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.

Japan

Nicholas J Benoit

Takuya Morozumi

Yusuke Shimizu

Kenta Takagi

Akihiro Yuu

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

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.

Tatsuo Kobayashi

Morimitsu Tanimoto

Takuya H. Tatsuishi

Modular -invariant flavor model in SU(5) grand unified theory

We study the spontaneous <math><mi>C</mi><mi>P</mi></math> violation through the stabilization of the modulus <math><mi>τ</mi></math> in modular invariant flavor models. The <math><mi>C</mi><mi>P</mi></math>-invariant potential has the minimum only at <math><mi>Re</mi><mo>[</mo><mi>τ</mi><mo>]</mo><mo>=</mo><mn>0</mn></math> or <math><mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>(</mo><mi>mod</mi><mtext> </mtext><mn>1</mn><mo>)</mo></mrow></math>. From this result, we study <math><mi>C</mi><mi>P</mi></math> violation in modular invariant flavor models. The physical <math><mi>C</mi><mi>P</mi></math> phase is vanishing. The important point for the <math><mi>C</mi><mi>P</mi></math> conservation is the <math><mi>T</mi></math> transformation in the modular symmetry. One needs the violation of <math><mi>T</mi></math> symmetry to realize the <math><mi>C</mi><mi>P</mi></math> violation.

Hikaru Uchida

<math><mi>C</mi><mi>P</mi></math> violation in modular invariant flavor models

We study a flavor model with A 4 symmetry which originates from S 4 modular group. In S 4 symmetry, Z 2 subgroup can be anomalous, and then S 4 can be violated to A 4. Starting with a S 4 symmetric Lagrangian at the tree level, the Lagrangian at the quantum level has only A 4 symmetry when Z 2 in S 4 is anomalous. We obtain modular forms of two singlets and a triplet representations of A 4 by decomposing S 4 modular forms into A 4 representations. We propose a new A 4 flavor model of leptons by using those A 4 modular forms. We succeed in constructing a viable neutrino mass matrix through the Weinberg operator for both normal hierarchy (NH) and inverted hierarchy (IH) of neutrino masses. Our predictions of the CP violating Dirac phase δ CP and the mixing sin2 θ 23 depend on the sum of neutrino masses for NH.

Takuya Tatsuishi

New A 4 lepton flavor model from S 4 modular symmetry

We study the modulus stabilization in an <math><msub><mi>A</mi><mn>4</mn></msub></math> model whose <math><msub><mi>A</mi><mn>4</mn></msub></math> flavor symmetry is originated from the <math><msub><mi>S</mi><mn>4</mn></msub></math> modular symmetry. We can stabilize the modulus so that the <math><msub><mi>A</mi><mn>4</mn></msub></math> invariant superpotential leads to the realistic lepton masses and mixing angles. We also discuss the phenomenological aspect of the present model as a consequence of the modulus stabilization.

<math><msub><mi>A</mi><mn>4</mn></msub></math> lepton flavor model and modulus stabilization from <math><msub><mi>S</mi><mn>4</mn></msub></math> modular symmetry

We study a flavor model that the quark sector has the <math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math> modular symmetry while the lepton sector has the <math><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub></math> modular symmetry. Our model leads to characteristic quark mass matrices which are consistent with experimental data of quark masses, mixing angles and the CP violating phase. The lepton sector is also consistent with the experimental data of neutrino oscillations. We also study baryon and lepton number violations in our flavor model.

Finite modular subgroups for fermion mass matrices and baryon/lepton number violation

We discuss the relation between the CP violation of the quark mixing and that of the lepton mixing by investigating a CP violating observable, the Jarlskog invariant, as well as the CP violating Dirac phase. The down-type quark mass matrix with three zeros is given in terms of the minimal number of parameters, while the up-type quark mass matrix is diagonal. These quark mass matrices leading to the successful CKM mixing angles and CP violation are embedded in both the Pati-Salam and SU(5) models. The leptonic Jarlskog invariant J  CP  l   (as well as CP violating Dirac phase) is examined for two cases: the neutrino mass matrix is diagonal or non-diagonal, where no additional CP violating phase is introduced apart from the Majorana phases. In the case of the diagonal neutrino mass matrix, the favorable sign of the leptonic CP violation is obtained, however, the magnitude of J  CP  l   is at most   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (10−4), which is too small compared with the expected value from the observation −0.02. In the case of the non-diagonal neutrino mass matrix where the tri-bimaximal mixing pattern is taken, we obtain the successful J  CP  l   up to its sign.

Shunya Takahashi

Sign of CP violating phase in quarks and leptons

We study the phenomenological implications of the modular symmetry Γ(3) ≃ A 4 of lepton flavors facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of modulus τ, which is the only source of modular invariance breaking. We introduce no flavons in contrast with the conventional flavor models with A 4 symmetry. We classify our neutrino models along with the type I seesaw model, the Weinberg operator model and the Dirac neutrino model. In the normal hierarchy of neutrino masses, the seesaw model is available by taking account of recent experimental data of neutrino oscillations and the cosmological bound of sum of neutrino masses. The predicted sin2 θ 23 is restricted to be larger than 0.54 and δ CP = ±(50°-180°). Since the correlation of sin2 θ 23 and δ CP is sharp, the prediction is testable in the future. It is remarkable that the effective mass m  ee  of the neutrinoless double beta decay is around 22 meV while the sum of neutrino masses is predicted to be 145 meV. On the other hand, for the inverted hierarchy of neutrino masses, only the Dirac neutrino model is consistent with the experimental data.

Naoya Omoto

Modular A4 invariance and neutrino mixing

Abstract We revisit the  model for leptons in light of the new result of NuFIT 3.2. We introduce a new flavon  transforming as an  singlet  or , which couples to both charged leptons and neutrinos in next-to-leading-order operators. The model consists of five parameters: the lightest neutrino mass , the vacuum expectation value of , and three CP-violating phases after inputting the experimental values of  and . The model with the  singlet flavon gives the prediction of  around the best fit of NuFIT 3.2 while staying near the maximal mixing of . Inputting the experimental mixing angles with the  error-bar, the Dirac CP-violating phase is clearly predicted to be –, which will be tested by the precise observed value in the future. In order to get the best-fit value , the sum of three neutrino masses is predicted to be larger than meV. The cosmological observation for the sum of the neutrino masses will also provide a crucial test of our predictions. It is remarked that the model is consistent with the experimental data only for the normal hierarchy of neutrino masses.

South Korea

Sin Kyu Kang

Revisiting the model for leptons in light of NuFIT 3.2 Latest oscillation results from NOvA. An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists

We discuss a supersymmetric model with discrete flavor symmetry . The additional scalar fields which contribute masses of leptons in the Yukawa terms are introduced in this model. We analyze their scalar potential and find that they have various vacuum structures. We show the relations among 24 different vacua and classify them into two types. We derive expressions of the lepton mixing angles, Dirac CP violating phase and Majorana phases for the two types. The model parameters which are allowed by the experimental data of the lepton mixing angles are different for each type. We also study the constraints on the model parameters which are related to Majorana phases. The different allowed regions of the model parameters for the two types are shown numerically for a given region of two combinations of the CP violating phases.

Hideaki Okane

Hiroki Sakamoto

Hiroyuki Umeeda

Phenomenological aspects of possible vacua of a neutrino flavor modelSupported by JSPS KAKENHI Grant Number JP17K05418 (T.M.). This work is also supported in part by Grants-in-Aid for Scientific Research [No. 16J05332 (Y.S.), Nos. 24540272, 26247038, 15H01037, 16H00871, and 16H02189 (H.U.)] from the Ministry of Education, Culture, Sports, Science and Technology in Japan. H.O. is also supported by Hiroshima Univ. Alumni Association