1. INTRODUCTION: CFTP CFTP, Centro de Fisica Teorica de Particulas, is the leading Portuguese research group in Theoretical Particle Physics with a high degree of internationalisation. In fact CFPT is the only Research Unit in Portugal whose research is entirely dedicated to this field. Although there exist several outstanding Theoretical Particle Physicists scattered in other Portuguese Research Units, CFTP dominates in the sense that the vast majority of publications, citations and training in this theoretical area of research is produced by members of FCT, often in collaboration with members from other Portuguese Centres or else in International Collaborations. As of the 31st of December 2024 there were 11 permanent members with a PhD and with tenure, working at CFTP, as well as three postdoctoral researchers. In addition CFTP has a strong record of training with a significant number of PhD students and MsC students. At the end of 2024 there were eight PHD students at CFTP and in addition a total of five students had finished their PhD during 2024. CFTP organises and carries out research in several topics, in the context of the Standard Model of Particle Physics and of Physics Beyond the Standard Model, such as Higgs Physics, Flavour Physics including Neutrino Physics, Symmetries, CP violation and Dark Matter studies. Its scientific program is driven by the most recent theoretical and experimental developments. It has a close scientific connection to CERN and is also involved in the organisation of several joint activities with LIP, the Portuguese Experimental Particle Physics Group. CFTP organises periodically International Workshops in Lisbon, Portugal. Examples are editions of some of the renowned international Conference series such as PLANCK and FLASY that took place recently in Lisbon. It has been a tradition of CFTP to invite experimentalists from LIP to give some of the experimental talks in LHC Physics together with other speakers from CERN. There is a strong connection between CFTP and CERN. Several members of CFTP have VISC contracts at CERN with close scientific ties to the CERN Theoretical Physics Department. They visit CERN frequently, sometimes on sabbatical leave, and participate in Workshops and Working Groups organised there. The CERN Neutrino Platform was created a few years ago in order to strengthen European participation in Neutrino Physics. Some members of our Centre participated at the CERN Theory Institute on Neutrinos that was the kick-off activity of this platform and continued participating in follow-up activities. Several members of CFTP participate in working groups at CERN and have contributed as co-authors to their reports. These reports are meant to provide guidance to experimentalists in their searches. Topics of Research of CFTP are described in point 4. Is should be clear from the given list of topics that the research done at CFTP is strongly related to LHC Physics as well as the Physics of FCC. 2. PRESENT SITUATION CONCERNING THE FCC CERN has been actively preparing the ground for the building of the FCC in the Geneva area. FCC was the recommendation of the European Strategy for Particle Physics Group in 2020 as the first choice for a new collider to be built at CERN. A lot of work has been done since then in all the fronts in order to make this project feasible. The 8th FCC Physics Workshop [is taking place/took] place at CERN from the 13th to the 17th of January 2025: https://indico.cern.ch/event/1439509/ with parallel sessions where several different aspects of FCC were discussed: - Physics Programme; - Physics Performance; - Software and Computing; - MDI (machine-detector interface); - EPOL (Energy Polarisation and Monochromatisation) - Detector Concepts; Fabiola Gianotti reported in the opening session of the Workshop that there is a growing interest in the FCC Project in the worldwide community. Gianotti urged the community to strongly support the FCC. Concerning feasibility, there was a talk on the first day by J. Gutleber (CERN) entitled "Future Circular Collider Study status and implementation aspects". Gutleber reported that the FCC feasibility study mid-term review has been successfully completed and on track to be submitted to Update of the European Strategy for Particle Physics: Host states provided feedback that was incorporated; Environmental initial state analysis, engagement with the public, socio-economic assessment and sustainability studies were carried out as pre-requisites for the authorisation processes; Demonstration of a committed engagement of a large and global science community is important; It was also mentioned that tunnelling is mainly in molasse, soft rock layer, well suited for fast, low-risk TBM construction. Management of excavated materials requires a specific plan that must be presented in a subsequent preparatory phase as part of the authorisation process. It was interesting to learn that environmental considerations play today a very important role in the planning of the future collider. These are, for instance, the type of construction used (green roofs, facade expression, innovative methods and materials, utilising terrain. etc) Grigorio Bernardi talking in both his name and in the name of Emmanuel Tsesmelis reported on the fact that the CERN Council just approved the Medium Term Plan for 2025-2029 including among other aspects funding for FCC new positions: - dedicated new positions (fellows, students, scientific associates, visitors) over the next three years having in mind the construction of the FCC. - many more engineers and detector physicists will be needed soon He pointed out that there is the need for coordination and structuring of the theoretical work needed to match the anticipated experimental precision of the FCC data and that this work had already started during the Feasibility Study. The CERN/TH FCC team currently consists of three physicists with the occasional participation of up to eight staff members and fellows. The scenario under consideration is the existence of 8 sites out of which 4 are interaction points. It is the aim of CERN to work with the scientific community, institutes, laboratories and funding agencies to ensure support and resources for four experiments, facilitating the exploitation of the full scientific potential offered by the large investment in the FCC-ee facility; CERN wants to give a strong signal to the HEP community worldwide of its commitment to FCC as host lab. 3. ADDRESSING THE QUESTIONS LISTED IN THE ECFA guidelines FOR THE INPUT OF THE EUROPEAN HIGH-ENERGY PHYSICS COMMITTEE Which is the preferred next major/flagship collider project for CERN? - In view of the CERN preference, the strong commitment and the smooth progress of all the planned preparatory steps it looks natural to strongly support the FCC. - Furthermore this was the direction chosen by the European Strategy for Particle Physics Group in 2020 and a lot of work has already been done towards this goal with very encouraging results. There does not seem to exist any reason to change gears. What are the most important elements in the response? i) Physics potential FCC will have two stages, first as an e plus e minus collider and next as a hadron collider. These two stages have very different characteristics and will give different contributions the study of the fundamental laws of nature; ii) Long-term perspective The lifetime of these two consecutive Projects will take us till the end of the 21st century, ensuring that experimental Particle Physics will continue alive for a long time in the future. It is foreseeable that in the meantime other Projects will come to life in other parts of the globe which will be complementary to the FCC. At the moment it looks very likely that China will also build a Circular Collider. This should be seen as positive. iii) Financial and human resources: requirements and effect on other projects iv) Timing Unfortunately building new colliders is a very long term task. As pointed out by J. Gutleber any new project at CERN with territorial development needs requires studies and works akin to the ones carried out in the frame of the FCC studies (investment so far 8 years). v) Careers and training Long-term projects like a future collider present unique challenges and opportunities. These endeavours often aim to address complex scientific, societal, or technological questions that require sustained effort, creativity, and adaptability. A critical factor in the success of such long-term projects is strategic investment in advanced-level training and development, including PhD programs, research grants, and postdoctoral positions. Projects with extended timelines require a highly skilled workforce capable of addressing complex challenges and advancing cutting-edge research. Investing in advanced training opportunities, such as doctoral and postdoctoral programs, ensures a steady stream of experts who can contribute to the project’s long-term vision. These programs foster deep specialisation, critical thinking, and innovative problem-solving skills essential for the success of ambitious initiatives. Providing grants and fellowships for advanced study not only attracts top talent but also enables researchers to focus on groundbreaking work without financial constraints. This investment creates a virtuous cycle, as trained experts contribute to the project's goals while mentoring the next generation of researchers. Advanced-level training programs often involve multi-year commitments, aligning well with the timelines of long-term projects. By offering attractive research grants, scholarships, and career advancement opportunities, organisations can encourage researchers to dedicate significant portions of their careers to these initiatives. Long-term financial and institutional support ensures that experts remain engaged and focused on delivering meaningful outcomes. In recent years, the scientific community has faced a growing concern over the loss of talented researchers due to job instability and inadequate funding. This "brain drain" threatens the sustainability and progress of crucial scientific fields, especially in areas requiring long-term commitment. Talented individuals, unable to secure stable positions or sufficient financial support, often leave research for more secure and lucrative careers, resulting in a significant loss of expertise and innovation.To ensure the continuity and vitality of our scientific endeavours, it is imperative to address these challenges through strategic investment in advanced  training and career support within our field. vi) Sustainability Sustainability studies are taking place at the moment with as far as we understand with optimistic perspectives 4. TOPICS OF RESEARCH AT CFTP Work in progress at CFTP includes, but it is not limited to: - Study of extensions of the SM with isosinglet vector-like quarks (VLQs). In recent years, interest in VLQs has been increasing, due to their contributions to new physics effects that can be tested in experiments at LHC and High- Luminosity LHC. The discovery of VLQs can be within the reach of present or future colliders being planned. Models with VLQs have a very rich phenomenology. They provide one of the simplest solutions to the strong CP problem, they can also provide an explanation for the CKM unitarity puzzle. Their presence affects a plethora of flavour observables such as meson mixing, oblique and electroweak parameters, among others. - A new paradigm for the flavour problem: Precision experiments offer indirect avenues to probe new physics. Modular forms, previously applied in supersymmetric scenarios, may offer insights into fermion mass and mixing patterns even in non-supersymmetric contexts. Exploring this avenue requires innovative theoretical approaches to unravel the mysteries of particle flavour. - The DM-neutrino connection: Theoretical models involving light dark matter (LDM) often involve gauge bosons of dark U(1)X symmetries, potentially connecting with neutrino physics. We will investigate this interplay between particle and astroparticle physics which is crucial for understanding the Universe's hidden sectors, and to establish connections among different open questions in modern particle physics. - Taking leptogenesis to the precision era: Beyond the Standard Model (BSM) scenarios offer promising avenues to address cosmological enigmas such as neutrino mass generation, the baryon asymmetry of the Universe (BAU), and dark matter (DM). These phenomena might share a unified explanation, suggesting interconnected physics. Notably, theoretical frameworks have emerged linking neutrino physics, DM, and BAU, yet precise, model- independent tools for calculating BAU remain lacking. To bridge this gap, we propose developing a public code (LEPTOCalc) to compute leptogenesis observables in diverse SM extensions, aiming to enhance our understanding of the Universe's fundamental processes. - Models, signatures, and tools: Current tests of BSM models often rely on simplified assumptions, potentially overlooking crucial aspects of new physics. Our objective is to identify novel signatures to test existing and new BSM models at colliders. This includes searches for lepton number violating processes which arise due to the existence of almost degenerate heavy neutral particles, exploring left-right symmetric models (LRSMs) and gauged U(1) models, and leveraging machine learning techniques for enhanced signal discrimination and anomaly detection. - Special attention to extensions of the Higgs sector with two or with three Higgs doublets as well as extended models including scalar singlets. There are strong motivations for each one of these scenarios. Models with two Higgs doublets have been extensively studied in the literature. There is a growing interest in the study of models with three Higgs doublets (3HDM). Such models can only be realistic if there is a special mechanism that controls Higgs mediated flavour-changing neutral currents. This requires the introduction of symmetries. Symmetries increase the predictability of the models. We are interested in continuing to analyse the theoretical constraints that the extended scalar potential has to obey. These are constraints coming from imposing perturbative unitarity, and boundedness of the scalar potential from below (BFB). Another aspect under study are the implications of such models and how to test them experimentally at the LHC and future colliders. We have built codes for these models and we will interface them with the latest results from LHC. Models with an extended Higgs sector can provide good dark matter candidates. For the case of two doublets there is already a large literature but for the 3HDM it is starting now. We are exploiting several symmetries that can lead to and unbroken symmetry that gives rise to dark matter candidate. - Gravitational waves - a new BSM opportunity: Gravitational wave signals offer a novel window into new particle physics effects, particularly related to phase transitions in the early Universe. Investigating the connection between gravitational waves and baryogenesis mechanisms, including potential ties to dark matter, represents an exciting frontier in physics research.