<p>In a predictive modular invariant theory of flavor there should exist correlations between the quark and lepton observables, induced by the common modulus, by joint experimental constraints and possibly, in grand unified theories (GUTs), by common gauge multiplets. So far, in non-GUT bottom-up modular flavor models, these observables have been analyzed separately, making it impossible to investigate their interconnections. We perform a joint analysis of quark and lepton observables (22 altogether, 18 being measured) and of their correlations in a modular flavor model of the indicated class. The model is based on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>O</mi></math> flavor symmetry and, within its class, it is characterized by the minimal number of free parameters (14 real constants). The joint analysis shows that the model is in good agreement with the experimental data for normal neutrino mass ordering, while predicting the leptonic Dirac <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi><mi>V</mi></math>) phase (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>δ</mi><mrow><mi>C</mi><mi>P</mi></mrow></msub></math>), the two Majorana <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi><mi>V</mi></math> phases (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mn>2</mn></msub></math>), the lightest neutrino mass (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mn>1</mn></msub></math>) and the effective neutrino masses probed by beta and neutrinoless double beta decay (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>β</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mrow><mi>β</mi><mi>β</mi></mrow></msub></math>). A detailed comparison of the separate (lepton-only and quark-only) and combined (lepton and quark) fit results shows differences in best-fit values and jointly allowed regions, that reflect a nontrivial interplay between quark and lepton observables in the model. Most importantly, our analysis highlights the existence of significant correlations between various pairs of such observables. For instance, the ratio of the strange and bottom quark masses, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>r</mi><mrow><mi>s</mi><mi>b</mi></mrow></msub></math>, is strongly negatively correlated with each of the three lepton mixing angles and with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>δ</mi><mrow><mi>C</mi><mi>P</mi></mrow></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>β</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mrow><mi>β</mi><mi>β</mi></mrow></msub></math>, while being positively correlated with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mn>2</mn></msub></math>. These findings, that are missed in separate analyses of quark and lepton flavor sectors, as well as in (GUT and non-GUT) models with a relatively large number of free parameters, fall within ranges that can be tested by current and future experiments.</p>

China

Gui-Jun Ding

Italy

Eligio Lisi

Antonio Marrone

Japan

S.T. Petcov

Interplay and correlations between quark and lepton observables in modular symmetry models

In the formalism of the non-supersymmetric modular invariance approach to the flavour problem the elements of the Yukawa coupling and fermion mass matrices are expressed in terms of polyharmonic Maaß modular forms of level in addition to the standard modula forms of the same level and a small number of constant parameters. Non-trivial polyharmonic Maaß forms exist for zero, negative and positive integer modular weights. Employing the finite modula group as a flavour symmetry group and assuming that the three left-handed lepton doublets furnish a triplet irreducible representation of , we construct all possible 7- and 8-parameter lepton flavour models in which the neutrino masses are generated either by the Weinberg effective operator or by the type-I seesaw mechanism. We identify the phenomenologically viable models and obtain predictions for each of these models for the neutrino mass ordering, the absolute neutrino mass scale, the Dirac and Majorana CP-violation phases and, correspondingly, for the sum of neutrino masses and the neutrinoless double beta decay effective Majorana mass. We comment on how these models can be tested and conclude that they are all falsifiable. Detailed analyses are presented in the case of three representative benchmark lepton flavour scenarios.

Jun-Nan Lu

Bulgaria

Bu-Yao Qu

Non-holomorphic modular

The problem of normalisation of the modular forms in modular invariant lepton and quark flavour models is discussed. Modular invariant normalisations of the modular forms are proposed.

On the normalisation of the modular forms in modular invariant theories of flavour

We consider low-energy tests of low-scale leptogenesis based on the type I seesaw scenario with three right-handed singlet neutrinos <math><msub><mrow><mi>ν</mi></mrow><mrow><mi>l</mi><mi>R</mi></mrow></msub></math>. In this scenario, successful leptogenesis is possible for quasi-degenerate in mass heavy Majorana neutrinos <math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub></math>, <math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>≅</mo><mi>M</mi></math>, <math><mo>|</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>≪</mo><mi>M</mi></math>, <math><mi>i</mi><mo>≠</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math>, heavy Majorana neutrino masses <math><mi>M</mi><mo>∼</mo><mo>(</mo><mn>0.05</mn><mo>−</mo><mn>5</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>)</mo></math> GeV, and <math><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></math> charged current and neutral current weak interaction couplings as large as <math><mi>O</mi><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></math>. We derive the constraints on the corresponding leptogenesis parameter space from the existing data from low-energy experiments, including the limits from the experiments on <math><mi>μ</mi><mo>→</mo><mi>e</mi><mi>γ</mi></math> decay and on the rate of <math><mi>μ</mi><mo>−</mo><mi>e</mi></math> conversion in gold. We show also that the planned and upcoming experiments on charged lepton flavour violation with <math><msup><mrow><mi>μ</mi></mrow><mrow><mo>±</mo></mrow></msup></math>, MEG II on the <math><mi>μ</mi><mo>→</mo><mi>e</mi><mi>γ</mi></math> decay, Mu3e on <math><mi>μ</mi><mo>→</mo><mi>e</mi><mi>e</mi><mi>e</mi></math> decay, Mu2e and COMET on <math><mi>μ</mi><mo>−</mo><mi>e</mi></math> conversion in aluminium and PRISM/PRIME on <math><mi>μ</mi><mo>−</mo><mi>e</mi></math> conversion in titanium, can probe significant region of the viable leptogenesis parameter space, and thus have a potential for a discovery. Experiments on <math><mi>τ</mi><mo>→</mo><mi>e</mi><mi>e</mi><mi>e</mi><mo>(</mo><mi>μ</mi><mi>μ</mi><mi>μ</mi><mo>)</mo></math> and <math><mi>τ</mi><mo>→</mo><mi>e</mi><mo>(</mo><mi>μ</mi><mo>)</mo><mi>γ</mi></math> decays (e.g., BELLE II) also can probe a part of the leptogenesis parameter space, although a relatively small one.

A. Granelli

Belgium

J. Klarić

Tests of low-scale leptogenesis in charged lepton flavour violation experiments

Working within the Standard Model effective field theory approach, we examine the possibility to test charged lepton flavour violating (cLFV) new physics (NP) by angular measurements of the outgoing electron and positrons in the polarised <math><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>→</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup></math> decay. This decay is planned to be studied with a sensitivity to the branching ratio of <math><msub><mrow><mi>B</mi></mrow><mrow><mi>μ</mi><mo>→</mo><mn>3</mn><mi>e</mi></mrow></msub><mo>∼</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>15</mn></mrow></msup></math> by the Mu3e experiment at PSI. To illustrate the potential of these measurements, we consider a set of eight effective operators generating the decay at the tree level, including dimension-five dipole operators and dimension-six scalar and vector four-fermion operators of different chiralities. We show that if the polarised <math><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>→</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup></math> decay is observed and is induced by a single operator, data from three angular observables – two P-odd asymmetries and one T-odd asymmetry – can allow to discriminate between all considered operators with the exception of scalar and vector operators of opposite chirality. For these two types of operators, the P-odd asymmetry is maximal in magnitude; they can be distinguished, in principle, by measuring the helicities of the outgoing positrons. The observation of a non-zero T-odd asymmetry would indicate the presence in the decay rate of a dipole and vector operator interference term that is T (CP) violating.

P.D. Bolton

Measurements of → 3 Decay With Polarised Muons as a Probe of new Physics

We consider sub-TeV scale flavoured resonant leptogenesis within the minimal type-I seesaw scenario with two right-handed singlet neutrinos <math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math> forming a pseudo-Dirac pair, concentrating on the case of masses of the pseudo-Dirac pair having values <math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>≲</mo><mn>100</mn></math> GeV. The case when the CP violating asymmetries in the individual lepton charges <math><msub><mrow><mi>L</mi></mrow><mrow><mi>l</mi></mrow></msub></math>, <math><mi>l</mi><mo>=</mo><mi>e</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>τ</mi></math>, and in the total lepton charge L of the Universe are generated in <math><mn>1</mn><mo>↔</mo><mn>2</mn></math> decay processes is investigated. We show that successful leptogenesis is possible for <math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math> lying in the interval <math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>=</mo><mo>(</mo><mn>0.3</mn><mo>−</mo><mn>100</mn><mo>)</mo></math> GeV. Our results show also, in particular, that for vanishing initial <math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math> abundance, flavour effects can play an important role in the generation of the baryon asymmetry, leading to an enhancement of the asymmetry by a factor up to ∼300 with respect to the “unflavoured” leptogenesis scenario.

United Kingdom

K. Moffat

Flavoured resonant leptogenesis at sub-TeV scales

We develop the formalism of the finite modular group <math><msubsup><mrow><mi>Γ</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>≡</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math>, a double cover of the modular permutation group <math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>≃</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub></math>, for theories of flavour. The integer weight <math><mi>k</mi><mo>&gt;</mo><mn>0</mn></math> of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight (<math><mi>k</mi><mo>=</mo><mn>1</mn></math>) modular forms in terms of two Jacobi theta constants, denoted as <math><mi>ε</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math> and <math><mi>θ</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math>, τ being the modulus. We show that these forms furnish a 3D representation of <math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math> not present for <math><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub></math>. Having derived the <math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math> multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to <math><mi>k</mi><mo>=</mo><mn>10</mn></math>. These are expressed as polynomials in ε and θ, bypassing the need to search for non-linear constraints. We further show that within <math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math> there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.

P.P. Novichkov

Portugal

J.T. Penedo

Double cover of modular For Flavour Model Building

JINR

Canada

S.M. Bilenky

F. Capozzi

Corrigendum to “An alternative method of determining the neutrino mass ordering in reactor neutrino experiments” [Phys. Lett. B 772 (2017) 179–183]

The formalism of combined finite modular and generalised CP (gCP) sym-metries for theories of flavour is developed. The corresponding consistency conditions for the two symmetry transformations acting on the modulus τ and on the matter fields are derived. The implications of gCP symmetry in theories of flavour based on modular invariance described by finite modular groups are illustrated with the example of a modular S 4 model of lepton flavour. Due to the addition of the gCP symmetry, viable modular models turn out to be more constrained, with the modulus τ being the only source of CP violation.

A.V. Titov

Generalised CP symmetry in modular-invariant models of flavour

We construct phenomenologically viable models of lepton masses and mixing based on modular <math><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub></math> invariance broken to residual symmetries <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>T</mi></mrow></msubsup></math> or <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>S</mi><mi>T</mi></mrow></msubsup></math> and <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>S</mi></mrow></msubsup></math> respectively in the charged lepton and neutrino sectors. In these models the neutrino mixing matrix is of trimaximal mixing form. In addition to successfully describing the charged lepton masses, neutrino mass-squared differences and the atmospheric and reactor neutrino mixing angles <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>23</mn></mrow></msub></math> and <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>13</mn></mrow></msub></math>, these models predict the values of the lightest neutrino mass (i.e., the absolute neutrino mass scale), of the Dirac and Majorana CP violation (CPV) phases, as well as the existence of specific correlations between i) the values of the solar neutrino mixing angle <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>12</mn></mrow></msub></math> and the angle <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>13</mn></mrow></msub></math> (which determines <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>12</mn></mrow></msub></math>), ii) the values of the Dirac CPV phase δ and of the angles <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>23</mn></mrow></msub></math> and <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>13</mn></mrow></msub></math>, iii) the sum of the neutrino masses and <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>23</mn></mrow></msub></math>, iv) the neutrinoless double beta decay effective Majorana mass and <math><msub><mrow><mi>θ</mi></mrow><mrow><mn>23</mn></mrow></msub></math>, and v) between the two Majorana phases.