https://doi.org/10.1140/epjc/s10052-021-08845-y
Regular Article – Theoretical Physics
Towards unification of quark and lepton flavors in 
modular invariance
1
Asia Pacific Center for Theoretical Physics, 37673, Pohang, Republic of Korea
2
Department of Physics, Pohang University of Science and Technology, 37673, Pohang, Republic of Korea
3
Department of Physics, Niigata University, 950-2181, Niigata, Japan
Received:
22
June
2020
Accepted:
8
January
2021
Published online:
19
January
2021
We study quark and lepton mass matrices in the modular symmetry towards the unification of the quark and lepton flavors. We adopt modular forms of weights 2 and 6 for quarks and charged leptons, while we use modular forms of weight 4 for the neutrino mass matrix which is generated by the Weinberg operator. We obtain the successful quark mass matrices, in which the down-type quark mass matrix is constructed by modular forms of weight 2, but the up-type quark mass matrix is constructed by modular forms of weight 6. The viable region of 
is close to 
. Lepton mass matrices also work well at nearby , which overlaps with the one of the quark sector, for the normal hierarchy of neutrino masses. In the common region for quarks and leptons, the predicted sum of neutrino masses is 87–120 meV taking account of its cosmological bound. Since both the Dirac CP phase 
and 
are correlated with the sum of neutrino masses, improving its cosmological bound provides crucial tests for our scheme as well as the precise measurement of and . The effective neutrino mass of the 
decay is 
–31 meV. It is remarked that the modulus is fixed at nearby in the fundamental domain of SL(2, Z), which suggests the residual symmetry 
in the quark and lepton mass matrices. The inverted hierarchy of neutrino masses is excluded by the cosmological bound of the sum of neutrino masses.
© The Author(s) 2021
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