Our present knowledge of elementary particles and interactions is based on solid theoretical grounds which culminate into the Standard Model of particle physics. The mathematical construction of this theory is one of the most beautiful achievements in Physics since it provides a unified description of the strong, electromagnetic and weak interactions. Since its formulation in the 1970’s, the SM has passed several consistency and experimental tests. The agreement between SM predictions and the data from several experiments in the last decades is truly remarkable. Until recently, a crucial ingredient was missing to close the SM chapter: the discovery of the physical manifestation of the Higgs field, the entity responsible for providing mass to elementary particles, and for the breaking of the electroweak symmetry. Finally, in July 14th of 2012, the discovery of the Higgs boson at the CERN Large Hadron Collider (LHC) was announced by the CERN ATLAS and CMS collaborations.
Despite all the extraordinary achievements of the SM, there are three questions that this theory does not answer:
1. 1) Why is there an excess of matter over antimatter in the Universe? Although the SM contains all necessary ingredients (charge-parity symmetry and baryon number violation, and departure from thermal equilibrium in the early universe) to explain why there is more matter than antimatter in the Universe, the predicted value of the asymmetry turns out to be too small.

1. 2) What is dark matter made of? Several astrophysical observations reveal that approximately 85% of the matter in the Universe comes in the form of dark matter. None of the SM particles has the required properties to be the dark matter particle.

1. 3) Where are neutrino masses (necessary to explain neutrino oscillations) coming from? In the SM, neutrinos are strictly massless particles. However, due to the observation of neutrino oscillations, we know that neutrinos have mass and mix among themselves.

The fact that the SM is not able to provide an answer to these puzzles is an undeniable evidence that there must be physics beyond the SM, i.e. the SM may be an effective theory or, in other words, the low-energy limit of a more complete setup. This observation is supported by the existence of some anaesthetic aspects of the SM like the hierarchy problem, the lack of gauge coupling unification, the breaking of parity symmetry by weak interactions, the flavour problem, and many others. In the last decades, several SM extensions have been put forward, ranging from theories with larger symmetry like Grand Unified Theories (GUTs), to theoretical constructions based on the existence of extra dimensions or extra symmetries beyond the gauge and spacetime ones, as in Supersymmetric (SUSY) theories.

At CFTP we explore new theories beyond the SM, study their phenomenological implications and devise strategies to test them at current and future experiments.

In the Standard Model there is only one Higgs doublet. The Higgs sector is responsible for the electroweak symmetry breaking and for giving mass to fermions and gauge bosons. After several decades of search, the Higgs boson was discovered in 2012 by the LHC at CERN. However, its discovery leaves many questions unanswered.

Multi-Higgs models are one of the simplest extensions of the Standard Model, and are very well motivated. In particular they may provide new sources of CP violation which are needed in order to construct viable models for the creation of the Baryon Asymmetry of the Universe (BAU). They also allow for spontaneous CP violation thus putting the electroweak symmetry breaking and CP violation on the same footing. They may also provide a solution to the strong CP problem through the Peccei-Quinn mechanism as well as provide interesting scenarios for dark matter. These models have a very rich phenomenology related to the new structures of scalar-scalar, scalar-gauge and scalar-fermion interactions. CFTP is actively involved in the analysis of new physics effects in such models, working in close contact with the international community.

Symmetries play a very important role in the study of multi-Higgs models since they reduce the number of free parameters, thus increasing predictability and helping to conform with the experimental constraints in a natural way. At CFTP a lot of work has been done, and is under way, on the role of symmetries in the scalar sector with emphasis on:
i) complying with the stringent experimental limits on Higgs flavour changing neutral currents;
ii) the possibility of generating a complex CKM matrix with spontaneous CP violation;
iii) finding realistic scenarios for fermion masses and mixing; iv) developing tools allowing to study the scalar potential in a basis independent way.

There is a strong interplay among symmetries in the scalar sector, mass degeneracies among scalar fields and CP violation. The set of all possible symmetries of two Higgs doublet models is already well known. In the case of three Higgs doublet models a lot of work along these lines is under way, in which CFTP is also actively involved, taking a leading role.

The members of CFTP have contributed to many publications in this field, in some of the best international journals. One of these, dated from 2011, already has close to 1 500 citations in the Inspire data base and is still being cited both by experimentalists and in theoretical works.

The Standard Model(SM) is a very successful theory which has been tested experimentally at an incredible level of accuracy. In spite of its great success, the SM leaves open many fundamental questions. The origin of fermion masses, mixing and CP violation constitutes a major puzzle in the field of elementary particle physics. The Standard Model (SM) can accommodate their observed pattern in the quark sector but does not provide an explanation for their origin. In the SM neutrinos are strictly massless therefore leptonic mixing and neutrino masses provide a clear evidence for Physics beyond the SM.

One of the approaches CFTP has followed in Flavour Physics involves the introduction of Family Symmetries (FS). Symmetries play a major role in Particle Physics and FS provide a very promising framework to understand flavour. In a significant number of papers, an extra family symmetry is introduced in the Lagrangian of the SM. The groups used as family symmetries include S3, S4, A4, delta27, etc. Some of our work has had a strong impact in the literature, with a large number of citations. The fields of CP violation and Flavour Physics are closely connected. This is reflected in some of the work of the CFTP team. For example, in the framework of the group delta 27, our team has investigated the question of having geometric CP violation.
One of the difficulties in extracting a possible family symmetry from the observed pattern of fermion masses and mixing, stems from the fact that the structure of Yukawa couplings depends on the chosen weak basis (WB). This has motivated the study of weak invariants, which do not change under a WB transformations. The CFTP team has given important contributions to the study of these invariants. A related question is the physical meaning of texture zeros. The CFTP team has shown that some of the texture zeros do not have any physical meaning, since they can be obtained from arbitrary couplings.

Often FS are combined with CP symmetries in order to extend their predictive power, either in the Yukawa sector or in the scalar potential of Multi-Higgs models. This can be illustrated when considering the mechanism of Leptogenesis which provides a natural explanation for the Baryon Asymmetry of the Universe (BAU) involving sources of CP violation coming from physics beyond the SM. Leptogenesis may occur in the context of the seesaw mechanism. In this framework CP violation at high energies responsible for Leptogenesis and CP violation at low energies can only be related in the context of a flavour model involving flavour symmetries. We have actively working along these lines.

In spite of the large differences between the leptonic and quark mixing, it is possible that they are related through a Grand Unified Theory based, e.g., on the gauge group SO(10). This relationship might allow for interesting predictions f or the neutrino masses while at the same time explaining the observed leptonic mixing. In the past our group has already looked into this possibility and we intend to pursue this research further.

CP violation plays a central role in Particle Physics and it has profound implications for Cosmology, since it is a required crucial ingredient for the generation of the Baryon Asymmetry of the Universe (BAU). In spite of the fact that CP violation has been observed experimentally for the first time more than fifty years ago, there are still many open questions, like for example what is the origin of CP violation, is there CP violation in the leptonic sector and what are the extra sources of CP violation beyond the Standard Model (SM) which are needed in order to generate sufficient BAU.

Some of the members of CFTP have been working for more than three decades in various aspects of CP violation. An important reference is the book on CP Violation written by three members of CFTP and published in the Oxford University Press. This book is considered one of the best references on this topic. There are many aspects of CP breaking which have been studied by t he members of CFTP. These include the construction of models with extra sources of CP breaking beyond the Kobayashi-Maskawa (KM) mechanism.

The SM and its KM mechanism for CP violation parametrizes by the CKM matrix is in general agreement with experiment as it is illustrated through the usual fitting of the vertex of the unitarity triangle. However, there is still plenty of room for significant physics beyond the SM, arising for example innew contributions for $B_d -_ \bar{B_d}$ and $B_s -- \bar{B_s}$ mixings. The members of CFTP have analysed in detail the phenomenological consequences of models with additional contributions to CP breaking. Models with spontaneous CP breaking have been suggested, some of them providing a possible solution of the strong CP problem.

The CFTP team has pioneered the study of CP-odd weak-basis invariants which play a very important role in the study of CP violation in the Standard Model and its extensions. We have done these studies in various contexts, including multi-Higgs doublets, extensions with vectorlike quarks, supersymmetric extensions, etc. We have also given important contributions to the study of CP violation needed for leptogenesis and also to the study of a possible relationship between leptonic CP violation at high energy and leptonic CP violation detectable at low energies through neutrino oscillations. A lot of work is still under way at CFTP along these lines.
The first model with geometric CP violation has been proposed some time ago by us and the subject has been extensively studied more recently by members of CFTP.

In 1930 Pauli proposed the existence of very light particles (latter dubbed as neutrinos by Fermi) as a “desperate” way out to explain why the nuclear beta decay spectrum is continuous. By the time of the SM formulation, neutrinos were introduced as being strictly massless. However, the experimental observation of neutrino oscillations (2015 Physics Nobel Prize), imply nonvanishing neutrino masses and lepton mixing. Thus, the SM is incomplete in the sense that it does not account for massive neutrinos.

Presently, the nature of the underlying neutrino mass mechanism remains a mystery. In the last decades, neutrino oscillation experiments have remarkably contributed to the knowledge of the neutrino mass and mixing pattern by achieving high precision in determining most of its parameters. Apart from some subtleties, the mixing angles and neutrino mass-squared differences are presently very well known. However, there are still fundamental questions to be answered about neutrinos like: Are neutrinos Dirac or Majorana particles? Is CP violated in the lepton sector? How are neutrino masses ordered?

From the theoretical viewpoint, a common approach to explain the observed neutrino mass and mixing pattern relies on extending the SM by introducing new flavour symmetries and/or new degrees of freedom. The most popular SM extensions accommodating massive neutrinos in a natural way are those in which neutrino masses are generated through the exchange of new particle, both at the classical or the quantum level. The existence of these “neutrino mass messengers” opens the door for a plethora of new phenomena which are unobservable in the framework of the SM. For instance, if the mass of these states is such that their production at colliders is viable, one can devise strategies for direct searches by looking at specific experimental signatures. Neutrinos can, therefore, play a crucial role at the energy frontier of particle physics. Moreover, once the SM is extended to accommodate massive neutrinos, new physical processes may occur like lepton flavour violating decays. The search for these signals is in the target of experiments at the intensity frontier.

A remarkable feature of a large class of neutrino mass mechanisms is that the physics responsible for providing mass to neutrinos also offers a simple explanation for why we live in a matter-dominated Universe. Moreover, in some models it is also possible to identify a plausible dark-matter candidate. The connection between neutrino phenomenology (intensity and energy frontiers) and cosmology (cosmic frontier) may thus be established.

At CFTP we look at these questions by exploring synergies among neutrino phenomena at the energy, intensity and cosmic frontiers of particle physics. This is done in the context of theoretical frameworks which account for massive neutrinos and which can predict other interesting phenomena which will be hopefully tested in current and future experiments.

Hadronic physics deals with quarks, which are the fundamental building blocks of ordinary matter. Three quarks are needed to form a proton or a neutron, which on their turn give rise to the many different atomic nuclei and so the periodic table of elements. Besides the proton and the neutron, many other three-particle combinations of the various quark types (``flavours''), the so-called ``baryons'', most of them very unstable, can be produced in experiment. Other simple hadrons are ``mesons'', made of one quark and one anti-quark, being all unstable to different degrees. Mesons are the mediators of the nuclear force, which holds protons and neutrons together in the nuclei.

The microscopic interactions between quarks inside hadrons are governed by Quantum Chromodynamics (QCD), a non-Abelian quantum field theory whose quanta are the massless gluons. Gluons carry a non-additive quantum number called ``colour'', just like the quarks. In Nature, only colourless objects can be observed as free particles, or colour ``singlets'' in group-theory language. At present, QCD can only be treated rigorously at high energies and through perturbative calculations, whereas at low energies, relevant for hadrons, approximative methods and models have to be used.

The main features of low-energy QCD are confinement, the property that impedes quarks from being observed in isolation, and chiral symmetry, which is an approximate symmetry of the QCD Lagrangian, owing to the very small masses of the light current quarks, as compared to hadron masses. Confinement is usually dealt with in models employing an ever rising interquark potential, which therefore does not allow quarks to become free. This potential results, in combination with the different quark flavours, in very diverse and discrete baryon and meson spectra in such models. Strong-decay processes should then give rise to the often very broad hadronic resonances observed in experiment. On the other hand, the dynamical breaking of chiral symmetry at low energies is inevitable owing to the Dirac spin structure of the confining interaction and the tiny masses of the light current quarks. This spontaneous chiral-symmetry breaking gives rise to dynamical quark masses that are roughly two orders of magnitude larger than the light current-quark masses, while also explaining the smallness of the pion mass. For spectroscopic purposes, these dynamical and momentum-dependent quark masses can be replaced by constant constituent quark masses, to a good approximation.

Our active research in hadronic physics is mostly focused on meson spectroscopy, by considering quark confinement, strong decay, and chiral-symmetry breaking. In doing spectroscopic calculations for light, heavy-light, and heavy scalar, vector, and axial-vector mesons, the dynamical effects on meson masses and widths are accounted for in the manifestly unitary and analytic Resonance-Spectrum-Expansion (RSE) model. As for chiral-symmetry breaking, the need for a covariant relativistic treatment is emphasised and made tractable via the so-called Covariant Spectator Theory (CST). The CST amounts to a covariant three-dimensional reduction of the four-dimensional Bethe-Salpeter equation, which always keeps one of the two particles on its mass shell in the intermediate state. In current research work, a new very accurate numerical method to solve the CST equations for a typical Coulomb-plus-linear confining potential is being developed, aimed at applying the formalism, for the first time, to the tensor-meson spectrum.

Our model results will be constantly compared to recent measurements at leading particle-physics facilities, such as LHC and SPS at CERN, Jefferson Lab (USA), Belle-II (Japan), BES (China), J-PARC (Japan), FAIR (Germany), and others.