We revisit a supersymmetric flavor model, based on the symmetries of <inline-formula><tex-math id='TM0001' notation='LaTeX'>$SU(2)_L \times A_4 \times Z_3 \times U(1)_R$</tex-math></inline-formula>, which extends the original Altarelli and Feruglio construction by introducing flavon and driving superfields responsible for the spontaneous breaking of the flavor symmetry in order to obtain a nonzero reactor angle. Vacuum alignments of the flavon fields are achieved through the minimization of the scalar potential derived from the superpotential. This setup leads to specific mass matrices for the charged leptons and neutrinos that are consistent with current experimental data, including the measured values of the lepton mixing angles and neutrino mass squared differences. We investigate whether the model can simultaneously accommodate successful thermal leptogenesis. In particular, we analyze the CP asymmetry generated by the decay of heavy Majorana neutrinos, the resulting lepton asymmetry, and its conversion to the baryon asymmetry through the electroweak sphalerons. We find, however, that the CP asymmetry is zero in the leading order of the model since the Dirac neutrino mass matrix has simple texture. Then we consider the next-to-leading order in Yukawa interactions of the Dirac neutrinos. As a result, we can realize the baryon asymmetry of the universe at the present universe. By numerically scanning the parameter space, we identify the regions consistent with both neutrino oscillation data and the observed baryon asymmetry. In the specific case such that one of the couplings for the right-handed Majorana neutrinos is a real parameter, the predicted lightest neutrino mass is at least 5 meV and 15 meV for the normal and inverted neutrino mass hierarchies, respectively. In addition, the range of the Majorana phases may be tested in future experiments.

China

Takaaki Nomura

Japan

Yusuke Shimizu

Towa Takahashi

Revisiting the Flavor Model and Leptogenesis

We have investigated an approach for determining Majorana-type phases using the time evolution of lepton family numbers. We show how the second-order time derivative of the expectation values for the lepton family numbers depends on the sum of the Majorana-type phases. Furthermore, others have connected the Majorana-type phases to the orientation of unitary triangles for the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix, and the usual Majorana phases <inline-formula><tex-math id='TM0001' notation='LaTeX'>$\alpha _{21}$</tex-math></inline-formula> and <inline-formula><tex-math id='TM0002' notation='LaTeX'>$\alpha _{31}$</tex-math></inline-formula>. Theoretically, this allows for the extraction of the orientation of the unitary triangles and the Majorana phases from lepton family numbers. We study three example situations: first, how to extract the Majorana-type phases and the lightest neutrino mass for three massive neutrinos, and when a neutrino is massless, i.e. <inline-formula><tex-math id='TM0003' notation='LaTeX'>$m_{1,3}=0$</tex-math></inline-formula>; second, the determination of the Majorana phase and the lightest neutrino mass for a two-generation toy model; and third, simplified realizations of the type I seesaw model with two gauge singlet neutrinos and two families of lepton doublets. We calculate how the Majorana phases and Majorana-type phases are related to CP violation for leptogenesis at high energies. At first, the effective Majorana mass matrix is parametrized with real and positive diagonal elements at low energies. In this basis, the phases of the off-diagonal elements are related to the CP-violating phases in the PMNS matrix. We explicitly show the relationship between the single Majorana phase and the phase of the effective Majorana mass matrix for the toy model with two generations of active neutrinos. Then, for the model with two gauge singlet neutrinos and two families of lepton doublets, we study the effective Majorana mass matrix generated by the seesaw model. In that model, we can show how the Majorana phase at low energy is related to the two CP-violating phases of the seesaw matrix. This relationship between the phases depends on the lepton number asymmetries of the heavy Majorana neutrino decays for the toy models.

Nicholas J Benoit

Yuta Kawamura

Saki Hamada

Takuya Morozumi

Kei Yamamoto

Determination of Majorana-Type Phases from the Time Evolution of Lepton Numbers

We study a relic density of the “flavino” dark matter in modified Altarelli and Feruglio A 4 model which is respecting the SU(2) L  × A 4 × Z 3 × U(1) R  symmetry. We calculate the Lagrangian from the superpotential in the model. In estimating the relic density, we consider the relevant interactions from the Lagrangian that realize the vacuum expectation value alignments and charged lepton masses where we assume that the supersymmetry breaking effects are small for “flavon” sector. As a result, we find the degenerate masses between the lightest “flavon” and “flavino”, and only two parameters in the potential determines the relic density. Then the allowed parameter space of these parameters are estimated from the relic density calculation taking a constraint from lepton flavor violation into account. We also briefly discuss other dark matter physics such as the direct detection, indirect detection, and collider search.

Flavino dark matter in a non-Abelian discrete flavor model

We present a formulation of lepton family numbers, based on quantum field theory, for neutrino oscillation phenomenology that can be applied to nonrelativistic and relativistic energies for neutrinos. It is formulated for both types of neutrinos, Dirac and Majorana. The formulation is constructed as the time evolution of a lepton family number density operator. Then, the time evolution of the lepton family number density operator becomes dependent on the mass and new features appear. The expectation value of the density operator is evaluated for the initial state with a Gaussian distribution for the momentum amplitude. This enables us to study wave-packet-like decoherence effects. We show in the nonrelativistic regime, the type of neutrino mass are distinguishable even under the presence of wave-packet-like decoherence effects.

Indonesia

Apriadi Salim Adam

Nicholas J. Benoit

Yamato Matsuo

Naoya Toyota

Spacetime evolution of lepton number densities and wave-packet-like effects for neutrino flavor and chiral oscillations in quantum field theory

We propose an SU(5) GUT model extended with two additional pairs of 10 representation vector-like fermions. The CDF collaboration W boson mass anomaly is explained by using the VEV of a real <math><mi>SU</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><mi>L</mi></mrow></msub></math> triplet scalar coming from the 24 representation Higgs. The vector-like fermions are decomposed partly into vector-like quark doublets. Those vector-like quark doublets acquire mass from two sources; through the Yukawa interaction with the real <math><mi>SU</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><mi>L</mi></mrow></msub></math> triplet via a type-II seesaw-like mechanism. And, they acquire mass from the 24 representation Higgs. We assume that the mass for the vector-like quark doublets is expressed in terms of the real triplets mass. By combining the constraints on the vector-like quark masses with those on the heavy Higgs boson masses, we can obtain the narrow allowed mass ranges for the vector-like quark doublet and the real triplet. Therefore, our model can be tested in searches for these particles in the near future. In addition, the two additional pairs of vector-like fermions allow the SM gauge couplings to unify successfully at <math><msub><mrow><mi>M</mi></mrow><mrow><mi>GUT</mi></mrow></msub><mo>≈</mo><mn>5.1</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>15</mn></mrow></msup></math> GeV. Our model is also testable by the future Hyper-Kamiokande experiment via the proton decay lifetime <math><msub><mrow><mi>τ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo><mo>&lt;</mo><mn>1.0</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>35</mn></mrow></msup></math> years.

Shonosuke Takeshita

boson mass and grand unification via the type-II seesaw-like mechanism

Abstract We propose a simple non-supersymmetric lepton flavor model with A$_{4}$ symmetry. The A$_{4}$ group is a minimal one that includes triplet irreducible representation. We introduce three Higgs doublets, which are assigned as a triplet of the A$_{4}$ symmetry. It is natural that there are three generations of the Higgs fields, as with the standard model fermions. We analyze the potential and find the vacuum expectation values for the local minimum. In the vacuum expectation values, we obtain the charged lepton, Dirac neutrino, and right-handed Majorana neutrino mass matrices. By using the type-I seesaw mechanism, we get the left-handed Majorana neutrino mass matrix. In the NuFIT 5.1 data, we predict the Dirac CP phase and the Majorana phases for the only inverted neutrino mass hierarchy. In particular, the Dirac CP phase and lepton mixing angle θ$_{23}$ are strongly correlated. If θ$_{23}$ is more precisely measured, the Dirac CP phase is more precisely predicted, and vice versa. We also predict the effective mass for neutrinoless double beta decay m$_{ee}$ ≃ 47.1 [meV] and the lightest neutrino mass m$_{3}$ ≃ 0.789–1.43 [meV]. This will be testable with our model in near-future neutrino experiments.

Yukimura Izawa

Hironori Takei

Non-SUSY lepton flavor model with the three Higgs doublet model

Abstract The type-I seesaw model is a common extension to the Standard Model that describes neutrino masses. The type-I seesaw introduces heavy right-handed neutrinos with Majorana mass that transform as Standard Model electroweak gauge singlets. We initially study a case with two right-handed neutrinos called the 3-2 model. At an energy scale above the right-handed neutrinos, the effective neutrino mass matrix is rank degenerate, implying that the lightest neutrino is massless. After considering renormalization effects below the two right-handed neutrinos, the effective neutrino mass matrix remains rank degenerate. Next, we study a model with three right-handed neutrinos called the 3-3 model. Above the energy scale of the three right-handed neutrinos, we construct the effective neutrino mass matrix to be rank degenerate. After solving for the renormalization effects to energies below the three right-handed neutrinos, we find that the rank of the effective neutrino mass matrix depends on the kernel solutions of the renormalization group equations. We prove that for the simplest kernel solutions the effective neutrino mass matrix remains rank degenerate.

Kenta Takagi

Akihiro Yuu

Renormalization group effects for a rank degenerate Yukawa matrix and the fate of the massless neutrino NuFit 5.1 webpage

We propose an A 4 modular invariant flavor model of leptons, in which both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ. The value of the modulus τ is restricted by the observed lepton mixing angles and lepton masses for the normal hierarchy of neutrino masses. The predictive Dirac CP phase δ  CP  is in the ranges [0°, 50°], [170°, 175°] and [280°, 360°] for Re [τ] &lt; 0, and [0°, 80°], [185°, 190°] and [310°, 360°] for Re [τ] &gt; 0. The sum of three neutrino masses is predicted in [60, 84] meV, and the effective mass for the 0νββ decay is in [0.003, 3] meV. The modulus τ links the Dirac CP phase to the cosmological baryon asymmetry (BAU) via the leptogenesis. Due to the strong wash-out effect, the predictive baryon asymmetry Y  B  can be at most the same order of the observed value. Then, the lightest right-handed neutrino mass is restricted in the range of M 1 = [1.5, 6.5] × 1013 GeV. We find the correlation between the predictive Y  B  and the Dirac CP phase δ  CP . Only two predictive δ  CP  ranges, [5°, 40°] (Re [τ] &gt; 0) and [320°, 355°] (Re [τ] &lt; 0) are consistent with the BAU.

South Korea

Hiroshi Okada

Morimitsu Tanimoto

Takahiro Yoshida

Modulus τ linking leptonic CP violation to baryon asymmetry in A 4 modular invariant flavor model

Abstract We revisit the time evolution of the lepton family number for an SU(2) doublet consisting of a neutrino and a charged lepton. The lepton family number is defined through the weak basis of the SU(2) doublet, where the charged lepton mass matrix is real and diagonal. The lepton family number carried by the neutrino is defined by the left-handed current of the neutrino family. For this work we assume that the neutrinos have Majorana mass. This Majorana mass term is switched on at time  and the lepton family number evolves. Since the operator in the flavor eigenstate is continuously connected to that of the mass eigenstate, the creation and annihilation operators for the two eigenstates are related to each other. We compute the time evolution of all lepton family numbers by choosing a specific initial flavor eigenstate for a neutrino. The evolution is studied for relativistic and non-relativistic neutrinos. The non-relativistic region is of particular interest for the cosmic neutrino background predicted from Big Bang models. In that region we find that the lepton family numbers are sensitive to the Majorana and Dirac phases, the absolute mass, and the mass hierarchy of neutrinos.

Yuya Tokunaga

Time evolution of lepton number carried by Majorana neutrinos

Abstract We present a flavor model with  modular invariance in the framework of SU(5) grand unified theory (GUT). The  modular forms of weights  and  give the quark and lepton mass matrices with a common complex parameter, the modulus . The GUT relation of down-type quarks and charged leptons is imposed by the vacuum expectation value (VEV) of the adjoint 24-dimensional Higgs multiplet in addition to the VEVs of  and  Higgs multiplets of SU(5). The observed Cabibbo–Kobayashi–Maskawa and Pontecorvo–Maki–Nakagawa–Sakata mixing parameters as well as the mass eigenvalues are reproduced properly. We discuss the leptonic charge–parity phase and the effective mass of the neutrinoless double beta decay with the sum of neutrino masses.