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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 $\mathbb{Z}_2$ 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 $U(1) \otimes U(1)$-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., Comment: 101 pages, 5 figures

The Two Higgs Doublet Model invariant under the gauge group SU(2)xU(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., Comment: 46 pages, minor changes, references added

CP violation plays a very important role in nature with implications both for Particle Physics and for Cosmology. Accounting for the observed matter anti-matter 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. 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 $\mathcal{O}(\text{MeV})$ scale., Comment: 42 pages, 2 figures, V2: version consistent with JHEP

Models with an extended scalar electroweak sector are well motivated. Such models could accommodate a dark matter candidate if there is an additional scalar representation with a vanishing vacuum expectation value and, in addition, there are no couplings between fermions and the dark matter candidate. The most natural way to have these conditions implemented is to consider models where an underlying symmetry is imposed. Governed by this, we consider a three-Higgs-doublet model with an $S_3$ symmetry. Within this framework there are different implementations which could possibly accommodate a dark matter candidate. The family of $S_3$-symmetric three-Higgs-doublet implementations arises due to different vacua and, as a result, different minimisation conditions. In this framework the dark matter candidate falls into the class of weakly interacting massive particles. The dark matter candidate is associated with an $\mathbb{Z}_2$ symmetry which survives spontaneous symmetry breaking and is a remnant of the $S_3$ symmetry. We explore two cases, they share many aspects of the Type-I two-Higgs-doublet model plus an inert SU(2) doublet. The main difference between these two cases is the presence of an irremovable phase, which leads to CP violation in one of the implementations. The two candidate cases differ from other previously studied models with three scalar doublets by the fact that they do not allow for heavy dark matter candidates, $\mathcal{O}(500)\text{ GeV}$. Valid dark matter regions were identified as $m_\mathrm{DM} \in [52.5,\,89]~\text{GeV}$ for a model without CP violation and $m_\mathrm{DM} \in [6.5,\,44.5]~\text{GeV}$ for a model with CP violation. In the present work we refine the parameter space by applying additional checks to our previous work coming from LHC data and from indirect detection data., Comment: 8 pages, 5 figures, to be published in the Proceedings of DISCRETE 2022: 8th Symposium on Prospects in the Physics of Discrete Symmetries, 7-11 November, 2022, Baden-Baden, Germany

The $\mathbb Z_2\times \mathbb Z_2$-symmetric 3HDM potential, sometimes called Weinberg's 3HDM potential, can accommodate both explicit and spontaneous CP violation as well as natural flavour conservation and is hence relevant for studies of CP violation coming from a realistic extended Higgs sector. The model has an interesting regime, controlled by a single parameter, where it acquires an approximate global $U(1) \times U(1)$ symmetry which causes CP violating effects to be small and two pseudo-Goldstone bosons to be present in the scalar spectrum. From parameter space scans, we find that, in a realistic implementation featuring an SM-like Higgs boson, this model predominantely leads to the existence of light additional Higgs bosons with a mass lower than the SM-like Higgs boson.

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., Comment: 40 pages, 1 figure, matches published version

We study the potential of Weinberg's $\mathbb{Z}_2\times\mathbb{Z}_2$-symmetric three-Higgs-doublet model (3HDM). The potential is designed to accommodate CP violation in the scalar sector within a gauge theory, while at the same time allowing for natural flavour conservation. This framework allows for both explicit and spontaneous CP violation. CP can be explicitly violated when the coefficients of the potential are taken to be complex. With coefficients chosen to be real, CP can be spontaneously violated via complex vacuum expectation values (vevs). In the absence of the terms leading to the possibility of CP violation, either explicit or induced by complex vevs, the potential has two global U(1) symmetries. In this case, spontaneous symmetry breaking would in general give rise to massless states. In a realistic implementation, those terms must be included, thus preventing the existence of Goldstone bosons. A scan over parameters, imposing the existence of a neutral state at 125 GeV that is nearly CP-even shows that, in the absence of fine-tuning, the scalar spectrum contains one or two states with masses below 125 GeV that have a significant CP-odd component. These light states would have a low production rate via the Bjorken process and could thus have escaped detection at LEP. At the LHC the situation is less clear. While we do not here aim for a full phenomenological study of the light states, we point out that the $\gamma\gamma$ decay channel would be challenging to measure because of suppressed couplings to $WW$., Comment: Minor changes, some references added. Matches Phys. Rev. D version

The full CMS Run 2 datasets with time-integrated luminosity of 137 fb$^{-1}$ in the diboson channels are used to probe benchmark models with extended gauge sectors such as $E_6$, left-right symmetric (LR) and the sequential standard model (extended gauge model, EGM), that predict the existence of neutral $Z'$- and charged $W'$-bosons decaying to a pair of bosons $WW$, $ZH$, $WZ$ and $WH$ in the semileptonic final state. These benchmark models are used to interpret the results. Exclusion limits at the 95% C.L. on the $Z'$ and $W'$ resonance production cross section times branching ratio to electroweak gauge boson pairs in the resonance mass range between 1.0 and 4.5 TeV are here converted to constraints on $Z$-$Z'$ and $W$-$W'$ mixing parameters and masses. We present exclusion regions on the parameter spaces of the $Z'$ and $W'$ and show that the obtained exclusion regions are significantly extended compared to those derived from the previous analysis performed with Tevatron data as well as with the CMS data collected at 7 and 8 TeV in Run 1. The reported limits are the most restrictive to date., Comment: 14 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:2012.13930, arXiv:1912.02106, arXiv:1904.01432

Models with an extended scalar electroweak sector can have vanishing vacuum expectation values in a basis where an underlying symmetry is imposed. Such extensions are very well motivated. If a symmetry prevents couplings between fermions and additional scalars, such scalars could become viable dark matter candidates if some additional criteria are satisfied. We catalogue $S_3$-symmetric three-Higgs-doublet models, also allowing for softly broken $S_3$-symmetric scalar potential terms, based on whether a specific model could possibly accommodate a dark matter candidate. The variety of the $S_3$-symmetric family models arises due to different possibilities to arrange vacuum expectation values. Such models can have vacua with one or two vanishing vacuum expectation values. In our study we assume that the dark matter candidate is stabilised by the $\mathbb{Z}_2$ symmetry. The $\mathbb{Z}_2$ symmetry is a remnant of $S_3$ symmetry which survived spontaneous symmetry breaking, and not superimposed over $S_3$. We explore two models; with an without CP violation. These models have a single dark and two active scalar sectors. The active sectors behave in many aspects like a Type-I two-Higgs-doublet model. The dark matter candidate masses, in two cases, are different from the known (previously studied) models with three scalar doublets. After investigating the models in detail, identifying parameters compatible with both theoretical and experimental constraints, we found that the dark matter candidate mass could be within the range of $[52.5,\,89]~\text{GeV}$ or $[6.5,\,44.5]~\text{GeV}$ for a model with CP violation., Comment: 6 pages, 3 figures, to be published in the Proceedings of DISCRETE 2020-2021: 7th Symposium on Prospects in the Physics of Discrete Symmetries, 29th November - 3rd December 2021, Bergen, Norway; v2: Minor changes

In spite of the success of the Standard Model of Particle Physics, there are some theoretical predictions which are not yet fully established experimentally as well as some experimental observations which cannot be fitted within its theoretical framework, thus requiring physics beyond the Standard Model. One of these is a hypothetical non-luminous form of matter - dark matter. Models with an extended scalar electroweak sector yield plausible dark matter candidates. In this paper we study a specific model, C-III-a, from a family of $S_3$-symmetric three-Higgs-doublet models. The model consists of two active SU(2) doublets and an inert one. The latter is inert due to a $\mathbb{Z}_2$ symmetry that survives the breaking of $S_3$, and would accommodate a dark matter particle. We explore the model numerically, based on theoretical and experimental constraints. After applying a number of successive checks over the parameter space we found a viable dark matter mass region in the range $[6.5;\,44.5]~\text{GeV}$. This region is drastically different from the Higgs-like dark matter states that have been proposed: the well-known Inert Doublet Model and models with three scalar doublets, with one or two inert doublets. Furthermore, the C-III-a model allows for spontaneous CP violation., Comment: 40 pages, 12 figures. v2: PRD version, minor clarifications. arXiv admin note: text overlap with arXiv:2108.07026

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 $V_\text{CKM}$ 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., Comment: 50 pages, 14 figures, V2: typos fixed, some plots changed, V3: minor text edits, V4: version consistent with JHEP

The full ATLAS Run 2 data set with time-integrated luminosity of 139 fb$^{-1}$ in the diboson and dilepton channels is used to probe benchmark models with extended gauge sectors: the $E_6$-motivated Grand Unification models, the left-right symmetric $LR$ and the sequential standard model (EGM). These all predict neutral $Z'$ vector bosons, decaying into lepton pairs, $\ell\ell$, or into electroweak gauge boson pairs $WW$, where one $W$ in turn decays semileptonically. 95% C.L. exclusion limits on the $Z'$ resonance production cross section times branching ratio to electroweak gauge boson pairs and to lepton pairs in the mass range of $\sim$ 1 - 6 TeV are converted to constraints on the $Z$-$Z'$ mixing parameter and the heavy resonance mass. We present exclusion regions on the parameter space of the $Z'$ which are significantly extended compared to those obtained from the previous analyses performed with LHC data collected at 7 and 8 TeV in Run 1 as well as at 13 TeV in Run 2 at time-integrated luminosity of 36.1 fb$^{-1}$ and are the most stringent bounds to date. Also presented, from a similar analysis of electrically charged $W'$ bosons arising in the EGM, which can decay through $W'\to WZ$ and $W'\to \ell\nu$, are limits on the $W$-$W'$ mixing parameter and the charged $W'$ vector boson mass., Comment: 18 pages, many figures. arXiv admin note: substantial text overlap with arXiv:1912.02106, arXiv:1904.01432, arXiv:1809.08933. Version published in PRD

Symmetries of the Two-Higgs-Doublet Model (2HDM) potential that can be extended to the whole Lagrangian, i.e. the CP-symmetries CP1, CP2, CP3 and the Higgs-family symmetries Z2, U(1) and SO(3) are discussed. Sufficient and necessary conditions in terms of constraints on masses and physical couplings for the potential to respect each of these symmetries are found. Each symmetry can be realized through several alternative cases, each case being a set of relations among physical parameters. We will show that some of those relations are invariant under the renormalization group, but others are not. The cases corresponding to each symmetry group are illustrated by analyzing the interplay between the potential and the vacuum expectation values., Comment: 58 pages, 7 figures. JHEP version. References added, clarifications and minor changes made

We report on the status of efforts to improve the reinterpretation of searches and measurements at the LHC in terms of models for new physics, in the context of the LHC Reinterpretation Forum. We detail current experimental offerings in direct searches for new particles, measurements, technical implementations and Open Data, and provide a set of recommendations for further improving the presentation of LHC results in order to better enable reinterpretation in the future. We also provide a brief description of existing software reinterpretation frameworks and recent global analyses of new physics that make use of the current data., Comment: 58 pages, minor revision following comments from SciPost referees

In this paper we analyse in detail an $S_3$-symmetric three-Higgs-doublet model with a specific vacuum configuration. This analysis allows us to illustrate important features of models with several Higgs doublets, such as the possibility of having spontaneous CP violation. We start with a real potential and pick a particularly interesting complex vacuum configuration, which does not violate CP before adding soft breaking terms to the potential. We study the r\^ ole played by different soft symmetry breaking terms. These are essential for our choice of vacuum in order to remove unwanted massless scalars which arise from the spontaneous breaking of an accidental continuous symmetry. We list scalar sector and scalar-gauge sector-couplings for the particular case we consider in detail in this work. Results presented in this paper will be useful for model building, in particular for implementations of models with $S_3$ symmetry and spontaneous CP violation, extensions of the fermionic sector with realistic Yukawa couplings and for Dark Matter studies., Comment: 33 pages. v2: PRD version, Minor clarifications. References added. Title compliant with APS style

The full ATLAS Run 2 data set with time-integrated luminosity of 139 fb$^{-1}$ in the diboson channels in hadronic final states is used to probe a simple model with an extended gauge sector (EGM), proposed by Altarelli et al., and often taken as a convenient benchmark by experimentalists. This model accommodates new charged $W'$ and neutral $Z'$ vector bosons with modified trilinear Standard Model gauge couplings, decaying into electroweak gauge boson pairs $WZ$ or $WW$, where $W$/$Z$ decay hadronically. Exclusion limits at the 95% C.L. on the $Z'$ and $W'$ resonance production cross section times branching ratio to electroweak gauge boson pairs in the mass range of $\sim$ 1 - 5 TeV are here converted to constraints on $W$-$W'$ and $Z$-$Z'$ mixing parameters and masses for the EGM. We present exclusion regions on the parameter space of the $W'$ and $Z'$ by using the full Run 2 data set comprised of $pp$ collisions at $\sqrt{s}=13$ TeV and recorded by the ATLAS detector at the CERN LHC. The obtained exclusion regions are significantly extended compared to those obtained from the previous analysis performed with Tevatron data as well as with LHC data collected at 7 and 8 TeV in Run 1 and are the most stringent bounds to date., Comment: 20 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1904.01432, arXiv:1809.08933. v2: clarifications, version published in EPJC

This is a review of Glauber's asymptotic diffraction theory, in which diffractive scattering is described in terms of interference between semiclassical amplitudes, resulting from a stationary-phase approximation. Typically two such amplitudes are sufficient to accurately describe elastic scattering, but the stationary points are located at complex values of the impact parameter. Their separation controls the interference pattern, and their offsets from the real axis determine the overall fall-off with momentum transfer. Asymptotically, at large momentum transfers, the stationary points move towards singularities of the profile function. I also include some reminiscences from our collaboration., Comment: 12 pages, contribution to SciPost Physics Proceedings: EFB 24. v2: minor changes

Symmetries play a very important r\^ole in Particle Physics. In extended scalar sectors, the existence of symmetries may permit the models to comply with the experimental constraints in a natural way, and at the same time reduce the number of free parameters. There is a strong interplay among internal symmetries of the scalar potential, its CP properties and mass degeneracies of the physical scalars. Some of these aspects were discussed in this talk., Comment: 8 pages, to be published in the Proceedings of DISCRETE2018: 6th Symposium on Prospects in the Physics of Discrete Symmetries, 26-30 Nov 2018. Vienna, Austria

New charged vector bosons $W'$ decaying into gauge boson pairs $WZ$ are predicted in many scenarios of new physics, including models with an extended gauge sector (EGM). Due to the large variety of models (other unification groups, models with Supersymmetry, Little Higgs Models, Extra Dimensions) the more general EGM approach is here considered. For what concerns $W'$-production, these models are parametrised by two parameters, the $W^\prime$ mass $M_{W^\prime}$ and the $W$-$W^\prime$ mixing parameter $\xi$. The diboson $WZ$ production allows to place stringent constraints on this mixing angle and the $W'$ mass, which we determine and present for the first time by using data from $pp$ collisions at $\sqrt{s}=13$ TeV recorded by the ATLAS detector at the CERN LHC, with integrated luminosity of 36.1 fb$^{-1}$. By comparing the experimental limits to the theoretical predictions for the total cross section of $W'$ resonant production and its subsequent decay into $WZ$ pairs, we show that the derived constraints on the mixing angle for the benchmark model are rather small, between $10^{-4}$ and $10^{-3}$, i.e., greatly improved with respect to those derived from the global analysis of electroweak data which yield $\xi\sim 10^{-2}$. We combine the limits derived from $WZ$ production data with those obtained from the $W'\to e\nu$ process in order to significantly extend the exclusion region in the $M_{W'}$-$\xi$ parameter plane and obtain the most stringent exclusion limits to date. We present the combined allowed parameter space for the EGM $W'$ boson after incorporating indirect constraints from low energy electroweak data, direct search constraints from Tevatron and from the LHC Run I with 7 and 8 TeV as well as at Run II with 13 TeV data., Comment: 13 pages, 5 figures. arXiv admin note: text overlap with arXiv:1809.08933. v2: References added, version to appear in PRD