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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>2</a:mn><a:mi>O</a:mi></a: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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>C</c:mi><c:mi>P</c:mi></c:math>-violating (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>C</e:mi><e:mi>P</e:mi><e:mi>V</e:mi></e:math>) phase (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>δ</g:mi><g:mrow><g:mi>C</g:mi><g:mi>P</g:mi></g:mrow></g:msub></g:math>), the two Majorana <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>C</i:mi><i:mi>P</i:mi><i:mi>V</i:mi></i:math> phases (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>η</k:mi><k:mn>1</k:mn></k:msub></k:math>, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>η</m:mi><m:mn>2</m:mn></m:msub></m:math>), the lightest neutrino mass (<o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>m</o:mi><o:mn>1</o:mn></o:msub></o:math>) and the effective neutrino masses probed by beta and neutrinoless double beta decay (<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>m</q:mi><q:mi>β</q:mi></q:msub></q:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>m</s:mi><s:mrow><s:mi>β</s:mi><s:mi>β</s:mi></s:mrow></s:msub></s: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, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>r</u:mi><u:mrow><u:mi>s</u:mi><u:mi>b</u:mi></u:mrow></u:msub></u:math>, is strongly negatively correlated with each of the three lepton mixing angles and with <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mi>δ</w:mi><w:mrow><w:mi>C</w:mi><w:mi>P</w:mi></w:mrow></w:msub></w:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>m</y:mi><y:mn>1</y:mn></y:msub></y:math>, <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>m</ab:mi><ab:mi>β</ab:mi></ab:msub></ab:math>, and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>m</cb:mi><cb:mrow><cb:mi>β</cb:mi><cb:mi>β</cb:mi></cb:mrow></cb:msub></cb:math>, while being positively correlated with <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msub><eb:mi>η</eb:mi><eb:mn>1</eb:mn></eb:msub></eb:math> and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>η</gb:mi><gb:mn>2</gb:mn></gb:msub></gb: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.