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We study the prospects of thermal leptogenesis in the framework of the minimal flipped $SU(5)$ unified model in which the RH neutrino mass scale emerges as a two-loop effect. Despite its strong suppression with respect to the unification scale which tends to disfavor leptogenesis in the standard Davidson-Ibarra regime (and a notoriously large washout of the $N_1$-generated asymmetry owing to a top-like Yukawa entry in the Dirac neutrino mass matrix) the desired $\eta_B\sim 6\times 10^{-10}$ can still be attained in several parts of the parameter space exhibiting interesting baryon and lepton number violation phenomenology. Remarkably enough, in all these regions the mass of the lightest LH neutrino is so low that it yields $m_\beta \lesssim 0.03$ eV for the effective neutrino mass measured in beta-decay, i.e., an order of magnitude below the design sensitivity limit of the KATRIN experiment. This makes the model potentially testable in the near future., Comment: 18 pages, 6 figures

We classify all different composite Higgs models (CHMs) characterised by the coset space $\mathcal{G}/\cal{H}$ of compact semi-simple Lie groups $\mathcal{G}$ and $\mathcal{H}$ involving up to 13 Nambu-Goldstone bosons (NGBs), together with mild phenomenological constraints. As a byproduct of this work, we prove several simple yet, to the best of our knowledge, mostly unknown results: (1) Under certain conditions, a given set of massless scalars can be UV completed into a CHM in which they arise as NGBs; (2) The set of all CHMs with a fixed number of NGBs is finite, and in particular there are 642 of them with up to 13 massless scalars (factoring out models that differ by extra $U(1)$'s); (3) Any scalar representation of the Standard Model group can be realised within a CHM; (4) Certain symmetries of the scalar sector allowed from the IR perspective are never realised within CHMs. On top of this, we make a simple statistical analysis of the landscape of CHMs, determining the frequency of models with scalar singlets, doublets, triplets and other multiplets of the custodial group as well as their multiplicity. We also count the number of models with a symmetric coset., Comment: List of models in the ancillary file, 19 pages, 2 figures, 2 tables

In recent years, theoretical and phenomenological studies with effective field theories have become a trending and prolific line of research in the field of high-energy physics. In order to discuss present and future prospects concerning automated tools in this field, the SMEFT-Tools 2022 workshop was held at the University of Zurich from 14th-16th September 2022. The current document collects and summarizes the content of this workshop., Comment: 116 pages, 10 figures. Based on the contributions to SMEFT-Tools 2022 (https://indico.icc.ub.edu/event/128/). v2: version published in EPJC

We study a non-supersymmetric SO(10) Grand Unification Theory with a very high energy intermediate symmetry breaking scale in which not only gauge but also Yukawa coupling unification are enforced via suitable threshold corrections and matching conditions. For gauge unification, we focus on a few symmetry breaking patterns with the intermediate gauge groups ${\rm SU(4)_C \times SU(2)_L \times SU(2)_R}$ (Pati-Salam) and ${\rm SU(3)_C \times SU(2)_L \times SU(2)_R\times U(1)_{B-L}}$ (minimal left-right symmetry) assuming an additional global U(1) Peccei--Quinn symmetry, and having the Standard Model supplemented by a second Higgs doublet field at the electroweak scale. We derive the conditions as well as the approximate analytical solutions for the unification of the gauge coupling constants at the two-loop level and discuss the constraints from proton decay on the resulting high scale. Specializing to the case of the Pati-Salam intermediate breaking pattern, we then impose also the unification of the Yukawa couplings of third generation fermions at the high scale, again at the two-loop level. In the considered context, Yukawa unification implies a relation between the fermion couplings to the 10- and 126-dimensional scalar representations of the SO(10) group. We consider one such possible relation which is obtainable in an ${\rm E_6}$ model where the previous two scalar fields are part of a single multiplet. Taking into account some phenomenological features such as the absence of flavor changing neutral currents at tree-level, we derive constraints on the parameters of the low energy model, in particular on the ratio of the two Higgs doublets vacuum expectation values $\tan\beta$., Comment: 40 pages, 7 figures, Fig. 7 added compared to v1, comments are welcome

A vector boson $W_{1}^{\mu}$ with the quantum numbers $\left(\boldsymbol{3},1\right)$ under $SU\left(2\right)_{L}\times U(1)_{Y}$ could in principle couple with the Higgs field via the renormalizable term $W_{1}^{\mu*}HD_{\mu}H$. This interaction is known to affect the $T$ parameter and, in so doing, it could potentially explain the recent CDF measurement of the W-boson mass. As it is often the case with vectors, building a viable model with a $W_{1}$ gauge boson is non-trivial. In this work I will describe two variations of a minimal setup containing this field; they are based on an extended $SO(5)\times SU\left(2\right)\times U(1)$ electroweak group. I will nevertheless show that interactions such as $W_{1}^{\mu*}H\partial_{\mu}H$ are never generated in a Yang-Mills theory. A coupling between $W_{1}$, $H$ and another Higgs doublet $H^{\prime}$ is possible though. Finally, I will provide an explicit recipe for the construction of viable models with gauge bosons in arbitrary representations of the Standard Model group; depending on the quantum numbers, they may couple to pairs of Standard Model fermions, or to a Standard Model fermion and an exotic one., Comment: 16 pages

Vacuum stability requires that the scalar potential is bounded from below. Whether or not this is true depends on the scalar quartic interactions alone, but even so the analysis is arduous and has only been carried out for a limited set of models. Complementing the existing literature, this work contains the necessary and sufficient conditions for two SU(n) invariant potentials to be bounded from below. In particular, expressions are given for models with the fundamental and the 2-index (anti)symmetric representations of this group. A sufficient condition for vacuum stability is also provided for models with the fundamental and the adjoint representations. Finally, some considerations are made concerning the model with the gauge group SU(2) and the scalar representations $\boldsymbol{1}$, $\boldsymbol{2}$ and $\boldsymbol{3}$; such a setup is particularly important for neutrino mass generation and lepton number violation., Comment: 19 pages, 4 figures

We do not know why there are three fermion families in the Standard Model (SM), nor can we explain the observed pattern of fermion masses and mixing angles. Standard grand unified theories based on the SU(5) and SO(10) groups fail to shed light on this issue, since they also contain three copies of fermion representations of an enlarged gauge group. However, it does not need to be so: the Standard Model families might be distributed over distinct representations of a grand unified model, in which case the gauge symmetry itself might discriminate the various families and explain (at least partially) the flavor puzzle. The most ambitious version of this idea consists on embedding all SM fermions in a single irreducible representation of the gauge group., Comment: Contribution to the proceedings of the 40th International Conference on High Energy physics (ICHEP2020), 28 July -- 6 August 2020, Prague, Czech Republic (virtual meeting)

GroupMath is a Mathematica package which performs several calculations related to semi-simple Lie algebras and the permutation groups, both of which are important in particle physics as well as in other areas of research., Comment: Describes version 1.1 of the program. Code available on http://renatofonseca.net/groupmath

We present a model where all fermions are contained in a single irreducible representation of an SU(19) gauge symmetry group. If there is only one scalar field, Yukawa interactions are controlled by a single number rather than by one or more $3\times3$ matrices of couplings. The low-energy concept of flavor emerges entirely from the scalar-sector parameters; more specifically, entries of the Standard Model Yukawa matrices are controlled by several vacuum expectation values., Comment: 9 pages

Motivated by the experimentally observed deviations from standard model predictions, we calculate the anomalous magnetic moments $a_\alpha = (g-2)_\alpha$ for $\alpha=e,\mu$ in a neutrino mass model originally proposed by Babu-Nandi-Tavartkiladze (BNT). We discuss two variants of the model, the original model plus a minimally extended version with an additional hypercharge zero triplet scalar. While the original BNT model can explain $a_\mu$, only the variant with the triplet scalar can explain both experimental anomalies. The heavy fermions of the model can be produced at the high-luminosity LHC and in the part of parameter space, where the model explains the experimental anomalies, it predicts certain specific decay patterns for the exotic fermions., Comment: 23 pages, 14 figures

The simplest possibility to generate small Majorana neutrino masses is the seesaw mechanism. However, the smallness of the observed neutrino masses can also be understood, if neutrino masses are generated by higher-dimensional operators and/or at higher loop level. In this talk recent work on systematic classifications of higher-dimensional and radiative neutrino mass models is summarized. Two particular classes of special genuine loop diagrams, i.e. diagrams which can lead to genuine neutrino mass models only under some specific, well-defined conditions, are also discussed., Comment: Contribution to the Proceedings of TAUP-2019; Toyama, Japan

Until recently little was known about the high-dimensional operators of the standard model effective field theory (SMEFT). However, in the past few years the number of these operators has been counted up to mass dimension 15 using techniques involving the Hilbert series. In this work I will show how to perform the same counting with a different method. This alternative approach makes it possible to cross-check results (it confirms the SMEFT numbers), but it also provides some more information on the operators beyond just counting their number. The considerations made here apply equally well to any other model besides SMEFT and, with this purpose in mind, they were implemented in a computer code., Comment: Version published in the Physical Review D journal

The number of leptons may or may not be a conserved quantity. The Standard Model predicts that it is (in perturbative processes), but there is the well known possibility that new physics violates lepton number in one or two units. The first case ($\Delta L=1$) is associated to proton decay into mesons plus a lepton or an anti-lepton, while the second one ($\Delta L=2$) is usually associated to Majorana neutrino masses and neutrinoless double beta decay. It is also conceivable that leptons can only be created or destroyed in groups of three ($\Delta L=3$). Colliders and proton decay experiments can explore this possibility., Comment: Contribution to the proceedings of the 6th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2018), Vienna, Austria, 26-30 November 2018

We study systematically the decomposition of the Weinberg operator at three-loop order. There are more than four thousand connected topologies. However, the vast majority of these are infinite corrections to lower order neutrino mass diagrams and only a very small percentage yields models for which the three-loop diagrams are the leading order contribution to the neutrino mass matrix. We identify 73 topologies that can lead to genuine three-loop models with fermions and scalars, i.e. models for which lower order diagrams are automatically absent without the need to invoke additional symmetries. The 73 genuine topologies can be divided into two sub-classes: Normal genuine ones (44 cases) and special genuine topologies (29 cases). The latter are a special class of topologies, which can lead to genuine diagrams only for very specific choices of fields. The genuine topologies generate 374 diagrams in the weak basis, which can be reduced to only 30 distinct diagrams in the mass eigenstate basis. We also discuss how all the mass eigenstate diagrams can be described in terms of only five master integrals. We present some concrete models and for two of them we give numerical estimates for the typical size of neutrino masses they generate. Our results can be readily applied to construct other $d=5$ neutrino mass models with three loops., Comment: Erratum added, published version in JHEP

For Majorana neutrino masses the lowest dimensional operator possible is the Weinberg operator at $d=5$. Here we discuss the possibility that neutrino masses originate from higher dimensional operators. Specifically, we consider all tree-level decompositions of the $d=9$, $d=11$ and $d=13$ neutrino mass operators. With renormalizable interactions only, we find 18 topologies and 66 diagrams for $d=9$, and 92 topologies plus 504 diagrams at the $d=11$ level. At $d=13$ there are already 576 topologies and 4199 diagrams. However, among all these there are only very few genuine neutrino mass models: At $d=(9,11,13)$ we find only (2,2,2) genuine diagrams and a total of (2,2,6) models. Here, a model is considered genuine at level $d$ if it automatically forbids lower order neutrino masses {\em without} the use of additional symmetries. We also briefly discuss how neutrino masses and angles can be easily fitted in these high-dimensional models., Comment: Coincides with published version in JHEP

We discuss the experimental prospects for observing processes which violate lepton number ($\Delta L$) in four units (or more). First, we reconsider neutrinoless quadruple beta decay, deriving a model independent and very conservative lower limit on its half-life of the order of $10^{41}$ ys for $^{150}$Nd. This renders quadruple beta decay unobservable for any feasible experiment. We then turn to a more general discussion of different possible low-energy processes with values of $\Delta L \ge 4$. A simple operator analysis leads to rather pessimistic conclusions about the observability at low-energy experiments in all cases we study. However, the situation looks much brighter for accelerator experiments. For two example models with $\Delta L=4$ and another one with $\Delta L=5$, we show how the LHC or a hypothetical future pp-collider, such as the FCC, could probe multi-lepton number violating operators at the TeV scale.

We discuss lepton number violation in three units. From an effective field theory point of view, $\Delta L=3$ processes can only arise from dimension 9 or higher operators. These operators also violate baryon number, hence many of them will induce proton decay. Given the high dimensionality of these operators, in order to have a proton half-life in the observable range, the new physics associated to $\Delta L=3$ processes should be at a scale as low as 1 TeV. This opens up the possibility of searching for such processes not only in proton decay experiments but also at the LHC. In this work we analyze the relevant $d=9,11,13$ operators which violate lepton number in three units. We then construct one simple concrete model with interesting low- and high-energy phenomenology., Comment: 8 pages, 5 figures, Published Version

Model builders often need to find the most general Lagrangian which can be constructed from a given list of fields. These fields are actually representations of the Lorentz and gauge groups (and maybe of some discrete symmetry group as well). I will describe a simple program (Sym2Int) which helps to automate this task by listing all possible interactions between Lorentz/gauge group representations., Comment: Contribution to the proceedings of the Fifth Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2016), Warsaw, Poland, 28 November - 3 December 2016. The computer program is available on http://renatofonseca.net/sym2int.php

We discuss contributions to neutrinoless double beta ($0\nu\beta\beta$) decay involving vector bosons. The starting point is a list of all possible vector representations that may contribute to $0\nu\beta\beta$ decay via $d=9$ or $d=11$ operators at tree-level. We then identify gauge groups which contain these vectors in the adjoint representation. Even though the complete list of vector fields that can contribute to $0\nu\beta\beta$ up to $d=11$ is large (a total of 46 vectors), only a few of them can be gauge bosons of phenomenologically realistic groups. These latter cases are discussed in some more detail, and lower (upper) limits on gauge boson masses (mixing angles) are derived from the absence of $0\nu\beta\beta$ decay., Comment: Matches the version accepted for publication in Physical Review D

Different models based on the extended $SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}$ (331) gauge group have been proposed over the past four decades. Yet, despite being an active research topic, the status of lepton number in 331 models has not been fully addressed in the literature, and furthermore many of the original proposals can not explain the observed neutrino masses. In this paper we review the basic features of various 331 models, focusing on potential sources of lepton number violation. We then describe different modifications which can be made to the original models in order to accommodate neutrino (and charged lepton) masses., Comment: Matches the version published in Physical Review D

Models based on the extended $SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}$ (331) gauge group usually follow a common pattern: two families of left-handed quarks are placed in anti-triplet representations of the $SU(3)_{L}$ group; the remaining quark family, as well as the left-handed leptons, are assigned to triplets (or vice-versa). In this work we present a flipped 331 model where this scheme is reversed: all three quarks families are in the same representation and it is the lepton families which are discriminated by the gauge symmetry. We discuss fermion masses and mixing, as well as $Z'$ interactions, in a minimal model implementing this idea., Comment: Version published in the Journal of High Energy Physics

Adding a scalar triplet to the Standard Model is one of the simplest ways of giving mass to neutrinos, providing at the same time a mechanism to stabilize the theory's vacuum. In this paper, we revisit these aspects of the type-II seesaw model pointing out that the bounded-from-below conditions for the scalar potential in use in the literature are not correct. We discuss some scenarios where the correction can be significant and sketch the typical scalar boson profile expected by consistency., Comment: 14 pages, 5 figures, references added, minor changes to match the published version

The vacuum of the Standard Model is known to be unstable for the measured values of the top and Higgs masses. Here we show how vacuum stability can be achieved naturally if lepton number is violated spontaneously at the TeV scale. More precise Higgs measurements in the next LHC run should provide a crucial test of our symmetry breaking scenario. In addition, these schemes typically lead to enhanced rates for processes involving lepton flavour violation ., Comment: 9 pages, 4+2 figures; some references added, some textual modifications: 2 figures added, appendices added. Results unchanged. Matches published version

The short-range part of the neutrinoless double beta amplitude is generated via the exchange of exotic particles, such as charged scalars, leptoquarks and/or diquarks. In order to give a sizeable contribution to the total decay rate, the masses of these exotics should be of the order of (at most) a few TeV. Here, we argue that these exotics could be the "light" (i.e weak-scale) remnants of some $B-L$ violating variants of $SU(5)$. We show that unification of the standard model gauge couplings, consistent with proton decay limits, can be achieved in such a setup without the need to introduce supersymmetry. Since these non-minimal $SU(5)$-inspired models violate $B-L$, they generate Majorana neutrino masses and therefore make it possible to explain neutrino oscillation data. The "light" coloured particles of these models can potentially be observed at the LHC, and it might be possible to probe the origin of the neutrino masses with $\Delta L=2$ violating signals. As particular realizations of this idea, we present two models, one for each of the two possible tree-level topologies of neutrinoless double beta decay., Comment: Small changes made; matches the published version in Physical Review D

The Standard Model (SM) is a chiral theory, where right- and left-handed fermion fields transform differently under the gauge group. Extra fermions, if they do exist, need to be heavy otherwise they would have already been observed. With no complex mechanisms at work, such as confining interactions or extra-dimensions, this can only be achieved if every extra right-handed fermion comes paired with a left-handed one transforming in the same way under the Standard Model gauge group, otherwise the new states would only get a mass after electroweak symmetry breaking, which would necessarily be small ($\sim100\textrm{ GeV}$). Such a simple requirement severely constrains the fermion content of Grand Unified Theories (GUTs). It is known for example that three copies of the representations $\mathbf{\overline{5}}+\mathbf{10}$ of $SU(5)$ or three copies of the $\mathbf{16}$ of $SO(10)$ can reproduce the Standard Model's chirality, but how unique are these arrangements? In a systematic way, this paper looks at the possibility of having non-standard mixtures of fermion GUT representations yielding the correct Standard Model chirality. Family unification is possible with large special unitary groups --- for example, the $\mathbf{171}$ representation of $SU(19)$ may decompose as $3\left(\mathbf{16}\right)+\mathbf{120}+3\left(\mathbf{1}\right)$ under $SO(10)$., Comment: Minor changes; matches publication in Nuclear Physics B

The physics responsible for gauge coupling unification may also induce small neutrino masses. We propose a novel gauge mediated radiative seesaw mechanism for calculable neutrino masses. These arise from quantum corrections mediated by new $SU(3)_c \times SU(2)_L \times U(1)_X$ (3-3-1) gauge bosons and the physics driving gauge coupling unification. Gauge couplings unify for a 3-3-1 scale in the TeV range, making the model directly testable at the LHC., Comment: 5 pages, 4 figures. Matches version accepted for publication in Phys.Rev.D. Rapid Communications

Flavour symmetries have been used to constrain both quark and lepton mixing parameters. In particular, they can be used to completely fix the mixing angles. For the lepton sector, assuming that neutrinos are Majorana particles, we have derived the complete list of mixing patterns achievable in this way, as well as the symmetry groups associated to each case. Partial computer scans done in the past have hinted that such list is limited, and this does indeed turn out to be the case. In addition, most mixing patterns are already 3-sigma excluded by neutrino oscillation data., Comment: Contribution to the proceedings of the ICHEP 2014 conference, Valencia, Spain, July 2014

Assuming that neutrinos are Majorana particles, we perform a complete classification of all possible mixing matrices which are fully determined by residual symmetries in the charged-lepton and neutrino mass matrices. The classification is based on the assumption that the residual symmetries originate from a finite flavour symmetry group. The mathematical tools which allow us to accomplish this classification are theorems on sums of roots of unity. We find 17 sporadic cases plus one infinite series of mixing matrices associated with three-flavour mixing, all of which have already been discussed in the literature. Only the infinite series contains mixing matrices which are compatible with the data at the 3 sigma level., Comment: Correction of a wrong identification (there are consequently 17 sporadic lepton mixing cases); change in the title to include the word "finite"; newer neutrino data used; minor corrections and clarifications; version matches publication in JHEP

There are reasons to believe that the Standard Model is only an effective theory, with new Physics lying beyond it. Supersymmetric extensions are one possibility: they address some of the Standard Model's shortcomings, such as the instability of the Higgs boson mass under radiative corrections. In this thesis, some topics related to the renormalization of supersymmetric models are analyzed. One of them is the automatic computation of the Lagrangian and the renormalization group equations of these models, which is a hard and error-prone process if carried out by hand. The generic renormalization group equations themselves are extended so as to include those models which have more than a single abelian gauge factor group. Such situations can occur in grand unified theories, for example. For a wide range of SO(10)-inspired supersymmetric models, we also show that the renormalization group imprints on sparticle masses some information on the higher energies behavior of the models. Finally, in some cases these theories introduce charged lepton flavor violating interactions, which can change the ratio $\Gamma\left(K\rightarrow e\nu\right)/\Gamma\left(K\rightarrow\mu\nu\right)$. In light of experimental bounds on other observables, our analysis shows that any change over the Standard Model prediction must be smaller than the current experimental sensitivity on this observable., Comment: PhD thesis. Instituto Superior T\'ecnico, Universidade de Lisboa. 253 pages

We work out a set of simple rules for adopting the two-loop renormalization group equations of a generic gauge field theory given in the seminal works of Machacek and Vaughn to the most general case with an arbitrary number of Abelian gauge factors and comment on the extra subtleties possibly encountered upon matching a set of effective gauge theories in such a framework., Comment: 6 pages; v2) correcting powers of g in eqs. (23), (24), (27) and (28)

We generalize the two-loop renormalization group equations for the parameters of the softly broken SUSY gauge theories given in the literature to the most general case when the gauge group contains more than a single abelian gauge factor. The complete method is illustrated at two-loop within a specific example and compared to some of the previously proposed partial treatments., Comment: 18 pages, 10 figures; v2: matches published version; v3: corrected eqs. (21) and (35), changed text in sec. 3.B accordingly

Susyno is a Mathematica package dedicated to the computation of the 2-loop renormalisation group equations of a supersymmetric model based on any gauge group (the only exception being multiple U(1) groups) and for any field content., Comment: Describes version 1.1 of the program (matches published version). The program's webpage is in http://web.ist.utl.pt/renato.fonseca/susyno.html