sufficient BFB conditions for this case.

Three-Higgs-doublet models (3HDM) allow for a novel, physically distinct form of CP invariance: CP symmetry of order 4 (CP4). Due to the large basis change freedom in 3HDM, it is imperative to recognize the presence of a possibly hidden CP4 in a basis-invariant way. In the present work, we solve this problem and establish basis-invariant necessary and sufficient conditions for a 3HDM to possess a CP4 symmetry. We also derive a basis-invariant criterion to decide whether or not a CP4 symmetric 3HDM possesses any additional CP symmetry, as well as a criterion to decide whether or not CP4 is spontaneously broken.

We explore the origin and Yukawa interactions of the scalars with peculiar $CP$ properties which were recently found in a multi-Higgs model based on an order-4 $CP$ symmetry. We relate the existence of such scalars to the enhanced freedom of defining $CP$, even beyond the well-known generalized $CP$ symmetries, which arises in models with several zero-charge scalar fields. We also show that despite possessing exotic $CP$ quantum numbers, these scalars do not have to be inert: they can have $CP$-conserving Yukawa interactions provided the $CP$ acts on fermions by also mixing generations. This paper focuses on formal aspects---exposed in a pedagogical manner---and includes a brief discussion of possible phenomenological consequences.

Models beyond the Standard Model (bSM) often involve elaborate Higgs sectors, which can be a source of CP-violation. It brings up the question of recognizing in an efficient way whether a model is CP-violating. There is a diffuse belief that the issue of explicit CP invariance can be linked to the existence of a basis in which all coefficients are real; with even a theorem proposed a decade ago claiming that the scalar sector of any multi-Higgs doublet model is explicitly CP-conserving if and only if all of its coefficients can be made real by a basis change. This is compounded by the fact that in all specific multi Higgs models considered so far, the calculations complied with this claim. Here, we present the first counterexample to this statement: a CP-conserving three-Higgs-doublet model for which no real basis exists. We outline the phenomenological consequences of this model, and notice that the extra neutral Higgs bosons are neither CP-even nor CP-odd but are "half-odd" under the generalized CP-symmetry of the model.

We study the scalar sector of the general N-Higgs-doublet model via geometric constructions in the space of gauge orbits. We give a detailed description of the shape of the orbit space both for general N and, in more detail, for N = 3. We also comment on remarkable analogies between NHDM and quantum information theory.

We consider the evolution of the ground state in the two Higgs doublet model during cooling down of the Universe after the big bang. Different regions in the space of free parameters of this model correspond to different sequences of thermal phase transitions. We discuss different paths of thermal evolution and corresponding evolution of physical properties of the system for different modern values of the parameters.

We study the masses of the Higgs bosons in the most general two-Higgs-doublet model in a basis-independent approach. We adapt the recently developed Minkowski-space formalism to this problem and calculate traces of any power of the mass matrix in a compact and reparametrization-invariant form. Our results can be used to gain insight into the dynamics of the scalar sector of the general two-Higgs-doublet model.

We continue to explore the consequences of the recently discovered Minkowski space structure of the Higgs potential in the two-Higgs-doublet model. Here, we focus on the vacuum properties. The search for extrema of the Higgs potential is reformulated in terms of 3-quadrics in the $3+1$-dimensional Minkowski space. We prove that 2HDM cannot have more than two local minima in the orbit space and that a twice-degenerate minimum can arise only via spontaneous violation of a discrete symmetry of the Higgs potential. Investigating topology of the 3-quadrics, we give concise criteria for existence of noncontractible paths in the Higgs orbit space. We also study explicit symmetries of the Higgs potential/Lagrangian and their spontaneous violation from a wider perspective than usual.

The Higgs potential of 2HDM keeps its generic form under the group of transformation $GL(2,C)$, which is larger than the usually considered reparametrization group $SU(2)$. This reparametrization symmetry induces the Minkowski space structure in the orbit space of 2HDM. Exploiting this property, we present a geometric analysis of the number and properties of stationary points of the most general 2HDM potential. In particular, we prove that charge-breaking and neutral vacua never coexist in 2HDM and establish conditions when the most general explicitly $CP$-conserving Higgs potential has spontaneously $CP$-violating minima. We also define the prototypical model of a given 2HDM, which has six free parameters less than the original one but still contains all the essential physics. Our analysis avoids manipulation with high-order algebraic equations.

We obtain tree-level unitarity constraints for the most general Two-Higgs-Doublet Model (2HDM) with explicit $CP$-violation. We briefly discuss correspondence between possible violation of tree-level unitarity limitation and physical content of the theory.

We reanalyze the issue of the PQCD factorization scale in the exclusive \ensuremath{\rho} production in diffractive deep inelastic scattering from the ${k}_{t}$-factorization point of view. We find that this scale differs significantly from the widely used value ${(Q}^{2}{+m}_{\ensuremath{\rho}}^{2})/4,$ and it is a much flatter function of ${Q}^{2}.$ With these results in mind, we discuss the ${Q}^{2}$ shape of the \ensuremath{\rho} meson production cross section and comment on the recent ZEUS observation of energy-independent ratio $\ensuremath{\sigma}({\ensuremath{\gamma}}^{*}\stackrel{\ensuremath{\rightarrow}}{p}\ensuremath{\rho}p)/{\ensuremath{\sigma}}_{\mathrm{tot}}({\ensuremath{\gamma}}^{*}p).$

The use of the differential gluon structure function of the proton ${\cal F}(x,Q^{2})$ introduced by Fadin, Kuraev and Lipatov in 1975 is called upon in many applications of small-x QCD. We report here the first determination of ${\cal F}(x,Q^{2})$ from the experimental data on the small-x proton structure function $F_{2p}(x,Q^{2})$. We give convenient parameterizations for ${\cal F}(x,Q^{2})$ based partly on the available DGLAP evolution fits (GRV, CTEQ & MRS) to parton distribution functions and on realistic extrapolations into soft region. We discuss an impact of soft gluons on various observables. The x-dependence of the so-determined ${\cal F}(x,Q^{2})$ varies strongly with Q^2 and does not exhibit simple Regge properties. None the less the hard-to-soft diffusion is found to give rise to a viable approximation of the proton structure function F_{2p}(x,Q^2) by the soft and hard Regge components with intercepts \Delta_{soft}=0 and \Delta_{hard}\sim 0.4., Comment: 37 pages, 25 figures

If next generations of heavy quarks and leptons exist within the standard model (SM), they can manifest themselves in Higgs boson production at the Tevatron and the LHC, before being actually observed. This generation leads to an increase of the Higgs boson production cross section via gluon fusion at hadron colliders by a factor 6-9. So, the study of this process at the Tevatron and LHC can finally fix the number of generations in the SM. Using the $WW^*$ Higgs boson decay channel, the studies at the upgraded Tevatron will answer the question about the next generation for mass values 135 GeV $\lsim M_H\lsim 190$ GeV. Studying the $\tau\bar{\tau}$ channel we show its large potential for the study of the Higgs boson at the LHC even in the standard case of three generations. At the Tevatron, studies in this channel could explore the mass range 110-140 GeV., Comment: 13 pages, 4 figures, LaTeX/RevTeX, final version accepted for publication

We discuss the potentialities of the non-standard interaction study via the Higgs boson production at photon ($\ggam$ and $e\gamma$) colliders. We estimate the scale of New Physics phenomena beyond the standard model that can be seen in the experiments with Higgs boson production. In particular, the effect of new heavy particles within the standard model is shown to be quite observable., Comment: 16 pages, 8 eps figures, RevTeX, content changed