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A bstract We explore three-Higgs-doublet models that may accommodate scalar Dark Matter where the stability is based on an unbroken U(1)-based symmetry, rather than the familiar ℤ 2 symmetry. Our aim is to classify all possible ways of embedding a U(1) symmetry in a three-Higgs-doublet model. The different possibilities are presented and compared. All these models contain mass-degenerate pairs of Dark Matter candidates due to a U(1) symmetry unbroken (conserved) by the vacuum. Most of these models preserve CP. In the CP-conserving case the pairs can be seen as one being even and the other being odd under CP or as having opposite charges under U(1). Not all symmetries presented here were identified before in the literature, which points to the fact that there are still many open questions in three-Higgs-doublet models. We also perform a numerical exploration of the U(1) × U(1)-symmetric 3HDM, this is the most general phase-invariant (real) three-Higgs-doublet model. The model contains a multi-component Dark Matter sector, with two independent mass scales. After imposing relevant experimental constraints we find that there are possible solutions throughout a broad Dark Matter mass range, 45–2000 GeV, the latter being a scan cutoff.

A bstract We explore the phenomenology of Weinberg’s ℤ 2 × ℤ 2 symmetric three-Higgs-doublet potential, allowing for spontaneous violation of CP due to complex vacuum expectation values. An overview of all possible ways of satisfying the stationary-point conditions is given, with one, two or three non-vanishing vacuum expectation values, together with conditions for CP conservation in terms of basis invariants. All possible ways of satisfying the conditions for CP conservation are given. Scans of allowed parameter regions are given, together with measures of CP violation, in terms of the invariants. The light states identified in an earlier paper are further explored in terms of their CP-violating couplings. Loop-induced CP violation in WWZ couplings, as well as charge-asymmetric scattering are also commented on.

The Two Higgs Doublet Model invariant under the gauge group S U ( 2 ) × U ( 1 ) is known to have six additional global discrete or continuous symmetries of its scalar sector. We have discovered regions of parameter space of the model which are basis and renormalization group invariant to all orders of perturbation theory in the scalar and gauge sectors, but correspond to none of the hitherto considered symmetries. We therefore identify seven new symmetries of the model and discuss their phenomenology. Soft symmetry breaking is required for some of these models so that electroweak symmetry breaking can occur. We show that, at least at the two-loop level, it is possible to extend some of these symmetries to include fermions.

A bstract Soft breaking of a symmetry requires an invariance of the dimension-4 part of the Lagrangian and non-trivial variation of the lower-dimensional part. However, in general, separation between the dim-4 and lower-dimensional Lagrangian is not invariant with respect to basis transformations of fields. Therefore, a natural question of the physical meaning of soft symmetry breaking arises. This problem is addressed here in the framework of two-Higgs-Doublet Models (2HDM). It has been shown, within these models, that in spite of the ambiguity corresponding to the separation between dim-4 and the lower-dimension Lagrangian, implications of the soft symmetry breaking could be formulated in terms of observables, i.e., they are physical and measurable. There are six global symmetries that can be imposed on the scalar sector of the generic 2HDM. Necessary and sufficient tree-level conditions for soft breaking of all of them have been formulated in terms of observables.

The goal of this report is to summarize the current situation and discuss possible search strategies for charged scalars, in non-supersymmetric extensions of the Standard Model at the LHC. Such scalars appear in Multi-Higgs-Doublet models, in particular in the popular Two-Higgs-Doublet model, allowing for charged and additional neutral Higgs bosons. These models have the attractive property that electroweak precision observables are automatically in agreement with the Standard Model at the tree level. For the most popular version of this framework, Model II, a discovery of a charged Higgs boson remains challenging, since the parameter space is becoming very constrained, and the QCD background is very high. We also briefly comment on models with dark matter which constrain the corresponding charged scalars that occur in these models. The stakes of a possible discovery of an extended scalar sector are very high, and these searches should be pursued in all conceivable channels, at the LHC and at future colliders.

A bstract We explore some aspects of models with two and three SU(2) scalar doublets that lead to mass degeneracies among some of the physical scalars. In Higgs sectors with two scalar doublets, the exact degeneracy of scalar masses, without an artificial fine-tuning of the scalar potential parameters, is possible only in the case of the inert doublet model (IDM), where the scalar potential respects a global U(1) symmetry that is not broken by the vacuum. In the case of three doublets, we introduce and analyze the replicated inert doublet model, which possesses two inert doublets of scalars. We then generalize this model to obtain a scalar potential, first proposed by Ivanov and Silva, with a CP4 symmetry that guarantees the existence of pairwise degenerate scalar states among two pairs of neutral scalars and two pairs of charged scalars. Here, CP4 is a generalized CP symmetry with the property that (CP4) n is the identity operator only for integer n values that are multiples of 4. The form of the CP4-symmetric scalar potential is simplest when expressed in the Higgs basis, where the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields. The symmetries of the model permit a term in the scalar potential with a complex coefficient that cannot be removed by any redefinition of the scalar fields within the class of Higgs bases (in which case, we say that no real Higgs basis exists). A striking feature of the CP4-symmetric model is that it preserves CP even in the absence of a real Higgs basis, as illustrated by the cancellation of the contributions to the CP violating form factors of the effective ZZZ and ZWW vertices.

Abstract CP violation plays a very important role in nature with implications both for Particle Physics and for Cosmology. Accounting for the observed matter-antimatter asymmetry of the Universe requires the existence of new sources of CP violation beyond the Standard Model. In models with an extended scalar sector CP violation can emerge either explicitly, i.e., at the Lagrangian level, or spontaneously. Spontaneous CP violation occurs in the framework of the electroweak symmetry breaking whenever the Lagrangian conserves CP and the vacuum breaks it. This requires that not all vacuum expectation values be real. In the context of multi-Higgs extensions of the Standard Model imposing the existence of a scalar basis where all couplings are real is a sufficient condition for CP to be explicitly conserved. We discuss a three-Higgs-doublet model with an underlying S 3 symmetry, allowing in principle for complex couplings. In this framework it is possible to have either spontaneous or explicit CP violation in the scalar sector, depending on the regions of parameter space corresponding to the different possible vacua of the S 3 symmetric potential. We list all possible vacuum structures allowing for CP violation in the scalar sector specifying whether it can be explicit or spontaneous. It is by now established that CP is violated in the flavour sector and that the Cabibbo-Kobayashi-Maskawa matrix is complex. In order to understand what are the possible sources of CP violation in the Yukawa sector we analyse the implications of the different available choices of representations for the quarks under the S 3 group. This classification is based strictly on the exact S 3-symmetric scalar potential with no soft symmetry breaking terms. The scalar sector of one such model was explored numerically. After applying the theoretical and the most important experimental constraints the available parameter space is shown to be able to give rise to light neutral scalars at the O 𝓞 (MeV) scale.

Abstract Models with two or more scalar doublets with discrete or global symmetries can have vacua with vanishing vacuum expectation values in the bases where symmetries are imposed. If a suitable symmetry stabilises such vacua, these models may lead to interesting dark matter candidates, provided that the symmetry prevents couplings among the dark matter candidates and the fermions. We analyse three-Higgs-doublet models with an underlying S 3 symmetry. These models have many distinct vacua with one or two vanishing vacuum expectation values which can be stabilised by a remnant of the S 3 symmetry which survived spontaneous symmetry breaking. We discuss all possible vacua in the context of S 3-symmetric three-Higgs-doublet models, allowing also for softly broken S 3, and explore one of the vacuum configurations in detail. In the case we explore, only one of the three Higgs doublets is inert. The other two are active, and therefore the active sector, in many aspects, behaves like a two-Higgs-doublet model. The way the fermions couple to the scalar sector is constrained by the S 3 symmetry and is such that the flavour structure of the model is solely governed by the V CKM matrix which, in our framework, is not constrained by the S 3 symmetry. This is a key requirement for models with minimal flavour violation. In our model there is no CP violation in the scalar sector. We study this model in detail giving the masses and couplings and identifying the range of parameters that are compatible with theoretical and experimental constraints, both from accelerator physics and from astrophysics.

A bstract In Two-Higgs-Doublet models, the conditions for CP violation can be expressed in terms of invariants under U(2) rotations among the two SU(2) Higgs doublet fields. In order to design a strategy for measuring the invariants we express them in terms of observables, i.e., masses and couplings of scalar bosons. We find amplitudes directly sensitive to the invariants. Observation of the Standard-Model-like Higgs boson at the LHC severely constrains the models. In particular, in the model with ℤ 2 symmetry imposed on dimension-4 terms (in order to eliminate tree-level flavour-changing neutral currents), CP violation is strongly suppressed. On the other hand, the most general Two-Higgs-Doublet model (without ℤ 2 symmetry) is compatible with the LHC data, and would still allow for CP violation to be present in the model. Consequently, also flavour-changing neutral currents would in general be expected. We briefly sketch a strategy for measuring the remaining CP violation.

A bstract For models with several Higgs doublets we present an alternative method to the one proposed by Branco, Gerard and Grimus, in 1984, to check whether or not CP is spontaneously violated in the Higgs potential. The previous method is powerful and rigorous. It requires the identification of a matrix U corresponding to a symmetry of the Lagrangian and verifying a simple relation involving the vacuum expectation values. The nonexistence of such a matrix signals spontaneous CP violation. This approach may be far from trivial as complexity grows with the number of Higgs doublets. In such cases it may turn out to be easier to analyse the potential by going to the so-called Higgs basis. The transformation to the Higgs basis is straightforward once the vacuum expectation values are known. The method proposed in this work is also powerful and rigorous and can be particularly useful to analyse models with more than two Higgs doublets and with continuous symmetries.