UNIVERSIDAD NACIONAL AUTÓNOMA DE MEXICO POSGRADO EN CIENCIAS FISICAS INSTITUTO DE FISICA FISICA DE ALTAS ENERGIAS, FISICA NUCLEAR, GRAVITACION y FISICA MATEMATICA PHENOMENOLOGY OF NEUTRINO AND DARK MATTER ORIENTED EXTENSIONS OF THE STANDARD MODEL TESIS QUE PARA OPTAR POR EL GRADO DE: DOCTOR EN CIENCIAS (FISICA) PRESENTA: LEON MANUEL GARCIA DE LA VEGA TUTOR O TUTORES PRINCIPALES DR. EDUARDO PEINADO INSTITUTO DE FISICA EN SU CASO, MIEMBROS DEL COMITÉ TUTOR DR. ERIC VAZQUEZ JAUREGUI INSTITUTO DE FISICA DR. CESAR FERNANDEZ RAMIREZ INSTITUTO DE CIENCIAS NUCLEARES CIUDAD DE MEXICO , AGOSTO DEL 2023 UNAM – Dirección General de Bibliotecas Tesis Digitales Restricciones de uso DERECHOS RESERVADOS © PROHIBIDA SU REPRODUCCIÓN TOTAL O PARCIAL Todo el material contenido en esta tesis esta protegido por la Ley Federal del Derecho de Autor (LFDA) de los Estados Unidos Mexicanos (México). El uso de imágenes, fragmentos de videos, y demás material que sea objeto de protección de los derechos de autor, será exclusivamente para fines educativos e informativos y deberá citar la fuente donde la obtuvo mencionando el autor o autores. Cualquier uso distinto como el lucro, reproducción, edición o modificación, será perseguido y sancionado por el respectivo titular de los Derechos de Autor. Phenomenology of Neutrino and Dark Matter oriented extensions of the Standard Model Leon Manuel Garcia de la Vega of Instituto de Física, UNAM A dissertation submitted to the Universidad Nacional Autónoma de México for the degree of Doctor of Physics ii iii Abstract Two of the most promising avenues for discovering new fundamen- tal physics are dark matter and neutrino masses. In this thesis we study these issues from a phenomenological point of view, with the aim of discovering the possible behavior of new physics through the construction of theoretically consistent models of dark matter and neutrino masses. iv v Declaration This dissertation is the result of my own work, except where explicit reference is made to the work of others, and has not been submitted for another qualification to this or any other university. This dissertation does not exceed the word limit for the respective Degree Committee. Leon vi vii Acknowledgements Of the many people who deserve thanks, some are particularly prominent, such as my supervisor Dr. Eduardo Peinado, my collaborators, my colleagues and friends in the Department, and my parents. viii ix Preface This thesis presents the results of the research carried out during my PhD studies under the supervision of Prof. Ediardo Peinado. These research was also published in the following research articles • [1] Fermion Dark Matter and Radiative Neutrino Masses from Spontaneous Lepton Number Breaking Cesar Bonilla(Munich, Tech. U. and Catolica del Norte U.), Leon M.G. de la Vega(Mexico U.), J.M. Lamprea(Mexico U.), Roberto A. Lineros(Catolica del Norte U.), Eduardo Peinado(Mexico U.) • [2] Dirac neutrinos from Peccei-Quinn symmetry: two examples Leon M.G. de la Vega(Mexico U.), Newton Nath(Mexico U.), Eduardo Peinado(Mexico U.) • [3]Neutrino phenomenology in a left-right D4 symmetric model Cesar Bonilla(Catolica del Norte U.), Leon M.G. de la Vega(UNAM, Mexico), R. Ferro-Hernandez(Mexico U.), Newton Nath(Mexico U.), Eduardo Peinado(Mexico U.) • [4] Flavored axion in the UV-complete Froggatt–Nielsen models Leon M.G. de la Vega(Mexico U.), Newton Nath(Mexico U.), Stefan Nellen(Mexico U.), Eduardo Peinado(Mexico U.) • [5]Complementarity between dark matter direct searches and CEvNS experi- ments in U(1) models Leon M.G. de la Vega(Mexico U.), L.J. Flores(Mexico U.), Newton Nath(Mexico U.), Eduardo Peinado(Mexico U.) • [6] Neutrino masses and self-interacting dark matter with mass mixing Z-Z￿ gauge portal Leon M.G. de la Vega(Mexico U.), Eduardo Peinado(Mexico U.), Jose Wudka(UC, Riverside) x • [7]Closing the dark photon window to thermal dark matter Leon M.G. de la Vega(Mexico U.), Alejandro Garcia-Viltres (Mexico U.) Eduardo Peinado(Mexico U.), R. Ferro-Hernandez(JGU-Mainz) and Eric Vázquez-Jáuregui(Mexico U.) • [8] Dark Z Bosons and Parity Violating Observables Leon M.G. de la Vega(Mexico U.), Jens Erler (JGU-Mainz) Eduardo Peinado(Mexico U.), R. Ferro-Hernandez(JGU-Mainz) Contents 1. Neutrino Physics in the SM and beyond 1 1.1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. The Standard Model neutrino . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1. PMNS matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2. Majorana and Dirac neutrinos . . . . . . . . . . . . . . . . . . . 8 1.2.3. Seesaw mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. Dark Matter Physics 11 2.1. The need for dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1. Galactic rotation curves . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2. Galaxy clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.3. Cosmic Microwave Background . . . . . . . . . . . . . . . . . . 13 2.2. Particle Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1. WIMP Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2. Axion Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3. Flavor Symmetric models of neutrino masses and dark matter 19 3.1. Flavor symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2. D4 as a symmetry group . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.1. The discrete non-abelian D4 symmetry group . . . . . . . . . . 20 3.2.2. Left-Right gauge symmetry . . . . . . . . . . . . . . . . . . . . . 22 3.2.3. A GLR ⇥ D4 ⇥ Z2 model of lepton masses and mixing . . . . . . 23 3.3. Gauged U(1)X symmetry as a flavor symmetry . . . . . . . . . . . . . . 34 3.3.1. Relic density and direct detection . . . . . . . . . . . . . . . . . . 41 3.3.2. Dark matter direct detection and CEnNS complementarity . . . 43 3.3.3. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4. Global U(1) as a flavor symmetry . . . . . . . . . . . . . . . . . . . . . . 50 3.4.1. Froggat-Nielsen mechanism and Nearest-Neighbour-Interactions 52 3.4.2. Fermion masses in flavored DFSZ axion models . . . . . . . . . 53 xi xii Contents 3.4.3. Type-I Seesaw UV-completion . . . . . . . . . . . . . . . . . . . 54 3.4.4. The UV-completion: DFSZ type-II Seesaw . . . . . . . . . . . . . 57 3.4.5. Lepton Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4.6. Scalar sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.7. Axion couplings and mass . . . . . . . . . . . . . . . . . . . . . . 61 3.4.8. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4.9. Flavor Violating decays with axions . . . . . . . . . . . . . . . . 65 3.4.10. Flavor Violating Higgs couplings . . . . . . . . . . . . . . . . . . 67 3.4.11. Flavored Axion as Dark Matter candidate . . . . . . . . . . . . . 71 3.4.12. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4. Dark and Lepton symmetries 73 4.1. Lepton number and darkness . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2. Neutrino masses and dark matter from the breaking of a global U(1) Lepton symmetry with a dark Z2 . . . . . . . . . . . . . . . . . . . . . . 73 4.2.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.2. Mass spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.3. Summary of constraints . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.4. Searches of new physics . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.5. Dark matter searches . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.6. Neutrino oscillation parameters . . . . . . . . . . . . . . . . . . 79 4.2.7. Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2.8. Viable dark matter mass regions . . . . . . . . . . . . . . . . . . 82 4.2.9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3. Global U(1)PQ symmetry as the source of neutrino Diracness . . . . . . 84 4.3.1. Dirac Neutrinos from PQ symmetry . . . . . . . . . . . . . . . . 85 4.3.2. An Alternate Loop Model . . . . . . . . . . . . . . . . . . . . . . 95 4.3.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4. Neutrino seesaws and Self- Interacting DM from Z-Z’ mixing . . . . . 97 4.4.1. Model for Z-Z’ mixing with a dark gauge symmetry . . . . . . 98 4.4.2. Neutrino sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.4.3. Dark Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4.4. Gauge sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4.5. Scalar Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4.6. Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4.7. Dark matter relic density . . . . . . . . . . . . . . . . . . . . . . 105 Contents xiii 4.4.8. Dark matter direct detection . . . . . . . . . . . . . . . . . . . . . 106 4.4.9. Dark Matter Self-Interactions . . . . . . . . . . . . . . . . . . . . 109 4.4.10. Neutrino masses and U(1)D breaking scale . . . . . . . . . . . . 112 4.4.11. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.5. Low energy constraints on a Dark Z boson . . . . . . . . . . . . . . . . 113 4.5.1. The General Dark Z model . . . . . . . . . . . . . . . . . . . . . 114 4.5.2. Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.5.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.6. Direct detection constraints on Secluded Dark Photon mediated dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.6.1. Dark Photon mediated secluded WIMP model . . . . . . . . . . 121 4.6.2. DM relic density . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.6.3. Experimental constraints on a dark photon . . . . . . . . . . . . 124 4.6.4. Direct Detection constraints on the dark photon mediated spin independent cross section . . . . . . . . . . . . . . . . . . . . . . 124 4.6.5. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5. General Summary 129 A. Appendices 131 A.1. Correlations in the MIII models . . . . . . . . . . . . . . . . . . . . . . . 131 A.2. Phase redefinition of quark mass matrices . . . . . . . . . . . . . . . . . 132 A.3. Scalar potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.4. Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Bibliography 139 List of figures 155 List of tables 159 xiv Contents Chapter 1. Neutrino Physics in the SM and beyond 1.1. Generalities Neutrinos are elementary particles, part of the Standard Model of Particle Physics (SM). They are produced in the beta decay in nuclei A Z N ! A Z+1N0 + n̄e + e. When beta decay was discovered, the emission of the electron and the transmutation of the nucleus into an element with the same atomic mass, but larger atomic number were quickly identified. The observed process seemingly consisted of A Z N ! A Z+1N0 + e. Subsequently, after the spectrum of the emitted electron was measured, an apparent violation of the conservation of energy was observed. The spectrum of beta electron is continuous [9], rather than a sharp peak, as expected from a two body decay. In 1930 Wolfgang Pauli proposed a solution to save the universality of the law of energy conservation, by postulating the production of the neutrino in the beta decay process, with a small mass and no electric charge. Enrico Fermi formalized this theory in 1933 by proposing a Hamiltonian describing the beta decay of a neutron [10]. In the modern language of Relativistic Quantum Effective Field Theories, Fermi’s theory is an effective Lagrangian for the four-point interaction of the up quark, down quark, electron and neutrino. The dimension-6 operator describing this interaction is Lb = GFp 2 ȳdgµ (1 g5) 2 yuȳngµ (1 g5) 2 ye , (1.1) where yi is the spinor describing the i fermion field, and GF is a constant. This interaction fixes the problem of the missing energy of the electron emmitted in the beta decay, as the invisible neutrino carries this energy and, because it is electrically neutral, leaves no ionisation trace in detectors. The neutrino seemed to be an undetectable 1 2 Neutrino Physics in the SM and beyond particle to Pauli and Fermi, and in 1934 Hans Bethe and Rudolf Peierls [11] estimated the free-streaming length of a MeV neutrino in the Inverse Beta Decay (IBD) process to be of the order O(1016) km in water. Thus, they (incorrectly) assumed that MeV neutrinos would never be detected through the IBD process. In a followup work [12] Bethe and Peierls proposed 3 physical processes to confirm the Fermi theory of the neutrino: • The precise measurement of the electron spectrum of artificial beta decay, if the masses and mass defects of all involved nuclei were known with sufficient precision, would reveal the mass of the neutrino from the endpoint of the electron energy. • The precise measurement of the recoil energy of a daughter light nucleus pro- duced in artificial beta decay would also reveal the neutrino mass, from the missing nucleus momentum. • The non-observation of the capture of an orbital electron by a proton in the hydrogen atom, with the transmutation of the proton into a neutron and the emission of a neutrino provides a limit on the masses of the neutron (which was not well measured at the time) and the neutrino. IBD turned out to be a viable detection process (for large and sensitive enough de- tectors and neutrino fluxes) and the measurement of the tritium beta decay electron spectrum is currently a leading possibility for the determination of the neutrino mass scale. In 1937 Luis Alvarez measured electron capture in Vanadium [13], further confirming Fermi theory. The first direct confirmation of the existence of the neutrino came from the Cowan- Reines experiment [14], performed in 1956 at the Savannah River plant in North Carolina. The detection strategy at first planned to detect the annihilations of positrons emmitted in IBD in an organic scintillator tank, carefully placed underground near the site of a nuclear test. A nuclear detonation was chosen over a nuclear reactor for the neutrino source, because the much larger neutrino flux from a nuclear explosion would result in a larger number of events, which was hoped would counter the large background gamma ray background from the source. The experiment was approved for construction at the Nevada Test Site, where nuclear devices were routinely det- onated until 1992, both under- and above-ground. During the construction of the detector and the underground shaft digged to house it, Reines and Cowan figured out a way to reduce the gamma ray background from the neutrino source by doping Neutrino Physics in the SM and beyond 3 the scintillator with Cadmium. Cadmium has a relatively large neutron absorption cross section, so it would be capable of absorbing the neutron produced in the IBD, producing an excited isotope which decays by gamma emmission. The gamma ray emmitted by Cadmium would subsequently be detected. The coincidence between the gamma ray pair from the positron annihilation and the gamma ray from the Cadmium de-excitation would enable the discrimination between errant gamma rays from the neutrino source and gamma rays produced by the IBD positron. This insight made the experiment at a nuclear reactor more viable than at a nuclear test and the experiment was moved from Nevada to the Savannah River nuclear reactor. The experiment measured the neutrino flux as predicted from Fermi theory, confirming the existence of the neutrino, and serving as a self-consistency check on beta decay theory. Further developments shaping the modern understanding of the neutrino and its interactions come from the observation of parity violation by the experimental groups of Chien-Shiung Wu [15] and Leon Lederman [16], the measurement of the active neutrino helicity by Goldhaber [17], and the discovery of the muon neutrino by Leder- man [18]. 1.2. The Standard Model neutrino The theoretical framework describing particle physics as measured by experiments is the Standard Model. It consists of a Lorentz invariant gauge field theory, in which the observed fundamental particles are assigned a representation of the Lorentz and gauge group. The gauge group of the Standard Model is GSM = SU(3)C ⇥ SU(2)L ⇥U(1)Y. The fermion and scalar content of the Standard Model, in the interaction basis, is shown in table 1.1. The SU(3)C subgroup of the SM describes the strong interaction, which holds quarks together inside a nucleon. It is mediated by the 8 gluons associated to the 8 generators of the Lie group SU(3). Gluons are massless in the SM, as there is no Spontaneous Symmetry Breaking of SU(3)C built into the SM. The SU(2)L ⇥U(1)Y subgroup is known as the Electroweak gauge sector. It has four gauge bosons associated to the four generators of the subgroup. In the SM, a scalar sector triggers SSB of this subgroup resulting in one massless and three massive gauge bosons. The gauge sector 4 Neutrino Physics in the SM and beyond QUARKS LEPTONS SCALAR Q = uL dL ! uR dR L = nL eL ! eR Φ = G+ Φ 0 ! SU(3)C 3 3 3 1 1 1 SU(2)L 2 1 1 2 1 2 U(1)Y 1/6 2/3 -1/3 -1/2 -1 1/2 Table 1.1.: Fermion and Scalar content of the SM, in the interaction basis. For each fermion type, there are three generations in the SM. The SU(2)L components of the fields are shown explicitly, while the SU(3)C components and indices are not shown. Lagrangian is given by Lg = 1 4 Ga µnGµn a 1 4 Fb µnF µn b 1 4 HµnHµn , (1.2) where we denote Ga for the SU(3)C gauge bosons, Fb for the SU(2)L gauge bosons and F for the U(1)Y gauge boson1. In eq. 1.2 the field strength tensor Ka µn for a non-abelian gauge boson K is given by Ka µn = ∂µKa n ∂nKa µ + g ∑ b,c fabcKb µKc n , (1.3) where g is the non-abelian gauge group gauge coupling and fabc is the group’s Lie algebra structure constants. In the SM, the color gauge bosons are already mass eigenstates, as there is no spontaneous breaking of SU(3)C. The electroweak gauge subgroup is however broken by the scalar Higgs doublet as follows. The scalar potential and kinetic terms are LΦ = (Dµ Φ)†DµΦ µ 2 Φ † Φ l(Φ† Φ)2 . (1.4) Given the assigned representations of H given in table 1.1, the first term of eq. 1.4 is DµΦ = ∂µΦ igLFa µTa Φ igY/2HµΦ (1.5) where Ta are the generators of the SU(2)L associated algebra2. The scalar quadratic and quartic interactions lead to spontaneous symmetry breaking of SU(2)L, with the 1In this notation a gauge vector field is denoted Aµ, while its strength tensor would be written as Aµn. 2A convenient parametrization is the Pauli basis, where Ta = sa are the Pauli matrices. Neutrino Physics in the SM and beyond 5 correct choice of parameters. We can parametrize the neutral part of the scalar field as Φ 0 = 1p 2 (vSM + h + iGZ) , (1.6) we have chosen < Φ >= 0 @ 0 vSM 1 A to be the real vacuum expectation value(VEV) of the Higgs3. From the tadpole equation of the potential we find ∂V ∂vSM GZ,G+,h=0 = 0 =) v2 SM = µ 2 l . (1.7) The stability of the Higgs potential requires that l > 0, so the solution for nonzero VEV requires µ 2 < 0. Once the conditions for a nonzero VEV are met, the stage for SSB is set. For convenience we choose to work in the Unitary gauge, where the Goldstone bosons G+ and GZ are absorbed into the gauge fields. In this gauge we have Φ T = ⇣ 0, 1/ p 2(vSM + h) ⌘ . After EW symmetry breaking by the Higgs vev we obtain mass terms for gauge bosons, we can identify the mass eigenstates W+() µ = F1 µ (+)iF2 µp 2 , (1.8) Aµ = cos qW Hµ + sin qW F3 µ , (1.9) Zµ = sin qW Hµ + cos qW F3 µ , (1.10) with respective masses M2 W = 1 4 g2 Lv2 SM (1.11) M2 A = 0 (1.12) M2 Z = 1 4 (g2 L + g2 Y)v 2 SM (1.13) and neutral mixing angle tan qW = gL/gY. (1.14) 3In the SM with one Higgs doublet any choice of VEV alignment leads to the same Physics. Different choices of the alignment can be cast in the form chosen here through a SU(2)L gauge transforma- tion. Extensions with higher multiplets or additional doublets are constrained to leave the U(1)em subgroup unbroken. For a pedagogical introduction to this issue see [19]. 6 Neutrino Physics in the SM and beyond Eqs. 1.11-1.14 hold at tree level in the SM, together they imply cos qW = MW MZ . (1.15) Non-singlet and doublet scalar multiplets of SU(2)L, as well as loop corrections to the parameters modify this relationship. A way of quantifying the deviation of the true parameters from the SM tree-level prediction is through the r parameter, defined as r = M2 W cos2 qW M2 Z . (1.16) From eqs. 1.10 and the gauge transformation properties of fermions (table 1.1), we can write down the couplings of fermions to the gauge boson mass eigenstates. The photon couples to the electric current JQ µ LQ = eJQ µ Aµ = e[∑ i (eigµei) + ∑ j 2 3 (ujgµuj) ∑ k 1 3 (dkgµdk)]A µ . (1.17) The Z boson couples to the neutral current Jnc µ Lnc = gZ 2 p 2 Jnc µ Zµ = gL 2 cos qW [∑ f (I f 3 Q f sin2 qW)( f gµ fi)]Z µ , (1.18) where the sum runs over all chiral fermions, I f 3 is the value of the third component of weak isospin and Q f is the electric charge operator. Lastly, the charged W boson couples to the charged current Jcc µ Lcc = gLp 2 [∑ i (nLigµeLi) + ∑ j (uLjgµdLj)]W µ . (1.19) 1.2.1. PMNS matrix The gauge interactions of the gauge boson mass eigenstates need to be further rotated to the fermion mass eigenstate basis. For the electric and neutral current interaction, the result of the transformation does not change the form of the interaction, it only changes the index from the fermion interaction basis to the mass basis. For the charged current interaction, this is not the case. Take for example the quark sector interactions. Neutrino Physics in the SM and beyond 7 Quarks acquire mass after symmetry breaking by the Higgs doublet Φ. LY q = Yd ijdLivSMdRj + Yu ij uLivSMuRj + h.c. (1.20) = Md ijp 2 dLidRj + Mu ijp 2 uLiuRj + h.c. (1.21) The matrices Md ij and Mu ij are diagonalized with the unitary transformations diag(md, ms, mb) = (Ud R) †Md ijU d L diag(mu, mc, mt) = (Uu R) †Mu ijU u L , (1.22) where U are unitary matrices. The effect of this transformation on the W interaction vertex with quarks is the appearance of the CKM matrix UCKM [20, 21] Lcc q = gLp 2 [∑ ab (UCKM ab uLagµdLb)]W µ UCKM = (Uu L) †Ud L, (1.23) where ua, db are the quark mass eigenstates. For the lepton sector we can similarly write the analogue PMNS matrix UPMNS [22–24] Lcc l = gLp 2 [∑ ab (UPMNS ab eLagµnLb)]W µ UPMNS = (Ul L) †Un L, (1.24) where Ul L is used to diagonalize charged leptons and Un L diagonalizes neutral leptons. Notice the mismatch between the definitions of UCKM and UPMNS, which follows historical convention. The mixing matrices UCKM and UPMNS can be parametrized in the PDG convention UPDG = 0 BBB@ c12c13 s12c13 s13eid s12c23 c12s23s13eid c12c23 s12s23s13eid s23c13 s12s23 c12c23s13eid c12s23 s12c23s13eid c23c13 1 CCCA , (1.25) where the shorthand cij = cos qij, sij = sin qij has been used. The existence of the PMNS matrix is a consequence of the mass terms of charged and neutral leptons, one of its predictions is the observation of neutrino oscillations. Neutrino oscillation is the phenomenon where a neutrino produced in a charged current interaction possesses an oscillating probability of interacting with each flavor of charged lepton as it propagates through space. The probability of detecting a neutrino of flavor b after a neutrino of flavor a and energy E was produced at a 8 Neutrino Physics in the SM and beyond distance L is Pa!b = |∑ j (UPMNS aj )⇤UPMNS bj ei m 2 j L 2E |2 , (1.26) where mj is the mass of the neutrino j. Neutrino oscillations have been measured [25], showing that neutrinos are indeed massive and that the SM must be enlarged to accommodate neutrino masses. 1.2.2. Majorana and Dirac neutrinos After electroweak symmetry breaking, the U(1)em symmetry associated to the photon and electric charge remains unbroken. The consequences of this is that the photon is exactly massless and that electric charge is a conserved quantity. Fermion masses of the Dirac type, which couple left and right chiral fermions mDyLyR (1.27) do not violate electric charge, which makes them acceptable for charged fermions. On the other hand, Majorana masses violate electric charge by two units 1 2 mMyC P yP , (1.28) and it couples a fermion chiral projection P to itself. Conservation of electric charge prohibits this type of mass for charged fermions, but it is acceptable for neutral leptons. Majorana masses violate any U(1) symmetry. In particular, if neutrinos are Majorana particles, the lepton number U(1)L symmetry would be violated. This allows the occurrence of processes such as neutrinoless double beta decay (A, Z)) ! (A, Z + 2) + 2e or lepton number violating meson decays M 1 ! 2l + M+ 2 . The search for neutrinoless double beta decay is crucial to determine the Dirac or Majorana nature of neutrinos. This is because the black box theorem [26] ensures that the observation of neutrinoless double beta decay implies that neutrinos are Majorana particles. This fact has motivated an array of experiments searching for this decay mode, with no signal detected as of today [27]. Neutrino Physics in the SM and beyond 9 1.2.3. Seesaw mechanisms If neutrinos are Dirac particles, it would require the existence of sterile right handed neutrinos nR. These fields are called sterile because they transform as nR ⇠ (1, 1, 0) under the SM gauge group, they do not couple to gauge bosons. With this matter content we can write the following Yukawa terms for neutrinos LD = Yn ij LiΦ̃nRj + h.c. (1.29) This terms result in the Dirac mass matrix mD = YijvSMp 2 . (1.30) The introduction of right handed sterile neutrinos, without any additional symmetries forces us to consider the Majorana mass terms for them L MN ij 2 nC R inRj . (1.31) The nL nR Dirac mass term combined with the Majorana nR Majorana mass terms induces Majorana masses for all neutrinos. The mass scale of the Majorana mass terms MN is not defined a priori, while the mass scale of Dirac mass terms mD is bound by the electroweak scale. In the limit where the eigenvalues of MN are much larger than the electroweak scale, the diagonalization of the neutrino mass eigenstates can be performed in two steps. First the heavy degrees of freedom are separated from the light ones. This is done perturbatively, at first order the mass matrices of light and heavy states are mlight = mT D(MN)1mD mheavy = MN . (1.32) In this way, light neutrino masses are of order mlight ⇠ v2 SM MN , (1.33) which naturally explains the smallness of neutrino masses with a large enough scale of MN. This framework is known as the type-I seesaw, and it can be seen as a UV- 10 Neutrino Physics in the SM and beyond completion of the dim-5 Weinberg operator for neutrino masses [28] CW Λ LC Φ̃ † Φ̃L . (1.34) Other UV-completions of this operator exist, at the tree and loop level. Only three tree-level completions exist, they are denominated type-I, -II and -III seesaws. If there is a symmetry prohibiting the sterile neutrino Majorana masses, then neutrinos can be Dirac particles. The challenge of explaining small neutrino masses remains, as the limit of mlight < 1 eV means that the Yukawa couplings of eq. 1.30 are of order Yn < 1011. We can extend the Weinberg operator scheme of neutrino masses to the Dirac case, proposing that neutrino masses are originated at the effective operator level CD Λ n LΦ̃nRfn , (1.35) where fn stands for a gauge invariant combination of scalar fields, which obtain a nonzero vev. This Dirac Weinberg operator must be made invariant over the symmetry prohibitting Majorana neutrino masses, and the breaking of the symmetry by fn must leave a remnant symmetry still prohibitting Majorana masses. Chapter 2. Dark Matter Physics 2.1. The need for dark matter The nature of dark matter is one of the greatest unresolved mysteries of contemporary physics. Astrophysical observations provide the framework for our current under- standing of the features of dark matter. We understand dark matter as the observed non-baryonic component of the matter content of the Universe. There is a large body of observations supporting either the existence of this non-baryonic component of matter, or a deficiency of Einsteinian gravity in the description of galactic and larger scale systems. Comprehensive descriptions of the observational evidence of dark matter, with plenty of references to historical and modern research, can be found in textbooks [29–31] or reviews [32–35]. Here we summarize very briefly a few of these observations. 2.1.1. Galactic rotation curves One of the simplest arguments for the existence of dark matter comes from the mea- surement of the rotational velocity of visible matter inside spiral galaxies [36,37]. Most of the visible matter in galaxies exists in the form of Hydrogen, and its velocity can be measured with the Doppler shift of the Hydrogen 21cm line. At the length scale of galaxies, one can expect from GR that the dynamics of matter can be well described by Newtonian gravity. Then, using Gauss’ law for gravity, the expected velocity of bound 11 12 Dark Matter Physics matter orbiting at a distance r from the galactic center is v(r) = r GM(r) r , (2.1) where G is the Gravitational constant, and M(r) is the total galactic mass inside the sphere of radius r. Beyond the point rd where the bright, central galactic disk ends, it is expected that M(r > rd)⇠ M(rd) and therefore the velocity profile becomes approximately v(r > rd)⇠ r1/2. The actual observation indicates that the velocity profile at large distances actually becomes constant, v(r > rd)⇠ v0, which implies that M(r > rd)⇠ r and in turn that the matter density profile beyond the disk follows r(r > rd)⇠ r2. This matter distribution does not appear to emit EM radiation in any frequency, signaling its non-baryonic nature. 2.1.2. Galaxy clusters The largest gravitationally bound structures in the Universe are galaxy clusters, which are self-gravitating collections of galaxies, usually containing hundreds of individual galaxies and x-ray emitting gas. Galaxy clusters provide several independent indica- tions of the existence of dark matter. One of these indications comes from the measurement of the total mass of clusters using the virial theorem [38, 39]. The virial theorem states that for stable systems of discrete particles, bound together by potential forces of the form V ⇠ rn the average total Kinetic energy T and average total Potential energy V satisfy 2T nV = 0 . (2.2) These two parameters can be inferred from measurements of the speeds and positions of galaxies in the cluster. The total mass of the cluster can be extracted using the relation hv2i = 2T M . (2.3) The typical mass to luminosity ratio obtained from this methods is Mc Lc ⇠ 400 Ms Ls (2.4) Dark Matter Physics 13 where the subscript c denotes cluster properties and the subscript s denotes solar properties. This means that the mass to luminosity ratio of galaxy clusters is much larger than that of stars, which are comprised of baryonic matter. Another method to observe the fraction of baryonic matter in galaxy clusters, fb, is through the measurement of the X-ray spectrum and intensity of the hot baryonic gas of a cluster [40]. Assuming thermodynamic and hydrodynamic equilibrium of the gas, the typical baryonic mass fraction of clusters that is observed [40] is fb = Mb Mtot = 0.144± 0.005 , (2.5) where Mb is the baryonic mass of the cluster and Mtot is the total matter content of the cluster. The observation of strong gravitational lensing of background galaxies by clusters is useful for determining the total mass distribution of clusters. The dark matter mass distribution can be extracted by comparing the total mass distribution to the luminous mass distribution. With this method clusters with a baryon fraction as low as fb = 0.02 have been observed [41]. 2.1.3. Cosmic Microwave Background One of the most reliable sources of cosmological data is the Cosmic Microwave Back- ground (CMB). The CMB is EM radiation produced at the moment of photon decou- pling from baryonic matter. This decoupling happens when the temperature of the coupled electron-photon thermal bath drops enough that electrons can form stable bound hydrogen atoms, resulting in a sudden drop of the free electron density. This leads to a drop of the photon collision rate, and consequently an increase in the mean free path of photons. These free photons propagate in the expanding Universe, red- shifting from the visible to the microwave spectrum, where they are detected today. The CMB temperature and its angular dependence has been measured to high preci- sion, for example by the Planck Spacecraft [42] with an angular resolution of ∆q = 5.0 arcmin and a temperature precision of ∆T/T = 2.5⇥ 106. The amplitude of the temperature anisotropies, and the angular power spectrum provide a determination of the cold dark matter density of the Universe. The analysis of Planck data show that 14 Dark Matter Physics the baryonic and dark matter densities are [42] Ωbh2 = 0.02237± 0.00015 (2.6) Ωcdmh2 = 0.1200± 0.012 . (2.7) 2.2. Particle Dark Matter All of the observations of sec. 2.1 point to the existence of a non-luminous component of matter in several structures of the Universe, and that it dominates the matter proportion of its matter-energy budget. All of the observations made of this matter component are based on its gravitational influence on baryonic matter, and therefore do not hint at any characteristic of dark matter other than its mass distribution. The nature of dark matter is one of the most interesting open questions of contemporaneous physics. A vast number of dark matter models have been proposed throughout the last half century, comprising candidates as dissimilar as black holes produced from the collapse of primordial energy fluctuations [43], ultra light bosons that form Bose- Einstein condensates [44], or a exotic, stable uuddss multiquark state [45], among many, many others. All of these models can be roughly categorized by their production mechanisms or experimental search strategies. In this thesis we will focus on the DM categories of Weakly Interacting Massive Particles (WIMPs) and QCD axions, which we will describe in this section. For a description of compelling, viable alternative models the reader can consult [43, 46–49] and the references therein. 2.2.1. WIMP Dark Matter The WIMP dark matter paradigm is based on the dynamics of the thermal history of standard Cosmology, ΛCDM. An electrically neutral, stable particle in thermal equi- librium with the SM bath in the early Universe can be a good dark matter candidate, if the dynamics of its thermal decoupling predict the observed relic density of dark matter of eq. 2.7. The decoupling mechanism of WIMPs is called freeze-out, and it happens in three steps [50, 51]. 1. Dark Matter and the Standard Model are in thermal equilibrium thanks to DM , DM $ SM , SM processes. Their temperatures are equal and the den- sity of all particles in this equilibrium is given by Fermi-Dirac / Bose-Einstein Dark Matter Physics 15 distributions. The equilibrium temperature lowers as time passes, in accordance with Cosmological theory. 2. As the Universe expands and the thermal bath cools, it reaches a threshold where the particles of the bath no longer have enough energy for the process SM , SM ! DM , DM to occur efficiently. DM falls out of thermal equilibrium, with only the DM , DM ! SM , SM process occurring. This depletes DM density from its thermal comoving density. 3. The Universe continues to expand, lowering the DM density until the rate of the annihilation process DM , DM ! SM , SM becomes negligible. This happens because DM has annihilated into SM particles and because the expansion of the Universe has diluted DM. The remaining density of DM is called the relic density and, for a model to be acceptable it must be equal or lower than the observed value [42]. This description of the process conforms to the results of the formal description of thermal decoupling, which is based on solving the Boltzmann equation of the DM species in the expanding Universe. This theory is stated as a differential equation which can be solved through analytical approximations1 or numerical means. The crucial feature of this framework is the existence of interactions between the dark and visible sectors, which keeps them in equilibrium, and eventually determines the DM relic density. A WIMP dark matter model that can predict this relic density will show explicitly how such an interaction looks like. This can be through gauge bosons or scalar particles mediating the interactions between the sectors, for example. Proposing an interaction between the dark and visible sector such that the freeze-out process successfully replicates the observed relic density inevitably leads to predictions for dark matter direct and indirect detection processes. Direct detection experiments aim to detect the elastic or inelastic scattering of the local cosmological density of dark matter with a target material in a detector. For example, the XENON1T experiment [53] searched for dark matter-nucleon elastic scatterings DM N ! DM N, by placing a volume of target material in an underground labora- tory to reduce cosmic ray flux. The target volume, in this case Xenon, is coupled to detectors which can observe the signatures of the induced recoil of nuclei from DM collisions. Generically, WIMP models that couple the dark sector to the SM will predict a cross section for the DM N ! DM N process, which is one of the observables that 1The derivation of the Boltzmann equation of freeze-out and its analytical approximation solutions can be found in most Cosmology textbooks, for example [29, 52]. 16 Dark Matter Physics direct detection experiments have placed constraints on. Another prediction of WIMP models is the indirect detection signal. Indirect detec- tion of dark matter consists on the search for signals of dark matter annihilation in astronomical objects. From the observation of the gravitational influence of dark matter in galaxies, we can infer that a dark matter distribution permeates all galaxies. WIMP models of dark matter need the existence of DM DM ! SM SM annihilation channels, so that thermal freeze-out can proceed. Inevitably, the annihilations can occur in galaxies today, which result in a production of SM particles in regions where there is no visible matter. These annihilations produce a detectable flux of primary or secondary photons [54], neutrinos [55] or positrons [56] on Earth. WIMP models predict the rate of DM annihilations in galaxies, and consequently the expected cosmic ray flux. These three observables, relic density, direct detection and indirect detection cross sections, are the main source of constraints on WIMP models. 2.2.2. Axion Dark Matter The QCD axion is a light (sub-KeV) dark matter category, a pseudo-Nambu Goldstone boson originally predicted by a method proposed to solve the strong CP problem [57, 58]. Its phenomenology is dominated by its anomalous couplings to photons and gluons. This category can be extended to include ALPs, axion-like particles, which do not solve the strong CP problem, but can be searched for in the same type of experiments. The QCD axion can be seen as a special type of ALP, with a strong correlation among its mass and its couplings. The QCD Lagrangian contains the CP violating q LCP = q as 8p Ga µnG̃µn a , (2.8) where as is the strong structure constant, Ga are the gluon field strength tensors and G̃a their dual. The value of the q parameter contains contribution from the nontrivial vacuum structure of QCD, as well as from the quark mass matrices. The q term has nonperturbative phenomenological consequences. Perhaps the most important of them, is that it induces a neutron electric dipole moment (nEDM), of magnitude dn = (2.4± 1.0)q ⇥ 103e f m , (2.9) Dark Matter Physics 17 while the experimental limit on it is [59] dexp n < 1.8⇥ 1013e f m , (2.10) which implies q < 0.8⇥ 1010 . (2.11) The strong CP problem lies in explaining why is q so small, when it is determined by two unrelated sectors of the theory. The Peccei-Quinn solution to the strong CP problem is based on a global symmetry, U(1)PQ, which is broken spontaneously at a high energy scale fA. The q term is effectively replaced by a dynamical field. QCD dynamics determine the potential of the field, in such a way that its vev is the CP-conserving value of zero [57]. The spontaneous breaking of a global symmetry induces the existence of a massless pNGB boson, the axion [58]. The coupling of the QCD axion to gluons induces axion mixing with pions and other mesons, inducing a nonzero mass for the physical pNGB. The mass of the QCD axion is mA = 5.70µeV 1012GeV fA , (2.12) and all axion-gauge boson couplings can be shown to scale as g⇠ 1/ fA. On the other hand, ALPs do not obey these relations, they do not solve the strong CP problem, but have a larger viable parameter space. Axions are a good dark matter candidate, if they are equipped with a viable produc- tion mechanism that can explain the observed relic density of dark matter. These production mechanisms are non-thermal, as the lightness of axions precludes them from being produced by a mechanism such as freeze-out. In general, axion production mechanisms rely on the details of the cosmological history of the axion field and the breaking of the PQ symmetry, and not on the couplings of the axion to SM particles. This means that unlike the WIMP scenario, the phenomenology of the axion as a dark matter candidate is not intricately linked to laboratory limits on its properties. ALPs and the QCD axion are searched for through astrophysical observations and laboratory experiments. These experiments probe the (mA, g) parameter spaces, where 18 Dark Matter Physics g can be the axion-photon coupling, or effective couplings to nucleons, mesons or fermions2. 2For a thorough review of the principle behind these experiments and current limits from them, see for example [60, 61]. Chapter 3. Flavor Symmetric models of neutrino masses and dark matter 3.1. Flavor symmetries The SM contains three copies of quarks and three copies of leptons with the same gauge transformation properties. These are called the three families of the SM. The Yukawa couplings of the fermions to the Higgs break the degeneracy among the families. After SSB of the SM gauge group by the Higgs VEV, the Yukawas lead to the observed masses and mixings of the fermions1. The Yukawa matrices of SM fermions have entries of incongruent magnitudes, as they must lead to masses of different scales and small quark mixings and large lepton mixings. A possible origin of these mass scales and mixings can be flavor symmetries. If we write the three generations of a gauge group representation as ya a a = 1, 2, 3 , (3.1) with a = Q, L, uR, dR, eR, NR, then a corresponds to the flavor index. A flavor symmetry consists of a group G f of transformations Tf that acts on the flavor index of fermions and scalars and leaves the Lagrangian density of the QFT invariant. 1Only in the quark sector this is entirely true. In the lepton sector neutral lepton masses must invoke additional fields and/or symmetries to characterize the mass generation mechanism. 19 20 Flavor Symmetric models of neutrino masses and dark matter To each ya we assign a matrix representation of Tf , then we have ya a ! y0a a = (Ta f )abya b , (3.2) L(ya) ! L(y0a a ) = L(ya), (3.3) where in the last equation we have transformed all fields under their assigned rep- resentation. A widely explored possibility is that the group G f of flavor symmetries is a finite discrete group. Examples of such groups are S3, A4 or Q6. A practical introduction to non-abelian discrete groups can be found in [62], with a large number of flavor models cited therein. For our purposes it suffices to know that we will work with flavor symmetries where the flavor group is a simple discrete group or a direct product of such discrete groups, the fermions of the SM will be assigned irreducible representations of such groups, or a direct sum of such representations, and flavon fields fb can be used to break the flavor symmetry. A flavon is understood to be a scalar field, gauge singlet of the SM. The motivation for studying non-abelian discrete group models is phenomenological. They can lead to the observed massses and mixings of SM leptons, but their origin can lie in well-motivated UV physics such as non-abelian continuous symmetries [63–71] or the compactification of extra dimensions in superstring theories [72–74]. We will take the approach of considering the non-abelian discrete symmetries to be valid at an intermediate scale, between the electroweak scale and a higher scale where a more complete theory may explain the origin of the flavor symmetry (or not). 3.2. D4 as a symmetry group 3.2.1. The discrete non-abelian D4 symmetry group In this section we will consider D4 as a symmetry group of the lepton sector2. D4 is the symmetry group of the square, all its elements can be generated by two operations: the rotation by p/2 S and the reflection T. These two generators satisfy S4 = e T2 = e TST = S1, (3.4) 2This section is based on [3]. Flavor Symmetric models of neutrino masses and dark matter 21 where e is the identity element. From these two generators, the eight unique elements of D4 can be obtained. D4 has five irreducible matrix representations, four are one- dimensional and we will denote them 1++, 1+, 1+ and 1. The fifth irrep is two-dimensional and we will denote it 2. The notation for the one-dimensional irreps is chosen like this, because the tensor product of singlets can be written 1ij ⌦ 1kl = 1mn, (3.5) with m = ik and n = jl. A basis for the two dimensional representation of D4 can be defined with the following complex representation for the generators S,T S = 0 @exp ip/4 0 0 expip/4 1 A , T = 0 @0 1 1 0 1 A . (3.6) In this basis we can write the tensor product of two doublets as 0 @a b 1 A 2 ⌦ 0 @c d 1 A 2 = (ad + bc)1++ (ad bc)1 (ab + dc)1+ (ab dc)1+ . (3.7) Finally, the tensor product of singlets with a doublet is (c)1++ ⌦ 0 @a b 1 A 2 = 0 @ca cb 1 A 2 , (c)1+ ⌦ 0 @a b 1 A 2 = 0 @cb ca 1 A 2 , (3.8) (c)1+ ⌦ 0 @a b 1 A 2 = 0 @ cb ca 1 A 2 , (c)1 ⌦ 0 @a b 1 A 2 = 0 @ ca cb 1 A 2 . With these relations, we can assign D4 irreps to fields, and write down gauge and flavor invariant Lagrangians. From eq. 3.7 we can immediately see the power of nonabelian discrete symmetry groups: by grouping together generations of a field in a flavor multiplet, only particular linear combinations of the fields will have definite irreps under the flavor symmetry, so flavor invariants are a particular combination of the fields of a theory. This can lead to a reduction of the parameters of the theory, resulting in correlations among observables, such as masses and mixings. 22 Flavor Symmetric models of neutrino masses and dark matter 3.2.2. Left-Right gauge symmetry Parity symmetry is explicitly violated by the SM at the level of gauge irrep assignment. Left-handed fermion fields transform as doublets under SU(2)L while right-handed fields transform as singlets. Because of this, the left projection of a fermion mass eigenstate couples differently than the right projection. This is measured in the parity asymmetries of beta decay [15] or electron scattering [75, 76]. Theoretically, it seems rather arbitrary that fundamental interactions are intrinsically parity violating. To reconcile the theoretical expectation of parity conservation and the experimental fact of parity violation, left-right gauge symmetric theories where parity is spontaneously broken have been explored [77–80]. The simplest setup for left-right gauge symmetry begins with the gauge symmetry group GLR = SU(3)C ⇥ SU(2)L ⇥ SU(2)R ⇥U(1)BL. (3.9) The fermion content of the model is that of the SM with the addition of right handed neutrinos, so that the right handed fermions can be arranged in SU(2)R doublets. Gauge coupling Parity is then restored, when the gauge coupling of SU(2)L and SU(2)R are set equal to some high energy scale. This, in principle, is suggestive of a higher gauge symmetry as the origin of parity symmetry. Indeed, GLR is a viable intermediate step towards SO(10) unification. The minimal scalar content to obtain the SM gauge group from the LR gauge group is a bi-doublet Φ⇠ (1, 2, 2,1) and left and right triplets cL(R) ⇠ (1, 3(1), 1(3), 2) . The extended gauge boson content of GLR consists of a neutral ZR bosons, as well as charged WR bosons. These bosons will mix with the SU(2)L bosons, leading to physical light W and Z bosons coupling primarily to left handed fermions and heavy W 0 and Z0 bosons coupling primarily to right handed fermions. As the model also needs right handed neutrinos, it can lead naturally to type-I seesaw masses for light active Majorana neutrinos. Experimentally, the limits on the mass scale of the heavy gauge bosons coupling to right handed fermions are of the order MR > 600 1000 GeV, the lower range obtained from direct searches in proton-proton collisions and the upper range from electroweak fit results [32]. Flavor Symmetric models of neutrino masses and dark matter 23 lLD lLS lRD lRS ∆L ∆R Φ h c SU(2)L 2 1 3 1 2 1 1 SU(2)R 1 2 1 3 2 1 1 U(1)BL -1 -1 2 2 0 0 0 D4 2 1 2 1 1 1 1 2 1 Z2 1 1 1 1 -1 -1 -1 Table 3.1.: Assigned gauge and flavor irreps of D4 lepton flavor model. As explained in the text, we omit the quark sector. 3.2.3. A GLR ⇥ D4 ⇥ Z2 model of lepton masses and mixing Consider the particle content of table 3.1. In this work we omitted the quark sector, as our focus is on neutrino masses. In Left-Right gauge symmetry models, the right handed quarks are analogously arranged in right gauge symmetry doublets. It is worth to note that if we want to preserve the possibility of total electroweak-color unification in a SO(10) model, the quark sector flavor representation must also be 2 1, so that all fermions can fit in a SO(10) 16 multiplet. The phenomenology of such model is a possible extension of this work. The scalar fields h and c are introduced to trigger SSB of D4 ⇥ Z2,while the scalars Φ, ∆L and ∆R assume the same role as in the unflavored Left-Right model. We write the scalars Φ, ∆L and ∆R in SU(2)L ⇥ SU(2)R space as Φ = 0 @f0 1 f+ 1 f 2 f0 2 1 A , ∆L(R) = 1p 2 0 @ d L(R) + p 2d L(R) ++p 2d L(R) 0 d L(R) + . 1 A (3.10) The scalar kinetic terms are LLR sk = Tr h (DµΦ)†(Dµ Φ) + (Dµ∆L) †(Dµ ∆L) + (Dµ∆R) †(Dµ ∆R) i (3.11) DµΦ = ∂µΦ + igLtb(W b L)µΦ + igRtb(W b R)µΦ Dµ∆L = ∂µ∆L + igL[tb(W b L)µ, ∆L] i2gBL∆LBµ Dµ∆R = ∂µ∆R + igR[tb(W b R)µ, ∆R] i2gBL∆RBµ . 24 Flavor Symmetric models of neutrino masses and dark matter The gauge symmetry breaking pattern of SU(2)L ⇥ SU(2)R ⇥U(1)BL models is SU(3)C ⇥ SU(2)L ⇥ SU(2)R ⇥U(1)BL (3.12) # ∆R SU(3)C ⇥ SU( 2 )L ⇥U(1)Y # Φ, ∆L SU(3)C ⇥ U(1)em where the fields next to the arrows denote which field develops a expectation value that partially breaks the symmetry. The electric charge operator, which should be left unbroken by the symmetry breaking chain, is set by the fermion sector to be Qem = T3 L + T3 R + QBL/2 . (3.13) This leads to the following allowed VEV alignment in the scalar sector hΦi = 0 @v1 0 0 v2 1 A , h∆L(R)i = 0 @ 0 0 vL(R) 0. 1 A (3.14) This symmetry breaking is assumed to have the hierarchy v2 L ⌧ v2 1 + v2 2 ⌧ v2 R , (3.15) where the first inequality follows from observational constraints on the r parameter, and the second one follows from limits on the right-handed gauge boson mass. As in the SM we define the charged gauge bosons W +[] L(R) = (W1 L(R) [+]iW2 L(R))/ p 2, (3.16) but here, these are not mass eigenstates yet. Taking into account the vev hierarchy of eq. 3.15, we cite the leading order contributions to gauge boson masses and mixing. The charged gauge boson eigenstates can be obtained as follows 0 @W+ 1 W+ 2 1 A = 0 @ cos x sin xeil sin xeil cos x 1 A 0 @W+ L W+ R 1 A (3.17) Flavor Symmetric models of neutrino masses and dark matter 25 with W+ 1 , W+ 2 having masses M1 and M2 respectively; the relevant parameters are given by v2 1 + v2 2 = v2 SM x = 2v1v2 v2 1+v2 2 ⇣ M1 M2 ⌘2 (3.18) M2 1 = 1 4 g2(v2 1 + v2 2) M2 2 = 1 4 g2(2v2 R + v2 1 + v2 2) eil = v1v⇤2 |v1v2| . The fermion couplings of the charged gauge bosons are then Lc.c. = gLp 2 ⇣ uLgµdL + nLgµeL ⌘ ⇣ W +µ 1 cos x + W +µ 2 sin x ⌘ (3.19) + gLp 2 ⇣ uRgµdR + nRgµeR ⌘ ⇣ W +µ 1 sin x W +µ 2 cos x ⌘ + H.c. with the fermion fields in the interaction basis. In the subsequent rotation to the fermion mass basis, the CKM and PMNS matrices of left handed fermions will appear. In the SM, right handed rotations are of no phenomenological consequence and can be ignored. In models with right handed charged current interactions, we can no longer ignore them, as they lead to observable interactions with charged bosons. The Left-Right symmetry of the Yukawa sector of quarks (leptons) leads to relationships between the CKM (PMNS) of the left handed quarks (leptons) and its equivalent in the right handed sector. One possibility is to take all scalar vevs as real. This scenario is known as Manifest Left-Right Invariance, and results in the equality of left and right handed lepton mixing. Another possibility, called Pseudo-manifest Left-Right Invariance is to take the scalar vevs as complex, and the Yukawa couplings as real. In this case, the mixing matrix in the right sector is equal to the product of mixing of the left sector with additional CP violating phases. Taking x ⌧ 1, W1 couples predominantly to left handed fermions and W2 to right handed fermions. The existence of both bosons leads to left and right handed charged current interactions, with effective coupling strengths GL F = g2 L( cos2 x 2M2 1 + sin2 x 2M2 2 ) , GR F = g2 L( sin2 x 2M2 1 + cos2 x 2M2 2 ) (3.20) respectively. At low energies, the left handed current dominates the right handed current due to the mass hierarchy M2 1 ⌧ M2 2 and small mixing angle x, both obtained from v2 1 + v2 2 ⌧ v2 R. Eq. 3.19, together with assumptions of Pseudo-manifest Left-Right 26 Flavor Symmetric models of neutrino masses and dark matter Invariance can be used to set the limit on the W2 mass MW2 > 2.5 GeV , (3.21) obtained from experimental limits on the contribution of W2 to K0 K0 mixing, neutron EDM, neutral B meson mixing, and CP-Violation in neutral B meson decays [81]. In the neutral gauge boson sector, the mass eigenstates can be obtained in two steps. First the massless photon is identified 0 BBB@ A ZL ZR 1 CCCA = 0 BBB@ sin qW sin qW p cos 2qW cos qW sin qW tan qW tan qW p cos 2qW 0 p cos 2qW cos qW tan qW 1 CCCA 0 BBB@ W3 L W3 R B 1 CCCA . (3.22) Then, ZL and ZR are rotated into the mass eigenstates Z1 and Z2 0 @Z1 Z2 1 A = 0 @ cos z sin z sin z cos z 1 A 0 @ZL ZR 1 A , (3.23) where the mixing angle and eigenstate masses are tan 2z = 2 p cos 2qW M2 Z1 M2 Z2 , (3.24) M2 Z1 = g2 L 2 cos2 qW (v2 1 + v2 2 + 4v2 L) , M2 Z2 = g2 L 2 cos2 qW cos 2qW (4v2 R cos4 qW + (v2 1 + v2 2) cos2 2qW + 4v2 L sin4 qW) . Taking z ⌧ 1,the couplings of the neutral bosons are at leading order Ln.c. = g cos qW [ Z1µ(J µ L zp cos 2qW + cos2 qW J µ R) (3.25) Z2µp cos 2qW (sin2 qW J µ L + cos2 qW J µ R)] + J µ Q Aµ . The neutral currents are J µ L/R = ∑ f f gµ(I3L/R Q f sin2 qW) f , J µ Q = ∑ f e f gµQ f f . (3.26) Flavor Symmetric models of neutrino masses and dark matter 27 In both charged and neutral gauge boson sectors, we can see that the limit x, z ! 0 makes the gauge bosons W1, Z1 behave as in the SM. That is to say, W1 couples only to left handed currents and Z1 couples to the SM neutral current. This limit can be achieved as vR/vSM ! ∞. The lepton kinetic terms of the flavor symmetric model are LLR lk = ∑a=D,S lLa gµ ⇣ ∂µ igLtb(W b L)µ + igBLBµ ⌘ lLa (3.27) lRa gµ ⇣ ∂µ igRtb(W b R)µ + igBLBµ ⌘ lLa . (3.28) The Yukawa couplings of the lepton sector of the model, up to dimension-5 operators, are LLR Y = lLD ⇣ y1 c ΛF Φ + ỹ1 c ΛF Φ̃ ⌘ lRD + lLD ⇣ y2 h ΛF Φ + ỹ2 h ΛF Φ̃ ⌘ lRS (3.29) +lLS ⇣ y3 h ΛF Φ + ỹ3 h ΛF Φ̃ ⌘ lRD + lLS ⇣ y4 c ΛF Φ + ỹ4 c ΛF Φ̃ ⌘ lRS + YL1 2 lT LD C(is2)∆LlLD + YL2 2 lT LS C(is2)∆LlLS + YL3 2 lT RD C(is2)∆LlRD + YL4 2 lT RS C(is2)∆LlRS + h.c. , where we have to take into account the G f flavor symmetry in order to form the flavor invariant combination of the fermions and scalars. More explicitly, from eqs. ?? we can see that flavor invariants are formed from the tensor product of two singlets or two doublets. Therefore, the lepton Yukawa couplings with Φ that are invariant under G f must be formed from the tensor product lL(R)D ⇥ lR(L)D , lL(R)D ⇥ h or lL(R)S ⇥ lR(L)S . The Yukawa couplings of the triplets ∆L(R) can only be formed from lL(R)D ⇥ lL(R)D and lL(R)S ⇥ lL(R)S . The Z2 symmetry is responsible for the exclusion of dimension-4 Yukawa operators with Φ, and allows us to use c and h as flavons, without compli- cating the gauge symmetry breaking dynamics at the price of needing dimension-5 operators to describe the mass generation of lepton masses. We can write in D4 space the D4 doublets as yD = 0 @y1 y2 1 A , y = lL, lR, h. (3.30) Because the scalar fields with non-trivial gauge representations have trivial flavor representations, the flavor symmetry is broken only due to c and h. We adopt the hierarchy where flavor symmetry is broken at a higher scale than the gauge symmetry, 28 Flavor Symmetric models of neutrino masses and dark matter that is to say hci = vc, hhii = vhi > v1, v2, vL, vR . (3.31) Additionally, we choose the flavor VEV alignment hhi = (vh1 , 0). After SSB of the gauge and flavor groups, the charged lepton mass matrix can be read from eq. 3.30 as MCh.l = 1p 2 (Y0 Lv2 + Ỹ0 Lv1) (3.32) Y0 L = 1 ΛF 0 BBB@ 0 y1vc 0 y1vc 0 y2vh1 0 y3vh1 y4vc 1 CCCA , Ỹ0 L = 1 ΛF 0 BBB@ 0 ỹ1vc 0 ỹ1vc 0 ỹ2vh1 0 ỹ3vh1 ỹ4vc 1 CCCA . Left-Right symmetry requires the following relations between Yukawa couplings y2 = y⇤3 , ỹ2 = ỹ⇤3 , y1 = y⇤1 . (3.33) With this in mind, the charged lepton mass matrix can be parametrized through three real parameters, a, b and c and a complex phase fCh.l as MCh.l = 0 BBB@ 0 a 0 a 0 beifCh.l 0 beifCh.l c 1 CCCA . (3.34) The complex phase fCh.l of this matrix can be removed through the transformation MCh.l ! PMCh.lP 0 , Mn ! PT MnP (3.35) thereby transferring the phases to the neutrino mass matrix with two rephasing diagonal matrices P and P0. As the charged lepton mass matrix now contains three real parameters, we can completely determine the magnitude of its entries through the use of its eigenvalues and matrix invariants. The eigenvalues of the matrix are of course the masses of the charged leptons as measured in experiments, and the matrix Flavor Symmetric models of neutrino masses and dark matter 29 invariants we use are Tr(MCh.l) = c = me + mµ + mt , (3.36) Tr(M2 Ch.l) = 2a2 + 2b2 + c2 = m2 e + m2 µ + m2 t , Det(MCh.l) = a2c = memµmt. Solving for the parameters a, b and c we find 3 a = ± p memµmtp me mµ + mt , (3.37) b = ± pmµ + mt q m2 e + memµ memt + mµmt p me mµ + mt , c = me mµ + mt , where a and b have a sign ambiguity. We classify the four possible solutions (sgn(a), sgn(b)) as A = (+,) , B = (,) , C = (+,+) and D = (,+) . (3.38) The charged lepton mass matrix is diagonalized through an orthogonal rotation diag(me, mµ, mt) = OT MCh.lO . (3.39) Numerically, the orthogonal transformation matrix O is O = 0 BBB@ 0.998 sgn(a)0.070 sgn(ab)0.001 sgn(a)0.068 0.969 sgn(b)0.236 sgn(ab)0.017 sgn(b)0.235 0.972 1 CCCA , (3.40) where the matrix O is shown to be explicitly dependent on the choice of sign for a and b. 3The determinant of the matrix is negative, we choose the muon mass parameter to be negative, with no phenomenological consequence. This is equivalent to absorbing a 1 phase in the muon wavefunction. 30 Flavor Symmetric models of neutrino masses and dark matter In the neutrino side, the Dirac neutrino mass matrix is given by mD = 1p 2 (Y0 Lv1 + Ỹ0 Lv2) . (3.41) The Majorana neutrino mass matrices are obtained by the scalar triplets ∆L,R Yukawa couplings, and are mL = p 2YLvL , (3.42) mR = p 2YRvR . The full neutrino mass matrix in the (nL, nC R) basis is Mn = 0 @mL mD mT D mR 1 A . (3.43) The vev hierarchy of Left-Right gauge models v2 L ⌧ v2 1 + v2 2 ⌧ v2 R guarantees the seesaw limit mL ⌧ mD ⌧ mR. The neutrino mass eigenstates are then obtained through the type-I seesaw mechanism. After decoupling the light from the heavy neutrino mass eigenstates, the light and heavy neutrino mass matrices are mlight = mL mT Dm1 R mD (3.44) mheavy = mR . The light neutrino mass matrix results in the complex two-zero texture matrix mlight = 0 BBB@ 0 an 0 an dn bn 0 bn cn 1 CCCA (3.45) The diagonalization of this matrix by a Hermitian matrix Un leads to the determination of neutrino masses and the PMNS matrix through the relations diag(mn1 , mn2 , mn3 ) = U† n mlightUn , (3.46) UPMNS = OTUnP . (3.47) Flavor Symmetric models of neutrino masses and dark matter 31 The 0 entries of the neutrino mass matrix of eq. 3.45 are a powerful prediction of the D4 flavor symmetry. Although these zeroes do not appear in the diagonal charged lepton basis as it has been extensively studied in the literature [82–84], we still show that in this setup the texture can be phenomenologically viable. This is to be expected, as the texture in the diagonal charged lepton basis, known as A2 in the Frampton- Glashow-Marfatia classification, is phenomenologically viable. In the basis of our D4 symmetric model, the charged lepton diagonal basis is reached through the rotation by O, which is numerically close to the identity. In a sense, this means that in the D4 symmetric model the texture A2 of the neutrino mass matrix appears in a basis "close" to the diagonal charged lepton basis. To determine the allowed parameters of the model, we take as the starting point eqs. 3.45 and 3.46. We determine viable non-physical angles in Un, along with possible neutrino masses that are consistent with the neutrino mass matrix texture. The possible neutrino masses are obtained from taking the lightest neutrino mass as a free parameter and deducing the other two masses with the observed squared neutrino mass differences m2 = q m2 1 + ∆m2 12 , (3.48) m3 = q m2 1 + ∆m2 13 . (3.49) We assume Normal Hierarchy because the A2 texture is only compatible with it, and we do not expect that the O transformation is large enough to make the Inverted Hierarchy viable. After obtaining Un, we extract the physical lepton mixing angles using eq. 3.47. The least-well established parameters of lepton mixing are the q23 mixing angle and the dCP CP violating phase. We take these two parameters as predictions of our model, and demand that the model is consistent with all other oscillation parameters as described in the global fit of [85]. Furthermore, we analyze which predictions are excluded already by global fit data, and the reach of the DUNE experiment, which is expected to further constrain q23 and dCP. To obtain an estimation of DUNE’s capacity to determine oscillation parameters, we used the GLoBES package [86–91], assuming the then-current best fit data points for oscillation parameters, and a running time of 3.5 years for neutrino and antineutrino modes. The results of the numerical fit are shown in Fig. 3.1, for the 4 solutions as defined in eq 3.38. The top figure shows the predicted areas for the 4 solutions and the experimental constraints on the parameter space (sin2 q23, dCP) taking all other oscillation parameters to their best fit points. The bottom figure corresponds to the 4 solutions and the projection for DUNE’s results, assuming the oscillation global fit best fit point as the 32 Flavor Symmetric models of neutrino masses and dark matter Figure 3.1.: Model Predictions for (sin2 q23, dCP) with experimental data. First figure shows model predictions with the model nomenclature of eq. 3.38 together with oscilla- tion data global fit results. Second figure shows model predictions with DUNE sensitive projections assuming the oscillation best fit point as the true value of oscillation parameters. Flavor Symmetric models of neutrino masses and dark matter 33 true value for oscillation parameters. We can readily see that current data excludes the solution A. DUNE 3.5 + 3.5 year results will exclude the D solution, assuming the oscillation bfp does not migrate to higher values of dCP and sin2 q23 with newer data. This shows the potential of DUNE to shed light on the flavor problem in the lepton sector. Another interesting prediction of the flavor symmetry of the model is the value of the neutrinoless double beta decay effective electron neutrino mass hmeei hmeei = 3 ∑ i=1 mi(UPMNS) 2 ei . (3.50) This parameter controls the rate of neutrinoless double beta decay. In nucleus where this decay mode is available, the half-life of the process can be written as (T0n 1/2) 1 = G0n|M0n(A, Z)|2|hmeei|2 (3.51) where G0n is a nucleus-dependent phase space factor, M0n(A, Z) is a nucleus depen- dent nuclear matrix element. In the D4 basis we chose to work in, the entry ee of the neutrino mass matrix is exactly 0. The rotation to the diagonal charged lepton is close to the identity, so we expect a relatively small value for this parameter. In figure 3.2 we show the model predictions for (|hmeei|, mlight), using the same color code of fig. 3.1. Numerically, we predict 103 < |hmeei| eV < 2⇥ 103 , 103 < |hmlighti| eV < 102 . (3.52) The current limits on both |hmeei| and hmlighti are too weak to further constrain our model. The limit on |hmeei| come from the nonobservation of neutrinoless double beta decay, which puts limits on the half-life of the process. Using eq. 3.51, |hmeei| can be extracted from this half-life limit. An important source of uncertainty on the determi- nation of the |hmeei| limit comes from the theoretical uncertainty of M0n(A, Z). The many methods for calculating M0n(A, Z) yield results spanning an order of magnitude, which propagates as a theoretical uncertainty on the limit of hmeei 34 Flavor Symmetric models of neutrino masses and dark matter Figure 3.2.: Model Predictions for |hmeei|. We show the model predictions for the 3 cases 3.1 that are consistent with current data, using the color code of figure. Also in the figure, constraints on |hmeei| from the KamLAND-ZEN experiments from 2016 and constraints on the lightest neutrino mass from Cosmological data analysis. We see that the model predicts values for (|hmeei|, mlight) too small for current and near future experiments to detect. 3.3. Gauged U(1)X symmetry as a flavor symmetry This section is based on the research presented in [5] We can consider a local U(1)X symmetry that acts on a field j carrying X charge QX j j(xn) ! j0 = eigXQX j q(xn)j(xn) . (3.53) The local nature of the symmetry calls for the introduction of a vector field Z0 trans- forming as Z0 µ(xn) ! Z0 µ(xn) 1 gX ∂µq(xn) . (3.54) Flavor Symmetric models of neutrino masses and dark matter 35 To conserve gauge invariance, the kinetic terms of fermion and scalar fields are recast through the replacement ∂µ ! Dµ = ∂µ + igXQXZ0 µ . (3.55) This leads to a theory with a new gauge interaction mediated by Z0. Scalar and fermion X charges are in principle arbitrary, with the constraint that the full fermion content of the theory leads to vanishing anomalies. Three anomalies are dangerous to field theories. The chiral gauge anomalies arising from triangle diagrams spoil the renormalizability of the theory [92, 93], the global non-perturbative SU(2) anomaly must vanish in order to define gauge invariant path integral with fermions [94], and the mixed gauge-gravity anomaly must vanish in order to ensure covariance of the theory [95]. In the SM these anomalies are cancelled for the gauge symmetries and the fermion content it contains [96]. Any extension of the SM enlarging the fermion content and/or gauge group of the model must address the cancellation of these undesirable anomalies in order to define a theoretically consistent framework. In the case we will consider here, a U(1)X extension of the gauge group of the SM with sterile right handed neutrinos, the cancellation of anomalies imposes the following set of 36 Flavor Symmetric models of neutrino masses and dark matter constraints on the X charges of SM fermions: [SU(3)C] 2U(1)X : 3 ∑ i=1 [F(Qi) F(ui) F(di)] = 0 (3.56) [SU(2)L] 2U(1)X : 3 ∑ i=1 [3F(Qi) + F(Li)] = 0 (3.57) [U(1)Y] 2U(1)X : 3 ∑ i=1 [F(Qi) + 3F(Li) 8F(ui) 2F(di) 6F(ei)] = 0 (3.58) U(1)Y[U(1)X] 2 : 3 ∑ i=1 [F(Qi) 2 F(Li) 2 2F(ui) 2 + F(di) 2 + F(ei) 2] = 0 (3.59) [U(1)X] 3 : 3 ∑ i=1 [6F(Qi) 3 + 2F(Li) 3 3F(ui) 3 3F(di) 3F(ei) F(Ni) 3] = 0 (3.60) gauge-gravity : 3 ∑ i=1 [F(Qi) + F(Li) F(ui) F(di) F(ei) F(Ni)] = 0. (3.61) In these eqs. we have adopted the notation F(ei) to denote the X charge of the field ei for example, as to avoid confusion between the notation of the charge and the left handed quarks Qi. We do not include the global SU(2)L anomaly cancelation condi- tions, it is already met in the SM and we will not entertain introducing SU(2)L fermion multiplets. We can proceed without worrying about pure SM anomalies, because we will only consider adding fermions with trivial SM gauge transformation properties : fermions transforming as SU(3)C and SU(2)L singlets with no hypercharge. One of the most popular gauged U(1)X is the X = B L, as it is anomaly-free in the SM, when 3 right handed neutrinos are added. It is also phenomenologically interesting because it can be embedded in Left-Right symmetric models that seek to address the parity vio- lation of the SM. It can also be obtained after the breaking of larger GUT groups. In the X = B L gauge symmetry, quarks carry 1/3 charge (F(Qi) = F(ui) = F(di) = 1/3 ) and leptons carry 1 charge (F(Li) = F(ei) = F(Ni) = 1). Generation-dependent solutions also exist. A simpler set of charges, motivated by the approximate mu- tau symmetry of lepton mixing, is the X = µ t, where µ leptons carry +1 charge Flavor Symmetric models of neutrino masses and dark matter 37 Qi ui di Li ei Ni B L 1/3 1/3 1/3 -1 -1 -1 µ t 0 0 0 {0,1,-1} {0,1,-1} {0,1,-1} B 2La Lb 1/3 1/3 1/3 {-2,-1,0} {-2,-1,0} {-2,-1,0} B 3La 1/3 1/3 1/3 {-3,0,0} {-3,0,0} {-3,0,0} Table 3.2.: Simple anomaly-free U(1)X symmetries. Anomalies are cancelled when the gauge symmetry is GSM ⇥U(1)X and no other fermions are considered. If more fermions are added to the theory, the anomaly cancellation conditions of eqs ?? must be met. and t leptons carry 1 charge, while all other fermions are uncharged. There are also flavor-dependent X = B + ∑i F(Li)Li anomaly-free charge assignments, with F(Li) = {2,1, 0}, {3, 0, 0}, or any permutation of these charges. In table 3.2 we write the anomaly-free models we have discussed. When the gauge symmetry has flavor-dependent charges, the scalar content of the theory can lead to flavor mixing predictions. An interesting consequence of BSM particles coupling to neutrino and quark fields is the enhancement of the Coherent Elastic neutrino-Nucleus Scattering (CEnNS ) [97]. This process (nN ! nN) is predicted by the Standard Model, mediated by the Z boson. The SM differential cross section is dsSM dER = G2 F 2p MNQ2 W 2 MNER E2 n ! , (3.62) where MN is the target nucleus mass , QW is the nuclear weak charge, ER is the nuclear recoil energy and En is the incoming neutrino energy. In the SM, the nuclear weak charge is given by QW = Zg p V Fp(Q 2) + Ngn V Fn(Q 2) , (3.63) here Z and N are the target nucleus’ proton and neutron numbers, g p V and gn V are the vector couplings to the Z of the proton and neutron, and Fp and Fn are the nuclear form factors, which are functions of momentum transfer squared Q2. The baryon weak 38 Flavor Symmetric models of neutrino masses and dark matter couplings are g p V = 1 2 2 sin2 qW , (3.64) gn V = 1 2 . (3.65) Given the value of the Weinberg angle at low energies, sin2 qW = 0.238± 0.0024 the CEnNS cross section is dominated by the interaction of the Z with neutrons, and therefore it scales roughly as N2. The CEnNS process has been detected and its cross section measured by the COHERENT experiment at Oak Ridge National Laboratory, using neutrinos produced from the decay of pions produced at the Spallation Neutron Source [99, 100]. The results of this measurement can be used to confirm the SM prediction, extract information on the nuclear neutron skin or the Weinberg angle, constrain the neutrino charge radius, magnetic moment or electric charge, among other SM and BSM physics [101]. Current and future experiments seek to observe this process using reactor neutrinos. Nuclear reactors are an intense source of electron an- tineutrinos (n̄e). As a representative example of these type of experiments, we consider the Scintillating Bubble Chamber experiment (SBC) [102]. This collaboration seeks to construct a bubble chamber, consisting of a target volume of 10kg of liquid Argon (LAr), with a small amount of Xenon dissolved in it (10-1000 ppm). The target material is enclosed in a fused silica jar, surrounded by liquid CF4, which serves as cryogenic material and hydraulic fluid. This system is pressure and temperature controlled to superheat the target material, making it susceptible to produce a detectable bubble when a particle interaction deposits enough localized energy to overcome a nucleation threshold. The device is also outfitted with photomultipliers to detect scintillation signals. This technology is capable of having an energy deposition threshold as low as 100 eV [103]. The construction of two initial experimental setups is planned, one for dark matter direct detection, to be placed in an underground laboratory, and another for CEnNS to be placed near a nuclear reactor. For concreteness, we consider that the SBC-CEnNS experimental phase will consist of 10 kg of detector material placed at 3 m from a 1 MW nuclear reactor. The neutrino flux at this distance [104, 105] is approximately 105 times larger than the flux from the SNS. The objective of this work is to investigate possible interplay between CEvNS and DM direct detection experiments in a complete model of BSM quark-lepton interactions 4This value is derived from the measurement of the weak charge of the proton through the measure- ment of the parity violating polarized electron-proton scattring by Qweak [98]. Flavor Symmetric models of neutrino masses and dark matter 39 with dark matter. We consider the solutions corresponding to the well-known B L model, the B 2La Lb models previously studied [106]. Finally, we consider B 3La models [107–109]. We avoid the well-motivated and widely studied Lµ Lt model, be- cause BSM CEnNS signatures arise at the one-loop level, and therefore are suppressed compared to the other models. It is easy to see that ZN is a discrete subgroup of the U(1) continous symmetry. Simply consider a U(1) rotation by 2p/N. This rotation can be used to generate the group ZN. A U(1)0 model can therefore be constructed to have a remnant ZN with an appropriate set of fermion and scalar charges and a spon- taneous symmetry breaking mechanism [110]. This ZN can be used to stabilize a dark matter candidate, and furthermore, it can determine the Majorana or Dirac nature of neutrinos [111,112]. We consider introducing a vector-like pair of fermions, with trivial SM gauge group representations and nonzero U(1)X charges. As all fermion X charges are multiples of 1/3, and quarks are SU(3) multiplets, dark matter can be stable when it has X charge 1/3 and scalar singlets are integerly charged. Any renormalizable operators of DM and SM particles must contain DM in particle-antiparticle pair. This is sufficient to prove that DM is perturbatively stable. The stabilizing remnant symmetry is Z3. For a Dirac fermion vector-like pair (cL, cR), with 1/3 X charge the resulting dark matter mass eigenstate is given by c = cL + cR. We can write the dark sector Lagrangian as follows LD = cgµ(∂µ + i g0 3 Z0 µ)c ! + Mccc + h.c. (3.66) Therefore, c only couples to the new gauge boson Z0. This interaction must account for the relic density of dark matter, if we propose that the observed dark matter relic density has a thermal origin. A popular scenario is the freeze-out of dark matter, where the dark matter that is initially thermally coupled to the SM bath falls out of equilibrium as the Universe cools. The dark matter density is determined by the amount of cc ! SMSM annihilations that can take place before the expansion of the Universe dilutes dark matter beyond a point where annihilations are too rare to affect its density. On the other hand, this interaction leads to direct detection signatures through the process cN ! cN, elastic dark matter-nucleon scattering. The dark matter relic density determined by freeze-out and the direct detection cross section are both only dependent on the Z0 mass, the dark matter mass Mc and the new gauge coupling 40 Flavor Symmetric models of neutrino masses and dark matter strength g0 5. The parameters MZ0 and g0 completely control the BSM CEnNS cross section. It has been shown that the dark matter relic density and direct detection constraints are in tension for mc < O(10 TeV) in U(1)0 models [113–119]. Beneath a dark matter mass of around 10 GeV direct detection experiments become poorly sensitive to the induced nuclear recoil of the elastic scattering. In this mass range, with a light Z0 a sizeable gauge coupling of order O(0.1) is required to produce the correct relic density. CEnNS experiments exclude this low Z0 mass - high g0 parameter space. One way of relaxing this constraint is too consider resonant dark matter annihilation. The Z0 mediated dark matter annihilation into SM fermions possesses a resonance at s = M2 Z0 , which lowers the value of g0 required to reproduce the observed dark matter relic density. At rest, the Mandelstam variable s is s = (2Mc) 2. We parametrize the resonant condition with the dRes parameter defined by the relation Mc = MZ0 2 (1 + dRes) . (3.67) We implemented the U(1)0 models in LanHEP [120] and micrOMEGAs [121] to calculate the dark matter observables, scanning over the ranges (103 50) GeV for the Z0 mass, (106 101) for the g0 gauge coupling and to ensure the resonant annihilation (0.45 0.55) MZ0 for Mc, varying |dRes| between (0.001 0.1). In the models we will consider, we have three right handed neutrinos, which will participate in the type-I seesaw mechanism. For each model, we will use a different set of scalar fields, which will couple to the RHN and develop a VEV. The charges of the scalars are obtained by demanding phenomenologically viable, but minimal, right handed neutrino Majorana mass matrices, taking into account that the Dirac neutrino mass matrix is diagonal. We introduce the terminology MI-MIV to designate the models with the respective gauge symmetry and scalar content needed for a succesful type-I seesaw. In Table 3.3 we write down the gauge symmetry and scalars of each model, with the nomenclature fi for a scalar field, singlet under the SM gauge model and with U(1)X charge i. The Z0 boson, after U(1)0 breaking by the scalars, obtains the mass MZ0 for different models as shown in Table 3.3. For MI there are no correlations in the active neutrino mass matrix, for MII there are four cases with 5From here on out, we take the scalar content of the model to be decoupled from the theory at the dark matter freeze-out temperature. This can be ensured by setting Mc ⌧ Mf for any flavons f. We can consider lowering these masses to obtain contributions to the dark matter annihilation cross section from more exotic dark matter annihilation channels, like Higgsstrahlung processes. In the spirit of simplicity, we will not take these processes into account. Flavor Symmetric models of neutrino masses and dark matter 41 U(1)0 models Scalar Fields Masses of Z0 (M2 Z0) MI U(1)BL f2 g02(4v2 2) MII U(1)B2LaLb f1, f2 g02(v2 1 + 4v2 2) MIII U(1)0B2LaLb f1, f2, f4 g02(v2 1 + 4v2 2 + 16v2 4) MIV U(1)B3La f3, f6 g02(9v2 3 + 36v2 6) Table 3.3.: Singlet scalar fields fi having charges i under U(1)0. good phenomenology and predictions as investigated in [106], for MIII there is one correlation, see Appendix A.1 and for MIV there are no correlations. 3.3.1. Relic density and direct detection As stated before, the main objective of this work is to investigate the DM phenomenol- ogy of a DM candidate embedded in the U(1)X gauge extension of the SM. A DM candidate that only couples to the Z0 can populate the early Universe through the freezeout process. The relevant annihilation cross section in the resonant regime MZ0 ⇠ 2Mc is obtained from the first diagram of Fig. 3.3. The final states of this anni- hilation are all the SM fermions that are charged under the new gauge symmetry, and are kinematically allowed (M f < Mc). This cross section depends on the DM mass, the Z0 mass, the new gauge coupling strength, but also on the Z0 decay width. We calculate this cross section and the relic density of dark matter with the micrOmegas code [121, 122]. This code also calculates the Z0 decay width, which we have tacitly assumed does not have any contribution from particles beyond the SM fermions and the DM candidate. It is worthwhile to note that in this model it is possible for DM to annihilate into Z0 pairs or Z0+ f final states. We do not take into account these annihilation channels. The first one is not permitted in the resonant regime, and the second one depends on the details of the scalar sector, which we have assumed has been integrated out 6 at the scale of the dark matter mass. The gauge coupling of dark matter also is responsible for the leading order contri- bution to dark matter direct detection. The couplings of the dark matter candidate and quarks to the vector mediator are not chiral, so the interaction leads to a Spin- 6Except for the 125 GeV SM-like Higgs. 42 Flavor Symmetric models of neutrino masses and dark matter Independent cross section. The Feynman diagram corresponding to the process is shown in the left hand side of Fig. 3.3. Figure 3.3.: Feynman diagrams leading to dark matter annihilation in the thermal freeze-out process (left) and dark matter direct detection (right). Flavor Symmetric models of neutrino masses and dark matter 43 The scattering cross section resulting from this diagram is given by [123] sSI ⇡ µ 2 cn p (Z fp + (A Z) fn) 2 A2 , (3.68) where, µcn is the WIMP-nucleon reduced mass, Z, A are the atomic number, atomic mass of the target nucleus, respectively. fp and fn are the U(1)0 charges of the proton and neutron, and are given by fp ⇡ gc M2 Z0 (2gu + gd), fn ⇡ gc M2 Z0 (gu + 2gd) . (3.69) In this study, gu = gd = gc = g0/3, hence fp = fn ⇡ g02 3M2 Z0 . Therefore, the spin- independent cross-section reduces to sSI ⇡ µ 2 cn p g04 9M4 Z0 . (3.70) Low energy constraints depend on the free parameters g0 and MZ0 , dark matter con- straints depend on g0, MZ0 and Mc. The resonance condition allows us to reduce the number of free parameters by one. This in turn allows us to constraint the dark matter parameter space (Mc, g0) using dark matter observable, the low energy parameters (MZ0 , g0) using low energy observables and project the constraints from one set of observables into the other set of observables. 3.3.2. Dark matter direct detection and CEνNS complementarity The numerical results of the constraints discussed before are presented here: con- straints on Z0 interacting with quarks and leptons and constraints on Z0 mediated DM-SM interactions. First, we can analyze the well-known U(1)BL model, shown in Fig. (3.4). We show the region excluded by the results of the CEnNS COHERENT-CsI data in purple. The SBC-CEnNS [125, 126] sensitivity projection is shown with a black dashed line (The region above the line is projected to be excluded). The COHERENT exclusion and SBC-CEnNS projection are calculated at the 95% C.L. using COHERENT-CsI data [124] and SBC-CEnNS [125, 126] experimental details, respectively. We used the exclusion 44 Flavor Symmetric models of neutrino masses and dark matter Ωh2 > 0. 11 98 SB C -C E νNS B eam dum pB B N + C M B 10-3. 10-2. 10-1. 100. 101. 10-6. 10-5. 10-4. 10-3. 10-2. 10-1. MZ′ [GeV ] g B - L COHERENT-CsI BABAR LHCb SBC-DM XENON1T PandaX B e a m D u m p B B N + C M B SB C- CE ⋁NS ν - flo o r Ωh2>0.1198 10-3. 10-2. 10-1. 100. 101. 10-50. 10-48. 10-46. 10-44. 10-42. 10-40. 10-38. Mχ [GeV] σ SI[c m 2 ] COHERENT-CsI LHCbBABAR SBC-DM XENON1T PandaX Figure 3.4.: Left panel: Exclusion regions in the (MZ0 , g0) plane for the B-L model. Right panel: Exclusion regions for the spin independent cross-section in the (Mc, sSI) plane for the B-L model. The light-purple shaded area corresponds to the constraint set by the current COHERENT-CsI data [124]. The limits set by the future reactor-based CEnNS experiment SBC [125, 126], is presented using the black long dashed line (SBC-CEnNS ). Cosmological constraints [127], beam dump experiments [128–138], BABAR [139] and LHCb dark photon searches [140] are presented using the red, light-green, light-brown and sky-blue regions, respectively. Limits set by the dark matter experiments are presented using the light- orange, light-cyan, and light-yellow regions for the SBC-DM [125], XENON1T [53], and PandaX-II [141] experiments, respectively (see text for more details). In the right panel, the argon n-floor background [142] has been marked using dotted-magenta curve. Flavor Symmetric models of neutrino masses and dark matter 45 limits on the (Mc, sSI) parameter space from SBC-DM (projected limits), XENON1T and PandaX-II experiments. The excluded regions are shown in orange, cyan and yellow respectively. The light-red region corresponds to the region where the dark matter annihilation cross section in the early Universe is too small to result in the cor- rect relic density of dark matter or lower. Cosmological data can be used to constrain the (MZ0 , g0) parameter space, as a new gauge boson coupling to quark and leptons can be in thermal equilibrium with the SM at early times and contribute to ∆Ne f f if it is too light. This bound is calculated in [127], and is shown in red, as a lower bound in the mass of the Z0. Exotic particles coupling to quarks and leptons have been searched for in electron beam dump experiments (E141 [130], E137 [129], E774 [131], KEK [132], Orsay [136], and NA64 [134]), proton beam dump experiments like n-CAL I [133], pro- ton bremsstrahlung [138], CHARM [128], NOMAD [135], and PS191 [137]. To obtain the bounds these experiments set on our models, we used the code Darkcast [143], which recasts analysis made for dark photon models and obtains the bounds for more general vector mediators. These results are shown in green. Similarly, Darkcast can be used to recast LHCb [140] and BABAR [139] data. LHCb data is obtained from searches of Z0 produced in proton-proton collisions, assuming the Z0 subsequently decays into a pair of muons. BABAR is a electron-positron collider that, among other physics objectives, searched for Z0 production followed by decay into muon or electron pairs. Numerically, we have found that the BABAR muon channel results are weaker than either LHCb results or COHERENT-CsI, so we only use BABAR electron decay channel data. We consider limits set by the proton-proton collider. Constraints from LHCb data is presented in sky-blue and constraints from BABAR are presented in brown. Immediately we notice in Fig. 3.4 that SBC-CEnNS is projected to surpass any other experimental search for a Z0 in the (0.02 0.5) GeV mass range, excluding gauge couplings down to (105 104). This sensitivity improves upon the existing COHERENT-CsI constraints approximately by one order of magnitude in g0. For masses above 0.2 GeV, BABAR constraints surpass COHERENT-CsI results. BABAR constraints surpass the SBC-CEnNS projections above 0.5 GeV. BABAR constraints are also stronger than LHCb’s up until BABAR’s kinematical limit of MZ0 ⇠ 10 GeV. On the lower end of the Z0 mass range, cosmological limits and beam dump exper- iments largely rule out Z0 masses lighter than 10 MeV. Constraints from thermal production of dark matter set a lower bound on g0 g0 , which are shown in light-red. In this region, annihilation cross section is too small, leading to a dark matter abundance above the observed value. Finally, existing dark matter direct detection results set an upper bound, for the relatively large Z0 mass of around 10 GeV and above. For the 46 Flavor Symmetric models of neutrino masses and dark matter B B N + C M B Ωh2 > 0. 11 98SB C -C E νNS B eam dum p 10-3. 10-2. 10-1. 100. 101. 10-6. 10-5. 10-4. 10-3. 10-2. 10-1. MZ′ [GeV ] g B - L e - 2 L μ COHERENT-CsI BABAR LHCb SBC-DM XENON1T PandaX B B N + C M B Ωh2 > 0. 11 98SB C -C E νNS B eam dum p 10-3. 10-2. 10-1. 100. 101. 10-6. 10-5. 10-4. 10-3. 10-2. 10-1. MZ′ [GeV ] g B - L e - 2 L τ COHERENT-CsI BABAR SBC-DM XENON1T PandaX B B N + C M B Ωh2 > 0. 11 98 10-3. 10-2. 10-1. 100. 101. 10-6. 10-5. 10-4. 10-3. 10-2. 10-1. MZ′ [GeV ] g B - L μ-2 L τ COHERENT-CsI LHCb SBC-DM XENON1T PandaX B B N + C M B Ωh2 > 0. 11 98 10-3. 10-2. 10-1. 100. 101. 10-6. 10-5. 10-4. 10-3. 10-2. 10-1. MZ′ [GeV ] g B - 2 L μ-L τ COHERENT-CsI LHCb SBC-DM XENON1T PandaX Figure 3.5.: Same as Fig. (3.4) but for MII U(1)0 models as given by Table (3.3). Flavor Symmetric models of neutrino masses and dark matter 47 B B N + C M B Ωh2 > 0. 11 98 SB C -C E νNS B eam dum p 10-3. 10-2. 10-1. 100. 101. 10-6. 10-5. 10-4. 10-3. 10-2. 10-1. MZ′ [GeV ] g B - 2 L e - L μ COHERENT-CsI BABAR LHCb SBC-DM XENON1T PandaX B B N + C M B Ωh2 > 0. 11 98 SB C -C E νNS B eam dum p 10-3. 10-2. 10-1. 100. 101. 10-6. 10-5. 10-4. 10-3. 10-2. 10-1. MZ′ [GeV ] g B - 2 L e - L τ COHERENT-CsI BABAR SBC-DM XENON1T PandaX Figure 3.6.: Same as Fig. (3.4) but for the MIII U(1)0 models as given by Table (3.3). B L case, collider limits dominate over direct detection. The SBC-DM sensitivity projection lowers the sensitivity to the Z0 mass down to 1 GeV, but will not be able to exclude beyond what is already excluded by collider searches. For this model a seesaw mechanism gives mass to neutrinos, after the scalar f2 spon- taneously breaks the new gauge symmetry. This scalar couples to all right handed neutrinos, filling the entire right handed neutrino Majorana mass matrix. The Dirac mass matrix is full as well, as there is no symmetry distinguishing neutrino flavor. This means that this model only contains a mechanism for light neutrino mass generation, but it has no predictions for mixing angles. The projection of these constraints to the (Mc, sSI) is shown in the right panel of Fig. 3.4. In this plot we can show the neutrino floor [142] as a magenta dotted curve. This floor is a limit on the practical sensitivity of dark matter direct detection, it arises from the unavoidable background CEnNS processes from solar, atmospheric and diffuse supernova neutrinos. Here we can see that the results from BABAR and LHCb reach down to the neutrino floor in this model. Now we move on to the next set of models, MII, which are defined as U(1)B2LaLb gauge models, which need only two BSM scalar fields to generate realistic neutrino phenomenology. Under this designation we have the U(1)BLe2Lµ , U(1)BLe2Lt , U(1)BLµ2Lt , and U(1)B2LµLt models. In these cases we find two-zero neutrino mass matrices. The neutrino phenomenology of these 48 Flavor Symmetric models of neutrino masses and dark matter scenarios has been explored previously [106], and has been shown to be consistent with neutrino oscillation data. The major difference in these scenarios, regarding the collider, beam dump and CEnNS phenomenology lies in the flavored nature of the gauge interaction. In these models, one lepton flavor does not participate in the gauge interaction, removing any experimental constraint that depends on this gauge coupling. Experiments depending on a non-zero electron gauge coupling are BABAR, reactor CEnNS and beam dump experiments. The only experiment depending on the muon coupling to the Z0 that we consider is the LHCb dark photon search. COHERENT CsI is sensitive to BSM ne and nµ interactions. Therefore, for gauge models with only one of these neutrinos coupling to the Z0 , COHERENT constraints are weaker. No Z0 searches have been performed using t leptons, so U(1)BLe2Lµ is the model that is most constrained among MII models. These results are shown in Fig. 3.5. Under the MIII designation we have grouped U(1)B2LaLb models which need three BSM scalar fields to yield realistic neutrino phenomenology. In this case, we have no zeroes in the neutrino mass matrix, but we do predict one correlation in the neutrino mass matrix, which is compatible with neutrino oscillation data (see A.1). The results for these models are shown in 3.6, and are similar to MII models. MIV models are defined as U(1)B3La gauge models. In this case we need three BSM scalar fields as well, in order to obtain realistic neutrino phenomenology. As in the U(1)BL model, there are no correlations or flavor predictions. In these mod- els, two lepton families are not charged under the gauge symmetry, so these models are comparatively less restricted. In the case of U(1)B3t in particular, there are no beam dump, collider or CEnNS constraints. Neutrino oscillation data can be used to constrain this model, however. A global analysis of neutrino oscillation data was performed in [144]. We use these results to constrain only MIV models, as they are not the leading laboratory constraints for the MI -MIII scenarios, but they are specially relevant for the U(1)B3Lt case. 3.3.3. Conclusions In this work, an analysis was performed on the viability of a gauged U(1)0 symmetry accounting for neutrino mass generation and dark matter phenomenology. The major findings are : Flavor Symmetric models of neutrino masses and dark matter 49 Figure 3.7.: Same as Fig. (3.4) but for MIV U(1)0 models as given by Table (3.3). Oscillation limits are obtained from the analysis made in [144]. 50 Flavor Symmetric models of neutrino masses and dark matter • A light Z0 boson mediating quark-lepton interactions is potentially discoverable in CEnNS experiments, with future experiments surpassing the sensitivity of collider searches, for masses below ⇠ 10 GeV. • The same Z0 boson can mediate DM-SM interactions, providing a channel for dark matter freeze-out and dark matter direct detection. • Collider, beam dump, cosmology, CEnNS and dark matter searches data can and should be used to search for DM and BSM interactions. • Despite all the current constraints on this simple type of DM models, they are still viable, but future experimental sensitivities will severely reduce the allowed parameter space. 3.4. Global U(1) as a flavor symmetry This section is based on the research presented in [4]. The QCD axion [57, 58, 145] is a particle predicted to exist as a consequence of the Peccei-Quinn solution to the Strong CP problem. The Strong CP problem consists of the non-observation of a neutron electric dipole moment (NEDM), which is predicted by the q term of the pure gauge part of the QCD Lagrangian. The experimental limits on the NEDM severely constrain the value of the q parameter. The Peccei-Quinn symmetry solution to the Strong CP problem addresses the smallness of q with a global symmetry. The breaking of the global symmetry dynamically sets the q parameter to zero. A byproduct of the mechanism is the existence of a pseudo Nambu-Goldstone boson, the QCD axion. This light scalar couples to the gluon and photon fields, which makes it detectable in laboratory experiments. The QCD axion can also constitute part or the whole of the dark matter of the Universe [146–150]. The QCD axion can be embedded in three types of models, the original PQWW, the DSFZ [151, 152] and the KSVZ [153, 154] models. Each model has distinct predictions for the relationship between the mass of the axion and its couplings to the photon and gluon [155]. The original PQWW predicts too large couplings, it already is an excluded scenario. The axion coupling to gluons and photons depends on two anomaly coefficients, the electromagnetic anomaly coefficient E and the color anomaly coefficient N. These coefficients are functions of the PQ charges of the fermion content of the model, and therefore highly model dependent. Flavor Symmetric models of neutrino masses and dark matter 51 The Froggatt-Nielsen (FN) explanation of the hierarchical flavor structure of SM fermion Yukawa couplings is another global U(1) extension of the SM. In this setup, a global symmetry is proposed, so that most fermion Yukawa couplings are forbidden at the dimension-4 level. Fermion mass terms are constructed with a scalar field f at the non-renormalizable level with dimension-5 operators and above. These non- renormalizable dimension-N operators lead to fermion mass terms with a (hfi/Λ)(4 N)) suppression. Assuming that Λ > hfi, this suppression can explain the diversity of scales in the entries of SM Yukawa matrices. When one considers UV completions of the FN mechanism, it is possible that the FN symmetry is anomalous, and its breaking also leads to the existence of a QCD axion. In recent years, several works have contemplated the identification of the FN symmetry with the PQ symmetry. The QCD axion of such a setup leads to rich axion-fermion flavor violating interactions, in addition to the traditional axion-photon and axion-gluon interactions. The flavored QCD axion has received the name of “flaxion" [156, 157], “axiflavon" [158–160], or “flavorful axion" [161–163]. The objective of this work is to study UV-complete implementations of the Froggatt-Nielsen-Peccei- Quinn symmetry. The UV-completeness of the anomalous Froggatt-Nielsen symmetry leads to predictions of the anomaly coefficients E and N, and therefore to predictions of the flavored axion anomalous coupling to gauge bosons. Furthermore, the flavored nature of the Peccei-Quinn symmetry results in flavor-violating axion couplings. This makes the implementation of the global U(1) symmetry a rich sandbox for axion and flavor phenomenology. We consider the DFSZ type axion models and show that one can construct a Nearest-Neighbour-Interaction (NNI) structure for both the up- and down-quark mass matrices Mu/d = 0 BBB@ 0 ⇥ 0 ⇥ 0 ⇥ 0 ⇥ ⇥ 1 CCCA , (3.71) which was originally studied in [164]. To obtain this mass matrix, adequate PQ charges must be assigned to the quark fields, to the Higgs doublets and flavons. At the non- renormalizable level, the operators for all entries of the mass matrix can be written, with free Wilson coefficients. To obtain a prediction for the relationship between these Wilson coefficients, we can build UV completions of the operators, by adding heavy mediator fields that give rise to the operators. In this work, two types of UV- completions are contemplated: type-I and II Dirac seesaws. Once we can calculate the 52 Flavor Symmetric models of neutrino masses and dark matter Wilson coefficients of the UV theory, the mass matrix can be studied. We show that the NNI matrix is obtained. For the neutral lepton sector, a UV completion consisting of the Majorana type-I seesaw [165–169] can be built. In the lepton sector an A2 neutrino mass matrix texture [82] is obtained, which is compatible with oscillation data [170–172]. The flavor violating nature of axion couplings is tightly related to the fermion mass generation mechanism. After showing the viability of the predictions of fermion masses and mixings in this model, axion phenomenology is studied. Finally, due to the presence of two Higgs doublets in the model without a symmetry forbidding scalar neutral flavor changing currents, we must study Higgs flavor violating decays. 3.4.1. Froggat-Nielsen mechanism and Nearest-Neighbour-Interactions We begin the construction of the models by revisiting the basics of the Froggatt-Nielsen mechanism and the Nearest Neighbour Interacion mass matrices. The Froggatt-Nielsen [173] explanation of the hierarchy of mass scales in the fermion sector of the SM consists of a global U(1)FN symmetry. With the field content of the SM, the presence of the U(1)FN symmetry and a new scalar singlet s, the effective quark Yukawa couplings of the SM are written as LY yd ij ⇣ s Λ ⌘nd ij QiHdRj + yu ij ⇣ s Λ ⌘nu ij Qi eHuRj , (3.72) where nu/d ij are complex numbers, s is the flavon field and Λ is the scale of flavor dynamics. After the spontaneous breaking of the Froggatt-Nielsen symmetry by the vev of the s field, the mass matrices for the quarks are generated with a hierarchy of scales. This hierarchy is generated by the parameter hsi/Λ which is taken hsi/Λ < 1. The entries of the mass matrices can then be written as mij = yijhHi(hsi/Λ)n, (3.73) where all Yukawa couplings can be taken O(1) and deviations from this are absorbed in (hsi/Λ)n. A U(1) symmetry can also serve to generate texture zeroes in the mass matrices. One phenomenologically viable texture in the quark sector is the Nearest- Neighbour texture. For a N ⇥ N matrix, the NNI texture is defined with the constraints mij 6= 0 if i = j± 1 or i = N = j. The NNI texture scheme is compatible with quark Flavor Symmetric models of neutrino masses and dark matter 53 masses and mixings, but contains more free parameters than the physical observables of the quark sector. The NNI texture can be obtained by a Weak Basis transformation from any Yukawa matrix [164] and imposes no physical constraints in the SM. In the presence of the U(1) flavor symmetry and a extended scalar sector, the NNI texture becomes a predictive framework for scalar interactions [174]. Numerically, imposing the NNI texture in the quark sector leads to large (3,3) entries of the up and down matrices, while the rest of the entries are suppressed by factors 101-102 [175]. This motivates the use of a Froggatt-Nielsen symmetry to obtain these suppressions. 3.4.2. Fermion masses in flavored DFSZ axion models Here, the DFSZ realization of the Froggatt Nielsen models is presented. In DFSZ axion models two Higgs doublets and a singlet are used to break the U(1)PQ symmetry. Phenomenologically, the breaking scale of the PQ symmetry is overwhelmingly domi- nated by the singlet vev. The necessary fields for the DFSZ realization of the model are given in Table 3.4. The charges are chosen so that the NNI texture appears with renormalizable and dimension-5 operators. The Higgs doublet charge is normalized to unity. Fields/Symmetry QiL uiR diR Hu Hd s SU(2)L ⇥U(1)Y (2, 1/6) (1, 2/3) (1, -1/3) (2, -1/2) (2, 1/2) (1, 0) U(1)PQ (9/2, -5/2, 1/2) (-9/2, 5/2, -1/2) (-9/2, 5/2, -1/2) 1 1 1 Table 3.4.: Field content and transformation properties of the PQ-symmetry under the DFSZ type-I seesaw model, where i = 1, 2, 3 represent families of three quarks. With the charges of 3.4, the effective Lagrangian of the up quark sector is L Cu 11 Λ 8 Q1LHuu1Rs8 + Cu 12 Λ Q1LHuu2Rs + Cu 13 Λ 4 Q1LHuu3Rs4 + Cu 21 Λ Q2LHuu1Rs + Cu 22 Λ 4 Q2L eHdu2Rs⇤4 + Cu 23 Λ Q2L eHdu3Rs⇤ + Cu 31 Λ 4 Q3LHuu1Rs4 + Cu 32 Λ Q3L eHdu2Rs⇤ + yu 33Q3LHuu3R , (3.74) 54 Flavor Symmetric models of neutrino masses and dark matter where Cu ij are the Wilson coefficients of the operators, and Λ is the cut-off scale of the model. For the down-quark sector the effective Lagrangian is L Cd 11 Λ 8 Q1LHdd1Rs8 + Cd 12 Λ Q1LHdd2Rs + Cd 13 Λ 4 Q1LHdd3Rs4 + Cd 21 Λ Q2LHdd1Rs + Cd 22 Λ 4 Q2L eHud2Rs⇤4 + Cd 23 Λ Q2L eHud3Rs⇤ + Cd 31 Λ 4 Q3LHdd1Rs4 + Cd 32 Λ Q3L eHud2Rs⇤ + yd 33Q3LHdd3R . (3.75) In these Lagrangians, both Higgs doublets couple to both quark types. This will lead to flavor violating Higgs couplings, which will constrain the phenomenology of the theory. After spontaneous breaking of the flavor symmetry, the mass matrices obtained up to dimension-7 operators are Mu/d = 0 BBB@ 0 #vu/dCu/d 12 0 #vu/dCu/d 21 0 #vd/uCu/d 23 0 #vd/uCu/d 32 yu/d 33 vu/d 1 CCCA , (3.76) where, # = hsi/Λ or hsi⇤/Λ. For typical values of #⇠ 0.2, one can safely neglect terms proportional to #4 and #8 as has been pointed in Eqs. 3.74, 3.75. As anticipated, varepsilonn is the source of the hierarchy inside a quark mass matrices. As in the Two Higgs Doublet Models, one can explain the hierarchy between mt and mb by a hierarchy between vu and vd. This framework constitutes the effective part of a flavorful axion model, with a Froggatt-Nielsen explanation of quark hierarchies. The value of the Wilson coef- ficients C are arbitrary and, in principle, uncorrelated. In the next section we will build UV-completions, which explain the origin of these coefficients. 3.4.3. Type-I Seesaw UV-completion We consider a type-I Dirac seesaw origin of the operators of eqs. ??. The simplest UV-completion consists of vector-like quarks. A vector-like pair of quarks does not contribute to the anomalous couplings of the axion. The new fields and their PQ charges are shown in 3.5. With this matter content, the Lagrangian that leads to the Flavor Symmetric models of neutrino masses and dark matter 55 Fields/Symmetry F12 uC F21 uC F23 uC F32 uC F12 dC F21 dC F23 dC F32 dC U(1)Y 2/3 2/3 2/3 2/3 -1/3 -1/3 -1/3 -1/3 U(1)PQ 7/2 -7/2 -3/2 3/2 7/2 -7/2 -3/2 3/2 Table 3.5.: Vector like fermions and their transformation properties of the PQ-symmetry under the DFSZ type-I seesaw model, where C = L, R. QiL Hq qjR σ F ij qR F ij qL Figure 3.8.: UV complete diagram within the DFSZ type-I seesaw framework as apparent from Eqs. 3.77 and 3.78. effective operators of eq. 3.74 are LUV u Yu 12Q1LHuF12 uR +Mu 12F12 uRF12 uL + Y 0u 12 F12 uLsu2R + Yu 21Q2LHuF21 uR +Mu 21F21 uRF21 uL + Y 0u 21F21 uLsu1R + Yu 23Q2L eHdF23 uR +Mu 23F23 uRF23 uL + Y 0u 23 F23 uLs⇤u3R + Yu 32Q3L eHdF32 uR +Mu 32F32 uRF32 uL + Y 0u 32F32 uLs⇤u2R . (3.77) For the down-quark sector we can write LUV d Yd 12Q1LHdF12 dR +Md 12F12 dRF12 dL + Y 0d 12 F12 dLsd2R + Yd 21Q2LHdF21 dR +Md 21F21 dRF21 dL + Y 0d 21F21 dLsd1R + Yd 23Q2L eHuF23 dR +Md 23F23 dRF23 dL + Y 0d 23 F23 dLs⇤d3R + Yd 32Q3L eHuF32 dR +Md 32F32 dRF32 dL + Y 0d 32F32 dLs⇤d2R . (3.78) The type-I Dirac seesaw mechanism Feynman diagram that these Lagrangians generate is shown in 3.8. The scalar fields acquire vevs, breaking the FNPQ and EW symmetries. The mass matrices generated by the symmetry breaking in the (QL, F(u/d)L)⇥ (qR, F(u/d)R) basis 56 Flavor Symmetric models of neutrino masses and dark matter are Mu/d = 0 @MQLqR MQLFR MFLqR MFLFR 1 A 7⇥ 7 , (3.79) where the 4 submatrices of Mu/d are given by MQLqR = 0 BBB@ 0 0 0 0 0 0 0 0 yu/d 33 vu/d 1 CCCA , (3.80) MQLFR = 0 BBB@ Yu/d 12 vu/d 0 0 0 0 Yu/d 21 vu/d Yu/d 23 vd/u 0 0 0 0 Yu/d 32 vd/u 1 CCCA , (3.81) MFLqR = 0 BBBBBB@ 0 Y 0u/d 12 vs 0 Y 0u/d 21 vs 0 0 0 0 Y 0u/d 23 v⇤s 0 Y 0u/d 32 v⇤s 0 1 CCCCCCA , (3.82) MFLFR = diag(Mu/d 12 ,Mu/d 21 ,Mu/d 23 ,Mu/d 32 ) . (3.83) Using the seesaw formula, the light quark mass matrices obtained are mu/d = MQLqR MQLFR M1 FLFR MFLqR , (3.84) Flavor Symmetric models of neutrino masses and dark matter 57 at leading order. With the operators of eqs. ??, the light quark mass matrices are mu/d = 0 BBBB@ 0 Yu/d 12 Y 0u/d 12 M12 vu/dvs 0 Yu/d 21 Y 0u/d 21 M21 vu/dvs 0 Yu/d 23 Y 0u/d 23 M23 v⇤d/uv⇤s 0 Yu/d 32 Y 0u/d 32 M32 v⇤d/uv⇤s yu/d 33 vu/d 1 CCCCA , = 0 BBB@ 0 Au/d 0 Bu/d 0 Cu/d 0 Du/d Eu/d 1 CCCA . (3.85) where A, B, C, D, and E are complex entries. This shows that the UV-completion successfully generates the mass matrix we desire. 3.4.4. The UV-completion: DFSZ type-II Seesaw The type-I seesaw UV-completion is not unique, we can consider extending only the scalar sector to construct a type-II seesaw UV-completion. In this case, the effective operators of eqs. 3.74, 3.75 can be generated by the inclusion of two additional Higgs doublets Φu(2,1/2, 2), and Φd(2, 1/2, 2). The up-quark sector Lagrangian is LUV u Yu 12Q1LΦuu2R + Yu 21Q2LΦuu1R + kuHuΦ † us + Yu 23Q2L eΦdu3R + Yu 32Q3L eΦdu2R + kd eHdΦds⇤ . (3.86) The SM ⇥U(1)PQ charges for the remaining fields are given by Table 3.4. In the down-quark sector we have LUV d Yd 12Q1LΦdd2R + Yd 21Q2LΦdd1R + kuHdΦ † ds + Yd 23Q2L eΦdd3R + Yd 32Q3L eΦdd2R + kd eHuΦds⇤ . (3.87) 58 Flavor Symmetric models of neutrino masses and dark matter QiL Hq σ qjR Φq Figure 3.9.: UV complete diagram within the DFSZ type-II seesaw framework as apparent from Eqs. 3.86, and 3.87. From these Lagrangians, the quark mass matrices obtained are mu/d = 0 BBB@ 0 Yu/d 12 vΦu/d 0 Yu/d 21 vΦu/d 0 Yu/d 23 vΦd/u 0 Yu/d 32 vΦd/u yu/d 33 vu/d 1 CCCA , (3.88) where the vevs vΦu/d of the additional doublets are determined by the scalar potential 7. The scalar vevs are approximately given by vΦu/d ⇡ ku/dvsvu/d M2 Φu/d (3.89) where MΦu/d are heavy scalar masses. Eq. 3.88 shows that this matter content also succesfully generates the desired NNI textures. 3.4.5. Lepton Sector For the lepton sector, a type-I Majorana seesaw can lead to realistic neutrino masses and mixings. The Yukawa couplings of the lepton sector are Ll y yeLeLHd`eR + yµLµL eHu`µR + yt LtLHd`tR +yn 1LeLHuN1 + yn 2LµL eHdN2 + yn 3LtLHuN3 . (3.90) The charge assignments for the leptonic fields are given in Table 3.6. 7In [176], the fermion mass hierarchies and the strong CP problem with four Higgs doublets along with the PQ symmetry have been discussed. Flavor Symmetric models of neutrino masses and dark matter 59 Fields/Symmetry LiL `iR Ni s0 SU(2)L ⇥U(1)Y (2, -1/2) (1, -1) (1, 0) (1, 0) U(1)PQ (1, -3, 0) (0, -2, -1) (0, -2, -1) 2 Table 3.6.: Field content and transformation properties of the leptonic fields and the scalar field s0, where i = 1, 2, 3 represent the three lepton families. This Lagrangian results in diagonal charged lepton and Dirac neutrino mass matrices. The right handed neutrino Lagrangian is LMajorana M1Nc 1 N1 + yN 12Nc 1 N2s0 + yN 13Nc 1 N3s + yN 33Nc 3 N3s0 . (3.91) This Lagrangian was studied in [106] and was shown to be phenomenologically viable. The resulting right handed Majorana mass matrix is MR = 0 BBB@ ⇥ ⇥ ⇥ ⇥ 0 0 ⇥ 0 ⇥ 1 CCCA . (3.92) The type-I seesaw formula yields the light neutrino mass matrix mn = 0 BBB@ 0 ⇥ 0 ⇥ ⇥ ⇥ 0 ⇥ ⇥ 1 CCCA , (3.93) which corresponds to the type A2 [82] neutrino mass matrix. This setup is phe- nomenologically viable, generating neutrino masses and lepton mixing consisten with oscillation data, with the correct choice of parameters. 3.4.6. Scalar sectors The study of the scalar sector is crucial for axion models, as it is here where the axion arises as a pseudo-Goldstone boson. In a prototypical DFSZ model, the singlet scalar field s couples to Higgs doublets with either a cubic eH† u Hds term or a quartic eH† u Hds2 term [151, 177]. Whether one coupling or the other is PQ invariant depends on the choices of the scalar PQ charges. In this work we have two singlets, one will have a cubic coupling and the other will have the quartic coupling. The full scalar potential 60 Flavor Symmetric models of neutrino masses and dark matter can consulted in Appendix A.3. We define the scalar fields present in the first DFSZ model as Hu = 0 @h0 u + iAu hu 1 A , Hd = 0 @ h+d h0 d + iAd 1 A , s = S + iA , s0 = S0 + iA0 , (3.94) whereas the second DFSZ model contains two additional SU(2)L doublets Φu and Φd, which we define as Φu = 0 @f0 u + iA0 u f u 1 A , Φd = 0 @ f+ d f0 d + iA0 d 1 A . (3.95) The Goldstone boson that gives mass to the Z gauge boson can readily be identified in each model AZ = ∑i Yivi Aiq ∑i Y2 i v2 i , (3.96) where Yi is the hypercharge of each scalar field, vi its vev, for the type-I seesaw model Ai 2 {A, A0, Au, Ad} and Ai 2 {A, A0, Au, Ad, A0 u, A0 d} for the type-II seesaw. The Goldstone boson related to the PQ symmetry is likewise given by APQ = ∑i Xivi Aiq ∑i X2 i v2 i , (3.97) where Xi is the PQ charge of the scalar field to which Ai belongs. The physical axion, however, is orthogonal to the Z-boson Goldstone, so to obtain it from these two equations, we must perform the following subtraction [155] a = APQ 2 4 ∑i XiYiv 2 iq ∑i Y2 i v2 i q ∑i X2 i v2 i 3 5 AZ . (3.98) The EW mass scale is set by the doublet field vevs ( q ∑i Y2 i v2 i = 246 GeV). The vevs of the singlet scalars are bounded by their relationship to the axion decay constant fa. Phenomenologically fa is expected to be much larger than the EW scale. From this, we can deduce using Eqs. 3.96 - 3.98 that the axion is constituted primarily of the singlet fields. The s0 field only couples to leptons. To enhance the quark flavor Flavor Symmetric models of neutrino masses and dark matter 61 phenomenology of the axion, we choose to adopt the hierarchy v0s << vs. This makes the axion primarily consisting of the pseudoscalar part of s, maximizing the axion- quark coupling strength. With these considerations in mind we extract the axion-quark couplings by setting a⇠ A. 3.4.7. Axion couplings and mass Now that we have made the necessary phenomenological considerations to identify the axion. The axion couplings to the gluon and photon can be written as an effective Lagrangian L as 8p a fa (FeF)C + E N aEM 8p a fa (FeF)EM , (3.99) with E and N being the electromagnetic and color anomaly factors,(FeF)C = 1 2 #µnrsFb µnFb rs, Fb µn is the color field strength (b = 1, ..., 8), and (FeF)EM = 1 2 #µnrsFµnFrs, Fµn is the electromagnetic field strength. The anomaly factors E and N are given by the expres- sions [155] N = ∑i XiT(Ri) , E = ∑i XiQ 2 i D(Ri) , (3.100) where the sums run over all fermions, Qi is the electric charge, D(Ri) is the dimension of Ri, Xi is the PQ charge of fermion i, T(Ri) is the index of the SU(3)c representation Ri of fermion i (in particular T(3) = 1/2, T(6) = 5/2, T(8) = 3). The axion decay constant is given by [178] fa = q ∑i X2 i v2 ip 2N , (3.101) where this sum runs over all scalars with PQ charge Xi and vev vi. Due to the vev hierarchy chosen between s and s0, we have that fa ⇠ 2vs/N. The axion mass in terms of the decay constant is [178] ma = 5.70 ¯eV 1012 GeV fa ! . (3.102) 62 Flavor Symmetric models of neutrino masses and dark matter Finally, the coefficients of the anomalous axion-gluon coupling and axion-photon coupling are given by [178] gag = aEM 2p fa ✓ E N 1.92 ◆ , gag = as 8p fa . (3.103) In the SU(5) Georgi-Glashow model [179] of gauge unification, using the fact that SM fermions must fit in 10 and 5 irreps, one can show that E/N = 8/3. With this result, the axion-photon coupling can be expressed as gSU(5) ag = 1.53 1016GeV ✓ ma µeV ◆ . (3.104) In the models presented in this section the SM fermions are the only contributors to the anomalies, resulting in N = 5 and E = 28/3. |gSU(5) ag | |gDFSZ ag | = 14 . (3.105) The axion photon coupling can have interesting cosmological and astrophysical con- sequences. A summary of these constraints can be found in [180]. Additionally, laboratory probes of the axion photon coupling can be consulted , for example, in [61]. These constraints are derived from haloscope, light shining through a wall or microwave cavity resonator experiments. The Axion Dark Matter eXperiment (ADMX) [181], for example, sets a stringent bound on axion dark matter with a halo- scope experiment. In the mass range of 2.66 to 3.1 µeV, it has probed down to the SU(5) axion photon coupling prediction of O(1015) (GeV)1. The next phase of this project will improve this sensitivity by one order of magnitude in the (1.8, 8)⇥ 106 eV axion mass range [181]. In the flavored axion models discussed here, by choosing fa ⇠ 1012 GeV, and ma ⇠ 106 eV, we find |gag|⇠ 1018 (GeV)1 using Eq. 3.103 and therefore the suppression of this coupling places our model beyond the reach of projected ADMX Phase IIa/Gen-2 sensitivity [181]. Flavor Symmetric models of neutrino masses and dark matter 63 3.4.8. Numerical Results Masses and mixings of fermions Now we can proceed to set the free parameters of the PQFN so that we obtain the observed values of fermion masses and mixing. This will allow us to calculate the scalar couplings to fermions, and then study axion-fermion and Higgs-fermion BSM interactions. A c2 analysis was performed on the entries of the mass matrices we obtained. For the quark sector we have c2 = ∑ (µexp µ f it) 2 s2 exp , (3.106) where the sum runs over all observables. µ f it represent the masses and mixings calculated in the model using fitting parameters, as given by Eq. 3.107, we list them in Table 3.7. µexp and sexp are the observables and their standard deviation [182, 183], we list them in Table 3.7. We still have the freedom to eliminate all but two phases from the NNI quark mass matrices. In the Appendix A.2 this calculation is presented. The free parameters to minimize the c2 function over are then mu/d = 0 BBB@ 0 Au/d 0 Bu/d eiau/d 0 Cu/d eiau/d 0 Du/d eibu/d Eu/d eibu/d 1 CCCA . (3.107) In the Appendix A.2 we also show that the physical phase to consider is a difference between up and down phases, so we can set ad and bd to zero with no physical consequence. The final count of free parameters in the fit is 12, 10 absolute values and two phases. On the other hand we hace 10 observables, 6 quark masses, 3 CKM angles and one CP phase. The fit was performed using the MPT code [184] to extract the observables from a proposed numerical form of the NNI matrices. The c2 was minimized with the resulting minimum presented in Table 3.7. In the lepton sector, the charged lepton matrix is diagonal from the start, it contains only the charged leptons 64 Flavor Symmetric models of neutrino masses and dark matter Parameter Best fit Au/(102 GeV) 1.519 Bu/(102 GeV) 5.368 Cu/ GeV 3.004 Du/(101 GeV) 3.562 Eu/(102 GeV) 1.679 Ad/(102 GeV) 1.233 Bd/(102 GeV) 1.228 Cd/(101 GeV) 3.074 Dd/(101 GeV) 4.782 Ed/ GeV 2.793 au/ 96.56 bu/ 98.23 Observable Global-fit value Model best-fit Best-fit value 1s range q q 12/ 13.09 13.06! 13.12 12.988 q q 13/ 0.207 0.202! 0.213 0.2000 q q 23/ 2.32 2.29! 2.37 2.381 dq/ 68.53 66.06! 71.10 68.720 mu/(103 GeV) 1.288 0.766! 1.550 1.2743 mc/(101 GeV) 6.268 6.076! 6.459 6.2592 mt/ GeV 171.68 170.17! 173.18 171.687 md/(103 GeV) 2.751 2.577! 3.151 2.7330 ms/(102 GeV) 5.432 5.153! 5.728 5.4311 mb/ GeV 2.854 2.827! 2.880 2.8501 c2 q 1.0901 Table 3.7.: Best-fit values of the model parameters in the quark sector are shown in the upper table. The global best-fit as well as their 1s error [182, 183] for the various observables are given in the second and third columns of the lower table. Also, the best-fit values of the various observables are listed in the last column of the lower table. mass eigenvalues. The fit is performed over the A2 texture matrix for light neutrinos mn = 0 BBB@ 0 a eifa 0 a eifa b eifb c eifc 0 c eifc d eifd 1 CCCA . (3.108) A similar c2 function to the one in Eq. 3.106 is defined. Here, only the neutral lepton sector needs to be fitted. The leptonic masses and mixings obtained from the fit, which are compatible with the latest global fit data, can be seen in Table 3.8 at c2 l = 2.0053. Now that we have numerical values for the entries in the mass matrices, we can calculate the scalar couplings to fermions. Flavor Symmetric models of neutrino masses and dark matter 65 Parameter Best fit a/(103 eV) 9.933 b/(102 eV) 2.646 c/(102 eV) 2.475 d/(102 eV) 2.264 fa/ 29.87 fb/ 91.88 fc/ 3.03 fd/ 109.97 Observable Global-fit value Model best-fit Best-fit value 1s range ql 12/ 34.5 33.5! 35.7 34.85 ql 13/ 8.45 8.31! 8.61 8.432 ql 23/ 47.7 46.0! 48.9 48.11 dl/ 218 191! 256 258.8 a/ 65.27 b/ 265.08 ∆m2 21/(105 eV2) 7.55 7.39! 7.75 7.571 ∆m2 32/(103 eV2) 2.424 2.394! 2.454 2.4221 ∑ mn/(102 eV) 6.453 me/ MeV 0.4865763 0.4865735! 0.4865789 - mµ/ GeV 0.10271897 0.10271866! 0.10271931 - mt/ GeV 1.74618 1.74602! 1.74633 - c2 l 2.0053 Table 3.8.: Best-fit values of the model parameters in the lepton sector are shown in the upper table. The global best-fit as well as their 1s error [182, 183] for the various observables are given in the second and third columns of the lower table. Also, the best-fit values of the various observables are listed in the last column of the lower table. 3.4.9. Flavor Violating decays with axions The models we have constructed in this section contain no symmetry to forbid flavor changing neutral currents. A consequence of this is the presence of flavor violating axion couplings. These couplings lead to quark decays with final state axions qi ! qja. The observables related to this processes are meson decays into final states with axions. This is a generic feature of FNPQ models. Starting from eqs. 3.74 and 3.75 the Yukawa couplings of s are Ls = yu ijujLsuiR + yu0 ij ujLs⇤uiR + yd ijdjLsdiR + yd0 ij djLs⇤diR , (3.109) 66 Flavor Symmetric models of neutrino masses and dark matter where the Yukawa coupling matrices y u(0) ij and y d(0) ij are yu/d = 1 vs 0 BBB@ 0 Au/d 0 Bu/d 0 0 0 0 0 1 CCCA , yu0/d0 = 1 vs 0 BBB@ 0 0 0 0 0 Cu/d 0 Du/d 0 1 CCCA . (3.110) We rotate to the physical quark mass basis, Lsq0 = yu ijU † ikLu0 kLsUjlRu0 lR + yu0 ij U† ikLu0 kLs⇤UjlRu0 lR + yd ijV † ikLd0kLsVjlRd0lR + yd0 ij V† ikLd0kLs⇤VjlRd0lR , (3.111) where u0/d0 denote quark mass eigenstates, UL and UR diagonalize the up quark mass matrix and VL and VR diagonalize the down quark mass matrix, respectively. By defining lu(0) = U† Lyu(0)UR and ld(0) = V† L yd(0)VR we may write Lsq0 = lu iju 0 iLsu0 jR + lu0 ij u0 iLs⇤u0 jR + ld ijd 0 iLsd0jR + ld0 ij d0iLs⇤d0jR . (3.112) By identifying the axion as the pseudoscalar part of s, which is consequence of the vev hierarchy imposed on the singlets, we have Laq0 = ia(eu iju 0 iu 0 j + ed ijd 0 id 0 j + e0uij u0 ig5u0 j + e0dij d0ig5d0j) , (3.113) where eu,d ij = (lij l† ij)/2 and e0u,d ij = (lij + l† ij)/2 . This Lagrangian is what was needed to calculate meson decay rates to final states with axions. A sensitive test of neutral flavor violation with axions is the search for the K+ ! p+a process. The decay ratio for the Kaon decay to axion and pion is given by Γ(K+ ! p+a) ⇡ mK 64p |ed 21| 2B2 S 1 m2 p m2 K ! , (3.114) where BS is a non-perturbative parameter BS ⇠ 4.6 [185]. Using the results of the c2 minimization, as shown in Table 3.7, in Eqs. 3.111 - 3.112 to calculate |ed 21| 2 we obtain Γ(K+ ! p+a) ⇡ 1.9⇥ 109GeV3 v2 s . (3.115) Flavor Symmetric models of neutrino masses and dark matter 67 The E949 collaboration [186] set the limit on the branching ratio BR(K+ ! p+a) = Γ(K+ ! p+a) ΓTotal(K +) < 7.3⇥ 1011 . (3.116) With this result we obtain the constraint vs > 2.5⇥ 1010 GeV. In terms of the axion decay constant, we have fa > 7⇥ 109 GeV , (3.117) using that in our framework N = 5. Another channel where axions can mediate a FCNC is in B+ ! K+a decay, where the strange-bottom-axion coupling is probed. The decay width of this process is given by Γ(B+ ! K+a) ⇡ mB 64p |ed 32| 2( f K 0 (0)) 2 ✓ mB mb ms ◆2 1 m2 K m2 B !3 , (3.118) with f K 0 (0)⇠ 0.33 [187]. The experiment Belle-II may constrain the branching ratio of this process to BR(B+ ! K+a) < 106 108 [158], which would lead to vs > 1.8⇥ (107 108) GeV . (3.119) In our framework, vs ⇡ p 2N fa, the bound above translates to fa > 6⇥ (106 107) GeV. (3.120) The relationships between decay constant, mass and photon coupling of the axion in our model set a limit for the last two observables. With the stronger constraint coming from the charged kaon decay limit we can write ma < 0.7⇥ 103eV , |gag(GeV1)| < 0.8⇥ 1014 (3.121) These bounds are stronger than astrophysical limits, for example see figure-1 of [180]. 3.4.10. Flavor Violating Higgs couplings In theories with more than one Higgs doublet a symmetry must be introduced to forbid the presence of FCNC mediated by scalars. By flavoring the PQ symmetry, we 68 Flavor Symmetric models of neutrino masses and dark matter have not taken care to forbid this phenomenon. In fact, the models depend crucially on two Higgs doublets coupling to both up and down quarks to create the quark mass operators. From this consideration we can conclude that flavored DFSZ axion models will generically contain problematic scalar FCNC, unless the Higgs PQ charges are carefully curated, or an additional symmetry is invoked to forbid them. Scalar FCNCs are experimentally tightly constrained, through numerous processes such as KL ! µ µ + or top decays such as t ! hc, hu [188], to name a few. The nonobservation of these processes can be used to set constraints on the flavor violating couplings of scalars and the mass of any BSM scalar mediating them. In the context of flavored axion models, the high PQ breaking scale introduces a decoupling of the singlet scalar degrees of freedom from the Higgs doublets neutral components. At leading order we can assume h ⇡ hu 0 cos a + hd 0 sin a , H ⇡ hu 0 sin a + hd 0 cos a (3.122) with h being the 125 GeV SM-like Higgs boson and H a heavy neutral CP-even scalar. The couplings of these two particles to SM fermions may be obtained from the effective Lagrangian and read as follows L Cu 1 vu uLuRhu 0 + Cu 2 vd uLuRhd 0 + Cd 1 vd dLdRhd 0 + Cd 2 vu dLdRhu 0 , (3.123) where the matrices Cu/d i are given by Cu/d 1 = 0 BBB@ 0 Au/d 0 Bu/d 0 0 0 0 Eu/d 1 CCCA , Cu/d 2 = 0 BBB@ 0 0 0 0 0 Cu/d 0 Du/d 0 1 CCCA . (3.124) In terms of the physical scalar and quark fields we have L hu0 Lu0 R C0u 1 vu cos a C0u 2 vd sin a ! + Hu0 Lu0 R C0u 1 vu sin a + C0u 2 vd cos a ! + (u ! d) , (3.125) where the matrices C0u/d i are defined as C0u i = U† LCu i UR , C0d i = V† L Cd i VR . (3.126) Flavor Symmetric models of neutrino masses and dark matter 69 Figure 3.10.: Exclusion region plot in the (tan a tan b) plane obtained from the non-observation of the t ! hc flavor violating decay. The gray colored region is excluded by ATLAS data [189], the purple colored region is expected to be probed in the future by the HL-LHC experiment [190] and the orange region will be further probed by ILC or CLIC [191]. The uncolored region (see white thin band) predicts a branching ratio beyond the sensitivity of these experiments. The dashed line indicates limit of no flavor violation in light Higgs Yukawa couplings. Generally, the Yukawa couplings of the scalars are not proportional to the quark mass matrices. Nevertheless, there are limits when this happens. For example for vu/vd = cot a (3.127) the Yukawa coupling matrices of the SM-like Higgs h becomes proportional to the mass matrices. When this happens, this Yukawa couplings are guaranteed to be diagonal in the physical quark basis, making FCNCs mediated by this scalar vanish. We call this the light Higgs flavor conserving limit. Note that these two parameters vu/vd and a can be moved independently, and therefore are not a prediction of our model. Any deviation of a from the flavor conserving limit will introduce flavor violating couplings of the light Higgs, which can be probed with observables such as t ! hc in the up-sector and h ! bs in the down sector. The t ! hc decay channel currently has 70 Flavor Symmetric models of neutrino masses and dark matter an upper bound set by ATLAS [189] BR(t ! hc)LHC < 1.1⇥ 103 , (3.128) while future experiments HL-LHC [190], ILC and CLIC [191] project the following sensitivities to this process BR(t ! hc)HLLHC < 2⇥ 104 , (3.129) BR(t ! hc)ILC/CLIC < 105 . (3.130) The branching ratio for this process can be obtained from Eq. 3.125 as [192] Γt!hc = C2 tcmt 16p rh 1 (Rc Rh) 2 i h 1 (Rc + Rh) 2 i h (Rc + 1)2 R2 h i , (3.131) where the coupling Ctc is defined as Ctc = h (C0u 1 )23 + (C0u 1 )32 i cos a vSM sin b h (C0u 2 )23 + (C0u 2 )32 i sin a vSM cos b , (3.132) mt is the top quark mass, Rh is the higgs to top mass ratio Rh = mh/mt, and Rc is the charm to top mass ratio Rc = mc/mt. Using the experimental value for the total width of the top quark [183] we derive the constraints on the free parameters tan a and tan b as shown in Fig. 3.10. We use the best fit point given in Table 3.7, to obtain a numerical value of Eq. 3.131 and we derive the approximate constraint cos a sin b (1 tan a tan b)  17 Γ Exp t!hc [GeV] , (3.133) for a given experimental input of the decay width Γ Exp t!hc. We see from Fig. 3.10 that, as expected from Eq. 3.133, small values for tan b allow only large values of tan a and vice versa. It can also be seen that ATLAS data has already ruled out a large portion of the parameter space (gray-region) and HL-LHC (purple-region) and CLIC (orange-region) will leave only a small region around the tan b = cot a limit unprobed (see Fig. 3.10 caption for details). Finally, we would like to mention that the t ! hu and h ! cu decays can also place constraints on a and b, however we find these to be numerically much weaker than the constraints from t ! hc, given that the hierarchy in the Cu i (see Flavor Symmetric models of neutrino masses and dark matter 71 Eq. 3.124) matrices is preserved in the physical C0u i matrix (see Eq. 3.126). Moreover, currently there are no experimental constraints on h ! bs decays, although they are phenomenologically interesting in the context of two Higgs Doublet models [193]. 3.4.11. Flavored Axion as Dark Matter candidate The axion can be a good dark matter candidate, provided a sufficient amount of them was produced in the early universe. There are several production mechanisms for axions. The relic density produced by the misalignment mechanism is [194, 195] Ωah2 ⇡ 2⇥ 104 ✓ fa 1016GeV ◆7/6 hq2 a,ii , (3.134) where qa,i is the initial misalignment angle of the cosmological axion field and it lies in the range (0, 2p). Now, we notice that for the axion breaking scale 5⇥ 1010 < fa < 1⇥ 1015 (GeV), one can match the axion relic density to the observed dark matter relic abundance ΩDMh2 ⇠ 0.12 for 0 < qa,i < 2p. It is worthwhile to mention that the N > 1 prediction of DFSZ models induce the formation of stable domain walls in the universe. The existence of these stable domain walls is incompatible with the standard cosmology [196]. One way to avoid the effect of domain walls on the observed universe is to embed this type of models in a cosmological model where inflation happens after the formation of these walls, thereby inflating away the density of the walls. Another possible resolution of the domain wall problem is to destabilize the walls with a dynamical breaking of the PQ symmetry [197, 198]. 3.4.12. Conclusion In this section UV-complete models for flavored axion with a Froggatt-Nielsen sym- metry were constructed. We performed an analysis of the UV-model prediction for the Wilson coefficients of the Froggatt-Nielsen effective operators, as well as the pre- dictions of the Froggatt-Nielsen-Peccei-Quinn symmetric model for fermion masses, axion phenomenology and axion and Higgs FCNC. We have shown the viability of such models, in the fermion mixing observables, QCD axion coupling and quark flavor violating phenomenology. We have also commented on the possibility of the detection of these types of models from meson decays, axion searches and Higgs phenomenology in colliders. 72 Chapter 4. Dark and Lepton symmetries 4.1. Lepton number and darkness In this chapter, we will study models with a global or gauged U(1) symmetry, under which leptons are charged with a flavor universal coupling. The breaking of the symmetry will give rise to dark matter candidates, either fermions charged under this symmetry or the Goldstone boson of the symmetry breaking. We have studied similar setups in the preceding chapter, where in the gauged U(1)X models we introduce a dark matter candidate which only couples to the new gauge boson, or in the U(1)FNPQ model, where the flavored QCD axion is the dark matter candidate. Here, we focus on the dark matter phenomenology and its interconnectedness with the neutrino mass generation mechanism that is obtained through the symmetry breaking. 4.2. Neutrino masses and dark matter from the breaking of a global U(1) Lepton symmetry with a dark Z2 The Scotogenic model [199] is one of the most popular and well-studied scenarios where dark matter and neutrino masses arise in a unified mechanism. In this proposal, the right handed neutrinos transforming as singlets of the SM gauge group transform are part of the dark sector. A dark SU(2)L Higgs doublet is introduced, which does not participate in spontaneous symmetry breaking. The right handed neutrinos form Yukawa couplings with the SM lepton doublets and the dark Higgs doublet. The Z2 dark symmetry is left unbroken, and right handed neutrinos cannot mix with the 73 74 Dark and Lepton symmetries active neutrinos. Neutrino masses arise from one-loop contributions, with the dark sector particles running in the loop. The lightest of the dark sector fields is a dark matter candidate, it can be either a fermion or a scalar. Dark matter can be produced thermally in the early Universe, and the model provides a testable direct detection cross section at the tree or one loop level, depending on which field is the dark matter candidate. Light neutrino masses of sub-eV mass can be obtained with O(1) Yukawa couplings and dark sector particles of TeV scale mass or heavier, thanks to the loop suppression. When the dark matter candidate is a fermion, it has been noted that the couplings that determine the freezeout relic density also lead to charged Lepton Flavor Violating (cLFV) process like µ ! eg or µN ! eN [200, 201]. Current bounds on cLFV observables exclude a large part of the model parameter space where dark matter is not overabundant. In this section1, we define a model where the Scotogenic framework is extended by a global continuous symmetry corresponding to lepton number. This symmetry forbids the right handed neutrino terms, which are necessary to construct the neutrino mass loop. This symmetry is broken by a singlet complex scalar field with lepton number L(f) = 2. The singlet field couples to right handed neutrinos, giving them mass terms after lepton symmetry breaking. The singlet fields also possesses quartic couplings with the neutral part of the SM Higgs doublet. After EW symmetry breaking, these two fields mix, creating a new channel for fermion DM annihilation in the early Universe. The purpose of this extension of the Scotogenic model is to allow fermion dark matter with the correct relic density in the model with smaller Yukawa couplings. The new scalar-mediated s-channel of dark matter annihilation also results in a t-channel for dark matter detection. This enhances the direct detection cross section from a loop suppressed process to a tree level process. It is necessary to investigate whether this new scalar interaction can lead to the correct relic density, while avoiding current direct detection constraints. We also need to address the presence of a Goldstone boson, product of the spontaneous breaking of a global continuous symmetry. 4.2.1. The model The extension of the SM we will analyze in this section consists of a scalar singlet s, a SU(2)L scalar doublet h with hypercharge 1/2, and three generations of Majorana 1This section presents the results of [1] Dark and Lepton symmetries 75 Li `Ri Φ h Ni s SU(2)L 2 1 2 2 1 1 U(1)Y 1/2 1 1/2 1/2 0 0 U(1)L 1 1 0 0 1 2 Z2 + + + + Table 4.1.: Particle content and charge assignments of the model. fermions Ni (with i = 1, 2, 3). In addition to the SM gauge symmetry, we consider the Z2 symmetry of the Scotogenic, where only the N and h fields transform as 1, and we also impose the SM lepton number symmetry as global U(1)L symmetry. Under this symmetry, N is charged with lepton number 1, h with 0, and f as 2. The particle content and charge assignments of the model are shown in Table 4.1. The renormalizable lepton Lagrangian invariant under the gauge symmetry and U(1)L ⇥ Z2 is LY Y` ij L̄iΦ`Rj + Yn ij L̄ih̃Nj + 1 2 YN ij sN̄c i Nj + h.c., (4.1) where h̃ = it2h⇤, Li = (nLi , `Li )T with i, j = e, µ and t. We parametrize the compo- nents of the scalar fields as Φ = 0 @f+ f0 1 A and h = 0 @h+ h0 1 A . (4.2) The scalar potential V of the model is V = µ 2 1Φ † Φ + µ 2 2h†h + µ 2 3s⇤s + l1(Φ † Φ)2 + l2(h †h)2 + l3(h †h)(Φ† Φ) +l4(h † Φ)(Φ†h) + l5 2 h (h† Φ)2 + (Φ†h)2 i + l6(s ⇤s)2 (4.3) +l7(s ⇤s)(Φ† Φ) + l8(s ⇤s)(h†h). We define the vevs of the s and Φ fields by shifting the fields in the following manner s = vsp 2 + R1 + i I1p 2 and f0 = vΦp 2 + R2 + i I2p 2 , (4.4) 76 Dark and Lepton symmetries where va (with a = s, Φ) are the vacuum expectation values and vΦ = 246 GeV, Rj and Ij (with j = 1, 2) represent the CP-even and CP-odd parts of the fields. 4.2.2. Mass spectrum The scalar potential of eq. 4.3 determines the scalar mass spectrum. The neutral CP-even mass matrix M2 R and CP-odd mass matrix M2 I are obtained from the second derivative of the potential, evaluated at its minimum. In the CP-odd sector, there are two degrees of freedom. One corresponds to the Z-boson longitudinal mode. The remaining one is a physical Nambu-Goldstone boson J, expected to be present from the spontaneous breaking of the global Lepton number by s. We identify them simply as J ⌘ I1, G0 ⌘ I2 . (4.5) The spontaneous breaking of Lepton number by two units leads to Majorana neutrinos. Because of this, the Goldstone boson of Lepton number breaking is commonly called the Majoron [202, 203]. The Z2 dark symmetry is not broken by any mechanism, so the components of h do not mix with the Z2 even sector. In the CP-even sector there are two mass eigenstates, formed from the mixing of hi. The scalar mixing matrix is defined as OR, and is a real, orthogonal matrix, parametrized by a single mixing angle a: 0 B@ h1 h2 1 CA = OR 0 B@ R1 R2 1 CA ⌘ 0 B@ cos a sin a sin a cos a 1 CA 0 B@ R1 R2 1 CA . (4.6) The scalar mixing matrix diagonalizes the mass matrix M2 R OR M2 R OT R = diag(m2 h1 , m2 h2 ) . (4.7) from the scalar potential, we can write down the eigenvalues of M2 R m2 (h1,h2) = ⇣ l1v2 Φ + l6v2 s ⌘ ⌥ q l2 7v2 Φv2 s + (l1v2 Φ l6v2 s) 2, (4.8) where the “” (“+”) sign corresponds to h1 (h2). One of the scalar CP-even mass eigenstates needs to be identified with the observed SM-like fundamental scalar with Dark and Lepton symmetries 77 νLi νLjNk hΦi hΦi Nk ηR,I ηR,I hσi Figure 4.1.: One-loop Feynman diagram for neutrino mass generation. 125.09 GeV mass [204]. In the dark sector, the neutral CP-even and CP-odd masses are m2 (hR,hI) = µ 2 2 + l8 2 v2 s + l3 + l4 ± l5 2 v2 Φ. (4.9) The mass of the dark sector charged scalar field is given by, m2 h ± = µ 2 2 + l3 2 v2 Φ + l8 2 v2 s. (4.10) The squared mass splitting of dark neutral scalars is l5v2 Φ = (m2 hR m2 hI ) . (4.11) The N fields acquire the following mass terms with the breaking of U(1)L by s (mN)ij = p 2YN ij vs. (4.12) The Feynman diagram for one-loop neutrino mass generation is depicted in Fig. 4.1. The resulting light neutrino mass matrix is given by [199, 205] 78 Dark and Lepton symmetries (Mn)ij = 3 ∑ k=1 Yn ik Yn kjmNk 32p2 " m2 hR m2 hR m2 Nk log m2 hR m2 Nk m2 hI m2 hI m2 Nk log m2 hI m2 Nk # . (4.13) 4.2.3. Summary of constraints In this subsection we list the theoretical and experimental constraints we consider on the model. Boundedness and perturbativity conditions We limit ourselves to analyze this model as a perturbative theory, which restricts the size of dimensionless couplings |li|, |Y a jk| 2  4p with i = 1, ..., 8; j, k = 1, 2, 3 and a = n, `, N. (4.14) To ensure the stability of the scalar potential, the conditions to be fulfilled by the scalar couplings are [206] l1, l2, l6 0, l3 2 p l1l2, 4l1 l6 l2 7, 4l2 l6 l2 8 and l3 + l4 |l5| 2 p l1l2. (4.15) 4.2.4. Searches of new physics The physical spectrum of the scalar sector consists of three CP-even neutral bosons hi (i = 1, 2) and hR, two CP-odd neutral bosons hI and the Majoron J; and one charged scalar h ± . The Majoron is theoretically expected to be light (sub-MeV), but not massless due to nonperturbative effects. In this work, we consider the Majoron to be massless, because we will only study processes where other involved particles are significantly heavier than the Majoron. The scalar mass spectrum of the model is constrained by collider searches of scalars with gauge couplings and invisible decay of the SM Higgs [207, 208]. We set the experimental limit on invisible Higgs decay from the h ! J J and h ! N1N1 channels following the data in [208] Binv ⌘ BR(h! invisible) < 0.28 at 95% C.L. (4.16) Dark and Lepton symmetries 79 LEP studies on the W and Z decays can be used to constrain the masses of the Inert Higgses hR(hI) and h ± [209] mhR + mhI > mZ, mh ± > mZ/2 and mh ± + mh(R,I) > mW . (4.17) LEP results also constrain mass splittings among these scalars mhI mhR > 8 GeV if mhR < 80 GeV and mhI < 100 GeV. (4.18) The oblique parameters S, T and U [210, 211] are also sensitive to new physics, we calculate the one loop contributions of the new scalars to these parameters and limit them to lie in the experimentally alowed region [208] S = 0.02± 0.10, T = 0.07± 0.12 and U = 0.0± 0.09. (4.19) 4.2.5. Dark matter searches The objective of the model is to reconcile dark matter observables with cLFV predic- tions. Therefore, the model is strictly constrained by dark matter observables. We restrict the parameters of the model to reproduce the cosmological abundance of dark matter eq. (??). The new DM annihilation channel provides a new DM direct and indirect detection channels. For the constraints on the direct detection cross section, we consider the limits from the XENON1T experiment [212]. For the indirect detection bounds we take into account the limits on the indirect detection cross section hsvig from the analysis of Fermi-LAT satellite data [213]. 4.2.6. Neutrino oscillation parameters The Yukawa couplings of the Inert Higgs to the Lepton doublets determine the entries of the neutrino mass matrix, see eq. 4.13. The diagonalization of this matrix determines the lepton mixing matrix UL ⌘ U† `Un = UL(q12, q13, q23, dCP) (4.20) where U` diagonalizes the charged lepton mass matrix and Un diagonalizes the neu- trino mass matrix. The relation between Mn and the diagonal neutrino mass matrix is 80 Dark and Lepton symmetries given by, Mn = U⇤ n diag(mn1 , mn2 , mn3 )U† n (4.21) where mni are the neutrino masses. We fit the mass matrix to reproduce the results of the global fit of oscillation parameters given in [214]. We consider the results for the Normal Ordering (NO) of neutrino masses: |∆m2 sol| = 7.55+0.20 0.16 ⇥ 105 eV2, |∆m2 atm| = 2.50± 0.03 ⇥ 103 eV2, q12/ = 34.5+1.2 1.0, q13/ = 8.45+0.16 0.14, q23/ = 47.7+1.2 1.7, and dCP/ = 218+38 27.(4 22) 4.2.7. Numerical analysis For the numerical study, we will consider the DM candidate to be fermionic. Therefore all dark sector masses will be taken heavier than the N1 mass. In this model, DM annihilates via the t- and s- channels depicted in Fig. 4.2. When the t-channel is the only source of DM annihilation, it has been shown that the large Yukawa couplings required to avoid DM over-abundance can lead to excluded cLFV cross sections [200,201]. In this work we show that the contribution of the s-channel can keep the Yukawa couplings of the inert Higgs doublet suppressed, leading to experimentally acceptable cLFV rates while keeping DM abundance on or under the observed value. To achieve this, the scalar singlet-doublet mixing must be made large enough to account for DM relic density, while also controlling the Higgs observables related to it. The numerical analysis of the dark sector was performed using the MicrOMEGAS code [121]. For the dimensionless parameters in the scalar sector we took the following intervals: 106  |l2,...,8|  1, (4.23) and l1 being determined by the observed SM Higgs mass. We are taking h1 as the SM Higgs with mh1 = 125 GeV. On the other hand, we scan over the mass range of h2 within mh2 2 [20, 2000] GeV. For the second Higgs we take masses below the SM Higgs. This is not in contradiction with collider data, as this scalar is composed of doublet and singlet fields. The limit on the doublet-singlet mixing we consider is sin a < 0.2 [215]. For the masses of the inert scalars we considered the following ranges, mhR 2 [110, 5000] GeV, mh ± 2 [135, 5000] GeV, (4.24) Dark and Lepton symmetries 81 Ni Ni ⌥L , L ±L , L ±, R,I Ni Ni h2, J h2, J Ni Ni Ni h2 h1 SM SM Ni Ni h2 h1, h2 h1, J h1, J Figure 4.2.: Feynman diagrams for the annihilation channels of the fermion dark matter in the model. On the left annihilation into SM particles. On the right annihilation into Majorons and Higgses. and the mass of the CP-odd part hI is determined by using the relation l5v2 Φ = (m2 hR m2 hI ). For the lepton number breaking scale, namely the singlet’s vev vs, we have used vs 2 [500, 10000] GeV. Bear in mind that this vev provides the mass of the heavy Majorana fermions, Ni, whose masses (taken to be diagonal) are varied in the following ranges, mN1 2 [8, 1000] GeV and mN2,3 2 [100, 5000] GeV. (4.25) Since N1 is the DM candidate of the theory we impose mN1 < mN(2,3) < mh(R,I,± ) . These choices are made so that the constraints described in Sec. 4.2.3 are satisfied. Using the one loop results in [216,217] to calculate the S and T oblique parameters, taking U = 0. We proceed to compare these values to the correlated limits for S and T from [208]. We constructed the neutrino mass matrix in eq. 4.13, with experimentally allowed lepton mixing parameters and neutrino masses. The lepton mixing parameters are taken from the global oscillation data fit [214]. Viable neutrino masses are calculated by using the squared mass differences measured in oscillation experiments and constraining the neutrino mass scale with the neutrino mass sum bound from cosmological data [42, 218]. Finally, we take into account the strong limits on the production of thermal 82 Dark and Lepton symmetries Figure 4.3.: Dark matter mass sSI plane showing the solutions in the model that satisfy all theoretical and experimental constraints given in Section 4.2.3. The latest bound on direct dark matter detection is set by the XENON1T experiment [212] (top shaded area). The dashed lines represent the expected sensitivities in forthcoming experimental searches such as XENONnT [220], LUX-ZEPLIN (LZ) [221], DarkSide 20k [222], DARWIN [223] and PandaX-4T [224]. Majorons from Dark Matter annihilation 2, by demanding that the cross section of DM annihilation into Majorons is always a subdominant component of the total cross section. In a more recent work [219], the phenomenology of the Majoron in this model was expanded upon, including an analysis of cLFV decays with a Majoron. 4.2.8. Viable dark matter mass regions The points of the numerical scan that fulfill the constraints of the previous section (Sec. 4.2.3) are shown in the (mN1 , sSI) space in Fig. 4.3. There are three DM mass regions where DM is not overabundant : • The light DM region, where 8 GeV . mN1 . 20 GeV and bb̄ is the dominant annihilation channel; • The resonant DM region, where mN1 . mh/2 (with mh = 125.09 GeV); and 2These bounds come from the limit on the relativistic degrees of freedom in the early Universe Ne f f . A light Majoron population, thermally produced during dark matter freezeout would contribute to this parameter. Dark and Lepton symmetries 83 Figure 4.4.: Predictions for the velocity averaged cross section of dark matter annihilation into gamma rays hsvig as function of the dark matter mass mN1 . The dashed line represent the limit set by Fermi-LAT satellite results [213]. • The heavy DM region, for DM masses above 80 GeV where the fermion DM annihilates efficiently into the gauge bosons, i.e. N1N1 !VV with V = (Z, W). In Fig. 4.3 we show in red points where N1 accounts for 100% of the observed dark matter relic density, and in purple and pink points where it only makes up a fraction of the relic density. We also show in 4.3 the limits from the XENON1T experiment on the direct detection cross section [212], as well as the neutrino floor on Xenon. Notice that some points lie below the neutrino floor. Fig. 4.3 also displays the future sensitivities for dark matter searches in direct detection experiments such as XENONnT [220] and LUX-ZEPLIN [221], DarkSide-20k [222], DARWIN [223], and PandaX-4T [224]. For completeness we provide three benchmarks in Appendix A.4 within each mass neighbourhood and their corresponding outputs. In Fig. 4.4, the model predictions for the dark matter velocity averaged annihilation cross section are depicted. This cross section is constrained by indirect detection gamma ray detectors. The results from Fermi-LAT satellite [213] are shown alongside model predictions in the Figure. The model predicts that a portion of the heavy DM points have cross sections above the experimental limit. This is to be expected, as the model independent analysis of the data [213] excludes heavy DM annihilating solely to W boson pairs. On the other hand, the light DM region is largely unconstrained by this data. 84 Dark and Lepton symmetries 4.2.9. Conclusions In this section we have analyzed an extension of the Scotogenic model, where lepton number is a global symmetry spontaneously broken by a singlet scalar. This provides a dynamical mechanism for neutrino mass generation, which also opens up a new channel for DM annihilation. We have seen that this new channel relaxes the tension between dark matter and flavor observables. We identified three regions where DM relic density can be fully explained by the model, without saturating bounds from Higgs decay, dark matter direct and indirect detection, lepton flavor violating decays and collider limits on charged scalars and EW parameters. This model predicts interesting Majoron phenomenology. Particularly the effect of the DM annihilating into Majorons on cosmological observables is a phenomenon that is yet to be explored and could result in constraints on the Lepton breaking scale and the N1 Yukawa couplings. 4.3. Global U(1)PQ symmetry as the source of neutrino Diracness In this section3, we analyze a framework, where Neutrino masses are generated dynamically from the breaking of the Peccei-Quinn symmetry. The breaking of the continuous symmetry leaves a remnant discrete symmetry that guarantees the Dirac nature of neutrinos. The PQ symmetry solves the strong CP problem, simultaneously explaining the mass scale of neutrinos and determining a Dirac nature of neutrinos, with the adequate field content. The purpose of this work is to show explicit UV constructions of this mechanism. We show that in these constructions there is also the possibility that a WIMP dark matter candidate arises in the spectrum. This WIMP can be part or the whole of dark matter, complementing the relic axion density. The starting point of these models is the Peccei-Quinn solution to the strong CP problem [57], which is invoked to explain the smallness of CP violation in strong interactions in a dynamical way. This smallness is deduced from the nonobservation of a neutron Electric Dipole Moment (nEDM) [225]. This solution implies the existence of a pseudo-Nambu Goldstone boson, the axion [58, 145], which can be a component of the dark matter of the Universe [146–149], when a suitable mechanism for its production is considered. In QCD axion models, SM leptons can be charged under 3This section presents the results from [5]. Dark and Lepton symmetries 85 the PQ symmetry. The connection between the breaking of the PQ symmetry and the generation of neutrino masses was proposed early on, with models breaking lepton number by two [169, 226, 227]. More recently, it was shown that the breaking of PQ symmetry can lead to Dirac neutrinos [228], with a tree-level model for neutrino masses. In this work, the same prescription is followed, showing more tree level and one-loop level possibilities. These ideas are not entirely new, in [229, 230] several models of Dirac neutrinos linked to the PQ symmetry were presented. In this work, however, the symmetry approach allows us to explicitly show that Majorana mass terms are forbidden to any perturbation order, and to see how an extension of this implementations can keep this feature unmolested. 4.3.1. Dirac Neutrinos from PQ symmetry Consider the framework set forth in [228]. The matter content of this framework is shown in Table 4.2. With the fields and PQ symmetry of the model, ∆L = 2 operators cannot be constructed at any order of perturbation theory. This guarantees that Majorana mass terms for neutrinos cannot arise in this theory. Adding a sterile right handed neutrino (RHN) can lead to the generation of a Dirac neutrino mass term. With a RHN PQ charge of 1, the Yukawa Lagrangian of such theory would be LY = yu ijQ̄iHuuj + yu ijQ̄iHddj + yl ij L̄iHdlj + yn ij L̄iHunRj + h.c. . (4.26) This leads to Dirac neutrinos of acceptable mass scale if the Yukawa couplings yn ij are O(1012), if the direct limit of neutrino masses is followed Aker:2019uuj. On the other hand, a RHN with 5 PQ charge will forbid the dimension-4 direct Yukawa coupling. At higher order, one may write the effective coupling LD dim 5 = yn ij L̄iHunRj s ΛUV + h.c. , (4.27) where s is a complex scalar singlet, as defined in 4.2, and can be parametrized as s(x) = 1p 2 (r(x) + fa) eia(x)/ fa . (4.28) Following the phenomenological considerations on the PQ breaking scale in DFSZ models as explained in Sec. 3.4, we can identify a(x) as the QCD axion [58, 145], fa as the PQ breaking scale and r(x) as the radial part that will gain a mass of order of the 86 Dark and Lepton symmetries Fields/Symmetry Qi ui di Li li Hu Hd s SU(2)L ⇥U(1)Y (2,1/6) (1,2/3) (1,-1/3) (2,-1/2) (1,-1) (2,-1/2) (2,1/2) (0,0) U(1)PQ 1 -1 -1 1 -1 2 2 4 Table 4.2.: Quantum numbers in the DFSZ axion model [228]. PQ symmetry breaking scale. In [228], a UV completion of the operator in eq. 4.27 was presented. This completion is an implementation of the Dirac type-I seesaw. In this work we present alternatives to this UV completion. Type-II Dirac seesaw Here we extend the matter content of Table 4.2 to construct a type-II Dirac seesaw UV completion of the operator in Eq. 4.27. This is achieved by adding a RHN with PQ charge 5 and a scalar SU(2)L doublet with PQ charge 6, as described in Table 4.3. The scalar potential of the model is given by V ⇠ kHuHds⇤ + l0HdΦus⇤2 , (4.29) where k has mass dimension 1 and l0 is dimensionless. We can explicitly check that Majorana mass terms cannot be generated at any order of perturbation theory. First we write Weinberg operator analogues for Majorana masses, with all the possible Dark and Lepton symmetries 87 combinations of doublets Hu, Hd, Φu, keeping track of the PQ charge of these operators Operator PQ charge Ldim 5 ⇠ 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>: LLH̃u H̃u ΛUV LLH̃u Hd ΛUV LLHd Hd ΛUV LLΦ̃uΦ̃u ΛUV LLH̃uΦ̃u ΛUV LLHdΦ̃u ΛUV 1 + 1 + (4) = 2 1 + 1 + (0) = +2 1 + 1 + (4) = +6 1 + 1 + (12) = 10 1 + 1 + (8) = 6 1 + 1 + (4) = 2 . (4.30) From Eq. (4.30) we have that Dimension-5 operators are forbidden by the PQ symmetry. These operators could be generated by higher order operators after the breaking of the PQ symmetry, however. To see that this cannot happen, notice that the EW singlet field combinations that could couple to the operators at higher dimension transform with PQ charge 4n sn (4n); (s⇤)n (4n); (HuHd) n (4n); (HuHd) ⇤n (4n); (H† u Hu) n (0); (H† d Hd) n (0); (H† uΦu) n (4n); (H† uΦu) ⇤n (4n); (HdΦu) n (8n); (HdΦu) ⇤n (8n); (Φ† uΦu) n (0); (4.31) The PQ charges QO all satisfy mod (QO, 4) = ± 2 , (4.32) 88 Dark and Lepton symmetries Hu(2) σ(4) L(1) νR(−5) Φu(6) Figure 4.5.: Feynman diagram for Dirac neutrino masses in Type-II DFSZ scenario. with which we conclude that any higher order operators that could be formed with gauge invariant combinations of scalars and lead to Majorana masses are also forbid- den by the PQ symmetry. With the particle content of Table 4.3, the relevant terms of the Lagrangian for neutrino mass generation are LY yn ijnLiΦunRj + µHuΦ † us + h.c. (4.33) Symmetry/Fields Li nRi Hu Hd s Φu SU(2)L ⇥U(1)Y (2, -1/2) (1, 1) (2, -1/2) (2, 1/2) (0, 0) (2, -1/2) U(1)PQ 1 -5 2 2 4 6 Table 4.3.: Fields content and transformation properties under PQ symmetry in type-II seesaw frame- work. The scalar spectrum of the theory is determined by the potential allowed by the gauge and global symmetries. Without writing out the full potential, we can analyze the necessary scalar phenomenology to obtain light neutrino masses. The scalar potential contains the quadratic operator µ 2 ΦΦ † uΦu; . (4.34) In the limit where the components of Φu decouple from the rest of scalars, the mass scale of µ 2 Φ determines the mass scale of these mass eigenstates. The scalar mixing can be parametrized by a unitary transformation fi = KijSj, (4.35) Dark and Lepton symmetries 89 where K is the unitary transformation matrix,fi are the interaction eigenstates and S is the set of mass eigenstates. The Φu decoupling limit is obtained when the components of this field have small mixing with other fields, and their mass is of scale much larger than its vev M2 Φu >> v2 u . The largest contribution to the mixing between Φu and other fields is the µ term of Eq. 4.33, the large vev of s can induce a large mixing between Hu and Φu. This mixing can raise the mass of one of the light eigenstates above the EW scale, excluding the possibility that the scalar which is predominantly Hu is the 125 GeV Higgs boson. At leading order the mixing between these fields goes as sin q ⇠ µ fa M2 Φu . (4.36) Therefore, the smallness of q demands µ fa << M2 Φu . This condition is similar to the fine-tuning of k in Eq. (4.29), needed to separate the PQ scale from the EW scale, as mentioned in [227]. The neutrino masses resulting from the breaking of the SU(2)L ⇥U(1)Y and U(1)PQ symmetries by the scalar vevs hHui = vu and hsi = fa are [231] (mn)ij = yn ij µvu fa M2 Φu ⇠ yn ijvu sin q , (4.37) where in the last term we have used Eq. 4.36. In the type-I Dirac seesaw scenario, as pointed out in [228], a large hierarchy among the PQ scale and the mediator scale is needed in order to explain the tiny neutrino masses. However, here the dependency is on the inverse squared mass. This suggests that a smaller mass hierarchy than in the type-I seesaw [228] may be allowed. As can be seen from Eq. (4.37), the smallness of neutrino mass is required by the smallness of the scalar mixing angle q. The measurement of the tritium beta decay spectrum at KATRIN [232] yields a direct limit for neutrino masses of mn < 1.1 eV at 90% C.L. , while the indirect limit from Cosmological measurements [42, 218, 233] constrains them further to ∑ mn < 0.12 eV (at 95 % confidence level using TT, TE, EE + lowE + lensing + BAO). The bounds of tritium beta decay and cosmology are translated into bands for the allowed scales for fa and Mf as shown in Fig. 4.6. 90 Dark and Lepton symmetries μ y = 1 G eV μ y = 1 M eV μ y = 1 keV 10 8. 10 9. 10 10. 10 11. 10 12. 10 13. 10 14. 10 8. 10 9. 10 10. 10 11. 10 12. 10 13. 10 14. MΦ (GeV) f a (G e V ) KATRIN (2019) μ y = 1 G eV μ y = 1 M eV μ y = 1 keV 10 8. 10 9. 10 10. 10 11. 10 12. 10 13. 10 14. 10 8. 10 9. 10 10. 10 11. 10 12. 10 13. 10 14. MΦ (GeV) f a (G e V ) Planck (2018) Figure 4.6.: Exclusion region plots (colored regions are excluded) in (MΦu fa) plane for a type-II Dirac seesaw mechanism. Three benchmark values for µy = 1GeV, 1MeV, 1keV have been adopted, respectively. The plots are presented by using the limits on neutrino mass from KATRIN Aker:2019uuj which gives mn < 1.1 eV at 90% C.L. (left panel) and Planck [42] ∑ mn < 0.12 eV (at 95 % confidence level using TT, TE, EE + lowE + lensing + BAO) (right panel). Dark and Lepton symmetries 91 One-loop Dirac seesaw One may also obtain small Dirac neutrino masses from the effective operator of Eq. 4.27 from loop level mechanisms. In [112] all the possible one-loop topologies resulting in this operator were presented. The implementation presented here represents the most economical of these possibilities. The field content of this model is listed in Table 4.4, along with their gauge and global transformation properties. The PQ charge assignment and symmetry breaking results in a low energy remnant Z2 symmetry. Under the remnant symmetry, SM fermions, hu and z are odd. The lightest field among the N and odd scalars is automatically stable, thanks to the combined effect of the remnant Z2 and Lorentz invariance [111]. This makes the lightest of these fields, which we take to conform to the WIMP paradigm, a Dark Matter candidate, alongside the QCD axion. In the multicomponent Dark Matter scenario, the DM fraction Ωa composed by the axion and the fraction ΩWIMP composed by the stable WIMP, are such that [42] ΩCDMh2(= 0.12) (Ωa + ΩWIMP)h 2. (4.38) WIMP dark matter phenomenology, including its relic density, is determined by the parameters of the theory described thus far, assuming ΛCDM. Axion relic density is dependent on the cosmological scenario, as well as on the dynamics of the axion field in the early Universe. Axion relic density may be produced by the axion misalignment mechanism or from topological defects of the axion field [61, 234], the details of these mechanisms are beyond the scope of this work, but the model presented here has plenty of freedom to accommodate these scenarios. Now we write down the scalar Symmetry/Fields Li nRi Hu Hd NR NL s hu z SU(2)L ⇥U(1)Y (2, -1/2) (1, 1) (2, -1/2) (2, 1/2) (1, 0) (1, 0) (1, 0) (2, -1/2) (1, 0) U(1)PQ 1 -5 2 2 2 2 4 -1 7 ZPQ 2 -1 -1 +1 +1 +1 +1 +1 -1 -1 Table 4.4.: Field content and transformation properties under PQ symmetry in the alternative one-loop mechanism. 92 Dark and Lepton symmetries Hu(2) σ(4) L(1) NR(2) NL(2) νR(−5) ηu(−1) ζ(7) U(1)PQ Z2 Hu(+1) σ(+1) L(−1) NR(+1) NL(+1) νR(−1) ηu(−1) ζ(−1) Figure 4.7.: Feynman diagram for Dirac neutrino masses in alternative one-loop DFSZ scenario. Here left panel respects PQ charge assignment, whereas right panel respects the remnant ZPQ 2 charge assignment that arises due to PQ symmetry breaking. potential that is allowed by the SM ⇥U(1)PQ symmetries V = µ 2 uH† u Hu + lu(H† u Hu) 2 + µ 2 hu h† uhu + lhu (h† uhu) 2 (4.39) + µ 2 ss⇤s + ls(s ⇤s)2 + µ 2 zz⇤z + lz(z ⇤z)2 + l1 uhu (H† u Hu)(h † uhu) + l2 uhu (H† uhu)(h † uHu) + lus(H† u Hu)(s ⇤s) +luz(H† u Hu)(z ⇤z) + lhuz (h † uhu)(z ⇤z) + lhus (h† uhu)(s ⇤s) + lsz (s ⇤s)(z⇤z) +kHuHds⇤ + l1[Huh† uz⇤s + h.c.] . After gauge and global symmetry breaking, Hu acquires a vev vu. The scalars hu and z do not acquire vevs and mix only with each other, leading to the mass matrix M M = 0 @µ 2 hu + luhu v2 u + lhus f 2 a l1vu fa l1vu fa µ 2 z + luzv2 u + lsz f 2 a 1 A . (4.40) where fa is the PQ symmetry breaking scale set by s, and we used luhu = l1 uhu + l2 uhu . The neutrino Yukawa couplings are LY yn ijLihuNRj + MjkNRjNLk + yn0 ki NLknRiz + h.c. (4.41) The neutrino mass obtained from the one-loop diagram in Fig. 4.8 is given by [199, 235] mij n = 1 64p2 ∑ X=R,I l1vu fa m2 SX2 m2 SX1 ∑ k yiky0kjmNk 2 4F 0 @ m2 SX2 m2 Ni 1 A F 0 @ m2 SX1 m2 Ni 1 A 3 5 , (4.42) Dark and Lepton symmetries 93 where F(x) = x log(x)/(x 1) and SRi/Ii (for i = 1, 2) denote the CP even and odd mass scalar eigenstates, obtained from the hu z mixing. For the fermionic WIMP case, considering m2 S1 ⇠m2 S2 = m2 S >> l1 favu , and subsequently in the mNlight << mS limit, the neutrino mass matrix can be expressed as mij n ⇠ l1vu fa 32p2 ∑ k yiky0kj mNk m2 S . (4.43) We make a simple numerical estimation of the eigenvalues of eq. 4.40. We take vu ⇠ 100 GeV, fa ⇠ 1012 GeV and l1 = 107 in our calculation. We write down M as M = 0 @1024 + 105 ⇥ 104 104 ⇥ 1024 107 ⇥ 102 ⇥ 1012 ⇤ 1020 + 0.92⇥ 104 104 ⇥ 1024 1 A [GeV2] . (4.44) For simplicity, we assume lhus = lsz . We also consider µhu = 1012 GeV, µz = 1010 GeV together with luhu = 105, luz = 0.92 and lhus = 104. Now, considering these numerical values and using them in Eq. (4.43), one estimates O(1) eV active neutrino masses as follows: mij n ⇠ 1eV ✓ l1 107 ◆✓ vu 102GeV ◆✓ fa 1012GeV ◆✓ y 107 ◆ y0 0.82 !✓ mNk 130GeV ◆ (280GeV)2 m2 S ! . (4.45) This Dirac Scotogenic model can also have the problems with cLFV processes that the original Majorana model possesses (see Section 4.2 and the references on cLFV therein). In this model we can avoid the necessity of large cLFV Yukawa couplings for Dark Matter annihilation, by relying on the Dark Matter couplings to the right chirality of the Dirac neutrinos. This process is not possible in the Majorana case. Its thermally averaged cross section hsvi is given by [236] s ⇥ vrel = y04 32p2 m2 Nk (m2 S + m2 Nk )2 . (4.46) One can obtain the total DM relic density Ωh2 ' 0.12 [42] by setting y0 = 0.82, mS = 280 GeV, and mNk = 130 GeV. Alternatively, by setting the Yukawa coupling y0 ⇠ 1 lowers the relic density contribution of N to a quarter of the total dark matter relic density, ΩN ⇠ 0.4 ΩCDM. The remaining fraction of dark matter density may come 94 Dark and Lepton symmetries from the axion relic density. Therefore we find that this model can accommodate axions as a negligible, or dominant form of dark matter. If the lightest dark particle is a scalar, the neutrino mass matrix can be approximated as mij n ⇠ l1vu fa 32p2 ∑ k yiky0kj mNK " log m2 Nk m2 S ! 1 # . (4.47) We find O(1) eV order masses for neutrinos by setting l1 ⇠ 105, vu ⇠ 102 GeV, fa ⇠ 1012 GeV, y⇠ 105, y0 ⇠ 104, mN ⇠ 108 GeV and mS ⇠ 1 TeV. Here, the DM is a mixture of the neutral part of the electroweak doublet and a singlet, as in [237]. Two limiting cases are found in the small mixing regime, when DM is mostly composed of hu or of z. We parametrize the mixing of these scalars with the angle qX, which at leading order is given by sin 2qX ⇠ l1vu fa m2 SX2 m2 SX1 . (4.48) We can obtain a ⇠ TeV scale eigenstate from this mixing using the following assign- ment of parameters in Eq. (4.40): luhu = luz , lzs = 1014lhus, µhu = 1012 GeV, µz = 106 GeV together with luhu = 1 and lhus = 104. In the case where the WIMP candidate Slight ⇡ z is mostly a gauge singlet, relic density is obtained by the annihila- tion through scalar couplings [238, 239]. The annihilation channel that these couplings allow is SlightSlight ! SM SM, mediated by scalars. For example, the SlightSlight ! hh annihilation channel contribution to hsvi is given by [238] hsvihh = l2 uz 64pm2 S 1 m2 h m2 S !1/2 , (4.49) where h is the 125 GeV SM Higgs. The additional annihilation channels into the SM fermions and gauge bosons are also controlled by luz . A scalar coupling luz of order O(101) is required to account for ΩSh2 ⇠ 0.12 at a scalar WIMP mass mS of 1 TeV [240]. A scalar coupling luz ⇠ 0.2 4 yields a relic density of Slight approximately forty percent of the total dark matter density, leaving the axion as the dominant component. In the case where the WIMP candidate is mostly a doublet Slight ⇡ hu, its phenomenol- ogy is more similar to the Inert Higgs Doublet Model. Masses of O(1) TeV are 4While a luz ⇠ 0.2 coupling would saturate the direct detection limit from XENON1T [212] at ΩSlight = ΩCDM, the diminished contribution of S to the dark matter density relaxes this bound. Dark and Lepton symmetries 95 compatible with the relic density and direct detection constraints, using the quartic scalar couplings with the SM Higgs of order O(0.1) [241, 242]. Given the possibility of having mixed dark matter in the model, a major difference of this model with “pure" models is that the lower relic density of the WIMP DM candidate needed to acommodate the axion requires larger couplings to the SM. This results in a larger direct detection signals for their relatively smaller densities. For example, the DM direct detection experiments [243] constrain only the WIMP component of DM, while the axion detection experiments [244] constrain the axion component. On the other hand, the decreased abundance of the WIMP component of DM necessitates larger couplings for it to augment the annihilation cross section. This same couplings are involved in Lepton Flavor Violating (LFV) processes, such as the µ ! eg decay, which are strongly constrained. In this model, the additional interaction with nR may be exploited to enhance the annihilation rate while keeping the LFV inducing couplings low. For example, the µ ! eg branching ratio is given by [200] Br(µ ! eg) = 3aemBr(µ ! en̄enµ) 64pG2 Fm4 h+ ∑ k yeky⇤µkG 0 @M2 Nk m2 h+ 1 A 2 , (4.50) where GF is the Fermi constant, aem is the fine structure constant and h+ is the charged scalar from the hu doublet. The loop function G(x) is defined by G(x) = 1 6x + 3x2 + 2x3 6x2 ln x 6(1 x)4 . (4.51) In the fermionic WIMP case, using the parameters we have provided in Eq. (4.45) and assuming m+ h ⇠mS1 we obtain a branching ratio of the µ ! eg decay of order ⇠ 1033, well below the experimental bound Br(µ ! eg)  4.2⇥ 1013 [245]. The new annihilation channel of the fermionic WIMP dark matter can result in an increased production of right handed neutrinos in the early Universe, which may oversaturate the effective number of relativistic degrees of freedom, Ne f f [246]. This may disfavour the fermionic DM case. 4.3.2. An Alternate Loop Model In this section we present an alternate loop model to generate Dirac neutrino mass. The SM ⌦ PQ charges of all the necessary fields are presented in Table 4.5. 96 Dark and Lepton symmetries Hu(2) σ(4) L(1) NR(1/2) NL(1/2) νR(−5) ηu(1/2) ζ(11/2) U(1)PQ Z4 Hu(1) σ(1) L(!2) NR(!) NL(!) ⌫R(! 2) ⌘u(!) ⇣(!3) Figure 4.8.: Feynman diagram for Dirac neutrino masses in one-loop DFSZ scenario. Here left panel respects PQ charge assignment, whereas right panel respects remnant ZPQ 4 charge assign- ment, arises due to PQ symmetry breaking. Symmetry/Fields Li nRi Hu Hd NR NL s hu z SU(2)L ⇥U(1)Y (2, -1/2) (1, 1) (2, -1/2) (2, 1/2) (1, 0) (1, 0) (1, 0) (2, -1/2) (1, 0) U(1)PQ 1 -5 2 2 1/2 1/2 4 1/2 11/2 Table 4.5.: Fields content and transformation properties under PQ symmetry in the one-loop mecha- nism. In this case, the vevs of s, Hu and Hd break the PQ symmetry into a Z4 symmetry. The fields’ transformation rules are given in Table 4.6. Here, the particles inside the loop are automatically stable [111]. Symmetry/Fields Li nRi Hu Hd NR NL s hu z SU(2)L ⇥U(1)Y (2, -1/2) (1, 1) (2, -1/2) (2, 1/2) (1, 0) (1, 0) (1, 0) (2, -1/2) (1, 0) ZPQ 4 w2 w2 1 1 w w 1 w w3 Table 4.6.: Fields content and transformation properties under PQ symmetry in the one-loop mecha- nism. We notice from the right panel of Fig. (4.8) that all the particles inside the loop carry Z4 odd charges, whereas the SM particles are even under Z4. Therefore, one can see that any combination of SM fields will be even under the Z4 charges. Further forbidding all effective operators to dark matter decay. 4.3.3. Summary We have discussed the DFSZ model where neutrinos are Dirac particles due to the PQ symmetry [228]. In this context, we propose two different scenarios to generate Dark and Lepton symmetries 97 naturally small effective Yukawa coupling for neutrinos. In order to explain the small- ness of the Yukawa coupling, the tree-level coupling is forbidden by the PQ symmetry while an effective dimension-5 operator with the PQ field is allowed. This means that the Dirac neutrino mass is proportional to the PQ breaking scale. The first scenario is based on the Type-II Dirac seesaw where an extra heavy SU(2)L doublet allows Dirac neutrino mass when acquires a small vev once the PQ and the EW symmetry are broken. We summarize our results for this scenario in Fig. 4.6, considering the latest KATRIN Aker:2019uuj and the Planck data [42]. These constraints set limits on the PQ breaking scale fa and the mass of the heavy scalar. We also discuss the UV completion of the dimension-5 operator at one-loop level. In this context once the PQ is broken, a residual Z2 symmetry remains. The SM fermions are odd while the scalars are even, making the lightest field inside the loop stable by means of the residual Z2 and Lorentz invariance giving a potential DM candidate. This residual symmetry is crucial, otherwise the the scalar particles inside the loop (odd under Z2) acquire a vev, and the loop would be a correction to the type-I Dirac seesaw. Therefore, in this scenario we have a potentially rich phenomenology with two dark matter components, a stable WIMP running inside the neutrino mass loop and the axion. We have also discussed what are the parameters of such a dark sector in order to avoid an overclosed Universe. It is worth mentioning that in both cases, the UV scale is more relaxed that in the type-I case [228], where the KATRIN neutrino bound set it to be O(MGUT)O(MPLANCK). 4.4. Neutrino seesaws and Self- Interacting DM from Z-Z’ mixing In this section5 we present a model with a dark gauge symmetry U(1)D. Only fermions with trivial SM gauge group representation carry dark charge, making up the RHN and dark matter sector. A scalar singlet and a second scalar Higgs doublet are introduced, both carrying dark charge. The spontaneous breaking of a the dark gauge group and the electroweak symmetry by the second scalar leads to mass mixing between the SM Z boson and the dark Z’ boson. This introduces couplings of SM fermions to the dark Z’ and of dark sector fermions to the SM Z boson. The breaking of the U(1)D gauge symmetry leaves a remnant Z2 symmetry which stabilizes a dark fermion, prohibitting 5This section is based on [247] 98 Dark and Lepton symmetries its couplings to neutrinos, and making it a DM candidate. The DM phenomenology is driven by the gauge interaction of DM, which can lead to sizable communication to the SM, or to a secluded WIMP scenario, depending on the size of the Z-Z’ mixing. 4.4.1. Model for Z-Z’ mixing with a dark gauge symmetry L N N0 F H1 H2 f cL cc R SU(2)L 2 1 1 1 2 2 1 1 1 U(1)Y 1/2 0 0 0 1/2 1/2 0 0 0 U(1)D 0 1 1 0 0 1 1 QD QD Table 4.7.: Matter content of the dark matter with mass mixed U(1)D gauge boson. Quarks and right handed charged leptons, not shown in this table, are not charged under the U(1)D gauge symmetry. The model contains two dark charges, we have chosen to absorb one of them into the definition of the dark gauge coupling, leaving the dark matter charge QD free. We consider the model for the mass mixing of a new gauge boson with the Z boson, described by Table 4.7 6. The dark sector (stable after the SSB) consists of a vector-like pair of fermions cL and cR. Fermions charged under U(1)D transform trivially under the SM gauge symmetry, guaranteeing the cancellation of all mixed SM-dark anomalies. For each charged fermion under U(1)D, there is a fermion with an opposite charge, such that the pure U(1)D and the U(1)D-gravity anomalies vanish. The right-handed neutrinos N, N0, F, participate in the seesaw mechanism, with their U(1)D charges shaping the seesaw mass matrix. The scalar sector will induce mass mixing among the electroweak and dark neutral bosons, linking the SM fermions with the dark sector. The c fields can act as a dark matter candidate, interacting with SM fields through the mass-mixed dark gauge boson. In this way, we show that the U(1)D can drive the phenomenology of neutrino and dark matter. The neutral gauge boson mixing will impact the quark and lepton physics, such as parity violation in polarized electron-nucleon and electron-electron scattering. 6Note that the charge assignment under the new U(1)D for N and N0 could take values different value as far as they remain vector-like, but it is fixed by convenience. To have this, we add to the Standard Model(SM) a new dark gauge symmetry U(1)D, a scalar SU(2)L doublet H2 charged under the new symmetry, and a scalar singlet f to trigger the symmetry breaking. The right handed neutrinos are charged under U(1)D. Therefore, at least two different sets of vector-like fermions N and N0 are needed to cancel the anomalies. We also need the inclusion of an extra fermion singlet under the new symmetry, F, to break the lepton number. Dark and Lepton symmetries 99 4.4.2. Neutrino sector The RH neutrinos are charged under the U(1)D. To generate the Yukawa Lagrangian, their charges must match that of the H2 Higgs doublet. To avoid extra Goldstone bosons, in the scalar sector we must have a term such as H2H† 1 f or H2H† 1 f2. From these two conditions, we conclude that the charge of f equals one of the RH neutrinos charges. The two RH neutrinos N and N’ have a Dirac mass term. In contrast, there is no way to generate a Majorana mass for any of those fields through the f field. The only way to do so is to include an extra fermion with no U(1)D charge. In this way, the Lagrangian density of the neutrino sector is Ln = Yn 1 LH̃1F + Yn 2 LH̃2N + M1NcN0 + YN NcFf + YN0 N0CFf⇤ + MFFCF + h.c. (4.52) After spontaneous symmetry breaking (SSB) the resulting neutrino mass matrix in the (nL, N, N0, F) basis is M = 0 BBBBBB@ 0 mD 2 0 mD 1 (mD 2 ) T 0 M1 YNvf 0 (M1) T 0 YN0 vf (mD 1 ) T (YN)Tvf (YN0 )Tvf MF 1 CCCCCCA , (4.53) where mD i = Yn i vi are the Dirac mass matrices. The light neutrino mass matrix is given by mlight = (mD 1 ) TamD 1 + (mD 2 ) TbmD 1 + (mD 1 ) TdmD 2 + (mD 2 ) TemD 2 , (4.54) 100 Dark and Lepton symmetries where the a, b, d, e matrices are defined as a =(MF + YN(MT 1 ) 1(YN0 )Tv2 f + YN0 (M1) 1(YN)Tv2 f) 1, b =((YN)Tvf) 1 + ((YN)Tvf) 1M1(MT 1 ) 1 ⇥ ⇥ [Id + MT 1 (Y Nvf) 1(YN0vfM1 1 (YNvf) T MF)((Y N0vf) T)1]1, d = (MT 1 ) 1(YN0vf) T[MF + YN(MT 1 ) 1(YN0 )Tv2 f + YN0 (M1) 1(YN)Tv2 f] 1, e = (YNvf) 1[YN0vf(MT 1 ) 1[Id + MT 1 (Y Nvf) 1(YN0vfM1 1 (YNvf) 1 MF)((Y N0vf) T)1]1+ + MFb], (4.55) where Id is the Identity Matrix. The minimal field content leading to two massive light neutrinos is (Nr(N) = 2, Nr(N0) = 2, Nr(F) = 2). There are several familiar limits to this framework: 1. The type-I seesaw limit can be obtained when YN, YN0 ! 0 or YN, mD 2 ! 0 or YN0 , mD 2 ! 0. The magnitude of light neutrino masses is mn ⇠ v2 1 MF . (4.56) 2. When MF, mD 1 ! 0, the light neutrino masses take the form of the inverse seesaw mn ⇠ (mD 2 ) 2YN0 M1YN . (4.57) 3. When MF, mD 2 ! 0, the light neutrino masses take a inverse seesaw form mn ⇠ (m1 D) 2M1 v2 fYNYN0 . (4.58) . Dark and Lepton symmetries 101 4. When YN, mD 1 ! 0, the light neutrino masses take the form mn ⇠ (mD 2 ) 2(YN0)2v2 f M2 1 MF . (4.59) In this case, there is an extra suppression compared with the inverse seesaw from the light (heavy) scale vf (MF). We will examine the viability of each limit, depending on the scale of vf indicated by DM phenomenology in section (4.4.10). 4.4.3. Dark Sector We choose as dark matter candidate a Dirac fermion c = cL + cR with mass term Lmass c = 1 2 Mccc. (4.60) We choose QD so that Majorana mass terms are forbidden at any order in perturbation theory, with the scalar content of Table 4.7. The condition to keep the Dirac character of c is QD 6= m 2 , m 2 . (4.61) Since QD 6= 0, c couples to the dark gauge boson X; once the gauge symmetry is broken this induces a coupling to both the physical Z boson and the dark photon Z0, see Eq. (4.70). For definiteness we will choose QD = 1/3 that satisfies Eq. (4.61). A similar dark matter model is described in [248]. 4.4.4. Gauge sector The U(1)D charges of the new fields, except for c, are equal in magnitude. Therefore, we may redefine the gauge coupling and field charges such that Q = ± 1 for these fields. With this in mind, the kinetic terms of the scalar fields in the model defined in 102 Dark and Lepton symmetries Table 4.7 are LSK = 2 ∑ i=1 h (DµHi) †(DµHi) i + h (Dµf)†(Dµf) i , (4.62) where the covariant derivatives for the SU(2)L Higgs doublets are DµHi = (∂µ + ig 2 ~t · ~Wµ + ig0 2 Bµ + ig D QiXµ)Hi , (4.63) with Q1 = 0 and Q2 = 1. The corresponding covariant derivative for the SU(2)L scalar singlet is Dµf = (∂µ ig D Xµ)f . (4.64) After electroweak and dark symmetry breaking, H1,2 and f acquire vacuum expecta- tion values and we write H1 = 0 @ H+ 1 (v1 + h1 + ia1)/ p 2 1 A ; H2 = 0 @ H+ 2 (v2 + h2 + ia2)/ p 2 1 A ; f = vf + hf + iaf . (4.65) The SM vacuum expectation value is v2 SM = v2 1 + v2 2, and we define tan b = v2/v1. There are five vector bosons, W ± = (W1 ⌥ iW2)/ p 2 correspond to the usual charged pair with mass gv SM /2, one neutral gauge boson, A = s w W3 + c w B (where s w = sin(qw) and tan(qw) = g0/g) remains massless. For the remaining fields, let Z̃ = c w W3 s w B, then the mass matrix for {Z̃, X} becomes m2 Z̃ X = 1 4 0 @ g2 z v2 SM 2v2 2 g D g z 2v2 2 g D g z 4g2 D (v2 2 + v2 f)) 1 A , (4.66) where g z = q g2 + g02. The Z̃-X mixing angle, qX, is given by tan 2qX = g z g D v2 2 1 4 g2 z v2 SM g2 D (v2 2 + v2 f) . (4.67) Dark and Lepton symmetries 103 Denoting the mass eigenstates by Z and Z0, the corresponding masses are given by M2 Z/Z0 = 1 8 g2 z v2 SM + 1 2 g2 D (v2 2 + v2 f)± (4.68) 1 8 ⇢h g2 z v2 SM 4g2 D (v2 2 + v2 f) i2 + ⇣ 4g z g D v2 2 ⌘2 1/2 . (4.69) The Z̃-X mixing induces a coupling of the Z0 to the electroweak neutral current JNC, and a coupling of the Z to the dark current JDC proportional to sin qX; explicitly, LNC = eJ µ EM Aµ Zµ(cos qX g z 2 J µ NC + sin qXg D J µ DC) Z0 µ( sin qX g z 2 J µ NC + cos qXg D J µ DC), (4.70) where e = gs w . The currents in Eq. ( 4.70) are J µ EM = ∑ r QEM r frgµ fr , J µ NC = ∑ r t3 L(r) f rgµ(1 g5) fr 2s2 W J µ EM , J µ DC = g D 3 cgµc + g D (NgµN (4.71) where QEM r is the EM charge of the fr fermion and t3 L(r) its weak isospin. 4.4.5. Scalar Sector. The scalar potential for the model in Table 4.7 is given by Vi =µ 2 1H† 1 H1 + µ 2 2H† 2 H2 + µ 2 ff⇤f + kf⇤H† 1 H2 + l1(H† 1 H1) 2 + l2(H† 2 H2) 2 + l3(f ⇤f)2 + l4(H† 1 H1)(H† 2 H2) + l5(H† 2 H2)(f ⇤f) + l6(H† 1 H1)(f ⇤f) + l7(H† 1 H2)(H† 2 H1), (4.72) where the only complex coupling is k, however by a field redefinition it can be made real. From the 10 real scalar degrees of freedom, four goldstone bosons are absorbed in the vector boson masses; the remaining six correspond to a charged pair H ± , a pseudoscalar A, and three neutral scalars. Using the notation of Eq. (4.65) the first three and their masses are given by H+ ⇠ sin b H+ 1 cos b H+ 2 , M2 H+ = 1 2 sv2 f 1 2 l7v2 SM , A⇠ vf sin b a1 vf cos b a2 + 1 2 v SM sin(2b) af , M2 A = 1 2 sv2 f + sin2(2b) 8 sv2 SM ,(4.73) 104 Dark and Lepton symmetries while in the {h1, h2, hf} basis the CP-even mass matrix is given by M2 E = 0 BBBB@ 2l1v2 1 kvfp 2 tan b v1v2(l4 + l7) + kvfp 2 v1vfl6 + kv2p 2 v1v2(l4 + l7) + kvfp 2 2l2v2 2 kvfp 2 cot b v2vfl5 + kv1p 2 v1vfl6 + kv2p 2 v2vfl5 + kv1p 2 2l3v2 f kv1v2p 2vf 1 CCCCA . (4.74) To simplify the expressions we defined s = p 8 k vf sin(2b) , (4.75) which is positive since in our conventions k < 0. The scalar potential must be bounded from below, leading to restrictions on the scalar couplings. We have collected these restrictions in Appendix ??. We note the existence of a decoupling limit, where the scalar masses become much heavier than the electroweak scale, save for the Higgs seen at LHC. This limit is achieved when v2 ! 0 and k ! 0, with µ 2 2 setting the heavy scalar scale. The decoupling limit drives the gauge boson mixing to zero, making the Z0 and dark matter invisible. 4.4.6. Phenomenology General constraints on light dark Z 0 bosons The induced couplings of the dark Z0 boson to the SM fermions can be probed by a variety of experiments [143,249–253]. The observables measured by those experiments, constrain the parameter space in the qX MZ0 plane. The most relevant experimental constraints are the following: • Atomic Parity Violation. As noted above, the mass mixing among the SM and X bosons induces a Z0 couplings to SM fermions (cf. Eq. (4.71)), which inherit the parity violating nature of the SM Z couplings. The Z0 parity violating couplings may induce observable effects on low energy experiments, when the mass of the Z0 is comparable to the energy scale of the experiment [254–256]. This parity- violating interaction of quarks and leptons mediated by the Z0 has been probed in atomic transitions of Yb, Cs, Tl, Pb, and Bi. The measurement of the nuclear weak charge in these experiments can be used to constrain the Z0 couplings Dark and Lepton symmetries 105 to the SM fermions as a function of its mass [249]. The resulting constraint is approximately [143, 247, 253] sin qX . 5⇥ 105, for MZ0 < 40 MeV ; and sin qX MZ0 . 106 MeV , for 40 MeV < MZ0 < 100 GeV. (4.76) • Collider searches. As the Z0 couples to the SM fermions, it can be produced in a multitude of collider experiments. The Z0-mediated Drell-Yan production of muons in hadron colliders yield some of the strongest constraints on the Z0 couplings for masses below MZ. Neutral gauge boson production and decay to leptons, in association with photon production has been searched for in e+e collisions, at BaBaR [257], CMS [258], LHCb [259], among others. The constraint from colliders in the mass region, 100 MeV < MZ0 < 80GeV, is roughly [143, 253] sin qX . 5⇥ 103. (4.77) • Beam dump experiments. The production of neutral bosons in electron bremsstrahlung processes in beam dumps has been probed, for example at the NA64 [260], E141 [130], E137 [129],E774 [131], KEK [132] and Orsay [136] experiments. The beam dump limits on sin qX in the Z0 gauge boson mass region, 1 MeV < MZ0 < 500 MeV are roughly [143, 253] sin qX . 3⇥ 108 or sin qX (MZ0/GeV)1.2 & 3.3⇥ 107. (4.78) Using the DarkCast code [143, 253], we derive the constraints from the APV, Collider and beam dump experiments mentioned. We show this results in Figure 4.15. 4.4.7. Dark matter relic density We consider the thermal freeze-out to determine the DM relic density, Ωc. We identify three scenarios which can result in a relic density of dark matter in accordance with cosmological measurements, Ωch2  0.1198: 106 Dark and Lepton symmetries 1. When Mc > MZ0 , the annihilation t-channel, c̄c ! Z0Z0, is kinematically allowed. This leads to a relic density which only depends on the Z0 boson mass MZ0 , the dark gauge coupling g D , and the dark matter mass Mc. Numerically we find that for each value of MZ0 there is a minimum value of g D for which there is no dark matter overabundance. We show this behavior in Figure 4.13. This scenario is well-studied and is known in the literature as Secluded WIMP Dark Matter [261–263]. 2. When Mc > M f , f being a SM fermion, the ZZ0 mixing allows the s-channel an- nihilation of dark matter into f , c̄c ! f̄ f . In the resonant regions Mc ⇠ MZ/Z0/2 the annihilation cross section can be enhanced enough to reach the required value to result in an allowed relic density, while keeping the value of qX in the allowed region discussed in section 4.4.6 [5]. In this channel in addition to MZ/Z0 , Mc and g D , the Z Z0 mixing angle qX is a crucial parameter. 3. When 2Mc > MZ(Z0) + MS, where S is one of the four neutral scalars in the model, the s-channel Higgsstrahlung channel (c̄c ! SZ(Z0)) is kinematically allowed. In this channel, the scalar masses and mixing angles become relevant to the relic density calculation. For dark matter masses above the W boson mass, the channel c̄c ! WW is open and can contribute significantly to the dark matter relic density. We illustrate the tree-level Feynman diagrams of these processes in Figure 4.12. To calculate the relic density for this model, we have implemented the model in SARAH [264] and micrOmegas [121, 122], scanning over a range of the free parameters. We study the first two cases, as they lead to an interesting interplay between the dark sector, the light gauge boson parameters and the neutrino sector. 4.4.8. Dark matter direct detection After excluding the parameter space where the relic density of c does not correspond to that of dark matter, we look at the Spin-Independent cross section of dark matter with nucleons. The Z Z0 mass mixing leads to tree-level dark matter-nucleon elastic scattering, a process searched for in direct detection experiments. The scattering is mediated by Z and Z0 exchange. In the case where relic density is determined by t-channel c̄c ! Z0Z0 annihilations, there is no correlation between dark matter relic density and the direct detection cross Dark and Lepton symmetries 107 .25 Figure 4.9.: t-channel annihilation into Z0/Z0 .25 108 Dark and Lepton symmetries Figure 4.13.: Case 1: t-channel annihilation into a Z0 pair. Parameter space in the Mc gc plane (where gc = g D cos qX ⇡ g D ) excluded by relic density overabundance. Area shaded in green results in relic density overabundance, while the green line corresponds to ΩDM = 0.1195. The area indicated in Magenta with the label “SIDM" corresponds to the region where dark matter self-interactions are consistent with astrophysical observations. Within this area, the blue region corresponds to a mediator mass MZ0 of 10 MeV, considering the uncertainty as estimated in the text. Figure 4.14.: Spin Independent direct detection cross section (sSI) as a function of dark matter mass, for the Z0 resonant case (s-channel). In the green lines the product g D sin qX is fixed to values between 108 and 103 as indicated in the caption. The purple lines show the experimental limit on sSI set by the LZ [265] and CRESST-III [266] experiments. The dashed magenta line shows the neutrino floor on Argon [267]. Note that the LZ experiment target nucleus is Xenon, and it has not yet reached the Xenon neutrino floor. Dark and Lepton symmetries 109 section, as the relic annihilation cross section is independent of the Z Z0 mixing angle at leading order. However, a bound on the Z Z0 mixing angle may be derived from the limits on this cross section. For the M2 Z0 << M2 Z limit, the spin independent c-nucleus elastic scattering cross section is approximately [268] sZ,A SI = µ 2 cN sin2 2qXg2 XQ2 c 4pM4 Z0 h Z(2gZu SM + gZd SM) + (A Z)(gZu SM + 2gZd SM) i2 , (4.79) where µ 2 cN is the reduced c-nucleus mass, g Zq SM is the vector coupling of q = u, d to the Z boson in the SM, and Z and A are the electric charge and atomic mass number of the nucleon respectively. Using the most stringent limits on dark matter direct detection we can list the following constraints • 1 2 |gXQc sin 2qX| . 107 at MZ0 = 10 GeV, and Mc = 30 GeV from LZ [265]. • 1 2 |gXQc sin 2qX| . 106 at MZ0 = 100 MeV, and Mc = 1 GeV from CRESST- III [266]. For the resonant Z0 channel case we observe a clear correlation between the parameter product g D sin qX and the SI cross section in Fig 4.14. For dark matter masses above 1 GeV and below 10 GeV, CRESST-III results exclude models where g D sin qX > 104. For dark matter masses above 10 GeV LZ results exclude models with g D sin qX > 106 up to 30 GeV, after which the constraint relaxes. In Figure 4.15 we show direct detection constraints alongside Z0 searches constraints, for a choice of g D = 1, 0.1. We see that for Z0 masses above 20 GeV, direct detection constraints are stronger than low energy constraints for these choices of g D . For lighter Z0 masses CRESST-III results are stronger than Z0 searches, down to 1 GeV, with g D = 1. For g D = 0.1 CRESST-III results are weaker than collider or APV constraints. 4.4.9. Dark Matter Self-Interactions Estimations of the DM distribution in dwarf galaxies indicate that the DM density at the core does not exhibit a spike, as would be expected if it behaved as an ideal gas; this is known as the “core vs. cusp” problem [269–271]. This problem can be alleviated [272, 273] by including self-interactions within the dark sector 7; such 7There are other puzzles associated with the standard cold-DM scenario, such as the “too big to fail” puzzle [274, 275]. In this paper we will not consider the extent to which the present model can address such issues. 110 Dark and Lepton symmetries Figure 4.15.: Low energy constraints and resonant dark matter direct detection constraints. In this figure we show the low energy constraints from Atomic Parity Violation [254], collider (BaBar [257],CMS [258] and LHCb [259]) and beam dump (NA64 [260], E141 [130], E137 [129],E774 [131], KEK [132] and Orsay [136]) experiments on the MZ0 sin qX parameter space. These constraints are obtained as discussed in Section 4.4.6. We also show in the same space the constraints derived from direct detection experiments (LZ [265] and CRESST-III [266]) in the resonant dark matter models, as discussed in Section 4.4.8. interactions must be relatively strong and velocity-dependent. Models of this type are often referred to as self-interacting dark matter (SIDM) models. In this section we determine the extent to which the present model can address the core-vs-cusp problem. Existing data constraints the SIDM cross section for galaxy clusters and for dwarf and low-surface-brightness galaxies; since the typical velocity in each environment is different, the cross section must have an appropriate velocity-dependence. Dark matter self-interactions in this model consist of cc ! cc scatterings mediated by the Z0 boson. Defining bc = s 1 4M2 c s , gc = g D cos qX, (4.80) Dark and Lepton symmetries 111 we obtain the self-interaction cross section sSIDM [276] sSIDM Mc = g4 c 4psMc ( (2s + 3M2 Z0)sb2 c + 2(M2 Z0 + 2M2 c) 2 2M2 Z0(M2 Z0 + sb2 c) (sb2 c + 2M2 Z0)(3M2 Z0 + 4M2 c) + 2(M2 Z0 + 2M2 c) 2 4M4 c sb2 c ⇣ 2M2 Z0 + sb2 c ⌘ ln 1 + sb2 c M2 Z0 !) ; (4.81) The requirements for SIDM are met in this model when this cross section is enhanced by the kinematic condition Mc MZ0 , in the small relative velocity bc regime.The SIDM cross section magnitude is determined by astrophysical data, namely dwarf and low surface brightness galaxies. A velocity-dependence of the cross section is obtained by the different typical velocities of dark matter in each environment. The central values of the cross sections and velocities are [277] sSIDM Mc galaxy = 1.9 cm2 gr , sSIDM Mc cluster = 0.1 cm2 gr ; bc galaxy = 3.3⇥ 104 , bc cluster = 5.4⇥ 103 . (4.82) The galaxy data were derived from dwarf and low surface brightness galaxies; the estimates of the average collision velocity bc were obtained using a generic halo model and a Maxwellian velocity distribution. Fitting to these values we find MZ0 = Mc 566 , gc = ✓ Mc 75 GeV ◆3/4 . (4.83) The errors in the data (Eq. (4.82)) are large; to take these into account we allow a factor of 2.5 in the numerical coefficients in Eq. (4.83), (e.g., the first coefficient ranges from 566/2.5 to 2.5 ⇤ 566). We find that these conditions can be met when dark matter relic density is obtained through annihilation to Z0 pairs. In Figure 4.13 we show the band where SIDM is viable. We note that there is an overlap between the SIDM band and the line where c accounts for dark matter completely. In this scenario, the mass of the Z0 is of order ⇠ 10 MeV, which is where low energy experiments are most sensitive. 112 Dark and Lepton symmetries 4.4.10. Neutrino masses and U(1)D breaking scale From the constraints on the dark sector parameters from the dark matter phenomenol- ogy, we can infer the following possibilities for the neutrino mass mechanism. For all dark matter relic density channels, the inverse seesaw-like scenario mn ⇠ (m2 D) 2YN0 M1YN , (4.84) and the type-I seesaw scenario ml ⇠ v2 1 MF , (4.85) are viable with heavy neutrinos of canonical seesaw scale. Of special interest are the cases where the neutrino masses are linked to the U(1)D breaking scale, namely the limits considered in eqs. (4.58) and (4.59). In the limit considered in eq. (4.59) mn ⇠ (m2 D) 2YN0v2 f M2 1 MF , (4.86) a light neutrino masses of the order O(eV) can be achieved with O(TeV) heavy neu- trinos, as in the canonical inverse seesaw. In this way the smallness of neutrino masses is linked to the smallness of the Z0 boson mass. Lets consider now the limit in eq. (4.58). For the t-channel annihilation case, the correct relic density is determined for values of g D larger than 3⇥ 104 for a dark matter mass of ⇠ 1 MeV, or g D larger than 2⇥ 101 for a dark matter mass of ⇠ 100 GeV (see Figure 4.13). In the light Z0 paradigm, with small Z Z0 mixing we have MZ0 > g D vf, (4.87) and the t-channel dominated annihilation has the kinematic condition Mc > MZ0 . (4.88) Dark and Lepton symmetries 113 These two equations rule out the inverse seesaw-like limit of neutrino masses mn ⇠ (m1 D) 2M1 v2 fYNYN0 , (4.89) as the low scale of vf needed to obtain a light MZ0 would result in either ⇠ 1GeV scale sterile neutrinos with large mixings with the active neutrinos, or neutrino Yukawa couplings much smaller than the electron Yukawa coupling of the SM, calling into question the necessity of the seesaw scheme. 4.4.11. Conclusions In this section we have studied an economical extension of the SM, with a new dark sector equipped with a dark abelian gauge symmetry. This simple setup has been shown to contain a viable dark matter candidate and a mechanism for light neutrino mass generation. Dark matter phenomenology can be dominated by the dark gauge boson leading to the self-interacting dark matter scenario, or by Z-Z’ mixing leading to strong direct detection signals. In both of these scenarios the Z’ boson can be light (MeV - EW scale), making it amenable to detection in low energy observables, such as in parity violating asymmetries. In the neutrino sector we obtained a generalized type-I seesaw mechanism for neutrino masses, with several limitting cases, such as the canonical type-I scenario or the Inverses seesaw scenario. The U(1)D breaking scale can be crucial in determining the neutrino mass scale, linking the dark sector to the neutrino sector. 4.5. Low energy constraints on a Dark Z boson In this section8 we revisit the general Dark Z model, where the gauge boson of a dark U(1)D gauge symmetry mixes with the SM Z boson through kinetic and mass mixing terms. The presence of both types of mixings can lead to interesting experimental signatures, as the mass mixing induces parity violating couplings of the dark Z to the weak neutral current and the kinetic mixing induces the coupling of the dark Z to the 8This section is based in [8]. 114 Dark and Lepton symmetries electric current. This type of new gauge boson in the 1 MeV - 1 TeV mass range can be searched in electroweak precision tests, parity violation observables [249], CEvNS experiments, collider experiments, among others. The focus of the work presented here is to analyze the sensitivity of current and future experimental data to the presence of a light Dark Z. To obtain the influence of the Dark Z on experimental observables, we develop two methods. One is based on the direct approach of calculating the couplings of the new gauge boson to SM fermions by obtaining the Lagrangian in the gauge boson mass eigenstate basis with canonical kinetic terms. The second method is based on calculating observables where the dark Z is strictly a virtual particle mediating an interaction of SM fermions, by working in the non-diagonal gauge boson basis. The propagators connecting the different matter currents of the SM can be found by perturbatively inverting the bilinear gauge boson matrix operator. We analyze in particular the sensitivity of experiments measuring parity violation that are planned to occur in the near future. 4.5.1. The General Dark Z model The setup for the appearance of the Dark Z boson [249] is based on the U(1)D gauge symmetry extension of the Standard Model. No SM fermions carry dark charge, but a Higgs SU(2)L doublet with dark charge is introduced. The simultaneous spontaneous breaking of the electroweak and dark symmetry by the darkly charged Higgs doublet induces mass mixing among all neutral gauge bosons. Additionally, a nonzero kinetic mixing term among the U(1)Y and U(1)D gauge bosons is allowed, its value depend- ing on the details of the UV part of the theory. We consider scalar EW multiplets fi, that carry hypercharge yi and dark hypercharge y0i. We consider that these fields have nonzero vevs vi in the t3 = yi component, which guarantees the conservation of the U(1)Q gauge symmetry. The mass matrix for the W3, B, and Zd fields is then given by L(3⇥ 3) mass = 1 2 ⇣ W3, B, Zd ⌘ ∑ i 0 BBB@ g2y2 i v2 i ggYy2 i v2 i ggdyiy 0 iv 2 i ggYy2 i v2 i g2 Yy2 i v2 i gdgYyiy 0 iv 2 i ggdyiy 0 iv 2 i gdgYyiy 0 iv 2 i g2 dy0 2 i v2 i 1 CCCA 0 BBB@ W3 B Zd 1 CCCA , (4.90) where g, gY and gd are the SU(2)L, U(1)Y and U(1)d coupling constants respectively. The photon, Â, and the Ẑ boson (which are not in the diagonal mass basis and hence Dark and Lepton symmetries 115 we use a hat to distinguish them) are obtained as the linear combination of W3 and B, defined by the weak mixing angle, tan qW ⌘ gY/g, 0 BBB@ W3 B Zd 1 CCCA = 0 BBB@ sin qW cos qW 0 cos qW sin qW 0 0 0 1 1 CCCA 0 BBB@  Ẑ Zd 1 CCCA , (4.91) which implies Lmass = 1 2 ⇣ Â, Ẑ, Zd ⌘ 0 BBB@ 0 0 0 0 m2 Z MZ Mdd 0 MZ Mdd M2 d 1 CCCA 0 BBB@  Ẑ Zd 1 CCCA , (4.92) where M2 Z = ⇣ g2 + g2 Y ⌘ ∑ i y2 i v2 i , (4.93) M2 d = g2 d ∑ i y0 2 i v2 i , (4.94) and d = ∑i yiy 0 iv 2 iq ∑i y2 i v2 i q ∑i y0 2 i v2 i =) |d|  1. (4.95) In this basis, the Lagrangian for the gauge bosons is L = 1 4 µn µn 1 4 ẐµnẐµn 1 4 ZdµnZ µn d e 2 cos qW BµnZ µn d + (4.96) dMZ MdẐµZ µ d + M2 d 2 ZdµZ µ d + M2 Z 2 ẐµẐµ, while the gauge interaction Lagrangian is Lcurrent = ieµ J µ Q + igZẐµ J µ Z + igdZdµ J µ D. (4.97) 116 Dark and Lepton symmetries where the electromagnetic and neutral currents J µ Q and J µ Z are the same as in the SM. We arrive at the diagonal mass basis with no kinetic mixing via the following transformation 0 BBB@ A Z Z0 1 CCCA = TaTW Te 0 BBB@ W3 B Zd 1 CCCA . (4.98) The diagonalization process can be split into three steps. Kinetic mixing is removed with Te. In order to arrive at canonical kinetic terms, Te is a non-unitary transformation and takes the form: Te = 0 BBB@ 1 0 0 0 1 e/ cos qW 0 0 q 1 e2/ cos2 qW 1 CCCA . (4.99) The second transformation TW = 0 BBB@ sin qW cos qW 0 cos qW sin qW 0 0 0 1 1 CCCA , (4.100) is the SM weak mixing angle rotation. As a result, the photon A remains massless. Finally, Ta identifies the last two eigenstates of the sector, namely, Z and Z0. Ta = 0 BBB@ 1 0 0 0 cos qa sin qa 0 sin qa cos qa 1 CCCA . (4.101) The mixing angle a obtained after eliminating the kinetic mixing term is given by tan 2qa = 2(dMZmd + sin qWeM2 Z) p 1 e2 M2 Z[e 2(1 + sin2 qW) 1] + M2 d + 2d sin qWeMdMZ . (4.102) Dark and Lepton symmetries 117 Using Eq. (4.97), the neutral gauge bosons couple with the SM fermions as follows Lint = ⇣ ieJQ, igZ JZ, igd JD ⌘ TW T1 e T1 W T1 a 0 BBB@ A Z Z0 1 CCCA . (4.103) In this basis, it is straightforward to calculate scattering cross sections. However, all SM cross sections must be calculated from scratch, considering the induced couplings of the Z to the electric current and of the Z0 to the neutral and electric currents. Another method to calculate the effect of the mass and kinetic mixing terms consists in considering the mixing terms as corrections to the propagators that couple to the JQ and JZ currents. In Classical Field Theory, the propagators can be found by writing the Lagrangian as a bilinear form of the gauge bosons, i.e., ZµOµnZn, and computing the inverse Dµn of the operator Oµn in momentum space. The mixing parameters are taken to be small, so that we can invert the nondiagonal matrix Oµn perturbitavely. The resulting propagator matrix is paramterized as Dµn(p2) = hµn 0 BBB@ DÂÂ(p2) DÂẐ(p2) DÂZd (p2) DÂẐ(p2) DẐẐ(p2) DẐZd (p2) DÂZd (p2) DẐZd (p2) DZdZd (p2) 1 CCCA . (4.104) The entries relevant for observables of SM fermion fields are DÂÂ(p2) = e2 p2 M2 d + 1 p2 , (4.105) DÂẐ(p2) = deMdMZ e2p2 tan qW⇣ p2 M2 d ⌘ ⇣ p2 M2 Z ⌘ , (4.106) and DẐẐ(p2) = 1 p2 M2 Z " 1 + d2M2 d M2 Z 2edMdMZ p2 tan qW + e2p4 tan2 qW (p2 M2 d)(p2 M2 Z) # . (4.107) With this result we can compute observables based on our knowledge of one of the loop corrections to an observable, by adding to the SM vacuum polarisation functions 118 Dark and Lepton symmetries the effects of these insertions. Rewriting the propagators as DVV0(p2)⇠ PropVdVV0 + PropV(ΠVV0)PropV0 we can read off Π mixing ẐẐ (p2) = ⇣ dMdMZ ep2 tan qW ⌘2 p2 M2 d , (4.108) Π mixing ÂẐ (p2) = ep2 dMdMZ ep2 tan qW p2 M2 d , (4.109) and Π mixing  (p2) = e2p4 p2 M2 d . (4.110) With these two methods we can calculate experimental observables. Depending on the particular process, one or the other method can be more convenient to use. 4.5.2. Observables CEvNS As an example of the observables that are affected by this model, and the methods we can use to calculate cross sections, we can take a look at the Coherent Elastic neutrino Nucleus Scattering. Following the analysis of [126], we calculate the effects of the Zd on a 1 MW reactor neutrino measurement of CEnNS with a 10 kg year exposure of a scintillating bubble chamber with a 100 eV threshold located at 3 m from the reactor. To calculate the cross section of the process with the effect of the Dark Z, we can take the SM differential cross section (see eq. 3.62) and make the replacements 1. We correct the Fermi constant measured in charged current interactions by the effective contribution of the mixing terms to the r parameter r ! rSM Π mixing ẐẐ (Q2) Q2 + M2 Z = rSM + ⇣ dMdMZ + eQ2 tan q̂W ⌘2 (Q2 + M2 d)(Q 2 + M2 Z) . (4.111) Dark and Lepton symmetries 119 2. We take the low energy Weinberg angle prediction of the SM, based on the running of the angle measured at high energies and correct it with ∆ sin2 qW(p2) = ŝĉ Π mixing ÂẐ (p2) p2 + ŝ2ĉ2 ĉ2 ŝ2 Π mixing ẐẐ (M2 Z) M2 Z . (4.112) To extract the exclusion limit potential for this experiment we use the c2 function as defined in [126], which takes into account the uncertainties in the neutrino flux, neutron backgrounds and the uncertainty in the threshold energy. We found that SBC limits on the mass mixing parameter for dark Zd mass between few MeV to 100 MeV goes from ⇠ 103 to 102. 4.5.3. Summary This work aims to clarify and simplify calculations of observables in the dark Z model. We outlined two methods to calculate cross sections, showing a practical application in a novel sensitivity study. The resulting sensitivity for future CEvNS experiments show that this observable can complement parity violation constraints on the model. 4.6. Direct detection constraints on Secluded Dark Photon mediated dark matter The Secluded WIMP model [261, 262, 278] is a relatively simple WIMP model which requires only a new gauge boson and dark matter candidate. The model is defined in the parameter space of this setup, where the dominant annihilation channel of dark matter in the freezeout process is into the new gauge boson. A possible identity for the new gauge boson is the dark photon, which is a massive state which has kinetic mixing to the photon and couples to the electric current of the SM. In this setup, the free parameters of the model are the strength of the kinetic mixing e, the mass of the dark photon MD, the dark gauge coupling gD and the dark matter mass Mc. Thermal freezeout can determine the relationship between gD and Mc, leaving e, MD and Mc to be constrained by experimental limits. In this work9 we analyze the Secluded WIMP 9This section presents preliminary results from [7] 120 Dark and Lepton symmetries Figure 4.16.: Projected constraints for future CEnNS experiment, assuming a 1 MW reactor neutrino source at 3 meters from a 10kg Argon scintillating bubble chamber with a 1 year exposure. Exclusion limits for ∆c2 = 2.7. First plot corresponds to d Md plot for future CEvNS. Second plot corresponds to e Md plot for future CEvNS. Dark and Lepton symmetries 121 model, taking into account the latest direct detection constraints. We find that low energy constraints on the dark photon, dark matter direct detection constraints and thermalization constraints exclude a large part of the parameter space of this scenario. Unless non standard physics are at play in the Cosmological history of the dark sector, this model seems to be disfavored by current data. 4.6.1. Dark Photon mediated secluded WIMP model The minimal Lagrangian for the dark photon mediated secluded WIMP scenario is given by L = 1 4 B0 µnB0µn e 2 cos qW B0 µnBµn + c̄(gµDµ + Mc) , (4.113) where Dµ = ∂µ + iQcgDB0 µ is the covariant derivative, where B0 µ is the new gauge boson, gD is the dark gauge coupling and Qy is the dark charge of the field y. In the absence of any other particles charged under the dark symmetry, we can set Qy = 1 without any phenomenological consequence. The source of the mass of the dark gauge boson is not of relevance to the scenario, it can come from the spontaneous breaking of the new gauge symmetry by a singlet scalar or from the Stueckelberg mechanism. In any case, the mass of the dark photon will be taken as a free parameter in this work. The kinetic mixing will induce interactions between the SM fermions and the dark photon. After removing the kinetic term with the appropriate transformation of the gauge fields and diagonalizing the neutral gauge boson sector, the couplings of the dark photon to SM fermions are g f V = ee " Q f + 1 2 cos2 qW M2 d M2 Z M2 d ⇣ t f 3 2 sin2 qWQ f ⌘# , (4.114) g f A = ee " 1 2 cos2 qW M2 d M2 Z M2 d t f 3 # , (4.115) where Q f is the electric charge of the fermion and t f 3 is the third component of weak isospin of the fermion. The photon and the Z boson couple to the SM fermions as usual, and they do not acquire couplings to the dark current. 122 Dark and Lepton symmetries 4.6.2. DM relic density When the dark matter candidate is heavier than the dark photon, the dominant annihilation channel of dark matter in the early Universe is cc ! gDgD [261, 278]. This annihilation can lead to the freezing-out of dark matter at the observed dark matter relic density. The scattering amplitude is obtained at the tree level from the Feynman diagram in Fig. 4.17. The annihilation cross section relevant for the thermal freeze-out of dark matter is [261] sv = g4 d 16pm2 y vuut1 m2 gD m2 y . (4.116) The resulting dark matter relic density turns out to be insensitive to the dark photon mass unless the masses of the dark photon and dark matter are finely tuned to a close value. This allows us to determine a relationship between gD and my where dark matter has the observed relic density, subject to the conditions my > mgD and m2 gD /m2 y ⌧ 1. Through the use of micrOMEGAs [121], we have obtained a numerical estimate of this relationship [6], which we have crosschecked with the analytical ap- proximation of the Secluded WIMP scenario [261]. The numerical solution is shown in Fig. 4.18. In the region of interest for dark matter mass 1 GeV < my < 100 GeV, the dark gauge coupling is in the range 0.01 < gD < 1. The freeze-out annihilation channel of dark matter produces dark photons which could constitute a relativistic degree of freedom in the early Universe. The amount of relativistic degrees of freedom in the radiation-dominated Universe control the rate of expansion, and Big Bang Nu- cleosynthesis is highly sensitive to them. To avoid contradicting BBN measurements of Ne f f we can impose a Dark Photon mass larger than 10 MeV and that the lifetime of the dark photon is smaller than O(1s), which sets a lower bound on #. A stronger constraint, however, is obtained from the requirement of thermal equilibrium of dark photon decays to electrons, which thermalizes the dark sector in the early Universe, leading to the freeze-out scenario assumed in this work. Requiring that the decay rate is larger than the Hubble rate at freeze-out leads to [261] |#| & 106 s✓ 10GeV Md ◆✓ mc 500GeV ◆ . (4.117) For mc = 316 GeV, md = 103 GeV a constraint of |#| & 104 is obtained, and for larger dark photon masses, this constraint is relaxed. Dark and Lepton symmetries 123 Figure 4.17.: Tree level Feynman diagram of dark matter annihilation in the Early Universe. 124 Dark and Lepton symmetries Figure 4.18.: Numerical relationship between my and gD. In the green line, relic density corresponds to the observed value. 4.6.3. Experimental constraints on a dark photon The dark photon is subject to experimental constraints that are independent of dark matter physics. The strongest experimental constraints come from beam dump experi- ments [129–131, 138, 279–282], collider experiments [257, 258, 283] and cosmological observations [284]. 4.6.4. Direct Detection constraints on the dark photon mediated spin independent cross section We consider limits from three direct detection experiments, PICO-60 [285], XENON-1T [53] and PANDAX-4T [286]. The direct detection cross section depends on (Md, Mc, e, gD). To set limits on the dark photon parameter space (Md, e) we fix the parameter gD by requiring that the freezeout process leads to the correct relic density value of dark matter, and we fix Mc to discrete values. Dark and Lepton symmetries 125 Figure 4.19.: Exclusion plot for the kinetic mixing given a value of the dark matter My = 10 GeV. 4.6.5. Results In figs. ?? we show the constraints on the dark photon parameter space in the secluded WIMP scenario. We integrate the constraints from dark photon searches, thermal freezeout and direct detection experiments. The combined effect of these three con- straints rules out a large part of the parameter space, with a smaller allowed region for larger dark matter masses. Direct detection constraints seem to dominate dark photon constraints at all dark photon masses, for the dark matter masses we consider. We expect this not to be true for dark matter masses below 10 GeV, as direct detection ex- periments quickly become less sensitive in this mass range. Thermalization constraints depend on the dark photon being in thermal equilibrium with the SM at some point after cosmological reheating. It is possible for freezeout to occur in the dark sector without it being in equilibrium with the SM, if the dark sector reheats independently of the SM. This means that the thermalization constraint can be ignored if we consider a more complicated reheating mechanism. 126 Dark and Lepton symmetries Figure 4.20.: Exclusion plot for the kinetic mixing given a value of the dark matter My = 100 GeV. Dark and Lepton symmetries 127 Figure 4.21.: Exclusion plot for the kinetic mixing given a value of the dark matter My = 1000 GeV. 128 Dark and Lepton symmetries 4.7. Summary In this section we presented the latest constraints on the Secluded WIMP model, when the dark matter is connected to the SM by a dark photon. We have obtained that direct detection constraints are stronger than dark photon limits for dark matter masses above 10 GeV. We also observed that if the dark photon was at thermal equilibrium at some point after reheating, heavy dark matter masses are heavily constrained by the combined effect of all observables. Chapter 5. General Summary The objective of the research presented in this thesis is the study of massive neutrinos and dark matter through the construction and analysis of QFT models. The Standard Model is the starting point for any particle physics model, it has been shown to be consistent with virtually every experimental test made so far. Never- theless, there seems to be phenomena that cannot fit in the SM framework. The construction and analysis of extensions of the SM is motivated by many theoretical and observational considerations. We have focused on the flavor problem, neutrino masses and dark matter, briefly touching on other subjects such as the strong CP problem and Grand Unified Theories. One of the main goals of this work is to show that neutrino mass and dark matter models can arise from very economical extensions of the Standard Model in a theoretically consistent manner. Using symmetries as the underlying structure of physical models has produced the picture of fundamental Physics we have today. From the cosmological scale down to the subnuclear scale, symmetries dictate the behavior of physical systems. The philosophy we have em- braced in this work adheres to this fact, by searching for symmetries that can unify our understanding of diverse physical phenomena. The use of large symmetries can lead to ambitious unifications, such as gauge group unification, or the unification of particle physics and gravity. These types of constructions, while having low energy constraints, seem to be based on dynamics of very massive fields, far from our current experi- mental reach. For this reason, the models we have considered are grounded on the dynamics of not-so-heavy physics, which we can thoroughly look for at past, current and near-future experiments. We have seen that indeed, the framework explaining neutrino masses and dark matter simultaneously can lie at or below the electroweak scale. This motivates us to look for dark matter hints in places beyond dark matter experiments. In the models we have studied with a dark matter candidate we have found constraints coming from parity-violation observables, beam dump experiments or CEvNS, among other experiments not planned explicitly to probe dark sectors. Given the lack of hard data that could be used to characterize dark matter interactions, the purpose of studying dark matter models is to get a sense of the phenomenology of complete, theoretically consistent models. The information we can obtain from this 129 130 General Summary exercise is to elucidate correlations among observables that we should expect given a certain feature of dark matter. The endgame of this strategy is to eliminate possible dark matter models with current data and guide the experimental search for dark matter. On the neutrino front, over the past few decades experimental efforts have yielded steady results. The observation of neutrino oscillations, the measurement of PMNS angles and the observation of the CEvNS process are perhaps the salient of these results. In the near future it is hoped that the CP violating phase of the PMNS matrix can be measured, and the neutrino mass hierarchy determined. This experimental progress has allowed us to search for BSM physics in the neutrino sector, while also confirming SM predictions. Maybe the most important unanswered question that can be resolved in future decades is the nature (Dirac or Majorana) of neutrinos. To answer this and other fundamental questions, it is important to merge theory, models and observations from particle physics, astronomy and cosmology. One of the main conclusions we can derive from the work presented in this thesis is that neutrino physics continues to be one of the most promising avenues for searching for new physics. Appendix A. Appendices This section contains the appendices for all sections, in the order they are reffered to in the main text. A.1. Correlations in the MIII models In the models B 2Le Lµ and B 2Le Lt, the charged lepton and Dirac neutrino mass matrices are diagonal by construction. While it is not enough to include the f1 and f2 to reproduce neutrino masses and mixings [106]. This can be alleviated by the inclusion of a third flavon field f4. In this way the right-handed neutrino mass matrix takes the form MB2LeLµ = 0 BBB@ y4v4 0 y2v2 0 y3v2 y1v1 y2v2 y1v1 M1 1 CCCA , MB2LeLt = 0 BBB@ y4v4 y2v2 0 y2v2 M1 y1v1 0 y1v1 y3v2 1 CCCA , (A.1) for U0 B2LeLµ , U0 B2LeLt models, respectively. Given the Dirac and Majorana neutrino mass matrices, one can construct the low-energy active neutrino mass matrices using type-I seesaw mechanism. We write the active neutrino mass matrix as mn = 0 BBB@ m11 m12 m13 ⇤ m22 m23 ⇤ ⇤ m33 1 CCCA . (A.2) 131 132 Appendices From the parameters in Eq. (A.1), we find following correlations m12m33 = m13m23 , m12m23 = m13m22 , (A.3) for U(1)B2LeLµ and U(1)B2LeLt , respectively. We have found that these correla- tions are compatible with current oscillation data, with the correct choice of neutrino masses and Majorana phases. For the MIV models there are no correlations in the mass matrix, there is enough freedom to fit neutrino oscillation data. A.2. Phase redefinition of quark mass matrices The up- and down-quark mass matrices as given by Eq. 3.85 can be written as mu/d = 0 BBB@ 0 Au/d eiau/d 0 Bu/d eibu/d 0 Cu/d eigu/d 0 Du/d eidu/d Eu/d eieu/d 1 CCCA , (A.4) with A, B, C, D, E, a and b as real parameters. The above mass matrix mu/d can be diagonalized by bi-unitary transformation of the form mdiag = V† L mVR = OTP† LmPRO , (A.5) where L and R depict the left- and right-chiral fields, respectively. Also, VL = PLO and VR = PRO are the unitary matrices that diagonalize m†m and mm† and PL = diag(1, eia, eib), PR = diag(eir1 , eir2 , eir3) are the diagonal phase matrices, respectively. Notice that we drop subscript (u/d) to demonstrate phase redefinition as the following formalism is same for both the up and down sector. Now, given the form of PL and PR, one can construct a real solution of the quark mass matrix (Eq. A.4) following the transformation P† LmPR as mentioned by Eq. A.5. Thus, the quark mass matrix in terms Appendices 133 of real parameters can be written as m = 0 BBB@ 0 A 0 B 0 C 0 D E 1 CCCA . (A.6) Notice that for most of the application VR is irrelevant, it is the VL that enters in the CKM parameterization. Therefore, one can write quark mixing matrix as VCKM = (V† L )u(VL)d = (OTP† L)u(PLO)d , = OT u diag(1, ei(auad), ei(bubd))Od . (A.7) We see here that it is the difference in the diagonal-phase matrix that enters in the VCKM. Thus, in our numerical an analysis we have used the quark mass matrix of the form P† Lm = 0 BBB@ 0 A 0 B eia 0 C eia 0 D eib E eib 1 CCCA . (A.8) A.3. Scalar potentials Here, we give a full scalar potential for both the type-I and -II DFSZ seesaw models corresponding to their charge assignments as given by Tables 3.4, 3.6, respectively. For the type-I DFSZ seesaw model the full scalar potential is given by V = µ 2 uH† u Hu + µ 2 dH† d Hd + µ 2 1ss⇤ + µ 2 2s0s0⇤ + lu(H† u Hu) 2 + ld(H† d Hd) 2 + l(ss⇤)2 + l0(s0s0⇤)2 + l1(H† u Hd)(H† d Hu) + l2(H† u Hu)(H† d Hd) + l3(H† u Hd)(H† d Hu) + l4(ss⇤)(H† u Hu) + l5(ss⇤)(H† d Hd) + l6( eH † u Hd)(s ⇤)2 + l7(s 0s0⇤)(H† u Hu) + l8(s 0s0⇤)(H† d Hd) + l9(s 0s0⇤)(ss⇤) + k(s2s0⇤ + (s⇤)2s0) + l10( eH † u Hd)s 0⇤ . (A.9) For the type-II DFSZ seesaw model we enlarge the previous model by two addi- tional Higgs Doublets transforming as Φu ⇠ (1/2, 2) and Φd ⇠ (+1/2, 2) under (U(1)Y, U(1)PQ). Therefore the scalar potential contains the following terms, in addi- 134 Appendices tion to those in A.9: V = µ 2 Φu Φ † uΦu + µ 2 Φd Φ † dΦd + ku(H† uΦu)s ⇤ + kd(H† d Φd)s ⇤ + l11(eΦ † dΦu)(s 0⇤)2 + l12( eH † d Φu)(s 0⇤s⇤) + l13( eH † uΦd)(s 0⇤s⇤) + l14(Φ † uΦu) 2 + l15(Φ † dΦd) 2 + l16(Φ † dHd)(H† d Φd) + l17(Φ † uHd)(H† d Φu) + l18(Φ † dHu)(H† uΦd) + l19(Φ † uHu)(H† uΦu) + l20(Φ † dΦu)(Φ † uΦd) + l21(ss⇤)(Φ† uΦu) + l22(ss⇤)(Φ† dΦd) + l23(s 0s0⇤)(Φ† uΦu) + l24(s 0s0⇤)(Φ† dΦd) . (A.10) A.4. Benchmarks Here we present three benchmarks (BM1, BM2, BM3) corresponding to the different mass regions described in Section 4.2.8 where the fermion DM satisfy all experimental and theoretical constraints summarized in Section 4.2.3, see Tables A.1 and A.2. Addi- tionally, these representative points are such that the DM particle N1 constitute 100% of the relic abundance in the Universe, see Table A.3. The values for the dimensionless parameters in the Lagrangian as well as the dimensionful parameters in the scalar potential are shown in Table A.1. Notice that the neutrino Yukawas Yn i (with i = 1, 2, 3) are small and as a result LFV processes are suppressed. We include as example in Table A.2 the branching fractions of µ! eg for each benchmark. Appendices 135 m N 1 [G eV ] m N 2 [G eV ] m N 3 [G eV ] m h R [G eV ] m h I [G eV ] m h ± [G eV ] m h 2 [G eV ] v s [T eV ] µ 2 2 [G eV 2 ] B M 1 10 11 9 31 6 52 0 53 6 48 6 20 .7 10 .4 2. 1 ⇥ 10 5 B M 2 59 .1 18 4 41 0 66 6 67 5 64 5 14 9 1. 08 4. 60 ⇥ 10 5 B M 3 70 7 92 4 94 0 11 32 11 19 11 19 14 98 5. 80 6. 98 ⇥ 10 6 si n a l 1 l 2 l 3 l 4 l 5 l 6 l 8 B M 1 1. 31 ⇥ 10 1 1. 27 ⇥ 10 1 8. 93 ⇥ 10 1 -6 .7 ⇥ 10 1 1. 4 -2 .8 2 ⇥ 10 1 3. 17 ⇥ 10 6 8. 19 ⇥ 10 4 B M 2 1. 57 ⇥ 10 1 1. 30 ⇥ 10 1 1 -1 .4 ⇥ 10 1 1. 1 -1 .9 9 ⇥ 10 1 9. 35 ⇥ 10 3 -6 .7 6 ⇥ 10 2 B M 3 1. 63 ⇥ 10 1 9. 23 ⇥ 10 1 1. 63 1. 63 6. 47 ⇥ 10 1 -3 .4 0 ⇥ 10 1 3. 24 ⇥ 10 2 -1 .4 8 ⇥ 10 1 Y N 1 Y N 2 Y N 3 Y n 1 Y n 2 Y n 3 B M 1 6. 78 ⇥ 10 4 8. 04 ⇥ 10 3 2. 14 ⇥ 10 2 -1 .6 1 ⇥ 10 4 -4 .9 9 ⇥ 10 5 -4 .1 8 ⇥ 10 5 B M 2 3. 86 ⇥ 10 2 1. 20 ⇥ 10 1 2. 68 ⇥ 10 1 -3 .2 2 ⇥ 10 5 -2 .7 5 ⇥ 10 5 -4 .7 0 ⇥ 10 5 B M 3 8. 61 ⇥ 10 2 1. 12 ⇥ 10 1 1. 14 ⇥ 10 1 -6 .3 3 ⇥ 10 6 -8 .8 8 ⇥ 10 5 -3 .1 7 ⇥ 10 5 T a b le A .1 .: T h e v al u es fo r th e d im en si o n fu l p ar am et er s ar e sh o w n o n th e to p ta b le . T h e v al u es fo r th e d im en si o n le ss p ar am et er s in th e sc al ar se ct o r an d th e n eu tr in o se ct o r ar e al so g iv en . 136 Appendices Binv BR(h2 ! J J) BR(h2 ! h1h1) Γ(h1)[MeV] BM1 3.5⇥ 102 9.8⇥ 102 - 3.9 BM2 9.7⇥ 102 9.2⇥ 101 - 4.2 BM3 4⇥ 103 1.6⇥ 102 2.3⇥ 101 3.8 BR(µ ! eg) S T BM1 4.9⇥ 1023 4.3⇥ 103 3.0⇥ 103 BM2 6.8⇥ 1028 2.7⇥ 104 3.2⇥ 103 BM3 3.8⇥ 1028 6.9⇥ 104 3.2⇥ 103 Table A.2.: The top table shows that (BM1, BM2, BM3) satisfy current experimental constraints in the Higgs sector. The bottom one is used to illustrate how these solutions are in agreement with the bounds coming from LFV processes as well as the electroweak precision tests. BR(N1N1 ! bb̄) BR(N1N1 ! J J) BR(N1N1 ! h2h2) BR(N1N1 ! ZZ) BM1 7.8⇥ 101 9.8⇥ 102 - - BM2 5.9⇥ 101 9.9⇥ 102 - 2.34⇥ 102 BM3 1.4⇥ 105 1.4⇥ 102 - 2.37⇥ 101 BR(N1N1 !W+W) sSI[pb] hsvig Ωch2 BM1 - 9.4⇥ 1012 1.7⇥ 1034 1.21⇥ 101 BM2 2⇥ 101 3.4⇥ 1012 1.0⇥ 1032 1.20⇥ 101 BM3 4.7⇥ 101 2.2⇥ 1010 4.5⇥ 1029 1.19⇥ 101 Table A.3.: Main DM annihilation channels in the model. Finally, Table A.3 shows the main DM annihilation channels in the model and prediction in the DM sector. Colophon This thesis was made in LATEX 2# using the “hepthesis” class [287]. 137 138 Bibliography [1] C. Bonilla, L. M. G. de la Vega, J. M. Lamprea, R. A. Lineros, and E. Peinado, New J. Phys. 22, 033009 (2020), 1908.04276. [2] L. M. G. de la Vega, N. Nath, and E. Peinado, Nucl. Phys. B 957, 115099 (2020), 2001.01846. [3] C. Bonilla, L. M. G. de la Vega, R. Ferro-Hernandez, N. Nath, and E. Peinado, Phys. Rev. D 102, 036006 (2020), 2003.06444. [4] L. M. G. de la Vega, N. Nath, S. Nellen, and E. Peinado, Eur. Phys. J. C 81, 608 (2021), 2102.03631. [5] L. M. G. de la Vega, L. J. 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Feynman diagrams leading to dark matter annihilation in the thermal freeze-out process (left) and dark matter direct detection (right). . . . . . . . . . . . . . . . . . . . 42 3.4. Left panel: Exclusion regions in the (MZ0 , g0) plane for the B-L model. Right panel: Exclusion regions for the spin independent cross-section in the (Mc, sSI) plane for the B-L model. The light-purple shaded area corresponds to the constraint set by the current COHERENT-CsI data [124]. The limits set by the future reactor-based CEnNS experiment SBC [125, 126], is presented using the black long dashed line (SBC-CEnNS ). Cosmological constraints [127], beam dump experiments [128–138], BABAR [139] and LHCb dark photon searches [140] are presented using the red, light- green, light-brown and sky-blue regions, respectively. Limits set by the dark matter experiments are presented using the light-orange, light-cyan, and light-yellow regions for the SBC-DM [125], XENON1T [53], and PandaX-II [141] experiments, respectively (see text for more details). In the right panel, the argon n-floor background [142] has been marked using dotted-magenta curve. . . . . . . . . . . . . . . . . . . . . 44 3.5. Same as Fig. (3.4) but for MII U(1)0 models as given by Table (3.3). . . . . . . . . 46 3.6. Same as Fig. (3.4) but for the MIII U(1)0 models as given by Table (3.3). . . . . . . 47 3.7. Same as Fig. (3.4) but for MIV U(1)0 models as given by Table (3.3). Oscillation limits are obtained from the analysis made in [144]. . . . . . . . . . . . . . . . . . . . 49 155 156 LIST OF FIGURES 3.8. UV complete diagram within the DFSZ type-I seesaw framework as apparent from Eqs. 3.77 and 3.78. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.9. UV complete diagram within the DFSZ type-II seesaw framework as apparent from Eqs. 3.86, and 3.87. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.10. Exclusion region plot in the (tan a tan b) plane obtained from the non-observation of the t ! hc flavor violating decay. The gray colored region is excluded by ATLAS data [189], the purple colored region is expected to be probed in the future by the HL-LHC experiment [190] and the orange region will be further probed by ILC or CLIC [191]. The uncolored region (see white thin band) predicts a branching ratio beyond the sensitivity of these experiments. The dashed line indicates limit of no flavor violation in light Higgs Yukawa couplings. . . . . . . . . . . . . . . . . . 69 4.1. One-loop Feynman diagram for neutrino mass generation. . . . . . . . 77 4.2. Feynman diagrams for the annihilation channels of the fermion dark matter in the model. On the left annihilation into SM particles. On the right annihilation into Majorons and Higgses. . . . . . . . . . . . . . . . 81 4.3. Dark matter mass sSI plane showing the solutions in the model that satisfy all theoretical and experimental constraints given in Section 4.2.3. The latest bound on direct dark matter detection is set by the XENON1T experiment [212] (top shaded area). The dashed lines represent the expected sensitivities in forthcoming experimental searches such as XENONnT [220], LUX-ZEPLIN (LZ) [221], DarkSide 20k [222], DAR- WIN [223] and PandaX-4T [224]. . . . . . . . . . . . . . . . . . . . . . . 82 4.4. Predictions for the velocity averaged cross section of dark matter anni- hilation into gamma rays hsvig as function of the dark matter mass mN1 . The dashed line represent the limit set by Fermi-LAT satellite results [213]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5. Feynman diagram for Dirac neutrino masses in Type-II DFSZ scenario. . . . . . . . 88 LIST OF FIGURES 157 4.6. Exclusion region plots (colored regions are excluded) in (MΦu fa) plane for a type-II Dirac seesaw mechanism. Three benchmark values for µy = 1GeV, 1MeV, 1keV have been adopted, respectively. The plots are presented by using the limits on neutrino mass from KATRIN Aker:2019uuj which gives mn < 1.1 eV at 90% C.L. (left panel) and Planck [42] ∑ mn < 0.12 eV (at 95 % confidence level using TT, TE, EE + lowE + lensing + BAO) (right panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.7. Feynman diagram for Dirac neutrino masses in alternative one-loop DFSZ scenario. Here left panel respects PQ charge assignment, whereas right panel respects the remnant ZPQ 2 charge assignment that arises due to PQ symmetry breaking. . . . . . 92 4.8. Feynman diagram for Dirac neutrino masses in one-loop DFSZ scenario. Here left panel respects PQ charge assignment, whereas right panel respects remnant ZPQ 4 charge assignment, arises due to PQ symmetry breaking. . . . . . . . . . . . . . . 96 4.9. t-channel annihilation into Z0/Z0 . . . . . . . . . . . . . . . . . . . . . . 107 4.10. Z/Z0 mediated resonant annihilation into f̄ f . . . . . . . . . . . . . . . 107 4.11. Higgstrahlung annihilation . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.12. Dark matter annihilation channels for the determination of the freeze- out relic density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.13. Case 1: t-channel annihilation into a Z0 pair. Parameter space in the Mc gc plane (where gc = g D cos qX ⇡ g D ) excluded by relic density overabundance. Area shaded in green results in relic density overabun- dance, while the green line corresponds to ΩDM = 0.1195. The area indicated in Magenta with the label “SIDM" corresponds to the region where dark matter self-interactions are consistent with astrophysical ob- servations. Within this area, the blue region corresponds to a mediator mass MZ0 of 10 MeV, considering the uncertainty as estimated in the text. 108 158 LIST OF FIGURES 4.14. Spin Independent direct detection cross section (sSI) as a function of dark matter mass, for the Z0 resonant case (s-channel). In the green lines the product g D sin qX is fixed to values between 108 and 103 as indicated in the caption. The purple lines show the experimental limit on sSI set by the LZ [265] and CRESST-III [266] experiments. The dashed magenta line shows the neutrino floor on Argon [267]. Note that the LZ experiment target nucleus is Xenon, and it has not yet reached the Xenon neutrino floor. . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.15. Low energy constraints and resonant dark matter direct detection con- straints. In this figure we show the low energy constraints from Atomic Parity Violation [254], collider (BaBar [257],CMS [258] and LHCb [259]) and beam dump (NA64 [260], E141 [130], E137 [129],E774 [131], KEK [132] and Orsay [136]) experiments on the MZ0 sin qX parameter space. These constraints are obtained as discussed in Section 4.4.6. We also show in the same space the constraints derived from direct detection experiments (LZ [265] and CRESST-III [266]) in the resonant dark matter models, as discussed in Section 4.4.8. . . . . . . . . . . . . . . . . . . . . 110 4.16. Projected constraints for future CEnNS experiment, assuming a 1 MW reactor neutrino source at 3 meters from a 10kg Argon scintillating bubble chamber with a 1 year exposure. Exclusion limits for ∆c2 = 2.7. First plot corresponds to d Md plot for future CEvNS. Second plot corresponds to e Md plot for future CEvNS. . . . . . . . . . . . . . . 120 4.17. Tree level Feynman diagram of dark matter annihilation in the Early Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.18. Numerical relationship between my and gD. In the green line, relic density corresponds to the observed value. . . . . . . . . . . . . . . . . 124 4.19. Exclusion plot for the kinetic mixing given a value of the dark matter My = 10 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.20. Exclusion plot for the kinetic mixing given a value of the dark matter My = 100 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.21. Exclusion plot for the kinetic mixing given a value of the dark matter My = 1000 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 List of tables 1.1. Fermion and Scalar content of the SM, in the interaction basis. For each fermion type, there are three generations in the SM. The SU(2)L compo- nents of the fields are shown explicitly, while the SU(3)C components and indices are not shown. . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1. Assigned gauge and flavor irreps of D4 lepton flavor model. As ex- plained in the text, we omit the quark sector. . . . . . . . . . . . . . . . 23 3.2. Simple anomaly-free U(1)X symmetries. Anomalies are cancelled when the gauge symmetry is GSM ⇥U(1)X and no other fermions are consid- ered. If more fermions are added to the theory, the anomaly cancellation conditions of eqs ?? must be met. . . . . . . . . . . . . . . . . . . . . . . 37 3.3. Singlet scalar fields fi having charges i under U(1)0. . . . . . . . . . . . . . . . . 41 3.4. Field content and transformation properties of the PQ-symmetry under the DFSZ type-I seesaw model, where i = 1, 2, 3 represent families of three quarks. . . . . . . 53 3.5. Vector like fermions and their transformation properties of the PQ-symmetry under the DFSZ type-I seesaw model, where C = L, R. . . . . . . . . . . . . . . . . . . 55 3.6. Field content and transformation properties of the leptonic fields and the scalar field s0, where i = 1, 2, 3 represent the three lepton families. . . . . . . . . . . . . . . . 59 3.7. Best-fit values of the model parameters in the quark sector are shown in the upper table. The global best-fit as well as their 1s error [182, 183] for the various observables are given in the second and third columns of the lower table. Also, the best-fit values of the various observables are listed in the last column of the lower table. . . . . . . 64 159 160 LIST OF TABLES 3.8. Best-fit values of the model parameters in the lepton sector are shown in the upper table. The global best-fit as well as their 1s error [182, 183] for the various observables are given in the second and third columns of the lower table. Also, the best-fit values of the various observables are listed in the last column of the lower table. . . . . . . 65 4.1. Particle content and charge assignments of the model. . . . . . . . . . 75 4.2. Quantum numbers in the DFSZ axion model [228]. . . . . . . . . . . . . 86 4.3. Fields content and transformation properties under PQ symmetry in type-II seesaw framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4. Field content and transformation properties under PQ symmetry in the alternative one-loop mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5. Fields content and transformation properties under PQ symmetry in the one-loop mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.6. Fields content and transformation properties under PQ symmetry in the one-loop mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.7. Matter content of the dark matter with mass mixed U(1)D gauge boson. Quarks and right handed charged leptons, not shown in this table, are not charged under the U(1)D gauge symmetry. The model contains two dark charges, we have chosen to absorb one of them into the definition of the dark gauge coupling, leaving the dark matter charge QD free. . . 98 A.1. The values for the dimensionful parameters are shown on the top table. The values for the dimensionless parameters in the scalar sector and the neutrino sector are also given. . . . . . . . . . . . . . . . . . . . . . . . 135 A.2. The top table shows that (BM1, BM2, BM3) satisfy current experimental constraints in the Higgs sector. The bottom one is used to illustrate how these solutions are in agreement with the bounds coming from LFV processes as well as the electroweak precision tests. . . . . . . . . . . . 136 A.3. Main DM annihilation channels in the model. . . . . . . . . . . . . . . 136